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CAN THE TOPEX/POSEIDON ALTIMETER MAKE USEFUL OBSERVATIONS OF EQUATORIAL INERTIA GRAVITY WAVES ? by SHERRYL A GILBERT A t hesis submitted in partial fulfillment of the requirements for the degree of Master of Science College of Marine Science University of South F lorida May 2001 Major Professor: Gary T Mitchum Ph.D
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Examining Committee: Office of Graduate Studies University of South Florida Tampa Florida CERTIFICATE OF APPROVAL This is to certify that the thesis of SHERR YL A GILBERT in the graduate degree program of Marine Science was approved on December 13, 2000 for the Master of Science degree Major Professor/ Gru) \ T. MVchum Ph D. Member: Mark E. Luthei, Ph.D Member: Robert }f.:WeiJberg Ph D
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TABLE OF CONTENTS List of Tables List of Figures Abstract Introduction Theory of Equatorial Waves Previous Research Remaining Question and Objectives TIP Altimetry and Specific Objectives for This Study A Revisit of the In Situ Observations Data Collection and Pre-Processing Tide Gauge Results Velocity Results Application of Altimetry Sampling Issues Choosing the Zonal / Temporal Box Size Significance Testing In Situ Comparisons lll IV Vll 3 24 40 43 45 46 48 57 60 62 69 70 74
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Preliminary Results 81 Summary 89 Tables and Figures 92 References 142 11
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Table 1 Table 2 LIST OF TABLES In situ locations of hourly data Summary of results from TIP and in situ Ill 92 94
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LIST OF FIGURES Figure 1 Dispersion curves for equatorial waves. 95 Figure 2 TIP coverage map. 96 Figure 3 Map of in situ data locations 97 Figure 4 Response of the filter applied to the hourly data 98 Figure 5 Autospectra of tide gauge sea levels 99 Figure 6 Example of mode 2 amplitude time series from comp l ex demodulation at Kanton Island 103 Figure 7 Autos p ectra of amplitude time series from complex demodulation at the tide gauges. 104 Figure 8 Autospectra in monthly bins 108 Figure 9 Autospectra ofT AO velocity data at 4 locations. 1 1 1 Figure 10 Autospectra of COARE velocity data at 3 locations 112 Figure 11 Estimated mode 2 squared amplitude versus time and depth at the COARE sites 113 Figure 12 Method used to select wavenumber range to search in fit. 114 Figure 13 TIP zonal and temporal sampling. 115 Figure 14 Schematics showing aliasing problems. 116 Figure 15 Wavenumber, frequency pairs where the first type of sampling problem occurs. 117 lV
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Figure 16 Check of the simplified model using the full fit. 118 Figure 17 Inertia-gravity wave responses to low frequency aliases. 119 Figure 18 Response of the filter used to eliminate low frequency signals. 120 Figure 19 Fit response for varying zonal box width. 121 Figure 20 Illustration of why does phase matters. 122 Figure 21 Effect of temporal box size. 123 Figure 22 Example of off equatorial data used in significance testing. 124 Figure 23 Choosing which meridional offsets to use in significance testing. 125 Figure 24 Comparison of A/( o0. 09)2 and x / distributions. 126 Figure 25 Dependence of ex on the parameters of the fit. 127 Figure 26 Distribution of ex-ex fit and scatterplot of ex and ex fi 128 Figure 27 Example of PE factor calculation at Tarawa. 129 Figure 28 Tide gauge and TIP intercomparisons. 130 Figure 29 Velocity intercomparisons. 134 Figure 30 Global picture of mode 2 PE variations from TIP. 135 Figure 31 Global picture of mode 2 PE with significance testing 136 Figure 32 Global seasonal variations of mode 2 PE. 137 Figure 33 Comparison of TIP seasonal structure with analogous tide gauge series 138 Figure 34 Zonal modulation of mode 2 PE from TIP. 139 v
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Figure 35 Zonal PE structure in Pacific from TIP Figure 36 Wavenumber distribution in the Pacific VI 140 141
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CAN THE TOPEXIPOSEIDON ALTIMETER MAKE USEFUL OBSERVATIONS OF EQUATORIAL INERTIA-ORA VITY WAVES? by SHERRYL A GILBERT An Abstract of a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science College of Marine Science University of South Florida May 2001 Major Professor : Gary T. Mitchum Ph. D VII
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Inertia-gravity waves, which make up the high frequency portion of the equatorial wave spectrum, have been observed in all ocean basins using tide gauges, current meters, and inverted echo sounders However, the large-scale zonal and temporal modulations of these waves have been difficult to quantify The ability of the TOPEX/Poseidon altimeter to measure these waves has been evaluated Due to the regularity of the sampling and the associated aliasing problems, only meridional mode 2 can be estimated. For mode 2, however, we find good agreement between results obtained from in situ locations and from the altimeter In the Pacific Ocean we find a clear but intermittent, temporal structure with higher energy levels late in the calendar year From significance testing, we conclude that energy levels in the Atlantic and Indian basins are below the altimeter's detection limit. Pacific inertia-gravity wave energy shows an interesting zonal structure with little energy at the western boundary, an increase of energy at a region of low energy at 160W, another increase at 1300W, and low energy at the eastern boundary. These sub-regions also appear to have different dominant wavenumbers as well, and a corresponding zonal variation in the direction of energy propagation Abstract Approved: Maj or T Mitchum, Ph D Associate Professor College of Marine Science Date Approved: /.J/3-cc viii
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INTRODUCTION One effect of the Earth's rotation on ocean physics is that the equator can serve as a waveguide, trapping energy at the equator. Inertia-gravity waves are one type of wave that can exist in this equatorial waveguide. Matsuno ( 1966) and Blandford (1966) were the first to theoretically describe quasi-geostroph i c motions in this region, and some time later Wunsch and Gill (1976) provided supporting evidence for the existence of equatorially trapped inertia-gravity waves. These waves are a global phenomenon supported by observations in all three equatorial oceans. The equatorial region is dominated by strong zonal currents forced by the trade winds Instabilities arise from the shear produced by these strong currents and these instabilities contribute to the equatorially trapped, zonally propagating energy (Cane and Sarachik, 1976). The oceans adjustment to these disturbances can be thought of as a two part process First, inertia-gravity waves are radiated producing flow that is nearly geostrophic. This stage occurs rapidly and corresponds to the higher frequency waves The low frequency portion of this energy is responsible for the large scale quasi-geostrophic balance and is associated with planetary waves The high frequency part of the adjustment allows energy not involved in the large scale processes to propagate away (Gill, 1982) 1
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Although many have commented on their existence and behavior, there are still some characteristics of inertia-gravity waves that remain unclear. Their zonal structure has been difficult to observe due to the spatially sparse observation sites. The in situ data can detect temporal modulations of inertia-gravity wave energy at one location but being able to compare how different areas of a basin behave in time or how different basins behave has been difficult. Poor spatial resolution has also made it difficult to determine whether the waves correspond to zero wavenumber or zero group velocity. This study is unique in that we evaluate the ability of the TOPEX!Poseidon (hereinafter, TIP) altimeter to study inertia-gravity waves. To our knowledge, using remotely sensed data to study inertia-gravity waves has not been done before. This is very important because most of the remaining questions arise because of poor spatial sampling. The purpose of the present study is to determine whether the use of altimetric data might help We start with a general description of inertia-gravity wave theory followed by a summary of previous research We then analyze in situ tide gauge and current meter data for inertia-gravity wave signals, with the emphasis being on providing "ground truth" for the altimetric analyses. A detailed description of how the altimetric data are used to study inertia-gravity wave potential energy is then given, followed by preliminary results addressing the remaining questions concerning inertia-gravity waves. Finally, we summarize the capabilities and limitations of the TIP altimeter for observing inertia-gravity waves 2
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THEORY OF EQUATORIAL WAVES The study of equatorial wave dynamics began with Matsuno (1966) using a d i vergent barotropic model. Only a brief summary of the model is described here and readers are referred to Gill (1982 ), Matsuno ( 1966) or Blandford ( 1 966 ) for a more c omplete discussion of equatorial wave dynamics. The basic equations we start with are the momentum equations without friction on a rotating earth the conservation of mass equation, and the equation of state for an incompressible fluid which are 'V u = 0 and Dp/ Dt = 0 respectively These are 1 u +uu +vu +wu = --p +jv I X y Z p X ( 1 ) 1 v +uv + w +wv = p -ju I X y l p y ( 2 ) 1 ( 3 ) w +uw +vw +ww =--p -g I X y l p l u + v +w = 0 (4) X y Z p +up +vp +wp =0 I X y Z (5) where u v, and ware the v elocity components in the x y and z directions with x positive eastward, y positive northward and z positive upward from zero at the 3
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position of the undisturbed sea surface. The density of the water is p, f is the Coriolis parameter, and g is the acceleration due to gravity. The following derivation may be somewhat laborious, but we consider it important to outline the steps involved in getting to the linearized vertical and horizontal equations by a separation of variables technique as opposed to simply assuming the correct solution. This is to allow a fuller understanding of the derivation Those reader s who wish to see a more straightforward treatment are referred to Moore and Philander (1976), Wunsch and Gill (1976), or Cane and Sarachik (1976) These equations are linearized by taking the particle speed (advective velocity) to be small relative to the wave speed in the x, y, and z directions. In equation ( 1) the non-linear t erm, uu x can be compared to the local acceleration term, u1 and scaled as U2T/LxU and reduced to U/C where C = L,/T, or the zonal wave speed. Assuming that U / C << 1 allows us to neglect uu x The same logic can be used to eliminate the other non-linear terms in (1), uu y and WU2 assuming that LjV and LfW scale similarly to LjU. This is a correct assumption by (4), which states U/Lx-V/L Y W/Lz. The same arguments can be used to eliminate the non-linear terms in (2) and (3). The hori zo ntal advection terms in equation (5) can be similarly scaled and neglected but the wpz is retained since the vertical length scale for th e density variations is not assumed to be the same as Lz above. The above equations thus become 4
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1 u -jv=--p t p X (6) 1 v +ju=-p t p y (7) 1 (8) w =-p -g t p z u +v +w = 0 (9) X y z p +wp =0 t z (10) It is convenient to divide the pressure and density fields into a static part and a dyna m ic part as, I p(x,y,z,t) =p (z) +p (x y,z,t) 0 (11 -a) I p (x,y,z,t) = p (z) + p (x,y,z,t) 0 (11 -b) where p'/p0 << 1 and p'/ p o << 1. In the above expressions, Po and Po are the static contributions of pressure and density that are, by definition, in hydrostatic balance dp 0 -=-p g dz o ( 1 2) while the primed parts are the dynamic or perturbation fields Using the above relationship when making the substitutio n s in (11) and (12), we obtain 5
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1 I u -jv=-p t p X 0 1 I v +fu=--p t p y 0 I I w p +p =-p g t 0 z u +v +w =0 X y Z I I p +wp +w(p ) =0 t z 0 z (13) (14) (15) (16) (17) We now define the buoyancy frequency as N2 = ( -g/ p0) Poz which allows us to write (17) as I 2 gp -wN p =0. t 0 (18) At this point in the analysis there are still five equations and five unknowns. We now eliminate p' by taking the time derivative of equation (15) and subtracting it from equation (18) to obtain 1 I u -jv=--p t p X 0 1 I v +fu=-p t p y 0 6 (19) (20)
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u +v +w =0 (21) X J l 2 p w +p +wN p =0. o n v o (22) In order to solve for the horizontal and vertical structure of the variables, we follow the usual procedure of dividing them into two parts, one with vertical dependence and one with horizontal dependence. Before we do this, we will make a low frequency approximation since inertia-gravity waves, which are the focus of this work, occur at a much lower frequency than the buoyancy frequency (Gill, 1982 ) This allows us to neglect the p0W11 term in (22) relative to wN2p0 As an aside, we note that taking W11 to be small is equivalent to taking w1p0 to be small in (15). When this is done, we are left with the dynamic parts of p and p satisfying the hydrostatic approximation. This means that not only is the static part of the pressure field in hydrostatic balance, but the total pressure field is as well. We also note that the hydrostatic approximation is not valid for waves that occur at frequencies comparable to the buoyancy frequency That is, for high frequency waves, the far left term in (22) may not be neglected However, because inertia-gravity waves occur at a much lower frequency than the buoyancy frequency, the hydrostatic approximation is valid for these waves. For the following vertical mode separation the simplified case where f = 0 will be examined, as we only intend to illustrate the method here. Gill ( 1982) derives the case where f 0 and it can be seen from his derivation that making this simplification will not change the vertical solutions The rotation term will be added for the 7
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horizontal solutions In order to separate these equations into a vertical and horizontal part we first eliminate u and v from the continuity equation. This is done by taking the x-derivative of (19), the y-derivative of (20), and by making the appropriate substitutions in the t-derivative of (21). Now we have two equations in p' and w, I I p +p =p w .xr yy 0 l1 (23-a) and (22) without the first term, which has been neglected due to the low frequency approximation, I 2 p +wN p =0. <.t 0 We now separate variables using the following definitions, w=w(x,y,t)h(z) p 1 =11 (x,y,t)ft(z). After the above substitutions are made, eqs. (23-a) and (23-b) become v+, v=w h p d Yf t<. o 2 1 A N hw=-p 11. p 4 t 0 In terms of a separation constant, c/, we thus have 8 (23-b) (24) (25) (26) (27)
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fj w, -2 ------c (28) hzY o V BTl n where cn2 has the units of m2/ s2 The resulting equations are A 2h p=c p n 4 o (29-a) w =c V 11 t n H (29 -b ) and similarly, (27) becomes (30) 11, It is appropriate to use 1 in this case as the separation constant because only one adjustable separation constant is needed Expanding (30) yields (31-a) (31-b) 9
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We focus initially on the z-dependent equations, (29-a) and (31-b). By taking the z-derivative of (29-a) and substituting into (31-b ), we obtain 2 2 2 c h p +c p h +p N h=O. nzoz n ozz o (32) At this point we make an additional simplification following Gill ( 1982), which is to assume that the depth scale for the density changes is much greater than the depth scale for the vertical velocity changes. Under this assumption, (32) becomes N2 h +h=O zz 2 (33) c n which is of the Sturm-Liouville form We thus know that there exists an infinite set of orthogonal eigenmodes, h with n = 1 2,3 ... and for each eigenfunction there is a corresponding eigenvalue, C0 The complete vertical structure also requires ft which n can be solved for by inserting the solutions for hn in (29-a). We turn now to the horizontal counterparts, namely (29-b) and (31-a) These will be used to simplify the horizontal momentum equations, still without rotation. Specifically, 1 I u=--p t p X 0 1 I v=--p t p y 0 10 (34) (35)
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become u = -g"l t X v =-g"l t y when the following definitions are introduced; p(z) u=u(x,y,t)-gpo p(z) v=v(x,y,t)-. gpo Making the same substitutions in the continuity equation ( 16) yields and by using d efi nitions of fi and w from (31-a) and (29-a), ( 40) becom es 2 c _n_( u +v ) +11 = 0 g X y t where cn2 is the eigenvalue for the n1 h vertical mode being considered. 11 (36) (37) (38) (39) (40) (41)
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At this point e n is simply a separation constant. In order to i nterpret it physically, we consider a wave propagating in the x-direction with no meridional velocity component and no change in the meridional direction. In this case ( 41) becomes 2 c 11 0 -u +11 = g X t ( 41-a) and this along with (36) and (3 7) describe surface gravity wave dynamics if en is taken to be the surface gravity wave speed (gH) 112 This naturally suggests identifying en in ( 41) as the internal gravity wave speed for the n1h baroclinic mode By analogy to the surface gravity wave case, an "equivalent" depth, Hn, is often defined via c / = Hn g Some typical values of Hn for the first five baroclinic modes in the Atlantic Ocean are 60 20, 8 4 and 2 em (Moore and Philander, 1977). The analogous internal gravity wave speeds are 2.42, 1.40 0 89, 0.63 and 0.44 rnls. Equations (36), (37), and (41) can be used to solve the meridional structure of these waves For convenience the geopotential, q>, will be used instead of vertical displacement 11 These variables are related via (42) The rotation terms will be now be added. The appropriate equations for the n t h vertical mode are then 12
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fi. jV+ q> =0 t X (43) v +jfi.+q> = 0 t y (44) c (u +v )+q> = O n x y t (45) where f = py. We have also left off the n subscripts on fi., v, and q> for simplicity. This particular definition of f is called an equatorial beta plane This approximation is derived from f = 2Qsinq> using the approximation sinq>:::::q>. However, f = 0 at the equator causing the Ross by number (U/tL ) defined as the ratio of the non-linear acceleration terms to the Coriolis acceleration, to break down. The Ross by number is a measurement of whether the equations can be linearized, meaning that the non-linear acceleration terms are small compared to the Coriolis acceleration. In the equatorial theory the ratio of the non-linear terms is compared instead to the local acceleration term, which leads to the scaling of U/C for the non-linear terms where U is the typical particle speed and C is the wave speed as stated previously Another difference between the equatorial and the mid-latitude theories is that the common "quasi-geostrophic" approximation can only be made in the mid-latitude regions In this case the mid-latitude Rossby number as defined above is small and for frequencies much less than f the local acceleration terms may also be neglected so that to first order the flow is purely geostrophic. A consequence of this is that at mid-latitudes the relative vorticity which is derived from the geostrophic part of the 13
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velocity field, is much smaller than the planetary vorticity, which is simply f In the equatorial region however, the scaling shows that the relative vorticity is of order I. Consequently, the relative vorticity cannot be neglected when compared to the planetary vorticity, and the equatorial region is not governed by quasi-geostrophic dynamics That is, the ageostrophic terms are important at lowest order Returning to the system (43)-(45), now that all of the variables have no vertical dependence, it is appropriate to non-dimensionalize the equations using length and time scales, L and T, respectively, which are unspecified at this point. Proceeding with the non-dimensionalization we defme L I u=-u T L I v=-v T I x=Lx y=Ly I t=Tt1 where the primes denote non-dimensional quantities After these substitutions are made, the above equations become I A I I u .... yv [TL]q> ,=0 t :c I I I v ,+pyu [TL]q> ,=0 t y 2 2 I I I L c (u +v ,)+q> ,(-] =0. n :c y t T2 14 (46) (47) (48)
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We now want to define LandT in tenns of P and en, the only dimensional quantities in the equations in such a way that these quantities do not appear in the non dimensional equations which leads to (49-a) 1 T=--. F. (49-b) This is convenient in t hat it allows us to so l ve the non-dimensional equations once, and then apply the solution for any vertica l mode by re dimensionalizing them with the appropriate value of the internal gravity wave speed. Using these definitions we have I I I =0 (50) u yv +q> t X I I I = 0 (51) v +yu +q> t y I I I = 0 (52) u +v +q> X y t as equations we will use to find the meridional structure of a given vertical mode for the equatorial waves. Following Matsuno's ( 1 966) procedure, we assume wave 15
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propagation in the east-west direction and define the meridional variation as which yields I (y) i(/cr (.)t) v = v e I (y) i ( /cr (.)t ) u = u e I (y) i(/cr (.)t)
k +--y )v=O. yy w This equation has meridionally bounded solutions if 2 2 k w k =2m+ l w the dimensional form being 16 (53-a) (53-b) (53-c) (54) (55) (56) (57) (58)
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w2 2 --k --=-(2m+l) c 2 w en (59) n (Gill, 1982) where w is the frequency and k is the wavenumber. The integer, m is the meridional mode number and Matsuno (1966) specifies the solutions for This gives two solutions for w for a given k Each frequency corresponds to one type of wave; the higher frequencies are the inertia-gravity waves that are the focus of this work, and the lower frequencies are westward-travelling Rossby waves (Matsuno, 1966) Figure 1 shows the dispersion relation for the Yanai wave, the Kelvin wave, the first three meridional modes of the Ross by wave and the first three meridional modes of the inertia-gravity wave, the latter being the only ones labelled. where For a given meridional mode number, m the solution for v is v (y) = w (y) m m 2 y exp( --)H (y) 2 m 17 (60) (61)
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is the normalized Hermite function and Hm is the Hermite polynomial of order m The solutions for um(Y ) and q> (y) are given by m i {2m lJ1 m 1 (y) +------2 w+k (62) and i {2m lJ1 m-l(y) 2 w+k (63 ) Our main objective here is to use the sea surface heights as measured by the TOPEX/Poseidon altimeter to determine potential energy (PE) variations of inertiagravity waves This parameter will be discussed throughout this thesis and therefore a brief discussion on the energetics assoc iated with these waves is necessary First, we derive the energy equation from the linearized momentum equations (50) and ( 51), to obtain 1 2 -(u +v J +uq> +vq> =0 2 t X J (64) Combining this with (52) results in 1 2 2 2 -(u +v +q> ) +(q>u) +(q>v) =0 2 t X J (65 ) 18
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which can also be written as (KE+PE) + 'V ii
, the second term on the right, the energy flux term, disappears and we can write 1 2 2 2 =(+< PE>) =-( + +<
) = 0 t t 2 t It is now appropriate to insert the expressions for the horizontal velocities and the geopotential into the expressions for the kinetic and potential energies to gain an understanding of the relative sizes of these terms When this is done, we obtain 2 1 =-2 2 1 m+ 1 m <
=[ + ] 4 ( w -k/ ( w +k)2 19 (68) (69) (70) (71)
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where <> indicates averaging over one cycle and integrating meridionally allowing us to take advantage of the following expressions: 00 (72) -co Upon manipulation and simplification, we can write 1 m+1 m =(+)=-[1 + + 2 ] 4 ( (i) -k)2 ( (i) +k) (73) where the first term on the right is the average KE contributed from the meridional velocity component (v), and the second two terms on the right represent equal contributions from the average KE due to the zonal component of velocity (u) and from the average PE (Moore and Philander, 1977) When we look at the ratio of these two terms, we are left with the expression 1 m+1 m [2( + )l ((i)-k/ (w+k) (74) 1 m+1 m [1 + 2( 2 + )l ( (i) -k) ( (i) +k) 20
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The quantity 1 m+1 m -[ + ] 2 ( w -k/ ( w +k)2 (75) can be rearranged by first factoring out w 2 from the denominator to obtain 1 m+1 m --2[ k 2 + k 2] 2w 1-(-) 1 +(-) (76) (J) (J) This can be simplified further by noting that for inertia-gravity waves lkl<0.5 and w>l.5, so the ratio klw 1/3, which is relatively small Consequently, a binomial expansion can be performed and (76) can be written as 2m+1 2k ] --[1 2w2 w(2m+ 1) (77) When we recall the dispersion relation for small wavenumber, which is w 2:2m+ 1, we can write (77) as 1 k 1 --(-)(-) 2 w 2m+ 1 (78) If we define the parameter k 1 E=--W 2m+ 1 (79) 21
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and recall that k/w 1 / 3 while also noting that l/(2m+1) 1 / 3 then E 1/9 This allows us to write the ratio of to as 1 -E 2 1 1-2E 1 4 = = [ ] :::: -(1--E] 1 3 2 3 3 1+-E 1-E 2 3 (80) meaning that there is about three times as much kinetic energy as potential energy for inertia-gravity wave dynamics This estimate is appropriate to 0(15%) for meridional mode 1 and 0(10%) for meridional mode 2 independent of w and k for 1 / 2 It will now be convenient to recall that the sea surface height, Tl, is proportional to the geopotential. In the following analyses we will be fitting sea surface heights from TIP in the form i(h -l.o) t) Tl (x,y,t) =Re[A e m m ] F (y) m m (81) where i (y) F (y)= m m co (82) co 22
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This renormalization of the meridional pressure function is convenient because now 00 f (83) -oo and we can then write the potential energy (PE,J of the mth meridional mode as PEm = 1Aml2/2. The potential energy is the variable we will be estimating from the TIP data in later sections of the thesis 23
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PREVIOUS RESEARCH Matsuno (1966) and Blandford (1966), independently were the first to provide a model that enabled the theoretical study of equatorially trapped waves Both analyses were strictly mathematical and suggested the need for atmospheric and oceanic evidence. The following paragraphs summarize observational evidence from the Pacific and Atlantic basins to support the presence of equatorially trapped inertia grav i ty waves using sea level and velocity measurements. Although the objective of this work deals with using remotely sensed data to observe inertia-gravity waves, it is important to evaluate what is presently known about these waves to understand the advantages of using altimetric data and to evaluate its performance The following is a chronological summary of inertia-gravity wave observations. Data from tide gauges Acoustic Doppler Current Profilers (ADCP) and Inverted Echo Sounders (IES) have been helpful in examining inertia-gravity waves in both the Pacific and Atlantic Oceans Evidence for the presence of inertia-gravity waves in the Indian Ocean currently exists in the literature from tide gauge autospectra (Luther 1980) and from velocity data (Eriksen, 1980). Atmospheric variables have also been examined to find local coherence and to possibly establish a forcing mechanism 24
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The first observations of sea level oscillations in the inertia-gravity wave frequency band were made by Groves (1956) using tide gauge measurements at a handful of locations in the Pacific, including Kanton. An oscillation with a periodicity of 4 days was observed at Kanton. Groves (1956) also looked at the meridional and zonal components of the wind and found the 4 day oscillation to be coherent with the meridional component of the wind. Groves and Miyata (1967) and Groves and Hannan (1968) provided additional evidence of energy in the inertia-gravity wave frequency band Both studies looked at Pacific sea level records from tide gauges in addition to both components of the wind Groves and Miyata (1967) found a large sea level response at 4 days with smaller responses at 2. 7 days and 5 days Both the 4 and 5 day periodicities were coherent with the zonal component of the wind Groves and Hannan ( 1968) found the same thing at two different Pacific tide gauges coherence between sea level and local wind. When Taft et al. (1974) looked at velocity records in the Pacific at 150 meters they also found a 4 day periodicity that was evident in both components of the wind as well. Although these four studies observed ocean responses in the inertia-gravity wave frequency band they did not attribute these oscillations to the presence of inertia-gravity waves. In the mid-1970's, Wunsch and Gill (1976) investigated sea level records at various locations in the equatorial Pacific, ranging from about 40S to 20
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Eniwetok Island located significantly further north than Canton and Ocean Island it was possible to see if there was any meridional dependence in inertia gravity wave energy. Eniwetok Island's energy spectrum lacked the 4 day oscillation that dominated Ocean and Canton Island's spectra suggesting to Wunsch and Gill (1976) these waves were equatorially trapped. When the 4 day energy was plotted as a function of latitude the drop off at higher latitudes suggested equatorial trapping demonstrating consistency with the appropriate Hermite functions Wunsch and Gill's (1976) analysis was based on the assumption that the observed sea level fluctuations were primarily due to the first baroclinic mode This was justified by noting that the vertical eigenvalue problem has an infinite number of solutions, the first being a surface solution the barotropic mode and the rest being baroclinic internal solutions The barotropic surface mode requires too high of a surface gravity wave speed to be equatorially trapped, so they concluded that the sea surface displacement must be caused by an internal mode This was because the gravity wave speed is calculated from the ocean depth from the barotropic mode and from the equivalent depth in the baroclinic modes Wunsch and Gill (1976) also noted as the baroclinic mode number increased, the displacement of an isopycnal must increas e to raise the sea surface 1 em. Thus, sea level was considered to be a natural filter of the higher modes, leading to the assumpt i on that sea level is dominated by the first baroclinic mode However, in more energetic sea level spectra higher vertical modes can be resolved (Luther 1980 ) The belief that sea level responds to all modes with the first vertical mode being responsible for a large portion of the sea 26
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surface displacement is commonly accepted. This can also be observed by combining the hydrostatic and geostrophic equations to obtain BTJ pyv=g ax When this is scaled with the appropriate length scale, which is a function of the internal gravity wave speed and thus a function of vertical mode we get TJ g (84) (85) where L=(c / P)112 This quantity is dependent on the vertical mode number, n. For higher vertical modes and thus a lower internal gravity wave speed, we obtain a lower value of the length scale. Consequently, the velocity must increase linearly to obtain the same sea level displacement. Wunsch and Gill (1976) provided evidence supporting the theory that the frequencies observed by Groves (1956), Groves and Miyata (1967), and Groves and Hannan (1968) were equatorially trapped inertia-gravity waves Until the late 1970's, little was said about the vertical structure of velocity in the inertia-gravity wave frequency band, except to say that the 4 day periodicity exists (Taft et al., 1974) Weisberg et al. ( 1979) and Weisberg ( 1979) used current meters in the Atlantic made available by the GARP Atlantic Tropical Experiment (GATE), to study the zonal (u) and meridional (v) velocity fluctuations at depth Weisberg et al. (1979) took measurements from four moorings located at 280W in a vertical line at 1.5"N, 0 and 27
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1.5S, and an additional one at 26W near the equator. The measurements were made at depths ranging from 200 m to 1 000 m with varying intervals while Weisberg ( 1979) did the same, but with a slightly different depth range, from 10 m to 700 m. Additional current meter data were examined by Weisberg (1979) located at 100W in a vertical line at 0 and 1 S. Having current meters that are located both on the equator and slightly north and south of it, it was possible for Weisberg (1979) and Weisberg et al. ( 1979) to compare the symmetry of u and v about the equator with the corresponding Hermite functions In both studies, there existed evidence of oscillations occurring at frequencies in the inertia-gravity wave frequency band between 0.2 day1 and 0.35 day 1 In the Weisberg (1979) study only the meridional component of velocity showed a peak in the autospectra at the appropriate frequency at 10 m This higher frequency energy decr e ased in d e eper mea s urements while the lower frequency energy increased The Weisberg et al. (1979) study found the appropriate oscillation in both components of velocity in the periodograms By looking more closely at the meridional structure of these waves, it was found that the meridional component of velocity was coherent in th e 1 "N and 0 moorings and in the 1 S and 0 moorings but wer e almost 180 out of phase This indicates a node between each of these pairs of moorings, which is consistent with high even meridional mode Hermite functions Aside from the meridional s tructure being consistent with theory, these fluctuations in the v-component of the velocity field are coherent zonally as well (Weisberg et al., 1979). Weisberg ( 1979) found this same phenomenon with the zonal coherence to be as large as 2000 km Because these 28
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studies were done in regions of strong background currents, the Equatorial Undercurrent, Weisberg (1979) commented on the influence of a strong mean current on these waves He stated that the vertical and horizontal shears of the mean current play an important role in wave propagation and waveguiding. Additional information was given by Philander (1979) and McPhaden and Knox (1979) on how the inertia-gravity waves are affected by background currents like the Equatorial Undercurrent or simply an eastward or westward jet. Philander ( 1979) constructed a model that enabled him to see the effects of a swift current like the Equatorial Undercurrent, with a velocity as high as 0.3 0.4 m/s however it s hould be noted that the Undercurrent has velocities as high as 1.5 m/s It was concluded that the first and second baroclinic mode inertia-gravity waves were not affe c ted by a background current with velocities in the 0.3 0.4 m/s range due to the high phase speeds of waves (Philander, 1979) McPhaden and Knox ( 1979) further examined the effect of background flows on small wavenumber waves by introducing five separate flows into their model. The five background flows they introduced were no current eastward jet, westward jet, antisymmetric jet, Pacific profile Again the background flows were not found to significantly influence the inertia-gravity waves However there was one exception to this In McPhaden and Knox's (1979) model for the Pacific profile, the periods for the inertia-gravity waves were 3-7 % higher than they were observed for no current, which might account for the fact that Wunsch and Gill's (1976) model gave periods that are approximately 10% lower than the observed periods. 29
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Groves and Hannan ( 1968) suggested the need for longer time series in order to increase the degrees of freedom. The sea level data in the global tropical oceans was revisited by Luther ( 1980) where tide gauges in the Pacific and Indian Oceans were examined for evidence of three types of waves: Rossby-gravity, Kelvin, and inertia-gravity. Through spectral analysis, data sets of hourly sea level measurements showed a higher amount of energy at frequencies corresponding to theoretical frequencies for inertia-gravity waves However, not all of this energy was equatorially trapped as Wunsch and Gill (1976) observed. In sea level autospectra, energy peaks at frequencies of 0.2-0.25 day1 were found as far as 10-20 away from the equator Further study involving correspondence with atmospheric variables indicated this energy was barotropic and basin wide. Although this barotropic wave is not the focus of this study, it still needs to be addressed because it appears to be a significant signal in sea surface displacement in the appropriate frequency band Luther (1980) conducted a more narrow frequency band study by comparing predicted and observed frequencies, noting that for the first vertical mode, each meridional mode has a corresponding frequency. Referring back to equations (33) and (59), in order to compute a predicted frequency, the separation constant, en or the internal gravity wave speed must be determined. Using buoyancy frequency profiles, Luther (1980) found the internal gravity wave speed for the first vertical mode to vary zonally and meridionally. The predicted frequencies of the inertia-gravity waves for varying values of c1 and the observed frequencies were in good agreement. Luther ( 1980) compared the theoretical meridional amplitude structure and frequency for the 30
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first five meridional modes with observations Using autospectral analysis he extended Wunsch and Gill's (1976) fmdings that inertia-gravity waves exist in the Pacific The summary of previous inertia-gravity wave observations thus far has also included analyses of atmospheric data for the possibility of fmding a forcing mechanism In Wunsch and Gill's (1976) study, Canton and Kwajalein Islands were the only two sites in the equatorial waveguide defmed to be between 1 O'N and 1 0 S where pressure and wind velocity were measured A majority of the analysis focused on Canton Island because that was where both atmospheric and oceanic data were collected simultaneously When Wunsch and Gill (1976) examined wind velocity spectra they found an absence of any statistically significant energy peaks in the appropriate frequency band However, the coherence between the wind and sea level data sets was found to be statistically significant, particularly at the 0.25 day frequency with the meridional wind Wunsch and Gill (1976) suggested that this oscillation was due to oceanic resonance rather than atmospheric resonance because the oceanic oscillations had higher energy than the atmospheric oscillations This was because these waves were free modes and the ocean shows preferential excitation at these frequencies even for broad band forcing. That is, broad band forcing can lead to narrow band energy peaks due to oceanic resonance. After summarizing some evidence of coherence between sea level and the local wind it is appropriate to address the consequences of using autospectra versus cross spectra Energy peaks in sea level autospectra do not necessarily coincide with energy 31
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peaks in autospectra of atmospheric data despite the possibility that w i nd i s a forcing mechanism The atmosphere could be forcing the ocean over a broad frequency band but if the ocean is responding resonantly this would cause a narrow frequency band response This would allow the autospectral amplitude of the atmospheric variables to be low while the autospectral amplitude of sea level remains high However when cross spectra are computed peaks would have to occur at the appropriate frequ e n c i e s in order to confirm that the wind is the for cing mechanism That i s it is necessary to find high energy in cross spectra a t inertia-gravity wave frequencies to suggest that e nergy is concentrated at certain preferred frequencies and to imply a forcing mechanism (Luther, 1980 ; Eriksen 1982) In order for these specific frequencies to exist they can either be for c ed at a specific wavenumber or frequency or the for c ing can correspond to a small group v elocity W e have omitted the possibil i ty of specific frequency forcing because of the absence of autospectral peaks in the wind data We also can assume that the specific wavenumber possibility will correspond to small k because the atmosphere bas such large scale variability Due to the high coherence between sea level and local wind Wunsch and Gill (1976) suggested the small group velocity possibility That is, the energy was locally introduced into the system and having no group velocity stayed there and accumulated causing peaks in the autospectra This problem of small w a venumber versus vanishing group velocity i s not quite resolved in the pres e nt literature Fork= 0 and c s = 0, Wunsch and Gill ( 1976) compared the frequenci e s for both of these situations and found that these differed by only 1 5 % for the first 32
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baroclinic mode, with this difference decreasing for higher baroclinic modes. Luther (1980) had similar difficulties but he agreed with Wunsch and Gill (1976). The dispersion relation enabled Luther (1980) to compute the theoretical wavenumber for a given meridional mode for the case of zero group velocity, dw/dk = 0. When the coherence amplitude and phase were computed for zonally separated stations, the results were inconclusive However, due to the close agreement in frequencies corresponding to zero group velocity and the observed frequencies, Luther (1980) remained in agreement with Wunsch and Gill (1976). By using only the first two baroclinic modes and forcing the peaks to be concentrated at zero zonal group velocity, Luther (1980) was able to simulate inertia-gravity wave characteristics observed in the central equatorial Pacific Wunsch and Gill (1976) found high energy associated with the oceanic inertia gravity waves frequencies that was not seen at the same frequencies in the atmospheric forcing variables, which was interpreted as oceanic resonant response. Luther's ( 1980) observations, however, suggested that this resonance was not unifonn over the entire equatorial Pacific basin. In the eastern and far western Pacific, there was a reduction in amplitude that implied changing of inertia-gravity wave dynamics with longitude, which was attributed to the changing topography of the Pacific. Luther (1980) postulated that the forcing may be constant throughout the entire basin, but the waves existing in regions of rough bottom topography were scattered into higher vertical modes and thus not observed in the sea level variations. The waves that were generated with near zero group velocity in regions of smooth bottom topography in the 33
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central Pacific, however, stayed in that region as low vertical mode waves that have a significant sea level expression. Suggestions on why zonal modulation exists is discussed later on in this section. Additional analyses were done in the Atlantic by Horigan and Weisberg ( 1981) who computed the internal gravity wave speed, c. The velocity data measured during GATE was analyzed in different ways in order to reach c, which gives the vertical and meridional mode number. All of the analyses resulted in frequencies consistent with inertia-gravity wave dynamics. Meridional modes 2, 3, 5 and 6 were observed, and the internal gravity wave speeds that were computed were consistent with vertical modes greater than 1. This is consistent with Weisberg et al. (1976) who observed a phase difference between the off and on equatorial velocity measurements indicating higher meridional mode numbers as well as a phase difference between the 200 and 300 meter measurements indicating higher vertical modes. In addition to tide gauges and current meters, inverted echo sounders (IES) have been used to make estimates of dynamic height by measuring the acoustic travel time for a sound pulse to the sea surface and back. The Garzoli and Katz ( 1981) study found higher frequency peaks in the Atlantic IES autospectra at-0.3 to 0.5 day-. Using equatorial symmetry properties similar to the Weisberg et al. (1979) study, they proposed that the lower frequency peak corresponded to a first baroclinic low meridional (2-3) mode wave. From cross spectra with both components of the wind there was evidence to suggest that this peak is forced by the meridional wind. Vertical propagation was hypothesized and vertical wavenumbers were computed for 34
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the -3.5 day oscillation It was found that theoretical forcing parameters such as a wavelength of 3000 km and a period of 3-4 days could indeed force a vertically propagating inertia-gravity wave with a meridional mode of 3 and a period of 3 5 days Another parameter that can be measured to give information about the vertica l structure of inertia-gravity waves is temperature. Eriksen (1982) measured both eastward (u) and northward (v) current in addition to the temperature field at depth in the Pacific In sea level autospectra from tide gauges, he also found evidence of inertia-gravity waves, consistent with Wunsch and Gill (1976) At all frequencies of interest, the surface data showed more energy than the deep temperature fluctuations which were recorded at 1100 meters. The frequencies of sea level and deep temperature fluctuations corresponding to the first vertical, first meridional mode were in phase while the frequencies corresponding to the second vertical, first meridional mode were out of phase These observations are consistent with theory. When the same signature was sought after for the velocity field, no coherence could be found because the depths at which horizontal currents were being measured was within meters of a theoretical node for the first baroclinic mode, which would imply a small horizontal current velocity. When the deep temperature fluctuations were measured zonally, Eriksen (1982) found zonal coherence within a horizontal distance of about 100 kilometers. As suggested by Luther (1980) and Moore and Philander (1977), the effects of a meridional boundary are important. Moore and Philander (1977) and Clarke (1983) 35
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further discussed inertia-gravity wave behavior at western and eastern boundaries In order for the zonal velocity to vanish at the boundary, waves must reflect from the boundary that cancel the zonal velocity component at the boundary. At the western boundary, for an incident inertia-gravity wave there are a finite number of eastward propagating inertia-gravity waves with modes less than or equal to that of the incoming inertia-gravity wave, along with the eastward propagating Y anai wave (corresponding to an even meridional mode inertia-gravity wave) and Kelvin wave (corresponding to an odd meridional mode inertia-gravity wave) (Clarke, 1983) If these waves do indeed correspond to zero group velocity, then reflected inertia-gravity wave energy would stay close to the boundary, while the Yanai and Kelvin wave energy, which have non-zero group velocities, would propagate away, allowing only the Y anai or Kelvin wave portion of the reflected wave to be seen far from the western boundary (Clarke, 1983) At the eastern boundary, the solutions are more difficult since there is no westward propagating Y anai or Kelvin wave to reflect the energy (Moore and Philander 1977). Reflected westward propagating inertia-gravity wave energy may be due to either incoming eastward propagating Kelvin waves or the inertia-gravity waves Inertia-gravity waves arising from the Kelvin wave consist of a finite number of Q modes, where Q is defined by Clarke (1983) to be a function of the incoming wave frequency Westward propagating inertia gravity waves (with mode, m) arising from reflection of eastward propagating inertia-gravity waves (with mode, M) consist of a finite number with m=M ... M+Q-1 where Q is defined previously. For both 36
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eastern boundary cases there also exists an infinite number of waves with larger mode number and complex wavenumber that are trapped near the boundary (Clarke, 1983). For lower mode incoming inertia-gravity waves that correspond to reflected modes between 3 and 6 there is a sharp increase in the reflectivity and for high frequency waves the only mode that is reflected is the M+2 mode. In this situation the amplitude of the reflected wave is equal and opposite to the amplitude of the incident wave, which satisfies the boundary condition (Clarke, 1983). This reflected energy could partly explain the reason why the total oceanic response decreases at the meridional boundaries Garzoli ( 1987) observed peaks in the energy density of autospectra of sea surface height in the Atlantic recorded by IES in the inertia-gravity wave frequency band She also provided zonal and meridional wind data from St. Peter and St. Paul Rocks where part of the mid-Atlantic ridge comes to the surface of the ocean The study focused on frequency peaks in the autospectra at 0.18 day -1 and 0 29 day -1 in the IES and the wind data taken from October 1979 to February 1980. Garzoli (1987) found that all frequency peaks in the IES were coherent with both components of the wind She also addressed some important questions about the zonal and vertical structure of inertia-gravity waves To address the question of k=O or c g=O, theoretical sea surface elevation for a first vertical, first meridional mode inertia-gravity wave c orresponding to vanishing group velocity and zero wavenumber were compared with observations The results from this analysis were inconclusive However, from high coherence and small phase lag in zonally separated stations in the 0.16 day -1 to 0 2 37
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day -1 frequency band, she concluded, contrary to previous observations that this observation corresponded to large zonal wavelength or very small zonal wavenumber With the wavelength and the frequency known, the separation constant, en, was estimated This particular frequency corresponded to a first baroclinic, first meridional mode inertia-gravity w a ve while the 0.29 day -1 frequency corresponded to a first baroclinic, third meridional mode inertia-gravity wave From the cross spectra between meridionally separated IES there was a peak at 0.18 day-1 and smaller peaks at 0.43 day -1 and 0. 29 day-1 (Garzoli 1984) Cross spectral studies indicated high coherence at the 0 22 day -1 frequency between the IES and the zonal wind. High coherence was also established between the equatorial IES and the meridional wind at the 0 .18 day -1 frequency while no peaks occurred in the cross spectra between the zonal wind and the equatorial IES In a study conducted at a different time of year no coherence existed between the meridional wind and the IES in this frequency band, although it did show coherence between these two data sets at the 0 .29 day -1 frequency (Garzoli, 1984). This contradiction suggests the possibility of seasonally varying amplitudes for inertia-gravity waves. Luther ( 1980 ) found a seasonal cycle in the atmospheric Rossby-gravity wave being stronger in the later summer and early fall. Perhaps the seasonality in the IES autospectra could be in response to the seasonal cycle of the atmospheric Rossby-gravity wave. Chiswell and Lukas (1989) also examined internal velocity data in add i tion to sea surface data. They looked at the autospectra from the tide gauges from the four Line Islands (Christmas Fanning Jarvis and Malden) and found evidence of inertia38
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gravity waves The zonal and meridional current were measured concurrently at depths ranging from 50 meters to 250 meters From the autospectra, a statistically significant peak was observed at 0 25 day1 in the sea level data and in the meridional component of velocity but was absent in the autospectra of the zonal component of velocity. Weisberg and Hayes (1995) also found a spectral peak at -0.25 day1 from the autospectra of meridional velocity measurements from ADCP at 50 meters and found this energy to be stationary over a three year period From comparisons with the appropriate Hermite functions for th e meridional structure in order for this to be considered a first baroclinic second meridional mode inertia-gravity wave theory demands that sea level should be high at Canton Island while the v-component of velocity should be large on the equator This agrees with the observation at the 90% confidence interval in Weisberg and Hayes (1995), and therefore agr e es with Wunsch and Gill's (1976) conclusion that this four day oscillation is a surface expression of a second meridional mode inertia-gravity wave. 39
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REMAINING QUESTIONS AND OBJECTIVES FOR THIS STUDY 1. Do the inertia-gravity waves in the tropical oceans show a seasonal modulation, and if so, what are the processes controlling these modulations? In several of the studies, evidence exists of a seasonal modulation in inertia gravity wave amplitudes. The Garzoli and Katz (1981 ), Garzoli (1984) and Garzoli (1987) papers all suggest a seasonal change in the amplitude of these waves. For example, the 2 1 day oscillation discussed by Garzoli and Katz (1981) is only found in the latter half of the year, and the 5-6 day oscillat ion showed different properties in the Garzoli (1984) paper than the Garzoli (1987) paper. These studies used data from different times of the year which could indicate seasonal modulation It is also interesting to note that tropical instability waves (TIW) also show a seasonal modulation in both basins with large energies from May to June in the Atlantic (Weisberg and Weingartner, 1988) and from August to December in the Pacific (Qiao and Weisberg, 1995; Qiao and Weisberg, 1998). Do all of the basins show a seasonal modulation, and if so, what are the processes controlling these modulations? 2. Do the inertia-gravity waves correspond to zero zonal wavenumber or zero zonal group velocity? Another possible question common to both the Pacific and Atlantic papers is 40
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the ability or inability to distinguish if the waves are excited at frequencies at vanishing zonal energy flux or at frequencies corresponding to vanishing zonal wavenumber For the first meridional mode the frequency difference between cgx = 0 and k = 0 is 1.5%, 0.5% for the second meridional mode, and 0.25% for the third meridional mode (Wunsch and Gill 1976) illustrating the difficulty of differentiating between the two. Garzoli (1987) and Eriksen (1982) believe that forcing leading to very low group speed is indistinguishable from forcing leading to zero zonal wavenumber However Luther (1980) gives evidence of the vanishing zonal group velocity possibility by using only the first two baroclinic modes to reproduce the observed sea level. Zero zonal group velocity implies locally resonating waves, suggesting that there should be high coherence between local forcing and sea level. Weisberg and Hayes ( 1995) show stationary results for a three year duration i n the equatorial Pacific that are consistent with the conclusions of Wunsch and Gill (1976) and Luther (1980) Most of the studies up to this point agree that the frequencies observed are consistent with zero zonal group velocity which is somewhat contrad i ctory to Eriksen's (1982) hypothesis Is it possible to rule out zero wavenumber as a possibility ? 3. How is the energy distributed in the vertical? That is, in a vertically standing mode decomposition, how does the energy partition with mode number? Can we say anything about standing vs. vertically propagating modes? A consistent question i n the Atlantic papers is whether or not these i nertia gravity waves are better described as vertically propagating modes or vertically 41
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standing normal modes. Of the preceding studies, Garzoli ( 1987) is the only one who shows supporting evidence for a more appropriate vertically standing first baroclinic mode composition while all others offer evidence of vertically propagating modes being more suitable. Also, all studies exhibit similar findings in that the sea surface displacement is due to the first baroclinic mode. If the correct way to interpret the inertia gravity waves is as vertically propagating then looking at sea level fluctuations may be deceptive in trying to resolve the energy structure. Is it possible to verify that the energy we are seeing in the sea level records is due to first baroclinic mode by studying the vertical distribution of velocity ? Might this be an interbasin difference ? Can we decipher whether the inertia-gravity waves are better described as vertically standing waves or vertically propagating waves? 4 Is there any zonal modulation to the energy of these inertia-gravity waves and if so, what are the processes determining these modulations? Are these trends similar in all ocean basins? In Luther's (1980 ) study it is observed that the energy associated with the inertia-gravity waves is not zonally uniform in the Pacific. He noticed a decrease of energy in the eastern and western Pacific suggesting that something is influencing these waves as they propagate zonally. One possible explanation for this zonal pattern is resonance The waves propagate over smooth topographical regions where there exists high resonance but as they encounter regions of rough topography and meridional boundaries, the waves are scattered into higher vertical modes and higher meridional modes, respectively (Luther 1980; Clarke, 1983) What is the zonal 42
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energy distribution of the inertia-gravity waves and what processes are responsible for their observed structure? TIP Altimetry and Specifi c Objectives for This Study The advent of satellite altimetry provided researchers with a new perspective on the ocean. Altimetry gives sea surface height information by calculating the travel time of an energy pulse from the satellite to the sea surface and back, and the advantage of this technique is obviously the spatial coverage Satellite altimeters lack the temporal resolution of instruments like tide gauges or current meters where the measurements are recorded at fairly short intervals TIP has a repeat period of 9.91564 days which is significantly long when looking at disturbances with periods of 3-7 days. However by using consecutive passes, the amount of time between zonal observations could be on the order of hours since the sampling is continuous over the 1 0-day period Figure 2 shows an example of this. The satellite itself was launched in August of 1992 and has been measuring sea surface height in the time since with unprecedented accuracy. The data go through corrections both at NASA and at individual institutions to correct for the geoid, orbit, pressure ocean circulation tides and water vapor. Water vapor content is estimated through the use of NASA's microwave radiometer and the altitude of the satellite can be determined within 5 em using independent techniques The satellite has recently gone through some rigorous 43
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independent testing in order to verify the altimeters accuracy Mitchum ( 1994) and Mitchum (1998) discussed the importance of comparing the TOPEX altimeter heights with in situ tide gauge sea level heights Much like a tide staff is used to check the accuracy of a tide gauge, Mitchum (1994) used the tide gauge data to in a similar fashion, check the accuracy of the satellite and also to see which tide gauge stations, if any were not representative of the open ocean. By using the closest two ascending and descending TOPEX passes to each tide gauge station it was found that the altimeter measurements were accurate to better than 4 em rms. The purpose of the following research is to assess the TIP altimeter's performance in observing equatorial inertia-gravity waves That is, the question that we are most concerned with is whether TIP can help in answering any of the above questions. In order to address this we will first revisit some observations from tide gauges and current meters in order to provide ground truth for the altimetric analysis We will then give a comprehensive description of how we are using the altimetric data to extract information about inertia -gravity waves, including the method that was used to assign significance to our results. The results from TIP are then compared with the in situ results from the tide gauges and current meters. Finally, we will pr e sent some preliminary descriptions and attempt to address some of the remaining science questions summarized in the preceding section. 44
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A REVISIT OF THE IN SITU OBSERVATIONS Whenever using remotely sensed data to characterize a ground based parameter, it is important to check the performance of the satellite against in situ data. In this case we use tide gauges and current meters to calculate inertia-gravity wave energies in order to check the altimeter's ability to measure the same quantity In regards to the tide gauge data, there are about twenty additional years of hourly heights available to us since the Wunsch and Gill (1976) and Luther (1980) papers These new data have not been analyzed for inertia-gravity waves and promise to be an important tool in providing target points to check the altimeter against. The following analyses focus mainly on the second meridional mode in anticipation of later results. This section will first describe how the tide gauge and current meter data were retrie y ed and pre-treated before analysis Both data sets were pre-whitened by taking out the tides, and then band-passed to isolate the appropriate frequency band The details of pre-processing are included below along w i th the i n situ results. The tide gauge and current meter locations are shown in Figure 3 and briefly described in Table 1. 45
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Data Collection and Pre-Processing The hourly tide gauge data used in this analysis were obtained from the University of Hawaii Sea Level Center (UHSLC). This data set is maintained by the Joint Archive for Sea Level (JASL), a collaboration between the UHSLC and the National Oceanographic Data Center (NODC) and consists of over 140 sea level time series distributed throughout the tropics. Sea level at these locations is measured with two or more sensors operating simultaneously to prevent the presence of gaps in the data and to allow quality control checks. Unlike the tide gauge data, the velocity data was retrieved from a variety of sources Data from mechanical current meters were obtained from Dr. Michael J McPhaden at the Pacific Marine Environmental Laboratories (PMEL) as part of the Tropical Atmosphere Ocean (TAO) project. Additional TAO data were obtained from Dr. Robert Weisberg at the University of South Florida. These moorings are recovered and re-deployed about every six months After recovery, processing and quality checks are performed. Hourly ADCP data from the Coupled Ocean-Atmosphere Response Experiment (COARE) equatorial array were also provided by Dr. Robert Weisberg from the University of South Florida. The moorings were located at 0 157E, oo 45'N 155 60'E, and 0 45'S 156 2'E and covered the time period from February of 1992 to April of 1994. An ADCP measures the Doppler shift of a released sound wave The water velocity can be determined from this shift. Following data retrieval, a correction for 46
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the speed of sound and magnetic deviation is applied. Each vertical profile is linearly interpolated between bins and then resampled at 1Om intervals, where each bin corresponds to the observation depth. The result is hourly sampled horizontal velocity components at depths from 30m to 270m in 1Om increments Following data retrieval the hourly heights and velocity measurements were pre-whitened by extracting one of the major energy sources the tides For this analysis, a routine provided by the JASL was used that includes M G G Foreman's tidal analysis and prediction routines This routine uses a least squares fit method to return a maximum of 45 astronomical constituents and 101 shallow water constituents if the length of the time series allows, however 77 of the I 01 shallow water constituents are insignificant and are not included in the standard package Both the amplitude and phase of each constituent is returned. The analysis creates the complete tides for that record by representing the tidal height at a particular station by the harmonic summation m h(t)= Lf(t)A cos[21t(V.(t)+u (t)-g.)] j=l j J J J J (86) where Aj and gj are the amplitude and phase lag of constituent j, uj are the nodal modulation amplitude and phase correction factors, and V j is the astronomical argument. The residuals are then computed by subtracting the tides from the raw data This program does not take out the tides completely, but it reduces their amplitude significantly Doing this pre-whitening initially makes it possible to remove the 47
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remaining tidal amplitudes with a relatively short numerical filter. Inertia-gravity waves occur in a specific frequency band allowing us to take advantage of band-passing the data. A convolution filter with a length (L) of 721 hours was applied to the data with a half amplitude period (T cnJ of 28 hours to take out high frequency signals An additional convolution filter with a half amplitude period of 470 hours was applied to the data to take out low frequency signals (Figure 4). The filtered time series was then subsampled every three hours. Tide Gauge Results To motivate further research we first need to verify that higher energies do exist at inertia-gravity wave frequencies In searching for these special frequencies, we initially look for energy in the autospectra at each tide gauge The detided bandpassed tide gauge data was divided up into yearly segments Yearly periodograms were calculated using 2 2 F( w ) =I FFT(X)I k T (87) (Bendat & Piersol, 1986), where Tis the length of segment, FFT is the fast Fourier transform defined by T < -rw n ) FFT= LX e k n=l n (88) 48
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is the segment of the time series and wk=27tk/T for The yearly periodograms were then averaged together to create a smoothed autospectrum at each tide gauge station. This gives a representation of how much variance is associated with each Fourier frequency. For each tide gauge, the number of degrees of freedom was computed by recognizing that for a given periodogram, each frequency has two degrees of freedom. If N periodograms went into the average, then the number of degrees of freedom was 2N. The degrees of freedom associated with the Pacific stations were initially much larger than in the other basins due to the longer time series. An attempt was made to force similar degrees of freedom for each tide gauge by smoothing the autospectra further with a boxcar spectral window. If the number of degrees of freedom was less than 50, we smoothed over an appropriate number of frequency bands in order to force a comparable number of degrees of freedom in each autospectra. In an attempt to compare the location of the peaks in the spectra with the basic theory we calc ulated the theoretical frequency using the dimensional dispersion relation 2 n w--2m+l (89) which is the same as before, except for now we have taken the wavenumber to be zero for simplicity Only first vertical mode, n= 1, will be used because we assume that sea level responds largely to the first vertical mode, as discussed above. The internal 49
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gravity wave speeds were those computed by Chelton et al. ( 1998). Figure 5 shows the spectra at various tide gauge stations as well as the theoretical frequencies for the first baroclinic, first four meridional modes as calculated from (89) noted by the solid red circles Later on we will want to investigate potential temporal modulations in inertia-gravity wave energy using complex demodulation. This technique will be applied at the locations designated by the solid black circles. The degrees of freedom for each autospectral calculation is also given for each tide gauge. One common feature that stands out in all of the autospectra is the high amount of energy .1 day1 By referring back to the full dispersion relation (59) that was derived in the theoretical section of this thesis which was vl 2 Pk P --k --=-(2m+l) c 2 w c n n (90) we note that if we take k=O and m=O, the result is (91) This corresponds to a frequency on the order of 8 X 1 o-6 s -1 which is about 9 days if we take c = 2.8 m/s On the frequency axis below, this is around the .11 day -1 mark. This is precisely where the energy lies. Gill ( 1982) identifies this as the Y anai or the mixed Rossby-gravity wave These waves have previously been observed in the far 50
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eastern Atlantic by Weisberg et al. ( 1979b) and in the Pacific by Legeckis ( 1977) and although it is a remarkable signal in the following ana l ysis, it is not the focus of this study and will not be discussed further. A possible way to decrease the amplitude of this signal would have been to change the critical period in the convolution filter This was not done, however, as it would have filtered out a portion of energy resulting from the inertiagravity waves. At some locations, the theoretical frequencies shown by the red circles match quite well with the in situ frequencies for the first few meridional modes. In the Pacific, Christmas (Figure 5-a), Kanton (Figure 5-d), and Tarawa Islands (Figure 5-m) have significant peaks at frequencies corresponding to first and second meridional mode inertia-gravity waves Less energetic first and second mode peaks exist at Fanning (Figure 5-b) and Majuro Islands (Figure 5-g) The Nauru autospectrum (Figure 5-h) shows a large mode 1 peak above the background energy but no mode 2 peak. Some locations exhibit uniform energy throughout the desired frequency band indicating the absence of inertia-gravity wave energy. Funafuti (Figure 5-c) Kwajalein (Figure 5-f) Rabaul (Figure 5-k) and Truk Islands (Figure 5-n) all have a somewhat uniform low energy level in the frequency region of interest. These locations could be near theoretical nodes for a particular meridional mode or they could be near oceanic boundaries where it has been observed that there is a decrease in inertia gravity wave energy. Ther e are also a few locations where it is difficult to isolate any particular frequencies Kapingama r angi (Figure 5 e), Pohnpei (Figure 5 i), Quepos (Figure 5-j) and Santa Cruz (Figure 5-l) all have slightly raised energy levels in the preferred frequency range, but the corresponding theoretical frequencies are inconsistent with the peaks. 51
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Generally speaking, the Atlantic and Indian stations (Figures 5-o to 5-s) have lower energy than the Pacific stations. Only two Atlantic stations show substantial energy at inertia-gravity wave frequencies The autospectrum at AbidjanVridi (Figure 5-o) shows elevated mode 1 and mode 2 energy while the autospectrum at St. Peter and Paul's Rocks (Figure 5-r) exhibits substantial meridional mode 1 and 2 peaks The remaining Atlantic stations, Ascension (Figure 5-p ), Fortaleza (Figure 5-q) and Takoradi (Figure 5-s), do not show any preferred frequencies. The Indian Ocean spectra appear to contain inertia-gravity wave energy as verified by the large mode 1 peaks at Colombo (Figure 5-t) and Diego Garcia (Figure 5-u) Additionally, Male (Figure 5-x) shows a mode 3 peak while and Pt. La Rue's spectrum (Figure 5-y) shows peaks at the first 3 meridional modes. As in the other two basins there are sites where very little energy exists aside from the 1 0-day energy discussed earlier The Hanimaadhoo (Figure 5-w), Port Victoria (Figure 5-z), and Zanzibar (Figure 5-aa) gauges are examples of this The reason for the low energy could be due to their coastal location Recall Luther's (1980) observations that inertia-grav ity wave energy decreased approaching oceanic boundaries. The coastal stations are included in this section to demonstrate this occurrence. As mentioned earlier, we will now investigate the potential for inertia-gravity wave energies to have any temporal modulation Some previous observations provided evidence for a seasonally varying component. If these temporal modulations are present in the data, they will be brought out using complex demodulation. The premise behind this is to isolate the frequency of interest by shifting the data in frequency space then to apply a low pass filter that passes the desired shifted frequency and eliminates other frequencies The result is a time series where any visible modulation is due to temporal modulations of the signals at the desired 52
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frequency. The frequencies that were chosen for this analysis exhibited high energy in the autospectra and are separated enough in frequency space so that a moderate length convolution filter would give little interference from neighboring high energy frequencies After the time series is shifted in frequency space by multiplying it by -iwt e where w is the frequency of the inertia-gravity wave of interest and t is time (92) (Bloomfield, 1976), the resulting complex time series is then low pass filtered with a half amplitude response at 1 00 days, allowing the time series to reflect modulations on time scales larger than 100 days This new shifted, low-pass filtered time series is further manipulated to extract information about how this frequency behaves in time. First, the time series is subsampled again to create daily values from 3-hourly values. To obtain the amplitude of the signal over time (Bloomfield, 1976), we take twice the absolute value of the demodulated, low-pass filtered time series, z(t) ; i e ., A(t) =21 z(t) l (93) An example of an amplitude time series from Kanton Island is shown in Figure 6. Temporal modulation is evident over this 50 year series, but due to the complex nature of the modulation, it is difficult to clearly see any periodic components By computing the periodogram of the amplitude time series we are able to see what these periodic components may be Periodograms of the amplitude time series from each station of interest were computed using the longest stretch in the time series without gaps Each periodogram was then smoothed over three frequency bands to 53
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retrieve a smoothed autospectrum. Figure 7 shows the results of this analysis The annual period 1 cpy and the semi-annual period, 2 cpy, are indicated on all spectra In the Pacific spectra (panels a-k) a majority of the stations have a spectral peak at the semi-annual frequency. This is quite large at Christmas for mode 1 (panel a), Fanning mode 3 (panel d), and still present but smaller for both modes 1 and 3 at Majuro (panel g i), mode 2 at Tarawa (panel k), and for mode 1 at Nauru (panel i). The annual signal is strong at mode 2 for Christmas (panel b) and Fanning (panel c), and strong for both modes 1 and 2 at Kanton (panel e, f). In some spectra, there is a recurring peak in the middle of these two frequencies, centered between 1.3 and 1 7 cpy corresponding to a periods of 215 and 280 days. This occurs at Christmas for mode 1 (panel a) at 250 days and for mode 2 (panel b) at 217 days At Fanning for mode 2 (panel c), the peak occurs at 220 days and at Kanton for mode 2 (panel f), at 236 days Mode 1 at Tarawa has this peak at 208 days (panel j). The autospectra of the amplitude time series for the Atlantic stations are not as revealing as the Pacific autospectra (panels 1 o). This is because the length of continuous amplitudes was longer for the Pacific amplitudes than the Atlantic With this being the case, the Atlantic peaks look much broader Abidjan-Vridi for mode 1 (panel 1) shows that peak in the middle of the annual and semi-annual frequencies centered around 220 days. Mode 2 for the same site shows a higher frequency peak corresponding to a 146 day periodicity (panel m). Modes 1 and 2 at St. Peter & Paul Rocks exhibit elevated broad energies at periodicities between the annual and semi annual (panel n o ) The broad peaks make it difficult to resolve the frequency 54
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structure. The amplitude spectra from the Indian ocean (panels p-u) also exhibit broad peaks. Mode 1 at Diego Garcia (panel q) and Male (panel u) are examples of this as well. Mode 1 at Colombo (panel p) and mode 3 at Pt. La Rue (panel t) show a peak at a frequency band slightly higher than the semi-annual frequency. The spectra at Pt. La Rue for modes 1 and 2 look similar (panels r, s) They both show a semi-annual peak in addition to a peak around 3 cpy However, only mode 1 at Pt. La Rue the same location gives evidence of an annual signal. Since seasonal signals have been identified in many of the tide gauges, it makes sense to look at how the tide gauge autospectra change over time The 3-hourly, detided, band passed tide gauge data were then divided into months by assigning each month to be equal to the last 15 days of the previous month, including the month to be analyzed, and the first 15 days of the following month. For example, a typical March is defined to be February 15th to April 15th The periodogram of each monthly time series is then computed and averaged with the others in that same month resulting in an average autospectrum for each month. The results are shown in Figure 8 The tide gauges presented here are the ones that showed a high level of energy in the complex demodulation results just discussed This figure makes it difficult to ignore the reality of a seasonal modulation in frequencies in the 0.15 to 0.4 cpd frequency band A majority of the gauges show elevated energy levels in the second meridional mode frequency band (0 25-0.29 cpd), corresponding to periods between 3 5 and 4 days. All Pacific stations shown here exhibit evidence of seasonal modulation except for Nauru (panel e) Kanton (panel c) 55
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exhibits higher energies for mode 2 in the winter months ( September to December) while other stations like Christmas (panel a), Tarawa (panel f), and Majuro (panel d) peak more than once a year Christmas and Majuro show an additional smaller peak in July however Majuro's winter peak begins in November and ends in February. Tarawa's mode 2 energy remains consistently high from June to February with the largest peaks in July, October, and January In the preceding examples second meridional mode energy is low in the spring months (March, April, and May) and large in the summer fall, and early winter months There is also first meridional mode energy at a majority of the tide gauges as well corresponding to 0 2 cpd. Kanton and Christmas both have slightly higher mode 1 energy in a broad band of time from May through March, however the amplitude of the modulation may be too small to be significant. Some Pacific stations show the same type of "double peaking" for mode 1 as for mode 2. Majuro's mode 1 peak peaks in September then again in January with almost no energy in March through July. Tarawa's mode 1 energy peaks in January and is comparable in size to the mode 2 energy The seasonal structure of the Atlantic stations are slightly out of phase with the Pacific energies AbidjanVridi in the Atlantic shows mode 1 energy elevated in May (panel g) while St. Peter & Paul's Rocks shows higher mode 1 energy in July (panel h) Indian stations Colombo (panel i), Diego Garcia (panel j) and Pt. La Rue (panel k) exhibit little modulation throughout the year while Male's mode 2 energy (panel 1) is slightly larger in the summer months These too are out of phase with the timing of the Pacific peaks. 56
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For the tide gauges discussed here, it is difficult to find any seasonally modulating energies in frequencies higher than mode 2 This absence was evident in the complex demodulation analysis as well although there was evidence of a semi annual modulation for Tarawa (mode 2), Fanning (mode 2) and Colombo (mode 1 ). This same pattern is verified in the seasonal contour plots V e locity Results Temporal structure in inertia-gravity wave energy has been verified in the tide gauges but the velocity data can provide additional information about the vertical structure of the waves For the velocity data, only the CO ARE and TAO 0' ACM) were used because the other sets either covered too short a period of time or they appeared to be too gappy. In this section we will only be looking at the autospectra of the meridional velocity component because the mooring sites are at the equator which is a theoretical node for the mode 2 zonal velocity component. The autospectra were computed in the same manner as the tide gauge autospectra Inertia-gravity wave frequency / depth structure is shown in Figures 9 and 10. Because of the relatively short velocity records, the periodograms were smoothed by applying a convolution filter to the autospectra There are 14 degrees of freedom for the CO ARE spectra while the number of degrees of freedom varies in depth for the TAO series It is however, always between 18 and 30 All locations show the high energy region around 10 days, cons i stent with the tide gauge spectra In the 57
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TAO sites at ll0W (Figure 9-a), 140W (Figure 9-b), and 170W, (Figure 9-d) there is inertia-gravity wave energy around 4 and 5 days corresponding to 0.2 and 0.25 cpd on the x-axis, but it appears to be lower at the 165 E site (Figure 9-c) The COARE spectra also show some signs of increased energy around 4-5 days, particularly at the 0 156 E site (Figure 10-c) There is also evidence of mode 1 energy at the and 156E sites In all of the velocity results, it is difficult to say whether or not the energy seen in the inertia-gravity wave frequency band is due to first or second baroclinic mode. Recall the statement made earlier that sea surface displacements are primarily due to the first baroclinic mode. This means that the altimeter will be seeing contributions made by the first baroclinic mode. The velocity data however can h a ve a more complex vertical mode structure The velocity data were also examined from another perspective Keeping in mind how the TIP data will be treated, we fit a sinusoid to the 3-hourly velocity data at the frequency corresponding to the frequency minimum for meridional mode 2. Again, mode 2 is the focus in anticipation of the TIP results The fit is done in 60 day bins overlapped by 10 days and the reason for this will be discussed later This is analogous to the complex demodulation technique applied to the tide gauge data Any temporal or depth structure in the fit amplitudes should only be due to the second meridional mode. The fit is only shown for the CO ARE velocities because the TAO data were too gappy to see any structure and Figure 10 shows the results of t his analysis Note that the squared amplitude is shown here. It is difficult to see any temporal modulation, however as we are limited by the length of the time series. In 58
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Figure 11-c there seem s to be a region of low energy at all depths in the summer months of 1993. This low period is not seen at COARE 1 (Figure 11-a) or at COARE 2 (Figure 11-b ) One general statement about the depth structure can be made which is that most of the energy occurs in the upper 150 meters as seen at the COARE 3 and COARE 2 sites This is not however, necessarily true for the COARE 1 site. 59
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APPLICATION OF ALTIMETRY As noted previously, the spatial resolution of the existing inertia-gravity wave observations is inadequate for detailed observation of zonal structure in their energies The small number of locations also makes it difficult to recover reliable information about the wavenumber structure It is in these two areas where we hope to demonstrate the importance and applicability of TIP. In this section, we will first provide a detailed description of how the altimetric data are fit to an inertia-gravity wave model and how this leads to an estimate of potential energy (PE). We will then investigate some difficulties we encountered as well as the solutions that we developed in carrying out this method The way in which significance is assigned to our results is then discussed, followed by the comparisons we made with the in situ data in order to evaluate our method. In order to quantify the amplitudes of inertia-gravity waves using TIP, we assume a model of the form h(x,y,t) = Re [A exp i(k x w t)] F (y) m m m m (94) In the above equation h are the TIP heights, Am is the complex amplitude k m is the zonal wavenumber, wm is the frequency, which is determined from the zonal wavenumber, and F m(Y) is the meridional pressure structure function defined earlier in 60
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the theory section. The m subscript refers to the meridional mode being fit. The two unknowns in this fit are the complex amplitude and the zonal wavenumber. The complex amplitude is obtained for a given k by minimizing (95) where hn is a TIP data point and hnm odel is the right hand side of (94) This results in a set of IAml2 as a function of km, and the fmal Am and km are then determined by maximizing IAml with respect to km. To reiterate, the pre-determined wavenumber is set, a frequency is then calculated from the dispersion relation, and the least squares fit is done. The wavenumber that is interpreted as the best wavenumber is the one that maximizes the amplitude, resulting in an amplitude and wavenumber for each central time and central zonal position The TIP data in a given zonal/temporal box are used to do fits at discrete longitudes and times. The choice of the box size to use will be discussed shortly. Note that the fit is performed over a pre-determined wavenumber range This wavenumber range was estimated by noting that in a majority of the tide gauge autospectra, the location of the peak on the frequency axis was always within a few percent of the minimum frequency for that meridional mode. The top panel of Figure 12 shows an example of a tide gauge spectra The mode 2 peak shown in red occurs at 0.25 cpd which corresponds to the minimum theoretical frequency for a mode 2 inertia-gravity wave. In looking at a number of spectra at various locations, we 61
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noticed that the autospectral peaks occurred at the theoretical minimum or within a few percent of the minimum In panel b, the dispersion curve for the mode 2 inertia gravity wave is shown. Horizonal lines are drawn at the minimum frequency and at the minimum frequency plus 5% The frequency range with a lower bound at the minimum frequency and an upper bound at a 5% increase corresponds to a wavenumber range shown by the two vertical lines. Specifically the wavenumber range that we will search is -0 9 to 0.5, non-dimensionally In doing the search we looked at wavenumbers from -1. 0 to 0 6 in i ncrements of 0 1 One additional bin was added on each side of the wavenumber range to allow a check for fits that did not have a maximum within the -0. 9 to 0 5 range That is, when the maximum occurred at -1.0 or 0 .6, the fit was rejected Sampling Issues Initially this method was tested by fitting to a known signal embedded in various noise models. If we could have recovered the known amplitude and wavenumbe r for the input signal then we would have concluded that the fit was reliable Instead, we found very unacceptable results for some of the modes at some wavenumbers. We were not able to account for these results in any simple fashion, which led us to explore the nature of the TIP sampling in greater detail. TIP samples along groundtracks that are inclined 66 to the equator and are spaced approximately 140 km apart on average at the equator These passes are 62
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repeated approximately every l 0 days Along the pass the sampling is once per second or at about 6-7 km spacing Consequently, the meridional resolution is quite high, whereas the zonal and temporal sampling is rather coarse (Figure 13) The temporal sampling, however, can be increased somewhat in our method if passes from a wider longitude range are used For observing inertia-gravity waves, the meridional structure should be well-resolved although because the tracks are inclined to the equator, the meridional and zonal characteristics of the waves are mixed In the zonal and temporal domains, however there is little that can be done to increase the resolution of the sampling Therefore, we focused on the potential sampling problems due to the zonal/temporal spacing and initially ignored the meridional sampling problem If we look at the zonal and temporal sampling (Figure 13) for a 30 wide box, the sampling is perfectly regular if we look at an obliquely rotated system in x' and t' This introduces the possibility of aliasing, which we will examine shortly. First however, we will take a closer look at the fitting procedure. Recall the form of the fit defmed in referring to (94). It would be ideal to expand the TIP heights onto the meridional structure functions taking advantage of the orthonormality of the structure functions meaning 00 i(k: x-w r) H (x,t)= f h(x,y,t)F (y)dy = Re[A e m m ] m m m (96) -oo because 63
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00 f (97) -oo However, because the TIP tracks are oriented 66 from the horizontal the meridional posit i on is a function of zonal position or y=y(x). This makes it difficult to look at the zonal and meridional sampling separately If we assume that the TIP heights can be written in the form I I h(x,y,t) = cos(k x-w t)F (y) m (98) where x is now the equatorial crossing longitude, the meridional dependence disappears. This is a pessimistic treatment because we are not using the meridional structure to distinguish inertia-gravity waves from other signals in the TIP heights In this simplified case, we are fitting cos(k1x-c.v1t) = a cos(k x-w t)+b sin(k -c.v t) m m m m m m (99) we have written Am= ibm Therefore if the sampling was perfect, then (100) fork' = and c.v'=wm and 0 otherwise. In looking fork', c.v' pairs that may be problemati c we look at IAml as a function of k' and c.v' for different meridional modes. This should re ve al any suspect results from our fit. 64
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It is possible to explicitly solve (1 00) for the coefficients, am and bm. Before writing these down, we first define the phases as 8 =k X -W t mn m n m n e I I I =k x -w t (101) n n n where n denotes a TIP data point at the equator and m, again, is the merid i onal mode. The least squares solution for a, and bm can now be written as a = m [L cos(6 1)cos(6 )L sin\e )-L cos(6 1)sin(6 )L cos(6 )sin(6 )] n n mn n mn n n m n n mn mn b = m -1 D [L cos(6 1)sin(6 )L cos\e )-L cos(6 1)cos(6 )L cos(6 )sin(6 )] D -1 n n mn n mn n n mn n mn mn with (1 02-a) (102-b) (103) These simplified equations will be used to investigate consequences of regular TIP sampling. Any questionable results found with these simplified expressions will be confirmed using the coefficients with the y-dependence included When a parameter is 65
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sampled regularly aliases can be a large problem and are two types of aliases that we are concerned with The first type of problem with perfectly regular sampling is aliasing to the mean We first note that regardless of the k' and w ', if e is a multiple of 1t then mn the denominator of the coefficients goes to zero and the fit can not be determined. In this case, the wave field of interest looks like a mean height signal in the TIP sampling regime. An illustration of this is shown in the top panel of Figure 14. Suppose that each time the altimeter samples, it samples the same phase of the wave as it did the last time On Figure 14 we have drawn a wave with a wavenumber and frequency such that they perfectly match the zonal and temporal sampling width The altimeter sees no change in the sea level as it samples over space and time, therefore the wave will look like part of the mean sea level height. In order to fmd out which wavenumber frequency pairs will alias to the mean, D from (103) was computed over a range of w and k pairs and Figure 15 shows the result. The w and k pairs that look like the mean sea level are the blue circles. Waves with these particular w and k pairs cannot be distinguished from the mean sea level height field due to the zonal/temporal sampling of TIP Overplotted are the dispersion curves for the first three meridional modes and the wavenumber range that we are interested in Notice that the dispersion curve for the third meridional mode passes directly through one of these w and k pairs. Recall that this check was done with a simplified set of equations using the north/south oriented TIP tracks Figure 16 shows the results for the complete set of equations and there is also a significant dip in the value of D for 66
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the third mode at the same wavenumber From these results, we conclude that it is not possible to retrieve any reliable results for meridional mode 3 The other type of alias occurs when the frequency and wavenumber of the wave that is being sampled looks like a wave with a different frequency and wavenumber due to the sampling regularity This occurs when 8 1 =8 +mt m mn (104) where n is an integer A one-dimensional example of this is illustrated in the bottom panel of Figure 14. The first series shows a wave in the height field and the black dots are places where we are sampling the data The second series shows what the higher frequency wave will look like in this regularly sampled height field Equation (1 04) above defines the two-dimensional analog of this problem. Because we have concluded already that it is not possible to look at mode three, we will now just focus on meridional modes one and two. To evaluate this problem we put in a pure w', k' wave with the same meridional structure as the inertia-gravity wave and checked the amplitude response of the fit. Theoretically we should only get a response at the parti c ular w' and k' that we put in. We did not fmd any serious problems with high frequency inertia-gravity waves masquerading as different inertia-gravity waves We did, however, find responses at lower frequencies, (Figure 17), meaning that low frequency signals will look like inertia-gravity waves in this regularly sampled regime. These correspond to signals with periods of 20 days or longer We have normalized the response so the amplitude ranges from 0 to 1. We see that the amplitude response 67
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for mode 1 is only 10% to 20% of the amplitude, but the response is as much as 50% for mode 2. If the low frequency region of sea level spectra did not contain much energy, then this would not be a significant problem. There are, however many high amplitude, low frequency waves in the equatorial sea level spectra The Y anai wave, the Kelvin wave, and the Rossby wave all lie in this part of the spectrum and have higher amplitudes than we are trying to resolve for the inertia-gravity wave In order to obtain reliable inertia-gravity wave results, we need to reduce the effect of these low frequency signals. One solution to this is to high pass filter the TIP data. This will decrease the low frequency energy substantially and retain the high frequency portion The response of the filter we used is shown in Figure 18. The response of the filter is shown at frequencies extending past the Nyquist frequency in order to show the response at the aliased frequencies Any energies at periods longer than 32 days is decreased, while anything in the 8-12 day period range is almost fully eliminated This is desirable since this is a highly energetic portion of the equatorial sea level and velocity spectra as seen from the in situ observations. A large section of the 4-5 day energy is also eliminated which is an advantage because Luther ( 1980) observed a large 4-6 day barotropic signal that could also contaminate our results The downfall to this is that meridional mode 1 has a period of about 5 days, which will also be removed by this filter Meridional mode 2, on the other hand, remains untouched by this filter It lies at full response, so this energy should be fully passed Because low frequency signals could mimic inertia-gravity waves due to aliasing, it is a necessity 68
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that we high pass the data, which removes meridional mode 1 energy. We therefore conclude that with the TIP data we can only look at meridional mode 2. Choosing the Zonal/Temporal Box Size When we first introduced the TIP data, we showed the coverage for a sample box width of 30. At this however, point the optimum zonal/temporal box size is still unknown. We would like to make the box as narrow as possible to obtain the best possible zonal and temporal resolution. On the other hand we need enough data in the box to obtain reliable fits. In looking for an ideal box size, we put in known mode 2 inertia-gravity waves at different wavenumbers and checked the amplitude response for a number of longitude and temporal widths At narrow box sizes for many of the input wavenumbers we see a temporal ringing. As an example, Figure 19 shows the response for varying zonal widths for an input mode 2 inertia-gravity wave with a non-dimensional wavenumber of -0.5 For example, at k=O, we see large responses at certain times and smaller responses at other times As the zonal box size increases, this time dependence becomes less obvious and at a zonal box width of 30, the ringing is mostly gone. This problem was traced to a dependence on the phase of the input wave. Figure 20 shows an example of how too few spatial points can lead to either observing unit amplitude or zero amplitude depending on the phase of the wave When we add more temporal points by increasing the box size, this problem goes away. When we look at the time 69
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dependence in Figure 21, increasing the number of cycles used in the fit has little effect on the results This is because each cycle samples at the same locations, so we gain little be using more cycles. Ideally we could set the temporal width to one T I P cycle, but this does not take into account the fact that there are gaps in the TIP data To provide some protection from the gaps we set the temporal width to two cycles or 20 days Significance Testing As discussed above, we have determined that we can only reliably fit meridional mode 2 waves to the TIP data. Further, we know that we need to use a zonal box width of at least 30 longitude But with these caveats we should be able to obtain results for mode 2. On obtaining these results, it is important that we have a significance test. For the significance testing, our initial idea was to use Monte Carlo simulations The goal is to test the null hypothesis that our results are consistent with data that has no inertia-gravity wave energy present. If we had a way to generate realiZations of such a data set, we could carry out our fits many times and thus determine the probability that any parameter we compute (e.g., the potential energy) could have arisen from data known not to contain inertia-gravity wave energy. The TIP data itself is ideal for significance testing because this allows the simulated data to retain the same sampling structure, error structure, and degree of independence in space and time To generate a simulated data set, we used off 70
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equatorial TIP data that has the same sampling structure zonally and temporally as the equatorial data we used for the fits (Figure 22) When we fit to these off equator i al data sets, treating them as if they were on the equator, we expect that the fitted amplitude should be zero, consistent with no i nertia-grav i ty wave energy present. The idea of using TIP data more than 30 away from the equator was rejected because we wanted to preserve similar signal and error structur e s as are found in the TIP equatorial data. To further ensure this, we se arched for offset locations that had the same zonal and temporal sampling as the "real" equatorial data. There were eight locations that met these requirements : , and For example the simulated data set at 8"N would extend north to 23 "N and south to 7 S or 15 in both directions. One important issue that warranted c o ncern was the possibility that at some of these offset locations, the meridional pressure function would be significantly correlated with the pressure function centered at the equator The top panel in Figure 23 shows the meridional pressure function for the second meridional mode at the e quator and at the 15 offset treated as if it were centered at the equator We were concerned that the meridional structures would map onto each other if they were corr e lated The bottom panel on the same figur e shows the correlation as a function of latitude offs e t The 4 shifted data would map onto the meridional structure and give a high amplitude for the fit with the simulated data. We chose the 8 and t he 15 offset because of their low correlation with the meridional structure at the equator Monte Carlo simulations, however require many realizations of the simulated data In this case however only four simulated sets are available This dilemma 71
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motivates the need for a random data model. The model that we choose though, can be checked against the simulations using the shifted data, and can even be calibrated if necessary using these results. Do we expect the off equatorial results to look a certain way? Recall that the quantity we are interested in quantifying is potential energy (PE). If we defme the fitted (complex) amplitude am to be a + ib, then the PE is proportional to the sum of the squares of the coefficients, a2 + b2 If we make the hypothesis that the data are white noise with zero mean and variance cr2 then the coefficients, a and b, are also normally distributed with zero mean and standard deviation a cr (Bendat and Piersol, 1986), where a depends on the details of the fit. This leads to the quantity (A/(acr)]2 being chi square distributed with 2 degrees of freedom (X/). Note that A=laml The only unknown variable in this relation is a An estimate of a can be made by noting that the mean of a X2 distribution with v degrees of freedom is v, so that (105) where < > denotes an expectation, or mean, value. If we use all of the amplitudes and standard deviations from the off equatorial sites we arrive at a preliminary estimate for ex of 0 09. Keeping in mind that this is a hypothesis to be tested Figure 24 shows the distributions for (A/(0 .09cr)f and X2 2 These distributions look very similar in their structure The distribution in the top panel has a slightly heavier tail than the bottom 72
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panel, though, which could lead to large discrepancies when looking for cutoff percentages at the 90% to 95% confidence levels. We therefore decided to make a more careful evaluation of a It makes sense that a could be a function of the parameters of the fit, which are zonal wavenumber, internal gravity wave speed, and the latitude offset. We first computed a set of a estimates from (1 05) above using each A and o as an approximation to the expectation value When we plot these a estimates versus the parameters, (Figure 24), we see a quadratic dependence with wavenumber and a linear dependence with internal gravity wave speed We then formed a model for a that was fit to all of the a estimates. The upper panel in Figure 26 shows the distribution of the difference between the calculated a from the shifted data and the a value from the model. We see a roughly normally distributed shape, but with some of the a model values being lower than the actual a values as seen from the excess of values on the right side of the distribution. The lower panel in Figure 26 shows a and a model plotted against each other. We see generally good agreement with some outliers seen above the diagonal that indicates perfect agreement. The a values that were not well represented by the model were examined in more detail. We found that many of the a values that were not well-modelled correspond to locations bordering the Indonesian Archipelago This is a region where the altimeter encounters potential problems from the land mass distribution. That is, the boxes of TIP data that were used for fits in this region would not have had very many data points. This suggests the need for allowing the percentage of points 73
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available in the box to be taken into account when performing the fit. Implementing this condition would eliminate fits in regions of missing data due to land mass distributions For present purposes however we will proceed with the x / no i se model using the a model described above In Situ Comparisons Now that we have a way to assign significance to our PE results w e need to compare the results with the in situ results from the tide gauge and the velocity data. In an earlier section of this thesis we showed spectra from a number of tide gauges that showed energy in the inertia-gravity wave frequency band From these sites we selected those that were the most appropriate for inferring meridional mode 2 variations. Recall that our TIP fits are in the form h(x,y,t)=Acos(kx-)F(y) (106) and for a tide gauge at a particular X0 Y o posi tion we can write h(x ,y ,t) =TJ (t) =A cos( ) 0 0 0 0 ( 107) where (108 ) The A0 value can be obtained by fitting to the time series at the tide gauge. To infer the PE value that TIP would see at a particular location from the tide gaug e A0 we 74
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use the relation where and A2 PEO PE =-= --2 F\y) 0 Az 0 PE= o 2 PE =F2(y ). fac o PE 0 PE fac (109) (11 0) (lll) An example of this is shown in Figure 27 for Tarawa Tarawa's meridional position is plotted in addition to the corresponding F(y J value followed by the PEfo c value These PEfac values were computed for all of the tide gauges that displayed inertiagravi ty wave energy in their autospectra Only tide gauges that had PEf a c values of 0.35 or larger were used for comparison to avoid amplifying random errors in the PE estimates In the figure, the region where F(y) is between -0 5 and 0.5 slopes steeply, so a very small change in latitude can lead to a large change in F(y) We wanted to avoid regions where uncertainty was involved in calculating the PEfacThe tide gauge data were treated in a similar manner as the TIP heights for the intercomparisons. We took the frequency at the meridional mode 2 peak in the 75
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autospectra and fit a sinusoid to the data with this frequency at 60 day intervals with a 10 day overlap The 60 day window was chosen in favor of the 20 day window because it was important to consider the bias error associated with both the tide gauge results and the TIP results For an estimate of the bias error associated with the tide gauge results, we look at two tide gauges at the same longitude but different latitudes recognizing that these two gauges should be seeing the same signal. Christmas and Fanning Islands are within 2 longitude degrees of each other but Christmas lies at 2'N and Fanning at From the theoretical meridional pressure structure function, the Fanning mode 2 inertia-gravity wave signal should be 25% of the same signal at Christmas If we define the PE at Christmas to consist of a signal and bias error, and the signal at Fanning to consist of the same signal reduced by 75% and the same bias error, then the bias error can be solved for and written as b1 8=(4PErannin8PE chnsanas)/3 In order to compare b1 8 to the bias error estimate from TIP (btp), we need to divide b1 8 by the PErac discussed earlier This is because when we divide by the PErac' not only is the signal amplified, but so is the bias error For a window size of 60 days the bias error at Christmas is about 45 mm2 as compared to 39 mm2 for TIP This particular window size was used at all the tide gauges For the three Pacific tide gauges that were used for intercomparisons (Christmas Island, Kanton, and Tarawa) (Figure 28), contemporaneous series were used. A summary of the intercomparisons for the tide gauge results is given in Table 2. The mean PE over both the TIP and tide gauge series is computed using points common to both series and compared with that of the 80% confidence level for the 76
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mean PE computed from the simulated data. The same is done for the PE range of seasonal variation for both instruments. Although an attempt was made to compensate for the differences in the bias errors for these two methods, it should be noted that there still may be some differences in the mean PEs from the bias error All of the tide gauges used for intercomparisons agree with the TIP estimates to within 15% and are significant at the 80% level. The seasonal variations shown in Table 2 also indicate generally excellent agreement, which is particularly encouraging since this represents a much more stringent test than simply getting the mean PE values correct for reasons previously mentioned. Figure 28 show the PE time series at the Pacific tide gauge sites. During most of the high PE events, these two series correspond well. Below each PE time series, the PE series are plotted against each other. The correlation coefficient is then calculated in addition to an estimate of the 95% value from a null test that the true value of the correlation is zero. For all stations the correlation coefficient exceeds the 95% confidence level. In the Christmas Island intercomparisons, Figure 28-e shows the PE time series from both methods as in the others, however, plotted in green is a large peak in the tide gauge series not present in the TIP series. This peak occurs in the middle of 1994 while all of the other peaks occur at the end of the year. The following Christmas Island results will take this peculiarity into consideration. Figure 28-f shows the correlation coefficient between the two methods twice, once for the TIP series and the tide gauge series, and once for the TIP series and the tide gauge series excluding this mid-1994 peak. 77
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For the velocity data intercomparisons, the data were treated in a consistent manner as the tide gauge heights. A simpler criteria went into deciding which velocity data to use. For meridional mode 2, the meridional velocity component is maximum at the equator since it is proportional to the second Hermite function. Consequently, equatorial velocity data are best suited for intercomparison and only these data were used (Figure 3). In doing the fits to the velocity data, instead of taking the frequency of the inertia-gravity wave to be fit from the autospectra, we took the frequency to be at the theoretical minimum. A sinusoid was fit to the data at each depth at 60 day intervals with a 10 day overlap. Bias error estimates calculated from the velocity results were difficult to obtain resulting in the choice to use the 60 day window selected from the tide gauge analyses. In order to compare these amplitudes to the TIP results, we now need to infer the meridional velocity structure from the sea surface height amplitudes To do this we first recall from the theory section that non-dimensionally, the meridional component of velocity and the geopotential can be written as v -alJI (y) 2 2 (112) where v2 is the second meridional mode component of the meridional velocity component, lJ12 is the second Hermite function, q>2 is the geopotential, and Q>(y) is the meridional pressure function before we renormalized, and a is the non-dimensional wave amplitude. The dimensional meridional velocity components that we have computed ( a0 ) are therefore interpreted as 78
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a =ac 111 (0) 0 1 2 (113) where a is the non-dimensional wave amplitude and c 1 is the first baroclinic wave speed if we assume that the velocity field is also dominated by the first vertical mode This may not be true, of course. Not also that a0 is estimated at a range of depths, and is therefore a function of z. To form the analogous quantity from the TIP amplitude estimates we first recall that the dimensional TIP heights were fit in the form 11 = AF(y) Ai!J>(y) where a = 4> dy. 2 f 2 Also, since the dimensional 11 is related to the non-dimensional geopotential by 2 2 c c 11 =-q> =-a!J>(y) g g We can write, from (114) and (115) gA a=--2 c a 79 (114) (115) (116) (117)
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Combining (98) and (94), we see that the TIP estimate of the surface meridional velocity component amplitude at the equator, a'0 should be I gA a =-ljl (0) o ca 2 (118) We can then use the first baroclinic mode vertical structure function to project this value downward In order to make the intercomparison we computed the typical amplitude in each time series as the square root of the temporal mean of the squared amplitude. Figure 29 shows the depth profiles computed at the equatorial velocity moorings along with the TIP estimates computed with the assumption that the variability is first vertical mode We see that the shape in the vertical agrees well at all of the moorings, which reinforces our assumption that we have extracted the first vertical mode portion of the variability. The amplitudes agree well at 156", but the TIP inferred amplitudes at the surface are smaller than the in situ estimates at the other three sites by 20-30% Given that the velocity and height series likely have very different signal and noise characteristics, though, we interpret this level of agreement as supporting the conclusion that the TIP data are fairly representing true second meridional mode inertia-gravity wave signals. 80
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PRELIMINARY RESULTS Now that we have assessed the performance of the TIP altimeter in measuring mode 2 inertia-gravity wave signals and have found very encouraging results, we will now briefly address some of the remaining science questions posed earlier Several authors, (e.g ., Garzoli, 1984; Garzoli 1987; Garzoli and Katz, 1981; Comejo Rodriguez and Enfield, 1987) suggested the presence of a seasonally varying component associated with inertia-gravity wave energies The in situ measurements at the tide gauges presented here support this as well. The waves have also been observed to exhibit zonal modulation. For example, Luther (1980) found inertia gravity wave energies highest in the middle of the Pacific basin with energies decreasing towards the boundaries. There also has been dispute about the zonal wavenumber structure between two possibilities, k=O and cg=O. Most studies support the group velocity theory, but the k=O possibility has not been ruled out. The primary purpose of this study was to assess the altimeter's capabilities and limitations, therefore the following results are only preliminary attempts to address the remaining science questions The power of the altimetric analysis is that we can see the zonal and temporal structure in the inertia-gravity wave energy, which is impossible when working strictly from the available in situ data. Figure 30 shows the zonal/temporal energy structure 81
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of the meridional mode 2 inertia-gravity energy. Large bursts of energy occur late in 1993 from 160W to 11 0 W and this location is a region of moderate energy levels for much of the time period observed High energy events also occur in late 1996 and late 1997 in the vicinity of 160 E to 160W. Another burst develops in late 1998 and spans almost the width of the basin decreasing rapidly at llOOW. In fact, at all times, the energy decreases a great deal as we approach the eastern boundary. The timing of the bursts is quite intermittent, in that these do not occur every year. But when the energy does appear, it is consistently late in the calendar year. This could be the reason for some of the inconsistencies found in the literature. The TIP analysis suggests that when and where the in situ measurements are taken will play a large role in determining the results found The Indian basin also exhibits a small sporadic temporal modulation, but confmed to the eastern boundary There is an absence of any clear temporal structure to this energy at first glance The bulk of the energy appears to occur close to the eastern boundary This is an area of concern because of the difficulties in the TIP analysis discussed earlier due to the land mass distribution in the Indonesian Archipelago. The difficulties in obtaining reliable fits in this region was addressed during the discussion of the differences between a and a tit and we are not confident in any results close to the eastern boundary of the Indian basin. The Atlantic basin shows little zonal and temporal energy structure Inertia-gravity wave energy here rarely exceeds the 80% confidence level. 82
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Figure 31 shows only the global PEs that were estimated to be significantly different from zero at the 80% confidence level. In this figure the large Pacific bursts discussed earlier remain while a majority of the Atlantic basin is shown to not be significant. The general structure of the Indian basin remains unaffected In general ignoring the values that are below the 80% s i gnificance level does not chang e the basic patterns that we noted in the previous figure. The temporal energy structure in the Pacific PE results merits further study The following analysis was carried out in all basins, however to examine the possibility of a global temporal structure The temporal modulation was examined by dividing the inertia-gravity wave PEs into monthly bins, as was the case when we were searching for a seasonal modulation in the tide gauge data Each monthly PE t i me series was averaged with others in the s ame month to get an average PE for each month from the TIP time series Figure 32 shows the results from this analysis It is difficult to ignore the higher energies late in the calendar year in the middle of the P a cific. This increase of energy begins in November and decreas e s by January and February There is also a zonal structure to the temporal modulation Around 1600f: the PE increases but then decreases near the dateline. It then increases again near 140W The Indian basin also shows a slight seasonal variation, but slightly out of phase with the Pacific temporal modulations At the western boundary e xtending towards 75 E the energy increase occurs between August and September. There is also a temporal structure at the far eastern boundary. As was discussed earlier, these 83
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results may be inaccurate as a result of the problems near the Indonesian Archipelago The Atlantic temporal structure is seemingly non existent save for a small increase occurring in February until June near the western boundary Again we remain skeptical of the results near land boundaries due to potentially inadequate fits from land interference The absence of any other temporal or zonal modulations in the Atlantic is another indication that the inertia-gravity wave signals in this region are below our detection limit. As stated earlier, we want to check the altimeter's performance against the in situ data where possible The seasonal TIP results just discussed were evaluated using the tide gauges at Tarawa, Kanton, and Christmas Island. These were chosen because contemporaneous time series were available at these locations A mean seasonal signal was computed from the PE time series computed from the same time series presented in Figure 28 These time series were divided into monthly bins as was done for the TIP series and averaged. The results from these intercomparisons are shown in Figure 33 and the Tarawa series (Figure 33-a) and Kanton series (Figure 33-b) are seen to agree closely. However in the Christmas Island series (Figure 33-c) there is an additional peak that occurs in the summer months in the tide gauge series If we exclude the mid-1994 data from the analyses (Figure 33-d) these semi-annual peaks d i sappear indicating that the mid-1994 peak is the sole contributor to this additional tide gauge peak in the summer months At all stations, even the phase of the temporal modulation is in agreement. Energies are consistently h i ghest in November and December in both the in situ data and the TIP data 84
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In addition to temporal modulations inertia-gravity wave energy has also been reported to vary zonally as well (Luther, 1980). To investigate this possibility, the PE was averaged over time resulting in a mean PE for each central longitude (Figure 34). Also plotted on the figure is the 80% confidence level computed from the x 2 noise model. The Pacific basin shows an interesting zonal structure A low energy region exists in the mid-Pacific near 1600W with energy peaks on both sides centered 1650W and 135W. This structure is not present in the 80% confidence level energies, which suggests that this zonal structure is not simply due to random noise or increased variance in the TIP heights. The energy also decreases approaching the eastern and western boundaries, consistent with the findings of Luther (1980). It should be noted that the tide gauges do not agree with this PE zonal structure. The tide gauge series show Tarawa having a lower mean PE than Kanton indicating that the western peak may not be real. However the mean PEs at Kanton and Christmas for the tide gauge series are consistent with the eastern peak seen by the altimeter. Again, it is difficult to say anything conclusive about the western peak because of the bias errors assoCiated with these two series and how sensitive the mean PEs are to these errors The high noise levels in the Atlantic basin relative to the observed signals (Figure 34) make it difficult to describe any real zonal structure The 80% confidence level not only has comparable temporal means across the Atlantic, but the general shape is the same as well We will not be making any statements about the zonal PE inertia-gravity wave structure in the Atlantic In the Indian ocean the PE80o;. levels are 85
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well below the TIP results, but the structure of the PE8 0o;. curve is strikingly similar. This likely indicates that we have underestimated the noise level in this region, which is consistent with the fact that our model for ex discussed earlier was apparently poor in just this region. In Figure 35 we focus on the Pacific PE estimates and only plot the region from 150E to 11 0W. Each subregion, the western high in blue, the central low in black and the eastern high in pink, was examined for a preferred wavenumber structure Recall that choosing an optimum wavenumber is part of the fitting procedure, although to this point we have only discussed the PE results derived from the fitted wave amplitudes. The wavenumbers that best described the PE, those that gave the maximum PE, are shown in Figure 36 in histogram form. Only wavenumbers associated with PE values that exceed the 95% null test are used and the distribution is shown in blue The red bars are an estimate of the wavenumber distribution from fits to random data generated by the x/ noise model. Again, we only consider the simulated wavenumbers corresponding to the locations where PE>PE9 5o;.. Each bar in the histograms thus gives the percentage of the PE values that were .. deemed significant that occurred at that wavenumber value. The top panel shows the wavenumber distribution for the entire region from 1500W to llOOW. The bottom three panels show the wavenumber distribution for each of the three sub-regions identified on the previous figure For example, Figure 36-b indicates that for the western peak, only about 7% of the PE values in this region were best described with zero zonal 86
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wavenumber while about 6% of the PEs obtained from the random noise model were best described with zero zonal wavenumber as well. In Figure 36 in the upper panel, the distribution is largest at a non-dimensional wavenumber of -0.1, which is between the c8=0 point and the k=O point, but the distribution is skewed slightly towards the c8=0 point. In the lower left panel on the same figure, the wavenumber distribution at the western energy peak is clearly shifted to the left of the c8=0 point, suggesting wavenumbers corresponding to c8<0 and thus westward propagating energy. The wavenumber distribution at the central energy low shown on the lower middle panel indicates wavenumbers close to or slightly greater than the c8=0 point, suggesting stationary energy or energy propagating slowly eastward. On the other side of the low, the distribution at the eastern energy peak shown on the lower right panel is clearly shifted to the right of c8=0, suggesting eastward group velocity. To summarize briefly, the only basin in which we found significant results was the Pacific There exists temporal structure with high energies ranging from late September to late January confmed to the middle of the basin. In looking at the temporal mean structure, there is also evidence of a zonal structure. The energy is low at the boundaries and high in the middle. The area of high energy ranging from 150E to 11 0W shows additional structure with a central low and peaks on either side. The wavenumber distribution associated with each of these regions suggests a zonal variation in energy propagation. The cause for the wavenumber distribution is unknown at present. 87
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One possible explanation for the structure might be undetected errors in to the TIP method described here Although a thorough investigation has been done concerning the capabilities and limitations of this method, there is still the possibility of poor fits for unknown reasons. In searching for an explanation for this structure, it is interesting to note that the TIW also have a seasonal and zonal modulation that is somewhat consistent with the inertia-gravity wave modulations presented here (Weisberg and Weingartner, 1988: Pullen et al., 1987) indicating that the dynamics governing these two types of waves may be linked Perhaps the forcing is different in these three regions of the Pacific. The wind field needs to be examined in these longitudinal bins in order to examine whether the forcing is a contributing factor. Another parameter that varies zonally is the internal gravity wave speed. The internal gravity wave speed is a function of the vertical density structure, a variable that varies zonally as well. Perhaps this is related to the mid-Pacific structure. In addition to the possibilities just mentioned, there are surely others. We leave this as a starting point for future work. 88
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SUMMARY Equatorial inertia-gravity waves have been observed in all ocean basins from tide gauge velocity, and inverted echo sounder measurements. These instruments give clear pictures of how inertia-gravity waves behave in particular locations, but becau s e of the sparse spat i al coverage of these in situ sites, it has been difficult to make any conclusive statements concerning large-scale zonal and temporal structure In this thesis, the TIP altimeter has been evaluated for its ability to measure equatorial inertia gravity waves and its ability to answer the remaining science questions in the literature The TIP heights were fit to an inertia-gravity height model that included a zonally propagating part and a meridional structure function. The parameters that were fit were the wavenumber and the amplitude The fit was done with TIP data in a box 30 wide and a 20 day or 2 cycle temporal box. It was found that the TIP sampling regularity gave rise to aliasing problems By looking at the amplitude response for many wavenumber and frequency pairs, it was determined that meridional mode 3 could not be fit because it aliases to the mean We also found low frequency aliases ; that is low frequency signals in the data can look like inertia gra v ity waves The was remedied by high-passing the data but it also eliminated the meridional mode 89
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1 energy. This leaves us with the ability to only fit to meridional mode 2 inertia gravity waves Significance testing was perfonned using simulations with a X 2 noise model that was calibrated using the characteristics of fits done using off equatorial TIP data. We chose off equatorial data that had the same temporal and zonal sampling regime as the equatorial data that were used for the fits This was done i n order to retain similar error structures to what we expect for the equatorial data that were used in our fits Potential energies (PE) computed from the fit amplitudes to the TIP heights were compared to data from in situ locations The mean PE structure at the tide gauges where contemporaneous measurements were available agrees to within 10-20% of the TIP PE while the PE at the velocity sites also agrees to within 10-20% Temporal modulation at all the tide gauges agrees quite remarkably with the phase of the temporal structure being reproduced as well as the seasonal range. The agreement between the TIP results and the in situ results prompted us to make a preliminary attack on the outstanding science questions Most of the interesting and significant structures occurred in the Pacific We con c luded that fits from some of the Indian Ocean were unreliable due to the land mass distribution The remaining Indian and Atlantic results were below the 80% confidence level. In the Pacific Ocean, we fmd a clear, but intennittent temporal structure with higher energies in the late calendar year when they do occur. Pacific inertia-gravity wave energy shows an interesting zonal structure with little energy at the western boundary, an increase of energy at 170 E, a region of low energy at 90
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160 W, and another increase at 130 W These sub-regions also appear to have a distinct wavenumber structure as well and associated zonal variations in the inferred energy fluxes 91
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Table 1. Locations of in situ hourly data used. Also shown is the length of time series that was available. The last six entries are velocity data, and the others are all tide gauge sea levels. Name Latitude Longitude Length of Time Series Christmas 01 59'N 157 29'W 1955-1998 Fanning 03 54'N 159 23'W 1972-1990 Funafuti 08 32'S 179 12'E 1977-1998 Kanton 02 49'S 171 43'W 1949-1998 Kapingamarangi 01 06'N 154 47'E 1978-1998 Kwajalein 08 44'N 167 44'E 1946-1999 Majuro 07 06'N 171 22'E 1968-1998 Nauru 00 32'S 166 54'E 1974-1998 Pohnpei 06 59'N 158 14'E 1969-1998 Quepos 09 24'N 084 10'W 1961-1994 Rabaul 04 12'S 152 11'E 1966-1997 Santa Cruz 00 45'S 090 19'W 1978-1998 Tarawa 01 22'N 172 56'E 1974-1998 Truk 07 27'N 151 51'E 1963-1991 Abidjan-Vridi 05 15'N 004 OO'W 1982-1988 Ascension 07 55'S 014 25'W 1993-1997 Fortaleza 03 43'S 038 28'W 1995-1998 St. Peter & Paul Rocks 00 55'N 029 21'W 1982-1985 Takoradi 04 53'N 001 45'W 1983-1986 Colombo 06 56'N 079 51'E 1953-1965 Diego Garcia 07 14'S 072 26'E 1988-1998 (continued on next page) 92
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Table 1 (continued) Gan 00 41' S 073 09'E 1987-1998 Hanimaadhoo 06 46'N 073 IO'E 1991-1998 Male 04 11 'N 073 31'E 1988-1998 Pt. La Rue 04 40'S 055 32'E 1993-1998 Port Victoria 04 37'S 055 28'E 1977-1992 Zanzibar 06 09'S 039 11' E 1984-1998 COARE 1 (velocity) 0 156 E 1992-1994 COARE 2 (velocity) 0 157 E 1992-1994 TAO 1 (velocity) 0 llOW 1985-1998 TAO 2 (velocity) 0 140 w 1988-1999 TAO 3 (velocity) 0 156 E 1991-1993 TAO 4 (velocity) 0 165 E 1986-1998 TAO 5 (velocity) 0 170 w 1989-2000 93
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Table 2. Summary of intercomparisons between in situ tide gauge and TOPEX!Poseidon results. PE is the potential energy (A2/2 where A is amplitude), and has units of mm2 Christmas Kanton Tarawa Period Observed 1992-1997 1992-1998 1992-1998 Position 1 59'N 2 49'S l020'N 157'W 171'W 173l'E PE Factor 0 5253 0.4968 0 3580 PE in situ 127 105 99 118 (w/ o 1994) PE from TIP 108 96 114 -----------------80% Null Test 46 48 47 PE Range of in situ Season a l Variation 112 142 113 PE Range of TIP Seasonal Variation 94 103 132 -------------------80% Null Test 25 20 17 Time of in situ Seasonal Apr May May Min Time of TIP Seasonal Min May May May Time of in situ Seasonal Nov Nov Nov Max Time of TIP Seasonal Max Nov Nov Oct 94
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8 2.5 ctS c: .o en 2 E 1:J I c: 0 c: 1 5 1 0.5 Dispersion Curve -1 0 1 2 3 -3 -2 non-dimensional k Figure 1 Dispersion curves for equatorial waves The frequency and wavenumbers shown here were non-dimensionalized using the planetary vorticity gradient. and c the internal gravity wave speed. The inertia-gravity waves are the upper branches labelled m=l.2. 3 Note that these waves can propagate phase and energy east or west. 95
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\C 0"1 TIP Sampling 5 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 4 3 2 I . Q)1 1,. v \[ '0 I 6 ::l I 0 I t ; 1 \ 8 ....J -1 -2 -3 5 I I \ I I \ I \ I I \ I \ I I \ I \ I I \ I I I \ I I 150W 180 175W 170W 165W Longitude 160W 155W Figure 2. TIP coverage map Thi s show s the spatial sampling of the altimeter. The average temporal sampling at the equator in the 30 box s hown here i s about 10 hours B e tt e r temporal resolut i on can be obtained by extending the zonal width The alongtrack s pacing is approximately 6-7 km which makes the curves appear continuous
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\0 -...l Q) 'C ::::J -.:::; ro .....J Loca t ions of i n sit u sites 100.----,.--------.---------.---------.--------,--------.---------.---, 80 I Sea Surfa c e H e i g ht V elocity 60 4 0 20 0 - 20 -40 -60 80 1 0 0 5 0 E 100 E . / 150 E 1 60 W Longitude 110W 60 W 10W Figure 3 Map of in sit u locations. This is a visual summary of all locations that were investigated for inertia-gravity wave signals, including both tide ga u ges and current meters Most Pacific tide gauge stations had 20 years of hour l y data avai l able, with shorter series available f r om the Indian and Atlantic stations. The velocity series were also shorter. Additional information is presented i n Tab le 1.
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\0 00 Amplitude Response of Both Filters 1.2 1 Q) C/) c 0 8 0 a. C/) i i 1 0/o Amplitude at 45.66 days and 1.13 days 50/o Amplitude at 19.84 days and 1.17 days 0.4 90/o Amplitude at 12.79 days and 1.20 days a. E <( 0 2 ;:::g 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Frequency day-1 Figure 4. Response of filter applied to hourly data. This was done in two steps First the data were low-passed to eliminate the high frequency signals The low-passed data were low passed again to eliminate periodicities shorter than 20 days This series was subtracted from the first low-passed series to isolate the desired frequency band The response shown reflects both filtering operations
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\0 \0 a) x 1 04 .--3 "'C fr2 C\IE E 1 Christmas d of=138 b) X 104 r-----3 2 1 Fanning dof= 5 4 0 1 0 2 0 3 0.4 0 0 1 0 .2 0.3 0 .4 frequency (cpd) f req u ency (cpd) c) x 104 .--3 2 1 d) x 1 04 Kanton e) x 1 04 Kapingamarangi f) x 1 04 Funafut i d o f= 8 4 0 1 0 2 0 3 0 4 f r eque n cy (cpd) Kwajalein 6 6 2 .----..------..------..------..---------. I ft IIA 11. 5 "'04 4 a. u C\1dof =72 E E 2 dof =6 2 0 I<" ' I -""' 0 0 1 0. 2 0 3 0 .4 0 0 1 0.2 0 3 0 .4 0 0.1 0.2 0.3 0 .4 fr e quency ( cpd) fr e quenc y ( cpd) f r e q u e n cy ( cpd) Figure 5 Autospectra of tide gau ge sea leve l s. Eac h tide gauge, its l ocation and the length of the time series is shown i n Tabl e 1. Periodograms were computed for each year and then averaged to create a s moothed spectrum. Additional frequency band smoothing was don e whe r e necessary in order to make the degrees of freedom approximately equal for all spectra shown. Red circles indicate theoretical freq u encies for the first four meridional modes and black circles denote locations w h ere complex demodulation calculations were also done (continued on next page)
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X 104 Majuro h) X 104 Nauru i) X 1 04 Pohnpei 5 1 5 5 1} 1 : 1 dof = 102 41 I I A 1 ..ll 11. 0 d of= 96 3 C\1r r \ 1 2 I '1/ L dof = 66 10. 5 1 0 1 / . . 0 0 1 0 2 0 .3 0 4 0 0.1 0 2 0 .3 0.4 0 0 1 0 2 0.3 0 .4 freque n cy (cpd) frequency (cpd) f r equency (cpd) j) X 1 04 Quepos k) X 1 04 Rabaul I) X 1 04 Santa Cruz 1} :1 2 8 ,Ak 0 0 0 I : I 1 1.5 dof = 96 J dof"'102 1 H (1/ d of= 78 C\12 E ]Jv. 0 0 1 0.2 0 3 0 4 0 0 1 0.2 0 .3 0 .4 0 0 1 0 2 0 3 0 4 f requency (cpd ) freque ncy ( cpd ) fre q ue ncy ( cpd ) Figure 5 (cont i nued)
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0 Tarawa "8_2 0 C\1E E 1 dof=72 0 0 1 0 2 0 3 0 4 frequency (cpd) p) x 104 Ascension 1 5 a. Truk oJ x 1o4 Abidjan-Vridi 1 5 1 5 1 dof=96 1 I I dof=50 0 5 0 5 0 0 1 0.2 0.3 0 4 0 0 1 0 2 0 3 0 4 frequency ( cpd) frequency (cpd) q) x 104 Fortaleza r) x & Paul Rocks 2 1 5 1 5 0 C\11 dof= 52 1 dof=26 1 dof=26 E E 0 5 0 1 0 2 0 3 0 4 frequency ( cpd) Fi gu re 5 (c o ntin u ed) 0.5 0 0.1 0 2 0 3 0 .4 frequency ( cpd) 0 5 0 0 0.1 0.2 0 3 0.4 frequency day -1
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0 N s) x 104 r-----3 "'0 (\J_g 2 E E 1 Takoradi dof=52 0 1 0 2 0 3 0 4 frequency ( cpd) v) x 104 Gan 2 .---..--..----1.5 "'0 a. t) X 104 5.---4 3 2 1 Colombo dof=56 0.1 0 2 0.3 0.4 frequency (cpd) w) x 104 Hanimaadhoo 1.5 1 (\JE dof=60 1 dof=54 E 0 5 0 0 1 0 2 0 3 0 4 frequency (cpd) Figure 5 (continued) 0 5 0 1 0 2 0.3 0 4 frequency (cpd) u) x 104 Diego Garcia 1 0 0.5 dof=54 0 1 0 2 0 3 0 4 frequency (cpd) x) x 104 Male 1 5 .---.--.--.--.-----. 1.0 dof=54 0.5 0 .__. __ __._ __ ___._ __ ____. ____ ...._____, 0 0 1 0.2 0.3 0 4 frequency (cpd)
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Y.) X 104 Pt. La Rue z) x 104 Port Victoria aaJx 104 Zanzibar 1.5 2 "&. 1 0 1. 5 dof=54 2 Q C\1 1 dof=26 E n r \ A dof = 60 E 0.5 0 5 0 0 0 0 0.1 0 2 0 3 0.4 0 0 1 0 2 0 3 0.4 0 0 1 0 2 0 3 0.4 f r equency (cpd) frequency (cpd) frequency (cpd) 0 w Figure 5. (con tinued )
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Mode 2 Magnitude at Kanton Island 50 45 40 --35 -E S3o ...., Q) "C 25 :::J c 20 g> :E 15 10 5 0 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Time (yr) Figure 6. Example of mode 2 amplitude time series from complex demodulation at Kanton Island This figure shows an ampl i tude mean of about I 0 mm with modulations extending up to about 3 times that amount.
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a) Christmas Mode 1 b) Christmas Mode 2 c) Fanning Mode 2 3000[ : l: ' 16000[ : ' 12500 I Ill A I 2000 >. .fr 2000 I .n I I I \ I I I f II' 1 I I I 1500 C\j 100:t 20:l W '! v 0 1 2 3 4 50 1 2 3 4 50 1 2 3 4 5 frequency (cpy) frequency (cpy) frequency (cpy) d) Fanning Mode 3 e) Kanton Mode 1 f) Kanton Mode 2 0000 2000 t I l A I 1 Q 1000 / ( 1\J\ J\ I 8000 VI I I 1500 I I. 1\/1 I I 6000 500 : /\ ; !\ A 1000 I Ill I I {\ I A. I 4000 500[ J I"" w 0 0 I I 0 0 1 2 3 4 50 1 2 3 4 50 1 2 3 4 5 frequency (cpy) frequency (cpy) frequency (cpy) Figure 7. Autospectra of amplitude time series from complex demodulation at the tide gauges. The autospectra were computed as described previously and smoothed over 3 frequency bands The annual and semi-annual frequencies are indicated by vertical dashed line s. (continued on next page)
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g) Majuro Mode 1 h) Majuro Mode 2 i) Nauru Mode 1 2000t 1\ ' 12500 I I 2000 >-1500 NE 1000 E 500 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 frequency (cpy) frequency ( cpy) frequency (cpy) j) Tarawa Mode 1 k) Tarawa Mode 2 I) Abidjan-Vridi Mode 1 ..... 4000 0 I I 4000 0\ J 1000 I I I 3000 I I 3000 I 800 >-MM I 0 600 2000 2000 E \ 400 1000 J.,,..., 1000 200 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 frequency (cpy) frequency (cpy) frequency (cpy) Figure 7 (continued)
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m) Ab idjan-Vridi Mode 2 n) S t. Pe t e r & Pau l Rocks Mode 1 St. Pet e r & Pa ul Roc k s M ode 2 10001 t 1500 I I I I '\ I 400 >. 800 I c. I A 1ooo r 1 / I \ 1 300 600 \!\ I I '"' j 5001 I I I I \,vi I I I I 0 I 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 frequency ( cpy ) f r equency (cpy ) freque nc y ( cpy ) p) Co l ombo Mode 1 q) Diego Garcia Mode 1 r) Pt. La Rue Mode 1 0 .....:1 5000 aool I : \ J 600 >. 4000 6001 c. y I \ "" I 400 3000 E 400 E 2000 I v \ I 20 0 I I ul I 20:1 ' I I I I I I I 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 f r eque ncy ( cpy ) fr eque n c y ( cpy ) freq u ency (cpy) Figu re 7. (continued)
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s) Pt La Rue Mode 2 t) Pt La Rue Mode 3 u) Male Mode 1 5001 ' l 500 600 I 1\ [\ I 400 400 fl: I 300t 'I \ J \ 1 300 400 I E \_I \ /1 I \ I .. i 200 E /""'\ 200 01 i V iV v v 1 10:r I I \d I I ' I I 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 frequency (cpy) frequency (cpy) frequency (cpy) -0 00 Figure 7. (continued)
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a) nov sep jul may mar -. r::: Jan o nov :::!: sep jul may mar jan c) nov sep jul may mar c jan 0 nov :::!: sep jul may mar jan I I I I I I I I I I I I Christmas 0.1 0.2 0 3 0 4 frequency (cpd} Kanton \ /'. I I / 0 1 0 2 0 3 0.4 frequency ( cpd) x 104 dot 3 5 b) nov sep jul may mar jan 3 N-. 2 1 5 1 0 5 0 5 0 nov sep jul may mar Jan 100 x104dot d) 4 or ,3 N"" E E l 2 1 0 5 0 nov sep jul may mar jan nov sep jul may mar Jan 100 I I I I I I I I I I I I Fanning 0.1 0.2 0.3 0.4 frequency (cpd) Majuro t.EJ ... \\
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e) Nauru x1o4dot f) Tarawa X 10 4 dot 12 nov 2 10 sep jul1 "8. 1 5 8 may1 , C\1 6 Jan 1 ( E 1 nov E 4 sep jul 0.5 2 may mar 0 1 0 2 0.3 0.4 0.5 0 100 jan 0.1 0.2 0 3 0.4 0 5 0 100 frequency (cpd) frequency (cpd) g) Abidjan-Vridi x 104 dot h) St. Peter & Paul Rocks X 10 4 dot nov 3 5 nov 2 -sep 3 sep 0 jul 2.5 jul 1 "8. 1 5 may1 ... mar 2 jan C\1 Jan 1 E 1 0 nov 1 5 nov E sep 1 sep jul jul o.5 may 0 5 may mar mar jan 0 1 0.2 0.3 0.4 0.5 0 100 jan 0 1 0.2 0 3 0 4 0 5 0 100 frequency ( cpd) frequency (cpd) Figure 8 (contin ued)
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i) Colombo x104dof j) Diego Garcia dot nov 8 nov sep sep 15000 jul 6 jul .may .Q may 0 -mar 10000 c:::: jan ce4 Jan 1 '" 0 nov E nov1 E sep sep 1 -jul 2 jul 5ooo may may mar mar jan 0.1 0 2 0.3 0.4 0.5 0 100 jan 0 1 0.2 0 3 0 4 0 5 0 100 frequency ( cpd) frequency (cpd) k) Pt. La Rue x 104 dot I) Male X 10 4 dof ...... __ ... 2.5 ['!""""""'""" -2 5 ...... ...... -r- E E 1 E l 1 0 5 0 5 I 0 1 0 2 0 3 0.4 0 5 0 0.1 0 2 0.3 0 4 0.5 0 100 frequency ( cpd) frequency (cpd) Figure 8. (continued)
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-N a) TAO 110W 50 _100 .s I : I I N 150 200 250 0 0.1 0.2 0 3 0.4 frequency cpd c) TAO 165E 50 100 -150 E 0 1 0 2 0.3 0.4 frequency cpd 0.5 0 5 50 :[ 100 N 150 200 0 d) 0 50 _100 E 200 250 L 0 TAO 140W 0.1 0.2 0.3 frequency cpd TAO 170W 0 1 0 2 0 3 frequency cpd 7000 6000 0.4 0.5 .Q 5000 N.._ 4000 E 0 113000 2000 1000 0.4 0.5 Figure 9 Autospectra of TAO velocity data at 4 locations. These were computed in the same fashion as the tide gauge heights, by taking the periodogram for each year and averaging these together to create a smoothed autospectrum The time series span 3-12 years, but contain many gaps. This perspective gives information on the vertical structure that cannot be obtained from the tide gauge spectra
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w COARE 0 157E COARE 45 N 156E 0 0 1 0 2 0.3 0.4 0 5 0 1 0.2 0 3 0 4 0.5 frequency cpd frequency cpd COARE 0 156E 1500 1000 500 0 1 0 2 0.3 0 4 0 5 frequency cpd Figure 10 Autospectra of CO ARE velocity data at 3 locations. These, too, were computed in the same fashion as the tide gauge spectra These time series were only about 18 months long, but contain no gaps
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COARE 1 : 0 157E 50 100 e 150 a. Q) 0 200 250 1992 .8 1993 1993.2 1993.4 1993.6 1993.8 1 994 COARE 2 : 0 .7 5N 156E 50 100 s 150 c. Q) 0 200 250 1992 8 1993 1993.2 1993.4 1993.6 1993 8 1994 COARE 3 : 0 156E 50 100 :[ 150 a. Q) 0 200 250 1992.8 1993 1993. 2 1993.4 1993.6 1 993 .8 1994 Time (yr) 0 5 cm2/s2 10 15 Figure 11. Estimated mode 2 squared amplitude versus time and depth at COARE sites. The results are analagous to the complex demodulation analysis presented for the tide gauge sea levels Only the meridional component is shown here because the theoretical zonal velocity component is zero at the equator. 114
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Christmas Island Autospectra 8 dot= 138 "'C c. 6 C\J.Q 4 2 .1 .2 .3 .4 .5 Frequency ( cpd) Dispersion Curve for Mode 21GW 3 8 ctS c: o 2 5 en c: (J) E "'C 2 I c: 0 c: 1.5 -2 -1.5 -1 -0.5 0 0.5 1 non-dimensional k Figure 12. Method used to select wavenumber range to search in fit. Panel a shows the Christmas Island spectrum. Note that the large peak corresponding to meridional mode 2 occurs within a few percent of the theoretical minimum. Panel b shows how this frequency range can translate to an appropriate wavenumber range to search through in the TIP fit. 115
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Regularity of T/P Sampling Geometry 315 t' 310 305 - (/) as "0 3 00 c: :::::J - .c: 295 Q) E t- 290 285 280 x 275 170E 180 170W 1 60W 150W 14()W 130W Longitude, x Fig u re 13. TIP z on al a nd tem por al sam p ling In 2-dimensi o na l space, sam p lin g i n the o b li qu e l y r o tate d x', t' co or d inat e syste m is e x a c tl y re gular 116
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290 "0 c:: as 3285 0 Q) E i= 280 170E 172E 174E 176E 178E 180 178W 176W 174W 172W 170W Longitude 0 10 20 30 40 50 60 70 80 90 100 Figure 14. Schematics showing aliasing problems Top pane l shows how waves with certain wavenumber and frequency pairs can look like the mean height field. The bottom panel shows how regular sampling can lead to aliasing in one dimension In the sampling regime donated by the red circles, the higher frequency wave looks like a low frequency signal. The altimetric problems discussed in the text are the two dimensional analog of th i s 117
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A measure of over grid for ro, k input wave 6 1 5 4 8 ctS c: Q en 553 E I c: 0 c: 2 1 0 -6 0.9 0 8 0 7 0 6 0 5 0.4 0 3 0 2 0.1 0 -4 -2 0 2 4 6 non-di mensional k Figure 15. Waven u mber, freq u e n cy pairs where the first type of sampling prob lem occ u rs. These r esults are from the simplified fit discussed in the text. This is from th e aliasi n g p ro b lem shown in Fig u re 14 -a. The non-dimensional d ispersion curves for the first three meri d io n al modes are overp lotted. Note that meridiona l mode 3 passes thro u g h one of these wavenumber, frequency pairs within the wavenumber range of interest, w h ic h is show n by the vertical lines. 118
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Wavenumbers Where Fit is lndeterminite for Actual TIP Tracks 3000 Mode 1 2000 1000 -2 -1. 5 -1 -0. 5 0 0.5 1 1.5 2 4000 S 3000 Mode2 0 S 2000 E g 1000 Q) Cl 0 -2 -1. 5 -1 -0.5 0 0.5 1 1 5 2 4000 ......... 3000 Mode3 2000 1000 -1 -0.5 0 0 5 1 1 5 -2 -1.5 2 Non-dimensional Wavenumber Figure 16. Check of the simplified model using the full fit Each mode was analyzed over the appropriate wavenumber range to check the results using the simplified case. The significant decrease for mode 3 near k=-0.6 is the analog of the pattern seen on Figure 15. 119
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Response at Mode 1 k=O : low ro aliasing 0.5 OS 0 -3 -2 -1 0 1 k of input wave 2 3 Response at Mode 2, k=O : low ro aliasing 0 0.5 OS 0 -3 -2 -1 0 1 k of input wave 0 2 3 Figure 17. Inertia-gravity wave response s to low frequency aliases We input signals with a known wavenumber and frequency and fit mode 1 and mode 2 This shows that low frequency s ignals can loollike inertia-gravity waves in our fit and corresponds to the aliasing problem illustrated in Figure 14-b. Non-dimensional frequency axis corresponds to periods of 20 days and longer 120
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-N -Filter Response 'Q) 32 qays I CJ) 1t......... ; ... .-ro!roo. I ..s:: .Q> ..s:: 0 Q) CJ) c:: 0.5.......... -........ .... 0 CJ) 0 .._--.....JL.____.__----L-_--lllol 0 0.5 IG#2 1 1.5 2 2.5 non-dimensional ro Figure 18. Response of filter used to eliminate low frequency signals. The filter was applied to each TIP pass. The response is shown past the Nyquist frequency to see how filtering affects aliased frequencies This shows the undesirable consequences of using this filter, the most important being that it eliminates the frequency band corresponding to mode 1 inertia-gravity waves. Mode 2 is unaffected.
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250 200 150 100 50 -0.5 0 0 5 -0. 5 0 0 5 -0. 5 0 0.5 non-dimensional k non-dimensional k non-dimensional k Figure 19. Fit response for varying zonal box width We input a mode 2 inertia gravity wave with a wavenumber of k=-0.5. Notice that is it possible to get a large response at other wavenumbers depending on the phase of the input wave. This effect decreases as the zonal width increases. 122
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1 0.5 0 -0. 5 1 0.5 0 -0. 5 a) 5 b) Wave in Height Field 1 0 15 20 25 30 35 40 Sampled at Maxima and Minima 45 50 -1 0 5 1 0.5 0 -0. 5 c) 10 15 20 25 30 35 Sampled at Node 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Figure 20. Illustration of why phase matters Panel a shows a hypothetical wave, panel b shows the effect if the wave is sampled only at the peaks, and panel c shows the effect if the wave is sampled only at the nodes The amplitude inferred from a fit to these subsamples would clearly be different. 123
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100 50 -0. 5 0 0.5 non-dimensional k -0. 5 0 0.5 non dimensional k 0 -1 -0.5 0 0 5 non-dimensional k Figure 21. Effect of temporal box size. A mode 2 wave with a wavenumber of k=-0.5 was input to the height field and retrieved by our fitting procedure. The response was checked for many different cycle sizes. There is little change for all three panels, corresponding to temporal box widths of 1 cycl e 2 cycles, and 3 cycle s A cycle is approximately 10 days. 124
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(1) "0 ::J 1a Sample Equatorial Location 20 _J -10 -20 180 175W 170W 165W 160W 155W 150W Longitude 15 Off-Equatorial Locations for Significance Testing 175W 170W 165W 160W 155W 150W Longitude Figure 22 Example of off-equatorial data used in significance testing These data preserve the same zonal and temporal sampling structure as the equatorial data 125
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Meridional Pressure Function at 0 and 15 -15 -10 -5 0 Latitude 5 10 15 0 5 10 15 Latitude Offset Figu r e 23 Choosing which meridional offsets to u s e in sign i ficance testing The top panel shows the merid i onal pressure fun c tion for mode 2 and the profile when this function is shifted by 15 The bottom panel shows the corr elations between curve s such as these as a function of latitude o f fset. Also shown are the candidate offsets, in red circles where the zonal sampling characterist i cs are conserved. 126
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Distribution of [AI(cr*0.09)2 ] For All Off-Equatorial Data 8000 a) 6000 *4000 2000 2 4 6 8 10 Distribution of 6000 =**= 4000 2000 0 0 2 4 6 8 Figure 24. Comparisons of (Ala 0 09)2 distributions The top b) 10 panel shows results from the off-equatorial sites and the bottom panel shows a Notice the similarities except for the heaviness in the tail in panel a. 127
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N 00 Distribution of a 50 a) =#: 0 0 05 0.1 0 15 0 2 a. a vs Wavenumber avsc .... ... = 1 0 15 c) 0 1 ., 1 0 5 0 0 5 1 2 2 5 3 3 5 Non d i me n si o nal Wave n umbe r c (m/s ) Figure 25 D ependence o f a o n the parameters of the fit The t op panel shows that the a distributi o n is roughly normal, b ut with o utliers in the large a tail. The bottom panels show that a has a qua d ratic relations h ip with wavenumber an d a linear relationship with the i n ternal gravity wave speed These depe n dencies were not accou n ted for in the initial check shown in Figure 24 an d on the t op pane l of this figure.
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60 50 40 30 20 10 -0.1 Distribution of a.-a.11t -0.05 0 0.05 a Scatterplot of a. and a.11t 0.2 r-------..--:-w--------::11 0.15 0.1 0.05 0 0 2 0.1 Figure 26 Distribution of a-afi and scatterplot of a and afi The a model It It and the calculated a match well except for some outliers on the right of the distribution These values mainly correspond to locations near the Indonesian Archipelago. Most of the points in the scatterplot fall along the diagonal, which indicates perfect fit, with the few outliers showing up as points above the line 129
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The PE factor for mode 2 at Tarawa 1r------.-------.-------.------.-------.------. . . o 8 ........... F(Y0 ) 77 ... PEtkc ::::. 0 . 35. .......... ......... . . . 0 . . . . : : Tarawa : : 0 .61----...... ___ ....., ___ ....,.. ... 0 0.4 .. = ...... = .... . : . . . . . . . . . . . . . 0 0 . . .................... . . . 0 2 . . . . . . . . . . . . . . >-No LL . : .............. . . . . -0. 2 . , o o o, o, o,,, o o o o o o o o o o o t o o . 0 0 0 -0. 4 -0.6 -0. 8 ......................................................................... . 15 -10 -5 0 5 10 15 Latitude (i.e., y) Figure 27 Example of PE factor calculation Tarawa. Shown are the mode 2 meridional pressure function, the meridional position of Tarawa (vertical line) and the associated F(y) value (horizontal line). The calculation of the PE factor from this F(y) is desc ri bed in the text. 130
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Tarawa Island Intercomparisons I = Gauge I a) "'E 800 E -e> 600 Q) c w as 400 c Q) 0 200 a.. 1994 1996 1998 Year Scatter of Potential Energies r = 0.76 600 r95% = 0 40 a.. t::::: 400 200 400 600 Tide Gauge b) 800 2000 Figure 28. Tide gauge and TIP intercomparisons. Panels a, c, and e show PE time series from TIP and from three contemporaneous tide gauge series Scatterplots are shown in panels b, d, and f. The correlations between the TIP PE series and the tide gauge PE series are shown on the scatterplot along with the value the correlation coefficient must exceed to be significantly different from zero at the 95% confidence level (r95% ) (continued on next page) 131
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Kanton Island Intercomparisons I = Gauge I c) 'E 8oo E E> 600 Q) c w S 400 c Q) 0 200 a.. 1994 1996 1998 2000 Year Scatter of Potential Energies r = 0.79 600 r95% = 0 29 200 d) 0 200 400 600 800 Tide Gauge Figure 28 (continued) 132
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Christmas Island Intercomparisons I = Gauge I e) C\IE 800 E C) 600 Q) c w n1 400 +=i c Q) 0 200 CL vq 1992 1994 1996 1998 2000 Year Scatter of Potential Energies r = 0. 77 (0.85) w/o 600 r95% = 0 12 200 f) 0o 200 400 600 800 Tide Gauge Figure 28 (continued) 133
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RMS Amplitude of Meridional Mode 2 Velocity 110W 140W I = I I = I 100 e -N -200 300 0 0 -100 -.s N 200 -300 0 0 100 .. N -200 10 20 30 40 500 rms amplitude (mm/s) 170W 10 20 30 40 500 rms amplitude (mm/s) 165 E 0 1 0 20 30 40 50 rms amplitude (mm/s) 10 20 30 40 rms amplitude (mm/s) 156 E -TAO T/P COARE156E COARE157E 10 20 30 40 rms amplitude (mm/s) 50 50 Figure 29 Velocity intercomparisons Vertical profiles of rms amplitude inferred from TIP results and a first baroclinic vertical structure (blue curves) are compared with the analogous fits for the meridional veloc ity component at the in situ sites (black, red, green curves) The 156 site had three in situ measurements 134
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'-1996 1503 3 1'1.) 1995 1994 1993 50E 100E 150E160W110W 60W 10W 50E Longitude 100 Figure 30. Global picture of mode 2 PE variations from TIP PE estimates were made each 5 of longitude and for each TIP cycle (10 days) Note, however that these estimates are from a box that is 30 wide and 2 TIP cycles (20 days) long. Gray areas are land and white areas are where there were not enough data to attempt the fit. 135
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Potential Energies From T/P Exceeding 80o/o Null Test '-ca Q) >300 2000 1999 250 1998 200 1997 1503 3 1996 1995 1994 1993 50E 100E 150E 160W110W 60W 10W 50E Longitude 100 50 0 Figure 31. Global picture of mode 2 PE with significance testing As in Figure 30 except that PE values that are not significantly different from zero at the 80 % confidence level are removed. Note that the major zonal and temporal features remain in the global picture 136 ,_.,
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Temporal Structure of Mode 2 IGW Potential Energy dec 200 nov oct sep aug jul jun may apr mar feb ..c. c Jan dec nov oct sep aug jul jun may apr mar feb jan 50E 100E 150E 160W110W SOW 10W 40E Longitude 180 160 140 120 100 3 3 8Q N 60 40 20 0 Figure 32 Global seasonal variations of mode 2 PE PE series from TIP are d ivided into monthly bins and averaged, much like the tide gauge spectra. As before, the PE seasonal series are plotted twice in time to aid visualization of seasonal signals 137
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Tarawa Island seasonal signal vs T/P jan marmay jul sep nov jan marmay jul sep nov Kanton Island seasonal signal vs T/P ;;z j jan marmay jul sep nov jan marmay jul sep nov Christmas Island excluding 1994 tide data i 0 jan marmay jul sep nov jan marmay jul sep nov Figure 33 Comparison of TIP seasonal structure with analogous tide gauge series PE series at the tide gauges were divided into monthly bins and averaged as was done for the TIP series. Shown are comparisons for the three Pacific tide gauges seen in Figure 28. Christmas Island results are shown with and without the 1994 event shown in green in panel e of Figure 28 138
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-w \0 Zonal structure of temporal mean PE 160 140 120 100 w a. 80 60 red is 95o/o null test 20 50E 100E 150E 160W 110W 60W 10W 40E Longitude Figure 34. Zonal modulation of mode 2 PE from TIP. TIP energies were averaged in time to obtain the zonal structure. The PE at the 80% confidence level for the null test is also shown in order to investigate zonal structures that may be due to noise.
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...... w 0.. Zonal structure of temporal mean PE 120 100 80 60 160E 170E 180 170W 160W 150W 140W 130W 120W 110W Longitude Figure 35 Zonal PE structure in Pacific from TIP. As in the previous figure, but only the Pacific region is s hown. The mid-Pacific structure was divided up inot three regions : a western region of high PE (blue), a middle region of low PE (black) and an eastern reg i on of high PE (pink)
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...... ...... Distribution of k fo r Pacif i c PE 15 cg ro -1 0 0 eft. 5 0 1 0 5 0 0 5 knd PE: 160 E-180 PE: 175W-155 W PE : 1 50W-1 20W 15 l c9=0 15 l c9= 0 1 5 1 l c9= 0 i i i 10 1 0 1 0 0 0 0 0 (/?. 0 5 5 5 0 0 0 -1 0 5 0 0 5 1 0 5 0 0 5 1 0 5 0 0 5 Figure 36 Wavenumber distri b uti o n in the Pacific Only th o se wavenumbers that correspond t o p o tentia l energies exceed i ng the 9 5 % confidence level are u sed and these are sh own in bl u e The red bars s h ow the n um ber of wavenumbers that fall into that bin when the data input into the fit was fr o m the x2 noise model. The upper panel is for the entire P acific regi on ; th e lower panels are for the s ub regio n s i d entified o n Figure 35.
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REFERENCES Bendat, J. S., and Allan G. Piersol, Random Data ; Analysis and Measurement Procedures, John Wiley & Sons New York 566 pp, 1986 Blandford, R., Mixed gravity-Rossby waves in the ocean, Deep Sea Res. 13 941 961, 1966 Bloomfield, P ., Fourier Analysis of Time Series : An Introduction John Wiley & Sons New York, 258 pp, 1976 Cane, M A., and E S Sarachik Forced baroclinic ocean motions I. The linear equatorial unbounded case J. Mar. Res., 34, 629-665, 1976. Chiswell S.M., D Randolph Watts, and M Wimbush, Inverted echo sounder observations of variability in the eastern equatorial Pacific during the 19821983 El Nino, Deep Sea Res., 34, 313-327, 1987. Chiswell, S. M ., and R. Lukas Rossby-Gravity waves in the central equatorial Pacific Ocean during the NORP AX Hawaii-toTahiti Shuttle Experiment, J. Geophys Res., 94, 2091-2098, 1989. Clarke A. J., The reflection of equatorial waves from oceanic boundaries, J Phys Oceanogr., 13, 1193-1207 1983. CornejoRodriguez M ., and D. B Enfield, Propagation and forcing of high-frequency sea level variability along the west coast of South America, J. Geophys Res., 92, 14,323-14,334, 1987 Eriksen, C ., Evidence for a continuous spectrum of equatorial waves in the Indian Ocean J. Geophys Res., 85, 3285-3303, 1980 Eriksen, C ., Equatorial wave vertical modes observed in a western Pacific island array J Phys Oceanogr ., 12 1206 1227 1982. 142
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