
Empirical studies of equatorial ocean dynamics from the tropical instability wave experiment
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 Permanent Link:
 http://digital.lib.usf.edu/SFS0040014/00001
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 Title:
 Empirical studies of equatorial ocean dynamics from the tropical instability wave experiment
 Creator:
 Qiao, Lin
 Place of Publication:
 Tampa, Florida
 Publisher:
 University of South Florida
 Publication Date:
 1996
 Language:
 English
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 xii, 158 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Ocean circulation  Pacific Ocean ( lcsh )
Upwelling (Oceanography) ( lcsh ) Dissertations, Academic  Marine Science  Doctoral  USF ( FTS )
Notes
 General Note:
 Includes vita.
Thesis (Ph. D.)University of South Florida, 1996.
Includes bibliographical references (leaves 138144).
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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.
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 022498413 ( ALEPH )
35689095 ( OCLC ) F5100198 ( USFLDC DOI ) f51.198 ( USFLDC Handle )
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 Book

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EMPIRICAL STUDIES OF EQUATORIAL OCEAN DYNAMICS FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT by 1.JNQIAO A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Marine Science University of South Florida May 1996 Major Professor: Robert H. Weisberg, Ph.D.
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Graduate School University of South F l orida Tampa, Florida CERTIFICATE OF APPROVAL Ph.D. Dissertation This is to certify that the Ph.D. Dissertation of LIN QIAO with a major in Physical Oceanography has been approved by the Examining Committee on February 20, 1996 as satisfactory for the dissertation requirement for the Ph.D. degree Examining Committee: Major Professoi: Robert H. Weisperg, Ph.D. Member: Mark E. Luther. Ph.D. Member: Boris Gal perin, Ph. D. Member. Julian P McCreary, Ph.D. Member: TswenYung Tang, Ph D.
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DEDICATION I wish dedicate this dissertation to my father, who has given me so much encouragement. Due to many reasons other than his capability, he was not even able to apply to a University.
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ACKNOWLEDGMENTS This dissertation is supported by the Ocean Sciences Division National Science Foundation, Grant Numbers OCE8813378 and OCE9302811 I would like to thank R. Cole and J. Donovan for their assistance with the field work and analyses. Without them we would not get the perfect 100% return of data. I would thank the officers and crew of the RN WECOMA and ALPHA HELIX for facilitating the sea going operations. The TOGATAO array wind stress and temperature data were kindly provided by M. McPhaden and L. Mangum of the NOAA/PMEL. I am greatly indebted to my major profe sso r Dr. Robert H Wei s berg, without who se concerned guidance and support, this dis se rtation would not have come to fruition I would also like to thank my committee members, Drs. Luther, Galperin, McCreary and Tang for their helpful comments on my dissertation draft. I would thank students in Ocean Circulation Group for their efforts to help each other in our "high pressure environment. Special thanks are due to my wife Huining Zhang, who gave me a special gif t my lovely so n Zhicheng Qiao before I finished this work She brought me joy to ease the tension s involved with working toward this degree
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TABLE OF CONTENTS LIST OFT ABLES iii LIST OF FIGURES IV LIST OF SYMBOLS AND ABBREVIATIONS ix ABSTRACT X CHAPTER 1. GENERAL INTRODUCTION CHAPTER 2 TROPICAL INST ABll..JTY WAVE KINEMATICS: OBSERVATIONS FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT 9 Introduction 9 Background 1 0 Field Program and Data 15 Variance Distribution and Modulation 22 Wave Kinematics 31 Discussion and Summary 38 CHAPTER 3. TROPICAL INSTABILITY WAVE ENERGETICS: OBSERVATION FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT 41 Introduction 41 Perturbation Energies 44 Reynolds stress tensor and Reynolds density flux 48 Mean divergence of PKE: right handside of line 1 a 54 Energy conversion between the waves and mean flow: line 1 b 57 Divergence of PKE by perturbation velocity : line lc 65 Pressure work: line 1 d 65 Energy conversion between PKE and PPE: line 1e 70 Summarizing the righthandside of equation 1 72 Advection of PPE by mean velocity: righthand side of line 2a 73 Energy conversion between the waves and mean density : line 2b 76 Advection of PPE by perturbation velocity: line 2c 76
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En e r g y co nversion betw e en PKE and PPE : lin e 2d 78 Summarizin g right hand side o f equati o n 2 78 Di s cu ss ion and Conclusion 78 CHAPTER 4 THE ZONAL MOMENTUM BALANCE OF THE EQUATORIAL UNDERCURRENT IN THE CENTRAL PACIFIC 88 Introduction 88 Background 89 Dat a and method s 93 The mean zonal momentum balance l 03 Material a cce l e ration b y flux div e r gen ce f o rmulation l 06 Mat e rial ac ce l e ration by adv ec tiv e f o rmulati on Ill Z onal pressur e g radi e nt and wind str e s s 114 V e rti c ally int e grated zo nal m o m e ntum balan c e 116 The implied vertical s tr e ss distributi on 118 Temporal evolution of the zonal momentum balance 121 Summary and dis cus s ion 130 SUMMARY AND CONCLUSIONS 135 REFERENCES 138 APPENDICES 145 APPENDIX 1. ERROR ANALYSIS FOR PHASE 146 APPENDIX 2 ERROR ANALYSIS FOR WAVENUMBER 147 APPENDIX 3. MEAN ZONAL MOMENTUM FLUX DIVERGENCE RANDOM ERRORS 151 APPENDIX 4 ADVECTIVE AND FLUX DIVERGENCE SCHEME COMPARISON 156 VITA 11
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LIST OFT ABLES Table I TIWE equatorial array mooring positions, record len gths (begi nning 0000 UT on 5/12/90) and instrument depths 6 Table 2 A summary of period, wavelength, a nd phase speed estimates made for tropical instability waves 12 Table 3 Covariance of mean velocity pairs used in the standard error estimates 155 111
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LIST OF FIGURES Figure 1. The location of the TIWE equatorial array in relation to the tropical Pacific Ocean's climatological SST distribution for September 5 Figure 2. Schematic diagram of the TIWE equatorial array subsurface ADCP moonngs 7 Figure 3 Time series of the hourly sampled vcomponent at 30 m depth from the TIW 4 mooring I7 Figure 4 The ucomponent as a function of depth and time at moorings TIW2 TIW4 and TIW5, low pass filtered to exclude fluctuations at time sca l es shorter than I 0 days 18 Figure 5 The vcomponent as a function of depth and time at moorings TIW2 TIW4 and TIW5, low pass filtered to exclude fluctuations at tim e scales shorter than 10 days 19 Figure 6 Isotherm depths as a function of time from a TOGATAO array mooring at 0 I40 W upon which are superimposed the EUC core depth defined by du/dz = 0 21 Figure 7. Log variance densitie s as functions of depth and frequency for th e a. vcomponent and the b. ucomponent along I40 W at 1 S, 0 and I 0N 23 Figure 8. Semiminor to semimajor axes ratio for the ve l ocity hodograph ellipse as a function of depth and frequency along I40 W at 1 S, o o and 1 N 25 Figure 9. Velocity hodograph ellipses at 30m depth at each of the 5 TIWE equatorial array mooring locations 26 Figure 10 The complex demodulation amplitude for the vcomponent as a function of depth and time a l ong 1 40W at 1 os, 0 and 1 o N 28 IV
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Figure 11. The complex demodulation amplitude for the ucomponent as a function of depth and time along 140 W at 1 S, oo and I oN 29 Figure 12. The first (frequency domain) EOF mode for the vcomponent a. amp litude s as a function of depth at each mooring location, b. coherence squared between the first mode and the data as a function of depth at each mooring locat ion, c. the phase (rad) as a functio n of longitude and depth along the eq uator and d the phase (rad) as a function of latitude and depth a lon g 140 W 33 Figure 1 3. The zonal and meridional wavenumber vector components as a function of depth along with their 90% confidence intervals for random errors cal cu l ated at 0 140 W u s ing the v component first EOF mode and independent plane fits by linear regression at each depth 36 Figure 14 Reynolds stress components and as functions of time and depth at 0, 140 W 49 Figure 15. Reynolds stress compone nt as a function of time and depth at 1 S, o o and I 0N a long 140W 50 Figure 16. Reynolds stress compone nt s and as functions of time and depth at 0 140 W 52 Figure 1 7. Meridional constituent of Reynolds density flux as a function of time and depth at I 0S, oo and I 0N a l ong 140W 53 Figure 1 8. Vertical con s tituent of Reynolds density flux as a function of time a nd depth at 0 140W 55 Figure 1 9. Mean zonal divergence of perturbation horizontal kinetic energy at 141 W, 140 W and 1 39W along equator 56 Figure 20. Mean meridional divergence of perturbation horizontal kinetic energy at 0.5 S 0 and 0.5N along l40 W 58 Figure 2 1 Mean vertical divergence of perturbation horizontal kinetic energy at 0 l40W 59 Figure 22. Mean divergenc e of perturbation hori zo ntal kinetic energy at 0, l40 W 59 v
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Figure 23. Deformation work constituent U y as a function of time and depth at 0.5S, 0 and 0.5N along 140 W 61 Figure 24. Deformation work constituent  V y as a function of time and depth at 0.5 S, 0 and 0.5N along 140 W 62 Figure 25. The ( < u'v '>Uy + Vy) as a function of time and depth at 0.5S 0 and 0.5 N a lon g 140 W 64 Figure 26. Deformation work constituent Uz and  V z as functions of time and depth at 0, 140 W 66 Figure 27. The Y:z <(u'2+v'2)v'>y as a function of time and depth at 0.5S, 0 a nd 0.5N along 140 W 67 Figure 28 The meridional pressure work  a s a function of time and depth at 1 S, 0 and 1 N a lon g 140 W 69 Figure 29. The energy conversion between the wave kinetic energy and wave potential energy through buoyancy work g at 0, 140 W 71 Figure 30. Mean meridional advec ti on of perturbation potential energy at I 0S and 1 N a l ong 140 W 74 Figu re 31. Mean vertica l advection of perturbation potential energy at 0 t40W 75 Fi g ur e 32. The baroclinic energy conversion between the wave potential ener g y and the mean den s ity field throu g h meridiona l den s ity gradient gpn y )IPnz a t 1 S and 1 N a long 140W 77 Figure 33. The meridional shear of zonal velocity component a s a function of time and depth at 0.5S 140 W and 0.5 N, 140 W 82 Figure 34. The weekly mean sea s urface temperature distribution in the region between 1 0 S and I 0N from 1 60 W to I 00 W 85 Figure 35. Horizontal velo c ity components as a f un ct i o n of depth a nd time for moorings along 140 W : a. ucomponent wit h westward flow denoted by lig ht s tip pling and eas tward flow greater than 80 em s1 denoted by dark s tippling and b vcomponent, with northward flow d e noted by s tippling 94 95 VI
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Figure 36. Horizontal velocity components as a function of depth and time for moorings along the equator: a. ucomponent, with westward flow denoted by light stippling and eastward flow greater than 80 em s1 denoted by dark stippling and b. vcomponent, with northward flow denoted by s tippling 9697 Figure 37. Recordlength mean vertical profiles for the uand vcomponents at the five mooring locations 99 Figure 38. Thewcomponent as a function of depth and time estimated at 0 140W 101 Figure 39. The recordlength mean vertical profiles of the u(thin solid) v( dashed) and w(thick solid) components at o o 140 W I 02 Figure 40. Isotherm depths as a function of time from moorings at o o 125 W ; 0 140W and 0 170 W 104 Figure 41. Vertical profiles of the individual constituents comprising the recordlength mean zonal momentum flux divergence estimated about0, 140 W 107 Figure 42. Vertical profiles of the record length mean zonal momentum flux divergence by the mean circulation, the resolvable Reynolds stresses and their sum I I 0 Figure 43. Vertical profiles of the individual constituents comprising the recordlength mean advection of zonal momentum estimated at 0 140 W 112 Figure 44. Vertical profiles of the record length mean advection of zonal momentum by the mean circulation, the resolvable fluctuations and their sum 113 Figure 45. Vertical profiles of mean zonal pressure gradients estimated between 170 W and 140 W, 140W and 125 W and 170 W and 125 W using a reference level of 250 m I 15 Figure 46. Mean vertical profiles for vertically integrated constituents of the zonal momentum balance at oo, 140 W I 17 Vll
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Figure 47. Mean vertical profiles at 0, 140 W of left panel: the three terms comprising the zonal momentum balance where is obtained as the residual of the ZPG and ; center panel: <'tx> and right panel: Av I 19 Figure 48. The nonlinear acceleration (vVu), the local acceleration (duldt) and individual constituents comprising the nonlinear acceleration 122(uCJu/CJx, vCJu/CJy, and wCJu/CJz) as functions of depth and time 123 Figure 49. The ZPG estimated between 170 W140 W, 140 Wl25 W and 170 W 125W as functions of depth and time 125 Figure 50. Log variance densities for duldt and v Vu as functions of depth and frequency 127 Figure 51. The vertically integrated zonal acceleration cfDu/Dt dz) and its z local cfau/CJt dz) and nonlinear cf vVu dz) constituents as z z functions of lower limit of integration and time 128 Figure 52. Time series of the vertically integrated (0250 m) ZPG, local acceleration, total acceleration and surface zonal wind stress 129 Figure 53. The 90% confidence intervals for random errors on phase a. as a function of longitude and depth alon g the equator and b. as a function of latitude and depth along 140 W for comparison with the phase estimations of Figure 12c, d 146 viii
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p v 't ADCP Ah,Av CTD EOF EUC ENSO f g GATE JGOFS NOAA NSF PKE PMEL PPE SEC SEQUAL so SST U, V, W TIWE TOGA TOGATAO UNOLS x y z ZPG LIST OF SYMBOLS AND ABBREVIATIONS Seawater density Horizontal gradient operator Stress Acoustic Doppler current profiler Horizontal and vertical eddy viscosity coefficient SEACAT conductivity, temperature and depth recorder Empirical Orthogonal Function Equatorial Undercurrent El Nino and Southern Oscillation Coriolis parameter Gravity GARP Atlantic Tropical Experiment Joint Global Ocean Flux Study National Oceanic and Atmospheric Administration National Science Foundation Perturbation Kinetic Energy Pacific Marine Experimental Laboratory Perturbation Potential Energy South Equatorial Current Seasonal Response of the Equatorial Atlantic Southern O s cillation sea surface temperature velocity components in (x, y, z) direction Tropical Instability Wave Experiment Tropical Ocean Global Atmosphere Tropical Ocean Global Atmosphere Tropical Atmosphere Ocean University National Oceanographic Laboratory System conventional coordinate s with positive eastward, northward and upward Zonal Pre s sure Gradient lX
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EMPIRICAL STUDIES OF EQUATORIAL OCEAN DYNAMICS FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT by LINQIAO 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 M ay 1996 Major Profe ss or: Robert H. Wei s berg Ph D. X
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Velocity measurements from an array of acoustic Doppler current profilers deployed in the equatorial Pacific during the Tropical Instability Wave Experiment are analyzed in three ways to determine the kinematics and the energetics of planetary waves originating from instability of the near surface equatorial currents, and the zonal momentum balance of the upper equatorial currents. A distinctive wave season was observed from August to December 1990, with wave energy confined primarily above the core of the Equatorial Undercurrent (EUC). Particle motions in the horizontal plane are described by eccentric ellipses oriented toward the north, but tilting into the cyclonic shear of the South Equatorial Current (SEC). The tilt is maximum near the surface just north of the equator and decreases to the south and with depth. The distribution of wave variance is narrowband in both frequency and zonal wavenumber, with central period, zonal wavelength and westward directed phase propagation estimated to be 500 hr, 1060 km and 59 em s1 respectively. Neither the meridional nor the vertical wavenumber components are statistically different from zero. These results generally agree with previous findings on tropical instability waves from the Atlantic and Pacific Oceans and, in the under sampled arena of geophysical measurements, they are an example where statistical inference is supported by an ensemble of independent measurements. The observed instability waves are generated by barotropic instability arising from the meridional shear of the mean zonal velocity component within the SEC just north of the equator Wave energy is maintained by this shear instability for the first half of the wave season, and by the sum of this shear instability and the barotropic instability arising XI
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from the meridional gradient of mean meridional velocity component just north of the equator for the later half of the wave season. The diagnosis of the upper ocean zonal momentum shows that the flow field and associated zonal momentum flux divergence are fully threedimensional over the upper 250 m, consistent with the earliest descriptions and theoretical ideas of EUC. Estimates of the vertical stress divergence show dynamical flow regimes that change between the s urface and the base of the EUC, being essentially linear (modified by non linearity) near the surface, weakly nonlinear at the EUC core and fully nonlinear below the core. The vertical stress divergence is much larger over the lower portion of the EUC than previously reported, but this is consistent with the observed downstream deceleration of the EUC and the necessity for vertical mixing to maintain the thermostad and account for the distribution of other material property isopleths Nonlinearity becomes increasingly important with decreasing frequency, but tends to cancel upon vertical integration Abs tract Approved: Major Professor: Robert H. Weisberg, Ph D Professor, Department of Marine Science Date Approved: xii
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CHAPTER 1. GENERAL INTRODUCTION Nowadays, the equatorial ocean is one of the most important regions in marine research. Due to high biomass and new productivity, the equatorial ocean is an interesting region for marine biologists, chemists and geologists and, of course the physical factors controlling the biological activity are important subjects for physical oceanographers. It is also believed that the key to understanding interannual climate variation is linked to the large scale airsea interaction between the ocean and the atmosphere in the equatorial region It has been known for over a century that the equatorial ocean provides a source of fish for fishing industry Whalers have followed the line of high biological productivity in the equatorial region. Early investigations on Equatorial Pacific Ocean which provided meridional sections of material property distributions, were supported by the U. S. Fish and Wildlife Service (Sette, 1949 ; Cromwell 1953) Motivated by the data, Cromwell ( 1953) developed a theory to interpret the equatorial upwelling that brings nutrient rich s ubsurface water to the surface and therefore, result s in high biological productivity in the Equatorial Pacific. The high biological activity lead s to a high C02 exchange between the ocean and the atmosphere, and high sedimentation in that region Currently the increase of C02 l eve l into the atmosphere by human activity has attracted more and more attention. It is
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believed that the equatorial ocean plays an important role in global C02 budget which may have an important effect on the future climate change. For physical oceanographers, the equatorial ocean is a unique region because the the horizontal components of the Coriolis acceleration vanish. As a consequence, the equatorial motions are different from most other l argescale oceanic motions. For example, the vanishing of Coriolis force makes Ekman theory invalid on the equator; on the equator a nonrotational homogenous wind may cause a vertical motion, whereas in midlatitudes a windinduced vertical motion is possible only if the wind field is rotational. The time and spatial scales of motion in the equatorial region are also generally different from those in the other ocean regions. 2 One prominent phenomenon in equatorial oceans is the co ld sea surface temperature (SST) on the eastern sides of the basins (both Pacific and Atlantic). The cold SST results from the equatorial upwelling driven by the southeasterly wind. It is the upwelling that makes equatorial region biogeochemically significant. The upwelling increases the photosynthesis rate by moving subsurface nutrientrich water to the surface. It has a l so suggested that the upwelling in the Pacific Ocean has important implications on the Southern Oscillation (SO), an interannual climate variability by Bjerknes ( 1969) who first proposed that SO was related to El Nino showing that it was associated with interannual variability of SST and the equatorial currents in the eastern equatorial Pacific. Numerous studies have confirmed that the SO and El Nino result from the interactions between the tropical ocean and atmosphere, with variations in two media being within the same frequency band and generally in phase
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3 In contrast to El Nino and Southern Oscillation (ENSO), there are large amplitude oscillations in the ocean current and temperature field of frequencies that are not prominent in the wind field. Those oscillations have been observed in all equatorial oceans by a variety of means, and are narrowband in frequency and wavenumber, with time and zonal length scales centered about 3 weeks and 1000 km, re spective ly They h ave been labeled "tropical instability waves" because the likely so urce s of the waves are the instabilities of the equatorial currents (Ano nymou s, 1988 ). It is believed that the in stab ility waves play an important role in upper ocean mass, momentum and heat balances; therefore, it is essential to understand them in order to understand completely the physics of the equatorial current system, as well as to be able to construct better theoretical and realistic numerical models. There are two major research programs sponsored by the National Science Foundation (NSF), that are directed at the issues of biogeochemical and climate implications of the equatorial ocean dynamic s The biogeochemical processes related to upwelling are being s tudied by the National Oceanic and Atmospheric Administration (NOAA)/NSF Joint Global Ocean Flux Study (JGOFS) Program and the coupled ocean atmospheric interactions related to SST and surface winds are being studied by the NOAA/NSF Tropical Ocean Global Atmosphere (TOGA) Program. The Tropical In stability Wave Experiment (TIWE) is one of the re search projects under the a u sp ice s of the TOGA. TIWE i s a multiinstitutional, multiinvestigator program aimed at furthering our underst a nding of tropical instability waves.
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4 Planning for the TIWE began at "The U S. TOGA Workshop on the Dynamics of the Equatorial Oceans held in Honolulu, ill during August 1115 1986. Subsequently a group of ocean scientists drafted a program prospectus (Anonymous, 1988), introducing the program concept and discussing the various elements to be proposed by particip ati ng principal investigators. One such element contributed by University of South Florida is an array of five subsurface moored acoustic Doppler current profilers (ADCPs), centered on the equator at 140 W Additional components of TIWE are under the direction of principal investigators from the University of Hawaii, the Scripps Institution of Oceanography and the University of Wa s hington. The duration of the TIWE array was planned to be about one year with the goal of observing the seasonal evolution of the tropical instability waves During May 1990, five subsurface moorings, each containing an RDInstruments Inc. 150kHz ADCP and a SeaBird Electronics Inc SEACAT conductivity, temperature and d ept h recorder (CTD), were deployed from the University National Oceanographic Laboratory System (UNOLS) vessel R!V WECOMA operated by Oregon State University. The five moorings, designated TIWl, TIW2, TIW3, TIW5 and TIW5, were deployed in a diamond shaped array centered upon 0 140 W This array, and its position in relative to the Tropical Pacific Ocean's climatological sea surface temperature distributions for September i s s hown in Figure 1. The five moorings were recovered 13 months later in June 1991 aboard the UNOLS vessel RIV ALPHA HELIX operated by the University of Alaska All of the instrument s were functional upon recovery with no visual signs of fouling (Weisberg etal. 1991)
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All five moorings were essentially the same. A schematic mooring diagram, representative of each of them, is shown in Figure 2 The setup parameters for all of the ADCPs and SEACATs were the same and the targeted depth for all of the moorings was 275m. Table 1 lists the mooring positions, the record lengths (in hours) of the time series and the instrument depths. 120 E 140 E e : UNIVERSITY OF SOUTH FLORIDA, TIWE MOORINGS Figure 1. The location of the TIWE equatorial array in relation to the tropical Pacific Ocean's climatological SST distribution for September (Courtesy of M. McCarty and M. McPhaden NOAAIPMEL). 5
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6 This dissertation is a report of analyses using current velocity data from the TIWE array. To achieve the purpose of TIWE, (which is to expand our understanding of the tropical instability waves) data have been analyzed to understand the kinematics and energetic of the waves. In addition, the recordlength mean and the temporal evolution of the zonal momentum balance are studied, because data from the TIWE equatorial array provides a unique opportunity to directly investigate the evolution of the threedimensional velocity field on the equator over a complete annual cycle that is pertinent to both TOGA and JGOFS. Table 1. TIWE equatorial array mooring positions, record lengths (beginning 0000 UT on 5112/90) and instrument depths Mooring Name Position (Lat/Long) Record Length Instrument Depth (m) (Hours) TIW1 0 Ol.4'N 9737 273 6 141 50.6'W TIW2 o o 57 8'S 9667 280.5 139 57.5'W TIW3 oo 02.4'N 9689 281.5 137 57 7'W TIW4 oo 03.2'S 9713 276 5 140 08.4'W TIW5 1 o Ol.5'N 9761 266.4 139 57.4'W
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NOVA TECH radio and flasher ADInstruments 150KHz SCADCP Flotation Technology 45 in syn ta ctic foam buoy 2.5 m of 1/2 in chain 3 ton Miller swivel SeaBird Electronics SEACAT 20m of 1 /2 in chain 3900 m NILSPIN jacketed wire rope with 36 16 in glass balls and hard hats on 1 /2 in chain EG&G 8202 acoustic releases (2} 3/4 in nylon rope cut to depth 10m of 1/2 in chain 4 railroad wheel anchor 7 Figure 2 Schematic diagram of the TIWE equatorial array subsurface ADCP moorings.
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8 The dissertation is organized as follows. Chapter 2 describes the kinematics of the instability waves from our observations. Chapter 3 provides the energetics of the waves. Chapter 4 gives the results of zonal momentum balance studies. Chapter 5 summaries all the important conclusions of these analyses. The main results in Chapter 2 to Chapter 4 have been reorganized and published or prepared for publishing (Qiao and Weisberg 1995; Qiao and Weisberg, 1996; Qiao and Weisberg, 1995).
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CHAPTER 2 TROPICAL INST ABllJTY WA VB KINEMATICS: OBSERVATIONS FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT Introduction 9 The near surface circulations of the equatorial Atlantic and Pacific Oceans may be described by a westward flowing South Equatorial Current (SEC) within the surface layer above an eastward flowing Equatorial Undercurrent (EUC) within the thermocline. These currents vary seasona lly with the trade winds that force them. Generally, as the southeast trade winds intensify from their boreal spri ngtime minimum, the SEC accelerates until a zonal pressure gradient is established to balance the wind stress. Prior to achieving this balance the equatorial currents become unstable, generating w av es of planetary sca le. Such wave observations, reported from the GARP Atlantic Tropica l Experiment (GATE) by Dtiing et al. ( 1975), motivated a set of s tability analyses by Philander (1976, 1978 ). Similar waves were reported in the Pacific by Legeckis (1977 ). While seasonally and interannually modul a ted, the se tropical instability waves are now recognized as ubiquitou s features of the tropical circulation field. Initial studie s of their interactions with the background fields u si ng in situ data from the Atlantic and Pacific Oceans by Weisberg (1984) and Hansen a nd Paul (1984), re spect ively s howed that these waves play important roles in the nearsurface heat and m omentum balances, confirmin g inferences drawn from the numerical model study of Cox (1980).
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10 With the objective of further defining the effects of these waves on the near surface mass, momentum, heat and mechanical energy balances, the Tropical Instability Wave Experiment (TIWE) was initiated in boreal spring 1990 near 140 W. The TIWE field program included arrays of moored instrumentation, shipboard hydrographic mapping and Lagrangian drifter tracking Reported herein are velocity observations from an equatorial array of subsurface moored, acoustic Doppler current profilers (ADCP) centered on 0 140 W Specific focus is on a description of the background currents, the observed variability and the wave's kinematics. Mass, momentum and energy considerations will be topics of future correspondence. Section 2 reviews previous instability wave observations. Section 3 introduces the field program and the data. Section 4 describes the instability wave variance, as observed by the equatorial array, and Section 5 provides a wavenumber analysis. The results are di s cussed in Section 6 where the available evidence suggests a hypothesis on the roles of the SEC and the EUC in tropical instability wave generation. Background Tropical instability wave observations have been reported by a variety of means. These include: (i) velocity measurements from a. moorings [Weisberg (1979) Wei s berg et al (l979a), Halpern et al. (1983), Weisberg (1984) Philander et al. ( 1985) Lukas (1987), Weisberg et al. (1987), Halpern et al. (1988), Weisberg and Weingartner (1988), Halpern (1989) Bryden and Brady (1989) and Weingartner and Weisberg (1991 ) ] b. drifters [Hansen and Paul (1984) and Reverdin and McPhaden (1986)] and c. shipboard
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11 profilers [Dtiing et al. (1975), Wil so n and Leetmaa (1988) and Luther and Johnson (1990)]; (ii) sea surface temperature (SST) measurements from a. satellite AVHRR imagery [Legeckis (1977), Brown (1979), Legeckis e t al. (1983), Legeckis (1986) Legeckis and Reverdin (1987), Pullen et al. (1987), Steger and Carton (1991 )] and b volunteer observing ships [Mayer, et al. (1990)]; (iii) sea lev el measurement s from a tide gauges [Wryrki (1978) and Mitchum and Lukas ( 1987)], b satellite altimetry [Malarde et al. (1987), Musman (1989) and Perigaud (1990)] and c inverted echo s ounders [Miller e t al (1985)]; and (iv) salinity measurements from shipboard CTD profiles [Dtiing et al. (197 5)] The subs urface expression of these instability waves by energy propagation into the aby ss ha s also been reported by Harvey and Patzert (1976) Weisberg et al.. (1979b) Weisberg and Horigan (1981), Luyten and Roemmich (1982) a nd Eriksen and Richman ( 1988 ). Many of th ese references provide estimates of zona l wavenumber and phase speed (Table 2) that genera lly show we s tward propagating waves with period, zonal wavelength and phase speed centered about three weeks, 1000 km and 50 em s1 re s pectively. The tropical instability wave s appear to be confined mainly to th e s urface lay e r with energy dropping precipitously through the thermocline. Thus, on the equator mo s t of th e wave energy, primarily in the form of perturbation kinetic energy, appears above the EUC core [Weisb e rg (1979, 1984), Philand er, e t al. ( 1985 ), Halpern et al. (1988 ), W e isberg and Weingartner (1988) and Luther and Johnson (1990)]. Below the EUC core a s p ec tral s hift to slightly lower frequency is also observed [Wei s berg ( 1979), Philander e t al. (1985) and Halp ern et al. (1988)].
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Table 2. A summary of period, wavelength, and phase speed estimates made for tropical instability waves 12 Reference Period Wavelength Phase Speed (days) (km) (crnls) Model Predictions Philander ( 197 6) 40 900/2000 Philander (1978) 30 1100 Cox (1980) 34 1000 Philander, et al. ( 1986) 2128 1000 McCreary and Yu (1992) 21 785 Velocity Measurements Dtiing, et al. (1975) 1621 2600 190 Harvey and Patzert (1976) 25 1000 50 Weisberg, et al. (1979a) 31 9901220 Weisberg, et al. (1979b) 16 1200 80 Weisberg (1979) 16/32 7601250 27 to 65 Luyten and Roemmich ( 1982)* 26 1400 Weisberg (1984) 25 1 140 53 Philander et al. (1985) 21 1000 Eriksen and Richman (1988)* 945 <4000 Halpern, e t al. ( 1988) 20 1320 1600 81 to 93 Wei s berg and Weingartner ( 1988) 25 10001200 50 to 55 Wi l son and Leetmaa (1988) 2030 1000 27 SST Measurements Legeckis ( 1977) 25 1000 Legeckis, et al. (1983) 25 1000 Legeckis ( 1986) 25 6001200 21 to 49 Legeckis and Reverdin ( 1987) 24 1000 Pullen, et al. ( 1987) 25 1000 Mayer et al. (1990) 2137 8251924 50 Sea Level Measurements Wyrtki (1978) 34 1100 Miller e t al. (1985) 2080 1000 Peri gaud ( 1990) 2840 10002000 These results are from current meter measurements below the thermocline.
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13 The instability waves are seasonally modulated, apparently in relationship to the westward flowing SEC (e.g., Halpern et al., 1988) Since the SEC is related to the intensity of the southeast trade winds and the zonal pressure gradient tending to balance these winds, the duration of the instability wave season differs betw ee n the Pacific and Atlantic Oceans [Legeckis (1986) and Halpern and Weisberg (1989 ) ]. Wei s berg and Weingartner (1988) attributed th e relatively short wave season in the Atlantic to the linear adjustment time of the zonal pressure gradient. The instability wave s may therefore be characterized as nonlinear features modulated by linear processes with their importance stemming from their non linear interactions with the background fields as shown in previous studies. Positive deformation work within the cyclonic shear of the EUC was implied from s hort duration measurements below the EUC core (Weisberg et al., 1979a) Near surface measurements were reported from the Se aso nal Response of the Equatorial Atlantic (SEQUAL) Experiment by Weisberg (1984) u s ing moored current meters and from the Pacific Ocean by Hansen and Paul (1984) u si ng surface drifter s Wei sberg (1984) found that the onset of large horizontal Reynold s stress values occurred synchronously with the instability waves and that the deformation work (barotropic conversion) calculated within the surface cyclonic shear region of the SEC just north of the equator was s ufficient to acco unt for the instability wave's growth. Both the Reynolds stress and the wave perturbation energy decreased below 50 m depth Wave energy was also found to progress eastward, opposite to the direction of pha se propagation Hansen and Paul ( 1984) with data covering a broad latitudinal extent, s howed that the l argest barotropic conversion of mean to wave energy at the surface occurred within the SEC Their
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14 observations implied that the waves extract energy from the mean flow just to the north of, and conversely just to the south of, the equator and that the instability waves cause heat to converge on the equator, opposing the cooling effects of Ekman divergence The Reynolds stress analysis of Lukas ( 1987) further supported the idea of near surface barotropic production just north of the equator. While the near surface horizontal Reynolds stresses associated with the instability waves in both the Atlantic and Pacific Oceans have been found to be largest on, with oppositely directed stress gradients to the north and south of, the equator the Reynolds stress gradients alone do not imply an interaction between the background currents and the waves (Charney and Drazin, 1961), since the Reynolds stress divergence may be balanced by a Coriolis force related to a stressinduced meridional circulation Using the SEQUAL data, Weisberg and Weingartner (1988) investigated the EP (EliasenPalm, 1960) flux vector divergence between the equator and 0 75N and 0.75 S and found that the waves decelerate the SEC to the north of the equator and accelerate it to the south. Thu s the instability waves tend to remove the negative near surface shear (cyclonic north of the equator). This finding, along with the associated energy balance on the equator and the distributions of the Reynolds fluxes observed off the equator at 1.75 N, 3N and 6 N led to the conclusion that the instability waves in the Atlantic gain their energy through barotropic instability within the cyclonic shear region of the SEC just north of the equator The Reynolds temperature flux distributions also supported the heat transport argument of Hansen and Paul ( 1984). Addition a l support for the near s urface confinement of the horizontal Reynolds stress was given by Wilson and Leetmaa (1988)
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15 Using repeated meridional sections of velocity and density from the Hawaii to Tahiti Shuttle Experiment, Luther and Johnson (1990) provided further insights on the in s tability mechanisms Three distinct sources of wave energy were identified ; the first being the barotropic mechanism described earlier but with empha s is on the shear region between the EUC at the equator and the SEC north of the equator, and the other two being baroclinic mechanisms occurring at the frontal regions between 3 N6N and at 5 N9 N respectively. The timing of these mechanisms differed from one another ; the first occurring in boreal summer/fall and the other two occurring in winter and spring, respectively. Instabilities deriving from these different mechanisms were suggested as accounting for the differences in the observed periods of oscillation There have been remark a ble agreements between tropical instability wave observations and numerical model simulations (e g. Philander et al., 1986) Recent progress on the mechanisms and regions of generation has been made by McCreary and Yu (1992) and Yu (1992 ) clarifying the importance of the cyclonic shear region of the SECIEUC, the relative unimportance of the North Equatorial Countercurrent, and introducing a new frontal instability mechanism related to the strong meridional temperature gradient in the surface layer north of the equator. Field Program and Data The TIWE equatorial array consi s ted of five subsurface moorings each with an RD Instruments Inc. 150 kHz ADCP and a Seabird Electronics Inc. SEACA T conductivity, temperature and depth recorder. The moorings de s ignated TIWI, TIW2,
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16 TIW3 TIW4 and TIW5, were deployed in a diamond shaped array nominally centered upon 0 140 W (Figure 1). The mooring locations record durations and nominal instrument depths are listed in Table 1. A 1 00% data return was achieved with the ADCPs providing vertical profiles of horizontal velocity vectors and the SEACATs providing for vertical positioning and sound speed corrections. The moorings were s table with vertical excursions of only a few meters Upon correcting for the sound speed at the transducers and for the mean ambient sound speed between the transducers and the surface (e. g., Johns, 1988), hourly velocity data from 5112/906118/91 were resampled by linear interpolation at 10 m intervals between 250 m and 30 m. An example of the hourly sampled meridional (v) velocity component from TIW4 at 30m depth is shown in Figure 3. From August through December, regular oscillations are observed with amplitude exceeding 50 em s1 and periodicity of about three weeks. These seasonally modulated, narrow frequency band primarily vcomponent oscillations are the tropical instability waves. After the abrupt end of the wave season the oscillations are smaller and with broader bandwidth The distributions with time, depth and latitude (1 oN to 1 as along 140W) of the u and vcomponent oscillations are s hown in Figure 4 and Figure 5 respectively. For this purpose the data were lowpass filtered (using a truncated Fourier transform) to exclude oscillations at time scales shorter than 10 days and the values at depths shallower th an 30 m were estimated by linear extrapolation using the vertical shear between 40 m and 30 m (the extrapolation is supported by the findings of McPhaden et al., 1991 ). The u component isotachs are representative of the zonally oriented currents found near the equator. Specifically, there is a highly variable, near s urface confined SEC overriding the
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100 50 0 50 100 M 1990 J J A s 0 N D J 1991 F M 17 A M J Figure 3 Time series of the hourly sampled vcomponent at 30m depth from the TIW4 mooring.
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18 1990 1991 Figure 4 The ucomponent as a function of depth and time at moorings TIW2, TIW4 and TIW5, low pass filtered to exclude fluctuations at time scales shorter than 10 days. Westward flow is denoted by light stippling, eastward flows faster than 80 em s 1 are denoted by dark stippling and the contour interval is 20 em I s
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0 50 ,...... <= Stoo .::1' = .c z Q. Q,j 150 0 .c 200 250 M J J 1990 1990 A s 0 N D J F 1991 D J 1991 19 M A M J Figure 5 The vcomponent as a function of depth and time at moorings TIW2, TIW4 a nd TIWS low pass filtered to exclude fluctuations at time scales shorter than 10 days. Northward flow is st ippled; so uthward flow is clear and the contour interval is 20 ems'.
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20 EUC with its high speed core located near the top of the thermocline (Figure 6). Discussions of these currents from longterm current meter records at this location are found in Halpern et al. ( 1988), McPhaden and Taft ( 1988) and Halpern and Weisberg (1989). Of the three latitudes sampled the EUC is maximum on, and nearly symmetric about, the equator while the SEC is maximum to the north of the equator. The primary variations in the EUC appear to be both annual and intraseasonal. Three major events of maximum EUC transport are observed. During July, 1990 and April, 1991 the EUC was shallow in contrast to December, 1990 when it was deeper. The former two maxima coincide with the annual cycle of the southeast trade winds which, with varying phase, tend to be weakest over the eastern half of the equatorial Pacific in boreal spring (e. g., Meyers, 1979 and Mitchell and Wallace, 1992), causing a concomitant eastward acceleration of the near surface currents by the eastward directed zona l pressure gradient force. This is in contrast to wintertime maxima that occur in response to bursts of westerly winds over the far western portion of the equatoria l Pacific (e. g., McPhaden and Taft 1988) In agreement with previous studies the SEC is observed to be most developed from August to December 1990 in between the first two EUC maxima, and again from February to April 1991, in between the second two EUC maxima. Other than those two periods the SEC appears to be relatively weak, and westward flow was even absent on the equator from April to June 1991. The SEC is also observed to penetrate deeper both to the north and south of the equator than on the equ a tor and to be strongest at the 1 N location. In contrast to u, the vcomponent consists of seasonally modulated, higher frequency oscillations. In particular, a series of regular, large amplitude oscillations are
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21 1990 1991 Figure 6. Isotherm depths as a function of time from a TOGATAO array mooring at o o 140 W (courtesy of McPhaden, NOAA/PMEL) upon which are superimposed the EUC core depth defined by au/az = 0 and the region of the westward flowing SEC denoted by stippling.
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22 observed at all of the sample locations beginning in August 1990 and lasting into December 1990. This instability wave season for the TIWE began with the seasonal acceleration of the SEC and ended with a wintertime pulse of eastward momentum (that propagated as a Kelvin wave from the western Pacific) which temporarily halted the SEC. The vcomponent oscillations during the wave season are largest on the equator and within the westwardflowing SEC with amplitudes decreasing across the thermocline to relatively small values at or below the EUC core Philander et al ( 1985) and Halpern et al. ( 1988) observed the instability waves to be energetic and regular when the surface flow was westward and absent when the flow was eastward. The TIWE observations are similar in that the instability wave season ended in December, 1990 with the appearance of an EUC maximum at that time. Subsequently whi l e the SEC reformed the instability waves did not, suggesting that westward flow locally is not a sufficient condition for instability. Luther and Johnson ( 1990) also made this point and suggested that a strongly developed EUC is also necessary Variance Distribution and Modulation Variance density spectra for the vand ucomponents support the description of the instability waves as narrowband, primarily vcomponent proces s es Figure 7 shows the logarithm of these spectral densities as functions of depth and frequency along 140 W a t I 0N, o o and 1 S. The highlighted values show a peak in the vcomponent centered on 500 hr periodicity, nearly symmetric about the equator and confined above the EUC core
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a. ,.. Q E .._, .s .4 c. Q .4 500 200 100 67.7 0 50 ,.. s 100 0 = .s .4 c. Q 150 "1 0 C> 200 250 0 .001 0.005 0.01 0.015 Frequency (cph) ;.:J c..,. Q a.. .... u 00 0 cii Q 0 .02 b. ,.. 0 s C> = .4 .... :Z c. 0 Q .4 Q .4 Q 50 100 Q .4 .s c. r5' Q ISO .4 200 23 Period (hrs) 0.001 0.005 0.01 0 015 0 02 Frequency (cph) Figure 7. Log variance den s ities as functions of depth and frequency for the a. v component and the b u c omponent along 140 W at I 0S 0 and I 0N With a contour interval of 0 5 the lightly stippled areas highlight densities between 104 5 to 105 (em s1 ) 2/cph and darkly stippled areas highlight den s ities greater than 105 (em s1 ) 2/cph The spectra were averaged over a bandwidth of 0.92 x 1 o 3 cph for approximately 18 degree s of freedom
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24 (located at 110m on average) On the equator there is a subtle suggestion of a shift to slightly lower frequency with depth. While this shift is not statistically significant, it is consistent with previous observations (Section 2). The ucomponent spectra do not show a similar spectral peak. Instead, the variance density increases with decreasing frequency except for local maxima about the EUC core at both 1 N and I 0S. These local maxima are consistent with meridional advection in the presence of the EUC mean meridional shear, as will be commented upon later. The kinematics of the velocity component oscillations may be described using a rotary spectral analysis. Figure 8 shows the semiminor to semimajor axes ratio for the velocity hodograph ellipse and the ellipse polarization as a function of frequency and depth along 140W at 1 N, 0 and 1 S. On the equator the motions tend to be rectilinear without preferred polarization while off the equator the ellipse eccentricity decreases and the oscillations tend to be polarized clockwise to the north and anticlockwise to the south, respectively. Similar findings, consistent with equatorial wave dynamics, have been reported for the equatorial Atlantic and Pacific Oceans by Weisberg et al. (1979a, b) and Halpern, et al., (1988), respectively. Additional properties of the hodograph description are the orientation of the semi major axis and the stability of this ellipse orientation (e. g., Gonella, 1972). Stable ellipses are generally limited in the spectra to the bandwidth of the instability waves centered on 2.00 x 103 cph (500 hrs periodicity) As an example, Figure 9 shows the velocity hodographs calculated at 30 m depth for each of the TIWE moorings with the spectral bandwidth increased to essentially encompass all of the instability wave variance. On average, the near surface instability wave oscillations consist of highly eccentric
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Period (hrs) 0.001 0.005 0.01 0.015 0.02 0.01 0 015 Frequency (cp h) Figure 8. Semiminor to semimajor axes ratio for the velocity hodograph ellip s e a s a function of depth and frequency along 140 W at I 0S 0 and 1 N. Stippled a nd clear regions denote anticlockwise and clockwise polarizations, respectively. The spectra were averaged over a bandwidth of0.92xl03 cph for approximately 18 degree s of freedom 25
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VELOCITY HODO G RAPHS : 0.00125 0 00275 cph 30m lN E Q lS 10 cmls 142W 140W 138W STA BILITY TIWl 0.56 T IW2 0 4 7 TIW3 0 .65 TIW4 0 .61 TIW5 0 37 Figure 9 Veloci t y h odog r ap h ellipses at 30m d e p th at each of the 5 TIWE equatorial array mooring locations T h e ellip s es are from spectral average s over a 1.58 x 1 o3 cph bandwid t h cen tered o n 2.00 x 1 o 3 cph with appr o ximately 29 degrees of f r ee d om Ell i pse stabi l ity is given on the right relative to a 90 % significance level of 0.15. 26
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27 ellip s e s, oriented northward but tiltin g toward the east, with this tilt increasing from nearly zero at I o s to maximum value s at I 0N. The direction of tilt into the s hear of the SEC (or equivalently into the shear between the SEC and the EUC a t slightly deeper depths) is consistent with barotropic instability extracting perturbation wave energy from the mean flow These tilts decrease with depth and the ellipse stability becomes s tati s tically insignificant by the EUC core depth. Thus where stable the hodographs provide kinematica l support for the hypothesi s of wave generation by barotropic instability The hodographs further sugges t that the maximum wavemean flow interaction occurs within the cyclonic shear regions of the surface SEC and the border between the SEC and the EUC, cons istent with the previous findings cited in Section 2. The narrowband nature of the instability waves suggests that an appropriate technique for examining the distribution of amplitude in time and space i s complex demodulation analysis (e g ., Bloomfield, 1976) This was performed over a 1 58 x I o 3 cph bandwidth centered upon 2.00 x I o 3 cph as in the hodograph description. The a naly s is consists of Fourier tran s forming the time s erie s s hifting the analysi s b a ndwidth to be centered upon zero frequency, bandpass fi l tering the s hifted Fourier tran sform to exclude all coefficients outside the analysis bandwidth and then inverting the Fourier tran s form The amplitude of the en s uin g complex valued time series give s the amplitude envelope of the fluctuation s and the rate of change of phase gives the deviation of the f r equency from the central frequency The amplitude re s ult s at I 0N o o and I o s along 140 W for v and u are shown in Figure 10 and Figure II, respectively For th e vcomponent, the amplitude shows the s e as onality and the near surface trapping of the instability waves. During the in s tability wave seas on the amplitude tend s to be largest on
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0 so .... 0 !.too "':t .c ..... z c. tSO Q 200 2SO M J J A s 0 N D J F M A M J 1990 1991 0 cS = so 0 .... !.too = 0 v "'0 .c "':t 0 ..... c. s tSO
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0 50 c 5100 = = 1""'1 z Q. 150 c Q 1""'1 200 250 M J J A s 0 N D J F M A M J ;::;> ""' 0 1990 1991 = 0 50 CIS 5100 = 0 "C = .s 0 1""'1 c. e 150 Q Q ll< 200 c. e 250 0 u M J J A s 0 N D J F M A M J 1990 1991 0 so Uu c 5100 = = 1""'1 Cl.i c. 150 c Q 1""'1 200 250 M J J A s 0 N D J F M A M J 1990 1991 Figure 11. The complex demodulation amplitude for the ucomponent as a function of depth and time along 140W at 1 a s o o and 1 o N computed over a 1.58 x 1 o 3 cph bandwidth centered on 2.00x 1 o 3 cph Stippling highlight s regions with amplitudes greater than 20 em s1 29
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30 the equator and at all locations it decreases rapidly below either surface maxima (or near surface maxima at 1 N) to background levels at the EUC core. The amplitude begins to build in June 1990, several peaks are observed thereafter with the maximum amplitude exceeding 60 em s1 on the equator. The ucomponent amplitude extends over a longer duration and depth range, and is not as well organized as the vcomponent. Larger ucomponent amplitudes are also observed off the equator than on the equator with the amplitudes being largest at 1 N The elevated offequator ucomponent amplitudes at the EUC core depth may be a kinematical consequence of advection in the presence of the mean meridional shear on either side of the EUC. A crossspectral analysis between autat and v at the offequator locations (using the same bandwidth as in the hodograph analysis) shows high values of coherence squared (greater than 0.6) at and below the EUC core for periodicities between the instability waves and about one week. Where coherent, the phase switches from being inphase north of the equator to 1t radians outofphase south of the equator and the transfer function amplitude has a magnitude consistent with the addition of the local mean meridional shear to the Coriolis parameter. By accounting for these deep, off equator ucomponent fluctuations kinematically, the similarities between the vand u component amplitudes (being largest north of the equator and near the surface) become consistent with a nearsurface instability mechanism.
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31 Wave Kinematics With 115 time series of horizontal velocity vectors (5 locations x 23 depths) an efficient analysis technique is required to determine their inter relationships. Empirical orthogonal function (EOF) analysis (e. g., Preisendorfer, 1988) provides a datadependent way of separating the coherent wave signatures from the extraneous background fluctuations. EOF analysis transforms a set of time series into a set of orthogonal modes that span the data space, with each mode accounting for a successively smaller portion of the total data set variance For a coherent array, most of the wave related variance will reside within one mode the structure of which describes the kinematical properties of the wave field. EOFs may be calculated in several ways: in the time domain using correlation or covariance matrices, in the frequency domain using coherence or cross s pectral matrices, or by a Hilbert transform which is a hybrid between the time and frequency domain approaches. For wave analysis, wherein information on phase propagation i s critical, analyses in the frequency domain or by Hilbert transform are the most u s eful. The present study us e d several of these approaches, including treating the v and the u co mponents separately as well as together. The calculations found to be most useful were by frequency domain and Hilbert transform applied separately to the v components, since the vcomponents are the mo s t coherent and energetic portion of the data set and they are not contaminated by the kinematical effects of background current shear advection. The EOF analyses were thus performed on the vcomponents from the five mooring locations each with time series at 23 depths between 30 m and 250 m.
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32 Extrapolated data above 30 m were omitted because they are not independent. The frequency domain analysis consisted of calculating the coherence (or crossspectral) matrices averaged with approximately 29 degrees of freedom (as in the velocity hodograph and the complex demodulation analyses), finding their eigenvalues and eigenvectors, projecting the Fourier transformed data onto the eigenvectors to get transfer functions for each mode (the principal components) and then determining the coherence between each mode and the data at each sample location. The Hilbert transform analysis employed the same bandwidth. The procedure was to bandpass filter and Hilbert transform the time series and then calculate the correlation (or covariance) matrices and their respective EOFs in the time domain. This results in a set of complex eigenvectors in space and principal components in time with similar information content as in the frequency domain EOF. The first two modes of the frequency domain EOF using the coherence matrix accounted for 53% and 21% of the total normalized variance, respectively These two modes are statistically separable following the criteria of North et al. ( 1982) and they are distinctly separate in space, with the first (second) mode primarily describing the variance distribution over the upper (lower) half of the sampled water column The first mode therefore captures the instability, and it is shown in Figure 12. Shown are: a. the mode amplitude as a function of depth at each of the mooring locations, b. the coherence s quared between this mode and the data as a function of depth at each of the mooring locations c. the phase distribution as a function of longitude and depth along the equator and d. the phase distribution as a function of latitude and depth along 140 W. The amplitudes are largest on the equator (note that these are rms values, as opposed to time
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30 30 50 50 100 100 s '" ..s 0...150 150 Q) 0 200 200 0 5 10 15 Amplitude (cm/s) 20 0 0.2 0.4 0.6 0.8 50 100 s '" ..s 0...150 Q) 0 200 Coherence squared 50 d. 1 00 1.2150 200 14 2" W 140" W 1 38" W 1 S Longitude Eq Latitude 33 Figure 12. The first (frequency domain) EOF mode for the vcomponent calculated from the coherence matrix ave raged over a 1.58 xl03 cph bandwidth centered on 2 .00xl03 cph (approximately 29 degrees of freedom). a. amplitudes as a function of depth at each mooring location (solid lines for the equator locations, dashed line for 1 N and dotted line for 1 S ), b coherence squared between the first mode and the data as a function of depth at each mooring location (90% significance level= 0.15 and the line designations are the same as a.), c the phase (rad) as a function of longitude and depth along the equator and d. the phase (rad) as a function of latitude and depth along l40 W The contour interval on the phase maps is 0.4 rad and phase has been interpolated to account for the moorings not being precisely on the equator or on 140 W The 90% confidence interval (see Appendix) is less than 0.2 rad above l 00 m and greater than 1 0 rad below I 80 m.
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34 varying amplitudes in the complex demodulation). At all but the I 0N location the amplitudes drop precipitously from their nearsurface maxima, reaching background levels by 130 m The amplitudes at I 0N also drop precipitously with depth, but below a maximum at 5080 m. The coherence distributions generally show largest values over the region where the amplitudes are above the background levels Thus, above the EUC core, where the instability wave oscillations are most readily observed some 80% of the v component variance is accounted for by this first EOF mode. The zonal and meridional slices through the first mode phase space provide phase propagation information. Where the mode and the data are coherent these phases have relatively narrow error bars (given in the Appendix). Above 100m the 90% confidence intervals are smaller than .2 rad which is much smaller than the zonal phase difference s These interval s increase with depth, but they remain s maller than the zonal phase differences along equator. To the contrary the vertical and meridional phase differences are smaller than the 90% confidence intervals at all locations. The zonal phase di s tribution along the e quator shows a very uniform phase gradient above 90 m. Here the direction of phase propagation is we st ward and downward. Below 90 m the zonal direction persi s ts while the vertical direction changes to upward and then downward again. In all cases the vertical phase differences are small and not statistically different from zero; however, it is curious that the changes in the sense of vertical phase propagation coincide with the upper and the lower portions of the thermocline within which the EUC is located. The meridional phase distribution along 140 W is not as well defined as the zonal phase distribution along the equator. Above the thermocline the meridional component of the phase gradient i s small and s tatistically
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35 indeterminate. The largest meridional phase gradient region lies between the equator and 1 6N at the EUC core depth, but again this is not statistically different from zero In summary, the phase distributions show very well defined zonal propagation, essentially no meridional propagation and interesting, but statistically insignificant vertical propagation With the spatial variations in phase, statistically signif icant at least in the zonal direction, estimates of horizontal wavenumber vectors were made by independently fitting planes to the phase information at each depth using linear least squares regression. The implied assumption is that the zonal and meridional wavenumber components at each depth are uniform over the array; deviations from this assumption result in error. Since none of the vertical phase differences are statistically significant similar analyses in the vertical were not performed. The procedure is as follows. At any specif ied depth let be the EOF phase at the ith samp le location (xi, Yi). A horizontal plane fit of the form
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50 70 90 110 130 150 170 190 210 230 9.42 1000 6.28 Wavelength (km) 2000 3.14 \ \ \ I I I I I I I 2000 I / I \ \ / \ ' \ \ I I / / .... .... .... 0.00 ' \ ) I 3 14 1000 6.28 Wavenumber (103 radlkm) 36 667 9.42 Figure 13. The zona l (so lid ) and meridional (das h ed) wavenumber vector co mponents as a function of depth along wit h their 90 % confidence intervals for random errors (see Appendix) ca lcul a ted at 0, 140 W using the vcomponent first EOF mode and independent plane fits by linear regre ssio n at each depth
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37 and between 9001300 km. Below 110m k begins to vary and the error bars increase, but the direction remains westward within the 90% confidence interval. In contrast to k, the meridional wavenumber component 1, while showing a small northward directed mean, is not statistically different from zero at any of the observed depths Repeating the analysis using the crossspectral matrix shows nearly identical results. Since the crossspectral matrix is weighted by the actual distribution of variance it follows that the region above the thermocline influences the first mode the most resulting in 82 % of the total variance being contained in this mode with phase estimates having slightly smaller (larger) error above (below) 110m. The spatial information from the Hilbert transform analysis was also the same. Additional information (not shown) pertains to the amplitude and frequency modulation of the fluctuations within the analyzed bandwidth Paralleling a complex demodulation, the amplitude and the rate of change of phase for the first mode give the amplitude and the frequency modulations, respectively, for the time variations of that mode An advantage over complex demodulation is that presumably the Hilbert transform EOF separates coherent wave motions from incoherent variability. The amplitude modulation results were similar to those found for the individual time series and the frequency modulation showed small variations about 2 0x103 cph, consistent with a periodogram analysis performed in choosing the analysis bandwidth. The main findings of this section are therefore independent of the various analysis techniques employed
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38 Discussion and Summary High vertical resolution profiles of horizontal velocity vectors obtained from an equatorial Pacific array of subsurface moored ADCPs were used to examine the kinematics of tropical instability waves observed during 1990. The distribution of variance showed a welldefined wave season lasting from August to December, with wave variance confined primarily to the near surface region above the EUC core. The onset of the wave season coincided with the acceleration of the SEC and the termination coincided with a strong eastward momentum pulse propagating from the west as a Kelvin wave The instability wave velocity fluctuations may be described by highly eccentric ellipses, oriented to the north, but tilting toward the east into the cyclonic shear of the SEC. Over the observational domain (1 o s to 1 N and 142W to 138 W) these tilts increased with latitude from essentially zero at 1 o s to maximum values at I 0N and decreased with depth from maximum values at the uppermost 30m measurement. By the EUC core at 110 m the wave variances hodograph tilts and ellipse stability's were all nil. The instability wave variance was contained within a narrow frequency band centered upon 500 hr periodicity. Averaged over this bandwidth the zonal wavenumber component was uniform with depth between 30 m and 110 m and directed westward with a magnitude and 90% confidence interval of 5.9xlo3 I.lx103 radkm 1 (or a wavelength of I 060 km and a 90% confidence interval between 900 to 1300 km) The corresponding westward directed phase speed was 59 em s1 Unlike the zonal component, neither the meridional nor the vertical wavenumber components were statistically different from zero.
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39 The systematic tilting of the velocity hodographs is consistent with previous findings of wave generation by barotropic instability within the near surface cyclonic shear region north of the equator. With the instability waves' onset tied to the acceleration of the SEC and the wave's kinetic energy confined primarily above the thermocline and hence above the EUC core, wave generation appears to be associated with the westward flowing SEC. However, the cyclonic shear of the SEC is related to the eastward flowing EUC, which has led some authors (e.g., Lukas, 1987; Wilson and Leetmaa, 1988; and Luther and Johnson, 1990) to argue for generation within the EUC/SEC shear region. From the hodograph tilts, the Reynolds stresses are largest nearest the surface, but the background shear is largest below the surface. Since the Reynolds stresses and the background shear both enter into the mean flow to wave energy conversion, it is difficult to label the specific current in which this occurs. The SEC appears to go unstable, but this would not occur if the entire flow field did not have the necessary meridional shear and curvature that the EUC, along with hemispherically asymmetric wind stress, provides. The eastward momentum associated with the EUC sharpens the positive curvature at the westward flowing SEC maximum north of the equator and causes cyclonic shear equatorward of this maximum. Owing to the positive curvature of the SEC may be destabilizing while the negative curvature of the EUC is stabilizing, with divergence further tending to destabilize the SEC (e. g., Philander, 1976). The hemispheric asymmetry in wind stress also accentuates the northern maximum, increasing the magnitude of the cyclonic shear north of the equator and maintaining the sign of the shear across the equator (for symmetric winds the shear would be zero on the equator). The resultant nearsurface current profiles given for the Pacific
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40 and Atlantic Oceans by Hansen and Paul (1984) and Richardson and McKee (1984), respectively thus show maximum positive curvature at about 2 3 N and negative (cyclonic north of the equator) shear from that point across the equator. From these findings, along with the Philander ( 1978) 1 112 layer stability analysis that utilized a realistic surface current distribution without a subsurface EUC, it may be hypothesized that the eastward EUC provides a catalyst for instability (by shaping the SEC) while itself being inherently stable The period and zonal wavelength estimates are consistent with previous results (Table 2) even including the earliest short duration statistically indeterminate estimates from the GATE Program. This provides an example in geophysics where the results from an ensemble of independent measurements may be compared within confidence intervals inferred by an ergodic hypothesis, and the comparison is very good. It may be concluded that the instability waves derive from a seasonally modulated, but stationary random process that is narrow band in frequency and zonal wavenumber. Barotropic instability as a mechanism for wave generation is qualitatively understood, but the selection of the central frequency and wavenumber along with additional mechanisms of instability requires more theoretical guidance.
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CHAPTER 3. TROPICAL INSTABILITY WAVE ENERGETICS: OBSERVATION FROM THE TROPICAL INSTABILITY WAVE EXPERIMENT Introduction This is the second part of the analyses of the tropical instability waves observed during the Tropical Instability Wave Experiment (TIWE) Chapter 2 (and Qiao and 41 Weisberg, 1995), reviewed previous work related to tropical instability waves, introduced the TIWE field program and described the kinematics of the observed waves. The instability waves observed during TIWE were found to be narrowband in both frequency and zonal wavenumber, with central period, zonal wavelength and westward directed phase propagation estimated to be 500 hr, 1060 km and 59 em s1 respectively. The distribution of variance showed a welldefined wave season lasting from August to December 1990 with wave variance confined primarily to the near surface region above the Equatorial Undercurrent (EUC) core The onset of the wave season coincided with the acceleration of the South Equatorial Current (SEC) and the termination coincided with a strong eastward momentum pulse propagating from the west as a Kelvin wave. Particle motions in the horizontal plane were described by eccentric ellipses oriented toward the north, but tilting into the cyclonic shear of the SEC. The tilt was maximum near the surface just north of the equator and it decreased to the south and with depth.
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42 Chapter 3 now examines the energetics of the waves observed during TIWE. Previous studies found that the seasonally modulated instability waves are related to the westward flowing SEC which is further associated with the intensity of the southeast trade winds and the zonal pressure gradient tending to balance these winds (e.g., Halpern et al., 1988 and Weisberg and Weingartner, 1988). These studies suggest that the instability waves interact with background fields and are modulated by linear processes controlling the background fields. Large horizontal Reynolds stress within the SEC was reported from near s urface measurement s by Weisberg (1984) using moored current meters and by Hansen and Paul ( 1984) using surface drifters. Weisberg ( 1984) found that the onset of horizontal Reynolds stress occurred synchronously with instability waves and that the deformation work (barotropic conversion) calculated within the surface cyclonic shear region of the SEC just north of the equator was sufficient to account for the waves growth. Hansen and Paul (1984) suggested that the waves extract energy from the mean flow just to the north of and, conversely, just to the south of the equator. The Reynolds stress analysis of Lukas (1987) and Wilson and Leetmaa (1988) provided further support of the nearsurface barotropic production just north of the equator. Weisberg and Weingartner (1988) investigated the EP (EliasenPalm, 1960) flux vector divergence between the equator and 0.75 N and 0.75S, the energy balance on the equator, and the distributions of the Reynolds fluxes off the equator at 1 75 N 3 N and 6N and concluded that the instability waves in the Atlantic gain their energy through barotropic instability within the cyclonic shear region of the SEC just north of the equator. Luther and Johnson (1990) analyzed the energy budget using repeated meridional sections of velocity and density from the Hawaii to Tahiti Shuttle Experiment.
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43 They identified three distinct sources of wave energy, each occurring at different times of the year. The first is the barotropic mechanism described earlier, but with emphasis on the shear region between the EUC at the equator and the SEC north of the equator, and it occurs in boreal summer/fall; the other two are baroclinic mechanisms occurring at the frontal regions between 3N6N in winter and at 5 N9 N in spring, respectively Wavelike oscillations of the zonal equatorial currents reported from numeric al model sim ulations (e.g. Philander et al., 1986) s how remarkable agreement with observations. Numerical models are also useful tools for studying the mechanisms and the regions of generation of the instability waves The s tability study by Cox ( 1980 ) u s ing a general circulation model confirms the importance of the barotropic instability due to the shear in the mean surface currents. Recent studies by McCreary and Yu (1992), Yu(l992) and Yu et al., (1995) using a 2 layer model clarify the importance of the cyclonic shear region of the SEC/EUC and the relative unimportance of the North Equatorial Countercurrent. They also introduce a new frontal instability mechanism related to the s t rong meridional gradient in the sea surface temperature (SS T ) n ort h of the equator and suggest that the SST front is an essential part of the tropical instability wave dynamics rather than being a passive consequence of advection by the instability waves In this analysis, the processe s that may affect the energy associated with the tropical instability wave are examined through the perturbation energy equations, u si ng the current velocity data from the TIWE array and temperature data (for estimating the density and meridional pressure gradient) from the Tropical OceanGlobal Atmosphere Tropical Atmosphere Ocean (TOGATAO) ar r ay. In Section 2 the results of the
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44 perturbation horizontal kinetic and potential energies are described and the results are summarized and discussed in Section 3 Perturbation Energies Consistent with the finding of Qiao and Weisberg (1995) that the instability waves observed during TIWE had a central periodicity of 500 hr, 500 hr is chosen as the basic time interval for averaging, as running means, the variances and covariance s in the perturbation energy equations. The perturbation energy includes both kinetic and potenti a l energies. The perturbation kinetic energy (PKE) equation follows directly from the momentum equation which in vector form is : ........ v ........ :;;t ,.. V1 + (v )v f XV=+ gk p where vis the velocity vector having components (u, v, w) in the conventional coordinates system (x, y, z) directed positive to the east, north and up, respectively;/ is the Coriolis parameter ; p is pressure ; p is density; V is a gradient operator; t is the s tress ,.. vector; g is the acceleration of gravity and k is the unit vector in the zdirection Performing a Reynolds decomposition the perturbation momentum equation is : v't + CVV)v' + (v'VYV + (v' V)v'fxv' = t' z p where angle brackets denote an average over 500 hr primes denote deviations of the individual variables about their 500 hr averages subscripted variables denote partial differentiation and capitalized variables are the 500 hr running means. Since p' is much
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45 smaller than p the pressure gradient term in the above equation can be further developed as: I Y'p + Y'pl I _!_ Y'p + Y'pl I _!_ 1 1 I ( p) ( p+pl) [p( l+pl/p )] _!_ ['\7 I '\7p 1/p] _!_ [ I I I] I [ II I] = p v P v P = p P x + P y + P z p 2 P P x + P P y + P P z The hydrostatic approximation 0 = p/p g gives: Dw/Dt = DW/Dt = Dw 1 /Dt = 0 and consequently, Pz = pg and P z 1 =pig Since the magnitudes of P x and py, respectiv e ly, will not be largely different from those of P 1 x and p 1y, the magnitude s of Px and j?y will not be l argely different from those of P x 1 and py' as well. being at the order of 103 12[P1Px + p 1j?y ] will be much smaller p p p [Px 1 + p y 1 ] and can be neglected. Thu s, _!_[ I+ I + I] J...plP x Py P z 2 P z p p p Takin g the dot product between vector form perturbation momentum equation and vi and then averaging over 500 hr results in the perturbation kin e tic energy equation: PKEt + + p + p [ .... 1 '\7 1 ] 1 .... 1 '\7] ... I ;;!1 =< V V p > = V p p p where PKE=p /2 Since W1 2 i s five orders s maller than ul2 and v 1 2
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46 The pressure work term can be further simplified as: I p = [ + + ] :p2 p p = [ + + ] + g Thus the pressure work includes two parts : the rate of work by the velocity vector against the pressure gradient redistributing PKE spatially (the first three terms) and the rate of work by the vertical velocity component against buoyancy (the last term). The integration of the first three terms over the domain beyond which the instability waves are evanescent is zero; therefore, these terms redistribute energy within the region To be consistent with previous studies, those terms hereafter will be referred to as the pressure work. The l ast term represents the energy conversion between PKE and perturbation potential energy ( PPE) Since this is the link between PKE and the density structure it will be considered separately from the pressure work. Note that since : + g=g + g = 0 the conversion of PKE from PPE is due to the pressure work by the vertical velocity component only Assuming a mean density that is constant at Po= I g cm3 and negle cti ng the stress term, the PKE equation may be rewritten as: PKEr=(UPKE) x(V PKE) y(WPKE)z Ux(Uy+ V x ) VyUc V z _th<(u'2 +v '2) u'>cV2<(u'2 +v'2)v'>yV2<(u'2 +v'2)w'>z ( ++) +g (Ia) (lb) (lc) (ld) (le) The roles of (ld) and (le) have been discussed. The roles of other lines are s uch that the righthandside of (la) is the divergence of the PKE by the mean flow; (l b) gives th e barotropic energy conversion between mean flow field and perturbation velocity through
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47 deformation work and (lc) represents the divergence of PKE by perturbation velocity field. The PPE equation follows from the perturbation density equation: Pt' + (v'V)p' + (VV)p' + (v'V)p<(v'V)p'> = o Taking the product of this equation with p' and averaging over 500 hr yields: (p'pt' + p'(v' V)p' + p '(V V)p' + p'(v' V)pp'<(v'V)p '>)g/ pOz = o Defining PPE= Yzg 1Poz, then given: or PPEt + CVV)PPE + Yzg<(v'V)p'2>1Poz + g( V)p/poz = 0 PPEt =UPPE x VPPEyWPPE z g(Pox gpoy )/Poz Yzg( + + )IPoz g The right hand s ide of (2a) is the mean advection of the PPE ; (2b) is the baroclinic (2a) (2b) (2c) (2d) energy conversion between the mean density field and the perturbations; (2c) gives the advection of the PKE by the perturbation velocity and (2d) is the conversion between PKE and PPE. Generally if g is small the net influence of the PPE on the PKE will be small and a detailed examination of PPE will not be necessary. An examination of both the PKE and the PPE are presented below. To reveal the temporal and vertical variation of the relevant quantities the format for presentation will consist of isopleth contours of each term as a function of time and depth at each location where estimates can be made.
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48 Reynolds stress tensor and Rey nolds density flux It is useful look at the distribution of the Reynold s stress tensor and the Reynold s density flux before comparing each of the terms in the PKE and PPE equations. The two diagonal components and at 0 140W are shown in Figure 14. They have the same structures as the complex demodulations of the zonal and the meridional velocity components at 0 140 W, respectively (see Figure 10 and Figure 11) but with s quared amplitudes Vertically, large stresses are generally confined to the upper 110m (the recordlength average depth of the EUC core) with a maximum at the surface. The is large only from August to December 1990 thus defining the instability wave season. The distributions with time, depth and latitude ( 1 S to 1 N along 140W) for the horizontal Reynolds stress term is shown in Figure 15. Positive values are observed during the instability wave season at all three latitudes with largest m a gnitudes at 1 N and smallest magnitudes at 1 S A large negative stress is also observed at all three latitudes in December 1990, coinciding with the termination of the instability waves. Vertically large positive values are confined to above 110m. The Reynolds stress components and are calculated at 0, 140 W, where the wcomponent is estimated from the observed uand vcomponent s using the continuity equation. A detailed discussion of w, including error estimates, comparisons with previous estimates and implications regarding the upper ocean heat balance are given in Weisberg and Qiao (1996), a nd the role of win the equatorial zonal momentum balance is discussed in Qiao and Wei s berg ( 1996). For purpo ses herein it is noted that w
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49 Oo, 140W 0 50 .. "' rrl .. ... s 100 8 C,J ..._. .c .I c. 150 = = v 200 250 MJ J A s 0 N D J F M A M J 1990 1991 0 50 .. "' rrl .. ... 8 s 100 C,J .c ..._. .I c. 150 .... .... v 200 250 MJ J A s 0 N D J F M A M J 1990 1991 Figure 14. Reynolds stress components and as function s of time a nd depth at 0 140 W Th e co n to ur interval is 200 cm2 s2
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50 50 ,..._ 5too' 0 = ..= ..... 00 0 150 Q J Figure 15. Reynolds stress component as a function of time and depth at 1 S, o o and 1 N along 140 W Po s itive values are s tippled and the contour interval is 200 cm2 s2
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appears to be well resolved at the instability wave frequency. The result s of and at 0 140W are shown in Figure 16. Both terms appear to be large from September 1990 to February 1991 and show different signs above and below the EUC core. Above the EUC core they tend to be negative and large and below the core they tend to be positive but with relatively small magnitude. 51 All three offdiagonal components of Reynolds stress tensor (, and ) appear to be related to instability waves in that they all tend to be largest during the wave season. The is strongly surface confined with relative maxima at the surface It is positive and large from July to September 1990 during the onset of the wave season. The and terms extend deeper than , with relative amplitude maxima oftentimes occurring at depths between 50 m and 80 m. To calculate the Reynolds density flux, the density distribution must first be estimated. The density distributions at 2S, 0 and 2N along 140 W are estimated from the equation of state of seawater using the available temperature data at those locations a nd a constant salinity of 35 ppt (salinity data is unavailable) The density distributions at 1 S, 140 W and 1 N, 140W are then calculated by linear interpolation. Since the temperature was sampled daily while the current velocity was sampled hourly, the velocity data was reduced to daily average for the Reynolds density flux calculation and the basic averaging length for calculating the mean variance and covariance in any of the terms involving density (and pressure gradient) was changed to 21 day s instea d of 500 hr. The Reynolds density fluxes by the vcomponent ( ) calculated at I o s oo and I 0N along 140W are shown in Figure 17. Large values of are observed before and
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52 "! !;/) ... 8 (,) '? Q .c: 1""'4 ..... '"' c.. 1\ Q) 150 Q a; = v 1990 0 I "! 50 !;/) ... 8 (,) _, 100 'b .c: @ 1""'4 ..... '"' c.. 1\ Q) 150 Q ;> v 200 1991 Figure 16 Reynolds stress components and as functions of time and depth at 0 l40W Positive values are stippled and the contour interval is 20 cm2 2 s
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0 50 0 100 Q .:1' = ....c z c. 150 0 Q ....c 200 200 250 MJ J A s 0 N D J F M A M J 1990 1991 0 50 0 Q ! 100 .:1' .c ....c .... Cl5' c. 0 150 ....c Q 200 250 MJ J 1990 Figure 17. Meridional constit u ent of Reynolds density flux as a function of time and depth at 1 as, 0 and 1 N along 140 W. Po sitive values are stip pled and the contour interval is l x l03 g cm2 s1 53
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54 during the instability wave season. After the wave season a large flux at the EUC core remains, but the flux above the core decreases The magnitude of this flux is large from the surface to the EUC core with the largest magn i tude generally located at the depth of EUC core The magnitudes of density fluxes increase to the north with maximum magnitude occurring at 1 N. Reynolds density flux by thewcomponent () at 0, 140W is shown in Figure 18. It is small except in September 1990 and again in January and February 1991. The large negative flux in September 1990 is confined between 70 m and 170 m with a maximum amplitude at 120m. The other large flux in January and February 1991 begins with negative values between 120m and 220m and is followed by positive values between 80 m and 180 m Mean divergence of PKE: righthand si d e of line la To examine the processes that control the growth and decay of the PKE the terms on the righthandside of the PKE equation are estimated Positive values act to increase the PKE and negative values act to decrease the PKE Central differences are generally applied in evaluating spatial differential operators The righthandside of line (1 a) in the PKE equation is the divergence of the PKE by the mean flow field. The mean zonal PKE divergence (UPKE)x at 141 W, 140W and 139W along the equator is shown in Figure 19. It has a magnitude of 2x103 cm2 s3 and its zonal variation is not large. Relatively large values are generally observed during the instability wave season and in the upper I I 0 m
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55 50 100 ;i c. Q 150 200 MJ J A S 0 N D J F M A M J 1990 1991 Figure 18. Vertical c onstituent of Reynolds density flux as a function of time and depth at 0 140 W Positive value s are s tippled and the contour interval i s 5x107 g cm2 s 1
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56 1990 1991 Figure 19 Mean zonal divergence of perturbation horizontal kinetic energy at 141 W, 140W and 139W along equator. Positive values are stippled and the contour 1 s 1 o4 2 J mterva IS x em s
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57 The mean meridional PKE divergence (VPKE)y at 0.5S, 0 and 0.5 N along 140 W is shown in Figure 20 It has the same magnitude as (UPKE)x. On the equator and at 0.5N it shows a welldefined pattern with a surface divergence and a su bsurface convergence The surface divergence and subsurface convergence are the consequence of the surface Ekman divergence and the subsurface geostrophic convergence of the mean flow field, respectively. Such tendency also occurs at 0 5 S, but the pattern is not as well defined near the surface At all locations this surface divergence and s ub surface convergence pattern is largest during the instability wave season and tends to inc re ase to the north. The mean vertical divergence of PKE, (WPKE)z evaluated at 0, 140 W is shown in Figure 21. It also shows a welldefined pattern but with a convergence near the surface and a divergence below. It, therefore tends to oppose the mean meridional PKE divergence at all depths, but with a relatively small magnitude (1.5xl03 cm2 s 3 ) compare d to the other two mean PKE divergence constituents. Summing the three mean PKE divergence constituents gives the total mean PKE divergence [the righthandside of (la)], as s hown in Figure 22 for location 0 140 W. Its magnitude (of order 10 3 cm2 s 3 ) is relatively small compared to each of its cons tituents. Energy conversion between the waves and meanflow: line lb The mean flow and the waves interact through the Reynold s stress tensor. Any positive term in line (1 b) implies that it converts kinetic energy from mean flow to instability waves and, conversely, a negative term implies a kinetic energy conversion
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Figure 20. Mean meridional divergence of perturbation horizontal kinetic energy at 0 5S o o and 0.5 N along 140 W Positive values are stippled and the 1 s 1 o4 2 3 contour mterva I S x em s 58
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0 50 .. ,, 100 .s: Q,. 150 200 59 1991 Figure 21. Mean vertical divergence of perturbation horizontal kinetic energy at 0 140 W Positive values are stippled and the contour interval is 5x104 cm2 s3 50 .. ,, 100 i Q,. 150 200 Figure 22. Mean divergence of perturbation horizontal kinetic energy at 0, 140 W. Positive values are stippled and the contour interval is 5 x 1 o4 cm2 s 3
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from waves to mean flow. The Ux and  V x terms are calculated on the equator at 141 W, 140W and 139 W and found to have magnitude generally smaller than 0.5 x 10 3 cm2 s3 60 The termUy, is calculated at 0 5 S, o o and 0.5N along 140 W and the result for each of the locations is shown in Figure 23. On the equator and at 0.5S, it is generally small except that at 0 5 S relatively large values are observed in November and December 1990. At 0.5N it is much larger than that at the other two locations Large positive values up to 2x10 3 cm2 s3 are observed in July and August 1990 in the upper 110 m, which implies that the instability waves are extracting energy from the mean flow field through the meridional gradient of the mean zonal velocity component. The average value of 103 cm2 s3 is large enough to support the initial wave growth (with an efold time scale of 20 days and an energy level of 1300 cm2 s2). This energy source, thus, acco unts for the generation of the instability waves during TIWE. Lar ge po s itive and negative values are observed in September, November and December 1990 but they generally last not longer than one month It is noteworthy to point out the large positive values in November followed by the large negative values in December 1990, bec a use those conversions occur just before the termination of the instability waves and they tend to be antisymmetric about the equator with large negative values followed by large positive values showing up at 0.5 S. The Vy result at 0 .5 S 0 and 0.5N along 140 W is shown in Figure 24. At all three locations, generally, negative value s are observed near the s urface and positive values are observed below the surface negative. Bec ause is po si tive
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0 50 .. ,.., = 100 .:!' 1""'i z .c ..... .. Q., ll'l 150 d 200 0 50 ,.., !.too .. = .c .:!' 1""'i ..... Q., .. Q,j 150 = Q 200 250 MJ J A s 0 N D J F M A M J 1990 1991 0 50 ,.., .. = !. 100 .:!' 1""'i v5' In Q., d 150 200 250 MJ J 1990 1991 Figure 23 Deformation work c o nstitu e nt < u'v '> U y a s a fun cti on of time a nd depth a t 0.5 S 0 a nd 0 5 N a long 140 W Po s itive v alues a re s tippl e d and th e 1 s 1 o 4 2 3 contour mterv a I S x e m s 61
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0 Q '? Ill Q 50 ,... SlOO = ..... Q., 150 200 50 200 250 MJ J 1990 Figure 24. Deformation work constituent < v'v'> V y as a function of time and depth at 0.5 S, 0 and 0.5 N along 140 W. Positive values are s tippl ed and the t 5 104 2 contour mterva ts x em s 62
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63 definite the sign of this term depends on the sign of Vy. Therefore, the negative surface and positive subsurface values result from the Ekman divergence and geostrophic convergence of the mean flow field, respectively. During the instability wave season the near surface negative values are large with magnitude of about 2x10 3 cm2 s3 and the magnitudes increase slightly to the south. Only at 0.5N are relatively large positive values observed in September and October 1990 between 30 m and 110 m. Thus, on and to the south of the equator this term generally tends to weaken the instability waves, but to the north of the equator it may act as both a source and a sink for the wave energy Because the magnitudes of both Uy and  V y are equally large and at times of different sign, the sum of these two terms is given in Figure 25 for the three locations along 140W. On the equator and at 0.5S values larger than lxl03 cm2 s 3 generally have a negative sign, occur only during the instability wave season and are a maximum at the surface. At these two locations, therefore, the combination of Uy and  Vy acts as an energy sink for the instability waves with most of the conversion from the waves to the mean flow occurring near the surface. At 0.5N, in contrast, large positive values are observed in July and August extending from surface to 120 m, representing a large kinetic energy conversion from the mean flow to the waves, as required for the initial instability wave growth. The negative values occurring in September 1990 coincide with the short decrease period of wave energy level (see Figure 10 or Figure 15). The corresponding positive and negative values from late September to December show good agreement with the increase and decrease of the wave amplitude during that time
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0 .... 0 _, 100 '<:!' z .c .... 0 c. ll) Q,l 150 = 200 250 MJ J 1990 Figure 25. The (U y +Vy) as a function of time and depth at 0.5 S 0 and 0.5 N along 140 W Po si tive values are stippled and the contour interval is 5 x 104 cm2 s3 64
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65 The results of Uz and Vz at 0 140W are shown in Figure 26. The term Uz tends to reverse sign across the EUC core. Large negative values with magnitude of2x103 cm2 s3 are observed from August 1990 to February 1991 above 110 m but below 110 m this term is generally small. The  V z is found to be small with magnitude hardly exceeding 0.5x103 cm2 s 3 In summary the three important terms in line (lb) are Uy, Vy and Uz. On the equator the combination of the first two terms and the third term all tend to weaken the instability waves. Only at 0.5N does the combination of the first two terms act as an energy source for the growth and maintenance of the instability waves. Estimates are not available for the term U z at 0 5N. Divergence of PKE by perturbation velocity: line lc Most of the triple product terms in (lc ) are generally smaller than 0.5 x 10 3 cm2 s 3 in magnitude except Yz<(u'2+v'2)v'> y whose magnitude may be up to 2.5 x 10 3 cm2 s3 off the equator. The result of Yz<(u'2+v'2)v'>y at 0 5S, 0 and 0.5 N along 140W is shown in Figure 27 It tends to be large in the instability wave season in the upper 110m. It is also larger off the equator than on the equator Since most of the large values are negative this term generally tends to weaken the instability waves. Overall the triple product terms are weak with respect to the deformation work conversion term s Pre ssure work : line 1 d The meridional pressure gradient at I 0S and I 0N along 140 W is estimated using the density data (calculated from the temperature data) along 140 W between 2 S and 0
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.;: ;:i Q.. ... 150 ... =' v I 200 MJ J 1990 .c > fr 150 ... ... > v I 200 A s 0 N D J F M A M J 1991 0 N D J F M A M 1991 J MJ J 1990 A s 66 Fig ure 26. D e formation work constituent < w'u'>Uz a nd  V z as function s of time and depth at 0 l40 W. Positive value s are s tippl ed and th e contour interv a l is 5x w 4 cm2 s 3
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0 50 ,.... = !too .:1' .c '? ..... Q.c II) 150 Q 200 250 MJ J A S 0 N D J F M A M J 1990 1991 Figure 27. The Y2 <(u'2+v'2)v'>y as a function of time and depth at 0.5S, 0 and 0.5 N along 140W. Positive values are stippled and the contour interval is 5x104 cm2 s3 67
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68 and between o o and 2N by referencing to 250 m The result of  at 1 S 140W and 1 N, 140 W is shown in Figure 28. Large values (with magnitude up to 4 5 x 103 cm2 s3 ) are observed at both locations (although generally larger at 1 N) during the instability wave season and they generally tend to be confined to the upper 110 m. The meridional pressure work is found to be generally negative except for one or two short positive events lasting less than a month. Therefore the meridional pressure work tends to weaken the instability waves by radiating perturbation energy meridionally out of the region being examined. The large positive and negative (antisymmetric about equator with negative and positive occurring at 1 S) pressure work in November and December 1990 is noteworthy to point out because it coincides with the termination of the instability waves. The alternation of positive and negative pressure work appears to be related to the upwelling and downwelling phases of Kelvin wave pulse that transited the array at that time. The antisymmetric about the equator for the pressure work is consisted with the antisymmetric structure of the meridional pressure gradient field associated with such Kelvin waves. This differs from other terms during the instability wave season since the v component and meridional pressure gradient for the instability waves are both symmetric about the equator. The  term equals g as pointed out earlier. It has the same values but opposite sign as g , which is shown in Figure 29 for 0 140 W It is negligible most of time except in September 1990 and again in January and February 1991 between 80 m and 220m. The positive values ob s erved in September 1990 imply that PKE is
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69 ,.._ 0 5100 = .:!' .c .... z c:l. 150 0 200 J F M 1990 1991 0 50 ,.._ 0 5100 = .:!' .c .... ... c:l. 00 150 0 200 D J 1991 Figure 28. The meridional pressure work as a function of time and depth at 1 S 0 and 1 N along 140W. Positive values are stippled a nd the contour interval is Sxl04 cm2 s 3
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convergent in the region between 80 m and 180 m. Overall it is not as important as  and relatively large values only exist for short duration. 70 With the available data  cannot be estimated Although the temperature data at 170W and 125W on the equator are available, the distance between those locations is too large compared to the zonal wavelength of the instability wave. From observation in the equatorial Atlantic Ocean Weisberg and Weingartner (1988) found that zonal and meridional pressure work terms are equally large and tend to oppose each other. Since the temporal variation of the zonal pressure gradient is large ( e .g Wilson and Leetmaa, 1988) there is no reason to expect that the zonal pressure work will be small, so it is likely that the  is offset by  as found by Weisberg and Weingartner (1988). Without the estimation of  it is not possible to further address the role of (1 d). Energy conversion between PKE and PPE: line le The term g at 0, 140 W is shown in Figure 29 During the wave season, relatively large negative values are observed in September 1990 between 80 m and 180m suggesting that PKE is being converted to PPE through buoyancy work Because the g term is the only term that links the PKE and the PPE, its negative values suggest that on the equator there is no net contribution to the large horizontal velocity oscillations (the PKE) by baroclinic instability. Locally the work against the buoyancy force tends to reduce the current fluctuations Thus, for understanding the cause of the large PKE near the equator it is not useful to examine the detailed development of the PPE
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g 50 200 MJ J A 1990 s 0 N D J F M A M 1991 J Figure 29. The energy conversion between the wave kinetic energy and wave potential energy through buoyancy work g at 0 140W. Positive value s are stippled and the contour interval i s 5 x104 cm2 s 3 71
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72 Nevertheless the PPE equation will be examined to further support of the conclusions and to compare with previous results. Summari z ing the righthandside of equation 1 The terms that are important for the kinetic energy balance of the instability wave th U < I I>V < I I>U lL ( 12 12) I I I I I I I d are ev y,v v Yu w z i'2< u +v v >y, , , an g The first three terms represent the energy conversion between the instability waves and mean flow field; the fourth one represents the flux divergence of the PKE by the perturbation themselves; the fifth to seventh represent the pressure work ; and the last one gives the energy conversion between the kinetic and potential energy The sum of the first two terms north of the equator in the upper 120m is large and positive during the onset of the instability wave season, and apparently accounts for the initial instability wave growth observed during TIWE. Its temporal variation is also generally in agreement with the amplitude modulation of the fluctuations associated in the v component. Thus, barotropic instability within the cyclonic shear region of the of the mean horizontal velocity field is an important energy source for the instability waves Near the surface  Vy is an energy sink for the instability waves However, owing to the geostrophic convergence of fluid just below the mixed layer, this term augments the wave growth just north of the equator. The calculation of Uz can only be performed on the equator and it tends to weaken the instability waves. Off the equator the role of Uz remains uncertain. The pressure work represents the propagation of the wave energy. The  term tends to weaken the instability wave by propagating the wave energy meridionally out of
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the region examined while a compensating energy propagation into the region through  can only be inferred. On the equator  converges wave kinetic energy within the region between 80 m and 180 m where it is converted to potential energy by g. Advection of PPE by mean velocity : right handside of line 2a 73 Of the three advection terms in the righthandside (2a) only the advection of PPE by mean meridional (VPPEy) and by mean vertical (WPPE z ) velocity components are able to be estimated. The VPPE y at 1 S, 140 W and 1 N I 40 W and the WPPEz at 0 140 W are shown in Figure 30 and Figure 31, respectively. The values of both terms are generally smaller than 0.5 x 10 3 cm2 s 3 except in January and February I991 when the la tter have values as large as 10 3 cm2 s 3 Qiao and Weisberg (1996) have found that the equatorial current is fully threedimensional with all three constituents of the zonal momentum divergence and their sum all being the same order of magnitude The result of the mean divergence of the PKE also suggests that the magnitude of mean divergence itself is not larger than that of each of it s constituents. Thus it is reasonable to assume that the total mean advection of the PPE generally has magnitude smaller than I o 3 cm2 s 3 as do the VPPEy and WPPE z though the UPPE x cannot be estimated. It may then be concluded that the mean advection of PPE is not of primary important in the instability wave energy balance.
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0 50 ..... 0 S1oo = .::1' .c .... '? c. 150 200 s 0 N D J F M A M J 1991 0 50 ..... S1oo = .::1' .c .... 00 c. 0 150 200 Figure 30. Mean meridional advection of perturbation potential energy at 1 S and 1 N along 140 W. Positive values are stipp led and the contour interv a l is 5 x 104 cm2 s 3 74
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75 0 50 g 100 .; g.. Q 150 200 Ou o MJ J A s M J 1990 Figure 31. Mean vertical advection of perturbation potenti a l e ner gy at 0, 140 W. Positive values are stipp led a nd th e contour interval is 5 xl04 cm2 s 3
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Energy conversion between the waves and mean density: line 2b The energy conversion between the wave potential energy and the mean density field through the mean meridional density gradient (gpoy IPoz) is calculated and 76 shown in Figure 32 for locations 1 S, 140W and 1 N, 140W. The energy conversion by this mechanism (baroclinic instability) is small (generally smaller than 1 x 10 3 cm2 s 3 ) at both locations although observed values are slightly larger during the instability wave season The potential energy conversion through mean zonal density gradient (gPox /paz) can not be estimated. Weisberg and Weingartner (1988) found that gPox IPoz had the same magnitude as gpOy )/paz and tended to oppose it. Since the value of Pox near the equator is generally smaller than that of Poy at 1 N (Poy tends to be zero on the equator), and since is not expected to be substantially larger than it is reasonable to assume that gpox /Poz is not larger than the estimated gpoy )/Poz Advection of PPE by p e rturbation velocity: line 2c Two of the three terms in (2c) representing the advection of PPE by the perturbation velocity of the meridional ( Y:zg IPoz) and the vertical (Y:zg IPoz) components, respectively can be estimated Both are found to be small with values never exceeding O.Sx l03 cm2 s3 Previous studies always found that these triple product terms are not important in the instability wave energy balance (e. g Weisberg and Weingartner, 1988 and Luther and Johnson, I 990).
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77 F ,.. 100 Q ..c 1"""1 00 c:l. 0 150 1"""1 200 250 1990 1991 Fi g ure 32. The baroclinic energy conversion between the wave potential energy and the mean den s ity field through meridional density gradient gp0y )IPoz at I 0S and 1 N along 140 W Positive values are stippled and the contour interva l is 5 x104 cm2 s3
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Energy conversion between PKE and PPE: line 2d The link between the PKE and the PPE equations g , shows that PPE is being generated at the expense of PKE. Thus, baroclinic processes are tending to decrease PKE. Summarizing righthandside of equation 2 78 The only important term in the righthandside of the equation (2) that has been estimated is the potential energy conversion from the kinetic energy through the buoyancy force. Therefore, on the equator the increase of potential energy is a consequence, as opposed to a cause, of the instability waves Discussion and Conclusion Current velocity data from the TIWE equatorial array have been used to study the energetics of the tropical instability waves observed in the central Pacific during 1990 The important terms in the perturbation energy equations are Uy Vy, Uz, Yz<(u'2+v'2)v'>y, , ,  and g (or g ). The term Uy, is large in the SEC and the upper portion of the EUC just north of the equator and it appears to provide the energy source for the onset of the instability wave season. After this initial growth, the instability waves appear to be maintained by a combination of Uy and  Vy just to the north of the equator. Contrasting then, the  Vy on and south of the equator, the Uz on the equator and the Yz<(u'2+v'2)v'>y at all locations all provide sinks for the perturbation energy. The former
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79 two act to convert the wave energy to the mean flow and the latter acts to weaken the waves through the meridional divergence of the PKE by the perturbation velocity The weakens the instability wave, but a countereffect by  may be inferred from previous studies Vertically  propagated kinetic energy to the region between 80 m and 180m, where the kinetic energy is converted to the potential energy by g Qiao and Weisberg (1995), from the tilt of the velocity hodographs that describe the particle motions in the horizontal plane, suggested that barotropic instability is an important source of the wave energy Here the results from wave energy balance support this idea The energy conversion from meridional shear of mean zonal flow Uy is found to be large during the onset of the wave at 0.5N, 140W. It decreases southward and at 0.5S it becomes much smaller, and this small positive value is only confined to the upper 30 m, which is in good agreement with the decease of the hodograph tilts to the south. Weisberg and Weingartner (1988) found that in the Atlantic Ocean the barotropic instability within the cyclonic shear region of the SEC is the energy source for the instability wave. The waves decelerate the SEC north of the equator and, thus reduce the shear of SEC. For a short wave season lasting le s s than one month, they found that the energy production by U y is large in the wave season which implies that the energy conversion from mean flow to waves generates and maintains the waves. During TIWE U y is found to be large during the onset of the waves and lasts le s s than two months, which is much shorter than the wave season that lasts for roughly four months.
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In the later half of the wave season, the instability waves are maintained by the sum of U y and  V y 80 It is apparent from the previous studies and the analyses of TIWE data that barotropic instability at and just north of the equator i s the main energy source for the instability wave. The existences of both the SEC and the EUC are important for the waves to develop; however, the role of the EUC in the wave energy balance needs to be further clarified Qiao and Weisberg ( 1995 ) hypothesized that the EUC helps the SEC go unstable by providing the nece ssary meridional shear and curvature, while the EUC itself is stable owing to the planetary vorticity gradient and meridional flow convergence. The result of the energy balance shows that maximum energy conversion from the mean horizontal flow field to the waves generally occurs within the SEC and the upper portion of the EUC above the EUC core. Since Uz is a si nk (negative) of the wave energy and it is generally significant in the upper part of the EUC, it tends to oppose the positive conversion from mean horizontal flow to wave at those depths and reduces the cont ribution of the EUC by leaving a s mall net effect. Thus, the whole SEC is involved in the onset of the wave, whereas only the upper part of the EUC makes a contribution to the waves through the meridional shear; this contribution may be reduced by the stability arising from the vertical s hear of the mean zonal velocity component. In contrast to th e conventional barotropic instability term < u v'>Uy, the finding of large instability due to Vy is new This term is related to the meridional converge nce of the mean flow fi e ld and the kinetic energy associated with the wave As conver ges on the equator the energy associated with the wave is also co nver ges. It seems unlikely for this term to initially generate th e instability waves because is small
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81 before the waves are fully developed and Vy is symmetric about the equator whereas the waves are not. Another requirement for this mechanism to be important is the strong eastward zonal pressure gradient, which is a necessary condition for a large geostrophic convergence on the equator. The termination of the instability waves during TIWE coincided with the passage of an eastward propagating Kelvin wave in December 1990. To verify if the Kelvin wave is responsible for the termination of the waves, we have examined the available TOGATAO current velocity records at 140 W. Those data suggest that almost every year the instability waves are terminated by the Kelvin wave. A possible explanation for this is because of its meridional s tructure, a Kelvin wave diminishes the meridional s hear of the zonal current and, therefore, cuts off the energy source for the instability waves. Further support is found from the meridional shear of the zonal velocity component (shown in Fi g ure 33), which shows that cyclonic shear is large in August and September 1990, relatively small in the rest of wave season and small after the Kelvin wave passes by until March 1991 from which the surface current is no longer westward. The role of the meridional pressure work is to radiate out the wave energy and weaken the waves; however, because the zonal constituent of pressure work  could not be estimated, the net effect of the pres s ure remains unclear (Relatively the vertical pre ss ure work is not as importance as meridional one as mentioned earlier). In the Atlantic Weisberg and Weingartner (1988) found that the meridional pressure work was negative. Their estimates of zonal pre ss ure work s ugge s ted that it was positive corresponding to a zonal energy propagation into the region. During the onset of the instability wave, the two constituents tended to cancel each other and the net wave energy
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82 Figure 33 The meridional shear of zonal velocity co mponent as a function of time and depth at 0 5 S, 140 W and 0 5 N, l40W. Cyclonic s hear (negative in the northern hemisphere and fositive in the so uthern hemisphere) is st ippled and the contour interval is 1 o s"1
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83 was generated by the deformation work. The zonal pressure work was of a shorter duration than the meridional one so the net effect of the two was negative during the later half of the wave season and wave energy decayed through meridional radiation Luther and Johnson (1990) similarly found a negative meridional pressure work south of I 0N but a large positive between I 0N and 3N. The zonal pressure work is the only term that could not be estimated in the perturbation kinetic energy equation and is probably the only significant term of all the other unavailable terms in the wave energetics. The zonal s eparation of the TOGATAO temperature array is too large to resolve fluctuations on the sca l e of the instabi l ity waves The mean divergence of the PKE is not significant for the wave energetics on the equator. However, its result again demonstrates that the mean flow fie l d in the equatorial region is fully threedimensional as concluded in the zonal momentum study by Qiao and Weisberg ( I996). Each of its individual constituents has the same magnitude with none of them being negligible. Since each of the con s tituents tends to cancel each other, the total mean PKE divergence is smaller than each of its constituents. On the equator the role of the vertical velocity component to the instability wave s is included in this analysis. Although th e vertical velocity component is estimated from the horizontal velocity components rather than directly measured it is well reso lved in the instability wave frequency band (Weisberg and Qiao 1996) due to the large zonal wavelength and meridional sca le of the instability waves compared to the distance of the moorings. There is a large vertical shear in the zonal velocity component that can lead to an interaction between the mean flow and the ins tability wave through the Reynolds s tr ess ten sor. The result from 0 140 W shows that the energy conversion is from the
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84 wave to mean flow. Thewcomponent also tends to convert the wave kinetic energy to the potential energy Those effects indicate that the wcomponent tends to stabilize the current on the equator. There are several terms in the perturbation energy equations that are unable to be estimated These terms are related to the zonal variation of either the density field or the pressure field. The contribution from the variation of the density field should be smaller than that from the pressure field because the pressure results from the vertical integration of the density and it is the vertical accumulation density contribution. This idea is supported by the result of the meridional pressure work and of the terms related to the meridional density field Further support may also be found in Weisberg and Weingartner (1988) and Luther and Johnson (1990). Therefore, the only significant term missing from this analysis is the zonal pressure work. McCreary and Yu (1992) and Yu et al. (1995) suggested that the SST front is an essential part of the tropical instability wave dynamics. They found that frontal instability, which is related to baroclinic instability associated with the horizontal temperature (density) gradient, is a main source for the wave energy. The results from the TIWE array show that the baroclinic instability due to the mean meridional temperature (density) gradient is small compared to the barotropic instability although it is larger during the wave season, and that instability due to the mean zonal temperature gradient i s also unlikely to be large because the zonal temperature gradient is generally smaller than the meridional temperature gradient. However, it should be noted that this analysis only covers the latitudes between 2S and 2N. In order to know the possibility of a large frontal instability further off the equator, we examine SST distribution from June 1990 to May 1991 (Figure 34 shows an example of weekly mean SST in June and September
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85 6/24/906/30/90 9/23/909/29/90 160W 150W 140W 130W 120W 110W 100W Longitude (I) '"0 1/13/911/19/91 0 j 160W 150W 140W 130 W 120W 110W 100W Longitude Figure 34. The weekly mean sea surface temperature distribution in the region between 1 o o s and 1 0 N from 160 W to l 00W The contour interval is 0 5C.
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86 1990 and January 1991). It is found that the temperature gradient is relatively strong from June to September 1990, when the shear barotropic instability is strong; it is relatively small in the rest of the wave season and even smaller after the wave season. The meridional variation of the temperature gradient is small between the equator and 5N and the gradient decreases outside this region. Thus, the frontal instability is not expected to increase significantly at other latitudes The evolution of the SST front and the cyclonic shear of SEC suggests they tend to vary in phase. It is known near the equator the SST front is linked to the Ekman divergence, which is further related to the strength of the SEC. Since the cyclonic shear of the SEC is also related to the strength of the SEC, the SST front and the cyclonic shear of the SEC could be two coupled elements of the upper equatorial current system requiring further study In summary, our results indicate that the tropical instability waves observed from August to December 1990 were generated by barotropic instability arising from the cyclonic shear of mean zonal velocity component mainly within the SEC just north of the equator. The instability waves were then maintained by this cyclonic shear instability for the first half of the wave season augmented and modulated by shear instability arising from the meridional gradient of mean meridional velocity component just north of the equator for the remainder of the wave season. The main sink of wave energy is the meridional pressure work that radiates energy meridionally out of the generation region (this effect may be canceled by the zonal pressure work). Another important sink is the deformation work performed by the meridional gradient of mean meridional velocity component on and to the south of the equator. This term provides an energy source for the instability waves north of the
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equator and converts the wave energy back to mean meridional shear of meridional velocity component on and to the south of the equator. The other three terms acting as small sinks to the instability wave energy are the deformation work performed by the vertical shear of the zonal velocity component, the mean divergence of perturbation kinetic energy by the meridional velocity component fluctuation and the conversion of kinetic to potential energy (to reduce the amplitude of velocity fluctuation) by the buoyancy force. 87 This study provides a detailed energy budegt of the instability waves in the central equatorial Pacific (within 1 latitude) However, due to the array location, the development of the waves at higher latitude can not be addressed. It should aslo be pointed out that 140W may not be the best location for wave energetics study because wave activit i es tend to be stronger and last longer further east of 140 W (e.g. see Figure 34 for 1990). Both the latitudinal and longitudinal variations of the wave energetics need further investigation, particularly the role of baroclinic sources/sinks within and across the frontal region north of the equator.
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CHAPTER 4. THE ZONAL MOMENTUM BALANCE OF THE EQUATORIAL UNDERCURRENT IN THE CENTRAL PACIFIC Introduction The discovery by Cromwell et al. (1954) of a very swift, subsurface current, 88 flowing eastward on the equator in opposition to the winds initiated an ongoing dialog on the dynamics of this remarkable Equatorial Undercurrent (EUC) Early descriptive studies by Knauss (1960, 1966) showed the EUC to be continuous along, symmetric about and tightly confined to the equator with transports comparable to other major ocean currents Early theoretical studies beginning with the nonlinear, inertial jet arguments of Fofonoff and Montgomery ( 1955) and the linear, frictional arguments of Arthur ( 1960) and Stommel (1960) were followed by numerous articles combining these arguments into more complete theories Central to all of these is the threedimensionality of the flow field, driven by a depth dependent zonal pressure gradient (ZPG) whose vertical integral tends to balance a westward surface wind stress This threedimensionality is what makes the circulation so important to contemporary climaterelated studies, because it largely determines the equatorial sea surface temperature distribution. But, it is also what makes a quantitative understanding of the EUC so difficult, since it requires resolving the circulation s divergence
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89 In May 1990, an array of five subsurface acoustic Doppler current profiling moorings was deployed about oo, 140W for 13 months as part of the Tropical Ocean Global Atmosphere (TOGA) Tropical Instability Wave Experiment (TIWE). The array provides estimates of the vertical circulation (Weisberg and Qiao, 1996) and thus a three dimensional view of the flow field. The present paper uses these data to estimate the zonal momentum flux divergence and, combined with other data from the TOGA Tropical Atmosphere Ocean (TOGATAO) array, to diagnose the upper ocean zonal momentum balance. Section 2 reviews previous work on this topic. Section 3 describes the data and methods. Section 4 attempts a quantitative diagnosis of the recordlength averaged, depth dependent zonal momentum balance and provides a description of how the dynamics change between the surface and the base of the EUC. Section 5 offers a more qualitative (owing to data limitations) view of the time dependent variations. These findings are then summarized and discussed in section 6 Background Knauss (1960, 1966) gives a comprehensive description of the EUC along with dynamical inferences. The flow is threedimensional with a meridional convergence upon the EUC core compensated by a vertical divergence away from the core Intense vertical mixing is surmised on the equator to account for both the ob s erved material property distributions in the meridional plane and the approximate geostrophic balance (meridionally) found for the near equator zonal currents Meridional convergence upon the EUC core is the essential element in Fofonoff and Montgomery ( 1955) where the
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90 speed of the EUC is accounted for by conservation of absolute vorticity. This convergence is attributed to an eastward directed ZPG owing to a westward wind stress over a bounded basin. Recognizing that the effects of the windinduced surface stress may extend vertically over the same region for which the ZPG is dynamically significant, Arthur ( 1960) calculated a velocity profile on the equator from the balance between the vertical stress divergence and the ZPG. Thus, the EUC core occurs where the stress is zero and the stress divergence crosses zero together with the ZPG. Charney (1960) and Charney and Spiegel ( 1971) combined these inertial and viscous effects in a constant density EUC model and found that the relative importance of these terms greatly affects the resulting threedimensional flow field. Theory and observations confirm that the zonal momentum balance on the equator must entail convergences of momentum flux and stress along with a ZPG, but further advances have been hampered by data limitations. Simultaneous data have been unavailable for estimating these constituents, leaving uncertainty in the zonal momentum balance for both analytical and numerical model results. In a diagnostic study, Bryden and Brady (1985) used historical hydrographic data for a box bounded by 5S and 5N, 150W and 110W and 500 db and the surface. The mean ZPG was referenced to 500 db, the horizontal velocity components were estimated from the ZPG and the climatological wind stress using geostrophic and Ekman assumptions and the vertical velocity component (w) was then calculated by mass conservation. The ZPG decreased monotonically to zero between the surface and the lower portion of the EUC and was slightly westward below. Its vertical integral balanced the surface wind stress to within about 80%. The vertical integral of the nonlinear
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91 accelerations overcompensated the surface wind stress/ZPG imbalance implying a significant stress (and stress divergence) at least to the base of the EUC. On the equator upwelling was found above 180 db with smaller downwelling below, and the flow was described as being primarily along isopycnals. The w profile of Bryden and Brady (1985) has been used in subsequent studies for estimating the vertical advection of eastward momentum. An example is given by Wilson and Leetmaa (1988), employing data from several shipboard velocity profile and hydrographic surveys on the equator roughly between 150W and 90 W The time dependent variations for the estimated terms were found to be as large as their means. Upon vertical integration the ZPG closely balanced the wind stress, but the lack of data on individual terms precluded analyses on the vertical profile. Another example is that of McPhaden and Taft ( 1988), wherein the zonal momentum balance is studied between 1400W and 11 0W using TOGATAO array moored current meter data The vertical profiles of the ZPG and the estimated nonlinear accelerations were very similar to those of Bryden and Brady (1985), as was the imbalance in the integrated ZPG and the surface stress The mean accelerations tended to oppose each other suggesting that nonlinearity redistributes momentum vertically in the zonal plane within the upper 250 m. Assuming a small, constant vertical eddy viscosity coefficient A v=l cm2 s1 (motivated by the microstructure measurements of Peters et al., 1988), the vertical stress at 250m was calculated to be 100 times smaller than the surface stress. Thus, the vertically integrated imbalance was not resolved. A time dependent analysis showed that on intraseasonal time scales the vertically integrated ZPG varied with the surface stress to within about the same imbalance as the mean.
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92 In an attempt to resolve the role of turbulent stress divergence, Hebert e t al., ( 1991 ) used shipboard microstructure and moored measurements collected between 140 W and 110 W in spring 1987 With zonal advection being the only calculable non linear acceleration term, correspondence s were not achieved between the estimated ZPG, acceleration and turbulent stress divergence It was suggested that annual averages are necessary for comparing estimates of turbulent stress divergence with the diagnostic calculation of Bryden and Brady ( 1985) Turbulent stress divergence occurs over synoptic as well as microstructure and intermediate scales. At synoptic scales, the tropical instability waves are particularly important. For example, the horizontal Reynolds stress divergence on the equator in the central Pacific is a significant fraction of the wind stressinduced body force (Hansen and Paul, 1984), and similarly for the Atlantic (Weisberg and Weingartner 1988) Additional s upporting evidence is found in Lukas (1987), Wilson and Leetmaa (1988), Bryden and Brady (1989) and Luther and Johnson (1990). In summary, the available data sets show an approximate balance on the equator between the vertically integrated ZPG and the surface stress. Estimates of the nonlinear acc eleration terms suggest that these tend to cancel but not completely. Since the non linear accelerations may be important in adjusting the flow field to external forcing their resolution is necessary for determining the vertical distribution of stress divergence and hence an improved understanding of the equatorial currents' zonal momentum balance.
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93 Data and methods The TIWE equatorial moorings, designated TIWITIW5, were deployed in a diamond shaped array centered upon 0 140 W (Figure 1). Hourly velocity profiles were sampled byRDInstruments 150kHz acoustic Doppler current profilers (ADCP) with 20 transducer configuration. Mooring performance and data editing procedures are given in Weisberg et al. (1991). With ambient sound speed correction, the instruments nominally sampled at 9 m vertical intervals Hourly velocity components resampled at I 0 m intervals between 250 m and the surface are used herein Resampling was by linear interpolation between 250 m and 30 m (the last bin unbiased by surface reflection) and by linear extrapolation over the upper 20 m using the 30 m40 m shear. The analysis period is 5/12/90 to 6118/91 and the mooring locations and nominal instrument depths are listed in Table 1. The time and depth variations of the zonal (u) and meridional (v) velocity components at the five mooring locations are shown in Figure 35 and Figure 36, where the data here and in subsequent time series plots are lowpass filtered to exclude oscillations at time scales shorter than 10 days The ucomponent isotachs are representative of the zonally oriented equatorial currents reported on by numerous precedent studies. Observed is a highly variable, near surface confined South Equatorial Current (SEC) overriding the EUC who s e high speed core is located within the thermocline (Figure 39). The five locations show an EUC that is maximum on, and nearly symmetric about the equator and a SEC that is maximum to the north of the equator. The primary variation s in the EUC are both annual and intraseasonal. Annually,
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94 a. 1990 1991 ,.., e 'b '../ ..,. '1""'4 00 =c '1""'4 1990 1991 Figure 35. Horizontal velocity components as a function of depth and time for moorings along l40W: a. ucomponent, with westward flow denoted by light stippling and eastward flow greater than 80 em s1 denoted by dark stippling and b. v component, with northward flow denoted by stippling. The contour intervals are 20 em s1 and all time series have been lowpass filtered to exclude fluctuations at time scales shorter than I 0 days.
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95 b Q 0 ...,. i ...... z =1 50 Q ...... 2 00 1990 1 991 0 5 0 ! 100 0 ..c ...,. ...... .... Q. 1 50 Q 2 00 A M J 1991 1990 1 991 Fi g ur e 3 5 (co nt i nu e d )
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96 a. 1990 1991 Figure 36. Horizontal velocity components as a function of depth and time for moorings along the equator: a ucomponent, with westward flow denoted by light stippling and eastward flow greater than 80 em s1 denoted by dark stippling and b. vcomponent, with northward flow denoted by stippling The contour intervals are 20 em s1 and all time series have been lowpass filtered to exclude fluctuations at time scales shorter than 10 days.
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97 b. Figure 36. (continued)
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98 the depth of the EUC's high speed core varies with the thermocline with maximum EUC speed observed in July, 1990 and April, 1991 when the core was relatively shallow. In contrast an intraseasonal maximum is observed in December, 1990 when the core was relatively deep In agreement with previous studies the SEC is most developed when the EUC is deepest, except when intervened upon by large intraseasonal events such as one in December 1990. When the EUC is shallow, the SEC is weak, and westward flow was generally absent on the equator from April to June 1991. In contrast to u, the vcomponent consists of seasonally modulated higher frequency oscillations. In particular a series of regular, large amplitude o sci llations are observed at all of the sample locations from August to December 1990 These are the tropical instability waves and a description of their kinematics during this time is given by Qiao and Weisberg (1995). The vcomponent oscillations are largest on the equator within the westwardflowing SEC, with amplitudes decreasing precipitously across the thermocline to relatively small values at the EUC core. Mean horizontal velocity component vertical profiles are obtained by averaging over the 13 month record length (Figure 37). For the ucomponent the three equatorial locations are similar, except that the s peed at the EUC core increases downstream as the core depth shoals to the east. The core speeds at the two offequator locations are nearly symmetric and about 60 % of the values on the equator. The SEC is shallow at all locations, but penetrates slightly deeper off the equator. For the vcomponent, the off equator locations indicate antisymmetric behavior consistent with a surface Ekman divergence and a subsurface geostrophic convergence The three locations on the equator all show similar vertical profiles intermediate between the offequator one s so the mean
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99 a. b. U (ems1 ) V (em s1 ) 0 0 ' ' ' ' 50 ' 50 ' ,. '. , ,., ..'" s 100 ,, 100 .. .. '' 1'. ..= I: I I : c.. I ell I 150 I 150 I I I I I I I I I 200 I 200 I I I ; I ; I : I ; I ; 250 I f 250 50 0 50 100 150 20 10 0 10 Figure 37. Recordlength mean vertical profile s for the uand vcomponents at the five mooring locations The solid lines denote the three equator mooring s ( the thick one being 140 W) and the dashed and dotted line s denote mooring s north a nd south of the equator, re s pectiv e ly.
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meridional divergence is nearly symmetric about the equator. At 250 m or the base of the EUC, all of the vcomponent profiles approach zero. Vertically integrating the continuity equation between the surface and depth z provides an estimate w: Io au av w(z) = w(O)z (dx. + dy)dz (1) 100 where x, y and z are positive to the east, north and up respectively. Central differences are used for the zonal and meridional derivatives followed by vertical integration using the trapezoidal rule such that: 0 u u v v w(z) = w(O)L( 3 I + s 2 )& z X3 xl Ys y2 (2) where the subscripts denote the station locations and & is 10 m. A rigid lid approximation is used for w(O) which is correct to a factor of about 1 o2 for the synoptic or longer time scales considered. The resulting (lowpass filtered) w at 0 140W is shown as a function of time and depth in Figure 38 and the corresponding recordlength averaged vertical profiles for u, v and ware given in Figure 39. The lowfrequency variations in ware of order 103 10 2 em s1 The mean w profile shows maximum upwelling of about 2.3xl03 em s 1 at 60 m depth and a zero crossing a t 140 mjust below the EUC core. The lower portion of the EUC is thus a region of downwelling on average The fluctuations in w are a factor of 5l 0 larger than the mean and these may be of the same sign over the entire region of the water column sampled. A specific example is the intraseasonal event in December 1990 where downwelling is observed at the leading edge of the eastward momentum pulse
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50 ,... SlOO .c: .... c. 150 200 250+.....'..ol....lii.::.:........:JU:.. M J J A S 0 N D 1990 J F M A M J 1991 Figure 38 Thewcomponent as a function of depth and time estimated at 0 140 W Upwelling is denoted by stippling and the contour interval is 5 x 103 em s 1 The time series have been lowpass filtered to exclude fluctuations at time scales shorter than 10 days 101
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50 5 100 ;: Q) Q 150 200 3.0 2.0 1.0 0.0 1.0 2.0 3 0 Speed W (10.3 em s 1), U V (em s 1 ) 102 Figure 39. The record l e ngth mean vertical profile s of the u(thin s olid) v(dashed) and w(thick solid) components at 0 140 W.
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103 followed by upwelling at the trailing edge. This is consistent with the pas sage of an equatorial Kelvin wave as will be commented upon later. A more detailed discu s sion of w including error estimates comparisons with previous estimates and implications regarding the upper ocean heat balance are given in Weisberg and Qiao (1996). It is noted here that the downwelling estim a ted below the EUC core is an essentia l element in the momentum arguments that follow Temperature and wind data used for estimating the ZPG and the surface wind stre s s respectively, are from the TOGATAO array Figure 40 show s (lowpa s s filtered ) temperature as a function of time and depth on the equator at 170 W, l40 W and l25 W Con s istent with previous observations the thermocline slopes up to the east giving rise to the depth dependent ZPG. Superimposed on the isotherms at 170W and l40 W is the EUC core depth which coincides and shoals with the thermocline Also the depth of penetration for the westward SEC decrease s eastward as the mixed layer becomes s hallower. The mean zonal momentum balance Beginning with the zonal momentum balance in the form: (3) where f is the Coriolis parameter ; p is pressure; p is density ; A v and Ah are vertical and horizontal eddy viscosity coefficients, respectively and Vis a horizontal gradient operator, and then averaging over the record length gives the mean equation :
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104 0 50 0 ,.._ 100 \" = ll') ;:: Q., = 150 = Q 200 250 M J J A s 0 N D J F M A M J 1990 1991 J A S 0 N D M J 1990 Figure 40. Isotherm depths as a function of time from moorings at 0 125W; 0 140 W and oo, 170 W (from the TOGATAO array courtesy of M. McPhaden, NOAA/PMEL). Superimposed at 0, 170 W and 0 140W are the EUC core depths defined by Ju/Jz=O and the westward flowing SEC regions denoted by stippling.
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105 (4) where the angle brackets denote the recordlength average and capitalized variables are the recordlength means Each of the non linear acceleration terms may be decomposed into mean circulation and Reynolds stress terms a s : (u ( v = W + ( w' where the primes denote fluctuations about the recordlength means. (Sa) (5b) (5c) The nonlinear acceleration terms in the above formulation are expres sed using an advective scheme Alternatively using the continuity equation, the s e may be expre s sed using a flux divergence scheme wherein their recordlength averages are : () < uu > () < uv > () < uw > (){) U dUV dUW dx + ()y + ()z = dx+ ()y+ ()z() < u'u' > () < u'v' > () < u'w' > + dx + ()y + ()z (6) In either case, central differences are used to estimate the horizontal derivatives; thus, for the advective and flux divergence schemes the acceleration terms become: or ()pq= p 3 q 3plql dx X 3 XI ()pq= P sqsP 2 q 2 ()y Y s Y z or
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106 where p and q represent the appropriate horizontal velocity components, and subscripts denote horizontal location s (Table 2). The vertical derivatives are calculated by forward differencing consistent with the w estimation. Given the mean circulation's symmetry about the equator the flux divergence sc heme is necessary to reveal the fully threedimensional nature of the nonlinear zonal momentum flux. Th is same sy mmetry however, biases the flu x divergence scheme rel a tive to the advective sc heme (as developed in Appendix). For these reasons results from both sc hemes are developed and the bias is accounted for. Estimation is then made of the ZPG and the zonal wind and the results are combined into vertically integrated and pointwise momentum balances from which the vertical stress divergence follows as a residual. Integrating the vertical stress divergence then results in a vertical profiles of stress and Av. Material acceleration by flux divergence formulation Vertical profiles of the nonlinear terms of (5) are shown in Figure 41. They are arranged s uch that the left central and right columns provide the divergence of means the divergence of the Reynolds fluxes and the divergence of their sums, respectively, and the rows from top to bottom provide the zo nal meridional and the vertical derivatives comprising these mean divergences Standard errors due to random variations (derived in Appendix) are indicated by dashed lines Rela tive to these, all of the estimated nonlinear terms are significantly different from zero at most depths (excepting (ku'v'>ldy) and the shapes and zero crossings of the vertical profiles are robust.
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107 ouutax olox O/Ox "' "' "' .. .. .. .. .. .. .. .. .. ] 100 100 100 .c 1lloy OIQy "' lO "' .. .. .. .. .. .. .. 10 .. ] 100 100 100 .c llloz oloz "' 10 "' .. .. .. .. .. .. .. .. ] 101 ... 100 .s 1l
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108 The mean zonal derivative oUU/ox is nearly zero above 40 m, positive from 40 m to 140m, and negative below 140m. The zerocrossing at 140m (about 30m below the EUC core) is consistent with the fact that the flow within the EUC core accelerates as the core shoals downstream (Figure 37) at this location This finding of a zerocrossing below the core differs from the Bryden and Brady ( 1985) and McPhaden and Taft ( 1988) findings of zerocrossings either at or above the core Zonal finite differences for these previous studies, however were over 30 of longitude a distance over which the vertical position of the core changed on a scale comparable to the measurement s vertical resolution. With higher zonal and vertical resolution oUU/ox is observed to be maximum and positive at the EUC core and negative below the EUC core The vertical profile of the mean meridional derivative oUV loy is negative with largest magnitudes at the surface and at the EUC core At the surface, this is due to the Ekman divergence of westward momentum away from equator and at the EUC core this is due to the geostrophic convergence of eastward momentum onto the equator. The minimum at 40 m marks the transition between Ekman divergence and geostrophic convergence dominance, and auvtay then approaches zero again below 220m as the geostrophic convergence goes to zero at the base of the EUC. The vertical profile of the mean vertical derivative oUW/oz varies in opposition to that of oUV/oy with relative maxima and minima of opposite sign occurring at the same depths In particular, within the EUC core the divergence of eastward momentum by the vertical circulation nearly cancels the convergence of eastward momentum by the meridional circulation Moreover, all of these mean derivative terms are of the same
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magnitude showing that the me a n divergence about the equator of the mean eastward momentum is fully three dimensional. 109 The mean divergences of the zonal momentum flux owing to the Reynolds fluxes (the three central panels in Figure 41) are generally sm aller than and tend to oppo s e, their corresponding mean circulation terms (the three left panels of Figure 41 ) Intuitively, the Reynolds fluxes act to smooth the mean gradient s An exception is the near surfa ce behavior of a/ax which adds to auu/ax between 30 m and 90 m a nd below 150 m Also (ku'u' >/ax is larger than auu/ax above 50 m Because the Reynolds flux divergences are relatively small, when added to the mean circulation momentum flux divergence term s, the total mean momentum flux divergence terms (the three right panels of Figure 41) retain their general shapes but with small changes in magnitude and extrema depths. With respect to the EUC, zonal momentum converges meridion a lly upon the core and divergence s vertically away from the core. Zonal momentum divergences zonally within and above the core where the EUC accelerates downstream and converges zonally below the core where the EUC decelerates downstream Adding the s e terms together gives the material rate of change of zonal momentum as shown in Figure 42 The divergence of the mean circulation zonal momentum flux, V Vu, is positive and maximum within the EUC core and negative near the surface and below the EUC core The divergence of the Reynolds momentum flux, V , is positive and maximum near the surf a ce, negative at the EUC core and relatively small but negative below the EUC core. Upon summation, the total divergence of the mean zonal momentum flux V < vu > is small near the surface, large and positive
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110 VVU lO .. 60 60 80 80 100 100 120 no 140 140 160 160 180 180 / ((( 43 0 l.O 1 0 0.0 1.0 1.0 J.O ..J.O 1.0 1.0 0.0 1 .0 1.0 3.0 .. 0 5.0 Acceleration em s ') Acceleration (1o ems') Acceleration em s ) Figure 42. Vertical profiles of the recordlength mean zonal momentum flux divergence by the mean circulation, the resolvable Reynolds stresses and their sum. The dashed lines represent standard errors by random fluctuations (Appendix).
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111 within the EUC core and equally large but negative below the EUC core. This development shows that the recordlength averaged divergence of the zonal momentum flux at 0 140W is fully threedimensional The contributions by the mean circulation are the largest, but the contributions by the Reynolds fluxes can not be ignored, particularly between the EUC core and the surface. The magnitude of the V profile of order 3xl05 em s2 is comparable to that of the ZPG Bias, due to the curvature in U is largest at the EUC core. However, this can be quantified (see Appendix) and corrected, giving flux divergence scheme results that are nearly identical to those from the advective scheme presented next. Mate rial acceleration by advectiv e formulation Using th e same format as Figure 41, Figure 43 shows the individual terms calculated by the advective scheme. Term by term the advective and flux divergence scheme results are different. The exception is the top row where the zonal derivative terms by the advective scheme are roughly half the magnitude of those by the flux divergence scheme as expected Zonally within and above the EUC core fluid accelerates downstream while below the core fluid decelerates downstream Vertically fluid decelerates as it diverges away from the core. However, it is not meaningful to compare the individual terms for these schemes, s ince it is only the summation of all terms that constitutes the material rate of change of zon a l momentum The sum of the three component terms are thus shown in Figure 44 Comp a red with the flux divergence scheme results the Reynolds stre s s contribution s are nearly identical. The mean circulation contribution, on the other hand, is smaller within the EUC core since that i s
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UiJU/dx l O 40 60 80 :g 100 ..: t20 a Q 140 t60 t80 lOO llO ViJU/iJy 20 !f It/ l 40 60 80 :: 100 ..: tlO a Q 140 160 180 lOO llO lO 40 60 .. too 12 0 t40 t60 tiO 200 220 lO 40 60 .. tOO tlO t40 t60 tiiO 200 220 <.u'au'tOx> I I ((( f 'I i 112 lO 40 .. .. too tlO t40 t60 110 200 220 lO 40 60 10 too tlO t40 t60 tiO 200 220 140 l40 l40 +.....+.,.....1 '.0 4.0 .J.O 0 1 0 0.0 1.0 1.0 3.0 4.0 l.O .J.O 0 0 1 0 1.0 .0 4.0 l.O 1.0 1.0 0.0 1 .0 1.0 l O wautaz 20 lO 40 40 60 60 10 10 :: 1 00 too .:; 110 c. Q 140 tlO t40 t60 t60 t 8 0 t&O lOG 200 llO llO .1.0 0 0.0 1.0 l.O 4.0 3..0 1.0 1 0 0 0 1 .0 1.0 J.O Acceleration em s ') Acceleration em s ') Acceleration (1o ems ') Figure 43. Vertical profiles of the individual constituents comprising the recordlength mean advection of zonal momentum estimated at 0 140 W. The d ashed lines represent standard errors by random fluctuations (Appendix).
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113 VVU ,. ,. lO .. .. .. .. .. .. ]: 1 00 100 100 .s llA) llA) 120 "" .. 1 40 140 ... Q 1 60 160 160 ... 180 ... lOO lOO lOO ,. ,. no l40 l40 ... 3.0 l.O 1.0 0.0 1.0 l.O J.O 4.0 5.0 0 l.O 1 0 1 0 l.O J O .... .J.O a 1.0 0.0 1.0 l.O u . 5.0 . Acceleration ( 10 em s"') Acceleration (10 em s ) Acce l eration (10 em s"') Figure 44. Vertical profiles of the recordlength mean advection of zonal momentum by the mean circulation the resolvable fluctu a tion s and their sum. The dashed lines represent standard errors by random fluctuation s ( Appendix ). 7 0
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114 where the velocity component curvature is largest. Elsewhere, the two forms are very similar. Upon correcting the flux divergence bias (Appendix) the discrepancy at the EUC core is accounted for, so consistent results are obtained from the two very different formulations. From this comparison between zonalmomentumflux diagnostic schemes it is clear that considering individual terms without the context of their total summation makes little sense, because of the threedimensionality in the material rate of change It is inappropriate to mix formulations between terms, because uninterpretable bias errors result and regardless of formulation the vertical circulation is critical. Zonal pressure gradient and wind stress The TOGATAO array temperature data on the equator at l25W 140W and 170W are used for estimating the ZPG. Salinity is held constant at 35 ppt for lack of data, and using constant salinity versus an historical TIS relationship does not result in significant error (e. g., Weisberg and Weingartner 1986). When referenced to 250m, the ZPG calculated for the three station pairs are shown in Figure 45 For the 125 W 170W and 140Wl70W pairs the mean ZPG force is eastward everywhere above 250m, while for the l25W140 W pair it reverses to very small westward values below 180m. The vertical profiles of the ZPG are consistent with previous measurements (e. g., Mangum and Hayes, 1984 ; Bryden and Brady, 1985; Weisberg and Weingartner, 1986; McPhaden and Taft, 1988). Following Large and Pond ( 1982) and in the same manner as McPhaden and Taft ( 1988), the zonal component of wind stress is calculated after transforming the buoy
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115 Pres s ure Gradient s o ,._ 1 0 0 a "' ..= C4 150 200 2 0 2 4 6 8 10 Figure 45. Vertical profile s of mean zona l pressure gradien t s e s timated between 170 W and 140 W (dotted) 140 W and l25 W (dashed) and 170 W and 1 25 W (solid) using a reference leve l of 250 m
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116 winds measured at 4 m to winds at the standard 10 m height (under neutrally stable conditions) using a drag coefficient of 1.2 and an air density of 1.2x103 g cm3 The resulting recordlength averaged zonal wind stress component is 0 .61 dyn cm2 Vertically integrated zonal momentum balance The large, vertically sheared currents on the equator suggest a balance between the ZPG and the vertical stress divergence (iJT!taz), with any imba lance resulting in a material acceleration To circumvent the problem of unknown a'tx'/(Jz, a vertically integrated analysis between the surface and any depth z is considered first. The diagnostic equation is: fo Jo J o laP J o dzf Vdz=< :1.. >dz+r;Av :L + V(AhVU)dz z z z pax U z z (7) where is the mean zonal wind stress The Coriolis term is zero on the equator and the horizontal stress divergence is assumed small relative to the vertical stress divergence. The integrated , the individual terms comprising it and the integrated ZPG are shown as functions of depth Figure 46 The integrated ZPG increases monotonically with depth, reaching the estimated value of within about 10 m of the EUC core (depending upon station pair). Below the core the integrated ZPG attains a nearly constant value. The zonal and meridional terms comprising tend to cancel with the vertical term, so upon integration the acceleration remains small until below the core where it reaches a relatively l arge negative value. Across the core, roughly between 80 m to 160 m the magnitude of the integrated is relatively constant at less than I 0% the magnitude of
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117 fz0(du/dt, .. )dz J z0 iJP/{Jx dz 0 0 I 'I I I \ I 50 \ 50 \ \ \ \ \ \ \ \ \ ...100 \ \ a \ \ 100 \ \ ...._, \ ': \ \ .::: \ \ .... Col 1\ QJ 150 J\ 150 li 1\ I i I I \ I I \ 200 I 200 I I I I I I l I : I i 250 250 40 30 20 10 0 10 20 30 20 40 60 80 100 (10.2 cm2 s 2 ) (102 dyn cm2 ) Figure 46. Mean vertical profiles for vertically integrated constituents of the zonal momentum balance at 0, 140W The left panel includes: J du/dt dz (thick z solid), jau/ot dz (thin solid), r uoulox dz (long dashed), r vouloy dz (short z z z dashed) and J wouloz dz (dotted). The right panel includes Jop/ox dz z z estimated between 170 W140W (dotted), 140W125 W (dashed) and 170 W125W (solid).
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118 Two important points follow. First when integrated vertically to the vicinity of the EUC core (located in the thermocline) the momentum balance is linear to within 10% This explains why linear, reduced g ravity models of the equatorial thermocline' s variability in response to perform well (e g., Busalacchi and O'Brien, 1981 and Weisberg and Tang, 1990) despite the pre se nc e of nonlinear currents. Second, the relatively small constant magnitude of the integrated near the EUC core i s consistent with the fact that <'t"> =A vautaz=O at the core. This follows from (5), showing that <'tx> equa l s the integrated where the integrated ZPG equals Since this occurs within 10 m of the core, or to within l 0 % we have an independent physical consis t ency check on the estimation of Along with quantifiable random errors (Appendix), it may now be argued that the finite differencing errors are not substantial either. The implied vertical stress distributi o n A ss uming a balance between the ZPG, and the latter term may be e st imated as a residual as shown in Figur e 47 Near the surface the balance is primarily between the eastward directed ZPG and the westward directed < a't"taz >. This i s modified by shaping the n ear surface stress divergenc e. Thus, at 50 m, where has a westward directed maximum becau se fluid particles decelerate as they move upward, there must be a corresponding maximum in vertica l friction to account for thi s deceleration. Over the upper 20m (the mean westward flowing SEC region), where the Reynolds stresses are po s itive (in particular < v'autay>) and s ufficiently l arge to accelerate the flow making it less westward downstream, there must be a corresponding
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, and  100 ISO 200 200 8 6 4 2 0 2 4 6 8 80 60 40 20 0 20 40 (10s dyn cm.3 ) (10.2 dyn cm.2 ) 119 Eddy coefficient A,. 0 so 100 ISO 200 2SO +rrr1 0 20406080 ( cm2 s) Figure 47. Mean vertical profiles at 0, 140 W of left panel: the three terms comprising the zonal momentum balance where is obtained as the residual of the ZPG and ; center panel: <'tx> and right panel: A v The dotted, dashed and so lid lines represent re su lts using the ZPG estimated between 170W140W, 140W125W, and 170 W125W, respectively
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120 decrease in over that at 50 m A similar near surface profile was reported by Wacongne (1989) in a numerical circulation model analysis applied to the equatorial Atlantic Ocean and such may be inferred from the Bryden and Brady ( 1985) results This behavior is now understood in terms of the processes that give rise to the observed . As the ZPG deceases and increases with depth toward the EUC core these two terms tend to offset resulting in a minimum westward directed at the core. Thus, the balance within the EUC core, where the flow accelerates downstream, is weakly nonlinear with the sum of similar magnitude and balancing the ZPG Below the EUC core, where the ZPG approaches zero, the balance becomes one between and . It is within this regime that the EUC decelerates downstream requiring a frictional retarding force in the absence of a sufficiently large westward directed ZPG force To summarize, the dynamic a l flow regimes change in going from the s urface through the base of the EUC. The zonal momentum balance goes from one in which the pressure gradient force balances the windinduced frictional force near the s urface (essentially a linear regime modified by nonlinearity), to a weakly nonlinear regime at the EUC core where the pressure gradient force drives a nonlinear acceleration equal in magnitude to the frictional retarding force, to a fully nonlinear regime below the core in which a relative maximum in the frictional retarding force decelerates the EUC downstream Integrating the residual de termined gives an estimate of the profile as shown in Figure 47 Consistent with the summary above is large, negative and
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121 monotonically increasing from the surface to the EUC core. A zerocrossing occurs within 10m of the core below which <f> is positive (as higher momentum fluid above is rubbing against lower momentum fluid below) and monotonically incr easi ng. This physically required change in sign at the core occurs independent of the estimation, providing some confidence that the errors, both random and finite difference, are not controlling. Dividing <f> by ()Uf()z provides an estimate of A v (except near the EUC core where is divided by d2U /()z2 since the shear is zero) as shown in Figure 47. This results in values of 4050 cm2 s1 in the near surface mixed layer decrea si ng to values around 3 cm2 s 1 within the EUC core and then increasing again below the core. Over the range 150m200m, where and are maximum, A v is estimated between 1020 cm2 s1 Temporal evolution of the zo nal momentum balance Since the relative magnitudes and spatial structures of the SEC and EUC are functions of time, the relative importance of the various terms comprising the zonal mome ntum balance may a lso be time dependent. Of particular interest are the time sca le s over which nonlinearity is important. Figure 48 s hows the terms comprising Du/Dt as a function of time and depth computed by advective formulation using raw hourly data. For presentation, these time ser ie s along with others in this sectio n (except the ZPG ) have been low pass filtered to exclude fluctuation s at time sca les shorter than 10 days. The same calculation u s ing low pass filtered time series gave n early identical result s s howing that the essential non linearitie s occur over time sca l es longer than 10 days.
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122 200 M J 1990 so Stoo l:l. Q 150 200 1991 Figure 48. The nonlinear acceleration (v V'u), the local acceleration (au/at) and individual constituents comprising the nonlinear acceleration (uau/ax vau/ay, and wauJaz) as functions of depth and time. Positive values are stippled, the contour interval is 5 xl05 em s2 and all time s erie s have been lowpass fi lt ered to excludes fluctuations on time scales shorter than I 0 days.
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123 ,.., ... ] ,.., 100 "' 0 1"""4 .c '" .... c. Q 150 = 1991 ,.., ... ] ,.., 8 "' 0 '" 1"""4 .c '" .... N c. Q 240 M 1990 1991 Fi g ure 48 (continued)
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124 The nonlinear acce leration V Vu is vertically inhomogenous and time dependent. It tends to be positive with large st variability within the EUC core (wh i ch migrated between 80 m150 m) and negative above and below the core Compared to v Vu, autat has largest magnitude near the surface and relatively uniform sign with depth. A notable event is the December 1990 eastward momentum pulse. While the local acceleration is welldefined and large, the nonlinear acceleration is as large at the EUC core. Such pul ses are generally identified as intraseasonal, linear, equatorial Kelvin waves (e. g., Knox and Halp ern, 1982; McPhaden and Taft, 1988) with phase lines traceable across the equatorial Pacific (Kessler and McPhaden, 1995) While linearity reasonably describes the vertical integral (Figure 51), v Vu may be as large as autat at individual depths. Of the terms comprising v Vu, wautaz has the l argest fluctuation s that tend to reverse sign across the EUC core, uautax tends to be large st in the core and vautay tends to be large st near the surface. These are all comparable in magnitude with autat. As with the record length average, the time dependence of V Vu is also fully threedimensional. The time dependence of Du/Dt sho uld reflect changes in external forcing Figure 49 s how s the ZPG referenced to 250m depth est imated between 170 W140W 140W125 0 W and 170W 125W as functions of time and depth. Th e l a rge data gaps and inadequ ate zonal resolution preclude analyses as performed for the recordlength mean, but some qualitative comparisons can be made. The two relative maxima and intervening minimum in the ZPG from July through October 1990 correspond to the relative maxima and intervening minimum in the EUC core speed s hown in Figure 35 and the ZPG maximum in December 1990 corresponds to the Kelvin wave pulse at that time. Thus,
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125 50 200 s 0 N D J F M A M J 1991 0 50 ..... 0 100 lr) N ...1 .c I ..... Q,) 150 0 Q ...1 200 F M A M J 50 200 M J J A S 0 N D J F M A M J 1990 1991 Figure 49 The ZPG estimated between 170W 140 W, 140W125 W and 170 W125 0 W as functions of depth and time. The small region of westward directed pre ss ure gradient force is stippled a nd the contour interval i s I xl05 dyn cm3
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126 the zonal momentum within the EUC responds to changes in the ZPG a s expected s ince an acceleration must occur until d'tYaz adjusts to strike a new balance However, with a complicated superposition of local (wind stresssee Figure 52) and remote (pre s sure gradient) forcing, the timedependent adjustments of d'txf()z can not be inferred from the present data set, primarily due to inadequate ZPG resolution. Over what frequency range does nonlinearity become important? To addres s this question, variance density spectra of 'du!'dt and v Vu are shown as a function of frequency and depth in Figure 50. At frequencies higher than those shown, the 'du!'dt spectral densities are a le as t an order of magnitude larger than those of v Vu The 'du!'dt (v V'u) spectra decrea s e (increase) with decreasing frequency Comparable magnitudes begin to occur in the EUC for frequencies between 0 .01 cph and the instability wave frequencies of about 0 002 cph. At lower frequencies the v Vu spectra exceed tho s e of 'dul'dt, especially in the EUC core. The EUC thus becomes increasingly nonlinear approaching the mean As with the means, the time dependent v Vu tend to cancel upon vertical integration This is shown in Figure 51, comparing the integrated Du/Dt, 'du!'dt and v V'u. The integrated v Vu is small above the EUC core and then increases in magnitude to maximum values just below the core (at about 150m) before decreasing again. Figure 52 provides a qualitative comparison between the vertically integrated (surface to 250m) Du/Dt, 'dul'dt, ZPG and The vertically integrated Du/Dt and 'dul'dt are similar and there is considerable correspondence between these accelerations and the fluctuation s. Limited correspondence with the ZPG is also evident ; but, s ince the TOGATAO array
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127 0, 140oW Period (hrs) 200 100 500 c bii j 200 0.001 0.005 0.01 0.015 0.02 c bii j 200 0 .001 0.005 0.01 0.015 0.02 Frequency (cp h) Figure 50. Log variance densities for c1u/c1t and v V'u as functions of depth and frequency The contour interval is 0 5 Light stippling highlight s densitie s betwe en 103 5 104 (em s2 ) 2 cph 1 a nd dark s tippling highlights den sit ie s greater than 104 (e m s2 ) 2 cph_1 The spec tra were averaged over a Q_92x 1 o3 cph bandwidth for approximately 18 degrees of f reed o m
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2 0 15 N "'.r' 1.0 c.= 0 5 0 0 0 5 3 0 2 0 N "0,.., 1.0 .... Ct)Cil ;g .. 8 0.0 1.0 2.0 3 0 2 0 N ., 1 0 Q !3 0 0 1.0 2 0 H 0 0 ltil0 5 .. 8 .... (,1 00 = 1.0 ="0 '' 1.5 M J J A s 0 N D J F M A M J 1 990 1991 Fi g ure 52 Time series of the vertically integrated ( 0250 m) Z P G, loca l acce l eratio n total acceleration an d surface zona l wind st r ess 129
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130 doesn't resolve synoptic fluctuations in the ZPG as observed in the wind and acceleration, the time varying zonal momentum balance at these scales can not be diagnosed. Summary and discussion Current velocity data from the TIWE equatorial array have been used to diagnose the upper ocean zonal momentum balance at 0 140W. The array resolves the large scale divergence of the upper ocean currents, thereby permitting an estimation of the zonal momentum flux divergence, the unknown factor in previou s studies Analyses are made of the recordlength (13 months) means and fluctuations In either case the flow is threedimensional, tending to converge meridionally upon, and diverge vertically away from the EUC core, consistent with the first comprehensive descriptions of the equatorial circulation by Knauss ( 1960, 1966) and the first theory for the EUC by Fofonoff and Montgomery (1955). The ensuing vertical circulation is found to be a critical element of the zonal momentum balance The recordlength average, dynamical flow regime changes between the surface and the base of the EUC. At the surface it is essentially linear with the ZPG balancing the windinduced frictional force modified by non linearity At the EUC core it is weakly nonlinear, with the ZPG driving a non linear acceleration equal in magnitude to the frictional retarding force. Below the core (where the ZPG is nominally small or s lightly westward) it is fully nonlinear, with the frictional retarding force accounting for the downstream deceleration of the EUC, consi s tent with the viscoinertial theory of Charney and Spiegel ( 1971 ).
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131 As with the flow field, the nonlinearities are threedimensional, requiring all three components to characterize the zonal momentum flux divergence The mean zonal momentum flux divergence also contains important Reynolds stress divergences found to occur primarily over the tropical instability wave scales. They effectively act to decelerate downstream both the surface SEC and the subsurface EUC, with the horizontal (vertical) Reynolds flux working against the SEC (EUC). The profile is largely affected by the mean zonal momentum flux divergence The mean vertical advection of eastward momentum combines with the divergence of the horizontal Reynolds fluxes to produce a maximum at 50 m depth. Without these nonlinear affects kd'tx/dz>l would be a maximum at the surface as recently found at midlatitude by Chereskin (1995). At the EUC core kd'tx/dz>l is a minimum owing to the downstream acceleration of the EUC, and it would even be smaller were it not for the resolvable vertical Reynolds flux divergence that reduces the magnitude of this acceleration. Below the EUC core kd'tx/()z>l increases again to a relative maximum necessary to account for the downstream deceleration of the EUC there The implied vertical profile of A v has near surface values of 4050 cm2 s1 decreasing to around 3 cm2 s 1 at the EUC core and increasing again to I 020 cm2 s 1 within the deceleration region below the core. Qualitatively, the minimum at the core is consistent with the thermocline inhibiting vertical mixing Quantitatively, the near surface results for Av agree reasonably well with results from numerous microstructure investigations (e. g Gregg, 1987 ; Peters etal. 1988; Dillon etal., 1989 ; Hebert, etal. 1991; Lien et al., 1995). However, they differ greatly at and below the EUC core where
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132 microstructure measurements suggest values at least an order of magnitude smaller. With the observed downstream deceleration of the EUC below the core demanding a sufficiently large westward directed force, three possible explanations are offered: 1) there exists on average a large (>2x105 dyn em\ westward directed ZPG force below the EUC core, 2) microstructure measurements within the EUC are incorrect or 3) microstructure measurements don't resolve the scales of motion that produce <'dT'/dz> across the EUC. Lacking observational evidence, the first seems unlikely, and this would also be inconsistent with the observed meridional convergence of mass which only goes to zero at the base of the EUC. There is no basis herein for discussion of the second, but the third may have merit. Friction ultimately occurs on molecular scale. Eddy friction is just a parameterization for unresolved scales. When resolved, these scales result in quantifiable Reynolds fluxes which for the present measurements occur on synoptic scales. So, it is possible that the gap between the synoptic and the microstructure scales may contain other physical processes causing eddy momentum flux divergence. Regardless of such conjecture, something is necessary to decelerate the flow and to account for the mixing of heat (the thermostad) and all other material properties below the EUC core, and it is noted that this data set does give consistent results for the eddy mixing of heat below the EUC core (Weisberg and Qiao, 1996) The analyses presented for the time dependent zonal momentum balance were more qualitative. Ironically, the ZPG, (the best determined of the terms in prior studies) is the limiting factor for the time dependent analysis here. Designed for large scale monitoring, the TOGATAO array does not resolve the spatial scale of the large amplitude synoptic variability. Consequently, there is a large mismatch in the
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133 fluctuations observed in either 'fa or Du/Dt with the fluctuations in the ZPG. TIWE lacked the resources to augment the TOGATAO array with closely spaced moorings for temperature and salinity. This was a mistake that future process experiments should avoid. Despite this shortcoming, the large and Du/Dt fluctuations showed considerable correspondence. Current nonlinearity is largest within the EUC core region, but upon vertical integration the nonlinear terms tend to cancel. Thus, for intraseasonal Kelvin waves, as an example, the local acceleration exceeds the nonlinear affects, especially upon vertica l integration. But, nonlinearity increases with decreasing frequency, becoming very important in the mean. This agrees with the observation that the mean meridional scale of the EUC is much less than an equatorial Rossby radius of deformation (the standard deviation of a meridional Gaussian distribution). For example, from Figure 37 an equivalent meridional scale of about 100 km is calculated, which is smaller by a factor of 23 than an equatorial Rossby radius of deformation. Thus, while the intraseasonal fluctuations may be described as nearly linear, the mean EUC is nonlinear. A similar conclusion was found for the equatorial Atlantic by Tang and Weisberg ( 1993) In summary, the TIWE equatorial array descriptions of the threedimensional circulation and zonal momentum flux divergence are consistent with the earliest descriptions and theoretical ideas of the EUC. Estimates of the vertica l stress divergence show dynamical flow regimes that change between the surface and the base of the EUC. The vertical stress divergence is much larger over the lower portion of the EUC than previously reported. Nonlinearity becomes increasingly important with decreasing frequency and in the mean the EUC circulation is nonlinear. Nonlinearities tend to cancel upon vertical integration consistent with reduced gravity models being able to
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134 account for equatorial thermocline variations Future process experiments on the equatorial circulation and heat balance will require higher resolution than provided by the present TOGATAO array.
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SUMMARY AND CONCLUSIONS High vertical resolution profiles of horizontal velocity vectors obtained from an equatorial Pacific array of subsurface moored ADCPs were used to examine the dynamics of current observed during 1990. The distribution of variance showed a welldefined wave season lasting from August to December, with wave variance confined primarily to the near surface region above the EUC core The on set of the wave season coincided with the acceleration of the SEC and the termination coincided with a strong eastward momentum pulse propagating from the west as a Kelvin wave The instability w a ve velocity fluctuations may be described by highly eccentric ellipses, oriented to the north, but tilting toward the east into the cyclonic shear of the SEC. Over the observational domain (1 o s to 1 N and 142 W to 138W) these tilts increased with latitude from essentially zero at 1 o s to maximum values at 1 N and decreased with depth from maximum values at the uppermost 30m measurement. By the EUC core at 110m the wave variances, hodograph tilts and ellipse stability's were all nil. The instability wave variance was contained within a narrow frequency band centered upon 500 hr periodicity. Averaged over this bandwidth the zonal wavenumber component was uniform with depth between 30 m and 110 m and directed westward with a magnitude and 90% confidence interval of 5 9 x103 1.1x103 rad km1 (or a wavelength of 1060 km and a 90 % confidence interval between 900 to 1300 km). The 135
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corresponding westward directed phase speed was 59 em s 1 Unlike the zonal component neither the meridional nor the vertical wavenumber components were statistically different from zero The calculation of the wave energetics supports the conclusion from hodograph tilts that the barotropic instability arising from the near surface cyclonic shear region north of the equator accounts for the generation of the waves Additional energy contribution by the barotropic instability arising from the meridional gradient of the mean meridional velocity component just north of the equator is also necessary for the waves to be maintained in the later half of the wave season Near the equator, the waves tend to lose their kinetic energy to the mean flow at the surface through meridional gradient of mean meridional velocity component and vertical shear of the mean zonal velocity component. The mean divergence of perturbation kinetic energy by the meridional velocity component fluctuation and the conversion of kinetic to potential energy by the buoyancy force al s o tend to weaken the waves The TIWE array resolves the large scale divergence of the upper ocean currents, thereby permitting an estimation of the zonal momentum flux divergence, the unknown factor in previous studies Analyses are made of the recordlength (13 months) means and fluctuations In either case the flow is threedimensional tending to converge meridionally upon, and diverge vertically away from the EUC core, consistent with the earliest descriptions and theoretical ideas of the EUC The ensuing vertical circulation is found to be a critical element of the zonal momentum balance Estimates of the mean vertical stre s s divergence show dynamical flow regimes that change between the surface 136
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and the base of the EUC. The vertical stress divergence is much larger over the lower portion of the EUC than previously reported. Nonlinearity becomes increasingly important with decreasing frequency and in the mean the EUC circulation is nonlinear. Nonlinearities tend to cancel upon vertical integration consistent with reduced gravity models being able to account for equatorial thermocline variations While the TIWE array provided adequate resolution for velocity, analyses were hampered by inadequate resolution for temperature and pressure gradient afforded by the TOGATAO array. Future process experiment will require improved spatial resolution for temperature, density and pressure relatied variable. 137
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145 APPENDICES
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APPENDIX 1. ERROR ANALYSIS FOR PHASE Consider the linear relationship between the input and output time series x(t) and y(t) with spectral estimates Gx(f) and Gy(f), respectively, and transfer function H (f) The phase of the transfer function is : where <1> = tan 1 lm{H(f)} Re{H(f) } A"' I r(f) Ll"'=sm , , H(f) 2 2 2 G Y (f) r (f)= n2 F2.n2;a (1Yxy ) G X (f) ( 10) (II) n are the number of degrees of freedom for the spectral estimates F2 n 2 ; a' is the 1 OOa percentage point for the F distribution with 2 and n2 degrees of freedom and Yx/(f) is the estimated coherence s quared between x(t) and y(t). Using the first EOF mode as input and the data at a given location as output, the relative phases and the confidence intervals between the mode and the data are e s timated using (1 0) and (11 ). Figure 53a, b s how s the resulting 90% confidence intervals for the phase determinations in the zonal and meridional planes of Figures 12c, d, respectively 146
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APPENDIX 1. (continued) 50 a. 50 b. 0. 2 100 100 !50 200 __ 142 W 140 W 138 W I o s Longitude Eq Latitude Figure 53 The 90% confidence intervals for random errors on phase a. as a function of longitude and depth along the equator and b. as a function of latitude and depth along 140 W for comparison with the phase estimations of Figure 12c, d. The contour interval is 0.1 rad. 147
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148 APPENDIX 2 ERROR ANALYSIS FOR WAVENUMBER The horizontal components of the wavenumber vector are estimated by fitting planes independently at each depth between 30 m and 250 m to the five phase estimates using linear least squares regression The basic assumption of the linear regression is that wavenumber vector is constant over the array Deviations from this assumption leads to imperfect fit and consequently error The following outlines the error analyses. Denote the wavenumber vector as (k, l), the EOF mode phases at each of the five locations as i (i= 1 ... ,5), and the phase function to be fit by linear regression as i computed by the EOF analysis has n degrees of freedom (29 in this case). Statistically, this is equivalent to expressing i as the average over n measurements i.j. U= 1 ... ,n). !i (i=1, ... ,5) n The linear least squares regression minimizes the sum of the squares of the 5 deviations between the phases i and the phase function i
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149 APPENDIX 2. (Continued) 1 5 1(5 J2 a22 = 5 25 =I =I and (xi, yi) is the ith station location The total number of degrees of freedom include s the degrees of freedom for each of th e original phase e s timate s combined with the additional number of degrees of freedom associated with the plane fits to these estimates. At each depth there are 5 phase estimates; however, the plane fit has three constraints: k, 1 and cpo. Therefore the linear regression u se d to estimate the wavenumber vector increase s the number of degrees of freedom by a factor of 2, rather than 5 The number of degrees of freedom for the (k, 1) determination i s thus (53)n, or 58. This is sufficiently larg e so that the probability di s tribution function fork and 1 may be approximated by a Student' st distribution regardless of their true distribution Given the Student'st distributi o n the errors for (k, 1) may be expressed as: (ell J = t(la,58) SD C22 (13)
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150 APPENDIX 2 (Continued) Where t( 1a,58) is the Student'st distribution at the ( 1a)% confidence level with 58 degrees of freedom and SD is the standard deviation of the phase q> t o be determined By definition, SD follows from : 1 5 n = [L (4>iq>i)2 + (4>ir 4>i)2 +2(4>ir 4>i)(4>ii)] i=l j=l (14 ) where it has b ee n assumed that 4>ii and $ij$i, the errors from lin ear regression an d EOF analyses, re s pectively, are unc o rrelated. Thi s i s a re asonab le assumption s ince these two calculation are performed independently The first p ar t of error in ( 14 ) the deviation s from the plane fit, is s traight forward. The s econd part of the error require s an evaluation of equation ( 11 ), which is the variance of the phase 4>i obta ined by linear sys tem s analysis. By definition: (15)
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APPENDIX 2 (Continued) where is the unknown probability density function for While is unknown an upper bound may be estimated by noting that for the interv al i::; ::;i, 2 Using this in equation (15) along with equation (11) results in: 11t12 ::; (1t12 (!!:.2) 2 sin2 M = 1t42 E{rl r(f) lj2 } 1t12 J ro2 H (f) Thus, we can useE{ [r(f)/H(f)] 2 } at each location as an upper bound on the variance of
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APPENDIX 3. MEAN ZONAL MOMENTUM FLUX DIVERGENCE RANDOM ERRORS It is assumed that the measured velocity components are stationary, Gaussian di s tributed random variables satisfying the ergodic hypothesis If their variations result from the superposition of several different physical processes and the record is long enough (has a sufficient number of degrees of freedom) to sample these, then the Gaussian assumption is supported by the central limit theorem. The wcomponent is estimated by linear operation upon the measured uand vcomponents, so its distribution is also Gausssian. The zonal momentum flux divergence (Vvu) and its constituents cauu/ax, auv/ay and auw/az) are estimated by nonlinear operation, so their distributions can not be specified a priori However, after forming time series of the velocity component products, their distributions follow the same argument given for the individual velocity components. The standard error for a stationary, Gaussian random variable x is d efi ned as the positive square root of its variance. Let the recordlength averaged estimate of the mean value of x be denoted by X and let the true mean value of x determined by the expectation operator E { x} be The variance of X is thus, which for large enough T (such that the autocovariance of x tends to zero) reduce s to (e g ., Bendat ad Piersol, 1972) : lJT Var{X}:::T Cx('t)dc T 152
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APPENDIX 3. (Continued) Upon drawing the equivalence between this variance of the mean estimate and the variance of the mean estimate for band limited noise this becomes, 'to Var{X}=:fCxCO) JT C x ('t)d't where 't0 TCx(O) is the integral time scale and N=T/'t0 is the equivalent number of degrees of freedom (e. g. Tennekes and Lumley, 1972 or Davis, 1977 ) The significance of the integral time scale is that statistically meaningful estimates of X may be obtained if T is a s ufficiently large multiple of 't0 and for T=N't0 (N> I) the time series may be considered as having N equivalent independent samples. Herein, 't0 was estimated using the smoothed spectrum at zero frequency [since S(O)= JT C x ('t)d't]. It is noted that T sufficiently large N supports the Gaussian assumption via the central limit theorem. In the present data set only u near the EUC core had N as small as 15. For all other variables, linear or nonlinear, N was generally larger than 30. The analysis of the momentum flux divergence i s split into two parts: V=V VU+V (17) The total and the Reynolds flux terms were analyzed using the above time series approach. With N larger than 30 for all of the non linear term s and their s um s the Gaussian assumption appears to be satisfactory. Estimating the s tandard error for the first term on the right hand side of ( 17) is more complex. In finite difference form this term may be written as: 153
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154 APPENDIX 3. (Continued) (18) Since this term and its constituents are not time series their random errors must be calculated from the properties of (assumed Gaussian) random variables. For the set of Gaussian random variables Xi (i= 1 . n) let the mean, the variance and the covariance be denoted by: E{xi}=Jl i Var{xi}=
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APPENDIX 3. (Co ntinued ) The s tandard error estimates for ilUU/ilx ilUV/ily and ilUW/11z then follow by replacing the variables Xi in (19) with the appropriate U, V and Wand then multiplyin g by a factors l/ilx2,1/ily2 or l/11z2. The covariance values Cij used in the estimations are shown in T ab l e 3 These were obtained from estimates of coherence at zero frequency ( by ave raging over the lowest frequency portion of the spectrum) with the understandin g that the s up e rp ositio n of variations h av ing mixed s ymmetry properties about the equator will decrea se covariances between component pair s at different locations Thus, high covariance between U pair s and W pairs are obtained compared with lower coherence between off equator V pair s. With no consistent se t of values to choose for lower coherence pairs the se were set to zero in the sta ndard error estimation. Since the sma ll, omitted terms have different sig ns in (19) they tend to cancel so the net effect of this omission i s relatively s mall. With thi s technique the s tandard errors for each of the individual terms on the right hand side of ( 18) were estimated. The s tandard error of the sum was not. A s a divergence the errors should tend to cancel like the terms compri s ing the sum, but there is no way of objectively determining this Simply adding the variances of each term together would provide a m eani ngle ss overestimate of the standard error for the divergence A similar development u s ing the expected values of pr oducts of the Xi was employed for the s tandard error analysis of the advective sc hem e terms. Since Var{x1x 2 } 155
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156 APPENDIX 3 (Continued) The standard error estimates for V and W then follow by replacing the variables Xi with the appropriate U, V, W and Note that the variances of the zonal meridional and vertical gradient of the ucomponent are also directly calculated from the time series. Table 3. Covaria n ce of mean ve l ocity pairs used in the standard error estimates ui vi wi u J y. J W J ui 0 0 0.8 0 0 vi 0 1 0 0 wi 0 0 0.8
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157 APPENDIX 4. ADVECTIVE AND FLUX DIVERGENCE SCHEME COMPARISON The central finite differences for the individual terms in advective and flux divergence schemes, respectively, are: and u3 + uJ u3uJ u 3 + uJ uJuJ _..::....._____:_ + =:.... 2 x 3x1 2 x 3x1 auu u 3 u 3 uJuJ ax X3XI v s + v 2 Usu 2 Us+ u2 v s v 2 _::....__.::.. + ___;;;...._.;:;.. _.:;....____.::... 2 YsY2 2 YsY2 auv U sUsu2u2 = dy Ys y2 _uuul uu+ul wuwl =w l1z + 2 l1z auw uuwuulwl dz l1z where the subscripts are either station numbers or upper and the lower layers, respectively; w is the average w between upper and lower layers and fl.z is the 10 m layer spacing. In differential form, these two schemes are equivalent via continuity. Is this true of the finite difference forms, or is one biased relative to the other? Note that with small acceptable error the central values u4 and v 4 may be approximated by averages of their horizontally adjacent values: This approximation allows the flux divergence form to be rewritten as:
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158 APPENDIX 4. (Continued) Thus, an equivalency is achieved between the flux divergence and the advective schemes if the term in parentheses on the right hand side is zero This occurs by conti nuity if U 3+U1 Uu+UJ Us+U2 2 2 and 2 are equal. The mean velocity component profiles of Figure 37 U3+U1 d Uu+UJ U s+U2 show that 2 an 2 are both approximately equal to u 4 ; however, 2 is Us+U2 generally less than U4. This discrepancy is largest with the EUC core where 2 is about uJ2. Thus, near the core U 3+Ul U 3Ul U s+U2 VsV2 Uu+Ul WuWI =:..,._...:... + + ='='2 x3x1 2 y5y2 2 /).z U 3UI V sV2 W uWI U4 VsV2 = u4 ( + + ) XJxl YsY2 /),:z 2 YsY 2 u4vsv 2 2 Y s y2 where the continuity equation has been applied Therefore, the flux divergence scheme is u v v biased relative to the advective scheme, with the bias 4 5 2 being due to the 2 YsY 2 Us+U2 meridional curvature of the ucomponent causing 2 to be different from u 4 Physically, the flux divergence formulation fails to conserve mass, but this is a quantifiable correctable error. For the TIWE equatoria l array data such bias applies primarily to the mean circulation. For the Reynolds fluxes, in contrast to the mean, the results from the two different schemes are nearly identical, since the meridional curvature u's +u'2 for the fluctuations is small relative to the mean and 2 = u'4.
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Personal Data Surname: First name: Birthday : Sex: Address Qiao Lin April 10, 1963 Male VITA 140 7th Ave. S., St. Petersburg, FL 33701 Education Department of Marine Science, University of South Florida, St. Petersburg, FL 33701 Time: September 1989 to May 1996 Major : Physical Oceanography Degree: Doctor of Philosophy, May 1996 Mathematics and Mechanics Department, Zhongshan University, Guangzhou, China Time: September 1983 to June 1986 Major: Hydrodynamics Degree: Master of Engineering, July 1986 Mechanics Department, Zhongshan University, Guangzhou, China Time: September 1979 to July 1983 Major : Fluid Mechanics Degree: Bachelor of Science, July 1983 Honors Second Prize for Excellent Graduate Student of Zhongshan University in 1985. Employment Institute of Water Resource and Hydroelectric Power, Hohai University, China, 19861989

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