
Numerical, analytical, and observational studies of tropical oceanatmosphere interactions
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 Permanent Link:
 http://digital.lib.usf.edu/SFS0040020/00001
Material Information
 Title:
 Numerical, analytical, and observational studies of tropical oceanatmosphere interactions
 Creator:
 Wang, Chunzai
 Place of Publication:
 Tampa, Florida
 Publisher:
 University of South Florida
 Publication Date:
 1995
 Language:
 English
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 xii, 184 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Oceanatmosphere interaction ( lcsh )
Oceanatmosphere interaction  Pacific Ocean ( lcsh ) Dissertations, Academic  Marine Science  Doctoral  USF ( FTS )
Notes
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 Thesis (Ph. D.)University of South Florida, 1995.
Includes bibliographical references (leaves 172177).
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 University of South Florida
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 University of South Florida
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 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 021578974 ( ALEPH )
33835057 ( OCLC ) F5100194 ( USFLDC DOI ) f51.194 ( USFLDC Handle )
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 Book

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NUMERICAL ANALYTICAL, AND OBSERVATIONAL STUDIES OF TROPICAL OCEAN ATMOSPHERE INfERACTIONS by /cHUNZAI WANG A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Marine Science University of South Florida August 1995 Major Professor : Robert H Weisberg, Ph. D
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Graduate School University of South Florida Tampa, Florida CERTIFICATE OF APPROVAL Ph.D. Dissertation This is to certify th a t the Ph.D Dissertation of CHUNZAI WANG with a major in Marine Science has been approved by the Examin in g Committee on May 12, 1995 as satisfactory for the dissertation requirement for the Doctor of Philosophy degree Examining Committee: Major Professor: Robert H. Weisberg, Ph.D. Member: Mark E. Luther, Ph D. Member: Boris Galperin, Ph.D Memb e r: Julian P. McCreary, Ph.D. Member: David T. Y. Tang, Ph.D.
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ACKNOWLEOOMENI'S I would like to express my sincere thanks to Professor Robert H. Weisberg, whose constant encouragements and suggestions and invaluable guidance have enriched my research experience, and who contributes essentially to the progress of the work. Without his help and support, I could never have finished this work. I gratefully acknowledge the discussions with Dr. Mark Luther on the numerical model development and the early comments of Dr. Julian M cCreary on the two papers which were published in Journal of Climate and Journal of Physica l Oceanography, respectively I would like to thank my Committee Memb e rs for their scholarly advice. I also wish to express my sincere appreciation to the staff and students of the Ocean Circulation Group, Department of Marine Science, University of South Florida, who have contributed to an excellent research environment. Finally, I would like to thank my wife (Yuan) and my daughter (Emilie) who always support and understand m e when I work in the evening, weekend, and on holidays.
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TABLE OF CONTENTS LIST OFT ABLES 111 LIST OF FIGURES 1 v ABSTRACT X CHAPTER 1. INTRODUCTION 1 1.1. Present State of ENSO Understanding 1 1 2. Motivations and Objectives 4 CHAPTER 2 THE SLOW MODE IN COUPLED OCEAN ATMOSPHERE MODELS 7 2 .1. Formulation of the Coupled Model 7 2.1.1. Model Ocean 7 2.1.2. Model Atmosphere 9 2 1.3. Coupling 10 2.2. Numerical Results 10 2.2.1. Description of the Coupled Response 11 2.2.2. Effects of Parameters 15 (a) The Parameters KQ, K s cr, a and y 15 (b) The Ocean Kelvin Wave Speed c 17 (c) The Ocean Basin Length 18 2 2 3. Interpretation of the Slow Mode 22 (a) Further Comments on the Delayed Oscillator 22 (b) Insights from the Energetics 23 2.3. Analytical Solution for the Slow Mode 27 2.4 Discussion and Summary 3 3 CHAPTER 3. EQUATORIAL WAVE MODES OF A COUPLED SYSTEM 3 7 3 .1. Coupled Equatorial Wave Modes 3 7 3 .1.1. Problem Formulation 3 7 3 1.2. General Solution 3 9 3.1.3. Special Solution 42 3.1.4. The Compl e te Dispersion Relationship 44 3.2. Horizontal Structure 4 7 3 .2.1. General Case 4 8 3.2.2. Rossby and Westward Slow Wave Mode s 54 3 2.3 Kelvin and Eastward Slow Wave Modes 61 3.3. Discu ss ion and Summary 66 CHAPTER 4. STABILITY OF EQUATORIAL MODES IN A COUPLED MODEL 72 4.1. The Model Formulation 72 4.2. Solutions of Coupled Equatorial Modes 7 4 4.2.1. Westward Propagating Modes 7 4 4.2.2. Eastward Propagating Modes 77 4.3. Analysis of the Solutions for the General Case 78
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4.3 1. The Basic Features 78 4.3.2. Dependence on Model Parameters 91 (a) Coupling Coefficient 91 (b) Warming Coefficient 93 (c) Zonal Mean SST Gradient 95 (d) Kelvin Wave Speed 95 (e) Zonal Phase Differen c e 98 (f) Rayleigh Friction/Newtonian Cooling 98 (g) Thermal Damping 102 4.4. The Coupled Modes in the FastWave and Fast SST Limits 105 4.4.1. The FastWave Limit 105 4.4.2. The FastSST Limit 109 4 5 Discussion and Summary 112 CHAPTER 5 LOW FREQUENCY VARIABILITY OBSERVED IN THE CENTRAL PACIFIC 118 5 .1. The General Description of Data Sets 118 5 2 Dynamics and Thermodynamics 131 5.2.1. Vertically Integrated Zonal Momentum Balanc 131 5 2 2 The ZPG in Relation to the EUC 136 5 2.3. SST and Thermocline 139 5 2.4. Hori z ontal SST Advection 142 5 2.5. Latent and Sensible Heat Fluxes 145 5 3 Discussion and Summary 162 CHAPTER 6. DISCUSSION AND SUMMARY 16 8 REFERENCES 1 7 2 APPENDICES 17 8 APP E NDIX A. OPEN BOUNDARY CONDITION FOR THE MODEL OCEAN 1 7 9 APPENDIX B METHOD OF SOLUTION FOR THE MODEL ATMOSPHERE 1 8 0 APP E NDIX C E IGENFUNCTION ORTHOGONALITY NORMALIZATION AND COMPLEfENESS 182 11
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Table 1 Table 2. LIST OFT ABLES Parameters for the Standard Experiment Values for the Basic Parameters 11 74
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Figure 1. LIST OF FIGURES The Evolutions on the Equator, as a Function of Longitude and Time, of the Coupled Model a) SST, b) Zonal Wind, c) Thermocline Thickness and d) Zonal Current Anomalies 13 Figure 2. The Evolutions on the Equator, as a Function of Longitude and Time, of the Coupled Model Global a) Atmospheric Pressure and b) Zonal Wind Anomalies 14 Figure 3 The Horizontal Structures of the a) SST and b) Wind Anomalies at Day 270; and c) SST and d) Wind Anomalies at Day 720 16 Figure 4 The Global Zonal Wind Anomaly on the Equator as a Function of Longitude and Time for Bounded Ocean Basin Widths of a) 160, b) 120, c) 100 and d) 60 20 Figure 5 A Schematic on the Evolution of the Wind Responses to Initial Ocean SST Perturbations for Large and Small Ocean Basins Figure 6. The Evolutions on the Equator, as a Function of Longitude and Time, of the Coupled Model SST with 21 an Open Western Boundary Condition 23 Figure 7. The Ratios of the Energy Sink Terms to the Energy Source Term for the Ocean Model as a Function of Time for the Standard Experiment 25 Figure 8. The Ratios of the Energy Sink Terms to the Energy Source Term for the Ocean Model as a Function of Time for the Experiment with the Ocean Kelvin Wave Speed Increased to 2.4 m/s 2 6 Figure 9 The Real (Solid Line) and Imaginary (Dashed Line) Parts of the Analytically Derived Dispersion Relationship for the Slow Coupled Mode 3 1 Figure 10. The thermocline height anomaly on the equator as a function of longitude and time for the analytical solution as given by equations (2.24), (2.25) and (2.29) 32 Figure 11 A Comparison Between the Dispersion Diagrams for a) Equatorially Trapped Wave Modes of a Coupled Oceaniv
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Atmosphere System and b) Conventional Uncoupled Equatorially Trapped Wave Modes 45 Figure 12. The Dispersion Diagrams for a) the Warming Coefficient cr=O and b) the Mean Zonal Temperature Gradient 11=0 46 Figure 13. The Meridional Scales a) L and b) LR as a Function of the Frequency for Different Values of the Coupling Coefficient 51 Figure 14. The Meridional Scales (L or LR) as a Function of the Frequency for Different Values of the Coupling Coefficient with the Warming Coefficient cr= 0 52 Figure 15 The Meridional Scales a) L and b) LR as a Function of the Frequency for Different Values of the Coupling Coefficient with the Mean Zonal Temperature Gradient 11=0 53 Figure 16. The Horizontal Htructures of the Eigenfunctions for the Gravest Mode (n=1) Equatorially Trapped Rossby Waves at the Frequency ro=12.6 Year1 55 Figure 17. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n=1) Equatorially Trapped Rossby Waves at the Frequency ro=6.28 Year 1 56 Figure 18. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n=l) Equatorially Trapped Rossby Waves at the Frequency ro=2.5 Year1 57 Figure 19. The Horizontal Structures of the Eigenfunctions for the Gravest mode (n=1) Coupled, Equatorially Trapped Rossby Wave at the Frequency ro=2.5 Year1 for Case of the Warming Coefficient cr= 0 59 Figure 20. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n=l) Coupled, Equatorially Trapped Rossby Wave at the Frequency ro=2.5 year1 for Case of the Mean Zonal Temperature Gradient 11= 0 60 Figure 21. The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency ro= 12 6 Year1 62 Figure 22. The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency ro= 6.28 Year1 63 Figure 23. The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency ro=2.5 Year1 64 v
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Figure 24. Figure 25 Figure 26. Figure 27. Figure 28 Figure 29. Figure 30. Figure 31. Figure 32. Figure 33 Figure 34. Figure 35 The Efolding Scale LK for the OceanAtmosphere Coupled Kelvin Wave as a Function of the Frequency for Different Values of the Coupling Coefficient 65 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with the Model Parameters of Table 2 8 0 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with 0=0. bt and Other Parameters of Table 2 81 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with 0=0.11t and Other Parameters of Table 2 8 2 The Horizontal Eigenfunction Structures for the Gravest Westward Propagating Modes with k=3 8 xl07 m 1 Denoted by the Solid Dots in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2 84 The Horizontal Eigenfunction Structures for Eastward Propagating Modes with k=3.0 xlo7 m1 Denoted by the Solid Dots in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2 8 5 The Horizontal Eigenfunction Etructures for the Gravest Westward Propagating Modes with k=3.0xlo7 m1 Denoted by the Open Circles in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2 8 7 The Horizontal Eigenfunction Structures for Eastward Propagating Modes with k=l.9xlo7 m1 Denoted by the Open Circles in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2 8 8 The Horizontal Higenfunction Structures for the Gravest Unstable Westward Propagating Modes with k=3.0 xio7 m 1 Denoted by the Solid dots in Figures 26 and 27 8 9 The Horizontal Eigenfunction Structures for the Unstable Eastward Propagating Modes with k=2 0 x l07 m1 Denoted by the Open Circles in Figures 26 and 27 90 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Coupling Coefficient J.L with k= .5xlo7 m 1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, respectively) and Other Model Parameters of Table 2 92 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Warming Coefficient cr with k=.0xio7 m1 (Positive and Negative Values of k vi
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Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 94 Figure 36. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Zonal Mean SST Gradient 11 with k=. 0 x 1o7 m1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 96 Figure 37. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Oceanic Kelvin Wave Speed c with k=.0x 1o7 m1 (Posit ive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 97 Figure 38. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Zonal Phase Difference 9 between the Zonal Wind Stress and SST Anomalies with k=. 6 x 1o7 m1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 99 Figure 39. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Rayleigh Friction/Newtonian Cooling Coefficient y with k=.0x1o7 m1 and 9=0.17t (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 1 00 Figure 40. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with 9=0.17t and y=O, and Other Parameters of Table 2 10 1 Figure 41 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Thermal Damping Coefficient a with k= 0 x 1o7 m 1 and 9=0.17t (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 103 Figure 42. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with 9=0 .17t and a=O, and Other Parameters of Table 2 104 Figure 43 Frequency and Growth Rate of the Coupled Equatorial Modes as a Function of k in the FastWave Limit (o=O) with 9=0 .17t and Other Parameters of Table 2 107 Figure 44. Frequency and Growth Rate of Eastward Slow Mode in the FastWave Limit Shown in Equations (4 38) and (4. 39) as a Function of Rayleigh friction/Newtonian Cooling Coefficient y with k=3 0 x 1 o 7 m 1 9=0.17t and Other Parameters of Table 2 108 vii
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Figure 45. Figure 46. Figure 47a. Frequency and Growth Rate of the Coupled Equatorial Modes as a Function of k for Values of o=3 with e=O .11t and Other Parameters of Table 2 Frequency and Growth Rate of the Coupled Equatorial Modes as a Function of the Relative Adjustment Time Parameter o with k=.0x 1 o7 m 1 e=0.11t and Other Parameters of Table 2 Contours of the 10Day Low Pass Filtered Zonal Velocity Component (cm/s) as a Function of Depth and Time with a Contour Interval of 20 cm/s Figure 47b. Contours of the 10Day Low Pass Filtered Meridional Velocity Component (cm/s) as a Function of Depth and 110 111 121 Time with a Contour Interval of 20 cm/s 123 Figure 47c Figure 48. Figure 49. Figure 50. Contours of the 10Day Low Pass Filtered Temperature (0C) as a Function of Depth and Time with a Contour Interval of 2C 125 The Vertical Distributions of Record Length Mean a) Zonal Velocity, b) Meridional Velocity, c) Temperature, and d) tan1(Ri) at 0, 170W 128 The 30Day Low Pass Filtered Time Series of the Vertically Averaged tan1(Ri) of the Mixed Layer at 0, 170W 129 The 30Day Low Pass Filtered Time Series of Zonal Wind Stress and Meridional Wind Stress at 0, 170W 130 Figure 51. The 30Day Low Pass Filtered Time Series of Zonal Wind Stress, Zonal Pressure Gradient Vertically Integrated from 0 to 300m, and Local Acceleration Vertically Integrated at oo, 170W 13 3 Figure 52. Ordinary Coherence Analysis between 'tx and ZPG and Multiple Coherence Analysis between both 'tx and the ZPG with the Local Acceleration 134 Figure 53. The 30Day Low Pass Filtered Time Series of the EUC Core Depth, the EUC Core Speed, and Zonal Pressure Gradient at the EUC Core Depth 137 Figure 54. Coherence Analysis between the EUC Core Speed and the Zonal Pressure Gradient at the EUC Core Depth 13 8 Figure 55 The 30Day Low Pass Filtered Time Series of the SST and the Thermocline Depth as Represented by the 20C Isotherm 140 VIII
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Figure 56. The 90Day Low Pass Filtered Time Series of Zonal Advection (the Solid Line) and Local Change of SST (the Bold Line) at 0, 170W 143 Figure 57. The 90Day Low Pass Filtered Time Series of Zonal Advection, Meridional Advection, and Sum of Zonal and Meridional Advections 144 Figure 58a. The Estimated Eensible Eeat Flux as a Function of Longitude and Time on the Equator 148 Figure 58b. The Estimated Latent Heat Flux as a Function of Longitude and Time on the Equator Figure 58c. The Bowen Ratio, Defined as a Ratio of Sensible Heat Flux and Latent Heat Flux, as a Function of Longitude 150 and Time on the Equator 152 Figure 58d. The SST and Air Temperature Difference as a Function of Longitude and Time on the Equator 154 Figure 59a. Coherence Analysis between the SST and Heat Flux 156 Figure 59b. Coherence Analysis between the Surface Wind Speed and Heat Flux 157 Figure 59c Multiple Coherence Analysis between both the SST and the Wind Speed with the Heat Flux 158 Figure 60 The 90Day Low Pass Filtered Time Series of SST Change (()T /at), Heat Flux (Q as Sum of Sensible and Latent Heat Fluxes), SST (T), and Surface Wind Speed (U8 ) at 0 170 W 159 IX
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NUMERICAL, ANALYTICAL, AND OBSERVATIONAL STUDIES OF TROPICAL OCEAN ATMOSPHERE INTERACTIONS by CHUNZAI WANG An Abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Marine Science University of South Florida August 1995 Major Professor: Robert H. Weisberg, Ph.D. X
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In this dissertation, numerical and analytical as well as observational studies have been performed to study tropical oceanatmosphere interactions. It is composed of four papers: Wang and Weisberg, 1994a; Wang and Weisberg, 1994b; Wang and Weisberg, 1995a; Weisberg and Wang, 1995b. The numerical model oscillates as a slow, eastward propagating, divergence mode, whose energetics are controlled by the ocean. Growth requires that the work performed by the wind stress minus the work required to effect the ocean divergence exceeds the loss terms The intrinsic scale of the atmosphere relative to the basin width is important. For sustainable oscillations, the ocean basin must be large enough so that oppositely directed divergence can develop on opposite sides of the basin. The global aspect of the atmospheric pressure field suggests that continental heating may provide either a direct source affecting adjacent oceans, or a connection between oceans. The important model parameters are the coupling and warming coefficients and the ocean Kelvin wave speed. The importance of the Kelvin wave speed derives from its specification of the background buoyancy state for the ocean. The neutral modes are analytically obtained over the full range of equatorial waves in a simplified coupled oceanatmosphere system with the assumption that zonal wind stress and SST perturbations are proportional without a zonal phase lag. In this system, inertial g ravity and Rossbygravity waves are unaffected by coupling while Rossby and Kelvin waves are affected, and in the low frequency limit, these Rossby and Kelvin waves transform to s lowly propagating modes. The primary modifications by airsea coupling are a decrease in phase speed and an increase in meridional scale. The stability and horizontal structure of equatorial modes in a coupled model, simplified by the assumption that zonal wind stress and SST perturbations are proportional with a zonal phase lag, are also examined analytically. The gravest coupled Ross by and Kelvin mod e s coexist with westward and eastward slow modes. Two of these four modes one propagating westward and the other eastward, are destabilized in each case depending upon the model parameters. For some particular parameter choices, coupled Rossby xi
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and Kelvin modes merge with westward and eastward slow modes respectively For other parameters, however, they separate and remain distinct from the slow modes Effects of the model parameters and the behaviors of coupled equatorial modes in the fastwave and the fastSST limits are also examined. Six years of ocean velocity, temperature and surface meteorological data collected in the central Pacific at 0, 170W show slow, interannual variability occurring in the dynamics of the ocean circulation and the thermodynamics of oceanatmosphere interactions. The implication is that the slow SST variations observed in transition from the 1988 La Niiia to the present protracted El Nifio are due to oceanatmosphere dynamical processes resulting from an SST zonal wind stress feedback through momentum flux. Surface heat flux, as controlled by wind speed, is important for intraseasonal SST variations, suggesting an SST wind speed feedback through surface heat flux on these time scales. Ab s tract Approved: Major Professor: Robert H Weisberg, Ph.D. Professor, Department of Marine Science Date Approved: .....;;Co;...J{Lz.._8,;;_L.,l9..:....S=Xll
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CHAPTER 1 IN'IRODUCTION Our planet' s climate varies in many ways that are well recognized but poorly unders tood. For example, the El NinoSouthern Oscillation (ENSO) phenomenon. w hich occurs every three to five years, appears to disrupt climate i n distan t p laces It has been accompanied by droughts in Australia, India and Afric a; floods in South America; and severe winter storms in the U nite d States. C limate variations such as ENSO have serious impacts on human affairs, including loss of life, crop failures and depletion of fisheries. Improving understanding and prediction may allow for effective actions by governments to mitigate adverse impacts. This chapter will review the 1 present state of E NSO understanding and give the motivations and objectives of t his work. L I. P resent State o f E NSO Understanding Since Bjerko e s (1969) identified unstable interactions between the tropical Pacific O cean and the atmosphere as a cause of the ENSO phenomenon, oceanographers and met e o r o logists have focused on the role of the coupled oc e aoalmosphe r e system upon ENSO A broad range of coupled oceanatmosphere models has been explored over the past decade beginning with the concep tual model of McCre ary ( 1983 ), the development of the coupling physic s b y P hilan d e r et al. ( 1 98 4 ) the s ubsequent applications of linear perturbation mode l s models linearized about different background states, and primitive equati o n general ci r c ulation model s (GCM). Mechanistic interpretations of these models have v aried, leading to three hypotheses : I) the delayed oscillator (Suarez and Scbopf 1 98 8 ; Battisti and Hirst, 1989), arguing for the importance o f equatorial ocean Rossby waves reflected at the western boundary; 2)
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external heating (Budin and Davey, 1990; Masumoto and Yamagata, 1991), arguing for the importance of continental heat sources; and 3) the slow 2 thermal or SST mode (Hirst, 1986, 1988; Neelin, 1991), arguing for the importance of a slowly propagating coupled mode distinctly different from the conventional equatorially trapped waves. The developments leading to these parad igms are reviewed by McCreary and Anderson (1991) The model ENSOs found in the linearized models of Zebiak and Cane (1987), Battisti (1988), Suarez and Schopf (1988) and Xie et al. (1989) have been attributed to the delayed oscillator of an analog model. The analog model is usually represented by a single ordinary differential delay equation with both positive and negative feedbacks. The positive feedback is the sum of all local processes due to the coupling, whereas the delayed negative feedback results from Kelvin modes generated at the western boundary as the reflection of Rossby modes. Therefore, this paradigm depends largely upon the stability properties of the Rossby and Kelvin modes. For example, damping of the Ro ssb y or Kelvin modes reduces the feedback necessary to change the sign of the airsea coupled instability in the eastern side of ocean basin. Wakata and Sarachik (1991) emphasized the importance of spatial variations in the mean thermocline depth and upwelling for determining the evolution of the SST anomalies. The meridional profile of the mean upwelling determines whether the SST anomaly is stationary or eastward propagating, and the delayed oscillator mechanism works only if the mean upwelling is narrowly confined to the equator. This also suggests a critical dependence of the delayed oscillator on the ocean background state. Masumoto and Yamagata (1991) argued that nonlinearities in the Anderson and McCreary (1985a b) model dramatically change the results predicted by the linear theory External heating (which they supplied at the western boundary of a coupled model for the Pacific Ocean) is then necessary to sustain the coupled oscillations. In the absence of land heating, the coupled sys tem se ttles into a perpetual El Nino state. These authors conclude that the mechanisms of oscillation are different from either the delayed oscillator or the slow mode.
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Depending upon the treatment of the ocean thermodynamics in models linearized about a background state, coupled modes may propagate eastward, westward or remain stationary. Hirst (1986) defined four models, each distinguished by the SST equation. With SST proportional to the thermocline thickness anomaly, the oceanic Kelvin wave is destablized. With the rate of change of SST proportional to zonal advection and thermal damping, the gravest oceanic Rossby wave is destablized. By considering thermocline thickness and thermal damping, with or without advection, a slowly propagating unstable mode occurs. Using a perturbation expansion, Neelin ( 1991) showed that this slow (SST) mode is distinctly different from the conventional equatorialocean wave modes, leading to the argument that the time delay by ocean wave propagation is not essential to the slow mode. Primitive equation, coupled ocean and atmopshere GCMs also produce ENSOlike phenomena. In a set of companion papers, Lau et al. (1992 ) and Philander et al. (1992) discuss the results for low and high resolution ocean models coupled with an atmopsheric GCM, respectively. These two models simulate El Nino episodes, but in very different ways. In the low resolution model SST anomalies first appear in the eastern tropical Pacific and migrate westward. In the high resolution model SST and zonal wind stress anomalies vary without propagation while the thermocline depth anomalies propagate eastward along the equator and westward off the equator. These authors 3 attribute the differences in the model evolutions to the effects of the ocean dynamics owing to model resolution; particularly equatorial upwelling. Despite this, the SST anomalies during the mature El Nino or La Niiia states are similar. In a series of papers, Jin and Neelin (1993a, b) and Neelin and Jin (1993) attempt to consolidate seemingly contradictory works by indicating how some of the relatively simple coupled models may relate to each other. Using an equatorial narrow band approximation in the SST equation, they argued that the unstable ocean dynamics modes of Cane et al. (1990) and the SST modes of Neelin (1991) represent a fast SST limit and a fastwave limit, respectively In the fastSST limit, SST adjusts much more rapidly than the ocean dynamics so
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4 SST change is mainly controlled by subsurface processes In the fastwave limit, the ocean dynamics adjust much more rapidly than SST so that SST change is mainly controlled by surface layer processes Ea s tward or westward propagating and stationary SST modes in the fastwave limit can mix with ocean dynamics modes away from the fast wave limit to form "mixed SST/oceandynamics modes". With this theory coupled modes, distinct in some regions can merge continuously across parameter space. 1.2. Motivations and Objectives Des pite the successful achievements of the coupled oceanatmosphere models related to ENSO, the present state of ENSO understanding remains somewhat pu z zled This dissertation work performs both numerical and analytical as well as obs ervational studies of the coupled oceanatmosphere system related to ENSO phenomenon. Motivations follow from several features, either ob served in the oceanatmosphere s y s tem, or arising in the hierarchy of ENSOrelated numerical models. Data from the TAO (Tropical Atmosphere Ocean) array of about 70 moorings reveal that another El Nino warming episode may be on its way Th e protracted warm conditions of 199195 indicate a detailed evolution of the coupled ocean atmosphere system unlike that of any event in the recent past. In particular, the continued major amplification of basin scale westerly wind and SST anomalies in 1993 and 1994/95 has no apparent analog in the past y e ars This unusual evolution of the coupled oceanatmosphere system in th e tropical Pacific during 1991995 emphasi z es the complicated ENSO phenomenon Although there are similarties for all events eventtoevent dissimilarit i es which are not well understood suggest the need for more re s earch on ENSO dynamics A better understanding of ENSO mechanism s which are responsible for the recent protracted 199195 El Nino will improve our ability to predict ENSO using coupled oceanatmopshere models. All numerical models of the coupled oceanatmosphere s ystem, from the s impl est linear perturbation models, e.g Hirst (1986, 1988) to the most
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5 complicated GCMs, e.g., Barnett et al. (1991) and Chao and Philander (1993) show meridional scales for coupled oscillations larger than the ocean's Rossby radius of deformation. This broadening of meridional scale with respect to ENSO is also an observed property of the ocean thermocline, e.g White et al. (1987, 1989) and Kessler (1990). In the Kessler (1990) analysis these thermocline displacements, found to be maximum at about l2N, propagated westward, but at speeds slower than the gravest mode equatorially trapped Rossby wave speed The role of these socalled offequatorial Rossby waves in the evolution of the ENSO cycle has been controversial with Graham and White (1991) contending that they are important and Kessler (1991), Battisti (1991) and Wakata and Sarachik (1991) contending that they are not. The mechanisms at work in coupled GCM simulations of ENSO remain unresolved. For example, similar El Niiio and La Niiia states are achieved when using either high or low resolution ocean model components (Philander et al., 1992 and Lau et al. 1992) despite the fact that the low resolution model can not resolve conventional equatorial waves and the authors' contention that the high resolution model shows no evidence for them. With conventional equatorially trapped Kelvin waves having been observed in the oceans (e.g., Knox and Halpern, 1982 and Katz, 1987) and having been used, along with Rossby waves, in describing the seasonal evolution of the Atlantic Ocean's equatorial thermocline (e.g., Weisberg and Tang, 1990), a question arises regarding the conditions, if any, by which conventional equatorially trapped waves are modified by oceanatmosphere coupling. Given the above facts, the goal of this dissertation is to improve our understanding of coupled oceanatmosphere system by numerical, analytical, and observational approaches The dissertation is formed by four papers (Wang and Weisberg, 1994a; Wang and Weisberg, 1994b; Wang and Weisberg, 1995a ; Weisberg and Wang, 1995b) which are presented in Chapters 2, 3, 4, and 5, respectively. Chapter 2 considers the simplest of linear perturbation models, to gain insights on the following questions: 1) the mechanism of the slow mode and its relation to the delayed oscillator; 2) the conditions for slow mode growth; 3) the ocean basin size necessary to sustain a slow mode; 4) the
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6 importance of continental heating; 5) the importance of the ocean background state; and 6) the importance of the ocean Kelvin wave speed. Chapter 3 analytically investigates the properties of equatorially neutral wave modes of a coupled oceanatmosphere system, simplified to use the same formalism as Matsuno (1966) for the case of conventional equatorially trapped waves. Chapter 4 extends the work of Chapter 3, showing how coupled Rossby and Kelvin modes may coexist with westward and eastward slow modes; how these modes may merge, separate and be destabilized by varying model parameters; and how their meridional scales broaden. In Chapter 5, the sixyear long data of upper ocean velocity, temperature, and surface winds collected at 0, 170W as part of the Tropical OceanGlobal Atmosphere TAO (TOGA TAO) array are analysed relative to the low frequency variability comprising ENSO. A summary with discussions is presented in Chapter 6
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7 CHAPTER2 THE SLOW MODE IN COUPLED OCEANATMOSPHERE MODELS As discussed in Chapter 1, several mechanisms have been developed to interpret ENSO phenomenon. The present chapter considers the simplest of models, namely a linear perturbation model of the form used by Hirst (1988), to numerically and analytically gain further insights on the slow modes, which are presented in Wang and Weisberg (1994a). In this chapter, we will focus on the following issues which are not fully resolved : (1) the mechanism of the slow mode and its relation to the delayed oscillator; (2) the conditions for slow mode growth; (3) the ocean basin size necessary to sustain a slow mode; (4) the importance of continental heating; (5) the importance of the ocean background state; and (6) the importance of the ocean Kelvin wave speed. Section 2 1 formulates a numerical model and Section 2.2 examines its behavior relative to the above issues. Upon simplification, an analytical solution is obtained in Section 2 3 showing the essential aspects of the model parameters. A discussion and summary then follow in Section 2.4. 2 .1. Formulation of the Coupled Model 2.1.1. Model Ocean The ocean dynamics are those of a linear, equatorial pplane, reducedgravity model forced by surface wind stress. The upper layer, which interacts with the atmosphere, has constant density and a mean depth Ho while the lower layer, isolated from the atmosphere, has slightly higher density, is infinite ly deep and motionless. The interface between these layers represents the tropical thermocline, separating the warm surface waters from the cold
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8 deep waters Momentum and mass conservation for the upper layer motions are governed by au A ,ah ,..x ..., yv = g+ _.__ yu, at ox pHo (2.1) av + = + _L_ yv' at ay pHo (2.2) (2.3) where u and v are the velocity components in the zonal (x) and meridional (y) directions, h is the upper layer thickness perturbation, t is time, g' is the reducedgravity, is the gradient in planetary vorticity, ,x and Y are the zonal and meridional surface wind stress components, and y is the coefficient of Rayleigh friction and Newtonian cooling. The ocean thermodynamics controlling the variations in the SST anomaly T consist of a trade off between ocean processes lUld surface heat fluxes. In their simplest form, these are represented by aT= crhaT, (2.4) at where cr is a warming parameter and a is a thermal damping coefficient. In th e present model crh and aT embody the ocean processes and the surface fluxes, respectively. Variations within the model ocean are evaluated using finite differe nce versions of the Eqs. (2.1)(2.4) on a staggered Arakawa Cgrid with a uniform g rid spacing of 55 km. A leapfrog scheme is employed for the time integration using a time step of 1 hour, with a forward scheme used every 199th time step to avoid the computational mode. The model domain is rectangular with boundaries at 20N, 20S, 120E and 80W. The western and eastern boundaries are rigid with no normal flow (u=O), while the northern and southern boundaries are open, permitting wave propagation to extratropical latitudes
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9 (Camerlengo and O'Brien, 1980). Appendix A summarizes the application of the open boundary condition. 2.1.2. Model Atmosphere The atmosphere model is a steady state, linear, equatorial reducedgravity model, forced by heating (Gill, 1980). The governing equations, with dissipation included in the form of Rayleigh friction and Newtonian cooling, are ap aua= , ax ap ava + = ay 2(aua ava) Q ap + Ca += a, ax ay (2.5) (2.6) (2.7) where the coordinate axes and velocity components are as defined in the ocean model, p is the pressure anomaly, a is the coefficient of Rayleigh friction and Newtonian cooling, Q. is proportional to the heating rate, and Ca is the reduced gravity wave speed. Similar to Zebiak (1982), variations within the model atmosphere are obtained by combining Eqs. (2.5)(2.7) into a single equation for v.. which, after Fast Fourier Transform (FFT) in x, may be expressed as a second order ordinary differential equation in y. The equation is solved numerically using a central finite difference scheme (Appendix B), after which the wind and pressure fields are obtained by inverse FFT. The atmospheric model domain is cyclic in x and ex t ends from 20 S to 20N, with a grid spacing of 440 km and 220 km in the zonal and meridional directions, respectively, and the atmo sphere is updated once per day. The northern and southern boundaries are taken to be rigid with boundary condition, Va = 0.
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10 2 1.3. Coupling The atmosphere depends upon the oceanic heating source for its evolution while the ocean depends upon the atmospheric wind stress for its evolution The coupling must therefore parameterize the thermal heating in terms of oceanic quantities and the wind stress in terms of atmospheric quantitie s. Following Philander et al. (1984) and Hirst (1986), the heating is taken proportional to the SST and the wind stress components are taken proportional to the wind velocity components as (2.8) (2.9) where KQ and Ks are the coupling coefficients for heat and momentum, respectively. As a model limitation it is noted that by omiting a mean zonal th ermoc line tilt along the equator the model thermodynamics may overestimate the oceanatmosphere coupling in the west. 2.2. Numerical Res ults The model oscillates over a broad range of param ete rs This section describes the behavior of the coupled oscillations, adds to the parameter s tudies of Hirst (1988) and offers further m ec hanistic insights. The experiments begin by perturbing a resting ocean and atmosphere with a positive, Gaussian shaped SST anomaly located symmetric about the equator m the ce nter of the basin. The coupled model then evolves without modification u s ing the standa rd parameters of Table 1
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Table 1. Parameters for the Standard Experiment. Parameter Value 200m 2m s1 60 m s1 ( 5 dayst 1 (100 dayst 1 (100 dayst 1 5.0x109 K m1 s1 5.0xlo3 K1 m2 s3 0.8x1o7 s 1 1.024x103 kg m3 2 .29x 10 11 m1 s1 2.2.1. Description of the Coupled Response Consistent with Gill (1980), heating symmetric about the equator 11 induces a primarily zonal wind response, with maximum convergence just east of the maximum heating region. The specified model thermodynamics causes an eastward shift in the SST anomaly and hence an eastward translation of the coupled response. The resulting evolutions of the SST, zonal wind, thermocline thickness and zonal current anomalies on the equator are shown in Figures 1 a, b, c, and d, respectively. For the standard parameters the anomalies oscillate at a periodicity of 2.5 years and grow exponentially as they propagate eastward at a speed of 0.28 m/s (which is much slower than the ocean Kelvin wave speed of 2.0 m/s). The SST anomalies lag the thermocline height anomalies and the zonal wind anomalies lag the SST anomalies. These lags result in a zonal phase difference between the SST and zonal wind anomalies at any given time which, as shown in Section 2.3, is important for coupled mode growth. In contrast to the similar zonal phase gradient for the SST, thermocline thickness and zonal wind anomalies, the phase gradient for the zonal current anomalies is smaller and these anomalies are largest in the east central portion of the basin. The atmosphere response is not limited to the ocean basin domain as shown in Figures 2a, b for pressure and zonal wind, respectively. The pressure re s ponse is global, with maximum values over the east central Pacific
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12
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Figure 1 The Evolutions on the Equator, as a Function of Longitude and Time, of the Coupled Model a) SST, b) Zonal Wind, c) Thermocline Thickness and d) Zonal Current Anomalies. Stippled (clear) regions denote negative (positive) anomalies. The contour intervals for SST, wind height and current are 0.6 K, 1.5 m/s, 20 m, 0.15 m/s, respectively.
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liD 1>. cd 0 a .... E< 14 288 288 Wind 216 216 180 180 144 144 108 108 72 72 36 36 0 180 120 60 0 60 120 180 0 180 120 60 0 60 120 180 Longitude Longitude Figure 2 The Evolutions on the Equator, as a Function of Longitude and Time, of the Coupled Model Global a) Atmospheric Pressure (Geopotential Height) and b) Zonal Wind Anomalies. Stippled (clear) regions denote negative (positive) anomalies. The contour intervals for pressure and wind are 10 m2 s2 and 1.5 m/s, respectively and the South American continent. The wind response ( proportional to the zonal pressure gradient) is large only over the model ocean, since SST provides the only atmosphere heat source. While the lack of continental heating is unr e alistic, several important points are noted. First, there is a difference in scale (both zonal and meridional) between the atmosphere and ocean responses. Second, since there is no a continental heat source, the pressure anomaly is blocked by the discontinuity in atmospheric heating between the ocean and the land. Large z onal pressure gradient and wind anomalies ther e fore develop near the eastern boundary except when the SST anomaly
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15 there goes to zero, at which time the pressure anomaly evolves across the land portion of the cyclic domain and the coupled model oscillates SST in this model is forced by ocean divergence (equations 2.3 and 2.4) Following the warm anomaly in Figure 1, it is observed that divergent (convergent) currents occur in the western (eastern) Pacific, resulting in upwelling (downwelling) and SST cooling (warming) The propagating SST anomaly induces a pattern of wind divergence and convergence that supports the eastward propagation. The culmination of the warmest and coldest SST anomalies in the eastern part of the basin may thus be likened to the El Niiio and La Niiia phases of ENSO. Prior to the first warm phase, the SST and wind anomaly fields for day 270 are shown in Figures 3a, b. Note the relative positions of the wind divergence and SST anomaly patterns with the wind divergence pattern centered just to the east of the SST pattern Strong winds on either side of the wind divergence alters the ocean divergence resulting in eastward propagation. During the mature phase of the model E1 Nifio the west central Pacific is a region of wind divergence, while the eastern Pacific is a region of wind convergence As the wind pattern continues to move eastward it reverses sign around day 570, resulting in warming of western Pacific and the cooling of eastern Pacific peaking as the cold phase of ENSO around day 810 Prior to this, the SST and wind anomaly fields for day 720 are shown in Figures 3c, d. Thus, the model ENSO may be summarized as a continuous propagation of ocean divergence and convergence patterns caused by the oceanatmosphere interaction. 2.2.2 Effects of Parameters (a) The Parameters KQ. Ks. cr, a andy. The Hirst (1988) Model IV explored the s ensitivity of the coupled oscillations to KQ, Ks. cr, and a, where KQ and Ks set the interactions between the ocean and the atmosphere and cr and a set the relative importance of ocean processes and surface fluxes in determining SST. As a test of our model formulation these studies were repeated with very
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Q) "0 ION SST Anomalies Day 270 3 EQ :;:; I'd ...J !OS 60 40 20 0 20 40 60 60 100 Surface Wind Day 270 20N _,_ _,_ ION t. Q) "0 ;:l ' EQ 1:;:l _______ __. __ j lOS 120S __ ____ ___ 60 40 20 0 20 40 60 60 100 Longitude SST Anomalies Day 720 8 ION EQ lOS 20S 8 60 40 20 0 20 40 60 60 100 Surface Wind Day 720 20N ION 1. EQ 1lOS t. 20S ____ ____ __ _ eo 40 2o o 20 40 60 60 10 0 Long itude Figure 3 The Horizontal Structures of the a) SST and b) Wind Anomalies at day 270; and c) SST and d) Wind A n omalies at day 720. Un i ts are K fo r SST and m/s for wind.
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17 similar results: the model exhibits evanescent, neutral or growing modes over a broad range of parameters. The behavior of the coupled oscillations relative to the parameters follows from the mechanism of the model. For example, in contrast to the Hirst (1988) finding of increasing frequency with increasing KQ and Ks, Wakata and Sarachik (1991) found the opposite result. They attributed their model oscillations to the delayed oscillator mechanism, wherein reflected waves interact with directly forced waves reversing the state of upwelling or downwelling. Increasing KQ and Ks increases the directly forced part of the solution, making it more difficult for the reflected waves to compete. By increasing the time required to change the sign of the ocean divergence this decreases the frequency. The slow mode mechanism is different. Increasing K Q and Ks increases the speed at which this mode propagates thereby increasing the frequency. Varying y relative to KQ and Ks shows that growth can occur even for large y, which also suggests a mechanism different from the delayed oscillator. If he combined effects of the coupling and the efficiency of the ocean processes in effecting SST are large enough then the coupled oscillations can grow despite dissipation. For the other parameters of Table 1, y = (80 days)" 1 yields a neutral mode, so for small growth y = (100 days)"1 was chosen for the standard parameter set. (b) The Ocean Kelvin Wave Speed c. To investigate the effects of varying the ocean Kelvin wave speed the standard experiment was repeated using c = 2.4 m/s, versus 2 m/s. This resulted in damping, versus growth, and an increase in period. The explanation follows from the change in the background state buoyancy as specified by c. Increasing c, by increasing buoyancy, decreases divergence, since the ratio of thermocline thickness perturbation to current perturbation is h / u = H0/c. Decreasing the ocean divergence decreases the rate of slow mode growth. In their analysis of the delayed oscillator, Battisti and Hirst (1988) found both the growth rate and period to decrease with decreasing delay time
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18 (increasing c). The decrease in period differs from our result due to the different mechanisms of these models. For th d I d 'I e e aye osc1 lator, varying c varies the delay time, while for the slow mode it varies the ocean divergence. Neelin (1991) argued that the time scale for equatorial wave propagation is not essential to the slow mode based upon distorted physics experiments with a GCM. These experiments, however, distorted the affects of wave speed independent of buoyancy. In contrast, the importance of c in our experiment is its effe c t o n buoyancy. This accounts for the sensitivity here, versus the insensitivity in Neelin (1991). (c) The O cean Basin Length. Experiments were performed with varying ocean widths. Figure 4 shows the zonal wind anomaly on the equator over the global domain for basi n widths of 160 120 100 and 60 using the parameters of T ab le 1. In each cas e the western boundary is at 60 longitude Only the 160 case shows growth; modes within the smaller basins decay. The distinguishing feature of the growing mode is the reversal of sign for the zonal wind, with st r o ng, opposite ly directed winds on opposite sides of the basin. For the 100 wide b as in ther e i s no appreciable oppositely directed anomaly on the western side of the basin and for the 60 case the anomaly does not change sign at all. Given the scales of the atmosphere relative to the ocean responses, these findings s ugge s t that for an ocean basin to oscillate it must be wide enough to co ntain th e atmo spheric response, as will be supported by the stabilit y a n a l ys i s of Section 2 3. The structure o f the wind perturbation issuing from an initial SST a nom a ly leads to the following hypothesis as illustrated by Figure 5 The wind re s p o nse to the ini tial ocean heat source con sists of a pattern of convergent winds th a t can promote the growth and propagation of the heat source (e.g., Philander et al., 1984). If the ocean bas in is large enough, an SST anomaly of opposi t e sig n can form on the we s tern s ide du e to a developing p a ttern of di v e rgent wind s there while a mature SST anomal y still exists on the eastern s ide of the basin. Once this occurs, the pattern of convergent and divergent w inds, s upported by warm and cold SST, may then be self sustaining. Smaller
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Figure 4. The Global Zonal Wind Anomaly on the Equator as a Function of Longitude and Time for Bounded Ocean Basin Widths of a) 160, b) 120, c) 100 and d) 60 In each case the oceans western boundary is located at 60 longitude Stippled (clear) regions denote negative (positive) anomalies.
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20 288 160 2 88 120 252 216 216 180 1 8 0 al II. cd "d 0 oo4 144 144 Q,) a .... 108 lOB Eo< 72 72 36 36 0 lBO 1 2 0 60 0 60 1 20 lBO 288 100 28 8 60 252 252 216 216 al lBO 180 II. cd "d 0 144 144 4 Q,) a .... lOB 108 Eo< 72 72 36 36 0 1BO 120 60 0 60 120 lBO Longitude Longitude
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21 .... __ .... ._ ._ a I E:) I I I ....... , E :? b \ I .... .... __ .... .__ ._ c I E:) I ._ ._ l' j d E I Figure 5. A Schematic on the Evolution of the Wind Responses to Initial Ocean SST Perturbations for Large and Small Ocean Basins. a) and b) show the initial and subsequent stages for the large ocean basin, respectively; and c) and d) show the initial and subsequent stages for the small ocean basin, respectively. For both cases, the initial SST perturbations are the same and located in the center of the basin as s hown. The arrows denote the winds and the dashed line denotes the ensuing SST perturbation of opposite sign. ocean wid th m akes it more difficult for patterns of opposite sign to evolve. If westwa rd directed currents are not large e nough to promote enough ocean divergence for a given y, patterns of opposite sign do not form, and the mode decays. What i s large enough is sensitive to the model parameters. Experiments with sma lle r y l ed to oscillations at smaller basin widths, but by t h e same
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22 mechanism (requiring a reversal in the ocean divergence owing to oppositely directed winds). The period of oscillation for smaller y, however, was several years longer. The smaller basin locked into a nearly steady state until sufficient divergence developed at the western boundary. Once this happened the patterns evolved at a rate similar to the standard case. The oscillations thu s appeared as warm and cold states with relatively rapid transitions in between. 2.2.3. Interpretation of the Slow Mode (a) Further Comments on the Delayed Oscillator. For the delayed oscillator, westward propagating Rossby waves generated in central Pacific are reflected at the western boundary as eastward propagating Kelvin waves that tend to change the phase of the warm or cold events. By eliminating the reflected Kelvin waves using an open western boundary we can determine whether or not this mechanism is operating within the model. The resulting SST anomaly on the equator for the standard parameters is shown in Figure 6 Compared with Figure la, the evolution of SST is nearly identical for the open or the closed western boundary; the only difference being a small increase in growth rate. Similar behavior between open and closed western boundaries was also found for the smaller r and smaller ocean basin experiments of the previous section. Therefore, the delayed oscillator is not an important mechanism in this model, regardless of y or basin size. Does this finding present an incompatibility between the delayed oscillator and the slow mode paradigms? The same physics are operant in each. Equatorial waves propagate, affecting the depth of the thermocline and the oceanatmosphere exchanges. For the slow mode these exchanges occur continuously in space and time, whereas the delayed oscillator has a regional dependence, owing to nonhomogeneous parameters. Both directly forced and reflected waves are present for both, but the oceanatmosphere coupling in the slow mode depends primarily on the directly forced waves. For example, waves directly forced in the western Pacific by strong easterly winds there were important in readjusting the thermocline in the eastern Pacific during
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216 180 144 ll'l cd "'=' 0 t 108 Q,l a ..... E< 72 36 0 Lon gitude Figure 6. The Evolutions on the Equator, as a Function of Longitude and Time of the Coupled Mode l SST with an Open Western Boundary Condition. Stippled (clear) region s d e note negative ( positive ) a nomalies The contour intervals are 0.6 K the 19821983 E l Nifio termination (e.g ., Tang and Weisberg, 1984) Where active coupling occurs in nature could therefore determine the relative impo rtance of the s e two m ec hani s ms without excluding either. 23 ( b ) In sigh t s fr om the Energetics Following Yamagata (1985) and Hirst (1988), the e qu a tions governing the atmosphere and ocean total perturbation e n erg ie s EaT and E0 T, are
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24 (2.10) (2. 1 1) where EaT and E0 T consist of perturbation kinetic and potential energies [EaT = EaK + E/ = (u/+v82)/2 + p2/(2c/) and E/ = E0K + E/ = (u2+v2)/2 + g'h2/(2Ho)] and the brackets indicate a model domain average. The first terms on the right hand sides of these equations are energy sources; the second and third term s represent r e distributions between the potential and kinetic energies, which combine to form the pressure work divergence and the last two terms are potential and kinetic energy dissipation t erms. A diagnosis of these equations provides several important points. First, th e kinetic energy for the atmosphere is much larger than the potential energy, and conversely for the ocean. It follows that di ssipation in the a tmosphere (ocean) is mainly due to dissipation of kinetic (potential) energy. Seco nd, the energy source terms  <"txu+"tYv> are always positive, regardless of whether the coupled model oscillates, o r not. This is a co ns equence of the simple coupling between the reducedgravity ocean model and the Gill (1980) atmosphere model used herein. It follows that the conditions of positive and <"tx u+"t Y v> are not useful a priori in determining coupled mode growth; they arc necessary, but not sufficient, co nditions for growth. Third, time seri es of E/ and E/ show that the total energy for the atmosphere is smaller than and lags that fo r the ocean, and t hat if E o T grows so does EaT, and conversely. The key to understanding the energetics of the coupled oscillations for this model therefore lies in the ocean energy equation. This is consistent with a conclusion of Barnett et al. (1991), fr o m analyses of uncoupled ocean and atmosphere GCM's, th a t the propagation of ENSO related anomalies is determined by the ocean. The energy equation for the ocean model consists of a source and three sink terms. If the source exceeds the sum of the sinks, the perturbations will g row; if the converse occurs, they will decay. The relative importance of the
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0 0 0. 5 1.0 1.5 0 12 24 36 60 72 Time (30 days) Figure 7. The Ratios of the Energy Sink Term s to the Energy Source Term for the Ocean Model as a Function of Time for the Standard Experiment. The solid, da s hed and dotted lin es represent the dissipation of perturbation potential energy, the pressure work div ergence and th e di ssipation of perturbation kinetic energy, respectively ; and the bold line is the sum of the se thr ee rat ios.
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0.5 0.0 0.5 1.0 1.5 0 12 24 36 48 60 72 Time (30 days) Figure 8 The Ratios of the Energy Sink Terms to the Energy Source Term for the Ocean Model as a Function of Time for the Experiment with the Ocean Kelvin Wave Speed Increased to 2.4 m/s The solid, da shed and dotted lines represent the dissipation of perturbation potential energy, the pressure work divergence and the dissipation of perturbation kinetic energy, respectively ; and the bold line is the sum of these three ratios.
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sink terms for the standard experiment are shown in Figures 7. Each line represents the ratio of one of the sink terms to the source; the dashed, solid 27 and dotted lines being the pressure work divergence, the potential energy dissipation and the kinetic energy dissipation, respectively, and the b o ld line is their sum. If the sum is greater (less) than 1 the source (sinks) exceeds the sinks (source) The largest sink term is the potential energy dissipation. This is followed by the pressure work divergence (equivalent to an energy transfer to midlatitudes through the open northern and southern boundaries) and the kinetic energy dissipation. The sum of the sink/source ratios in Figure 7 being greater than 1, on average, is consistent with growth. Figure 8 is a similar presentation for c=2.4 m/s. Here the sum of the sink/source ratios is Jess than 1, on average, and the coupled mode decays, as in Section 2 .2.2 b The transition from a growing mode to a decaying mode is subtle. All of the terms on the right hand side of ocean energy equation grow or decay together, and this is reflected, with a small phase Jag, in the growth or decay of the atmosphere energy. The sufficient condition for growth is that the source term for the ocean exceeds the sum of the sinks, but the set of conditions for which this occurs is very sensitive to the parameters. One factor affecting the source term is the difference in phase gradient between the evolution of the winds and currents. Owing to the relative importance of Rossby waves in the ocean current field, the phase gradients for these two quantities are different (Figures lc, d), and the larger this difference, the smaller the ocean energy source term. 2.3 Analytical Solution for the Slow Mode Noting that the numerically obtained velocity fields for both the ocean and the atmosphere are primarily zonal and in analogy to an equatorial Kelvin wave, a so lution is sought for the slow mode with v and Y both equal to zero. For the model ocean equations (2.1) (2.4) become (2. 12)
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28 (2.13) + Ho au = yh at ax (2.14) or_ crhaT. at (2 15) A closed form solution requires 'tx to be expressed in terms of ocean variables. Battisti and Hirst (1989), in their derivation of the delayed oscillator equation, assumed 'tx proportional to SST based on numerical model results averaged over the eastern equatorial Pacific. Schopf and Suarez (1990) assumed 'tx to have a specified meridional structure with magnitude proportional to SST at a fixed point on the equator. Cane et al. (1990) related 'tx to the thermocline height anomaly at the eastern boundary, also with specified meridional structure, and Neelin (1991) used a more general formulation. With application to a GCM, Barnett et al. (1993) coupled an OGCM with a statistical atmosphere wherein the winds were taken proportional to a linear combination of SST EOF modes Our model also shows a correlation between zonally averaged 'tx and SST that is maximum on the equator and diminishes to zero poleward of 14, and a zonal phase difference between 'tx and SST at any given time We, therefore, assume 'tx to be a linear function of SST lagged by a phase angle e and seek solutions of the form ( u(x y, t)) ( u(y) ) h(x y, t) = h ( y) eit>, T(x, y, t) T(y) 'tx(x, y, t) pHoJ.LT(y)eie (2.16) where Jl is a coupling coefficient, k is the zonal wave number, the real and imaginary parts of ro are the frequency and the growth rate, respectively, and the other parameters are as in Section 2.1. In reality, the coupling coefficient
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29 1..1. in Eq. (2.16) is not homogeneous, nonlinear effects are important and the correlation between 'tx and SST diminishes poleward from the equator. Thus, spatially uniform coupling between 'tx and SST is erroneous, but this model limitation is tempered by the fact that the gravest mode ocean. (atmosphere) equatorial waves are forced primarily by winds (heating) near the equator. Recently, Jin and Neelin (1993a) showed that the meridional structure of zonal wind stress did not play a crucial role in destabilizing oscillatory modes when thermocline feedbacks dominate. Substituting equations (2.16) into equations (2.12)(2.15) yields . k 'h T ie I(J)U = l g + 1..1. e yu, (2.17) = g'hy, (2.18) iroh + ikHou = yh, (2.19) iroT = crhaT. (2.20) The algebraic equations (2.17), (2.19) and (2.20) have solutions only when 'Y2 i2yro ro2 + k2c2 + iHo!..Lcrk eie = 0. (2.21) a iro This dispersion relationship, cubic in ro, has three roots It is anticipated that one of the these roots should correspond to the Kelvin wave, and one to the slow mode. First, consider oscillations at frequencies considerably higher than the slow mode. The dispersion relation simplifies to (2.22)
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30 By writing ro = OOr + iroi, we find roi = y and OOr = kc. The positive root is the same as a Kelvin wave, damped by roi. The negative root is rejected because i ts associated amplitude increases exponentially with y. the third root is that of the slow mode. In the low frequency limit, upon neglecting second and third order terms in ro equation (2.21) b ecomes (2.23) which has solution Ho11crkcose OOr = 2 k c 2 + y 2 + 2ya. (2.24) and OOi = Hwcrksine a.(k2 c 2 + y 2 ) k2c2 + y2 + 2ya. k2c2 + y2 + 2ya. (2.25) The real part resembles an equatorial Rossby wave dispersion relationship, but with eastward phase propagation. The imaginary part shows that the phase lag b etween 'tx and SST is necessary for instability. Slow mode growth requires that which, in turn, requires that k1 < k < k2, where and k1 = Hwcrsine ( 1 2 a.c 2 ( ) 2 ) 1 2ya.c Ho11crsine k 2 = Hwcrsine ( 1 + 1 ( 2 ) 2 ). 2a.c 2 HoJ.Lcrsme (2.26) (2.27) (2.28) The resulting slow mode dispersion relationship is shown in Figures 9, using the p a rameters of Table 1, with 11 = 1.92 x107 m s2 K1 and e = 0.37t. Growth occurs for 0 .1x1o7 m 1 < k < 3 .2 5 x1o7 m 1 the upper limit of which corresponds
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i J.4 Cl:l Q) t>.. 3 J.4 0 .. 3 5.0 4.0 3.0 2.0 1.0 0.0 1.0 2. 0 3. 0 0.0 1.0 2.0 3.0 ' ..... ..... ..... ..... .............. ............. ........ 4 0 5.0 ........ __ 6 .0 .... __ 7.0 31 8.0 Figure 9. The Real (Solid Line) and Imaginary (Dashed Line) Parts of the Analytically Derived Dispersion Relationship for the Slow Coupled Mode. Positive imaginary part denotes growth. to a wavelength of 176 Wavelengths longer than this value are required for instability. Given the dispersion relationship, the slow mode eigenfunctions are h(x, y, t) = A exjy2} ei(kltmt), 1 2kc2 u(x y, t) = A(ro+iy) exjy2) ei(klt. mt), kHo 1 2kc2 T(x, y, t) = _AsL exjy2) ei(kltrot), o.iro 1 2kc2 (2.29) (2.30) (2.31)
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32 where A is a constant of integration The meri d ional scale is L5 = = L(c/c5 )1 12, where L = is the equatorial radius of deformatio n and c5 = rorlk is the slow mode phase speed. Since Cs << c it follows that the meridional scale of the slow mode is larger than the equatorial radius of deformation, con sis tent with the numerica l results of Figure 3 and the findings of Wang and Weisberg (1994 b ). Using the parameters of Table 1, with J.L = 1.92x1o7 m s 2 K1 e = 0.3tt and k = 3.0x 1 o7 m1 h(x, 0, t) from equation (2 29) is shown in Figure 10. The appearance is very similar to t h e numerical result of F igure lc despite 288 216 180 Ill aS 0 144 4 a .... E< 108 72 36 0 60 20 20 60 100 Longitude Figure 10. The Thermocline Height Anomaly on the Equator as a Function of Longitude and Time for the Analytical Solution as Given by Equations (2 24), (2.25) and (2.29). The parameter values are as given in the text. Note that we have arbitrarily drawn these contours within a 160 wide basin (fo r comparison with Figure lc) without consideration of boundaries.
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the fact that we have arbitrarily drawn these contours within a 160 wide basin without consideration of boundaries The dependencies of ror and roi on the model parameters are similar to those found numerically Both ror and ro i increase with increasing coupling 33 and warming coefficients, decrease with increasing Rayleigh friction and thermal damping and decrease with increasing oceanic Kelvin wave speed The analytical solution points out the importance of a zonal phase lag between the 'tx and SST anomalies for instability and the narrow range of wavenumbers for which growth occurs 2.4. Discussion and Summary Motivated by the state of complexity in ENSO modelling, a model similar to Hirst (1988) was revisited for insights on the questions raised at the beginning of this chapter. With spatially homogeneous parameters and no background state dependence, other than the reducedgravity wave speed, the model oscillates as a slow, eastward propagating mode with evanescent, neutral or growing behavior dependent upon the parameters The mechanism of this slow mode differs from that of the delayed oscillator in that ocean waves reflected at the western boundary are not important. With SST governed by [equations (2.3) and (2.4)] a1' + (y + a)aT + yaT = + av), at2 at ax ay (2.32) which has an evanescent complementary solution, the wind forced ocean divergence determines the coupled mode growth For a given the ocean divergence is controlled by the ocean buoyancy (c2/H0 ) Thus, from equation (2 11), if the rate of work by minus the rate of work against buoyancy is sufficient to overcome the energy sinks, then the slow mode will grow With slow propagation speed, the zonal pressure gradient nearly balances 'tx on the equator, so an extremum in thermocline height anomaly coincides approximately with a zero in 'tx. Such an extremum requires that 'tx reverses
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34 sign across the ocean basin, consistent with the hypothesis on ocean basin size presented with Figure 5 The slow mode in this model therefore evolves as a directly forced mode relatively unaffected by reflected waves The results suggest the need for a more global view in ENSO modeling While the ocean in this model controls the growth and propagation of the slow mode through its divergence field, the atmosphere response is global, due to the atmosphere's large radius of deformation. As the coupled mode evolves, a discontinuity in the atmospheric heat source (proportional to SST) develops at the eastern boundary when the SST anomaly there is large This effectively blocks the atmospheric pressure anomaly until the SST anomaly goes to zero at the eastern boundary, allowing the atmospheric pressure anomaly to progress cyclically around the global domain. Two important points follow from these pressure variations. First, continental heat sources may be important for transmitting (or blocking) atmospheric pressure perturbations between ocean basins, so adjacent ocean basins may interact if continental heat sources and orographic effects allow for the communication of their pressure perturbations For example, the outofphase behavior observed between the Pacific and Atlantic Oceans during some ENSO events may be a consequence of this Second, gi ve n the large scale of the atmosphere response to a localized heat source, variations in continental heating may initiate a coupled mode within an adjacent ocean basin Just how important is the background state in determining coupled o s cillations and what is the role of the ocean Kelvin wave speed c? Beginning with the simplest analog model of McCreary (1983) through the more complete GCMs of Lau et al. (1992) and Philander et al. (1992), all models produce oscillations despite the fact that the background states are either neglected or specified to various levels of sophistication For the linear perturbation model considered herein slow mode oscill a tions are robust with respect to the parameters, the most important ones being the coupling and warming coefficients and c Other than c these coefficients are not well defined; particularly the warming coefficient, a lumped parameter covering all of the ocean processes that give rise to SST variations To the extent that these ocean
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35 processes are modelled incorrectly, thermal damping will provide a correction This is a major limitation of all coupled models. The importance of c lies in its specification of the background state buoyancy. Excepting c, the implication is that the background state is important to the extent that the mean conditions control the processes embodied within the warming and coupling coefficients, as in Zebiak and Cane (1987) This remains an issue for future research, including field experimentation to determine the actual role of the various ocean processes, relative to the surface fluxes, in controlling SST. In summary, the slow mode of a linear perturbation, coupled ocean atmosphere model with homogeneous coefficients of the form Hirst (1988) is an ocean divergence mode distinctly different from the delayed oscillator mode. This slow mode propagates eastward. Its energetics are governed by the ocean; the atmosphere merely follows. Growth occurs if the work performed by the wind minus the work required to effect the ocean divergence exceeds the sum of the ocean loss terms The ocean Kelvin wave speed is therefore an impo t nt parameter, since it sets the background buoyancy state of the ocean. For a given level of dissipation the intrinsic length scale of the atmosphere relative to the width of the ocean basin is important. For sustainable oscillations the ocean basin must be large enough so that oppositely directed divergences can develop on opposite sides of the basin. Analytical results support this with the requirement for sufficiently large wavelength and a zonal phase difference between the tx and SST anomalies. The global aspect of the atmospheric pressure component gives importance to both adjacent land masses and adjacent ocean basins. Continental heat sources may block or facilitate the propagation of the pressure perturbations, or they may provide a direct source affecting an adjacent ocean. Whether this model has relevance to nature depends upon several factors intrinsic to both the oceans and the adjacent continents Homogeneous coefficients omit both the background state influences of the ocean circulation and the nonlinear aspects of heat exchange. This facilitates the direct forcing versus the reflected wave behavior of the model which may be
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36 unrealistic. Omitting continental heat sources may also be unrealistic. These findings therefore suggest the need for 1) models incorporating all three oceans and intervening continents; and 2) field experiments defining the roles of the ocean processes (mean and perturbation) and surface heat fluxes in determining the evolution of SST
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37 CHAPTER3 EQUATORIAL WAVE MODES OF A COUPLED SYSTEM The present chapter investigates the properties of equatorially trapped waves of a coupled oceanatmosphere system, simplified to use the same formalism as Matsuno (1966) for the case of conventional equatorially trapped waves All wave modes discussed in this chapter are neutral, owing to omitting the zonal phase lag between 'tx and SST and eliminating dissipation. The chapter, which is the work of Wang and Weisberg (1994b), proceeds as follows Section 3 1 formulates the problem by assuming that the perturbations in 'tx and SST are proportional, gives the general solution, the solution for Kelvin waves as a special case when the meridional velocity component is zero, and describes the full dispersion relationship for the equatorially trapped waves of this simplified coupl e d oceanatmosphere system. Section 3.2 discusses scale modifications and describes the eigenfunctions for the coupled Rossby (Kelvin) and westward (eastward) propagating slow mode waves, comparing thes e eigenfunctions with their uncoupled oceanic counterparts. Section 3 3 then discusses these findings relative to the model limitations and the motivating factors. 3.1. Coupled Equatorial Wave Modes 3.1.1. Problem Formulation Most models of the coupled oceanatmo s phere sy s tem relative to ENSO have considered the ocean t o be driven by the surface winds and the atmosphere to be driven by SSTinduced heating Consistent with this, we seek the simplest set of equations solvable analytically, for the purpose of gaining in sight into the properties of coupled oceanatmosphere equatorial waves
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38 The ocean dynamical equations are those of a linear, equatorial reducedgravity model perturbed about a basic state of rest by the zonal component of surface wind stress 'tx, which are similar to those in Chapter 2. The ocean thermodynamics are also similar to the SST equation of (2.4), except for an additional term of zonal advection by the perturbation currents on a mean temperature gradient. and SST anomalies are The equations governing the momentum, mass au A ,ah ,..x pyV =g+"yu, at ax pHo (3.1) av + = g'ah yv, at ay (3.2) ah H (au av) + 0 += yh at ax ay (3.3) aT + TlU = crhif, at (3.4) where the symbols are as defined in Chapter 2; Tl is the mean zonal temperature gradient and y is a coefficient representing Rayleigh friction, Newtonian cooling and thermal damping, providing dissipation in the momentum, mass and SST equations, respectively. To form a coupled oceanatmosphere system, we need a way of specifying 'tx in terms of ocean variables. By assum ing 'tx and SST to be proportional everywhere, Wang and Weisberg (1994a) obtained a solution for eastward propagating wave modes. These waves show either exponential growth or decay depending upon whether or not there is a zonal phase difference between 'tx and SST. By om i tting the phase lag the neutral wave results may be obtained over the full range of equatorially trapped waves. Thus, we seek solutions of the form
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39 u(x, y, t) u(y) v(x, y, t) v(y) h(x, y, t) = h(y) ei(lcx cot) yt (3.5) T(x, y, t) T(y) 'tx(x, y, t) pJ.LHoT(y) where J.l is the oceanatmosphere coupling coefficient, k is the zonal wave number component and ro is the frequency. With ro taken to be positive the direction of zonal phase propagation is determined by the sign of k. The assumption of equal values for Rayleigh friction, Newtonian cooling and thermal damping allows the effects of dissipation to be factored out, resulting in a neutral wave dispersion relationship and an evanescent factor ert multiplying the eigenfunctions. The limitations imposed by the three principal assumptions: 1) proportionality between 'tx and SST, 2) spatial homogeneity in coupling and other parameters and 3) dissipation by equal values of Rayleigh friction, Newtonian cooling and thermal damping will be discussed in Section 3 3. 3.1.2. General Solution Substituting equation (3.5) into equations (3.1)(3.4) yields four equations with four unknowns u, v, h and T irou=ikg'h + J.LT, r:t dh IOOV + JJYU = g, dy iroh + ikHou + Hod.Y. = 0 dy iroT + T)U = crh. Eliminating u, h, and T results in the single equation for v (3.6) (3.7) (3.8) (3 .9)
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40 (3 1 0) where A= c2(ro2 (3.11) (3.12) B = 002 + k\:o2 k@ro c2 (3 .13) and c = (g'H0 ) 112 is the ocean equatorial Kelvin wave speed. Equation (3 10) may be transformed into canonical form using the substitution v(y) = AL2)]. 11_ 2 2 (3 .14) whence (3 .15) where = y/L, and L = (A2/4 + 02)"1 4 Equation (3 15) is the Hermite equation. Its solutions satisfy the boundary conditions, v(y) 0 as y oo, when the constant L2(B + A/2) equals an odd integer. This results in the dispersion relationship 2c2( ro2 11) [m..:_ k 2 ro 2 k@ ro + (k ro + ..@.)] rrV4ro2c2(ro2+ c2 ro2c.o2c2(ro22 = 2n + 1, (n = 0, 1 2, .. ) (3 .16) and the solution
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41 v(y) = C1NnexjLkn(y/L) = y\ (3 .17) where C1 is a dimensional constant, is the Hermite polynomial of .order n, N n = (2n! 1t 112)112 is the normalization factor for 'I' and LR = L/(1 AL2/2)112. The eigenfunctions = + corresponding to the eigenvalues of equation (3.10) are orthonormal, over the interval oo < < oo, with respect to the weighting function = i.e. (3 .18) where Omn = 1, if m = n and Omn = 0, if m '# n (see Appendix C). If the ocean is not coupled with the atmosphere = 0), then A = 0, LR = L = 112 and (3.16) reduces to the conventional dispersion relationship for equatorially trapped waves of either the ocean or the atmosphere (Matsuno, 1966) 2 2 kR k = 2n + 1, c2 ro (n = 0, 1, 2, ... ) (3 .19) which includes inertialgravity and Rossby waves, along with Rossbygravity and Kelvin waves as special cases. Analogous with these conventional equatorially trapped waves, the complete dispersion relationship (3.16) includes ocean atmosphere coupled inertialgravity, Ross by, Rossbygrav ity, Kelvin, and a modified class of Rossby and Kelvin waves at low frequency referred to as slow mode waves. The effects of the oceanatmosphere coupling upon these equatorially trapped waves vary inversely with frequency. Thus, the coupled inertialgravity and Rossbygravity waves are very similar to their conventional counterparts. For frequencies intermediate between the Rossbygravity and the slow mode waves (that is, ro2 small but much larger
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42 than Jlll). L2 == and the dispersion relationship for the coupled Rossby waves may be approximated by (3 .20) where roc = k2 + (2n + (3 .21) is the conventional dispersion relationship for the equatorially trapped Rossby waves At lower frequencies where terms quadratic or higher in ro may be neglected, the dispersion relationship (3 16) reduces to 2c2 ( 00 + HO)lcrkro + Ho)lcrp) = 2n + 1 c2 2c2 (3 .22) from which 00 = nHO)lcrp == nHo)lcr H o )lcr)k c2k (3 .23) Equation (3.23) is the approximate dispersion relationship for a westward propagating slow mode wave, shown later to be a continuation of the coupled Rossby wave dispersion relationship at low frequency 3 .1.3 Special Solution Similar to conventional equatorially trapped waves, we seek a solution with the meridional velocity component identically zero. With v(y) = 0 equations (3 .1)(3.4) are irou = ikg'h + J.1T, = gd h, dy (3 .24) (3 .25)
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43 iroh + ikHou = 0, (3 .26) iroT + 11u = crh. (3 .27) This set of algebraic equations has solutions for u, h and T only when which is the dispersion relationship for this special case. Combining equations (3 25) and (3.26) leads to the solution h(y) = C2 exJL 2kc2 (3 .29) where C2 is a dimensional constant. Note that the meridional length scale, which we will refer to as LK, is given by LK = The dispersion relationship, being cubic in ro, has three roots. It is anticipated that one of the these roots should correspond to the coupled Kelvin wave, and one to the coupled slow mode wave At relatively high frequencies the dispersion relationship reduces to (3 .30) and the negative root is rejected because the associated wave amplitudes for u, h and T increase exponentially with y At frequencies low enough so that the second order term in ro can be neglected, equation (3.28) reduces to 00 = HoJ.J.crk c2k2 + JlTI (3 .31) which is the dispersion relationship for an eastward propagating slow mode wave Furthermore, for JlTI << c 2 k 2 equation (3.31) becomes
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44 (3 .32) which is the same as the dispersion relationship for the westward propagating slow mode wave, given by equation (3. 23), with n set equal to 1 (conventional for equatorially trapped waves). Thus, both the coupled Rossby and Kelvin waves have westward and eastward propagating slow mode waves, respectively, in their low frequency limits 3.1.4. The Complete Dispersion Relationship A comparison between the dispersion relationships for the oceanatmosphere coupled and the conventional ocean equatorially trapped waves is given in Figures 11a and b Only the Rossby, Kelvin and the slow mode waves are considered, since the Rossbygravity and the inertial gravity waves are essentially unchanged by coupling, owing to their frequency range. Except at low frequency there is very little difference between the conventional and the coupled waves. However, once the physics of the coupling begins to dominate over the intrinsic p plane ocean wave propagation physics, the character of the dispersion relationship changes and a continuous transition occurs between Rossby (Kelvin) waves and westward (eastward) propagating slow mode waves. Since the local time derivatives are the only terms within this model yielding propagation, the phase speed for the coupled waves must decrease with decreasing frequency as these derivative terms become less important than the coupling term [using a different set of assumptions Lau (1981) also noted a reduction in phase speed for a coupled Kelvin wave]. As the phase speed decreases with decreasing frequency the zonal wavenumber magnitude must either increase, or remain constant (for the case CJ = 0) Where this transition occurs in the dispersion plane depends upon the parameters of the coupled system The parameters used in Figures lla and b are: J.1 = 0 .85x 1 o7 m s 2 K1 cr = 5 0 x 1o9 K m1 s1 11 = 5 0 x 1o7 K m1 H0 =200m, c =2m s1 and P = 2 .29x 10 11 m1
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3 70,rr, 60 50 ,/ I I I I I I , : // : / \ I ,. I \ I ; \ / \ : \ : ,,. \\ I I : 4 0 .. : / \ I . .. \ Ill 30 ........ \\ .. \ \\ .... \ \ \ : \ \ \ .. I I \ \ \ : jll__ 20 10 0 1 I I I I 150 125 100 75 50 25 0 I 25 50 Figure 11. A Comparison between the Dispersion Diagrams for a) Equatorially Trapped Wave Modes of a Coupled OceanAtmosphere System and b) Conventional Uncoupled Equatorially Trapped Wave Modes For the coupled waves, westward (eastward) propagating slow modes occur as low frequency transitions from the Rossby (Kelvin) waves The family of curves for negative k correspond to Ross by and slow wave modes n= 1 to 4 The dimensions for ro and k follow from the parameters given in the text.
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70 60 50 20 10 150 so 25 0 I I I I I I I I I I I I I I I I I I I I I I I 25 5 0 70 60 50 40 ... ..... . .. I I I I I 30 20 10 \, \ ....... '' \ \ \< ... \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \. \ I \ \':\ : \ \ : ) } : } ... .... ', ... 0 150 125 1 0 0 75 so 2s o 2s 5 0 Figure 12 The Disper s ion D i agrams for a) the Warming Coefficient cr=O and b) the Mean Zonal Temperature Gradient T1=0. The family of curves for negative k correspond to Rossby and slow wave modes n=l to 4
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47 s1 With these values, the warming and zonal advection terms are the same order (tending about 0 15 K month 1 for h and u perturbations of 10 m and 0.1 m s1 respectively) and the coupling coefficient results in an approximate 0 .02 N m2 wind stress perturbation per each 1 K perturbation in SST. Modifications to the Rossby and Kelvin waves become discernible between frequencies of 1020 radians year1 As frequency decreases the effects of coupling impacts the higher mode Rossby waves first with the gravest mode being modified the least. The transition to slow mode waves doesn t occur until the frequency is less than about 5 radians year1 and again the gravest modes (the first mode Rossby and the Kelvin waves) are the last to undergo this transition. Once the transition occurs between Rossby or Kelvin waves and their slow mode continuations, i.e., at point where ak/ac.o = 0, the wavenumbers no longer tend toward zero with decreasing frequency. Instead the magnitude of the wavenumber increases with decreasing frequency and the group velocity reverses to be opposite in direction to the phase propagation Also, unlike the Rossby waves, the frequency for the westward propagating slow modes increases with increasing meridional mode number. The dispersion curves are sensitive to the parameters Increas i ng J.l increases the frequency at which the coupling effects become evident. Figures 12a and b compare the dispersion diagrams for the cases i n which cr = 0 and 11 = 0, respectively, with all of the other parameters the same as in Figure 11a With cr = 0 there is no transition to the slow mode waves, whereas with 11 = 0 these transitions do occur. This implies that the slow modes are dependent upon those ocean processes that give rise to horizontal divergence and associated vertical advection and mixing, as opposed to horizontal advection in the presence of a mean zonal temperature gradient. For this reason the slow modes here are interpreted as horizontal divergence modes. 3.2. Horizontal Structure Along with modifying the dispersion relationship, oceanatmosphere coupling significantly alters the horizontal structure of the equatorially
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48 trapped waves described herein This section develops these structural modifications for the coupled Rossby and Kelvin waves, along with their slow mode transitions As was the case for the dispersion relationship, the structural modifications to the inertialgravity and the Rossbygravi ty waves are minor, so these are omitted from further discussion, although they are included within the general formulation that follows. 3 2.1 General Case Given the dispersion relationship of equation (3 .16), the eigenfunctions for v, u, h and T are v(y) = C1NnexJAL = C1NnexjL\H n(y/L), 2 2 1 Making use of the recurrence formulas for the Hermite polynomials, and defining = = J AL2 ] = NneXI[_2) (3.33) (3.34) (3.35) (3.36) (3.37)
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49 we can obtain the following relationships (3.38) (3 .39) which are used to express the eigenfunctions of equations (3.33)(3.36) m the more convenient forms v(y) = (3 .40) Two different scales, L and LR, appear in the eigenfunctions. The scale L nondimensionalizes the meridional coordinate y and LR is the efolding scale. For the case of uncoupled waves (J..L = 0) L and LR are the same constant and they are equal to the radius of deformation for conventional equatorially trapped waves. For coupled waves, however, these scales differ from one another and they are both functions of frequency as well as the coupling and
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50 thermodynamic parameters. Using the same parameters as in Figure 11 a (Ho = 200 m, c = 2 m/s, = 2.29x1011 m1 s1 cr = 5.0x1o9 K m1 s1 and Tl = 5.0x1o 7 K m 1), the frequency dependencies of L and LR are shown in Figures 13a and b respectively, for different values of the coupling coefficient J.l, with the solid lines corresponding to J.l = 0 85 x 107 m s 2 K1 as in Figure lla. Both L and LR increase with decreasing frequency from their high frequency limit of ( 1 2 This change in meridional scale is sensitive to the coupling coefficient. Increasing the coupling coefficient increases the frequency at which the coupling effects become important. These variations in the length scales parallel the behavior of the dispersion relationship wherein the coupled and conventional equatorially trapped waves differ only at low frequency Since these length scales set the horizontal structure of the eigenfunctions, an important effect of the coupling is to broaden the meridional extent of the low frequency waves When the warming coefficient cr is set equal to zero, LR equals L and LR is somewhat smaller than for non zero values of cr, as shown Figure 14. Curiously, when Tl is set equal to zero, L decreases to zero with decreasing frequency while LR increases to large values, as shown in Figures 15a, b. Despite the decreasing L, the meridional length scale for such coupled waves still exceeds that of the uncoupled waves because of LR. The broadening of the meridional length scale for low frequency coupled equatorially trapped waves follows from the wind stress curl induced by the coupling. Without wind stress curl in the vorticity equation, the meridional component of velocity must tend to zero at low frequency, leaving the intrinsic radius of deformation as the meridional scale. With wind stress curl, the meridional component of velocity adjusts to a slowly varying Sverdrup balance, thereby broadening the meridional scale. Thus, at low frequency, the meridional scale for coupled oceanatmosphere equatorial waves tends toward the scale of the overlying atmosphere which in this model is imposed by the ocean.
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so oo 375 s s ,!:.II 350 . w (year') 1000 ""TTIT'"':", 875 I 750 ! i i i I : I : I : I : I : 1 : I : I : I ; I 1 : 625 II \ \ 1 : 1 : 1 I 1 : I 1 I I I 5oo I \ 375 i I i \ i I I I i I i \ \ \ \ , .... ..... "". ............. : : : : : ._... 0. 250 0 10 20 30 40 50 60 7 0 Figure 13. The Meridional Scales a) L and b) LR as a F unction of the Frequency for D iffe r ent Values of the Coupling Coefficient. The dotteddashed, solid, dashed, and dott e d lines are for coupling coefficients Jl=0 .25xt07 m s2 K1 Jl=0.85 x107 m s2 K1 Jl=l.92xt07 m s2 K1 and Jl=5.0xto7 m s2 K1 respectively, with the solid line corresponding to the value used in all other plots of this chapter.
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1000 :. 875 I \ 75o!1: ,... II: d !1: II: u !1: ....... : ._. il: i I : a: i I : .....:I 625 i I: i 1. i I : 0 il: i I : i I : .....:I i I : i I : i 500i 1 i I i i I i \ i \ i \ .. i \ . i \ 375 \ \ \ \ ', '. ..... ...... __ ::=. .:::::._::.:::...;;:.;w. ........ ;.;.; . ............... .., ................ 250 0 10 20 30 40 50 60 70 52 Figure 14. The Meridional Scales (L or LR) as a Function of the Frequency for Different Values of the Coupling Coefficient with the Warming Coefficient cr=O. The dotteddashed, solid, dashed, and dotted lines are for coupling coefficients J..1.=0.25xl07 m s2 K1,J..1.=0. 85 xl07 m s2 K 1 J..1.=1.92xl07 m s2 K1 and J..1.=5.0xi07 m s2 K1 respectively
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350 .300!, .. /, ;..r' I t I I / 25o I / I 1 j I ; i I : I I I I : I I : S 200I : i .... I : e : : ...:I 150!! I I : I I I I : I I : j I : I I : 1 0 0 U' i l : 50 I I:' 1 : 1 : 0 10 20 30 <40 50 6 0 7 0 s 1000 rTI, : I : I : I I I I 875l I I I I I I I 750 : I I I I I I \ :: 625 I I : I I : i I : i I : i I i I : i \ I I 500 i I i I i \ .. I I i \ I I 375 \ \ i \ \ \ .... .... :::: _::: _ :::: __ ,; _ ;:,; ;;;_ ,._...,.....,;.;.;.;. ;.;.; o;.: ,;.;.;.;. ................ .... ............... "'""'"'! 250 0 1 0 20 .30 <40 50 60 7 0 Figure 15 The Meridional Scale s a) L and b) LR as a Function of the Frequency for Different Values of the Coupling Coefficient with the M e an Zonal Temperature Gradient 11=0. The dotted das hed, solid, dashed, and dotted lines are for coupling coefficients 11= 0 25 x I o 7 m s 2 K 1 11=0. 8 5 x 1 o 7 m s 2 K 1 J.1= 1.92 xi07 m s 2 K1 and J.1=5.0xi07 m s2 K1 re s pectively
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54 3 2.2. Rossby and Westward Slow Wave Modes Given the scale dependencies on the frequency and the parameters, we examine the effects of varying sca les on the eigenfunctions. Three sets of comparisons are made between the gravest mode coupled and uncoupled Rossby waves. These comparisons are for: 1) Rossby waves at the frequencies c.o = 12.6 year1 6 28 year1 and 2 5 year1 (approximate periodicities of 6 months, 1 year and 2.5 years, respectively), using the standard set of parameters as in F igu re 11a ; 2) Rossby waves at c.o = 2.5 year 1 for the extreme parameter choice of cr = 0 and 3) Rossby waves at c.o = 2 5 year1 for the extreme parameter choice of 11 = 0. The c.o = 2 5 year 1 frequency corresponds to the transition point between coupled equatorially trapped Rossby waves and westward propagating slow mode waves. A similar set of comparisons are developed in the next section for the Kelvin waves. The thermocline height and the horizontal velocity vector anomalies for the oceanatmosphere coupled and the conventional equatorially trapped, gravest mode (n=1) Rossby waves at the frequency c.o = 12.6 year 1 are compared in Figure 16. The meridional and zonal dimensions are in latitude and longitude degrees, respectively, with the zonal dimension spanning one wavelength. Qualitatively both waves appear very similar, with strong zonal flows confi ned close to the equator and with maximum height anomalies centered at about for the coupled Rossby wave and for the uncoupled Rossby wave. From Figures 11a, b the wavenumbers for the coupled and the uncoupled waves at this frequency are 6.677x107 m1 and 6 136 x107 m1 respectively, so the coupled waves have both larger meridional scale and smaller zonal scale than the uncoupled waves. Similar presentations for the coupled and the uncoupled Rossby waves at c.o = 6.28 year1 and for the westward propagating slow mode and the uncoupled Rossby wave at c.o = 2.5 year1 are shown m Figure 17 and Figure 18, respectively. For the uncoupled wave changing the frequency simply changes the wavelength according to the dispersion relationship (and the amplitude which is arbitrary). The effects on the coupled wave are much
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R Wave (cpl,n=l,w = l2.6 yr1 ) 20 ...,. 20 0 85 R Wave (cpl,n=l,w= l 2 6 yr1 ) 2 0 ..... 1 0 1. 0 1______ _. 10 1. 20 L_ ______ .....__ ______ __,__ ______ __._ __ __ __ __.J 0 85 Longitude R Wav e (uncpl,n=l,w= l2.6 yr1 ) 2 0 ..... 10 10 2 0 L_ ______ ______ _t_ ______ _L ______ 0 93 R Wave (uncpl,n= l.w= 12. 6 yr1 ) 2 0 ,.... 10 0 10 2 0 ________ .....__ ______ __,__ ______ __._ ______ __.J 0 Longitude Figure 16. T h e Hori zo ntal Structures of the Eigenfunctions for the Gravest Mod e (n =l) Equatorially Trapped Rossby Waves at the Frequency ro=12.6 Year1 (Corresponding to a Periodicity of about 6 Months) The left hand plates give the thermocline height and the velocity vector anomalies for the oceanatmosph e re coupled waves and the right hand plates give the counterpart fields for the conventional, uncoupled ocean waves The meridional and zonal dimensions are in latitude a nd longitude degrees, re spectively, with the zonal dimension spanning one complete wavelength The units for the height and current fields are m and m s 1 re spectively.
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R Wave (cpl,n=l,w=6.28 yr1 ) R Wave (uncpl,n =l,w=6.28 yr1 ) 2 0 ...., 20 10 r10 rIll 'C ;::J :::! 0 .. ro ..J 10 10 20 20 0 140 0 188 R Wave (cpl.n=1,w= 6 .28 yr1 ) R Wave (uncpl,n= 1 ,w=6.28 yr1 ) 20 ...r, 2 0 10 10 .. _______ ...... _______ ... 0 10 r10 20 2 0 0 140 188 Longitude Longitude Figure 17. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n= 1) Equatoriall y Trapped Rossby Waves at the Frequency ro= 6 .2 8 Year 1 (Corresponding to a Periodicity of about 1 Year) The left hand plates give the thermocline height and the velocity vector anomalies for the coupled waves and the right hand plates give the counterpart fields for the conventional, uncoupled ocean waves The meridional and zonal dimensions are in latitude and longitude degrees respectively, with the zonal dimension spanning one complete wavelength The units for the height and current fields are m and m s 1 respectively.
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Q) '0 ;j :::! ...,
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58 more noticeable. Changing the frequency from w = 12 6 year1 to w = 6.28 y e ar1 increases the meridional scale which results in a poleward shift of the maximum height anomaly to about and a poleward broadening of the equatorial zonal jet. Qualitatively, the coupled and the uncoupled Rossby waves at w = 6.28 year1 are now distinctively different. A further decrease in frequency to the slow mode transition point at w = 2.5 year1 further accentuates the structural differences between this coupled mode and its counterpart uncoupled Rossby wave. The uncoupled Rossby wave shows a further increase in zonal wavelength to the extent of no longer fitting within any realistic bounded ocean basin, while the coupled mode has a much broader meridional extent, with the maximum height anomalies centered at about 1 0 o and with the zonal jet expanding to become a tropical, rather than just an equatorial feature. A further decrease in frequency for the slow mode wave would further increase its meridional dimension and decrease its zonal dimension as contrasted with the uncoupled Rossby wave whose zonal dimension would continue to increase. The eigenfunctions for higher meridional mode coupled and uncoupled Rossby waves compare in a similar fashion as those for the gravest mode. For the westward propagating slow mode wave, choosing the thermodynamic parameters to have their extreme values of either cr = 0 or Tl = 0 changes the scales, but not the qualitative differences between the slow modes and their uncoupled Rossby wave counterparts. This is shown for the height and velocity anomaly fields in Figure 19 and Figure 20 for the cases of cr = 0 or Tl = 0, respectively, with all of the other parameters as given in Figure 11a. Recall that for the cr = 0 case there is no distinct transition to a westward propagating slow mode, and hence the meridional scale for this case is smaller than it would be at the same frequency for nonzero cr, or for the 'fl = 0 case For both cases, however, the meridional scale is substantially larger than that for the uncoupled Rossby waves, since the governing scale for the meridional structure is the e folding scale, LR, which is sensitive to the value of the coupling coefficient.
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RWave ( cpl,n= 1 ,w=2. 5 yr1 ,a=O) 10 0 ....... +> ro .....:l 10 20 0 172 Longitude RWave (cpl,n = l,w=2. 5 yr1,a=O) 20 10 f. 1_,. ... 10 f. 20 172 L ongitude Figure 19. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n=l) Coupled, Equatorially Trapped Rossby Wave at the Frequency ro= 2.5 Year1 (Corresponding to a Periodicity of about 2.5 Years) for Case of the Warming Coefficient cr=O. The meridional and zonal dimensions are in latitude and longitude degrees, respectively, with the zonal dimension spanning one complete wavelength. The units for the height and current fields are m and m s 1 respectively. 59
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60 RWave (cpl,n= 1 ,w=2.5 yr1,7]=0) H'l Q) "'0 ;::l ...,J 0 ...... ...,J ro .....:1 10 Longitude RWave ( cpl,n= 1 ,w=2.5 yr1 ,7]=0) 10 , __ ...__ __ 0 .,._. .,__ _. 10 f 202 Longitude Figure 20. The Horizontal Structures of the Eigenfunctions for the Gravest Mode (n = l ) Coupled, Equatorially Trapped Rossby Wave at the Frequency ro=2.5 Year1 (Correspond i ng to a Periodicity of about 2.5 Years) for Case of the Mean Zonal Temperature Gradient T\=0. The meridional and zona l dimensions are in latitude and longitude degrees, respectively, with the zonal dimension spanning one complete wavelength. The units for the height and current fields a re m and m s 1 respectiv e l y.
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61 3.2.3. Kelvin and Eastward Slow Wave Modes Given the dispersion relationship of equation (3 .28) for the special case of the equatorial Kelvin wave, the associated eigenfunctions are u(y) = C2 _JQ_ ex.JkHo 1 2kc2 h(y) = c2 ex.J1 2kc2 T(y) = c 2 i(crkHo roll) rokHo ex.J 1 2kc2 (3 .44) (3 .45) (3 .46) Similar in format to the Rossby wave comparisons, Figure 21, Figure 22 and Figure 23 compare the height and velocity anomaly eigenfunctions for the coupled and uncoupled Kelvin waves at the frequencies ro = 12.6 year1 6.28 year1 and 2.5 year1 respectively. For the ro = 12.6 year1 case there is very little difference between the coupled and the uncoupled waves Meridional scale differences become noticeable for the ro = 6.28 year1 case, and upon the transition to the eastward propagating slow wave mode for the ro = 2.5 year1 case the scale differences are pronounced, with the meridional (zonal) scale being much larger (smaller). Without the modifications due to coupling the zonal wavelengths for low frequency Kelvin waves are much too large for these waves to fit within any realistic bounded ocean basin. The coupled Kelvin wave meridional scale, LK = has a simpler form than that for the coupled Rossby waves Using the dispersion relationship for the coupled Kelvin wave LK may be expressed as a function of frequency and the model parameters which, in the low frequency limit, becomes LK = Thus, LK has the same functional dependence on ro as does LR as shown in Figure 24. Note that LK may also be expressed as LK = Lo[c/(ro/k)]112, where L0 = 112 is the ocean's equatorial Ross by radius of deformation, showing that the meridional scale for the coupled Kelvin wave is larger than that for the conventional ocean Kelvin wave since ro/k < c at low frequency.
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K Wave (cpl,w= 1 2 6 yr1 ) 2 0 10 Q) '0 :!:! 0 .... j 10 20 0 224 KW ave ( c pl.w=l2.6 yr1 ) 20 10 . . Q) g r... 0f:;l rj 10 ... . . 20 L_ ______ L_ ______ 0 224 Longitude K Wave (uncpl, w = 12 6 yr1 ) 20 r..,, 10 0 10 20 L_ ______ L_ ______ L_ ______ J_ ______ 0 282 K Wave (uncpl,w = 12 6 yr1 ) 2 0 r..., H'l 0!t10 r20 0 282 Longitude Figure 21. The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency w=l2.6 Year1 (Corresponding to a Periodicity of about 6 Months) The left hand plates give the thermocline height and the velocity vector anomalies for the coupled waves and the right hand plates give the counterpart fields for the conventional, uncoupled ocean waves. The meridional and zonal dimensions are in latitude and longitude degrees, respectively, with the zonal dimension spanning one complete wavelength The units for the height and current fields are m and m s1 respectively.
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K Wave (cpl,w=6.28 yr1 ) KWave (uncpl,w=6.28 yr1 ) 20 2 0 10 10 QJ '0 ;::1 ..... 0 :;::l 0 ., ....:l 10 10 20 20 L_ ______ L__ _____ _,___ ______ ..J.._ ___ _, 0 277 0 566 KWave (cpl,w=6.28 yr1 ) KWave (uncpl,w=6.28 yr1 ) 20 20 10 10 QJ '0 ;::1 :::! 0 .... j 1 0 '"277 566 Longitude Longitude Figure 22. The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency ro=6.28 Year1 (Corresponding to a Periodicity of about 1 Year) The left hand plates give the thermocline height and the velocity vector anomalies for the coupled waves and the right hand plates give the counterpart fields for the conventional, uncoupled ocean waves The merid ional and zonal dimensions are in latitude and lon g itude degrees, respectively, with the zonal dimensio n s panning one complete wavelength. The units for the height and current fields are m and m s 1 respectively.
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Cl) '0 ::s ..... :;:: "' ...J K Wave ( cpl,w=2. 5 yr1 ) 2 0 10 0 10 20 0 188 KWave (cpl,w=2. 5 yr1 ) 2 0 r,... 1 0 f1'101+11F110 12 0 L_ ______ ..J...._ ______ _,_ ______ __,_ ______ ___j 0 100 L ongitude K W ave (uncpl, w = 2 5 yr1 ) 2 0 r.... 10 0 10 2 0 L_ __ __ __ J_ __ __ __ 0 1 4 2 1 K Wave (uncpl,w= 2. 5 yr1 ) 2 0 r,.., 10 1f01+11 0 2 0 L_ __ __ __ ..J...._ __ __ __ _,_ ______ __,_ ______ __j 0 1 421 L o ngitude Figure 23 The Horizontal Structures of the Eigenfunctions for the Equatorially Trapped Kelvin Waves at the Frequency w=2.5 Year1 (Corresponding to a Periodicity of about 2.5 Years ). The left hand plates give the thermocline height and the velocity vector anomalies for the coupled waves and the right hand plates give the counterpart fields for the conventional, uncoupled o c ean waves. The meridional and zonal dimensions are in latitude and longitude degrees, re s pectively with the zonal dimen s ion spanning one complete wavelength. The units for the height and current field s are m and m s 1 respectively.
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1000 .._ 8 625 i I i I I I i I i I i I i I i I i I i I i I i I i I i I i I i I i I i i I i i I i I i I i I I I I I I I I i I i I i I i i I i I i I i I i I 875 750 500 i I i \ i \ i \ i \ i \ \ \ \ \ .... .... ....... . .... .. .. .. = 375 250 I I I I I I 0 10 20 30 40 50 60 70 Figure 24 The E f o lding Scale LK for the OceanAtmosphere Coupled Kelvin Wave as a Function of the Frequency for Different Values of the Coupling Coefficient. The dotted dashed, solid, dashed and dotted lines correspond to the coupling coefficient values J..L=0.25xl07 m s 2 K1 J..L= 0.85 xto7 m s 2 K1 J..L=l.92xl07 m s2 K 1 and J..L=5. 0 xl07 m s 2 K 1 resp e ctiv e ly, with the solid line corresponding to the value used in the previous plots. 65
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66 Furthermore, for the parameters of this study, as shown by equation (3. 31), the pha se speed for the eastward propagating slow mode depends primarily upon the coupling coefficient, the warming parameter and the background buoyancy state for the reducedgravity ocean (specified by the ocean's Kelvin wave speed); zonal advection in the presence of a mean zonal temperature gradient 11 is of secondary importance. 3.3. Discussion and Summary The preceding sections described analytical solutions for equatorial wave modes of a simplified coupled oceanatmosphere system. The simplifying assumptions, necessary to obtain a closed form solution, were those of homogeneous parameters, equal values for Rayleigh friction, Newtonian cooling and thermal damping, and spatially uniform coupling between 'tx and SST. Homogeneous parameters are consistent with a model having no mixed layer formulation. The spatially variable mixed layer of the tropics makes this assumption tenuous to the extent that perturbations owing to the included ocean dynamics are small relative to those owing to the omitted mixed layer processes. Choosing Rayleigh friction Newtonian cooling and thermal damping coefficients of equal value is an expediency in factoring out dissipation in the form ert. This facilitates a solution in terms of Hermite functions and results in the continuous transition between the coupled Kelvin or Rossby waves and their slow mode counterparts. An analytical solution may also be obtained using unequal values of a lumped Rayleigh friction/Newtonian cooling parameter and a thermal damping parameter (Wang and Weisberg, 1995a). The most restrictive assumption appears to be the proportionality between 'tx and SST Results from numerical coupled oceanatmosphere models that include a mixed layer formulation suggest that this assumption is good near the equator, at least when averaged over the eastern half of the Pacific Ocean (Battisti and Hirst, 1989). Our calculations, using a numerical coupled oceanatmosphere model without a mixed layer formulation, show that the
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correlation between zonally averaged tx and SST is maximum on the equator and diminishes to zero poleward of 14 degrees Thus, spatially uniform 67 coupling between tx and SST is erroneous but this weakness is tempered by the fact that the gravest mode ocean (atmosphere) equatorial waves are forced primarily by winds (heating) near the equator Local coupling between tx and SST (without a zonal phase lag) is also erroneous since it precludes growing modes in this model. Nevertheless, despite the above limitations, the present analytical results provide insight into features of coupled model behavior which may have a more general application One such feature of coupled model behavior observed in our results is the increase in meridional scale at low frequency For a more general coupled oceanatmosphere system, with separate ocean and atmosphere physics there exists two distinctly different meridional scales determined by the i ntrinsic parameters of the ocean and the atmosphere. For linear perturbation, reducedgravity ocean and atmosphere models these scales are due to the models' equivalent depths. With the atmosphere scale larger than the ocean scale, th e meridional scale of the coupled oscillations tends to approach that of the atmosphere. In the present case, the intrinsic scales of the ocean and the atmosphere are the same ; however, a meridional scale broadening for the coupled oscillations still occurs A physical explanation follows from the coupling induced wind stress curl in the model vorticity equation With the correlation between tx and SST decreasing away from the equator, the assumption of uniform coupling may overestimate the broadening, but the broadening would occur by the wind stress curl even if the coupling and other parameters were nonuniform. Consistent with this Sverdrup transport influence on the vorticity balance, the coupled waves differ from the uncoupled waves only over the low frequency portion of the wave spectrum. Thus, the equatorially trapped inertialgravity and Rossbygravity waves are unchanged by coupling, whereas the Rossby and Kelvin waves are changed The two primary modifications are in the dispersion relationship and in the meridional scale of the eigenfunctions As the zonal wavenumber decreases and then as the
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68 frequency tends toward zero, the coupled Rossby and Kelvin waves show a continuous transition in the dispersion plane from short waves, nearly indistinguishable from their uncoupled counterparts, to modified long Rossby and Kelvin waves which then transform to westward or eastward propagating slow mode waves, respectively. Unlike conventional equatorially trapped waves, these slow mode waves have zonal wavenumber increasing with decreasing frequency, and for the westward propagating slow modes at a given zonal wavenumber, frequency increases with increasing meridional mode number. Along with the dispersion relationship, the meridional scale for the coupled oceanatmosphere waves is a function of the model thermodynamic parameters and frequency, increasing with decreasing frequency. Of the thermodynamic parameters, the coupling coefficient and the warming parameter are the most important ones; horizontal advection of the mean temperature gradient is of secondary importance. Upon omitting the warming parameter, which embodies the effects of horizontal divergence and vertical mixing on SST, the dispersion relationship does not show a transition to the slow mode waves, nor is the increase in meridional scale as large As a dynamical parameter, the ocean Kelvin wave speed is also important since it sets the background buoyancy state thereby affecting the ocean divergence. The slow mode waves herein are therefore interpreted as directly forced, slowly varying, oscillations feeding back to the atmosphere via the induced ocean divergence. For the parameter values chosen, the modifications to the Kelvin and Rossby waves by the oceanatmosphere coupling become noticeable at periodicities exceeding 6 months. Thus, the intraseasonal Kelvin waves observed in the Pacific (e.g ., McPhaden and Taft, 1988 and Johnson and McPhaden, 1993) and the superposition of Kelvin and Rossby waves describing the seasonal evolution of the equatorial thermocline in the Atlantic (e. g., Weisberg and Tang, 1987, 1990) appear to be correctly characterized by conventional equatorially trapped wave theory However, for periodicities of a year or longer, the modifications due to the oceanatmosphere coupling
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69 suggest that conventional trapped wave theory may no longer be applicable. This is supported by the Philander et al. (1992) high resolution, coupled GCM experiment wherein they find no evidence for ocean Kelvin and Rossby wave s on interannual time scales while these waves are prominent on shorter time scales The structural differences between the coupled waves and their conventional equatorially trapped wave counterparts also provides for speculation on why the low and high resolution coupled GCM experiments of Lau et al. (1992) and Philander et al. (1992) respectively, arrive at similar El Nino and La Nina states but in different ways The high resolution model can resolve all of the equatorial waves while the low resolution model can only resolve broader meridional scale slow modes. Given the coupled wave modifications, the question of the importance of socalled offequatorial Rossby waves for ENSO may be reexamined In a series of communications summarized by Graham and White (1991) oceanic processes occurring poleward of the equatorial waveguide (defined by a multiple of the ocean's equatorial Rossby radius of deformation) are argued as being important elements for establishing the character of the ENSO cycle. Specifically, it is argued that 1) the meridional scale for ENSO variability exceeds that of the oceanic waveguide; that 2) this scale broadening is related to Ekman pumping and that 3) the Ekman pumping, in tum, is related to equatorial SST. Increased meridional scale and the importance of Ekman pumping has been inferred from historical temperature data. Kessler (1990) found that the region of maximum vertical displacement for the 20 C i s otherm, during the period 1970 1987, was at l2 N near the western boundary and that the progression of offequatorial Rossby waves during the 1972 El Nino was related to the wind stress curl distribution associated with equatorial westerly wind anomalies. Where controversy has ensued is with the importance of these offequatorial Rossby waves upon reflection at the western boundary and subsequent eastward propagation as Kelvin waves. Kessler (1991) Battisti (1991) and Wakata and Sarachik (1991) argue that the equatorial Kelvin wave results primarily from the reflection of the gravest mode Rossby wave, le a,!i ng the former two authors to define offequatorial as the region poleward of 8
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70 and concluding, based upon western boundary reflection arguments, that off equator variations should not be a major factor in ENSO In contrast, Graham and White (1991) contend that coupled model simulations of ENSO are greatly altered if effects from poleward of 8 are neglected. The result of increased meridional scale for equatorially trapped waves of a coupled oceanatmosphere system, owing to wind stress curl, is consistent with the findings of both groups of investigators With the meridional scale of the coupled modes increasing beyond that of the oceanic radius of deformation, offequator variability becomes important, but for reasons other than conventional equatorial wave reflection. In this sense our results agree with the conclusion of Battisti (1991) that : II... an argument for a significant role for the offequatorial signals in the ENSO cycle must incorporate exotic effects These effects may not be exotic; they may simply reflect the interactive II forcing between the ocean and the atmosphere occurring slowly at low frequency. Even in the simplest oceanatmosphere coupled, linear perturbation, r ed uced gravity models (e. g., Hirst, 1988 and Wang and Weisberg, 1994a) the evolving spatial structures appear more Kelvinlike (Rossbylike) in the eastern (western) half of the ocean basin. A single Kelvinlike or Rossby like wave mode cannot describe these results. For example, the evolving structure ac ross the ba s in of the Hirst (1988) U1 mode is different from the eastward propagating slo w mode herein. At the very least, a wave interpretation would require a of waves with different m e ridional structures, consistent with a forced wave problem How these waves might reflect at the ocean boundaries, however, is conceptually difficult since the ocean is bounded while the atmosphere is not. In summary, solutions have been presented for equatorially trapped waves of a simplified coupled oceanatmosphere system. The effects of co upling are found to be frequency dependent. Inertialgravity and Rossbyg ravity wave s are not modified while the Kelvin and Rossby waves are, and as frequency tends toward zero these latter waves transform into slow eastward o r westward propagating modes respectively. The transition region from
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71 conventional to slow mode waves occurs where the intrinsic dispersion properties of the ocean medium are supplanted by the coupling between the ocean and the atmosphere. For the parameters used, the modifications by coupling become noticeable at periodicities longer than 6 months. The primary modifications are a decrease in phase speed and an increase in meridional scale. The meridional scale increase is consistent with both hydrographic observations and results from ENSOrelated models, and a physical explanation follows the wind stress curl induced in the ocean's vorticity balance by the airsea coupling. However, achieving a closed form solution necessitated several restrictive assumptions on the model thermodynamics, airsea coupling and dissipation The effects of these on the general nature of the findings warrant further study.
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72 CHAPTER4 ST ABU..ITY OF EQUATORIAL MODES IN A SIMPLIFIED COUPLED MODEL By omitting the phase lag between the wind stress and SST and assuming equal coefficient values for Rayleigh friction/Newtonian cooling and thermal damping, neutral modes are obtained over the full range of equatorial waves in Chapter 3 The present chapter, which is presented in Wang and Weisberg (1995a), extends the work of the last chapter. It shows how coupled Rossby and Kelvin modes may coexist with westward and eastward slow modes, describes how these modes may merge, separate and be destabilized by varying model paramete rs, and finally discusses how their meridional scales broaden. These coupled equatorial modes are also analysed in the fastwave and fastSST limits of Jin and Neelin (1993a). The chapter is organized as follows Section 4 1 presents the coupled model by assuming that the zonal wind stress and SST perturbations are proportional and separated zonally. Section 4.2 develops analytical solutions for both westward and eastward propagating modes, the properties of which are explored in section 4.3. The behavior of these modes in the fastwave and fastSST limits is presented in section 4.4. A discussion and s ummary are then given in section 4.5. 4.1 The Model Formulation The ocean dynamical equations are similar to those in Chapters 2 and 3 except with a longwave approximation. Momentum and mass are governed by s: au A ,ah 'tx u f' yv = g+ yu, at ax pHo pyu =ay (4.1) (4.2)
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73 s: ah H (au av u+ 0 +) =yh. at ax ay ( 4.3) Following Jin and Neelin (1993a), the parameter o, measuring the ratio of the time scales of adjustment for oceanic dynamics and SST, is introduced in Eqs. (4.1) and (4.3), with the fastwave and fastSST limits being defined by this ratio being s mall and large, respectively. The long wave approximation, with the zonal flow in the meridional momentum equation in geostrophic balance is valid if the zonal scale is large compared with the meridional scale. Consistent with this, tY is neglected because ENSO related wind perturbations are primarily zonal and tY has a smaller effect at large zonal scales than does tx. The long wave approximation filters inertialgravity, Rossbygravity and short Ross by modes out of the system [shown to be unaffected by coupling in (Wang and Weisberg, 1994b)] while maintaining the important Kelvin and long Rossby modes (e.g., Hirst 1988; Wakata and Sarachik 1991, 1994; Jin and Neelin 1993a). Choosing equal values for the coefficients of Rayle igh friction and Newtonian cooling in Eqs. (4.1) and (4.3) has been shown to be reasonable for th e long wave approximation by Yamagata (1985). The th ermodynamic equation i s aT + l)ucrhaT, at (4.4) where cr is the warming parameter, 11 is the mean zonal SST gradient, and a is a thermal damping coefficient. Equation (4.4) with constant parameters omits background state processes that are generally thought to be important. This model limitation will be discussed later. To form a coupled oceanatmosphere system, we need a way of specifying tx in terms of ocean variables. As Gill (1980) noted using a simple atmosphere model, the response of the tropical atmosphere to SST induced heating has a s ignificant east west asymmetry with the westerly winds to the west of being more intense than the easterly winds to the east of the SST
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induced heating. This suggests that the anomalous 'tx due to heating may be parameterized proportional to the anomalous SST. Thus, similar to that in Chapter 2, we assume 'tx to be a linear function of T lagged by a zonal phase angle e i.e. 74 y, t) = PllHoei9 T(x, y, t), (4.5) where ll is the oceanatmosphere coupling coefficient and i = Y1. Given these assumptions, the basic model parameters (also used in Chapter 3), are shown in Table 2. Table 2. Values for the Basic Parameters Parameter Ho c p ll (J Tl ex 'Y e 0 Value 200 m 2m s1 2 29 x 10 11 m 1 s1 0 85 x 10 7 m s 2 K 1 5 0 x 10 9 K m 1 s 1 5.0 x 10 7 K m 1 1 year1 1 year1 0 1 4 2 Soluti o ns of Coupled Equatorial Modes 4 2 .1. Westward Propagating Modes With a closed form of Eqs. (4.1) (4.5), we assume wavelike solutions of the form u(x y, t ) u(y) v( x y, t) v(y) h(x, y, t) = h(y) ei(kx (l)t>, (4.6) T(x, y, t ) T(y) 't x (x y, t) PllHoeie T(y)
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75 where k is a real zonal wave number and w is a complex frequency with real (Wi) and imaginary ( wi) representing the frequency and the growth ra te, respectively With Wr taken to be positive the direct io n of zonal phase propagation is determined by the sign of k Substituting Eq (4.6) into Eqs. (4.1)(4.4) yields ( yio pyv = ikg'h + J..Lei9T, pyu = gd h, dy (y+ ikHou + Hod v = 0, dy (aiw)T+11u=oh. Eliminating u, h and T results in the single equation for v 2 di + Ayll + B(lCy2) v = 0, dy2 dy where B ikc2p(aiw) c2[(yiocq(aiw) + and c = (g'Ho) 1 2 is the ocean equatorial Kelvin wave speed Using the substitution v(y) = (4.7) (4.8) (4.9) ( 4 1 0) (4.11) (4.12) (4.13) (4.14) (4.15)
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where ;{A )112 = ho Y we can transform Eq. (4.11) into Equation ( 4 .17) has analytical solutions for D 2DB = 2n + 1 A 76 D= 1 ( ) I I 2' 1 + 4BC/A2 ( 4. 16) (4. 17) (4.18) n = 1, 2, ... (4.19) which is the dispersion relationship. Here C1 is a dimensional constant, is th e Hermite polynomial of order n Substituting Eq (4 18) into Eq. (4.15) and using the dispersion relationship ( 4 19), we obtain v(y) = Ctexp [nA + B y2]HJi(A/2D) 112y]. 2(2n + 1) (4.20) A wellbehaved solution of (4 20) [v(y) 0 as y oo] requires Re(nA+B) > 0 Using Eqs (4 1214) and (4. 16) in (4 19), the dispersion relationship is rewritte n as Hoa!Jeie i2kc2 ( a iro) ;:.======================== = 2n + 1, V 2 + 4c2 ( airo)( yio ro)[( airo) ( yio ro) + Jl11ei9 ] (4.21) and the ensuing meridional eigenfunctions subject to the constraint Re(nA+B) > 0 are v(y) = Ctexp [nA + B y2]HJi(A/2D) 112y ], 2(2n + 1) (4.22)
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77 u(y) = L{(aim)(y[Hocrlleieikc2(aim)] d v} II dy (4.23) h(y) =Ho {ik(a+ [Cyiom)(aim) + 1111eie] li}, II dy (4.24) T(y) = l{[iHook + l)(y+ [Hoo(yiom) + ikc 21)J.d..Y_} II dy (4.25) (4.26) To obtain the anomalies of velocity, thermocline thickness and SST, Eqs. (4 23) ( 4.25) are then multiplied by the factor exp[i(kxm t)]. 4 2.2. Eastward Propagating Modes Like the conventional equatorially trapped waves, eastward propagating modes can be obtained with the meridional velocity component identically zero. Setting v = 0 and substituting Eq. (4 6) into Eqs (4 .1)(4.4), we obtain ( yio + ikg h 11ei6T = 0, = g.d.h, dy ikHou + (yio o)h = 0, l)u crh + (a im)T = 0 (4.27) (4.28) (4.29) (4.30) The algebraic Eqs. (4 27), (4 29) and (4 30) have solutions for u, h and T only when ( s: k2 2 iHo!!okeie (yio 0 ylu UJ + C + + , a im a im (4.31)
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78 which is the dispersion relationship for the eastward propagating modes, and their associated meridional eigenfunctions are h(y) = C2exjp(oro + iy) y 2J. (4.32) 1 2kc2 u(y) = C 2 (oro + iy) exjp(oro + iy) y2], kHo 1 2kc2 T(y) = C:{HokcrTl(oro + iy)] exjp(oro + iy) y2]. kHo(a iro) 1 2kc2 where c2 is a dimensional constant. 4.3 Analysis of the Solutions for the General Case 4 3.1. The Basic Features (4.33) (4.34) The basic features of stability, periodicity, and horizontal structure for the coupled equatorial modes for the general case o= 1 are now explored The horizontal structure of the modes can be described by taking the real part of Eqs. (4.6). The meridional velocity component, for example, may be expressed as v(x y t) = Re [v(y)ei(kxmt>]=v'vr + vtcos(kxrot+ cp), (4.35) where Vr and vi are the real and imaginary parts of v(y) and cp = tan1v/vr. Unlike uncoupled equatorial waves both the wave amplitude and phase are functions of y. The frequency ror and growth rate roi of the coupled equatorial modes as a function of wavenumber k with the model parameters of Table 2 are shown in Figure 25. The negative (positive) values of k represent westward (eastward) propagating modes; the dashed lines represent the gravest coupled
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79 Rossby or Kelvin (R or K) modes and the solid lines represent the gravest westward or eastward slow (WS or ES) modes. For the westward propagat i ng modes, only the gravest (n=l) modes will be discussed since higher meridional modes behave similarly. In the dispersion plane the coupled Rossby and Kelvin modes are distinguished from the westward and eastward slow modes in that the wavenumber magnitude increases with increasing frequency for the former and decreases with increasing frequency for the latter. At the transition points where these modes merge there is a branch within which the frequencies of the coupled Rossby (Kelvin) and the westward (eastward) s low modes are the same, as shown by the dashed line overlapping the solid line The merge point for the westward propagating modes occurs at relatively lower frequency and larger k than for the eastward propagating modes. The decay rates of all equatorial modes are the uncoupled oceanic damping rate of 1 yr1 for I kl exceeding the merge point values. However, for smaller I kl the coupled Rossby (Kelvin) mode is unstable (stable) whereas the westward (eastward) slow mode is stable (unstable) Upon merging, the coupled Rossby (Kelvin) and westward (eastward) slow modes split into two branches : one growing and one decaying It is noted that the dispersion relationship does not extend to the origin for the westward propagating modes owing to the c onstraint on (4.20) that Re(nA+B) > 0 The effects of positive and negative zonal phase differences e between the 'tx and SST anomalies are shown in Figures 26 and 27, respectively, using the model parameters of Table 2 but with 9=0.11t or 9=0.bt. Positive or negative e indicates that the 'tx anomaly is located to the west or east of the SST anamoly, respectively. In comparison with Figure 25, the coupled Rossby and Kelvin modes no longer merge with the westward and eastward slow modes. The frequency of the modes is independent of the sign of e whereas the stability properties reverse sign with e Positive e tends to destabilize the westward and eastward slow modes, while tending to damp the coupled Rossby and Kelvin modes. Negative e reverses these tendencies; that is, the coupled Rossby and Kelvin modes are destabilized whereas the westward and eastward slow modes are d a mped This means that if westerly winds are to the west (east) of the SST
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80 cx=36 0 y=360, 8=0 18 I \ I \ \ 15 \ I \ I \ /K 1 2 I I ";''"' I I \ I 9 \ >. \ I '' \ I .. I 3 6 \ \ I 3 0 14 10 6 2 2 6 10 14 10 6 r 2 '"' 1\ R I >. '' 2 3 I I WS I 6 I ./K 10 I I I 14 10 6 2 2 6 10 1 4 Figure 25. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with the Model Parameters of Table 2. The negative (positive) values of k represent westward (eastward) propagating modes. The dashed lines represent the gravest coupled Ross b y or Kelvin (R or K) modes and the solid lines represent the gravest westward or eastward slow (WS or ES) modes. The solid dots and open circles denote the points at w hich horizontal eigenfunction structures are presented in the subsequent figu res.
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cx=360, 'f'=360, 8=0.1rr 18 ' I I I 15 ' I I IK 12 I I ... I I I q,j 9 I ...... ' I 3 I I 6 I I I I 3 I I I ws 0 14 10 6 2 2 6 10 14 10 6 .:;. ... 2 q,j WS 1\ /_ ...... 32 , ... I I I R I 6 I I f'" K 10 I I I I 14 10 6 2 2 6 10 14 Figure 26. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Wavenumber k with 9=0.11t and Other Parameters of Table 2. The negative (positive) values of k represent westward (eastward) propagating modes. The dashed lines represent the gravest coupled Rossby or Kelvin (R or K) modes and the solid lines represent the gravest westward or eastward slow (WS or ES) modes The solid dots and open circles denote the points at which horizontal eigenfunction structures are presented in the subsequent figures. 81
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.:"' cc >. .._ .. 3 :. "' cc >. .._ 3 a=36 0 y=360 8 = 0.17T 18 ' 15 ' 12 'lt ' ' 9 ' ' 6 ' \ 3 ' 0 14 10 6 2 10 62R "' J' 2 ws 6 10 I I 14 10 6 2 I I I I I /K I I I I I 2 ES 6 10 14 ES 6 10 14 82 Figure 27 Frequency Wr and Growth R ate wi of Coupled Equatorial Modes as a F un ction of Wavenu m ber k with e=0.11t and Othe r Parame t ers of Tabl e 2. The negative (positive) values of k represent westward (eastward) propagating modes. The dashed lines represent the gravest coupled Rossby or Kelvin (R or K) modes and the solid lines r epresent t h e gravest westwa rd or eastwa r d s l ow (WS o r ES) modes The soli d do t s and open ci rcles denote the po ints at which horizontal eigenfunct ion structures are presented in the subsequent figures.
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83 anomaly, westward and eastward slow modes (coupled Rossby and Kelvin modes) are destabilized. In nature, warm waters induce atmospheric convergence resulting in westerly (easterly) winds to the west (east) of a warm anomaly (e. g., Gill 1980) Therefore, westerlies to the west of a positive SST anomaly are more realistic than westerlies to the east, implying that destabilization tends to favor the slow modes The horizontal eigenfunction's structures for the gravest westward (with k=3 8 x 10"7 m1 ) and eastward (with k=3.0 xto7 m 1 ) propagating modes with the model parameters of Table 2 are shown in Figures 28 and 29, respectively. This choice of wavenumbers, with magnitudes larger than the merge point wavenumbers, allows us to compare the relative structures between the westward Rossby and westward slow modes and between the east w1 rd Kelvin and eastward slow modes The associated points in the dispersion plane of Figure 25 are denoted by solid dots; the frequency of the slow modes being 2.5 yr1 and all modes being neutral if the uncoupled damping rate is factored out. The left hand plates give the thermocline height and the velocity vector anomalies for the coupled Rossby (or Kelvin) mode and the right hand plates give the counterpart fields for westward (or eastward) slow mode. The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength. The structures are the same as those of the coupled modes at 2.5 yr1 frequency shown in Wang and Weisberg (1994b) Note that the meridional scales for the slow modes are much larger than the oceanic equatorial Rossby radius of deformation. For example, the maximum height anomalies for the westward slow mode are centered at 1 0 from the equator. Similarly, the zonal jets for these slow modes expand to become tropical, rather than equatorial, features The meridional scales of the coupled Rossby and Kelvin modes, while smaller than their slow mode counterparts, are larger than those for uncoupled equatorial waves (Matsuno 1966). Other interesting points for eigenfunction comparisons are those for which the coupled Rossby or Kelvin modes have merged with their slow mode counterparts; for example, at k=3.0 x107 m 1 and k=l.9xto7 m1 denoted by open
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2 0N 2 0 S 0 75 150 20N !ON Ill "0 ;:l ' E Q :;; j lO S 20S 0 7 5 L ongitude 150 20N 10N _, _______ .... E Q , lOS 2 0 S 0 75 Longitude 150 Figure 28. The Horizontal Eigenfunction Structures for the Gravest Westward Propagating Modes with k =3.8xl o 7 m 1 Denoted by the Solid Dots in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2. The left hand plates give the thermocline height and the velocity vector anomalies for the coupled Rossby mode and the right hand plates give the counterpart fields for westward slow mode The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength.
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QJ o :::I ..... :;:; j QJ o 20N ION EQ lOS 20S 0 20N ION :::I _. EQ :;:; 1j lOS 195 190 I ... . 20S '''1 ____ ..._ ___ ___, 0 95 190 Longitude 20N ION EQ lOS ._ '" ....... 20S 0 95 190 20N ...., ION :EQ ::1lOS 120S 0 95 190 Longitude Figure 29. The Horizontal Eigenfunction Structures for Eastward Propagating Modes with k= 3.0xlo7 m1 Denoted by the Solid Dots in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2. The left hand plates give the thermocline height and the velocity vector anomalies for the coupled Kelvin mode and the right hand plates give the counterpart fields for eastward slow mode. The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength.
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86 circles in the dispersion plane of Figure 25. The horizontal structures for the gravest westward and eastward propagating modes with the model parameters of Table 2 are shown in Figures 30 and 31, respectively. The left hand plates give the SST and the velocity vector anomalies for the coupled unstable Rossby (or stable Kelvin) mode and the right hand plates give the counterpart fields for stable westward (or unstable eastward) slow mode Again, all these modes have a much broader meridional scale than for uncoupled equatorial waves For the westward propagating modes, the maximum SST anomalies are centered at from the equator; additionally, with the meridional phase difference, flows now reverse with latitude. Note that the velocity vectors for the unstable Rossby mode have opposite sign to those for the stable westward slow mode. Also, owing to the meridional phase shift imparted by the imaginary part of ro, the SST contours for the unstable Rossby mode and the stable westward slow mode tilt in opposite directions with latitude. For the eastward propagating modes, the velocity vector (zonal only) and SST anomalies have similar structures except for a zonal phase shift and an oppositely directed tilt. By comparing the relative positions of the velocity and SST anomalies for the unstable eastward slow mode (and the stable Kelvin mode), it is seen that eastward (westward) SST anomaly shift and westward (eastward) velocity anomaly shift results in a positive (negative) correlation between the wind stress that is proportional to SST and the velocity. With westerlies (easterlies) overlying oceanic eastward (westward) velocity for the coupled Rossby and eastward slow modes the necessary condition for instability of positive correlation between the wind stress and current, as given by Yamagata is satisfied enabling these modes to grow In contrast, the correlation between the wind stress and current is negative for the coupled Kelvin and the westward slow modes, so these modes decay The effects of a on the horizontal structures of the eigenfunctions for the gravest unstable westward (with k=3.0 x 1 o7 m 1 denoted by solid dots in Figures 26 and 27) and unstable eastward (with k=2.0x 1 o7 m1 denoted by open circles) propagating modes with a either 0 .11t or O.ht and other parameters of Table 1 are shown in Figures 32 and 33 respectively The left hand plates give
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Cll "0 ;:l ' :;:::; j Cll "0 20N ION EQ lOS 20S 0 95 190 20N ..... T 1 ON 1_____ .... ..... f.... :I :::! EQ 1. ......, ...... Ill l .. _______ ... . .. lOS f, , 2 0S ___ ....L_ ___ ____ __.__ _ _, 0 9 5 Longitude 190 lON lOS 20S L_ ___ ...J....=:=._l ____ ..J....::::::=__J 0 95 190 20N .... lON 1. EQ 1... .... ____ _ F. .. ... ...... lOS 1. 20S L__ ....J..... ___ __ __._ .J 0 95 190 Longitude Figure 30. The Horizontal Eigenfunction Structures for the Gravest Westward Propagating Modes with k=3.0x 1 o 7 m1 Denoted by the Open Circles in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2. The left hand plates give the SST and the velocity vector anomalies for the coupled Rossby mode and the right hand plates give the counterpart fields for westward slow mode. The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength.
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20N lON Q) '0 ;:l EQ ...> :;::; j lOS 20S 0 150 300 20N I lON 1Q) g f..,J EQ :;::; f.j 1lOS 1. 20S 0 150 Longitude 300 20N lON EQ lOS 20S 0 150 20N I lON 1:.EQ . 1lOS 300 . 20S 0 150 300 Longitude Figure 31. The Horizontal Eigenfunction Structures for Eastward Propagating Modes with k= 1. 9x 1 o7 m1 Denoted by the Open Circles in the Dispersion Plane of Figure 25 and the Model Parameters of Table 2 The left hand plates give the SST and the velocity vector anomalies for the coupled Kelvin mode and the right hand plates give the counterpart fields for eastward slow mode The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength.
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20N !ON v o ;:l EQ _, :;:; ro ....J lOS e 20S 0 95 190 20N ION Iv g EQ r. :.;:; ... j lOS 120S 0 95 190 Longitude !ON EQ lOS 20S 0 95 190 20N _. .. .:. :_ : :: : : : _ .. tON :: : :: :;, _: :_ __ .. _ __ t_'. '......... ... ; ... EQ .... ... .... _____ .... ______ .; ..... lOS _........... 20S 0 . : : : :. :._ _ : : : : : ,95 Longitude 190 Figure 32. The Horizontal Eigenfunction Structures for the Gravest Unstable Westward Propagating Modes with k=3 .0xi07 m1 Denoted by the Solid Dots in Figures 26 and 27. The left hand plates give the SST and the velocity vector anomalies for unstable Rossby mode and the right hand plates give the counterpart fields for unstable westward slow mode. The meridional/zonal dimensions are in latitude/longitude degrees and the zonal dimension spans one wavelength.
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20N 10 N Q) "0 ;:l EQ _, :;:; j lOS 20S 0 14 3 286 20N I 10N . Q) ] f:::: EQ r_, fj lOS . 20S 0 143 266 Longitude 20N 10N EQ lO S 20S 0 14 3 20N I lON r:fEQ f. f!FlOS I286 .. . 20S 0 143 266 Longitude Figure 33. The Horizontal Eigenfunction Structures for the Unstable Eastward Propagating Mod es with k=2 0 x10"7 m 1 Denoted by the Open Circles in Figures 26 and 27. The left hand plat es g ive t he SST and the velocity vector anomalies for unsta ble Kelvin mode and the right hand plat es g ive the counterpart fields for unstable eastward slow mode. The meridi o nal/zonal dim e n s ions are in l atitude/longitude degrees and the zonal dimension spans one wavelength.
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91 the SST and the velocity vector anomalies for the unstable Rossby (or Kelvin) mode and the right hand plates give the counterpart fields for unstable westward (or eastward) slow mode The meridional scales are again larger than for uncoupled equatorial waves and the meridional scales for the slow modes are larger than for the coupled unstable Rossby or Kelvin modes This reflects the frequency dependence of the meridional scale for coupled modes as shown by Wang and Weisberg (1994b). As shown in the next section the f requencies of coupled Rossby and Kelvin modes (slow modes) increase (decrease) with increasing e. Therefore, the meridional scales of coupled Rossby and Kelvin modes (slow modes) will decrease (increase) with increasing e The anomaly patterns facilitate growth owing to a positive correlation between the winds (proportional to phase shifted SST) and the currents. 4.3 .2. Dependence on Model Parameters Given the dispersion relationship and eigenfunction dependencies upon illdefined model parameters, the following sensitivity studies are presented. (a) Coupling Coefficient. The coupling coefficient J.1 sets the interactions between the ocean and atmosphere The effect of J.1 on the frequency and stability of the coupled equatorial modes with k= .5x 1 o 7 m1 and other parameters of Table 2 is shown in Figure 34. The trivial result of zero frequency for the slow modes when J.i=O is consistent with the slow modes owing their existence to coupling. The frequency of the slow modes increases with increasing J.1 until merging with the coupled Rossby and Kelvin modes. The coupled Rossby and Kelvin modes, which begin as the conventional Rossby and Kelvin modes for J.i=O, are strongly modified by coupling The frequencies of the coupled Rossby and Kelvin modes decrease with increasing J.1 until merging with the slow modes. After merging, the frequency of th e se modes is independent of J.1 except for the coupled Rossby and the westward slow
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.. 3 :.... i>.. '' 3 cx=360, 7=360, 8=0, k3.5 20 16 12 8 ...... 4 0 10 62 2 6 1014 0 .... ......... 1 ... ... _.., ,"" .... ... I / E/ ..... 2 3 K ... . R . WS I 1 2 3 4 4 92 Figure 34. Frequency <.Or and Growth Rate roi of Coupled Equatorial Modes as a Function of Coupling Coefficient J.L (10 7 m s 2 K"1 ) with k=.5 xi07 m l (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 The solid and dashed lines represent westward slow (WS) mode and coupled Ross by (R) mode, respectively The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively.
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93 modes upon reaching the eigenfunction constraint Re(nA+B) > 0. Before merging, the decay rate for all modes is the uncoupled damping rate of 1 yr1 After merging, the coupled Rossby and Kelvin modes can be destabilized whereas the slow modes decay more rapidly Thus, upon merging, coupled Rossby (Kelvin) and westward (eastward) slow modes develop two branches : one growing and the other decaying. Destabilization of the coupled Rossby or Kelvin mode and stabilization of the slow modes by increasing result from the induced positive and negative correlations between the winds and currents of these modes, respectively, for this parameter choice. The effects of on the coupled mode's frequency and stability are largely dependent upon the zonal phase lag e between the wind stress and SST anomalies As an example, for 9=0 .11t (not shown) the coupled Ross by and Kelvin modes do not merge with the slow modes and it is the slow modes that become unstable with increasing while the coupled Rossby and Kelvin modes decay Growth or decay is determined by the correlation between the lagged SST (wind) and ocean currents. (b) Warming Coefficient. In the model thermodynamics, the ocean processes controlling the SST anomaly are lumped into a term proportional to the thermocline thickness anomaly. This is parameterized by the warming coefficient cr as in Eq. (4.4) The effect of cr on the frequency and stability of the coupled modes with k=.0x1o7 m1 and other parameters of Table 2 is shown in Figure 35 The frequency for the slow modes is zero for cr=O, implying that the slow modes do not exist without the warming processes (Wang and Weisberg 1994b) With increasing cr the coupled Rossby and Kelvin modes merge with the westward and eastward slow modes For the coupled Ross by mode and the slow modes frequency increases with increasing cr. Before merging with the eastward slow mode the coupled Kelvin mode frequency decreases with increasing cr. After merging, it increases slowly with cr. All modes decay at the uncoupled damping rate of 1 yr1 before merging. After merging the coupled Rossby and Kelvin modes can be destabilized whereas the slow modes decay more rapidly with increasing cr.
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a=360,y=360,9=0,k3 2016 K .... .. 12 89 12 15 18 62 R .. 2 6 100 3 6 9 12 15 18 94 Figure 35 Frequency Wr and Growth Rate wi of Coupled Equatorial Modes as a Function of Warming Coefficient cr (10 9 K m1 s1 ) with k= 0 xlo7 m 1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 The solid and dashed lines represent westward slow (WS) mode and coupled Rossby (R) mode, respectively. The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively.
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95 As with ll the effects of cr are largely dependent upon e. With 9=0.11t (not sho wn), the coupled Rossby and Kelvin modes separate from the westward and eastward slow modes, respectively, the slow modes are destabilized as cr increases while the coupled Ross by and Kelvin modes decay (c) Zonal Mean SST Gradient. The rate of change of SST anomaly is proportional to zonal advection by the perturbation currents on a zonal mean SST gradient 11 The effect of 11 on the frequency and stability of the coupled modes with k= .Ox 1 o 7 m1 and the other parameters of Table 2 is displayed in Figure 36 Unlike ll and cr the existence of the slow modes is not dependent upon 11, as shown by Wang and Weisberg (1994b). With increasing magnitude for 11 the coupled Rossby and westward slow modes merge. For the gradient smaller than this merging value, the frequency of the coupled Rossby (westward slow) mode decreases (increases) with increasing gradient. After merging, the frequency of the coupled Rossby mode is identical with that of westward slow mode, and decreases with increasing the gradient until reaching the eigenfunction constraint. In contrast, the coupled Kelvin mode does not merge with the eastward slow mode throughout the range shown. The frequency of the coupled Kelvin mode (eastward slow mode) decreases (increases) with increasing gradient. As the gradient increases, the coupled Rossby mode becomes unstable whereas all other modes are damped. Neither the frequency nor growth rate of the eastward slow mode is sensitive to 11 Again, the stability properties change with 9=0 .l1t (not shown) whence the westward (in particular) and eastward slow modes are destabilized while the coupled Rossby and Kelvin waves are damped (d) Kelvin Wave Speed. The effects of varying the oceanic Kelvin wave speed c on the frequency and stability of the coupled modes with k=.0xlo7 m 1 and the other parameters of Table 2 are displayed in Figure 37. The frequency of the coupled Rossby and Kelvin modes decreases with decreasing c and conversely for the westward and eastward slow modes prior to their merging After merging, the westward and eastward slow modes share the same
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.;'"' ..... 3 20 16128a=360,7=360,8=0,k3 K ........................................ ... _!4 ES 0 T I 10 8 6 4 2 0 33 WS 6 10 8 6 4 2 0 TJ 96 Figure 36. Frequency ror and Growth R ate c.oi of Coupled Equatorial Modes as a Function of Zonal Mean SST Gradient ll (l07 K m 1 ) with k=.0xlo7 m 1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2 The solid and dashed lines represent westward slow (WS) mode and coupled Ross by (R) mode, respectively. The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively.
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97 a=360,7=360 8=0,k3 20162 6 K 10.... ... 0 1 2 3 4 5 c Figure 37. Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Oceanic Kelvin Wave Speed c (m s 1 ) with k=.0xl07 m 1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2. The sol id and das h ed lines represent westward slow (WS) mode and coupled Rossby (R) mode, respectively. The dotteddashed and dotted lines represent eastward s low (ES) mode and coupled Kelvin (K) mode, respectively.
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98 frequency with the coupled Rossby and Kelvin modes, respectively. Before merging all modes decay at the uncoupled damping rate of 1 yr1 After merging, the coupled Rossby and the eastward slow modes may be unstable while the coupled Kelvin and westward slow modes become more damped with decreasing c. The results for the eastward slow mode are consistent with the numerical findings of Wang and Weisberg (1994a) They argued that decreasing c decreases buoyancy, thereby increasing the divergence for a given value of surface current. Increasing the oceanic divergence increases the eastward slow mode growth rate. (e) Zonal Phase Difference. The effects of varying the zonal phase difference e between the wind stress and SST anomalies on the frequency and stability of the coupled modes with k=.6x10"7 m 1 and the other parameters of Table 2 are displayed in Figure 38. The frequency of all modes is symmetric about 9=0. For the slow modes, frequency decreases with increasing e and they do not exist for e larger than some parameter dependent set of values, the implication being that large zonal phase differences do not favor slow modes. On the other hand, the coupled Rossby and Kelvin mode frequencies increase slowly with increasing e. The growth rate of all modes is antisymmetric about 9=0 relative to the uncoupled damping rate of 1 yr1 The growth rate of the slow modes increases with increasing e, and e must be positive for instability Conversely, the coupled Rossby and Kelvin modes require negative e for instability The magnitude and sign of e therefore play an important role in instability properties of the coupled equatorial modes. (f) Rayleigh Friction/Newtonian Cooling. The Rayleigh friction/Newtonian cooling coefficient y sets the mechanical energy dissipation rate for the wave modes. Figure 39 shows the frequency and stability as a function of y with k=.0xt07 m 1 9=0.11t, and other parameters of Table 2. The frequency of the coupled Rossby and Kelvin (westward and eastward slow) modes increases (decreases) slowly with increasing y. The westward slow mode does not exist
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.... '7;;.... '" .. 3 28 24 ... K 20 16 12 8 .!t_ 4 0 1.0 0.6 a=360, '}'=360, k3.6 .... .... ... ... ..... , 0.2 0.2 .... 0.6 1 0 1 K ........ _.,..__ ES .:......... /;.s, . "'' . :..." .. .... .. ;:.... ',/ / 1 _,.,.;;" .. / ,.i ;... ,_!"" \ .. \ .. .. _,,._"':. _..,_'"':. ': ... 2 3 1.0 0.6 0.2 0 2 0.6 1 0 9 99 Figure 38. Frequency C.Or and Growth Rate c.oi of Coupled Equatorial Modes as a Function of Zonal Phase Difference e (7t) between the Zonal Wind Stress and SST Anomalies with k=.6xlo7 m1 (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2. The solid and dashed lines represent westward slow (WS) mode and coupled Rossby (R) mode, respectively The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively
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100 a=360, 8=0.11r, k3 16......... ............. .......... .......... ..... ........ .................... ... .r 12"" .__ .. 8 3 4_______ !ws 0 1 2 3 4 5 3 WS 0 ES .... 0 1 2 3 4 5 Figure 39 Frequency ror and Growth Rate roi of Coupled Equatorial Modes as a Function of Rayleigh Friction/Newtonian Cooling Coefficient 'Y with k=. 0xlo7 m1 and 8=0.11t (Positive and Negative Values of k Represent Eastward and Westward Propagating Modes, Respectively) and Other Model Parameters of Table 2. The solid and dashed lines represent westward slow (WS) mode and coupled Ross by (R) mode, respectively The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively.
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cx=360, ')'=0, 9=0.17T 18 \ I \ I \ 15 \ I \ I \ /K 12 I .;\ I ... \ I \ I \ I 9 \ j;lo, \ I ..._, \ I .. \ I 3 6 \ I \ I I I I I I 3 I I I 0 14 10 6 2 2 6 10 14 10 6 .:;. ... 2 1\ ;s j;lo, ..._, 3 2 I ' I ' I I R I I 6 I 'K 10 I I I I I 14 10 6 2 2 6 10 14 Figure 40 Frequency C.Or and Growth Rate C.Oi of Coupled Equatorial Modes as a Function of Wavenumber k with 9=0. l7t and y=O, and Other Parameters of Table 2. The negative (positive) values of k represent westward (eastward) propagating modes The dashed lines represent the gravest coupled Rossby or Kelvin (R or K) modes and the solid lines represent the gravest westward or eastward slow (WS or ES) modes 101
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102 for y larger than 2.5 year1 due to the eigenfunction constraint. The growth rate of the coupled Ross by, Kelvin, and westward slow modes decreases with increasing y In contrast, the growth rate of the eastward slow mode increases slowly with y, which under this set of parameters is due to zonal temperature advection term in the SST equation While y effectively damps the coupled Rossby and Kelvin modes, the westward and eastward slow modes can be unstable even for large values of y (decreasing k increases the eastward slow mode growth rate). In order to further illustrate the role of y in the coupled equatorial modes, the frequency and growth rate as a function of k with y=O, 9=0.l7t, and other model parameters of Table 2 are shown in Figure 40. Comparison with Figure 26 shows that the growth rates of the slow modes are relatively unaffected at all wavenumbers whereas the decay rates of the coupled Rossby and Kelvin modes are decreased Thus, the effect of Rayleigh friction/Newtonian cooling is mainly to damp the coupled Rossby and Kelvin modes, since y enters the solution through the dynamical equations (g) Thermal Damping. The role of surface heat fluxes in this model enter through the thermal damping coefficient a.. Figure 41 shows the effects of a. on the frequency and stability with k=. 0 x107 m1 9=0.11t, and other parameters of Table 2 The frequencies of coupled Kelvin and eastward slow modes are not sensitive to a.. In fact, i t will be seen in next section that the frequency of eastward slow mode in the fastwave limit is independent on a. For the westward propagating modes, the frequency dependence on a. is opposite to that of y in that the frequency of the coupled Rossby (westward slow) mode decreases (increases) with a.. The coupled Kelvin and Rossby modes decay for all values of a. while the growth rates for the slow modes decrease with a.. In a parallel analysis with the effect of the mechanical damping by y, the effect of the thermal damping a. on the coupled modes is shown in Figure 42, using a.=O, 9=0 .11t, and other model parameters of Table 2. Comparison with Figure 26 shows that the thermal damping mainly affects the slow modes. The
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103 ')'=360, 9=0.11T', k3 16......... .......... ..... .. ..... ...... ....... ....... .... . ... . .. .......... ... . 128 WS 4 R = .. .......... __ 0 1 2 3 4 5 6,, 3 ...... ............ ... ...... .... .... ................... . 3 ES R . 6 0 1 2 3 4 5 Figure 41. Frequency Wr and Growth Rate Wi of Coupled Equatorial Modes as a Functio n of Thermal Damping Coefficient c:x with k=.0xlo7 m1 and 9=0.11t (Pos itiv e and Negative Values of k Represent Eastward and Westward Propagating Modes Respectively) and Other Model Parameters of Table 2. The solid and dashed lines represent westward slow (WS) mode and coupled Rossby (R) mode, respectively. The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively.
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104 a=O, 7=360, 8=0.1rr 18 I \ I \ I \ 15\ I \ I \ fK 12I ..\ I ""' \ I eo: \ I \ I 9 \ \ I ...._, \ I 3 \ I \ I 6\ I \ I I I 3' ,'/"' WS ES 0 I I 14 10 6 2 2 6 10 14 10 6 .r 2 ""' eo: \_ WS ...._, 2 3 .... ...' ... ... I I R I I 6 I "K 10 I I 14 10 6 2 2 6 10 14 Figure 42. Frequency Wr and Growth Rate wi of Coupled Equatorial Modes as a Function of Wa v enumber k with 8=0. l7t and a=O and Other Parameters of Table 2 The negative (positive) values of k represent we s tward (eastward) propagating modes The dashed lines represent the gravest coupled Rossby or Kelvin (R or K) modes and the solid lines repre sent the gravest westward or eas tward slow (WS or ES) modes.
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105 slow modes no longer decay, whereas the growth rates for the coupled Ross by and Kelvin modes are unaffected Thus, the thermal damping mainly affects the slow modes since they originate from the SST equation in which a enters the coupled system. 4.4. The Coupled Modes in the FastWave and FastSST Limits Neelin (1991) and Jin and Neelin (1993a) introduced fast wave and fast SST limits for the purpose of distinguishing ocean dynamical modes from slow SST modes. To investigate the effects of these limits on the solutions given here, a related analysis is performed with the time derivative of the oceanic dynamical equations artificially decreased or increased by the parameter o, which measures the ratio of adjustment time by ocean dynamics to the net time scale from the SST equation 4.4.1. The FastWave Limit The fastwave limit is one in which the time scale td for the ocean dynamical adjustment is much smaller than the time scale tT of change in SST equation. While the ocean adjustment time by equatorial waves may still be large, it is smaller than the net adjustment time in the SST equation. In terms of the parameter o=td/tT this limit is given by o=O. Introducing o in the time derivatives of Eqs. (4.1) and (4.3) is equivalent to using tT as a time scale in the dynamical equations since oo/ot=O(O/td)=O(l/tT). With tT large the fastwave limit implies that the ocean dynamics are in relatively steady state with the thermocline tilt along the equator balanced with zonal wind stress as SST varies on ENSO time scale. Upon substitution o=O into Eq (4.21), the dispersion relationship of the westward slow mode in the fastwave limit is (4.36)
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106 Similarly, by seting in Eq. (4.31), we can obtain the dispersion relationship of the eastward slow mode in the fastwave limit 2 H J..Lcrkeie ,., ie 'Y 2 + k c2 + 1 o + y J..L.1e O a. iro a. iro (4.37) By writing ro=ror+iroi> Eq (4.37) becomes (4.38) J..L(Hocrksine YT1 cose) (l}j = a. + y2 + k2c2 (4.39) The frequency and growth rates of the coupled modes in the fastwave limit are shown as a function of k in Figure 43, with 9=0.11t and other parameters of Table 2 The fastwave limit filters the coupled Rossby and Kelvin modes out of the system, leaving only the westward and eastward slow modes which are unstable at small wavenumber. The variations of these coupled modes in the fastwave limit depend upon the time derivative of the SST equation instead of that in the dynamical equations, demonstrating that instability can exist in the coupled system without Rossby or Kelvin modes. The parameter dependence of the slow modes in the fastwave limit is consistent with the results presented in section 4.3.2 Two of the most interesting results are that: 1) the frequency of the eastward slow mode is independent on a. whereas its growth rate is mainly affected by a. and 2) given zonal advection instability for the eastward slow mode is facilitated by large y. The first implies that thermal damping affects the instability of the slow mode, but not the slow mode propagation The second implies that Rayleigh friction/Newtonian cooling tends to destabilize the eastward slow mode through zonal temperature advection within the SST equation. The frequency and growth rate of (4.38) and (4.39) are shown as a function of y in Figure 44, with 9=0.11t, k=3.0 x 1o 7 m 1 and other parameters of Table 2. The frequency decreases with increasing y whereas the growth rate first increases with
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107 a=360, 7=360, 8=0.17T, o=O 18 15 12 ":'""' t'O 9 ...... .. 3 6 3 0 6 4 0 2 4 6 10 8 6 r ... 4 ...... 2 3 0 4 4 0 2 4 6 k (10 7 m 1 ) Figure 43 Frequency and Growth Rate of the Coupled Equatorial Modes as a Function of k in the FastWave Limit (o=O) with 9=0. bt and Other Parameters of Table 2. Westward and eastward slow modes are the only modes to remain in the system.
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1 08 a = 360 8=0.17T, k3, o=O 4 3 .;1.1 4.> .... 2 3 1 0 0 10 20 30 40 50 1 0 10 20 30 40 50 Figure 44. Frequency and Growth Rat e of Eas t wa rd Slow Mode in th e FastWave Li m it Shown in Equatio n s (4 38) and (4.39) as a F u nc t io n of Rayleigh Friction/Newtonian Coo l ing Coefficient y with k=3.0 xl07 m 1 8=0. l1t and Other Parameters of Table 2
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109 increasing y reaching a maximum at an unrealistically large value of y= 1 3. 6 year1 Instability requires y larger than 3 year1 From (4.39) the growth rate increase is associated with the zonal mean SST gradient parameter T'l Physically, increasing y increases the relative magnitude of currents for a given h since H0ou/ox=yh for the eastward slow mode in the fastwave limit. The increase in eastward currents then advects more warm water eastward facilitating the slow mode growth. 4.4 2 The FastSST Limit The fastSST limit is one in which the time scale for oceanic dynamical adjustment is much larger than the time scale of change in the SST equation This limit is represented by a large value of o. Paralleling the fastwave limit argument, introducing the relative adjustment parameter 0 in the time derivatives of Eqs (4. 1) and (4.3) is now equivalent to using tT as a time scale in the dynamical equations since oo/ot=O(O!td)=O(l/tT). Therefore, the fastSST limit implies that the time scale of change in SST equation is small and that wave propagation in the dynamical equations is important in setting the ENSO time scale. Using a value of o=3 as an approach toward the fastSST limit, Figure 45 shows the frequencies and growth rates of the coupled modes as a function of k, with 9=0.11t and the other parameters of Table 2 Unlike the fastwav e limit, the coupled Rossby and Kelvin modes coexist with the slow modes. However, the coupled Rossby and Kelvin modes are largely modified by increasing o The magnitude of both the frequency and growth rate is reduced compared to the o = 1 case, as shown in Figure 26. Increasing o increases the wavenumbers at which the slow modes are destabilized In fact, increasing artificially the time derivative of the oceanic dynamical equations (by increasing o) results in an increase in the time scale td of oceanic wave propagation or a decrease in the time scale tT of change from SST equation This accounts for a decrease in the frequency of the Kelvin and Rossby modes Even without the coupling, increasing o results in a decrease in the frequnecy of both the convention a l
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110 cx=360, ')'=360, 8=0.17r 6=3 18 I I I 1 5 I I I I I r 1 2 K l I '"' I I QJ 9 I ..... ... I ... ... I .. ',.a I 3 I 6 ... ... I ... I ' I I 3 ' ; WS ; ; 0 14 10 6 2 2 6 1 0 1 4 10 6 r 2 a QJ WS ........ 3 2 ' .... _R I ; f1( 6 10 I I I I I 14 10 6 2 2 6 1 0 1 4 Figure 45. Frequency and Growth Rate of the C o uple d Equatorial Modes as a Function of k for Val u es of 5=3 with 9=0.17t an d Other Parameters of Table 2 The negative (positive) values of k rep resent westward (eastward) propagating modes. The dashed lines represent the gravest coupled Rossby or Kelvin (R or K) modes and the solid lines represen t t h e gravest westward or eastward s lo w (WS or ES) modes.
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.;' I.e ...... "' .. 3 .;' I.e ...... "' 3 24 201612840 0 2 0 2 I I I I I I I I I I \R \ \ \ \ \ \ ' cx=360, 7=360, 9=0.17T, k4 .K ... .... ...... ...... ES I 2 3 ... ... ... ws I 4 .... 5 .. .;, ;,.R 4 ... /"' 6 8 10 0 I I I ' ' ' ' I 1 2 3 4 5 Figure 46. Frequency and Growth Rate of the Coupled Equatorial Modes as a Function of the Relative Adjustment Time Parameter o with 111 k=. 0 x l o7 m1 9=0 .11t and Other Parameters of Table 2. The solid and dashed lines represent westward slow (WS) mode and coupled Rossby (R) mode, respectively. The dotteddashed and dotted lines represent eastward slow (ES) mode and coupled Kelvin (K) mode, respectively
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112 Rossby and Kelvin modes since the frequencies of the uncoupled long Rossby and Kelvin modes are ror=kc/[5(2n+1)] and ror=kc/5, respectively The frequency and growth rate as a function of 5, with k= .0x107 m l, 9=0.11t, and other parameters of Table 2 are shown in Figure 46. The frequencies of the slow modes are insensitive to 5, consistent with Neelin (1991) wherein coupled oscillations in distorted physics experiments with a GCM were independent of wave propagation time scale This is accounted for by 5 distorting physics only in the dynamical equations, whereas the slow modes result through the SST equation. On the other hand, the frequency of the coupled Rossby and Kelvin modes do vary largely with 5. As 5 increases from fastwave limit values, frequencies for both the coupled Rossby and Kelvin modes decrease toward the fastSST limit, approaching those of the slow modes This may explain why the ocean dynamics modes and the slow SST modes merge into complicated mixed modes in Jin and Neelin (1993a). For this choice of parameters, the westward and eastward slow modes are destabilized with increasing 5, so large 5 facilites slow mode instability In contrast, the coupled Rossby and Kelvin modes decay 4.5 Discussion and Summary A simplified, analytically tractable, coupled oceanatmosphere model is employed to gain insights on the stability properties of coupled equatorial modes. The paper extends the work of Wang and Weisberg (1994b) by relaxing the assumption of equal coefficients for Rayleigh friction/Newtonian cooling and thermal damping and by providing for a zonal phase separation between SST and wind stress anomalies. In doing this the model provides an analytical diagnosis for the thermodynamics of Hirst (1986). The two primary restrictive assumptions remaining are: 1) the proportionality between the wind stress and SST anomalies and 2) the spatial homogeneity in thermodynamic parameters. Allowing a zonal phase difference between the wind stress and SST anomalies in this chapter makes the first assumption more realistic. However,
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113 spatially uniform coupling is still erroneous. In nature, the coupling varies both in space and time For example, Wakata and Sarachik (1994) argued that coupling occurs differently during the warm and cold phases of ENSO and numerous authors have argued that coupling is largest over the eastern portion of the Pacific owing to the largest SST anomalies there. The second assumption of spatial homogeneity in thermodynamic parameters renders the SST equation overly simplistic since it omits background state processes ; for example, the reason why the SST anomalies are largest over the eastern Pacific. Nevertheless this simplistic SST equation provides an analytical basis for diagnosing instabilities that may be applicable to more realistic systems. One property of the solutions obtained herein is that the meridional scales for all coupled equatorial wave modes at low frequency are larger than those associated with an oceanic Rossby radius of deformation. A similar property was obtained by Wang and Weisberg (1994b) for neutral modes under a much more restrictive set of assumptions. In general, for a coupled system with separate atmosphere and ocean physics, there should exist two intrinsically different meridional scales with the atmosphere scale being larger than the ocean scale Since the atmosphere and ocean are coupled it follows that the scale of the coupled modes should take on an intermediate value. Under the present set of assumptions there is only one intrinsic scalethat of the ocean; however, a meridional scale broadening still occurs A physical explanation follows from the couplinginduced wind stress curl in the vorticity equation. Without wind stress curl, the meridional component of velocity must tend to zero at low frequency, leaving the intrinsic radius of deformation as the meridional scale With wind stress curl, the meridional component of velocity tends toward a slowly varying Sverdrup balance, thereby broadening the meridional scale. In the present anomaly model, with the meridional scale of the atmosphere imposed by the ocean through the assumption that wind stress is locally (albeit with a zonal phase lag) proportional to SST with a constant coupling coefficient, the increase in meridional scale may be overestimated However, the analytically determined increase in meridional scale provides a basis for understanding why all
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114 numerical models of the coupled oceanatmosphere system show meridional scales for coupled oscillations larger than the ocean's Rossby radius of deformation (e g Barnett et al. 1991; Chao and Philander 1993) which is also an observed property of the ocean thermocline with respect to ENSO (e g White et al. 1987, 1989 and Kessler 1990). The increase in meridional scale and the modifications to the horizontal eigenfunction structures due to coupling at low frequency also raises the question of tropical, extratropical teleconnections occuring directly as a consequence of coupled modes. The P plane dynamics upon which this model is built break down at a certain frequency, parameter range for which the scale of the eigenfunctions is too large; however, the role of coupled modes in tropical, extratropical teleconnections remains an interesting topic for future research. By including different values for the dissipation constants and a zonal phase lag between the wind stress and SST anomalies the model solutions have complex eigenvalues and eigenfunctions with properties of growth or decay and meridional phase gradients. These solutions reduce to the neutral modes of Wang and Weisberg (1994b) as a special case. The solutions herein thus allow for the existence of westward and eastward slow modes along with coupled Rossby and Kelvin modes. The gravest Rossby mode and the Kelvin mode coexist with the slow modes and two of these four modes, one propagating westward and another propagating eastward, can be destabilized by varying the model parameters. If 9=0 and y=a., the coupled Rossby and Kelvin modes merge with the westward and eastward slow modes, respectively When they merge, the coupled Rossby (Kelvin) and westward (eastward) slow modes split into two branches: one growing and the other decaying. However, for other parameter choices, the coupled Rossby and Kelvin modes remain distinct from the slow modes. Thus, the coupled Rossby and Kelvin modes generally have different structure and phase speed than their counterpart slow mode. All unstable (stable) modes display a positive (negative) correlation between the wind stress and current to satisfy the necessary condition for instability.
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115 The model parameters are critical to the coupled mode stability Among the parameters, the zonal phase lag e between the wind stress and SST anomalies is the most interesting The frequency for all modes is symmetric about 9=0, whereas the stability is antisymmetric about 9=0 relative to the uncoupled damping rate. Positive and negative e represents wind anomalies located to the west and east of the SST anomaly, respectively Therefore, coupled mode instability is determined by the relative position between the wind and SST (e g Battisti et al. 1989) The slow modes and coupled Rossby or Kelvin modes can be destabilized if the wind anomaly is located to the west and east of the SST anomaly, respectively The simplest atmospheric models [e g Gill (1980)] show the former, suggesting that unstable slow modes are favored by this coupled system. However, th e relationsh i p between SST, atmospheric heating and the resultant winds, especially in the western Pacific warm pool region remains in question and curiously the unstable Kelvin wave frequencies for negative e correspond to the intraseasonal Kelvin wave frequenc i es observed in the western Pacific (e.g. McPhaden and Taft, 1988) The stability effects of the coefficients 'Y for Rayleigh friction/Newtonian and a for thermal damping are also interesting The coupled Rossby and Kelvin (slow) modes are mainly damped by 'Y (a), since the coupled Rossby and Kelvin (slow) modes originate from the time derivatives of the oceanic dynamical (thermodynamical) equation, in which 'Y (a) enters the coupled system. Subtle effects, such as an increase in growth rate for the eastward slow mode with increasing 'Y through the effect of zonal temperature advection, are also found Given that ro= ro(k, J.L, cr, 11. c e, y, a), the stability properties of coupled modes, even in this relatively simple model, becomes a very complicated problem. Generally, however, an increa s e in the coupling, warming, and zonal mean SST gradient coefficients and a decrease in the Kelvin wave speed will increase the instability of a particular unstable mode Neelin (1991) and Jin and Neelin (1993a) argued that slow unstable SST modes and unstable ocean dynamics modes represent the fastwave and fast SST limits, respectively and that in a realistic parameter space, the coupled modes may be charaterized as mixed SST/oceandynamics modes. Employing
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116 analogous limits it is found that in the fastwave limit the coupled Rossby and Kelvin modes are filtered out leaving only the slow modes while in the fastSST limit all coupled modes coexist, but with the Rossby and Kelvin mode frequencies greatly reduced to those of the slow mode frequencies For this simplified model coupled instability can exist in the fastwave limit without the effects of Rossby and Kelvin modes and in the fastSST limit all modes are present but they become mixed in frequency range. Observations of ocean and atmosphere variability on ENSO time scale and GCM simulations of the coupled ocean atmosphere system of the tropics all remain unclear on the details of evolution: propagation may be eastward westward or stationary and the locations and durations of anomaly fields differ from event to event. Paradigms of the delayed oscillator and the slow SST mode have been developed in the literature along with their potential for connectivity in parameter space. In the present model an attempt is made to analyse the stability of equatorial modes in a relatively simple, analytically tractable model with limiting assumptions on the ocean thermodynamics and airsea coupling (uniform ocean parameters and uniform coupling). The results show a broad range of coupled Rossby and Kelvin modes and eastward and westward slow modes which may merge in parameter space and for which instability generally favors the slow modes, especially when accounting for wind anomalies being positioned to the west of SST anomalies as generally observed in nature The mechanism of the delayed oscillator depends upon the propagation of Kelvin and Rossby modes which, if coupled, must not decay. Since the delayed oscillator is not a property of models with uniform thermodynamic and coupling coefficients such as the this one (even if boundaries are added), the finding that the coupled Rossby and Kelvin modes tend to decay should not detract from that paradigm The important point is that regardless of the limiting model assumptions, it may be concluded that coupling profoundly affects the structure propagation and stability properties of equatorial waves at low frequency. Determining just where within the equatorial waveguide and with what efficiency this coupling occurs (the two most limiting assumptions of this model) would therefore seem
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117 to be critical in understanding the coupled airsea interactions that comprise ENSO For example Mayer and Weisberg (1994), using COADS, found a regression coefficient between SST and surface pressure to be a factor of two larger in the western Pacific than in the eastern Pacific despite the SST anomalies being much larger in the east
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118 CHAPfER5 LOW FREQUENCY VARIABILITY OBSERVED IN THE CENTRAL PACIFIC The Tropical OceanGlobal Atmosphere (TOGA) Program has provided an improved understanding of ENSO as an oscillation of the coupled ocean atmosphere system of the tropical Pacific Many numerical models of the coupled system have been developed to study the ENSO mechanisms and to predict this phenomenon. The need for in situ observations motivated the establishment of a basin scale network of moorings called the TOGA thermal array for the ocean (TAO) As part of TOGA TAO, upper ocean velocity, temperature and surface meteorological measurements were initiated at 0 170W in May 1988 This chapter describes six years of measurements at 0 170W, discussing the data relative to the dynamics and thermodynamics of the low frequen c y variability (Weisberg and Wang, 1995b) Specific focus is on intraseasonal to interannual variability, and relationships among quantities relevant to the oceanatmosphere system. Possible physical processes controlling variability in the central Pacific on different time scales are also discus s ed. The chapter is organized as follows. Section 5 1 gives a general description of the data, Section 5.2 presents analyses of the dynamics and thermodynamics, and the results are discussed and summari zed in Section 5.3 5.1. The General Description of Data Sets The instrumentation and processing procedures used for the o o 170 W monitoring site are described by Weisberg and Hayes (1995) A subsurface moored acoustic Doppler current profiler (ADCP) is used for velocity and a surface ATLAS mooring is used for t e mperature and winds. SST, subsurface temperature, surface wind, and air temperature data from other sites of the TOGA TAO array are also used to calculate the zonal SST gradient and the
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119 vertically integrated zonal pressure gradient at 0, 170W, as well as the latent and sensible heat fluxes across the equatorial Pacific throughout the study period. To examine the role of horizontal SST advection, the Reynolds' blended SST product (Reynolds, 1988) is used to estimate zonal and meridional SST gradients. Contours of the 10day low pass filtered zonal (u) and meridional (v) velocity components (cm/s) and temperature (0C) are shown as a function of depth and time in Figures 47a, b, and c, respectively. For the zonal component light shading denotes westward flow and dark shading denotes eastward flow in excess of 80 cm/s; for the meridional component light shading denotes northward flow and dark shading denotes northward flow in excess of 20 cm/s. For temperature (T) shading denotes values larger than 21 C with the darkest region being larger than 29C. The contour intervals for the velocity and temperature plots are 20 cm/s and 2 C, respectively In general, the Equatorial Undercurrent (EUC) core speed has been decreasing since 1988. At the beginning of the record the core speed was greater than 80 cm/s, reaching 125 cm/s in June 1988. Toward the end of the record the core speed decreased to around 40 cm/s This behavior is associated with a decrease in the zonal pressure gradient at the EUC core and the surface easterly wind stress that occurred during the transition from the La Nina conditions at the beginning of the record to the protracted El Nino of 199193, as will be further studied in the next section. Generally, a deepening and shallowing of the EUC core occur in winter and summer, respectively. The deepest depth is about 220 m and the shallowest depth is 80 m, with the average depth 160 m at this location Annual cycles are also observed in the surface South Equatorial Current (SEC) During the winter, the SEC is more intense and penetrates deeply with the mixed layer and conversely during the summer. During summer the surface flow may reverse to be eastward as the EUC shoals. Thus, the eastward surface flows in May/June 1989 and in July/August 1990 are contiguous with the EUC, whose depth is shallower during these times. However at other times reversals in the SEC appear to be due to other processes. During wintertime reversals in the near surface zonal current are
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Figure 47a Contours of the 10 Day Low Pass Filtered Zonal Velocity Component (cm/s) as a Function of Depth and Time with a Contour Int e rval of 20 cm/s Light shading denotes westward flow and dark s h a ding denotes eastward flow in excess of 80 cm/s
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170 W ucomponent (em/sec) 0 50 ,..... 100 s .._., 5 150 4) Q 200 250 300 MJ J A M A M 1988 D J F M 1990 0 50 ,..... 100 s .._., 5 150 4) Q 200 250 MJ J A S 0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A M 1991 1992 1993 1994 ....... N .......
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Figure 47b. Contours of the 10Day Low Pass Filtered Meridional Velocity Component (cm/s) as a Function of Depth and Time with a Contour Interval of 20 cm/s. Light shading denotes northward flow and dark shading denotes northward flow in excess of 20 cm/s.
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170 W vcomponent (e m /sec) 0 UooilliIIUIIIllilliiiiHIIIIIIIHiilf.lltalllilllll&mtlallhli IIIIU I l111 II H IIIIVIIIIII IIIIF'HIIUIIiiiiiiiii'IIJI'i r 191 HIIIM Mi II II IIIIIU h 111 ill lllliilii'IIIIIVHUI 50 ,... 100 s = 150 a Q 200 250 MJ J A S 0 N D J F 1988 1989 0 1111::1 V I ,..\::1\? I I 1111. 1 I I fill I lh tkRIIJI:::I All Ulfii Cili 11.1 : 1 I: :::.::.:11:1 a 1:1 Ill l 50 ,... 100 s = 150 =Q 200 250 0 N D J 1990 0 D J F M A M J J A S 0 N D J F M A M 1993 1994 ..... N w
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Figure 47c. Contours of the 10Day Low Pass Filtered Temperature (0C) as a Function of Depth and Time with a Contour Interval of 2C Shading denotes temperature larger than 21 o c with the darkest region larger than 29C
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170 W Temperature CCC) 0 .... 50 ,.... 100 5 ..= 150 ..... aJ Q 200 250 MJ J A S 0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A M 1988 1989 1990 1991 0 50 ,.... 100 s ; 150 fr Q 200 250 MJ J A S 0 N D J F 1991 1992 D J F M A M 1994 ...... N Vt
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126 associated with Kelvin wave propagation from the west. Beginning in November 1991 there is a particularly pronounced set of reversing jets with the onset of the protracted El Nifio event. Such jets are noticeably absent during the fall 1993 and the winter 1994 seasons. Another interesting result is that in December 1991 and January and March 1992 layers of westward flow are located between eastward flow above and below. Although Hisard et al. (1970) and McPhaden et al. (1988) have also observed this behavior farther west at 0, 170E and 0, 165E, respectively, this flow regime at 0, 170W is unique to the protracted ENSO warm event. In contrast to the zonal velocity, the meridional velocity oscillates at a relatively higher frequency; in particular, during fall 1988 and winter 1989 a set of very regular, 3week period oscillations associated with tropical instability waves are observed (Weisberg and Hayes, 1995). This was the only year of record thus far that has shown a prominent instability wave season in the westcentral Pacific. Also noted is that preceding and during the relatively warm years the subsurface northward flows are much stronger than those during the relatively cold years. This suggests that meridional transport, in view of the hemispheric asymmetry in the central Pacific, may be partly responsible for the warming since the center of the warm pool region has a southern hemisphere bias in concert with the Southern Pacific Convergence Zone (SPCZ). The isotherms show both annual and interannual variations. The thermocline, as characterized by the 20C isotherm, appears to be shallower in summer and deeper in winter. For the protracted El Nino, the duration of warmest waters (greater than 29C) of the 199192 El Nifio was about a half year beginning late in 1991 with a second weaker warming event occurring during the following year. During the initial 1991/92 warming the thermocline depth varies from being deep to shallow and during the secondary warming the thermocline depth remained relatively steady. Like the interannual variation, the annual variation in the thermocline depth also do not show a correlation with SST since only the interannual variation is large for SST while both the annual and the interannual variations are large for the thermocline depth.
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127 The vertical distributions of record length mean u, v, T, and tan1(Ri) at 0; 170W are shown in Figures 48ad The Richardson number Ri defined as a ratio of buoyancy frequency squared and velocity shear squared is estimated as Ri=(ag()T/()z)/[(au,az)2+(av,az)2], where a=8.75x106(T+9) oc1 is the thermal expansion coefficient (Hayes et al., 1991). Note that tan1(Ri) is approximately equal to Ri for small Ri. The EUC core corresponds to the thermocline. Above and below the EUC core the velocity shears are highest and the temperature gradient is smallest, whereas in the EUC core the velocity and temperature shears are minimum and maximum, respectively. Therefore, maximum Ri occurs in the EUC core, and relative minima in Ri are located above and below the EUC core with the lowest values above the core Consistent with shipboard turbulence measurements (e.g., Peters et al., 1988), this suggests that mixing is most intense above the EUC core, weak near the EUC core, and moderate below the EUC core, since turbulence levels vary inversely with Ri. Such mixing modulated by the vertical gradients of velocity and temperature implies that variations of the EUC along with the thermocline may play an important role in the thermodynamics of the oceanatmosphere system. The 30day low pass filtered time series of the vertically averaged tan 1 (Ri) of the mixed layer is shown in Figure 49. The mixed layer is estimated by finding the depth at which the contoured temperature is 0.5C less than the SST. Since 1988 the Ri has been slowly increasing while the EUC core speed has been slowly decreasing (see Figure 53). This suggests that the slow decrease in the EUC reduces mixing. Near the surface above the minimum Ri, the mean flow is toward the southwest, whereas below the minimum Ri the mean flow reverses to northeast. It implies that, on average, the center of upwelling associated with surface poleward Ekman flows and subsurface equatorward flow is located slightly north of the equator. Time series of the zonal (tx) and meridional (tY) wind stresses are shown in Figure 50, where each time series has been low pass filtered to remove fluctuations at time scales shorter than 30 days. The wind is northeasterly during the warm event, whereas it is southeasterly during the normal situation. It seems that the northeasterly wind during the warm event is due
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... e .c Q, Q ... e .c Q, Q (a) 0 50 100 150 200 250 40 20 0 20 40 60 80 U (cmls) ( c ) 100 150 200 5 1 0 15 20 25 30 35 ... e .c Q, Q ... e .c Q. Q (b) 0 \ 50 \ 100 l 150 200 250 15 10 5 0 5 10 15 V. (cmls) (d) 50 100 1 50 200 250 0.25 0 .50 0 75 1 .00 1 .25 1.50 tan.'(Ri) Figure 48. The Vertical D istributions of Record Length Mean a) Zonal Velocity, b ) Merid i on a l Vel ocity, c) Temperature a n d d ) tan 1(Ri) at 0, 170W. 1 28
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1.20 1.00 """' 0 80 ::;' 0 60 s 0 40 0.20 0 .00 MJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFM 1988 1888 1880 1891 188 2 1993 1994 Figure 49. The 30 Day Low Pass Filtered Time Series of the Vertically Averaged tan 1(Ri) of the Mixed Layer at oo, 170W.
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0.0 8 0 .0 4 0 .00 .() ,04 .() ,08 .() .12 0 .04 0 .02 0 .00 1.() ,02 .() ,04 .().06 MJJASONOJFMAMJJASONOJFMAMJJASONOJFMAMJJASONDJFMAMJJASONOJFMAMJJASOND J FMAM 11188 1GG1 1GG2 1993 1994 Figure 50 The 30Day Low Pass Filtered Time Series of Zonal Wind Stress and Meridional Wind Stress at 0, 170W.
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131 to the northeast trade wind extending south of the equator. The southward extension of the northeast trade wind is associated with the positive wind curl which induces the northward transport according to the Sverdrup balance. This northward transport may affect the warming in the equatorial central Pacific since the SPCZ is located in the southern hemisphere. The zonal wind stress displays both annual and interannual variabilities; in particular, an interannual variation is apparent. The zonal wind is strong except during winter and warm ENSO event. Beginning with the cold ENSO event in 1988 through the warm ENSO event in 199192 zonal wind stress has become weaker and weaker. There are three westerly wind burst events during the 199192 El Nifio, which may play an important role in dynamics and thermodynamics of the oceanatmosphere system. After the westerly wind burst events, the zonal wind stress strengthens The concept underlying most ENSO models is that the zonal wind stress that affects interannual variations of SST which in turn affects zonal wind. This SST zonal wind stress positive feedback manifested through a dynamical response has been studied in detail [see McCreary and Anderson (1991) and Philander (1990) for reviews] in connection with ENSO. The key feedback element is that the atmosphere couples with the ocean through momentum flux, which emphasizes the effects of zonal wind stress on oceanic dynamics However, as will be seen in Section 5.2.4, an SSTwind speed feedback manifested through surface heat flux may also be responsible for the warming and cooling in the central Pacific. 5.2. Dynamics and Thermodynamics 5 .2.1. Vertically Integrated Zonal Mom e ntum Balance The TOGA TAO data allow for the evaluation of some important terms in the vertically integrated equatorial zonal momentum equation,
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0 0 0 f puldz + f pvVudz =f pxdz + 't\ 300 300 300 where the zonal wind stress r = PaCo1 ui + viua, with drag coefficient C0=1.43x 10 '3 and air density p8=1.225 kg/m3 The nonlinear advection of 132 (5.1) momentum in Eq. (5.1) can not be estimated by the present data set. By combining the data at 0, 170W with the temperature data at 0, 140W and 0 165E, we can estimate the other three terms at 0, 170W. The vertically integrated zonal pressure gradient (ZPG) is estimated by centered difference between 0, 140W and 0, 165E. Time series of the zonal wind stress ('tx), the vertically integrated ZPG, and the vertically integrated local acceleration are shown in Figure 51, where each time series has been low pass filtered to remove fluctuations at time scales shorter than 30 days. On interannual time scales, 'tx and the integrated ZPG are observed to covary with the same magnitude These time series show a secular decrease from maximum values during the 198889 La Nif'i.a to m i nimum values during the 199192 El Nino While both time series increased after the 199192 El Nino period of westerly winds and wind bursts their magnitudes have not yet recovered to the previous La Nina magnitudes. On intraseasonal time scales, covariation is observed in all three time series. In particular, the westerly wind burst events during the October 1991 to April 1992 period are associated with sign reversals of the integrated ZPG, with the ZPG lagging 'tx by about one month. This implies that the ZPG is responding to 'tx. Unlike the integrated ZPG, the integrated local acceleration oscillates with relatively larger amplitude at intraseasonal frequencies. The frequency mismatch between the integrated ZPG and local acceleration suggests that the TOGA TAO does not resolve the spatial structure of these higher frequency oscillations. A closer spacing b e tween TAO moorings would be required The covariability among these three time series may be further explored using a coherence analysis. If 'tx and the integrated ZPG balance, a s appears to be the case at interannual time scales, then there is no nee d for an integrated local acceleration. If such balance does not obtain as would be
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0 .08 0 .04 .;0 E o o ... .._ .0.04 .0.08 .0. 12 MJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASOND 1988 1989 1990 1992 1993 Figure 51. The 30Day Low Pass Filtered Time Series of Zonal Wind Stress, Zonal Pressure Gradient Vertically Integrated from 0 to 300m, and Local Acceleration Vertically Integrated at 0, 170W.
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134 P eriod (days) 167 83 56 42 33 1.0 o s o.o 0 000 0.0 0 6 O.OIZ 0 018 0 024 0.030 0.036 Frequency (cpd) Figure 52. Ordinary Coherence Analysis between 'tx and ZPG and Multiple Coherence Analysis between both 'tx and the ZPG with the Local Accel e ration From top to bottom are the ordinary coher e nce between 'tx and the ZPG, the multiple coherence between both 'tx and the ZPG with the local acceleration the p a rtial coh e rence between the ZPG and the local acceleration after removing the influence of 'tx from both of these time series and the partial coherence between 'tx and the local acceleration after removing the influence of the ZPG from both of these time series Averaging was performed over a bandwidth (6B) of 0 0036 for approximately 17 degrees of freedom
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135 expected for time scales shorter than annual (since it takes approximately one year for a Kelvin wave and its first mode Rossby wave reflection to transit the basin), then an integrated local acceleration would result as the residual between 'tx and the integrated ZPG. Two types of coherence analyses are thus presented: an ordinary coherence analysis between 'tx and the integrated ZPG, and a multiple coherence analysis between both tx and the integrated ZPG with the integrated local acceleration. The results are shown in Figure 52. From top to bottom are the ordinary coherence squared between 'tx and the integrated ZPG, the multiple coherence squared between both 'tx and the integrated ZPG with the integrated local acceleration, the partial coherence squared between the integrated ZPG and the integrated local acceleration after removing the influence of t x from both of these time series and the partial coherence squared between tx and the integrated local acceleration after removing the influence of the integrated ZPG from both of these time series The ordinary coherence squared shows prominent bands of coherence at interannual and intraseasonal time scales. While not shown, the phase is zero and the transfer function amplitude is near one over the interannual band, implying that these two terms are nearly in balance. At intraseasonal time scales the phase is positive with the ZPG lagging 'tx so these two terms can not be in balance. However, over intraseasonal time scales, the multiple coherence between both tx and the integrated ZPG with the integrated local acceleration is high. Comparing this with the partial coherences, either excluding the ZPG or tx, shows that both of these two time series are of equal importance in determining the vertically integrated local acceleration. These findings support the conceptual assertion given above. At interannual time scales 'tx and the integrated ZPG tend to balance, while at intraseasonal time scales the imbalance between these two time series gives rise to a vertically integrated local acceleration. The transfer functions (not shown) also support the assertion that the TOGA TAO fails to resolve the ZPG on intraseasonal time scales. The transfer function amplitude increases with increasing frequency, suggesting that the magnitude of the ZPG becomes increasingly
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underestimated as the spatial scales of variability decrease relative to the array spacing. 136 The inphase covariability between 'tx and the integrated ZPG at interannual time scales may be particularly relevant to ENSO. Physically, the density and pressure gradients associated with the eastwest slope of the thermocline in the Pacific are supported by an easterly wind stress. As the wind stress slowly decreases so will the zonal pressure gradient, and if this causes an increase in SST to the east then the easterly wind stress will further decrease. A continuation of this positive feedback within the coupled ocean atmosphere system leads to the mature phase of ENSO. 5 2 2 The ZPG in Relation to the EUC Since the ZPG is the driving force for the EUC, it must play a primary role in maintaining the EUC. Figure 53 shows 30day low pass filtered time series of the EUC core depth, the EUC core speed, and the ZPG at the EUC core depth. Both the core depth and speed show annual and interannual variations, with the speed tending to be small when the core is deep in winter and during the warm phase of ENSO, and conversely in summer and during the cold phase of ENSO. Interannually, there has been a slow decrease in the EUC core speed along with a slow increase in the EUC core depth associated with a slow decrease in the zonal pressure gradient at the EUC core depth, since the beginning of the record. The interannual variations of these three time series are consistent with the evolution of ENSO from the La Nifia condit ions of 1988 through the protracted El Nifio conditions of 199194. During this time the oceanatmosphere system associated with the western Pacific warm pool region has sidled eastward, resulting in reduced easterly wind stress, zonal pressure gradient and elevated thermocline over the central Pacific. A coherence analysis between the EUC core speed and the ZPG at the EUC core depth is shown in Figure 54. Bands of coherence are found at interannual and intraseasonal time scales with the ZPG at the EUC core
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... 240 8 210 u ;;::;, ,...._ 180 w s 'o .._.. 150 .c c. 120 ... eo 150 125 100 75 50 25 0 0 .60 0 45 o r o3 us o ('I z 0 15 0 .00 co I .0. 15 .(), 30 MJJASONOJFMAMJJASONOJFMAMJJASONDJFMAMJJASONDJFMAMJJASONOJFMAMJJASONOJFM 1eee 1eu 1eeo 1ee1 1ee2 1ee3 1ee4 Figure 53 The 30Day Low Pass Filtered Time Series of the EUC Core Depth, the EUC Core Speed, and Zonal Pressure Gradient at the EUC Core Depth.
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"0 eo: :I a' Cl:l CJ c= ..c: 0 u "0 :I "= Q. s < l:ll ,..._ 1.00 "0 Q, Cl:l u ;:J 0 50 l:ll > Q, co 0 00 '' 220 llO 0 3 14 _g 0 0 0 Q,.. 3 .14 Peri od ( d ays) 167 83 56 42 0 000 0.006 0.012 0 0 18 0 .07.4 0.030 Frequency ( cp d ) Figure 54. Coherence Analysis between the EUC Core Speed a n d the 138 28 0.036 Zonal Pre ss ure Gradient at the EUC Core Depth. From top to b ottom are the coherence squared, transfer fu nction amplit ude, and phase angle Averaging was performed over a bandwidth of 0 0036 for approximately 17 degrees of freedom and significant level on coherence squared is 0 .27.
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139 varying in phase with the EUC core speed. If we hypothesize that a linear momentum balance exists at the EUC core, i.e (1/p)ap/ax = Aa2u/az2 then we can estimate a vertical eddy viscosity coefficient from the transfer function amplitude that would be necessary to balance the ZPG Such eddy viscosity would have to be order 1 m2fs which is unrealistically large by a factor of 103 (e.g., Lien et al., 1995) We must therefore reject this hypothesis The inphase variability between the ZPG and the EUC core speed therefore suggests that the momentum balance at the EUC core must be nonlinear. This is consistent with the early theory for the EUC (e.g., Fofonoff and Montgomery, 1955) inferences from the Atlantic observations (e.g., Tang and Weisberg, 1993) and recent observations from the central Pacific (Qiao and Weisberg, 1995) While the vertically integrated momentum balance may tend to be linear, the EUC appears to be nonlinear. 5 2.3 SST and Thermocline In the eastern equatorial Pacific interannual variations in SST and thermocline depth are large and highly correlated, whereas the annual changes in SST and thermocline depth are not as well correlated (e.g Hayes et al., 1991). This may be due in part to the annual cycle in the thermocline depth being relatively weak compared to its interannual variability, whereas SST and wind stress both have large annual cycles. Thus, the annual cycle in SST appears to be more correlated with the local winds than with the thermocline depth. What is the relationship between SST and the thermocline depth in the central Pacific and what does this suggest regarding the maintenance of SST on annual and interannual time scales ? A comparison between SST and thermocline depth (as represented by the 20C isotherm) is shown in Figure 55, where the time series have been smoothed with a 30day low pass filter. Interannual variability exceeds annual variability for SST, whereas interannual, annual and intraseasonal variations are all prominently observed in the thermocline depth. Thus the thermocline depth does not appear to be well correlated with SST in the central Pacific. For
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30 29 28 27 26 25 24 210 liS 180 165 150 135 120 105 MJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAM 1i88 1i8i 1ii0 1K1 1i92 1993 19i4 Figure 55. The 30Day Low Pass Filtered Time Series of the SST and the Thermocline Depth as Represented by the 20C Isotherm.
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141 example, the thermocline was deepest in the winters of 1989 and 1990 as o ppos e d to during the warmest phase of the 199192 El Nino, It was also shallowest in April 1992 while SST was still very high Contrasted with the thermocline depth, the interannual variations of SST at this central Pacific location appear to be more correlated with the zonal wind stress (Figure 50). Specifically, the slow warming in SST observed from the 1988 La Nifia to the 199192 El Nino appears to be associated with the slow weakening in zonal wind stress and, after reaching warmest temperatures in 1992, the slow decrease in SST appears to be associated with the slow strengthening in zonal wind stress. The interannual correlation between the SST and zonal wind stress may be due to several factors including a positive SSTzonal wind stress feedback related to ocean dynamics The basis for this ocean dynamical feedback is that increasing SST decreases easterly wind stress which, in tum, affects the ocean through momentum flux to increase SST further through variations in thermocline depth due to either upwelling or entrainment. Interestingly, both the annual and interannual variations of thermocline depth are correlated with the annual and interannual fluctuations of the EUC core speed (Figure 53) In general, a deep (shallow) thermocline is asssociated with a weak (strong) EUC on both time scales Annually, the deep (shallow) thermocline corresponds to the weak (strong) EUC in winter (summer) Interannually, beginning with the 1988 La Nifia throughout the 199194 El Nifio, the EUC core speed slowly decreases as the thermocline deepens Varying the easterly wind stress varies the zonal pressure gradient. Since the zonal pressure gradient is the driving force for the EUC, there should be a correlation between the zonal pressure gradient, as reflected in the thermocline depth, and the EUC core speed As will be seen in Section 5 2 5 another factor that may affect warming and cooling in the central Pacific is an SST wind speed feedback operating through surface heat flux as contrasted with momentum flux As will be further developed, both SSTzonal wind stress and SSTwind speed feedbacks appear to be responsible for the slow variations observed in the central Pacific during this transitional period from the La Nina conditions of 1988 through the protracted El Nino conditions of 1991199 4
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142 5.2.4. Horizontal SST Advection The changes of SST are controlled by horizontal advection, vertical advection, vertical mixing (entrainment), and surface heat flux. The relative importance of these processes is difficult to estimate using TOGA TAO data since the processes are not fully resolved. The contribution to the local change of SST at 170W may be estimated using the velocity at 170W and the zonal SST gradient computed from the TOGA TAO moorings at 165E and 140W This is shown in Figure 56 where the time series have been further low pass filtered to remove oscillations at time scales shorter than 90 days owing to the scale mismatch between velocity and zonal SST gradient. The zonal velocity here is calculated as the vertically averaged velocity of the mixed layer. The local change of SST (the bold line) is found to lead the zonal advection (the solid line) contribution by 12 months. Thus, while they are the same order of magnitude, this phase lag suggests that the zonal advection can not cause the local change of SST in the central Pacific Since the zonal SST gradient may be poorly estimated by this largescale TOGA TAO SST difference the calculation was repeated using the Reynolds' blended SST product. The results for zonal advection, meridional advection and their sum are shown in Figure 57, each curve (the solid line) being compared with local change of SST (the bold line) Qualitatively, with this degree of smoothing, the zonal advection estimates using the TOGA TAO SST gradient and Reynolds' blended SST gradient are similar. The meridional advection calculated this way is smaller in magnitude and more out of phase with the local change of SST than the zonal advection During the onset of 199192 El Nifio the warming induced by the meridional advection does occur simultaneously with the local change of SST and in general while the magnitude of meridional advection is smaller than that of zonal advection, their sum does improve the comparison with the local SST change. However, the sum still lags the local change of SST, indicating that physical processes other than horizontal advection must be important in
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0 4 0.2 c 4.1 0 0 co. c.!) .().2 =o < Eo .().4 .().6 MJJASONDJFMAMJJASONDJFMAMJJASONDJF MAMJJASONDJFMAMJJASONDJFMAMJJ ASOND 1888 1888 1880 1881 1882 1883 Figure 56. The 90Day Low Pass Filtered Time Series of Zona l Advection (the Solid Line) and Local Change of SST (the Bold Line) at 0 170 W The zonal SST gradient is es timated from the TOGA TAO array SSTs at 0, 165E and 0, 140W.
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MJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFM 1G88 1G88 1880 1881 1882 1983 1984 Figure 57. The 90Day Low Pass Filtered Time Series of Zonal Advection, Meridional Advection, and Sum of Zon al and Meridional Advections. Each curve (t he solid line) is compared with local change of SST at 0, 170W from TAO array (the bold line). The SST gradients is calculated using the Reynolds' blended SST product.
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145 controlling SST variations on these intraseasonal time scales. Also note that, on average, urn aT J()x is negative, indicating the advection of cold water into the region and hence a cooling effect. This also suggests that the warming in the central Pacific must be controlled by processes other than just zonal SST advection on all time scales. 5 2.5. Latent and Sensible Heat Fluxes The net surface heat flux across the oceanatmosphere interface is composed of shortwave radiation, longwave radiation, sensible heat, and latent heat. Since the bulk formulae of the shortwave radiation and longwave radiation depend on the cloud cover, we can not estimate them using the present data set (which provides winds, SST and air temperature). Bulk formulae, similar to those of Hayes et al. (1991), are used to calculate the sensible (Qsen) and latent (Qr.81) heat fluxes Qsen = paCpCE Ua(TTa), QLat = PaLCe Ua[q,(T) RHq,(Ta)], (5.2) (5.3) where T and Ta are the SST and air temperature, p8=1.225 kg m3 is the air density, Ce=l.2x103 is the exchange coefficient, Cp=3.94x103 J kg1 oct is the heat capacity, L=2.44x 106 J kg1 is the latent heat of vaporization, RH is the relative humidity fixed at 0.8 and Ua= + vi is the surface wind speed. The saturated moisture content q5 is given by the ClausiusClapeyron equation (5.4) where q0=6.14x 1 o 4 and T is in degrees Kelvin. The estimated sensible and latent heat fluxes, the Bowen ratio (defmed as the ratio of the sensible to latent heat fluxes) and the SST, air temperature difference are shown, as a function of longitude and time on the equator, in
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146 Figures 58a, b, c and d, respectively. These quantities are calculated using TOGA TAO SST air temperature and surface wind data smoothed with a 10day low pass filter For the sensible heat flux the contour interval is 20 W /m2 and shading denotes positive values with the darkest regions in excess of 20 W/m 2 For the latent beat flux the contour interval is 40 Wfm2 and shading denotes values larger than 60 W/m 2 with the darkest in excess of 100 Wfm2. For the Bowen ratio the contour interval is 0 2 and shading denotes values larger than 0 2 Most of the time, the sensible heat flux is positive, reflecting the fact that SST is higher than air temperature. The largest values are found over the western half of the basin. While an annual cycle is not readily apparent at any longitude, there does exist a slow, secular increase in sensible heat flux across the entire equatorial domain begi nning with the 1988 La Nina conditions and continuing through 199192 El Nino, with the largest values extending into the westcentral portion of the basin Sensible heat flux remains elevated, but begins a slow decrease from the peak of the El Nino through the end of the record. The latent heat flux variability is very similar to the sensible heat flux Again there is a secular increase, peaking at the height of the 199192 warm event, followed by a slow decrease. This variability is also reflected in the Bowen ratio, showing that (in a relative sense) the interannual variations in the sensible heat flux exceed those in the latent beat flux across the entire basin. Since the sensible and latent heat fluxes enter the SST equation through the net surface heat flux, their increase (decrease) represents a cooling (warming) effect upon the ocean. During the warm phase of ENSO, the maximum heat loss from the ocean to the atmosphere, which had been located in the west [consistent with climatological data (Esbensen and Kushnir, 1981 ; Oberhuber 1988)], penetrates eastward into the central Pacific. The i ncrease in the sensible and latent heat fluxes over the central (and eastern) Pacific during El Nino requires that the water that has been warmed remains in contact with relatively cold, dry air. In fact, the largest difference between SST and air temperature along the equator during the transition from the 1988 La Nina to the peak of the warm event in 199192 was in the central Pacific
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Figure 58a. The Estimated Sensible Heat Flux as a Function of Longitude and Time on the Equator. The sensible flux is calculated by using the 10day low pass filtered SST, air temperature and surface wind of the TOGA TAO data Shading denotes positive flux with the darkest shading in excess of 20 Wfm2 and a contour interval of 20 Wfm2.
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Sensible Heat (W/m2 ) I ,,.,,,./,. ... @.1 ,,, ...; \1.1111 ,,\ J..Jt, ,JL,., ,tJ. ,,.,.,,,,,.,, fit Wffff.m!v /,IU.L .. IDIWAiJ, ,ifil'.:00
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Figure 58b. The Estimated Latent Heat Flux as a Function of Longitude and Time on the Equator The latent flux is calculated by using the 10day low pass filtered SST, air temperature and surface wind of the TOGA TAO data Shading denotes flux l arger than 60 Wfm2 with the darkest s hading in excess of 100 W Jm2 and a contour interval of 40 W Jm2.
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Latent Heat (W/m2 ) 170E 180 170W 160W 150W 140W 130W 120W llOW 'f'.FJ n,u,u, I " ,,. . ... ,n 'fn:n pr ; p;u,J::tpF"f n,uvt, u, pF::rntt f ," .... 'r j,FW 170E 180 170W 160W 150W 140W 130W MJ 1991 J A S 0 N D J F M A M J 1992 J A S 0 N D J F M A M J 1993 J A S 0 N D J F M A M 1994 Ul 0
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Figure 58c The Bowen Ratio, Defined as a Ratio of Sensible Heat Flux and Latent Heat Flux, as a Function of Longitude and Time on the Equator. Shading denotes ratio larger than 0 2 with a contour interval of 0 2.
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The Bowen Ratio c 0 llOW l1"11''f' 'l"ua'f'''lmr"'r'"''r"w1ml' 1 1 r"Mr'''r" 1 ul ,nl'"f' '\"'r' r ,'"1 MJ J A S 0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A M 1988 1989 1990 1991 170E 180 170W 160W ........... ..... ... ...., _,,_._,_._,_._,_._,_no .,___,. .&.<_,_,_, ___ ., _......_._,_,.., D
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Figure 58d The SST and Air Temperature Difference as a Function of Longitude and Time on the Equator. Shading denotes positive temperature difference with the darkest shading in excess of 1 oc and a contour interval of 1 C.
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170E 180 170W 160W lSOW 140W 130W 120W 110W I .I "'f' J 'JIIV!fll ,.,,r''H 'fWI I n I. h\ . ,.,. f'.,., .... (,.,.,. ( . . . . . . . . . MJ J A S 0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A M 1"0 1"1 170E 180 170W 160W 150W 140W 130W 120W ...... 1..11
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155 So, while the maximum interannual SST variability may occur over the eastern part of the basin, the maximum SST, air temperature difference, and hence the maximum interannual variabilty in the sensible and latent heat fluxes, occurs over the central part of the basin. This increase in the SST, air temperature difference accounts for the increase in the Bowen ratio showing that the sensible heat flux, while smaller than the latent heat flux, becomes increasingly more important in cooling the ocean upon going from the cold to the warm phase of ENSO, and conversely. To assess the effects of the sensible and latent heat fluxes on the coupled oceanatmosphere system; their origin and their cooling influence, we need to carefully examine the relationships between the SST (T), the heat flux (Q as the sum of the sensible and latent heat fluxes), and the wind speed (U8 ) at 0 170W Ordinary coherence analyses between T and Q, between Ua and Q, and a multiple coherence analysis (e. g Bendat and Piersol, 1972) between both T and Ua with Q are used for this purpose as shown in Figures 59a, b, and c, respectively Time series of aT /at, Q, T, and Ua are also shown in Figure 60, afler low pass filtering to remove oscillations at time scales shorter than 90 days The assumption behind these analyses are that the heat fluxes, via the bulk formulae are primarily due to T and U8 These variables result in Q which, in tum, affects T through the ocean's temperature equation. Ideally, an independent measure of Q would be used to correlate against aT/at to quantitatively determine the effect of Q on T Since this is not possible, we will examine the interrelatedness of these variables to determine within the constraint that the above assumption places upon the effect of Q on T. First consider the relationship between T and Q (Figure 59a). On interannual time scales they vary approximately in phase, whereas on intraseasonal time scales T and Q are approximately in quadrature. The quadrature relationship is consistent with aT /at responding to Q, implying that the local SST change on intraseasonal time scales is mainly due to the heat flux. The transfer function amplitude (increasing with increasing frequency because aT/at is proportional to Q) is consistent with the surface heat flux
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"'0 Q.l l.o :I a' r:J:J Q.l tJ c Q.l l.o Q.l ..c 0 u Q.l "'0 :I ..... Q. E < Q.l Ill 1.00 0 Ill 0.50 > E'" 0.00 120 60 0 3 1 4 _g 0.00 Q.,. 3 1 4 P e r io d ( d ays) 167 83 56 28 0.000 0 006 0.012 0.018 Fr eq u ency ( c pd ) Figure 59a Co h erence Analysis betwe e n the SST and Heat Flux. From top to bottom are t he cohe r ence squared t ransfer function amplitude, and phase angle Averaging was performed over a bandwidth (LlB) of 0 00 3 6 for approximate l y 1 7 degrees of f r ee d om and sign i ficant l e v e l on co h erence squ ared i s 0.27. 156
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1.00 "0 QJ "" :I ,.... 0 00 QJ rll 0 50 u j;lo c: QJ ;J "" '" QJ .c Q u 0 00 40 QJ "0 :I 20 Q.. E < 0 3 .14 0.00 3 .14 0.000 167 I I 0.006 83 I T 0 .0 1 2 Period ( da y s ) 56 I I 0 018 I T 0.024 F r e qu e nc y (c pd ) .:1B I I 0.030 157 28 0 036 Figure 59b Cohere nce A n alys i s between the S u rface Wind S peed and Heat Fl u x From top to b ottom a r e t he coherence squared, tra n sfer f u nction ampli tu de, and phase angle Averaging was performed over a ba ndwidth (.:1B ) o f 0 0036 f o r app r oxi m ate l y 1 7 degrees of free d o m a n d s i g nificant level on coherence squa r ed is 0.2 7
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158 Period (days) 0 0 1.0 0.0 1.0 0.5 0.0 0.000 0 006 0 012 0 018 0.024 0.030 0.036 Frequency (cpd) Figure 59c. Multiple Coherence Analysis between both the SST and the Wind Speed with the Heat Flux. From top to bottom are the multiple coherence between both the SST and the wind speed with the heat flux, the partial coherence between the wind speed and the heat flux after removing the influence of the SST from both of these time series and the partial coherence between the SST and the heat flux after removing the influence of the wind speed from both of these time series.
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G 0 ......., 0 .4 0 0 200 150 100 50 30 28 26 24 8 6 4 2 MJJASONOJFMAMJJASONDJFMAMJJASONOJFMAMJJASONDJFMAMJJASONDJFMAMJJASONOJFMAM 1oa8 10ill 1890 1891 1!1!12 1893 1994 Figure 60. The 90 Day Low Pass Filtered Time Series of SST Change (()T/dt), Heat Flux (Q as Sum of Sen s ible and Lat e nt Heat F l uxe s ), SST (T), and Surface Wind Speed (U8 ) at 0 170 W.
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160 mixing to a depth of about 25m. Refering to Figure 60 it is observed that Q and aTiat do indeed vary 1t radians out of phase on intraseasonal time scales. The flux Q, however, is derived from both T and u.. Which of these two variables are more important in determining Q? From Figure 59b it is observed Q is highly coherent and approximately in phase with Ua on intraseasonal time scales with a fairly constant transfer function amplitude. Such covariability is also evident in Figure 60 The multiple input linear systems analysis (Figure 59c) helps to dete rmine which input T or Ua is more important. The multiple coherence squared between both of these inputs and Q on intraseasonal time scales shows that over 80% of the Q variance may be accounted for by U a and T as inputs The large partial coherence squared between Ua and Q, after removing the influence of T from both of these time series, shows that Ua is the dominant input variable, since in comparison the partial coherence squared between T and Q, after removing the influence of the Ua from both of these time series, is relatively nil. This implies that the change in heat flux derived from the bulk formulae is mostly controlled by the change in wind speed, rather than by the change in the SST. While not shown, the results of the multiple coherence analysis also apply to higher frequencies using the original time series smoothed with a 10 day low pas s filter. In summary, these analyses suggest that on intraseasonal time scales the combined sensible and latent heat flux varies primarily due to changes in wind speed in the bulk formulae, versus SST, and that the rate of change of SST on intraseasonal time scales is then determined largely by this bulk formulae derived surface heat flux through the ocean's temperature equation. On interannual time scales the rate of change of SST must be due to other processes, since the combined sensible and latent heat flux appears to vary in phase with SST, acting to cool the ocean while SST increases This statistical analysis of the relative importance of Ua and T in controlling Q is supported analytically. For simplicity, since the latent heat flux exceeds the sensible heat flux, we assume that SST is equal to air temperature, allowing Q can be written as
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161 Q = QLat = paLCE(1 RH)Ua Qs(T) = QoUa Qs(T), (5.5) where Qo = PaLCE(l RH). The change in heat flux then comes from two parts: the changes in wind speed and moisture content (proportional to SST via the Clausius Clapeyron equation), where dQ = (dQ)u. + (dQ)q1 = dQ dUa + dQdqs, au. aqs (dQ)u. = dQ dUa = Qoqs(T)dUa, au. (dQ)q. = dQdqs = QoUaqs(T)(2353ln10/T2)dT. aqs (5.6) (5.7) (5.8) Thus, the ratio between the changes in moisture content and wind speed is (dQ)q. = 2353ln10UadT (dQ)u. T2dUa (5.9) The relative importance of these two quanuues in controlling the variation of 2 the heat flux depends upon U8 T, dU8 and dT. If T the change in heat flux is controlled by the change in wind speed. Applied to the central Pacific using representative values of T=28C=301K, U8=5 m/s, dT=1C, and dU8=1 m/s, it is found that (dQ)qJ(dQ)u.=0.3. Thus, the surface heat flux is mostly controlled by wind speed, consistent with the statistical analyses on intraseasonal time scales. The ratio does increase in the eastern Pacific where T and dU8 are smaller and dT is larger, but representative values are still less than unity. The observations supported by the above analysis suggests a mechanism of SSTwind speed feedback operating through the surface heat flux. This feedback, in which SST through surface heat flux affects wind speed which in tum affects SST, can be explained as follows. In response to an increase in SST, the atmosphere shows increases in cumulus convection and moisture content difference [q5(T)RHq5(T 8)]. Cumulus convection induces surface wind
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convergence with low wind speed at its center tending to decrease the heat flux [see Eqs. (5.2) and (5. 3)) On the other hand, the increase in moisture content difference tends to increase the heat flux If the heat flux is 162 controlled by wind speed, as shown above on intraseasonal time scales then the decrease in heat flux due to decreasing wind speed overcomes the increase in heat flux due to increasing moisture content, with the net result being a decrease in heat flux, less ocean cooling and hence the possibility for increasing SST. This positive feedback however, may be counteracted by a decrease in solar radiation owing to increased cloudiness This SST wind speed feedback through surface heat flux operates along with the SSTzonal wind stress feedback through the ocean's dynamical response. In nature, both of these feedbacks are likely to be important, offering explanation on why the relationship between SST and thermocline depth is so complicated in the equatorial central Pacific. 5.3 Discussion and Summary Six years of upper ocean velocity, temperature and surface meteorological data collected at o o 170 W shows slow, interannual variability occurring in the dynamics of the ocean circulation and the thermodynamics of oceanatmosphere interactions. This central Pacific location is the transition region between the warm surface waters found to the west (the western Pacific warm pool) and cold surface waters found to the east (the eastern Pacific cold tongue) It is a region of large interannual variability in the surface wind stress and it is the region through which the variability associated with ENSO in both the ocean and the atmosphere evolves Analyses have focused upon interannual and intraseasonal aspects of this variability which behave differently over these two broadly defined time scales. The measurements began in May, 1988 coincident with the La Nina (cold) phase of ENSO and they have continued through the present protracted El Nino (warm) phase of ENSO, which exhibited peak warming during the fall/winter of 1991 92 During this time the EUC core speed and 'tx has slowly
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163 decreased while SST has slowly increased These slow variations are associated with a slow decrease in the ZPG and a slow increase in the Richardson numb e r averaged across the mixed layer. The implication of these slow covariations in the ocean circulation dynamics is that the slow SST variations in the equatorial central Pacific are due to ocean dynamical processes. This may be described a s a positive SST zonal wind stress feedback In this feedback, the atmosphere interacts dynamically with the ocean through momentum flux Decreasing t x decreases the ZPG which, in turn, decreases the fully threedimensional upper ocean circulation. Since it is the ocean circulation that provides for cooling, either through horizontal advection, vertical advection (upwelling), or vertical mixing (entrainment), a relaxation of the circulation leads to an increase in SST and, in tum, a further decrease in tx. Analyses have ruled out horizontal advection as a primary factor controlling SST on interannual time scales. If anything, horizontal advection provides a cooling, rather than a warming, influence. This leaves upwelling and entrainment. Thus, a relaxation of the three dimensional circulation as seen directly in the EUC and indirectly in the ZPG (that drives the EUC) accounts for the slow ris e in SST by a downwelling of the thermocline and a decrease in entrainment (evidenced in the Richardson number) This spin down of the central equatorial Pacific circulation may occur in two ways Either the tx decreases locally, or since the central Pacific is the region through which the EUC normally accelerates and shoals downstream, a slow eastward shift of the whole system of winds and currents during the transition from La Nina to El Nino will result in a slow d e crease in the EUC core speed and a slow increase in the EUC core depth. These slow, interannual variations in the ocean circulation dynamics are mirrored by slow variations in the ocean atmosphere interaction thermodynamics. The transition from the 1988 La Nina to the 199192 peak El Nifio is characterized by a secular increase in both the sensible and latent h e at fluxes and in the Bowen ratio, with maximum values in all of these quantiti e s extending into the central Pacific from the western Pacific From the peak warming to the end of the present record (spring, 1994), these thermodynamic quantities have shown a slow decrease and retreat toward the western Pacific.
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164 The dynamical/thermodynamical relationships are different on intraseasonal and interannual time scales. While interannually, 'tx and the ZPG covary in phase, suggesting a slowly varying steady state, intraseasonally they exhibit a phase lag, necessitating a local acceleration to account for their imbalance A multiple coherence analysis indeed shows that both the local acceleration and the ZPG are equally coherent with 'tx on intraseasonal time scales. Both remote and local forcing are important, but the coarse zonal spacing between TOGA TAO elements does not resolve the ZPG on intraseasonal time scales adequately enough to bring closure on the momentum balance While interannually, SST and the surface fluxes covary out of phase (the fluxes tending to cool as SST increases) intraseasonally they covary in quadrature, consistent with the local rate of change of SST being forced by the net sensible and latent heat fluxes Moreover, the horizontal temperature advection appears to be more a consequence of SST variability than a cause, since the horizontal advection terms lag the local rate of change of SST. Thus, neither on interannual nor intraseasonal time scales does hori z ontal advection appear to be a causal factor for local SST variability at 0 170 W. The surface heat flux on intraseasonal time scales was analysed with respect to the relative importances played by SST and wind speed in the bulk formulae. The data suggest that wind speed is the dominant factor and this is supported analytically for the latent heat flux (which exceeds the sensible heat flux) Thus, the local rate of change of SST is proportional to wind spe e d with increasing wind speed providing a cooling effect on the SST. This SST wind speed feedback through surface h e at flux is different and opposite from the SST zonal wind stress feedback through momentum flux The SST wind speed feedback may provide an explanation of El Niiio warming in relation to westerly wind bursts. The fall/winter 1989 90 provides an example of a sequence events which look like a developing El Nino yet nothing happened. The Southern Oscillation Index was dropping SST was warming, reaching a peak at the end of February 1990 following a burst in westerly winds, and conditions shortly thereafter returned to normal. The wind speed, however, was high which through the SST wind speed feedback would tend to decrease
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SST rather than increase it. In contrast the westerly wind bursts m fall/winter 199091 had low speed in the central Pacific. 165 With SST appearing to be controlled by different physical processes on interannual and intraseasonal time scales and with horizontal advection acting as a consequence rather than a cause of SST variability, it is not surprising that SST variability in the central equatorial Pacific is not well correlated with the thermocline depth This is different from the eastern Pacific where the thermocline is shallow and SST and thermocline depth are correlated on interannual time scales. Such regional dependencies of SST on thermocline depth is consistent with different physical processes being controlling, e. g ., Chang (1993) and Koberle and Philander (1994). Here we have suggested two oppositely directed influences, one operating through momentum flux (on interannual time scales) and the other operating through heat flux (on intraseasonal time scales). The momentum flux argument for a slow increase in SST in response to a synchronous decrease in tx and the ZPG is th a t the ocean dynamics (through upwelling and entrainment) provides a cooling influence upon SST. A relaxation in the ocean dynamical effect th e refore leads to warming. This ocean dynamical effect, which is tied to th e strength of the EUC, is fully threedimensional. Decreasing upwelling decreases the threedimensional advective temperature flux imbalance, which decreases entrainment leading to warming by solar radiation (e. g ., Weisberg and Qiao, in preparation). While this occurs (on interannual time scales versus intraseasonal time sc a les) the sensible and latent heat fluxes tend to cool, not warm. The synchronous interannual variations in tx, the ZPG, the EUC core depth and speed, SST and the sensible and latent heat fluxes may therefore be described as a slow mode, with the oceans contribution being a fully threedimensional divergence mode tied to the overturning circulation of the EUC. The atmosphere's contribution also appears to be fully three dimensional and tied to the overturning Walker Circulation This may be deduced from the sensible and latent heat flux distributions. As these h eat fluxes both increase with increa sing SST and decreasing tx (and wind speed),
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166 the Bowen ratio also increases. This means that the effect of the SST air temperature difference outweighs the nonlinear effect of the Clausius Clapeyron equation. While the latent heat release from the ocean does exceed the sensible heat release, the overlying atmosphere (despite increasing SST) remains relatively cold and dry Indeed while the maximum interannual SST variability may occur in the east, the maximum SSTair temperature difference variability appears to occur in the central Pacific Since the ocean's influence on the overlying atmosphere is warming and moistening, the only way that the atmosphere can remain cold and dry is for air to descend from aloft. Thus, we deduce a counterpart threedimensional atmosphere divergence mode. Also important in the sensible and latent heat flux and the Bowen ratio distributions is that they have maximum values emanating from the west into the central Pacific and minimum values in the eastern Pacific Thus, the heat flux from the ocean to the atmosphere during the 1988 La Nina to the 199192 peak El Nino transition was maximum in the central Pacific, not in the eastern Pacific. So, while the climatological SST and surface pressure anomalies and their correlation may be largest in the east (e. g Mayer and Weisberg, in preparation), the heat flux from the ocean to the atmosphere is largest elsewhere. It follows that the local surface pressure anomalies in the east rather than being a consequence of local airsea exchange, are a consequence of the Walker Circulation response to heat exchange occurring elsewhere. This does not alter the finding of Lindzen and Nigam (1987) that 'tx anomalies are related to SST gradients through the surface pressure field, but it does bring into question the nature of the oceanatmosphere coupling that gives rise to the SST gradients In summary, six years of observations in the central Pacific, as part of TOGA TAO, show slow synchronous variations in the upper ocean circulation, winds SST and surface fluxes during the transition from the 1988 La Nina to the present protracted El Nino These observations suggest a coupled slow mode operating within the oceanatmosphere system having ocean and atmosphere elements that are fully threedimensional divergence modes The
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167 ocean portion is deduced as related to the overturning circulation of the EUC and the atmosphere portion is deduced as related to the Walker Circulation As conceived, these ocean and atmosphere modes differ from previous slow modes within reduced gravity models (e. g., Hirst, 1988; Neelin, 1991; Wang and Weisberg, 1994a) in that the internal workings of the layers as opposed to just their vertical integrals, are important in determining SST and the overlying air temperature and moisture content. Unlike the interannual variability where ocean dynamics appears to control SST the intraseasonal variability at this location appears to be controlled by wind speed related surface heat flux variations. Finally, while focus has been on zonal currents and winds, there are interannual variations also observed in the meridional currents and winds Given the meridional asymmetries in the surface temperature and th e atmospheric divergence fields (the ITCZ and the SPCZ), these interannual meridional variations also warrant further study
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CHAPfER6 DISCUSSION AND SUMMARY 168 The numerical and analytical models and observations have been used to study tropical oceanatmosphere interactions. The ocean component of the numerical coupled model is a linear, perturbation model of the Hirst (1988) the atmosphere component is Gill's (1980) steady state model. In the analytical models, the ocean component is similar to that in the numerical model the atmosphere component is assumed that the zonal wind stress is proportional to the SST without or with a zonal phase difference. In part of the observational studies, the 0 170 W data combined with other data of the TOGA TAO array and the Reynolds' SST product are used to describe low frequency variability of the oceanatmosphere system in the central Pacific The model limitations are the assumptions of 1) the proportionality between the wind stress and SST anomalies; and 2) the spatial homogeneity in thermodynamic parameters. In nature the coupling varies both in space and time For example, Wakata and Sarachik (1991, 1994) emphasized the importance of variations of coupling in meridional direction and time, respectively. The assumption of spatial homogeneity in thermodynamic parameters renders the SST equation overly simplistic since it omits background state processes; for example, the reason why the SST anomalies are largest over the eastern Pacific. Nevertheless, the simple models are useful to illustrate possible physical interactions between the ocean and atmosphere and provide analytical basis and results that may be applicable to more realistic systems. The coupled numerical model shows that the slow mode is an oce a n divergence mode distinctly different from the delayed oscillator mode. This slow mode propagates eastward Its energetics are governed by the ocean; the atmosphere merely follows Growth occurs if the work performed by the wind
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169 minus the work required to effect the ocean divergence exceeds the sum of th e ocean loss terms The ocean Kelvin wave speed is therefore an important parameter, since it sets the background buoyancy state of the ocean For a given level of dissipation the intrinsic length scale of the atmosphere relative to the width of the ocean basin is important For sustainable oscillations, the ocean basin must be large enough so that oppositely directed divergences can develop on opposite sides of the basin The global aspect of the atmospheric pressure component gives importance to both adjacent land masses and adjacent ocean basins. Continental heat sources may block or facilitate the propagation of the pressure perturbations, or they may provide a direct sourc e a ffecting an adjacent ocean. The neutral modes are analytically obtained over the full range of equatorial waves in a coupled oceanatmosphere system by omitting the phas e lag between the wind and SST and assuming equal coefficient values for Rayleigh friction/Newtonian cooling and thermal damping. The effects of coupling are found to be frequency dependent. Inertialgravity and Rossbygravity waves are not modified while the Kelvin and Ro s sby waves are and a s frequency tends toward zero thes e latter w a ves transform into eastward or westward slow modes, respectively The transition region from conventional to slow mode waves occurs wh e re the intrinsic dispersion properties of th e ocean medium are supplanted by the coupling between the ocean and the atmosphere. For the parameters used, the modifications by coupling become noticeable at periodicities longer than 6 months. The primary modifications are a decrease in phase speed and an increase in meridional scale The meridional scale increase is consistent with both hydrographic observations and results from ENSOrelated models, and a physical explanation follows the wind stress curl induced in the ocean s vorticity balance by the airsea coupling. The analytical model of allowing a zonal phase difference between the zonal wind stress and SST shows the stability, periodicity and horizontal structure of equatorial wave modes. The gravest coupled Rossby and Kelvin modes coexist with westward and eastward slow modes. Two of these four
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170 modes, one propagating westward and the other eastward, are destabilized in each case depending upon the model parameters. For some particular parameter choices, coupled Rossby and Kelvin modes merge with westward and eastward slow modes respectively. For other parameters, however they separate and remain distinct from the slow modes For all of these modes the primary modifications by coupling relative to uncoupled oceanic equatorial waves are a decrease in phase speed and an increase in meridional scale. Among the model parameter effects, those of the zonal phase lag between the wind stress and SST anomalies and the coefficients of thermal and mechanical damping are the most interesting. The frequency of all modes is symmetric about zero phase lag, whereas the growth rate is antisymmetric relative to the damping rate Wind anomalies to the west of SST anomalies favor slow (coupled Rossby and Kelvin) mode growth (decay) Slow (Rossby and Kelvin) mode damping is mainly thermal (mechanical) The fastwave limit filters coupled Rossby and Kelvin modes out of the system while in the fast SST limit all c oupl e d modes coexist, but with the Rossby and Kelvin mode frequencies greatly reduced to those of the slow mode frequencie s For this simplified model coupled instability can exist in the fast wave limit without the effects of Rossby and Kelvin modes and in the fastSST limit all modes are present but they become mixed in frequency range. Finally, it may be concluded that coupling affects the structure, propagation and stability properties of equatorial waves at low frequency. Determining where within the equatorial waveguide and with what efficiency coupling occurs would therefore seem to be critical in under s tanding the ENSO phenomenon. In the observational studies of the central Pacific it is shown that for all surface quantities interannual variations are large and annual cycle is weak, whereas for subsurface quantities both interannual and annual variabilities are large. A deep (shallow) thermocline is as s ociated with a w e ak (strong) EUC on both annual and interannual time scal es. Slow variations observed in all quantities since 1988 are due to air sea coupling during the transition from the 1988 La Nina to the protracted 1991 94 El Nino. The equatorial z onal momentum balance shows that zonal wind stress and
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171 vertically integrated zonal pressure gradient are the same order of magnitude, and they vary in phase on annual and interannual time scales. It also points out that the TOGA TAO array data can not resolve the oscillations of high frequency The slow decrease in the EUC core speed observed in the central Pacific since 1988 may be due to two reasons: 1) a slow decrease in the zonal pressure gradient at the EUC core; 2) an eastward shift of the whole system during the transition from La Nina to El Nino. In the central Pacific, zonal SST advection alone can not cause the local warming and SST is not simply related to thermocline depth. Surface heat flux seems to be important in regulating SST variation on intraseasonal time scales, whereas oceanic dynamics may be important in annual and interannual SST variations. The sensible and latent heat fluxes are mostly controlled by atmospheric dynamics manifested through wind speed instead of by thermodynamics manifested through moisture content (or SST). The relation between surface wind speed and SST suggests that an SSTwind speed positive feedback through surface heat flux may be responsible for the local warming and cooling on intraseasonal time scales.
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178 APPENDICES
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179 APPENDIX A. OPEN BOUNDARY CONDffiON FOR THE MODEL OCEAN The open boundary conditions, which permit all internal waves impinging on these walls to travel poleward, are used at north and south boundaries of the ocean The accurate way to prescribe the outflow at open boundary is to use Sommerfeld radiation condition av + at ay (A.1) The phase speed F is numerically evaluated at the interior points close to the open boundary B. For the leapfrog and forward time schemes the differencing of equation (A.1) gives n+1 n1 :.i n+1 n1 n+1 n 1) VB1 VBI + F VBI +VBI VB2 ; VB:2 = 0 2M !J.y 2 (A.2) n + l n VB I VBI + F (vn vn ) 0 "\ BI B 2 = (A.3) !J.t !J.y Solving equations (A.2) and (A 3), respectively, we can obtain the phase speeds for two schemes n+I nI F= VBI VBI ( n+1 n I) ( n+I nI) At VB2 + VB2 VBI + VB1 (A.4) n+ 1 n F= VBI VB1 VBI !J.t (A.5) The sign of phase speed determines whether the boundary condition i s outflow or inflow condition The outflow boundary condition is specified as F = !).yj!).t, and the inflow boundary cond i tion is set to F = 0 By representing the finite difference of equation (A.l) at boundary point B, we can get the following boundary conditions (1) If bound a ry is inflow (F ::; 0), then
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APPENDIX A. (Continued) for the leapfrog scheme for the forward scheme (2) If boundary is outflow (F > 0), then vit 1 = v:1 + VB1 {( n+ 1 n1)/2 VB1 for the leapfrog scheme for the forward scheme APPENDIX B. l\.1ETHOD OF SOLUTION FOR THE MODEL ATMOSPHERE 180 The set of equations (2.5)(2 7) in the model atmosphere can be reduced to a single equation for variable Va ( 2 2 ) 3 2 2 2 a v a a v a 2av a aQa aQa a Va + a(3 y Vaac +(3c =a(3y. ax 2 ay2 ax ay ax By performing the Fast Fourier Transforms (FFT) in x where kn = 21tn/L (n = 1, 2, ... I 1 and L is the zonal length of basin), and i = V1, we can get the compact equation (B.l) (B.2) This is a second order ordinary differential equation (ODE) Using the central difference scheme, that is
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APPENDIX B. (Continued) Vn v'J+12Vj + vJ1 yy' !:ly 2 <$ = OJ+1QJ1' 2!:ly one can obtain that where We seek a solution of (B .5) by writing where the Clj and l3j are to be determined Writing (B.8) for j1 as n n Vj1 =apVj + 1'j1 and substituting into (B .5) give V!l1 djf3j1 J n J+1 + n Clj1 + bj Clj1 + bj Comparing (B.8) with (B 10), we find Clj=1 n' Clj1 + bj 181 (B.4) (B.5) (B.6) (B. 7) (B.8) (B.9) (B.10) (B 11 )
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182 APPENDIX B. (Continued) A djPj1 1'J. n Clj1 + bj (B .12) Applying the boundary conditions V = 0 to equation (B.5) at southern wall (i.e., V 1 = 0) and using (B.8) for j = 2, we can get that
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183 APPENDIX C. (Continued) where (C.2) (C.3) (C.4) and (C.5) Equation (C.l) has the eigenfunctions = for the eigenvalues An. Consider any two eigenfunctions Vm and Vn and the corresponding the eigenvalues Am and An; then + = 0, (C. 6) and (C.7) After multiplying equation (C 6) by Vn, equation (C.7) by Vm, subtracting and integrating from oo to +oo, the terms containing cancel and the two derivative terms may be simplified by integration by parts, one of which becomes (C.8) Thus when the two derivative terms are subtracted, the integrals on the right hand side of the equation (C 8) cancel out. The boundary condition that v(y) 0 as y oo also allows the first term on the righthand side of the equation (C.8) to vanish The net result is thus,
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184 APPENDIX C (Continued) (C.9) which leads to the statement that eigenfunctions corresponding to different eigenvalues are orthogonormal with respect to the weight function r(l;) Therefore, we can conclude that where Omn = 1, if m = n and Omn = 0, if m "# n. Given the orthogonality property of the eigenfunctions, a function f(l;) may be represented by an infinite series expansion of these eigenfunctions f(l;) = L am'Vm(l;). (C.ll) m The expansion coefficients am are obtained by multiplying both sides of (C.11) by r(l;)w 0(l;) and integrating from oo to +oo, assuming that termbyterm integration is permissible, and employing the orthogonality property. From this operation, only the t e rm with n = m will remain on th e righ thand side, yielding an= r = r (1 + (C. 12)

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