Measuring gravity field variability, the geoid, ocean bottom pressure fluctuations, and their dynamical implications



[1] To obtain insight into bottom pressure variability and requirements for assimilating gravity data into an ocean general circulation model, we use recent results from a constrained global model. Bottom pressure variability, as contrasted to surface elevation variability, is examined at the annual, semiannual, Chandler-wobble, and 4-month periods. Surface elevation and bottom pressure variability occurs on all spatial and temporal scales resolvable by our analysis, and patterns can be quite different. A barotropic index, developed to quantify variability character, suggests the importance of steric effects at longer periods. Variability decomposition into standing, eastward, and westward traveling waves indicates a strong zonal component for both surface elevation and bottom pressure, with bottom pressure showing a greater overall tendency toward eastward propagation, and surface elevation showing a greater overall tendency toward westward propagation. Bottom pressure variability signals are estimated to be recoverable from space to spherical harmonic degree 27 for the annual cycle and 14 for the semiannual and Chandler-wobble cycles. Mass exchange, arising from model approximations, is detectable to about degree 7. Polar motion excitation by ocean seafloor loading is found to be quantitatively important. Spurious oscillations encountered in using spherical harmonics to represent oceanic fields are suppressed using Lanczos factors.

1. Introduction

[2] The primary motivation behind this paper is that new, entirely novel, space-based measurements of the time-varying components of the Earth's gravity field [Tapley and Reigber, 2001] together with the very high accuracies and precisions which appear possible, raise a myriad of new and interesting challenges for understanding and using the measurements for oceanographic purposes. The focus of the paper is on obtaining a quantitatively useful estimate of the expected variability of the oceanic bottom pressure field and a complementary surface elevation field on relevant timescales, using an ocean circulation derived from a constrained ocean model. The variability estimate is used to examine detectability by the gravity mission (including sensitivity to mass leakage by general circulation models) and to infer the barotropic nature of the motions. Estimates of the ocean's role in polar motion excitation are also described.

[3] We attempt to summarize the main scientific issues in using accurate static and time-variable gravity to improve oceanographic calculations, and to make the discussion accessible to both oceanographers and geodesists. The geostrophic and hydrostatic equations are used to illustrate concepts, while practice will employ general circulation models. In a second paper (F. Condi and C. Wunsch, Self-gravitation, load and filtering effects in time-dependent ocean gravity fluctuations, manuscript in preparation, 2004) (hereinafter referred to as Condi and Wunsch, manuscript in preparation, 2004), we take up some of the more technical aspects of using observations of time-varying gravity, including the important effects of self-gravitation and seafloor loading. This current paper complements and extends results of Wahr et al. [1998] and Wünsch et al. [2001].

[4] With the advent of accurate satellite altimetry, physical oceanography and geodesy have come to have many overlapping problems. The most fundamental of these problems concerns the detailed determination of the mean geoid. This gravitational equipotential of the Earth is central to a description of the solid Earth, and appears as the principal reference surface for computing oceanic currents [Wunsch and Gaposchkin, 1980; Wunsch and Stammer, 1998; Fu and Chelton, 2001; Tapley and Kim, 2001]. Great progress has occurred in recent years in determining the geoid with much improved accuracy [Lemoine et al., 1998], although much remains to be done [Ganachaud et al., 1997; LeGrand and Minster, 1999; LeGrand, 2001] for the result to be fully useful for oceanographic purposes, and there should be considerable future progress, some of which is described below.

1.1. Geoid

[5] A number of different definitions, both physical and mathematical, of the geoid exist [see, e.g., Heiskanen and Moritz, 1967; Lambeck, 1988]. For present purposes, we can define it as the particular gravitational equipotential surface of the rotating Earth that would coincide with the sea surface if the ocean were brought to rest by removing all external forces, and the density were everywhere set equal to its global mean value. With this definition, we include the apparent contribution to the Earth's gravitational attraction of the centripetal force of rotation.

[6] Local gravity is the result of integrating over the entire mass of the Earth, both solid and fluid, including core, mantle, crust, atmosphere, ice, and ocean. Because all of these components are in nonsteady motion to some degree, the geoid itself is time dependent and its definition must provide a statement of the instant in time being used to compute it (an instantaneous geoid), or the period over which the time-dependent motions are to be averaged (a static geoid). Note that until very recently, an understanding of the time-dependent contributions to the geoid was uninteresting, because they were unmeasurable, except on geological timescales or extremely locally.

[7] The exception to the statement that time-dependent contributions to the geoid were generally unmeasurable is the tidal contribution. For our purposes we will arbitrarily define the geoid as not including the tides. These are defined in turn as the purely periodic (and hence predictable) components at the standard tidal frequencies given by the Doodson numbers [see, e.g., Cartwright, 1999]. Again, with the advent of precise altimetry, the tides are known with an accuracy which would have seemed extraordinary 10 years ago, and they are today best treated as a separate subject [see, e.g., Le Provost, 2001]. Gravity disturbances by tides are discussed by, for example, Munk and Macdonald [1960] and Lambeck [1988]. Again, we must introduce two exceptions. The permanent presence of the sun and moon deforms the Earth on average (the zero frequency tide), and this component is best regarded as part of the geoid in our definition. To the extent that the tidal motions generate narrow-band random components as in atmospheric and oceanic internal waves [see, e.g., Egbert and Ray, 2000], these are partially unpredictable, and although part of the geoid, we will regard them as primarily measurement noise processes to be understood.

[8] To a good first approximation, the Earth is a rotating ellipsoid, and in practice, the contribution to the geoid of a reference ellipsoid is computed once and for all, and subtracted from the actual geoid. The geoid “elevation” or “height” or “undulation,” N(θ, ϕ), where θ, ϕ are co-latitude and longitude, is measured with respect to this reference surface [Lambeck, 1988]. For convenience, sometimes the modifier “height,” etc., is omitted, and the terminology “geoid” is used in this slightly incorrect way.

[9] Elevations and depressions of the geoid are caused by mass distribution deviations from the idealized ellipsoid. A map of the geoid height, for example, that derived from the EGM96 geopotential model and the WGS84 ellipsoid [Lemoine et al., 1998] (Figure 1), shows most values on the order of tens of meters. However, high and low extremes of ±80 m or greater exist, and are associated with processes such as tectonic subduction zones and deep mantle structures. In contrast, oceanographic processes induce maximum steady changes of the sea surface relative to the geoid of less than 2 m and time variable changes of less than 1 m; time variable changes of the geoid itself due to these processes are on the order of millimeters or less.

Figure 1.

Geoid derived from EGM96 geopotential model with respect to WGS84 ellipsoid [Lemoine et al., 1998] developed to spherical harmonic degree 90. Scale is in meters. Contours represent 10-m intervals.

[10] Figure 2 is a schematic of the relationship of geodetically important quantities. Mass redistribution causes the geoid position to change and the sea floor to deflect. Both atmospheric and oceanic mass redistributions affect geoid height. In general, the variability of geoid position due to both the atmosphere and ocean about a static or time-averaged value (the variability of N′ about N) is less than the variability of the sea surface S′ about S, but not necessarily dynamically insignificant. Errors in determining S, S′ are currently larger than the variability of N, N′. A spacecraft attempting to measure S′ or N′ would observe these quantities relative to the ellipsoid, but an Earth-based observer would experience the deflection of the sea floor as well. Accounting for this unstable platform is one of the major difficulties in determining the source of sea level changes from Earth-based measurements.

Figure 2.

Schematic of geodetically important quantities. The primed variables represent instantaneous quantities: unprimed, static, or time averaged. S, S′, sea surface; N, N′, geoid height; e, ellipsoid; h, h′, geodetic height; d, d′, depth of the sea floor. In general, the variability of geoid position due to both the atmosphere and ocean (schematicallly the variability of N′ about N) is less than that of S′ about S, but not necessarily dynamically insignificant.

[11] From the oceanographer's point of view, knowledge of the sea surface elevation relative to the geoid determines the absolute circulation of the ocean [Wunsch and Gaposchkin, 1980]. At the time altimetry first became available, the best estimates of the geoid had errors exceeding tens of meters and could not be used oceanographically. Thus one could generate a greatly improved oceanic static geoid estimate simply by setting the geoid height to the sea surface height, where both heights are relative to the same reference ellipsoid [Wunsch and Gaposchkin, 1980; Wunsch and Stammer, 1998]; the residual error is then primarily that of the ocean circulation alone. (Although we will usually omit the modifier oceanic, geoid determination over land is important, too.)

1.2. Time Dependence

[12] The novel element in the discussion of the geoid is the possibility of determining major components of its time dependence through the launch of a dedicated gravity mission, in particular the Gravity Recovery And Climate Experiment (GRACE) [Wahr et al., 1998; Tapley and Reigber, 2001]. This recently launched pair of spacecraft is sensitive to the time-mean gravity field, but is able to measure much of the spectral range of the time-dependent components. Because the Earth's gravity field is determined by a volume integral over the mass of the Earth, and because that mass involves the ocean and atmosphere, to the extent that the temporal variability can be spatially localized, time-varying atmospheric and oceanic mass distributions are equivalent, through the hydrostatic relation, to measurements of ocean bottom pressure changes. GRACE thus holds out the startling possibility of the measurement of ocean bottom pressure changes from space. Below, we will further explore this idea and how these data might be used.

[13] Because ocean bottom pressure has been so difficult to measure for long periods of time [see, e.g., Brown et al., 1975; Wearn and Baker, 1980; Luther et al., 1990; Woodworth et al., 1999], there is comparatively little experience available in interpreting it and in using it in tests of dynamical ideas. Our strategy here is to use a newly available estimate of the time-varying ocean circulation [Stammer et al., 2002, 2003] obtained from combining 6 years of altimetric, hydrographic, and other data with twice-daily wind fields, and other forcing in a least-squares optimization. These results have been compared with the few direct ocean bottom pressure measurements available (not shown here, but comparable to the results of Wahr et al. [2002]) so as to establish the degree of model skill. The fields are then used to describe the expected global variation of ocean bottom pressure. Because little attention has generally been paid to the use of ocean bottom pressure observations, we will discuss some of the approximations made in models, particularly the Boussinesq approximation, that influence the calculated values. At the level of precision anticipated for GRACE, one must also examine approximations not normally of concern in the large-scale general circulation, including the effects of self-gravitation, and loading of the seafloor, with their corresponding influence on gravity (although these are well-known problems addressed in accurate tidal models).

2. Bottom Pressure, Surface Elevation, and the Geoid

[14] The ocean is forced by winds, atmospheric pressure, and buoyancy, and its response to the forcing is reflected in the sea surface elevation and seafloor pressure. As is well-known [Pedlosky, 1987; Wunsch, 1996], on timescales longer than about 1 day, on spatial scales greater than about 10 km, and at latitudes farther from the equator than a few degrees, the ocean tends, to a very high degree of accuracy, to be in hydrostatic, geostrophic balance. This balance state can be written as

equation image
equation image
equation image

Here, θ is co-latitude, ϕ is longitude, ρ is density, u,v are southward and eastward velocities, p is the pressure, g is gravity, and a is Earth radius (the notation is completely standard). The Coriolis parameter is f = 2Ωcosθ, where Ω is the Earth rotation rate, and z is positive upward from the local geoid N. The spherical Earth and geoid approximation being used is completely adequate for all known dynamical computations [see, e.g., Phillips, 1966], but it is not adequate as a kinematic description. For example, the polar radius is about 21 km shorter than the equatorial radii and the difference dominates altimetric data.

[15] The hydrostatic equation (3) is used to find the pressure at depth, by integration of the density field,

equation image

where η is the sea surface height above the geoid. The sea surface does not in general coincide with the geoid, but ∣η∣ is very small relative to the typical depth of integration and over this distance the density can be treated as constant. The surface pressure, ps, (evaluated on the local geoid), is approximately

equation image

and pa is the surface atmospheric pressure. (Spatial and temporal dependencies of g are ignored because, in this particular context, they are negligible.) In geodetic applications, the sea surface is defined with respect to the Earth ellipsoid, while oceanographers refer the sea surface to the geoid (the dynamic topography). Also, different ellipsoids are used in practice, and care must be taken during data preparation to refer both geoid and sea surface to the same reference surface.

[16] If pa vanished, then η would be a proxy for sea surface pressure. In practice, η is in part dependent upon pa, and one seeks to understand how the sea surface responds to fluctuating atmospheric pressure (reviewed by Wunsch and Stammer [1997]). In one limit, the so-called inverted barometer response asserts that a 1-mbar increase (decrease) in atmospheric pressure leads to almost exactly a 1-cm depression (elevation) in the sea surface. The inverted barometer is, however, a good assumption over much of the range of oceanic time and space scales over the world ocean [Fu and Davidson, 1995; Wunsch and Stammer, 1997]. From the point of view of an oceanographer studying bottom pressure, the validity of the inverted barometer effect is irrelevant: Uncompensated atmospheric load effects will generally penetrate to the seafloor and will produce real oceanic motions that one wishes to determine. (In this present paper, atmospheric loading is not considered.)

[17] At the sea surface it follows that

equation image
equation image

demonstrating the tight connection between the surface flow, us,vs, and the slope of the sea surface relative to the geoid. By cross-differentiation, one obtains, at depth, the “thermal wind” equations, and then integrating in the vertical,

equation image
equation image

where the integration constants, c1,2 are a function of the reference level, zref, and are as yet undetermined. Historically, the apparent inability to determine these constants was one of the great obstacles to a quantitative description of the ocean circulation [Wunsch, 1996].

[18] On the geoid,

equation image
equation image

Evidently, the ability of altimetry together with a sufficiently accurate geoid to determine η permits the use of equations (6) and (7) to evaluate the constants directly. This result is one of the great benefits that accurate altimetry and geodesy have brought to the study of the ocean circulation, up to the remaining errors in the system. (See the discussions by Ganachaud et al. [1997], Legrand and Minster [1999], and Losch et al. [2002]). The value of more accurate gravity data can be seen by examining equations (10) and (11). The surface velocities on the left-hand side of the equations are constrained in present practice by altimetry. Because the variability of the geoid is neglected when considering changes in the sea surface height with respect to the geoid, a more accurate static geoid will improve the estimate of these terms. The first terms on the right-hand side, containing vertical integrals of horizontal density gradients, have traditionally been estimated from hydrography. More accurate time-variable gravity will improve the estimate of these terms as discussed below.

[19] The relationship between bottom pressure and gravity requires the connection between the mass distribution and the geoid height. In geodesy, one considers a disturbing potential which, when added to the reference potential, gives the actual potential on the geoid. For an (idealized) equipotential reference surface with a potential Uo, the value of the potential Up at a point P above the reference surface is determined to first order by the Taylor series expansion about the idealized surface. Let T be a small disturbing potential, then [e.g., Heiskanen and Moritz, 1967] one has the Bruns formula,

equation image

where N is the height above the reference surface to point P and

equation image

is normal gravity evaluated on the ellipsoid. Actual gravity, g, appears in the dynamical equations, but normal gravity is used in equation (12).

[20] Disturbing potentials arise from many causes. Astronomical ones, such as those caused by the gravitational attraction of the Sun and Moon, are externally imposed. Anomalous mass distributions, both static and time variable in the atmosphere, ocean, and interior, are internally derived. With externally imposed disturbances, there are dynamical responses causing internal mass redistributions that must be accounted for.

[21] For a given disturbing mass distributed throughout a volume, V,

equation image

where r′ denotes a source point, r is an observation point, and ρ is the density anomaly. For a thin shell, with sheet mass density σ distributed over the surface S,

equation image

[22] The oceanic contribution to the geoid can be obtained in a first approximation using bottom pressure because it is calculated from equation (4) with the lower limit of the integral at the sea floor, z = −h. It then reflects the total mass variation of ocean plus atmosphere. Under the hydrostatic assumption, treating the ocean as a thin spherical shell, the sheet mass density can be approximated by

equation image

[23] Write the disturbing potential as the sum of a static or quasi-permanent part, equation image, and a time variable part, T′,

equation image

where the overbar represents a time average. Depending upon the relative accuracies, one can determine the mass distribution from T, or vice-versa [Wunsch, 1996]. In this context, the geoid measurements could be used to constrain ocean models.

3. Model Results

[24] Model skill at estimating bottom pressure is of fundamental importance both because the more accurate the model, the more effective will be data constraints on it, and because high-frequency motions can lead to aliasing errors [Stammer et al., 2000] in real spacecraft missions. At this stage, we seek some understanding of the nature of bottom pressure fluctuations in the ocean. To obtain quantitative estimates, we rely upon the estimates described by Stammer et al. [2002, 2003] obtained by combining a 2° × 2° horizontal resolution general circulation model of the ocean with 6 years (1993–1998) of altimetric and other data. The procedure is called state estimation or data assimilation [see, e.g., Wunsch, 1996] in which the variability estimate is constrained by a large number of oceanic observations. A related calculation has been described by Wünsch et al. [2001], who used an unconstrained model that necessarily has a reduced skill compared to the constrained one [Stammer et al., 2002, 2003]. The use of a constrained model to improve ocean angular momentum estimates is discussed by Ponte et al. [2001].

[25] Ocean general circulation models (OGCMs) are not formulated with the geostrophic equations, but, commonly, with a more general set of equations called the hydrostatic primitive equations [Marshall et al., 1997]. The “ECCO-model” (an acronym for Estimating the Circulation and Climate of the Ocean [see Stammer et al., 2002, 2003]) used here employs the Boussinesq approximation, conserving volume instead of mass. When bottom pressure (reflecting the total mass of the system) is integrated globally and expressed as a spatially uniform equivalent height of water about a mean distribution, a seasonal mass exchange superimposed on a drift over time in the model is evident. The drift of about 3.5 cm over 6 years or about 0.58 cm/year (Figure 3) appears to be about twice that seen in sea-level observations made from altimetry [Nerem, 1995; D. Chambers, personal communication, 2001] and is evidence of unphysical mass leakage in the model. Possible corrections to Boussinesq models are now widely discussed [e.g., de Szoeke and Samelson, 2002; McDougall et al., 2002; Losch et al., 2002]. See also Dewar et al. [1998] for a discussion of problems with the equation of state. A later calculation [Stammer et al., 2003] done with 9 years of data and constraints on model drift shows a reduction in the extent of the drift.

Figure 3.

(top) Total mass variation represented as a uniform change in sea level before expansion of the bottom pressure in spherical harmonics; seasonal cycle is superimposed on a drift of about 5 mm/yr. Note the similarity of the trend to that in global mean sea level determined from altimetry (bottom panel). (bottom) Ten-day average global mean sea level (GMSL) from the TOPEX/Poseidon altimeter, computed using method described by Nerem [1995]. A 5-mm bias has been added to data after February 1999 (cycle 236), based on comparisons with tide gauges (D. Chambers, personal communication, 2002).

3.1. Surface and Bottom Pressure Analysis

[26] In a constant density (homogeneous) hydrostatic ocean, surface and bottom pressure variations are identical. In a stratified ocean, the two can be very different. To contrast the surface and bottom pressure variability structures over the duration of the ECCO estimate, we produced variability estimates from the estimated circulation as RMS deviations about the 6-year (72-month) mean using 30-day averages of the model output. Let S denote surface or bottom pressure time series values; then,

equation image
equation image

where n denotes the nth time average. The time-averaged bottom pressure contributes to the static geoid. Analogous values equation image, η′ are defined for surface elevation.

[27] A spherical harmonic representation is used (see Condi and Wunsch (manuscript in preparation, 2004) for details) in the form

equation image

where αnm(t) are expansion coefficients, Ynm(θ, ϕ) are the fully normalized harmonics, and n and m are the degree and order. The coefficients are determined using a quadrature method [Swarztrauber, 1979] exploiting the orthogonality properties of the spherical harmonics. The method requires regular spacing of grid cells, does not allow for variable weighting of the specified data, and employs triangular truncation (∣m∣ ≤ n) of the spherical harmonics used to represent the desired function. Other truncations, such as rhomboidal (∣m∣ ≤ nK, where K is a constant), are used in meteorology. An advantage of the triangular truncation is that it provides uniform resolution on the sphere. In this method, the number of known values of the function required (for example S in equation (20)) is about twice the number of coefficients to be determined, and an exact representation of the function cannot be achieved (unless one imposes solvability conditions that will not be met with real observations). The solution for the coefficients is then equivalent to a least-squares fit [Swarztrauber, 1979]. Because the ocean does not cover the globe, difficulties arise in the use of spherical harmonics, including the discontinuities incurred at boundaries. Spurious oscillations are introduced at ocean-continent boundaries because of the abrupt change of properties. (Note that this problem does not occur with the discrete Fourier transform [e.g., Bracewell, 1986] for which discrete frequencies are assumed in the formulation.) There is no concept of a discontinuity in the discrete transform case; the number of unknown coefficients is exactly equal to the number of points used, and the expansion agrees exactly with the specified data. The spherical harmonic analysis of computing coefficients consists of a Fourier transform in longitude and an operation in latitude, sometimes called a Legendre transform [see, e.g., Swarztrauber, 1979]. While the transform in longitude is implemented as a discrete Fourier transform, there is no discrete latitudinal equivalent.

[28] For an accurate representation of a function in terms of spherical harmonics, the spurious oscillations need to be removed or greatly attenuated without affecting the part of the spectrum containing the desired signal. We use an extension of the Lanczos method [Lanczos, 1961] for reducing the effects of spurious oscillations in a Fourier series representation of functions. The objective is to eliminate the overshoot and accelerate the convergence of the series. A moving average is applied to the series so that the oscillating function is replaced by its mean over some interval. The process can be extended to a sphere by averaging a function, S(θ, ϕ), over a spherical element, sin θdθdϕ. Let Δθ and Δϕ be one half of the grid angle size; then the averaged function, equation image(θ, ϕ) is given by

equation image


equation image

Substituting the equation for the expansion of an arbitrary function in spherical harmonics into equation (21) gives

equation image


equation image

Δϕ is the longitudinal size of the grid cell, image = sin mΔϕ/mΔϕ, equation image represents the normalized Legendre function (Appendix A), and A and B are real coefficients. We obtain acceptable results by replacing Inm(θ) in equation (21) by imageequation image(θ),

equation image

or in compact form, using S, as before, to represent the function,

equation image

where equation imagenm(t) = image are the expansion coefficients; details are described by Condi and Wunsch (manuscript in preparation, 2004). From now on, we drop the equation image with the understanding that the variables represent the filtered fields.

[29] The filtering operation attenuates the high wavenumber structures yet maintains the longer wavelength structures that should be detectable by GRACE. For a concrete example, consider the “ocean function,” which is defined as 1 over the ocean and as 0 over land. Its expansion in filtered spherical harmonics to degree 90 is shown in Figure 4. Land-sea boundaries are sharpened by the operation, and the oscillations have been considerably suppressed. Noticeable differences in the spectral content after filtering begin around spherical harmonic degree 20. The filtering effect on bottom pressure is much the same (Figure 5).

Figure 4.

(top) Difference in filtered (see text) ocean function expanded to spherical harmonic degree 90 from the pre-expanded ocean function. (middle) Same as above but unfiltered. (bottom) Degree variances corresponding to above.

Figure 5.

(top) Filtered bottom pressure in Indian Ocean region. Units are centimeters of equivalent water thickness. (bottom) Same as above but unfiltered.

[30] To treat the land gaps, various approaches can be used. To minimize the oscillations, Wunsch and Stammer [1995], in their analysis of sea surface height, assigned time-averaged zonal mean values to land areas. Here we first remove a mean and trend in the bottom pressure at each point, and then set the land values to zero. The resulting fields are expanded in spherical harmonics, and filtered to remove the oscillations. Expansion of the model fields derived from the 2° × 2° model grid is done to degree Ne = 90, using equation (20), and results in (90 + 1)2 = 8281 separate time series, αnm(t), of length 6 years for each of the fields. Fields reconstructed in the spatial domain after this operation (S′ now represents the reconstructed field) contain the same information as in the spectral domain and Parseval's relation,

equation image

applies; the coefficients completely represent the new spatial function. The post-expansion and filtering operation fields are then used in the variability calculations. A convenient measure of the variance is given by ∫∫equation imageimage ϕ, n)2sin θdθdϕ, where NT represents the number of months in a period; in the following discussion we refer to this quantity as the global variance.

[31] In general, the filtered surface and bottom pressure fields are different in a number of respects from their pre-expansion counterparts, but appear to be faithful enough representations to use for estimating variability. The global variance reduction owing to the filtering is roughly 43% for the bottom pressure (Table 1) and 29% for the surface pressure, while the large-scale structure is maintained. A map of the ratios of variances of filtered post- to pre-expansion bottom pressure (not shown) shows some increase in variability in several coastal areas because of the discontinuities. Some phase differences do occur, with the largest appearing in or near areas of high bottom pressure variability, but overall, the representation seems satisfactory, and the filtering does well at suppressing the oscillations.

Table 1. Global Variance ∫∫equation imageimage ϕ, n)2 sin θdθdϕ (cm2)
72 months (pre-expansion)40142
72 months (post-expansion)23101
Annual total7.9648.67
Semiannual total1.304.41
Chandler wobble total1.268.06
4-month total0.390.92

[32] Coefficients αnm′(t) were Fourier transformed to equation imagenm(σ), and used to form the frequency periodograms [Wunsch and Stammer, 1995],

equation image

Summing over the order, m, one has the frequency dependent degree variances,

equation image

Frequency-degree variance plots of surface and bottom pressure exhibit strong annual and semiannual signals over a broad wavenumber band (not shown); less prominent ridges are evident at other frequencies, such as the 4-month cycle. A drop in power at high wavenumbers is due to the filtering. While these plots have been constructed from 30-day averages of model output fields, they are still complex and indicate variability in both surface and bottom pressure on all spatial and temporal scales resolvable by our analysis, with no gaps in the spectrum. The frequency resolution is 0.1667 cpy; wavenumber resolution can be interpreted according to the rule by which half-wavelengths in kilometers are given by 20,000/n, where n is the spherical harmonic degree. Annual, semiannual, 14-month (Chandler-wobble frequency), and 4-month cycles have been extracted and mapped both as amplitude and phase and as time series. The analysis for the 14-month cycle used a 70-month mean (rather than 72) with a frequency resolution of 0.1714 cpy.

3.2. Variability

[33] One of the problems to be solved in the interpretation of the GRACE data will be the accurate separation into component atmospheric, oceanic, and hydrologic parts. Barotropic and baroclinic ocean models will be used as tools, and an understanding of the geographic patterns and magnitude of variability as a function of period and length scale will be important for effecting this separation. Below we examine the variability and the character of the motions noting that the fields were obtained from the model under the inverted barometer assumption, and so pressure forcing effects are not included.

[34] Regions of high variability (Figure 6) in pb occur in the northwest Pacific basin, southern Pacific (northwest of the Drake Passage), Southern (southwest of Australia), and high-latitude Atlantic Oceans (compare to Fukumori et al. [1998], Ponte [1999], Stammer et al. [2000], and Tierney et al. [2000]). Regions of high variability (Figure 6) in gη occur in the Indian Ocean, subtropical Atlantic, equatorial Pacific, and northwest Pacific basin. Notable common regions of variability occur in the Southern Ocean, high-latitude Atlantic, and northwest Pacific basin. Few areas appear to have very low variability in both surface and bottom pressure. Global variance (Table 1) is greater for surface pressure (101 cm2) than for bottom pressure (23 cm2); the ratio (4.4) is somewhat higher for post-expansion fields than for those before expansion and filtering (3.6).

Figure 6.

Variability estimates produced from the constrained model as RMS deviations about a 6-year mean for the period from 1993 to 1998. Time series values used for this figure and those which follow consist of 30-day averages of the model output. (top) Surface pressure variability and (bottom) bottom pressure variability are shown. Note the different scales. Regions of large bottom pressure variability occur in the northwest Pacific basin, southern Pacific (northwest of the Drake Passage), Indian, and high-latitude Atlantic Ocean.

[35] For both surface and bottom pressure, the frequency at which the largest fraction of total energy over the 72-month record resides is the annual (Table 1), including 48% of the surface, but only 35% of the bottom pressure, variance. Geographic distributions of annual variability for both surface and bottom pressure (Figure 7) are visually similar to the 6-year (total) variability estimates (Figure 6). There are, however, notable differences that can been seen in a map of the fraction of the energy residing at the annual period (Figure 8). For surface pressure, the fraction reaches, in some regions, about 80%, but nowhere more than 91%. Regions of annual dominance are apparent in the northern Pacific, an area southeast of Australia, the midlatitude Atlantic, and the Indian Ocean. The annual component in the high-latitude North Atlantic, while significant, appears to be generally less than 50% of the total. The annual cycle of surface pressure distribution shows a strong hemispheric asymmetry with time, and is maximum in the Northern Hemisphere in the second half of the year (Figure 9).

Figure 7.

Annual component of the RMS variability. Units are centimeters. (top) Surface pressure. (bottom) Bottom pressure. Note scale change from previous figure.

Figure 8.

Ratio of annual period variance to that of the 72-month period. (top) Surface pressure. (bottom) Bottom pressure.

Figure 9.

Phase of the annual cycle component of variability relative to January. Units are months. (top) Surface pressure phase. (bottom) Bottom pressure phase. Month 1 is January for phase plots.

[36] Prominent regions of annual bottom pressure variability occurring northwest of the Drake Passage, in the high-latitude Atlantic, and in the Southern Ocean (southwest of Australia) (Figure 7), for the most part, have variability near 2-cm equivalent water thickness. The annual band in bottom pressure energy is largest at 50 to 80% in two midlatitude bands extending across the Pacific, in the Indian, and in the western North Atlantic Oceans (Figure 8). It also is a significant part of the total in the eastern North Atlantic Ocean. A notable exception is the region northwest of the Drake Passage where the ratio ranges from about 10 to a maximum of less than 30%. Temporally (Figure 9), the same nearly fixed phase occurs in the North Atlantic, a large region in the North Central Pacific, and the Southern Ocean, just southwest of Australia. The high-latitude Atlantic and part of the southern Atlantic peak around the same time as do large parts of the Pacific and Indian Oceans. Our results, in a gross sense, confirm those of Ponte [1999], but with noticeable differences from his seasonal maps (Figure 10). Both our results and those of Ponte indicate high-variability regions near the Drake Passage and in the North Central Pacific. Unlike his results, the Atlantic Ocean in our analysis does not reach a spatially uniform maximum, and the extent and phase of the variability structures in the Pacific also differ.

Figure 10.

Anomaly of the annual cycle for bottom pressure in each season. (top left) December through February. (top right) March through May. (bottom left) June through August. (bottom right) September through November. Scale is in centimeters. Contour interval is 0.5 cm. Convention follows that of Ponte [1999].

[37] The semiannual period-band variability is shown in Figure 11. Regions of high semiannual surface pressure variability occur in the Indian Ocean (associated with monsoon activity), the northwest Pacific, and high-latitude Atlantic Oceans at the semiannual period with values on the order of 3-cm equivalent water height. In parts of the Indian Ocean this component contains up to 30% of the record variance (Figure 12); otherwise, the fraction rarely exceeds 20%. Large parts of the Southern Ocean have the same phase as the high-latitude North Atlantic (Figure 13), and, in general, there is no hemispheric asymmetry as there is in the annual cycle.

Figure 11.

Semiannual component of the RMS variability. Units are centimeters. (top) Surface pressure. (bottom) Bottom pressure. Note scale reductions from 72-month period.

Figure 12.

Ratio of semiannual period variance to that of the 72-month period. (top) Surface pressure. (bottom) Bottom pressure.

Figure 13.

Phase of the semiannual cycle component of variability. Units are months. (top) Surface pressure. (bottom) Bottom pressure.

[38] Large variability in bottom pressure (Figure 11) occurs in the high-latitude North Atlantic, the northwest Pacific, and the Southern Ocean (primarily in two regions just southwest of Australia). Bottom pressure appears to have a nearly fixed phase in the Atlantic and another fixed value in the Indian Ocean. The Pacific appears, as in the annual cycle, to have a large region, in the center, of different phase than regions to the north and south of it (Figure 13). One difficulty encountered in estimating the semiannual energy directly from TOPEX/Poseidon data is the aliasing due to the K1 tide [see, e.g., Parke et al. [1987]. The model does not include tidal forcing directly, but an aliased tidal effect can enter the estimate through the constraints on sea surface heights obtained from altimetry. We believe that this effect is suppressed in the model because the spatial structure of an aliased tide is likely to be inconsistent with the geostrophic balance dominating the model at these periods; the semiannual signal displayed here probably arises from wind effects. Semiannual fractional energy is largest in the Indian Ocean, northwest and equatorial Pacific, and the Southern Ocean near 50°E (Figure 12).

[39] Variability structures of both surface and bottom pressure at the Chandler-wobble period (Figure 14) appear in a gross sense to be similar to the annual cycle (Figure 7), although much reduced in magnitude and with exceptions, such as the eastern North Atlantic coastal area. The surface pressure variability displays substantial hemispheric asymmetry (Figure 15). Bottom pressure displays the same nearly fixed phase (Figure 15) in the Indian Ocean, Southern Atlantic, and most of the Southern Ocean (except an area southwest of Australia).

Figure 14.

Chandler-wobble cycle component of the RMS variability. Units are centimeters. (top) Surface pressure. (bottom) Bottom pressure. Note scale reductions from 72-month variability.

Figure 15.

Phase of the Chandler-wobble cycle component of variability. Units are months. (top) Surface pressure. (bottom) Bottom pressure.

3.3. Decomposition of Surface and Bottom Pressure Into Zonal, Eastward, and Westward Traveling Waves

[40] Surface and bottom pressure are decomposed into zonal (standing), and eastward and westward traveling oscillations according to

equation image

where Dnme,w are coefficients, n, m are the degree and order, ω is the frequency, ψ is the phase, and the superscripts e and w denote eastward and westward. Zonal oscillations are given by the m = 0 terms. Results are presented as degree amplitudes and degree correlations.

[41] In general, for the annual, semiannual, and Chandler-wobble periods, the global variance of the westward component of surface pressure is larger than the zonal or eastward components (Table 1). For bottom pressure, the global variance of the eastward component (Table 1) is largest. The global variance of the zonal component of bottom pressure is more nearly uniform in the three frequency bands (about 20%), while for surface pressure, the annual band has about 28% and the semiannual has about 11% of the variance.

[42] For the annual cycle in bottom pressure, the two largest coefficients are zonal: D3,0 and D1,0, with the magnitude of D1,0 being 88% of that of D3,0. Overall, the global variance (Table 1) of the eastward traveling oscillations is greatest (3.5 cm2) while that of the westward is somewhat less (3.0 cm2), followed by the zonal (1.4 cm2). The tendency toward dominance of eastward propagation over westward extends to about spherical harmonic degree 5. For degrees above 5, eastward and westward degree amplitudes are approximately equal. The zonal component shows significant energy at degrees 1–3, 5, 8, and 11, after which its importance diminishes (Figure 16), becoming effectively less than 20% of the eastward and westward degree amplitudes at degree 14.

Figure 16.

Annual cycle, zonal-eastward-westward wave decomposition. (a) Degree amplitude: surface pressure. (b) Degree amplitude: bottom pressure. (c) Degree correlation: surface and bottom pressure.

[43] The two largest coefficients for surface pressure (Figure 16) are also the zonal coefficients, but with D1,0 the largest and D3,0 the second largest with 60% of the largest value. The global variance is greatest for westward propagation at 22.5 cm2; zonal follows at 13.7 cm2, and eastward at 12.5 cm2. The surface pressure odd degree zonal component appears generally to be a significant part of the total out to degree 25 and therefore appears to play an important role in surface pressure variability at these length scales (degrees 20–25 correspond to half-wavelengths of approximately 1000–700 km).

[44] At the semiannual period (Figure 17), the degree amplitudes for the westward surface pressure component are generally slightly greater than those for the eastward component in degrees 1–20. The zonal component is relatively important at degrees 1–4 and 8. It then diminishes in importance, out to degree 19, where it increases somewhat.

Figure 17.

Semiannual cycle. Convention as in previous figure.

[45] The Chandler-wobble band (Figure 18) shows a structure similar to that at the annual. At the wobble period, the degree amplitudes for westward components of surface pressure are greater than those of the eastward components at degrees 4 and 6–40. The zonal component is significant for bottom pressure to degree 10.

Figure 18.

Chandler-wobble cycle. Convention as in previous figure.

[46] Additional information can be obtained from triangular diagrams of coefficients (not shown), Dnm, in which the distribution by both degree and order are presented. Strong zonal terms are evident in all four diagrams. The annual bottom pressure coefficient distribution appears to be the most symmetric in structure with a bimodal distribution consisting of zonals and eastward and westward sectorals, while that for the surface pressure shows primarily zonals and westward sectorals. The zonal coefficients describe pressure variability in the north-south direction and are associated with east-west geostrophic flows. The sectoral coefficients describe pressure variability in the east-west direction and are associated with meridional (north-south) geostrophic flows. At the semiannual period, the diagrams do not show this bimodal zonal-sectoral coefficient distribution, but rather a more even distribution including tesseral coefficients.

3.4. Implications for Barotropic Dynamics

[47] Ponte [1999] compared baroclinic model results to those of a barotropic model and found indications that the seasonal bottom pressure variability on basin scales is barotropic in nature. While a complete vertical analysis is required to determine the barotropic or baroclinic character of the motions [Wunsch, 1997], agreement of the relative phases for surface and botttom pressure is a necessary (but not sufficient) test of the degree to which the motions are barotropic. Thus an examination of relative phase and amplitude for each cycle gives some insight into the character of the motions.

[48] Of the energy bands mapped here, the 4-month periods show the closest agreement between surface and bottom pressure phases and amplitudes (not shown). Notable amplitude differences occur in the tropical Pacific and Atlantic, and the Indian Oceans. At the semiannual period, there is zero relative phase in the high-latitude North Atlantic, the central North Pacific, and parts of the South Atlantic, Indian, and Southern Oceans. Zero or small relative amplitudes are spread throughout many of the same areas. The Southern Ocean is evidently the region of greatest barotropic motion at the annual cycle. Significant departures from barotropic structure appear in the high-latitude North Atlantic.

[49] A simple measure of the change of relative energy with period can be obtained from the ratio of surface pressure to bottom pressure variances as a function of frequency (Figure 19). The ratio, which can be interpreted as a barotropic index, decreases with increasing frequency, indicating the importance of steric effects at longer periods. The greatest barotropic character is at 4 months where the ratio is 2.4. For the annual period, it is 6.1.

Figure 19.

Ratio (r) of surface pressure variance to bottom pressure variance as a function of frequency (cpy).

[50] Under the assumption of isotropy and homogeneity on a sphere, the covariance of a function S evaluated at point p, and a function T evaluated at point q, depends only on the angular distance between the two points [e.g., Heiskanen and Moritz, 1967; Kaula, 1967]. The corresponding covariance function, K, can be expanded as

equation image

where Pn is unnormalized,

equation image

and anm and bnm are the fully normalized coefficients. Here kn can be interpreted as the cross spectrum and, for S = T and p = q, the cross spectrum becomes the degree variances or power spectrum.

[51] Use of the Parseval relations produces the degree correlations [e.g., Bracewell, 1986],

equation image

[52] Barotropic dynamics would require the surface and bottom pressure degree amplitudes to be identical and the degree correlations to be +1. For all periods (Figures 1618), the degree amplitudes are quite different at the basin scales (degrees 4–40). For the semiannual period in particular, and the Chandler-wobble period to a lesser degree, there appears to be a tendency over most of the lowest spherical harmonic degrees (2–16) for surface pressure and bottom pressure to be positively correlated, especially in the zonal component.

[53] The ocean basin geometry influences ocean dynamics and so, in this sense, the variability as well as the mean are functions of the continent-ocean distribution. The distribution does not change with time, for our purposes. Since the coefficients are computed by quadrature from global integrals, all points with a non-zero value contribute to the integral; each coefficient is independent of the others. The coefficients will be affected by how the continents are treated, whether, for example, zero or mean values are used over land. We have not addressed the question of the homogeneity and isotropy of oceanic variability. To prove the assumption invalid, one would need to show that the results obtained using an isotropic covariance are significantly in error with respect to a full covariance.

3.5. Chandler Wobble and Polar Motion

[54] Polar motion exhibits a continuous spectrum with superimposed narrow-band peaks at the Chandler and annual periods. Mechanisms of both excitation and decay of the motion have long been controversial. The Chandler wobble is the 14-month (433 day) free nutation of the Earth [Munk and MacDonald, 1960]. For the three periods that we have mapped above, observations indicate a Chandler-wobble amplitude of about 200 milliarcseconds (mas), an annual wobble amplitude of about 100 mas, and a semiannual wobble amplitude of about 10 mas. The Chandler period is distinct because it is a free, rather than a forced, mode of a rotating body [Munk and MacDonald, 1960; Lambeck, 1980], and the response is that of a resonant system.

[55] Excitation of the Chandler wobble is related physically to the changing angular momentum of the Earth system and is formulated mathematically in terms of moments and products of inertia and velocities. A number of excitation mechanisms have been proposed, including earthquakes, core dynamics, and redistribution of the Earth's fluid envelope (atmosphere and ocean). Most recently, ocean bottom pressure fluctuations have been proposed as a dominant mechanism for exciting the Chandler wobble [Ponte et al., 1998; Gross, 2000], in the latter case using results of this same unconstrained model.

[56] The wobble excitation problem is formulated as a perturbation to the angular momentum of the Earth-atmosphere-ocean system [Munk and MacDonald, 1960; Barnes et al., 1983], in which the perturbations are separated into two terms reflecting how the angular momentum is changed. One, called the mass or moment-of-inertia term, accounts for the contributions to the angular momentum from the redistributed positions of mass of the system. The other, the motion term, accounts for the angular momentum of the moving fluids. The formulation is expressed mathematically through equatorial angular momentum (EAM) functions χi [Barnes et al., 1983],

equation image

where χp is the mass term and χw is motion. The indices i = 1, 2 indicate the equatorial directions; these equatorial terms are responsible for polar motion. (An axial function, χ3, influences the rotation rate, but it is not discussed here). The EAM functions are typically combined into a complex vector,

equation image

and the mass term is

equation image

where C and A are moments of inertia. The coefficient of 1.00 deserves some comment. Accounting for the Earth's rotational deformation produces a coefficient, ks/(ksk2) = 1.43, in the EAM function, where ks is the secular Love number and k2 is the degree 2 Love number [Barnes et al., 1983]. Under the thin shell approximation, bottom pressure affects the angular momentum and is treated as a surface load using the load Love number, k2′. The coefficient is then modified to (1 + k2′)ks/(ksk2) = 1.00 (for k2′ = −0.3); the surface loading response effectively cancels the rotational deformation contribution. This formulation also assumes that the Earth's crust and mantle are decoupled from the core. Under these assumptions, the EAM functions are referred to as “effective angular momentum functions” [Barnes et al., 1983].

[57] We have computed excitation amplitude and phase owing to the oceanic motions at the annual, semiannual, and Chandler-wobble frequencies using equation (36) (Table 2) assigning each to one Fourier harmonic, with a frequency resolution of 0.1667 cpy for the annual and semiannual, and 0.1714 cpy for the Chandler wobble, and then decomposed the EAM function into prograde and retrograde components according to

equation image

where the subscript denotes prograde or retrograde motion; the superscript p has been suppressed because we only consider the mass term. The Chandler wobble is not a pure frequency, and a bandwidth must be chosen for analysis. Gross [2000], with 5-day-averaged data over a period of somewhat more than 10 years (1985–1996), was able to resolve frequencies separated by 0.0913 cpy; three discrete frequencies fell within the Chandler band (0.730,0.822, and 0.913 cpy), and the power over the band was computed to be 3.4 mas2. In our case, only one discrete frequency (0.8571 cpy) falls within the band, and the computed power is 3.1 mas2. At this resolution, the annual and Chandler-wobble frequencies differ by a single harmonic and are resolvable though the time domain averaging does cause leakage. For the Chandler band analysis using 70 months as the record length, the annual occurs at 1.0286 cpy. Unlike astronomical forcing and its associated response, the forcing and responses that we examine are at best narrow band. Note that not all of the oceanic variability at the Chandler-wobble period forces the wobble. The prograde spherical harmonics of degree 2 and order 1 are the important components of the variability for excitation.

Table 2. EAM Function Amplitude (mas) and Phase (Degrees) for Annual, Semiannual, and Chandler-Wobble Periods
 Amplitude ProgradeAmplitude RetrogradePhase ProgradePhase Retrograde
Chandler Wobble1.750.93111.69−38.99

[58] A prograde-retrograde decomposition is convenient for analyzing the excitation because it is independent of the position of the equatorial axes. The χ1,2 components themselves can also be used, though they depend on the axis positions. The amplitude of χ2 annual EAM function is larger than that of χ1 (Figure 20), as was found by Ponte and Stammer [1999]. In contrast to the annual, the semiannual contribution to χ1 is larger than in χ2; at the Chandler-wobble period, ∣χ1∣ ≈ ∣χ2∣.

Figure 20.

(top) Values χ1 and (bottom) χ2 for annual, semiannual, and Chandler-Wobble periods as a function of time. Units for χ are radians; for time, units are months.

4. Detectability of Bottom Pressure Change From Space

[59] Sea surface variability (not including that due to pressure forcing) attains values up to 10 cm for the filtered fields. The oceanic contribution to the annual component of geoid variability, as measured by a space-based observer, can be estimated using the Bruns formula (12) and the expressions (15) and (16) relating bottom pressure to the disturbing potential. In these calculations, we have also included the effect of surface loading on the redistribution of mass within the Earth through load Love numbers (Condi and Wunsch, manuscript in preparation, 2004). Geoid variability can thus attain values in some regions on the order of a millimeter (Figure 21). High-variability regions occur in the Pacific, northwest of the Drake Passage, and in the North Atlantic. Because the geoid is an integral over the anomalous mass, its variability is a smoothed representation of the anomalous bottom pressure variability.

Figure 21.

Variability in the annual oceanic contribution to the geoid as observed from space. Scale is in millimeters.

[60] A major issue in discussing a satellite mission is the extent to which particular signals can be detected practically. Determination of the contributions from each of the mapped cycles can be made by comparing the geoid degree amplitudes for each of the cycles with the estimated GRACE errors [Tapley and Reigber, 2001]. The present bottom pressure analysis suggests possible recovery of the annual cycle to about degree 27, and the semiannual and Chandler-wobble cycle to degree 14 (Figure 22). The steric correction was calculated as a uniform layer of water over the oceans based on mass loss/gain of the GCM for both the annual cycle and full 72-month record and appears to be detectable to about degree 7 for both. Any dynamical correction would be concentrated at higher wavenumbers, and although mission errors become larger at higher wavenumber, the corrections could still be detectable. The bottom pressure cumulative variances (not shown) for each of the extracted cycles suggests that about 67% of the total filtered variance is achieved by degree 15 (half-wavelength of about 1300 km) and 95% is achieved by degree 40 (half-wavelength of 500 km). The cumulative variance plots for the time-variable geoid (not shown) show a much more rapid rise than for bottom pressure, with close to 99% of the total achieved by degree 15.

Figure 22.

Degree amplitudes of mapped ocean cycles and corrections for mass exchange, compared to estimated GRACE errors (thick line). Scale is in millimeters.

[61] A comparison using degree amplitudes (or variances) is oversimplified in a number of respects. The estimated mission error is for a single month and under ideal conditions [Tapley and Reigber, 2001]; realistic errors, including strong high-frequency aliases, will be larger. A frequency decomposition of this error estimate is not available for the periods of interest in this paper (longer than 1 month). Presumably, a longer period such as a year would have smaller errors. Note also the large error in degree 2 (Figure 22), a result of systematic errors in the mission. This error may preclude a strong constraint on the oceanic motions responsible for polar motion, requiring reliance on other space based methods for constraints. The errors drop dramatically for degree 3.

[62] The length scales at which GRACE will be able to detect changes are larger than typical widths of western boundary currents. A complementary mission, planned by the European Space Agency, expected to be launched in the year 2006, to partially address this shortcoming, is the Gravity Field and Steady-State Ocean Circulation Explorer Mission (GOCE) [Drinkwater et al., 2003], in which a gravity gradiometer will be used to map the static geoid at wavelengths shorter than GRACE will be capable of mapping.

5. Final Comments

[63] Making improved estimates of the time-average and time-varying geoid involves facing difficulties in oceanography, geodesy, and engineering. Contributions to gravity and the geoid come from all mass sources, the magnitude of which depend on the density and distance from the observing point. Mass redistributions from outside the spatial domain of interest such as underground fluid movement, sedimentation, and vertical and horizontal tectonic movement (in some cases due to post glacial rebound) can contribute to the signal as well as oceanic and atmospheric mass redistribution. In addition, the sampling period is long compared to much of the variability of barotropic energy in the ocean, and this higher-frequency energy can alias into the signal [Stammer et al., 2000; Tierney et al., 2000]. These aliased motions will have to be modeled and removed from the data stream.

[64] Gravity data and ocean modeling have dual roles. The spatial scale improvement from GRACE for the time-variable geoid is expected to reach spherical harmonic degree 70 or 80 (S. Bettadpur, personal communication, 2001). Degree 80 corresponds to a half-wavelength of 250 km. Improvement in the static field comes from removing the time-variable geoid (e.g., the annual and semiannual cycles) from the total. In our analysis of OGCM results, most of the bottom pressure variability for the periods mapped is contained below degree 40, and 99% of the geoid variability is achieved by degree 15. Considering that the estimates of bottom pressure variability recovery from GRACE are to degree 27 for the annual, degree 14 for the semiannual and Chandler-wobble periods, and somewhat higher degree for the total, the use of the time-variable gravity as a direct constraint will be limited, as it will exclude scales with wavelengths less than about 1500 km.

[65] For the geopotential data to be useful as a constraint for ocean studies, the errors must be sufficiently small on the scales of interest and the atmospheric contribution will have to be accounted for. The atmospheric contribution is larger than that of the ocean, and should be recoverable to about spherical harmonic degree 33. In a practical state estimation scheme, the ocean could be modeled over all resolvable scales, using data with spherical harmonic degrees of approximately 27 and below as direct constraints on ocean bottom pressure. Such a solution would improve the marine geoid. The ocean model then, must be consistent with the longer wavelength time-variable geoid, and it is in such a case that the capability of variable weighting by spectral component is of utility in ocean modeling. An accurate geoid could be used for discriminating changes in mass from steric changes. For example, an expansion of water through heating elevates the sea surface, but leaves the total mass, and hence the geoid, undisturbed.

Appendix A:: Spherical Harmonics

[66] For analysis using the real form of spherical harmonics, we use the following convention. The Legendre function is defined as [Heiskanen and Moritz, 1967]

equation image

where t = cos θ, and θ represents colatitude. The real, fully normalized spherical harmonics are then

equation image
equation image


equation image

and ϕ represents longitude. The functions are normalized so that

equation image

The normalized Legendre function is defined as

equation image


[67] This work was supported by the University of Texas, Austin, through the NASA GRACE Project, and by the ECCO Consortium funded by ONR, NASA, and NSF through the National Ocean Partnership Program (NOPP).