JULY 1997 SANCHEZ ET AL. 1371

The Calculation of the Dynamic Sea Surface Topography and the Associated Flow Field from Altimetry Data: A Characteristic Function Method

BRAULIO V. S ANCHEZ Space Geodesy Branch, NASA/Goddard Space Flight Center, Greenbelt, Maryland

WILLIAM J. CUNNINGHAM AND NIKOLAOS K. PAVLIS Geodynamics Group, Hughes STX, Greenbelt, Maryland (Manuscript received 2 October 1995, in ®nal form 4 December 1996)

ABSTRACT The quasi-stationary sea surface topography (QSST) and associated oceanic circulation is determined by means of a characteristic function technique. The method was originally implemented in an ideal simpli®ed case. The present application involves a 4Њϫ4Њgrid in spherical coordinates approximating the boundaries of the main basins. The data ®eld is provided by the ®rst year of altimetric data from the TOPEX/POSEIDON mission. The method requires the numerical determination of the eigenfunctions spanning the streamfunction ®eld and the associated characteristic functions from the balance equation. The former yields the ¯ow ®eld and the latter the surface height distribution, or QSST. These functions are determined by the geometry and topography of the ocean basins and satisfy the linear steady-state dynamical equations. They are de®ned within the basins only and avoid the problems encountered when using functions de®ned over the entire sphere. The velocity ®eld can be computed over the entire ocean area, including the equatorial regions. The coef®cients of the height functions have been determined by ®tting the surface height ®eld provided by TOPEX altimetry. Seasonal variations have been computed by subtracting a 32-cycle solution from each of the four 8-cycle seasonal solutions.

1. Introduction 1) the determination of the quasi-stationary surface to- This paper presents the results of the application of pography (QSST) a different method to an existing problem. It is not in- 2) the determination of the ¯ow velocity components tended to diminish or demerit the efforts of previous from the obtained QSST. investigations in any form. In what follows, a brief ac- The ®rst efforts by investigators were concerned count is given of some of the work previously done in mainly with step 1). Mather et al. (1978) estimated a the ®eld; it is not intended to be an exhaustive list. few zonal spherical harmonics of the QSST from mean Altimetric measurements from a satellite yield the sea sea surfaces derived from altimetry and independent surface height (SSH) topography on a global basis. The geoid measurements. Tai and Wunsch (1983) deter- SSH topography represents the sum of the marine geoid, mined the QSST in the Paci®c basin with a 20Њ reso- the departure of the ocean surface from the geoid due lution using a geoid based on earth's gravity ®eld model to the near-surface ocean circulations, and the variations GEM-L2 (Lerch et al. 1982a, 1985). Cheney and Marsh related to the determination of the satellite orbit, ocean (1982) used GEOS-3 and Seasat data with the GEM-L2 , instrument errors, errors due to the environment, geoid. Engelis (1985) used the GEM-L2 geoid and a etc. Detailed discussion of the many aspects of satellite mean sea surface of Rapp (1982, 1985) to produce a remote sensing and its applications to are model for QSST in spherical harmonics complete to a given by Stewart (1985) and Stewart et al. (1986), degree and order of 6. Tai and Wunsch (1984) combined among others. the geoid information from GEM-L2 with that from the The determination of the large-scale ocean circulation gravity ®eld model PGS-S4 (Lerch et al. 1982b). Tai from satellite altimetry involves two distinct steps: (1988) used the geoid based on the gravity ®eld model GEM-T2 (Marsh et al. 1987, 1988). Uncertainties in the geoid and the radial position of the satellite are the main Corresponding author address: Dr. Braulio V. Sanchez, Space Ge- errors in the QSST determinations by the investigations odesy Branch, Goddard Space Flight Center, Greenbelt, MD 20771. above. Tapley et al. (1988) demonstrated that altimeter E-mail: [email protected] data (Seasat) can be used in a joint solution to simul-

᭧1997 American Meteorological Society

Unauthenticated | Downloaded 09/28/21 02:21 PM UTC 1372 JOURNAL OF VOLUME 27 taneously estimate the QSST (degree and order 6) and geometry and topography of the basins; they need to be the gravity ®eld (geoid). computed only once. They are de®ned within the basins Recently, Marsh et al. (1990) used Seasat altimetry only and should be free of the dif®culties encountered to solve simultaneously for the gravity ®eld, the dy- when using functions de®ned over the entire sphere. namic topography, and the satellite position, and a mod- Implicit in the analysis is the capability of computing el of the QSST complete to degree and order 10 was the total ¯ow ®eld, not just the geostrophic component. obtained. Nerem et al. (1990) used Geosat altimetry to The ocean currents can be computed over the entire derive models of the ocean circulation. Hwang (1995) basins including the equatorial regions. used Geosat altimeter data to estimate the QSST; he made use of traditional global spherical harmonics, but 2. Mathematical preliminaries also implemented solutions in terms of orthonormal functions derived from spherical harmonics. The laws of conservation of mass and momentum The investigations mentioned above have expressed provide the basic equations, assuming steady-state con- the QSST in terms of spherical harmonics. This is to ditions, the wind-driven linear shallow-water equations be expected because the geoid uncertainties are given yield in spherical harmonics. Some problems arise with this (choice, such as high correlations between degree 1 terms 2⍀ϫVϭϪg١␨ϩ(1/␳h)␶ Ϫ cV (1 (V ϭ 0, (2´١ of the SST and the parameters of orbital corrections of 1 cycle per revolution terms, aliasing due to lack of orthogonality and edge smoothing (Gibbs phenome- where non). The spherical harmonic representation also creates ⍀ angular velocity vector problems in representing the current ¯ow near the con- V ¯uid velocity vector tinental boundaries since the current vectors do not obey g gravitational acceleration the proper boundary conditions. Another possible area ␨ surface height ¯uctuation of improvement in the investigations lies in the fact that ␳ density of the ¯uid the derivation of the ¯ow velocity components has been h depth of wind-driven circulation based on the geostrophic relations, which assume a bal- ␶ wind stress vector ance between the Coriolis and the pressure gradient c linear friction coef®cient. forces. This assumption is valid for most of the ocean areas, but breaks down in the equatorial regions. The For this investigation, the depth of the wind-driven tropical ocean circulation plays an important role in the circulation is assumed to be a constant. The large-scale atmosphere±ocean interaction that determines short- wind stress is assumed to be nondivergent; (␶ ϭ 0. (3´١ .term climate on a global scale The objective of this investigation is to test a different method (Rao et al. 1987) for the assimilation and anal- The velocity ®eld can be expressed in terms of a ysis of satellite altimetric data in order to produce mod- streamfunction (١⌿); (4a els of the global- and basin-scale ocean circulation. The V ϭ (1/h)(e ϫ technique was originally tried in an ideal rectangular r basin on a beta plane north of the equator with a syn- therefore, (␭)e , (4bץ/⌿ץ)(␪)e Ϫ (1/hr sin␪ץ/⌿ץ)(thetic data ®eld. This work will extend the technique to V ϭ (1/hr more realistic applications and will use a data ®eld pro- ␭ ␪ vided by TOPEX altimetry. where er, e␭, e␪ are unit vectors in the radial, north± The theory requires the numerical computation of the south, and west±east directions and ⌿ is a streamfunc- characteristic functions used in an expansion of the tion. The angle ␪ in Eq. (4b) stands for colatitude, so streamfunction ®eld, as well as the functions spanning that the singularity occurs at the poles and not at the the surface height ®eld. The coef®cients of both ex- equator; ␭ is longitude; and r denotes the radius of the pansions are the same and they can be estimated in a earth. Applying the divergence operator to Eq. (1) and least squares sense from a ®eld of satellite altimetry making use of Eqs. (2) and (3) yields data. The ¯ow ®eld components (currents) are obtained (⌿ϭgٌ2␨, (5´١ f´١ directly from the streamfunction ®eld. The method can be used in a simulation mode if a where f is the Coriolis parameter. wind ®eld is available from data. The theoretical forced The streamfunction can be expanded in terms of the -solutions can be used to provide simulated data to val- characteristic functions of the operator ٌ2; the imper idate the estimation procedure and to obtain an energy meability of the boundary requires it to provide a con- spectrum indicating which characteristic functions pre- tour of constant value for the streamfunction. The value dominate. is arbitrarily selected to be equal to zero in a simply The functions used in the expansion for the stream- connected domain. In a multiply connected domain, the function and surface height ®elds are determined by the boundary of the ``mainland'' can still represent a con-

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j j j stant contour with zero value, but the island boundaries surements; B ϭ {}B␥ ϭ␨(␪,␭), the height function represent contours of constant value that differ from matrix; P ϭ column (p1, p2, p3,´´´,pN) of the coef®- zero, and these values have to be determined by means cients. of line integrals around each of the islands. This problem The least squares solution to Eq. (10) is given by was addressed originally by Kamenkovich (1961), and P ϭ (BTB)Ϫ1BT␨. (11) more recently by Platzman (1979), in reference to the normal mode problem. The boundary conditions can be Given a ®eld of altimetry data for the QSST and a obtained from single-value considerations for the sur- numerical solution for the ␨␥ and ␺␥ from Eqs. (9) and face displacement, but they also follow naturally from (6), Eq. (11) will yield a least squares solution for the the self-adjointness requirement for the operator: coef®cients PÅ . Substitution into Eqs. (7) and (8) yields the total streamfunction and height ®elds over the entire 2 ⌿␥␥␥ ϭϪ␮⌿ (6a) basin area. Equations (4) give the velocity componentsٌ on the main boundary for the ¯ow expressed by Eq. (1), subject to the con- ditions given by Eqs. (2) and (3), not just the geostrophic ⌿ϭ␥ 0 (6b) components. on island boundary Equation (9a), including the associated boundary con- ditions, does not guarantee the orthogonality of the height function basis. The ®rst function can be used to (n) ds ϭ 0 (6cץ/ ⌿ץ) ⅜ ͵ ␥ initialize a Gram±Schmidt orthogonalization procedure, ci that will yield an orthogonal set. If the orthogonal set i ⌿ϭ␥␥b, (6d) is used in the estimation of the coef®cients by means

i of the equivalent of Eq. (11), it is necessary to develop where theb␥ are constants to be determined and the line the proper transformation that will yield the coef®cients integral is taken around the island boundaries; n denotes to be used in the velocity computations by means of the direction normal to the boundary. The ␮␥ are the Eqs. (7) and (4); let characteristics values associated with the vectors ␺␥. ␨ ϭ OQ. (12) The vectors ␺␥ form an orthogonal set. The total streamfunction ®eld is then given by Establishing the equality between Eqs. (10) and (12) ␺ ϭ p ␺ . (7) and recalling that matrices B and O are not necessarily ͸ ␥␥ ␥ square, we solve The height ®eld can be expanded in terms of the P ϭ (BTB)Ϫ1BTOQ, (13) expansion coef®cients p␥: where the Q are obtained from surface height data by ␨ ϭ p ␨ . (8) estimation, with the orthogonal basis, and the P are to ͸ ␥␥ ␥ be used in Eqs. (7) and (4) to compute the velocity ®eld. Substitution of Eqs. (7) and (8) into Eq. (5) yields 3. Numerical solution of the height and 2 ١␺␥, (9a) streamfunction equations f´١ gٌ ␨␥ ϭ with the following boundary conditions: The basic functions involved in the development of ␨ ϭ 0 on meridional boundaries (9b) the method are the streamfunctions obtained from the ␥ solution of Eqs. (6) and the height functions that satisfy Eqs. (9). These equations have been solved by means (␪) on zonal boundaries (9cץ/ ␨ץ)␪) ϭ gץ/ ␺ץ)f ␥ ␥ of ®nite differences expressed in spherical coordinates Equations (9) establish a relation between the expan- on a grid with a 4 degree angular resolution. As shown sion modes of the height ®eld and those of the stream- in Fig. 1, the grid does not include the Arctic Ocean, function ®eld. The meridional boundary condition is and the latitude limits are 67.2Њ and Ϫ72.8Њ. These limits postulated by noting the similarity in ␭ dependency for were imposed by the data coverage provided by TO- the operators on both sides of the equation. The zonal PEX/POSEIDON. boundary condition follows from the north±south mo- There are 1848 streamfunction points in the ocean mentum equation using the assumption of zero wind basins: the main boundary is de®ned by the continents, stress normal to the zonal boundaries. This is a reason- Antarctica is de®ned by 100 streamfunction boundary able assumption for the global . points, Australia and the Indonesian archipelago by 46 Now suppose that M altimetric measurements of the points, and New Zealand by 12. These are the ``islands'' sea surface topography are available and that the ex- of the grid where boundary conditions speci®ed by Eqs. pansion of Eq. (8) is truncated at N modes, then (6a±c) must be satis®ed. The numerical solution of the eigenvalue problem ␨ ϭ BP, (10) expressed by Eq. (6a) was obtained by means of the where ␨ is the column (␨1, ␨2, ␨3,´´´,␨M) of data mea- Lanczos method (Lanczos 1950). The depth was arbi-

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FIG. 1. The spherical coordinate grid used to solve the streamfunction and height function equations. Angular resolution is 4 deg. Latitude limits are 67.2Њ and Ϫ72.8Њ.

trarily set to a constant value of 1000 m. The nature of 4. The preparation of the data the solution is not affected by the particular constant The geocentric sea surface height to be used as data depth chosen, as long as the values of the coef®cients is de®ned as the difference between the geocentric po- in the expansions given by Eqs. (7) and (8) are obtained sition of the satellite and the altimeter range; that is, by estimation from a data ®eld; that is, the values of SSH ϭ ORBIT Ϫ ALTIMETER RANGE. The satellite the coef®cients will vary as a function of the chosen orbit used is that computed at the Goddard Space Flight depth, but the ®nal result remains constant. The case of Center from laser ranging and DORIS data, with an variable depth will be the subject of future investiga- estimated accuracy in the range 3±4 cm. The altimeter tions. range is subject to environmental and geophysical cor- The solution of Eqs. (9) for the height functions was rections, which include the solid earth and ocean tides obtained by means of mathematical subroutines avail- (Schwiderski's), the tropospheric (wet and dry), iono- able in LAPACK, which is the equivalent of the math- spheric, and inverted barometer corrections. The geoid ematical library LINPACK with CRAY vectorization, undulations have been modeled by a composite model the particular subroutines used employ the LU factor- to degree 360. This model is based on Goddard's gravity ization method. ®eld model JGM-2 up to degree 70 and Ohio State Equation (9a), including the associated boundary con- University's gravity ®eld model OSU91A from degree ditions, does not insure the orthogonality of the solution. 71 to 360. An orthogonal basis was created by means of the Gram± After the corrections, alongtrack averages were Schmidt process. Both orthogonal and nonorthogonal formed by means of line ®ts over 10-s intervals, creating functions were tried in the analysis of the data with no a ®le of normal points. A ®nal editing was applied by signi®cant differences in the results, although the or- rejecting points where the absolute values of sea surface thogonal basis exhibited a more robust numerical be- heights exceed 3 m, therefore avoiding areas with large havior. If the orthogonal solution is adopted, the ex- geoid omission errors. Points where the alongtrack slope pansion coef®cients must be properly transformed be- of the topography exceeds 10 s of arc were also rejected, in order to exclude areas with very abrupt changes due fore computing the ¯ow ®eld from Eqs. (7) and (4). The to bathimetry, such as trenches. Finally, any points transformation is given by Eq. (13). where the rms of line ®t exceeds 15 cm are also excluded The space structure of some of these functions is to avoid outliers. As an indication of the quantity of shown in Figs. 2 and 3. Positive and negative contours data included in the estimation, consider the 10-day re- of equal value are represented by solid and dotted lines. peat cycle number 17, the oceanwide number of points The streamfunction contours are also streamlines of the is 50 104. The time and location of the normal point ¯ow, since the land boundaries are de®ned as contours were de®ned by the observation time closest to the 10-s of constant streamfunction, and the ¯ow will always be bin's midpoint. tangential to them. The space structure of both the The design matrix B in Eq. (10) was formed by in- stream and height functions becomes more complex terpolating the height functions to the observation lo- with increasing wavenumber, approaching a cell struc- cation (␪j,␭j), using the following simple bilinear inter- ture de®ned by the individual grid points. polation scheme:

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FIG. 2. Streamfunction eigenfunction (a) number 1 and (b) number 50. Positive and negative contours of equal value are represented by solid and dotted lines. The bold solid lines denote the zero value contours.

(C␨ )swϩ (C␨ ) nwϩ (C␨ ) neϩ (C␨ ) se OSU91A gravitational model. This weighting scheme B(j, ␭) ϭ ␥␥␥␥, was chosen because the averaging into 10-s intervals sw nw ne se ΂΃CϩCϩCϩCwill be inaccurate where the sea surface slope is chang- where ing rapidly, that is, over trenches.

sw C ϭ (1 Ϫ S␭␪)(1 Ϫ S )

nw 5. Solutions based on TOPEX altimetry C ϭ (1 Ϫ S␭␪)S The ®rst question to be addressed by the investigation C ne ϭ SS ␭␪ is associated with Eqs. (10) and (11), that is, the de- se termination of the number of functions to be included C ϭ S␭␪(1 Ϫ S ) in the solution. To provide an answer, various data ®ts j S␭ ϭ(␭Ϫ␭)/4 were performed. Each ®t used a different number of coef®cients. The data covered a span of 36 repeat cycles, S ϭ (␪j Ϫ ␪)/4. ␪ which is close to a year. The results of the various ®ts Note that the C were set to zero if the corresponding to the altimetry data are shown in Fig. 4, with the rms grid point is on land, as determined by the grid. Bound- of the solution plotted as a function of the number of ary points are not considered as land. Here ␭ and ␪ are coef®cients. A rigorous analysis requires a simultaneous the closest meridian and parallel less than or equal to solution for the parameters representing the geoidal the observation ␭ j and ␪ j, respectively. height, as well as a complete analysis of the associated The weighting of the 0.1-Hz normal points was as covariance matrix. Such analyses will be done in a fu- follows: ture extension of this work; the present analysis indi- cates that much of the information in the data is retrieved ␴ ϭ [0.202 ϩ 0.10(N )2]1/2(m), hf by the ®rst 75 functions. Solutions based on this many where Nhf is the high-frequency geoid undulation con- functions (or less) will be referred to as long wave- tribution from N ϭ 71 to 360 as computed from the length.

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FIG. 3. Height function (a) number 1 and (b) number 50. Positive and negative contours of equal value are represented by solid and dotted lines. The bold solid lines denote the zero value contours. a. Various solutions based on cycle 17 cording to Eq. (4b); another set of solutions was ob- tained through the use of the geostrophic relations given To further investigate the methodology, cycle 17 of below: TOPEX/POSEIDON was used to provide the altimetry dataset used to obtain various solutions. (␪ץ/␨ץ)(The associated velocity ®elds were computed ac- u ϭ (g/ fr ,(␭ץ/␨ץ)(␷ ϭϪ(g/fr sin␪

where the height ®eld ␨ is computed from Eq. (8). The results from the two approaches are shown in Figs. 5±8. Figure 5 shows the associated velocity ®eld obtained from a solution using 50 coef®cients. Figure 5a presents the results for a velocity ®eld computed from Eq. (4b), and Fig. 5b is obtained by using the geo- strophic relations for the velocity. In this latter case, it is necessary not to include the grid points close to the equator in order to avoid the excessive numerical mag- ni®cation introduced in the geostrophic equations. This is accomplished by not including the grid points be- tween 3.2Њ and Ϫ4.8Њ latitude. The results corresponding to solutions based on 75 functions and 440 functions are presented in Figs. 6 and 7. The associated velocity ®eld calculated from the ®rst FIG. 4. The recovered weighted rms of the solution plotted as a 50 coef®cients of a solution expanded in terms of 440 function of the number of coef®cients. functions is presented in Fig. 8.

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FIG. 5. (a: top) The velocity ®eld computed from Eqs. (7) and (4b). Surface topography ®t with 50 functions to altimetry from cycle 17 of TOPEX/POSEIDON; (b: bottom) The velocity ®eld computed from the geostrophic equations. Surface topography ®t with 50 functions to altimetry from cycle 17 of TOPEX/POSEIDON.

From examination of panels a and b of Figs. 5±8, is expected from other physical considerations, and it is possible to conclude that both the velocity ®eld it should be considered a positive feature in the meth- computed according to Eq. (4b) and the geostrophic odology. velocity ®eld yield very similar results in the areas The character of the velocity ®eld changes consid- that do not include the equatorial band. This is what erably as the number of functions increases, cf. Figs.

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FIG. 6. (a: top) The velocity ®eld computed from Eqs. (7) and (4b). Surface topography ®t with 75 functions to altimetry from cycle 17 of TOPEX/POSEIDON; (b: bottom) The velocity ®eld computed from the geostrophic equations. Surface topography ®t with 75 functions to altimetry from cycle 17 of TOPEX/POSEIDON.

5 and 7, although the circumpolar current seems to Comparison of Figs. 5 and 8 indicates that the co- be a fairly constant feature. A solution based on 440 ef®cients are not highly correlated since both solutions coef®cients includes functions of very short wave- produce very similar velocity ®elds. length, and lacking an associated covariance analysis, The geostrophic velocity ®eld associated with a so- it should be considered of experimental interest only. lution obtained by ®tting the cycle 17 dataset with a

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FIG. 7. (a: top) The velocity ®eld computed from Eqs. (7) and (4b). Surface topography ®t with 440 functions to altimetry from cycle 17 of TOPEX/POSEIDON; (b: bottom) The velocity ®eld computed from the geostrophic equations. Surface topography ®t with 440 functions to altimetry from cycle 17 of TOPEX/POSEIDON. spherical harmonic expansion to degree and order 6 has the characteristic functions (Fig. 5b). Simple considera- been evaluated, Fig. 9; in other words, the height ®eld ␨ tions of mass conservation and coast impermeability used in the geostrophic equations has been expanded in would seem to indicate that long wavelength solutions spherical harmonics. This ®eld has notable differences should yield velocity ®elds similar to those based on the with respect to a solution based on a similar number of new methodology. The velocity ®elds in both ®gures

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FIG. 8. (a: top) The velocity ®eld computed from the ®rst 50 coef®cients, Eqs. (7) and (4b). Surface topography ®t with 440 functions to altimetry from cycle 17 of TOPEX/POSEIDON; (b: bottom) The velocity ®eld computed from the geostrophic equations. Surface topography given by ®rst 50 functions of ®t with 440 functions to altimetry from cycle 17 of TOPEX/POSEIDON. were computed by means of the geostrophic equations, harmonic expansion and those based on the streamfunc- demonstrating the advantages of the characteristic func- tion expansion (Fig. 5a), then the superiority of the new tions for the representation of the surface height ®eld. If approach is even more evident, since now the equatorial the comparison is made between the results of spherical band is also included in the solution.

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FIG. 9. Velocity ®eld computed from the geostrophic equations. Surface height (not shown) from a spherical harmonic expansion to de- gree and order 6. Data ®eld provided by cycle 17 of TOPEX/POSEIDON. b. Long wavelength seasonal variations sonal ``deltas'' in the sea surface height and velocity ®eld with respect to an average over the four seasons. A solution, using the ®rst 50 functions and Eq. (4b) These ``deltas'' correspond to a difference of averages: to compute the velocity ®eld, is shown in Fig. 10. The (8-cycle average)±(32-cycle average). surface height values are in the Ϫ1.5±1.5-m range, with Figure 11 shows the fall seasonal variations. The most Ϫ1 a maximum velocity of 0.359 m s . The data span noticeable features are the pronounced positive sea sur- covered cycles 1±19, 21±30, and 32±34, for a total of face heights in the North Atlantic and North Paci®c, 32 cycles. Cycles 20 and 31 did not provide TOPEX with values in the range of 10 cm. The south circumpolar altimetry and were not included. region exhibits two highs and two lows, in what could Seasonal solutions were computed by choosing the be interpreted as a 3/2 wave. Both and appropiate subsets of the data. The combined normal Kuroshio regions indicate an intensi®cation in the cir- equations matrices were formed and solved for the co- culation pattern, as well as the gyre in the Indian Ocean ef®cients of the height function/streamfunction ®elds. where the maximum change in velocity occurs: 3.2 cm The autumn solution was based on cycles 1±8, corre- sϪ1. sponding to the period September±December 1992. The Figure 12 portrays the winter seasonal variations. The winter solution encompasses cycles 9±16, from Decem- North Atlantic and Paci®c are now areas of low sea ber 1992 to February 1993. The spring solution uses surface heights, with the lowest values occurring in the cycles 17±19 and 21±25, which covers February±May Philippine Sea in the Ϫ10 cm range. The south circum- 1993. Finally, the summer solution combines cycles 26± polar region is slightly high all around. The Indian 30 and 32±34, for the period May±August 1993. shows a weakening of the average circu- Seasonal variations were computed by subtracting the lation with a maximum velocity change of Ϫ4.2 cm sϪ1. sea surface height obtained from the 32-cycle solution Figure 13 exhibits the spring seasonal variations. The from each of the four 8-cycle seasonal solutions men- most evident feature is the distribution of high sea sur- tioned above. The velocity ®elds corresponding to the face heights in the Southern Hemisphere and low sea variations were computed by subtracting the corre- surface heights in the Northern Hemisphere, with some sponding zonal and meridional components of the ve- minor exceptions (i.e., west of the Philippine Sea). The locity; the resulting ⌬us and ⌬␷s were combined to ob- highs and lows are not as pronounced as in other sea- tain sons, with values in the Ϯ5 cm range. The maximum Ϫ1 ⌬V ϭ⌬ue ϩ⌬␷e. velocity change is Ϫ2.5 cm s in the Indian Ocean. ␭ ␪ The Gulf Stream and the Kuroshio exhibit a weakening The results are shown in Figs. 11±14. They portray sea- in the circulation pattern.

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FIG. 10. Sea surface topography and associated velocity ®eld computed from Eqs. (7) and (4b). Surface topography ®t with 50 functions to altimetry from cycles 1±19, 21±30, and 32±34 of TOPEX/POSEIDON.

FIG. 11. Fall seasonal variations. Differences between a solution based on cycles 1±8 of TOPEX/POSEIDON and the solution shown in Fig. 10.

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FIG. 12. Winter seasonal variations. Differences between a solution based on cycles 9±16 of TOPEX/POSEIDON and the solution shown in Fig. 10.

FIG. 13. Spring seasonal variations. Differences between a solution based on cycles 17±19 and 21±25 of TOPEX/POSEIDON and the solution shown in Fig. 10.

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FIG. 14. Summer seasonal variations. Differences between a solution based on cycles 26±30 and 32±34 of TOPEX/POSEIDON and the solution shown in Fig. 10.

Figure 14 shows the summer seasonal variations. The teristic functions, indicates that the latter yields more Indian Ocean is the most energetic, exhibiting the largest realistic results for the ¯ow in the proximity of the highs and lows in the Ϯ10-cm range. In fact, it appears boundaries. This is true even if the geostrophic relations to accommodate a complete wavelength from north to are used to compute the velocities in both cases. south. The highest velocity change of 4.6 cm sϪ1 occurs The computed long-wavelength seasonal variations in the intensi®cation of the Indian Ocean gyre. The south (Figs. 11±14) are in general agreement with the circumpolar region is now a belt of low values. variations based on TOPEX/POSEIDON data, as re- ported by other investigators (Fu et al. 1994; Stammer 6. Conclusions and Wunch 1994; Tapley et al. 1994; Knudsen 1994; Cheney et al. 1994; Nerem et al. 1994). Generally, ac- In its original formulation by Rao et al. (1987), the cepted features can be detected, such as the higher characteristic function method was tested in an ideal Northern Hemisphere variations (produced by the ir- rectangular basin in a beta plane north of the equator, regular distribution of the continents), the very pro- with a ®eld of synthetic data. The present application nounced semiannual cycles of the Indian Ocean (due to involves a 4Њ grid in spherical coordinates approximat- the seasonal monsoon winds), and the zonal ¯uctuations ing the boundaries of the main ocean basins. The depth is assumed constant, but the presence of islands requires throughout the tropical regions of the oceans (connected the modi®cation of the boundary conditions for the with the seasonal cycles of the tradewinds). streamfunction problem. The data ®eld is very realistic Further re®nements and extensions of the method and state of the art, consisting of altimetry provided by could include the following: First, the 4Њ resolution is the TOPEX/POSEIDON mission. too coarse to allow proper resolution of features such The velocity ®elds computed with the data provided as the Gulf Stream, therefore, the adoption of a ®ner by cycle 17 demonstrate that the velocity formulation grid should be among the ®rst modi®cations to be im- of Eq. (4b) is consistent with the geostrophic approxi- plemented. Second, the method is not limited to constant mation in the nonequatorial areas while allowing the depth conditions; the case of variable bathimetry should velocity computation throughout the equatorial band. be investigated. Third, a more ambitious extension could Comparison of the velocity ®eld obtained from a surface try to accommodate vertical variations in the velocity height solution based on spherical harmonic functions ®eld, perhaps by a several-layer approach. Fourth, as and that obtained by a similar number of the charac- originally formulated, the technique allows forced so-

Unauthenticated | Downloaded 09/28/21 02:21 PM UTC JULY 1997 SANCHEZ ET AL. 1385 lutions to be calculated if a wind ®eld is available; there- , S. M. Klosko, C. A. Wagner, and G. B. Patel, 1985: On the fore, the creation of synthetic solutions based on avail- accuracy of recent Goddard Gravity Models. J. Geophys. Res., 90, 9312±9334. able realistic wind ®eld data should be of interest. Marsh, J. G., and Coauthors, 1987: An improved model of the Earth's Further applications of the TOPEX data should in- gravitational ®eld: GEM-TI. NASA Tech. Memo. TM-4019, 354 clude the estimation of multiannual variabilities by sub- pp. [Available from Space Geodesy Branch, Code 926, Goddard traction from appropriate long-term averages. Simul- Space Flight Center, Greenbelt, MD 20771.] , and Coauthors, 1988: A new gravitational model for the Earth taneous solutions involving the estimation of the geoid from satellite tracking data: GEM-T1. J. Geophys. Res., 93, and the sea surface topography should be of interest, 6169±6215. especially if different functional expansions are used for , C. J. Koblinsky, F. Lerch, S. M. Klosko, J. W. Robbins, R. G. each; that is, the geoid can be expanded in spherical Williamson, and G. B. Patel, 1990: Dynamic sea surface topog- harmonics, and the QSST in terms of the characteristic raphy, gravity, and improved orbit accuracies from the direct evaluation of Seasat altimetry data. J. Geophys. Res., 95, functions used in this investigation. 13 129±13 150. With respect to the value of the results from the stand- Martel, F.,and C. Wunsch, 1993: Combined inversion of , point of oceanography, an answer probably will have current meter data and altimetric elevations for the North Atlantic to come from a study such as that by Martel and Wunch circulation. Manuscr. Geodaetica, 18, 219±226. Mather, R. S., F. J. Lerch, C. Rizos, E. G. Masters, and B. Hirsch, (1993), which strives to combine sea surface estimates 1978: Determination of some dominant parameters of the global with models of the oceanic general circulation. Un- dynamic sea surface topography from GEOS-3 altimetry. NASA doubtedly, the more accurate the surface estimates, the Tech. Memo. TM-79558, 39 pp. [Available from Space Geodesy better the ®nal synthesis with a general circulation mod- Branch, Code 926, Goddard Space Flight Center, Greenbelt, MD el. 20771.] Nerem, R. S., B. D. Tapley, and C. K. Shum, 1990: Determination of the ocean circulation using Geosat altimetry. J. Geophys. Res., Acknowledgments. We thank the Space Geodesy 95, 3163±3179. Branch of the Goddard Space Flight Center and the , E. J. Schrama, C. J. Koblinsky, and B. D. Beckley, 1994: A TOPEX/POSEIDON Project for providing support for preliminary evaluation of ocean topography from the TOPEX/ this work. We thank the following individuals from the POSEIDON mission. J. Geophys. Res., 99(C12), 24 565±24 583. Platzman, G. W., 1979: Effects of multiple connectivity on a ®nite- Hughes STX Corporation: Brian Beckley for providing element barotropic model. J. Phys. Oceanogr., 9, 1276±1283. supporting software to access the altimeter database Rao, D. B., S. D. Steenrod, and B. V. Sanchez, 1987: A method of maintained at NASA/Goddard Space Flight Center and calculating the total ¯ow from a given sea surface topography. Dr. Mash Nishihama for preparing a number of graphics. NASA Tech. Memo. TM-87799, 19 pp. [Available from Space Geodesy Branch, Code 926, Goddard Space Flight Center, Greenbelt, MD 20771.] REFERENCES Rapp, R., 1982: A global atlas of sea surface heights based on the adjusted Seasat altimeter data. Dept. of Geod. Sci. and Surv., Cheney, R. E., and J. G. Marsh, 1982: Global ocean circulation from Ohio State University Rep. 333, 63 pp. satellite altimetry. Eos Trans. Amer. Geophys. 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