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 48348grid in spherical coordinates approximating the boundaries of the main ocean 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 208 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 tides, 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 oceanography 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- q1997 American Meteorological Society Unauthenticated | Downloaded 09/28/21 02:21 PM UTC 1372 JOURNAL OF PHYSICAL OCEANOGRAPHY 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 2V3V52g=z1(1/rh)t 2 cV (1) of the SST and the parameters of orbital corrections of =´V 5 0, (2) 1 cycle per revolution terms, aliasing due to lack of orthogonality and edge smoothing (Gibbs phenome- where non). The spherical harmonic representation also creates V 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 z surface height ¯uctuation of improvement in the investigations lies in the fact that r 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- t 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; term climate on a global scale. =´t 5 0. (3) 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 els of the global- and basin-scale ocean circulation. The V 5 (1/h)(e 3 =C); (4a) technique was originally tried in an ideal rectangular r basin on a beta plane north of the equator with a syn- therefore, thetic data ®eld. This work will extend the technique to V 5 (1/hr)(]C/]u)e 2 (1/hr sinu)(]C/]l)e , (4b) more realistic applications and will use a data ®eld pro- l u vided by TOPEX altimetry. where er, el, eu are unit vectors in the radial, north± The theory requires the numerical computation of the south, and west±east directions and C is a streamfunc- characteristic functions used in an expansion of the tion. The angle u 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; l 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 =´f =´C5g¹2z, (5) 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 ¹2; 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- Unauthenticated | Downloaded 09/28/21 02:21 PM UTC JULY 1997 SANCHEZ ET AL. 1373 j j j stant contour with zero value, but the island boundaries surements; B 5 {}Bg 5z(u,l), the height function represent contours of constant value that differ from matrix; P 5 column (p1, p2, p3,´´´,pN) of the coef®- zero, and these values have to be determined by means cients.