## 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.

[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.

[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).