Abstract

The nature of ocean bottom pressure () variability is considered on large spatial scales and long temporal scales. Monthly gridded estimates from the Gravity Recovery and Climate Experiment (GRACE) Release-05 and the new version 4 bidecadal ocean state estimate of the Consortium for Estimating the Circulation and Climate of the Ocean (ECCO) are used. Estimates of from GRACE and ECCO are generally in good agreement, providing an independent measure of the quality of both products. Diagnostic fields from the state estimate are used to compute barotropic (depth independent) and baroclinic (depth dependent) components. The relative roles of baroclinic and barotropic processes are found to vary with latitude and time scale: variations in at higher latitudes and shorter periods are affected by barotropic processes, whereas fluctuations at lower latitudes and longer periods can be influenced by baroclinic effects, broadly consistent with theoretical scaling arguments. Wind-driven Rossby waves and coupling of baroclinic and barotropic modes due to flow–topography interactions appear to be important influences on the baroclinic variability. Decadal simulations of monthly variability based on purely barotropic frameworks are expected to be in error by about 30% on average ( in the tropical ocean and at higher latitudes). Results have implications for applying GRACE observations to problems such as estimating transports of the Antarctic Circumpolar Current.

1. Introduction

Launched in March 2002, the twin Gravity Recovery and Climate Experiment (GRACE) spacecraft have been making nearly continuous measurements of mass redistribution in the climate system for more than a decade (Tapley et al. 2004). Estimates of ocean bottom pressure () derived from such observations are a powerful tool for studying ocean circulation and climate variability (Wahr et al. 1998). The data have been applied to quantify ice sheet and mountain glacier contributions to global sea level change and to estimate Antarctic Circumpolar Current transport variability (see the review by Chambers and Schröter 2011).

To interpret the GRACE estimates dynamically in the context of ocean circulation, it is important to understand what variations are—specifically, how much do they reflect depth-independent (barotropic) and depth-dependent (baroclinic) fluctuations? (We will refer to this question as the vertical structure of .) Despite its centrality, this topic has received little focused attention. The vertical structure of variability can be gleaned in some cases from investigations that interpret GRACE records using ocean dynamical frameworks (Boening et al. 2011; Cheng et al. 2013; Piecuch 2013; Piecuch and Ponte 2014; Ponte and Piecuch 2014) or satellite altimetry data (Wouters and Chambers 2010; Quinn and Ponte 2012; Landerer and Volkov 2013; Piecuch et al. 2013; Volkov 2014). However, since such studies tend to focus on specific geographic regions and/or frequency bands, general questions—for example, regarding the relative importance of baroclinic effects as a function of space and time—still remain open.

Gill and Niiler (1973), in their seminal study of the seasonal cycle, formulate theory for variation valid away from the equator on space scales larger than the deformation radius and at time scales longer than the inertial period. They submit that behavior becomes increasingly baroclinic as stratification and time scale increase and as latitude and space scale decrease. However, by and large, the veracity of their theoretical arguments pertaining to the vertical structure of variability remains to be assessed based on reliable estimates.

To test theoretical expectations, and also to inform the contemporary use of the GRACE data, a more global investigation is warranted. To this end, we address questions related to the vertical structure of variability by examining what are the relative roles of barotropic and baroclinic contributions as a function of geographic region, time scale, and latitude. For purposes of this study, we use estimates of based on GRACE Release-05 gravity data and the Release 1 of the new version 4 global bidecadal ocean state estimate of the Consortium for Estimating the Circulation and Climate of the Ocean (ECCO) (e.g., Wunsch et al. 2009).

2. Methods and materials

a. Satellite gravimetry

We use monthly estimates based on Release-05 GRACE spherical harmonic coefficients. Gravity data processed at the University of Texas at Austin’s Center for Space Research (CSR) following protocols described by Bettadpur (2012) are postprocessed by Don P. Chambers (University of South Florida) largely using methods detailed by Chambers and Bonin (2012): values for the degree 2 order 0 coefficient are taken from Cheng et al. (2011) and values for the degree 1 coefficients are taken from Swenson et al. (2008); leakage of terrestrial water storage is reduced following Wahr et al. (1998); glacial isostatic adjustment signals are removed using the model of A et al. (2013); a destriping filter is used to reduce meridionally oriented errors; a 500-km Gaussian spatial smoother is applied to the data to reduce short-wavelength errors. Gridded estimates are taken from the GRACE Tellus server (version RL05.DSTvDPC1401).

b. State estimate

Also employed is the ECCO-Production version 4 Release 1 (ECCO) ocean state estimate produced by Gaël Forget (Massachusetts Institute of Technology). The estimate constitutes an optimized solution of the Massachusetts Institute of Technology general circulation model (Marshall et al. 1997) that is constrained to myriad observational data (e.g., Wunsch and Heimbach 2013, Table 1). Using the method of Lagrangian multipliers, an objective function weighing the difference between model and data is minimized through iterative adjustment of a control vector populated with initial conditions (temperature and salinity), boundary conditions (zonal and meridional wind stress, air temperature, specific humidity, precipitation, etc.), and internal model parameters (diffusion coefficients) using the model’s adjoint (Heimbach et al. 2005; Heimbach 2008). For more details concerning the optimization procedure, the interested reader should consult Wunsch and Heimbach (2007).

The estimate is truly global, incorporating the Arctic Ocean and including an interactive dynamic and thermodynamic sea ice and snow model (Hibler 1980; Zhang et al. 1998; Zhang and Rothrock 2000), and covers two decades (1992–2011). The Boussinesq solution employs a fully nonlinear free surface with real freshwater exchanges (Campin et al. 2004) and a rescaled vertical coordinate (Adcroft and Campin 2004; Campin et al. 2008). The native grid is a latitude–longitude-cap configuration, with a nominal resolution of in the horizontal, isotropic in the middle latitudes, and telescoping to meridional resolution near the equator. Fifty levels are used in the vertical and topography is represented using partial cells (Adcroft et al. 1997). The surface forcing is in the form of bulk formulas (Large and Yeager 2004) based on the European Centre for Medium-Range Weather Forecasts interim reanalysis (ERA-Interim) described by Dee et al. (2011) and iteratively adjusted using methods described in the preceding paragraph. Parameterization schemes are used to include the effects of processes occurring on spatial scales smaller than the model grid scale—for example, geostrophic eddies (Redi 1982; Gent and McWilliams 1990; Griffies 1998), vertical mixing (Gaspar et al. 1990), and salt plumes (Duffy et al. 1999). Importantly, time-variable fields from GRACE are not employed in this solution, allowing for independent comparison.

3. Comparing estimates from GRACE and ECCO

Investigation of the vertical structure of variability in a continuously stratified ocean requires knowledge of density and pressure over the full ocean depth (see the next section). Continuous deep-ocean measurements are sparse, so a purely observational assessment over the global ocean is precluded. However, provided model-based estimates that agree well with GRACE records, one can make the assumption that the model must be capturing the relevant physics, including any deep-ocean processes. On the basis of this reasoning, we motivate the use of ECCO in subsequent sections by comparing it to GRACE in this section. To be consistent with the GRACE data, which are optimized for examination of regional variability, area-weighted global mean values are removed from the ECCO output and the fields are smoothed with a 500-km Gaussian filter. We consider monthly detrended behavior for the period 2003–11, quoting values in units of equivalent water thickness.1

Heterogeneous spatial patterns of variability are revealed by GRACE (Fig. 1a). A meridional gradient is generally evident, with amplitudes ranging from roughly 1–2 cm at lower latitudes to about 2–3 cm at higher latitudes. Elevated values relative to this background behavior are apparent in several Southern Ocean basins (Bellingshausen, Australian–Antarctic, Weddell–Enderby, and Argentine), the North Pacific Ocean, and the Arctic Ocean. These patterns reflect variability at intraseasonal, seasonal, and interannual periods (Quinn and Ponte 2012; Johnson and Chambers 2013; Piecuch et al. 2013). Monthly variations in these areas have been discussed previously based on numerical models (Fukumori et al. 1998; Ponte 1999), radar altimetry (Fu and Smith 1996; Fu 2003), and satellite gravimetry (Boening et al. 2011; Chambers 2011); the dynamics is generally thought to involve barotropic adjustment to winds under rotation and topography [but cf. Peralta-Ferriz and Morison (2010) for an exception to this rule].

Fig. 1.

RMS values (cm) of monthly detrended time series over 2003–11 from (a) GRACE and (b) ECCO. Time series are smoothed with a 500-km Gaussian filter. To reduce the likelihood of considering data contaminated by land leakage, values are not shown within a few hundred kilometers of land.

Fig. 1.

RMS values (cm) of monthly detrended time series over 2003–11 from (a) GRACE and (b) ECCO. Time series are smoothed with a 500-km Gaussian filter. To reduce the likelihood of considering data contaminated by land leakage, values are not shown within a few hundred kilometers of land.

Similar spatial patterns are simulated by ECCO (Fig. 1b). The correlation between the two maps in Fig. 1 is 0.89. The observed meridional gradient in amplitude and major centers of elevated variability are mostly captured by ECCO. However, compared to GRACE, ECCO values can be somewhat lower at low latitudes (tropical Pacific) and higher at high latitudes (Arctic and Southern Oceans). Higher values in GRACE over the tropical Pacific could reflect noise in the data—for example, owing to aliasing of tropical instability waves (Song and Zlotnicki 2004), which cannot be simulated by the coarse resolution of the background model used for processing the GRACE data. Higher values in ECCO over the Arctic and Southern Oceans are probably related to the fact that the wind forcing over ice-covered regions in this particular solution uses a drag coefficient that is too large; forthcoming state estimates, which are currently in production as of this writing, use a smaller value for the drag coefficient, effecting closer agreement between GRACE and ECCO in those regions.

A direct assessment of the correspondence between GRACE and ECCO is facilitated by the root-mean-square (RMS) differences and correlation coefficients shown in Fig. 2. RMS differences are mainly lower than observed RMS values (cf. Figs. 1a and 2a), meaning that ECCO can skillfully explain variance in GRACE. Exceptional regions are apparent in the Arctic and Southern Oceans, where RMS differences are larger than the observed variability, on account of the stronger simulated signals that were mentioned earlier. Correlation coefficients between GRACE and ECCO time series are statistically significant at the 95% confidence level over most of the ocean (Fig. 2b), demonstrating that ECCO captures the basic temporal behavior of GRACE. Statistically insignificant correlation coefficients are found near land and over parts of the tropical Atlantic Ocean and western tropical Pacific Ocean. These regional discrepancies between GRACE and ECCO could reflect noise in the data, for example, due to tectonic activity (e.g., along the Izu–Bonin–Mariana Arc in the western tropical Pacific), leakage of terrestrial water storage (close to land), or low overall levels of variability (over the tropical Atlantic), which could amplify the relative effects of random background noise in the data, rendering lower the signal-to-noise ratios. Differences might also reflect GRACE errors due to aliasing of high-frequency oceanic variability.

Fig. 2.

(a) RMS difference (cm) and (b) correlation between ECCO and GRACE time series over 2003–11. Black contours in (b) bound regions where the correlation is significant at the 95% confidence level.

Fig. 2.

(a) RMS difference (cm) and (b) correlation between ECCO and GRACE time series over 2003–11. Black contours in (b) bound regions where the correlation is significant at the 95% confidence level.

We judge from Figs. 1 and 2 that ECCO and GRACE are in sufficiently good agreement so as to justify further exploration of the vertical structure of variability based on ECCO.

4. Vertical structure of variability

a. Definitions

We now use ECCO to study the vertical structure of oceanic variability over 1993–2011. The vertical momentum balance in the model is given by the hydrostatic condition:

 
formula

where and are total pressure and density, respectively, and g is the acceleration due to gravity. Integrating Eq. (1) vertically and assuming no surface loading, variations in are seen (after removal of a constant summand) to result from changes in the anomalous mass of the overlying water column:

 
formula

where is a constant reference density, η is the sea level, H is the ocean depth, and is the density anomaly. We can define the barotropic bottom pressure as the vertical average of the hydrostatic pressure anomaly p:

 
formula

The baroclinic contribution to bottom pressure can be defined by taking the residual of and defined in Eqs. (2) and (3), respectively, and using the hydrostatic condition [Eq. (1)]:

 
formula

which is equivalent to the difference between the baroclinic sea level (Fukumori et al. 1998) and the steric height (Gill and Niiler 1973).2 Exploiting the full knowledge of the three-dimensional ocean state furnished by the ECCO estimate, we evaluate and based on Eqs. (3) and (4) using hydrostatic pressure and anomalous density state estimate diagnostics, respectively.

b. Results

Root-mean-square values of and are shown in Fig. 3. Clearest is close correspondence between the simulated total term and the barotropic contribution (cf. Figs. 1b and 3a); a first visual pass suggests that these two maps are identical. Interpretation of this result is that most of the prominent variability features (e.g., in the Southern, North Pacific, and Arctic Oceans) are barotropic in nature, consistent with previous model and data studies (Fukumori et al. 1998; Fu 2003; Vivier et al. 2005; Quinn and Ponte 2012).

Fig. 3.

RMS (cm) of monthly detrended ECCO over 1993–2011 due to (a) the barotropic component and (b) the baroclinic component ().

Fig. 3.

RMS (cm) of monthly detrended ECCO over 1993–2011 due to (a) the barotropic component and (b) the baroclinic component ().

Upon closer inspection, however, subtle differences between the and maps are made apparent (Figs. 1b and 3a)—for example, along the tropics, where baroclinic effects are implicated (Fig. 3b). In general, RMS values of are comparatively small (roughly 0.1–0.2 cm), but in some regions the amplitudes are relatively elevated. For instance, RMS values of in excess of 0.3–0.5 cm can be seen over the southern tropical Indian Ocean and the western tropical North Pacific, consistent with previous papers using linear models to study GRACE estimates (Piecuch 2013; Piecuch and Ponte 2014), as well as along the Australian–Antarctic and Bellingshausen basins, consistent with prior analysis of an earlier state estimate (Ponte and Piecuch 2014). There are also other areas of elevated variability that have not been discussed in the literature, such as the eastern tropical North Pacific.

To show more clearly where baroclinic effects are important compared to the total signal, the RMS value of is shown relative to the RMS value of in Fig. 4a. This ratio can be roughly interpreted as the relative error associated with the assumption that a given record is purely barotropic in nature. Ratios from Fig. 4a are shown in zonally averaged form in Fig. 4c. Values in Figs. 4a and 4c are around 0.3 on average, suggesting that the assumption of complete barotropic variability is generally associated with an error of roughly 30%. However, there is considerable spatial variation. Values are mostly at higher latitudes (e.g., Arctic Ocean, extratropical North Pacific Ocean, and Southern Ocean areas discussed previously) where overall variability is most pronounced, reflecting more barotropic behavior. In contrast, across the equatorial and tropical ocean, where the total variability is relatively small (Fig. 1), ignoring baroclinic effects renders larger relative errors; values close to the equator are mainly and there are even some areas (e.g., eastern tropical North Pacific) where RMS values of and are comparable to each other.

Fig. 4.

(a) Quotients of RMS values of ECCO detrended monthly and over 1993–2011. (b) Correlation between and . Black contours bound regions of correlations significant at the 95% confidence level. (c) Black curve represents zonal averages of the ratios shown in (a) taken over -latitude bands. Gray shading is plus/minus one corresponding standard deviation.

Fig. 4.

(a) Quotients of RMS values of ECCO detrended monthly and over 1993–2011. (b) Correlation between and . Black contours bound regions of correlations significant at the 95% confidence level. (c) Black curve represents zonal averages of the ratios shown in (a) taken over -latitude bands. Gray shading is plus/minus one corresponding standard deviation.

A map of correlation coefficients between and is shown in Fig. 4b to suggest the physical nature of the baroclinic signals. In areas where and time series are anticorrelated, fluctuations in can be enhanced relative to variations: the correlation coefficient between the two maps in Fig. 4 is (exceptions to this rule are apparent, for example, over the equatorial Pacific and Atlantic Oceans, where and are positively correlated). Along the eastern flanks of the tropical basins, there are elongated regions of significant negative correlations in Fig. 4b that progress to the west with increasing proximity to the equator, reminiscent of the bending of phase lines of Rossby waves with latitude (i.e., “beta refraction”). Indeed, assuming a barotropic Sverdrup response and a baroclinic Rossby adjustment driven locally by a small patch of wind stress curl, one would anticipate anticorrelation between and time series (see the appendix of Piecuch 2013).

Also noteworthy is the significant anticorrelation between and around regions of strong bathymetric gradients, for instance, island arcs and trenches including the Izu–Bonin–Mariana Arc and Izu–Ogasawara Trench in the North Pacific and the Kermadec Arc and Tonga Trench in the South Pacific (Fig. 4b). [While it is admittedly not obvious in Fig. 4b, which is based on model fields smoothed with a 500-km Gaussian filter, this correspondence between significant negative correlation coefficients and bathymetric features is strikingly apparent in a similar figure based on unsmoothed model output (not shown).] Recalling that, unlike normal modes in a stratified ocean with a flat bottom (e.g., Gill 1982, section 6.11), baroclinic and barotropic signals cannot be separated in an ocean with stratification and a sloping seafloor (since the kinematic bottom boundary condition involves both horizontal and vertical velocities), we suggest that the correspondence could signal coupling between the baroclinic and barotropic components due to interactions between the flow and topography.

Statistically significant relationships between and could also indicate vertically trapped signals. For example, given a surface-trapped wave, the vertical structure would be single signed, with the amplitude decreasing monotonically with depth (Philander 1978, Fig. 17; Frankignoul and Müller 1979, Fig. 1); in this case, the barotropic amplitude would be less than the value at the surface but greater than the value at the bottom; this requires the baroclinic amplitude at the bottom to be of the opposite sign in order to recover the true bottom amplitude, implying that the barotropic and baroclinic contributions at the seafloor [as defined in (3) and (4), respectively] must be out of phase with each other. An analogous line of reasoning in the case of a bottom-trapped wave leads to the conclusion that barotropic and baroclinic components at the seafloor will be in phase with one another. It is interesting to observe that the negative correlations around topographic features mentioned in the preceding paragraph would not appear to be indicative of bottom-trapped phenomena.

Definitive diagnosis of the respective roles of these processes (i.e., Rossby waves, flow–topography interactions, vertical trapping, etc.) is beyond our scope, and more work needs to be done in the context of detailed future studies to understand their relevance to regional signals. To consider more generally the relative importance of baroclinic and barotropic processes, ratios of and power spectral density estimates are shown as a function of latitude and frequency in Fig. 5a; the values of the spectral ratios have been averaged in space over discrete 5-latitude bins. Broadly speaking, for a given latitude, ratios mostly increase with decreasing frequency, signaling increasingly baroclinic behavior, whereas, for a given frequency, values mainly decrease with increasing latitude, signifying increasingly barotropic behavior. However, there are hints of some exceptions to these rules: at the lowest latitudes and frequencies, the ratios seem to plateau toward an asymptotic value that appears to be independent of both latitude and frequency; an annual spectral peak, which is most evident near and poleward of latitude, deviates from the generally red behavior.

Fig. 5.

(a) Shading is of the ratio of and power spectral densities [which were estimated based on Welch’s (1967) method using a Hamming window, a segment length of 64, and 50% overlap] as a function of frequency and latitude. White contours are shown at an interval of 1. Black dashes mark the annual band. Gray curve corresponds to the inverse of the time for a long baroclinic Rossby wave to transit an ocean basin 5000 km across as a function of latitude computed using an equivalent depth of 0.9 m (Gill 1982, section 12.3). (b) Shading is of the square of the scaling expression Eq. (8) due to Gill and Niiler (1973) as a function of frequency and latitude; see the text for details.

Fig. 5.

(a) Shading is of the ratio of and power spectral densities [which were estimated based on Welch’s (1967) method using a Hamming window, a segment length of 64, and 50% overlap] as a function of frequency and latitude. White contours are shown at an interval of 1. Black dashes mark the annual band. Gray curve corresponds to the inverse of the time for a long baroclinic Rossby wave to transit an ocean basin 5000 km across as a function of latitude computed using an equivalent depth of 0.9 m (Gill 1982, section 12.3). (b) Shading is of the square of the scaling expression Eq. (8) due to Gill and Niiler (1973) as a function of frequency and latitude; see the text for details.

c. Interpretation

How do we interpret the spectral behavior in Fig. 5a? While focused on the seasonal cycle, Gill and Niiler (1973) give an equation for appropriate for use more generally away from the equator on space scales larger than the deformation radius and at time scales longer than the inertial period, from which they derive the quotient of baroclinic and barotropic Fourier amplitudes. From the linearized and vertically integrated momentum and mass conservation equations, variations are affected by changes in wind and stratification, namely,

 
formula

where f is the Coriolis parameter (i.e., the vertical component of planetary vorticity), τ is the wind stress, is the vertical component of the curl operator, J is the Jacobian operator, and

 
formula

is the potential energy per unit area. Assuming that density variations are primarily due to vertical heave of the pycnocline due to Ekman pumping, in which case ρ scales according to

 
formula

where N is the Brunt–Väisälä frequency and ω is angular frequency, Gill and Niiler (1973) reason that the quotient of the first and second terms on the right-hand side of Eq. (5) scales as

 
formula

where β is the meridional derivative of f and L is a representative zonal length scale. Note that this expression assumes a transient baroclinic response as well as an equilibrium barotropic adjustment. Making the simplifying assumption of an ocean with a flat bottom, Eq. (8) provides a measure of the amplitude of the baroclinic relative to barotropic variability.3

To see whether the simple reasoning due to Gill and Niiler (1973) captures the spectral behavior in Fig. 5a, the square of Eq. (8) is plotted versus latitude and frequency in Fig. 5b. [Note that, since it is in terms of Fourier components, the square of Eq. (8) must be taken in order for it to be compared to power spectral densities.] For the purpose of the figure, we choose a representative midlatitude value of 20 m2 s−2 for following Gill and Niiler (1973), but note that this integral can range from roughly 30 m2 s−2 at lower latitudes to about 10 m2 s−2 at higher latitudes (not shown); given this range of integral values, it is interesting to note that, for fixed frequency, the variation in Eq. (8) over the global ocean is controlled more strongly by latitude than by stratification. We choose a zonal length scale of 1000 km for L, which corresponds to the shortest possible wavelength given the Gaussian smoothing.

For constant length scale and stratification, spectral ratios from theory increase with decreasing frequency and latitude (Fig. 5b); they capture the basic behavior of spectral ratios from ECCO: the correlation between the two panels shown in Fig. 5 is 0.90. But despite gross correspondence, there are clearly differences between spectral ratios based on theory and those derived from the state estimate. Theory seems to predict a stronger overall increase with decreasing frequency at fixed latitude (specifically at the lowest latitudes and frequencies) as well as a stronger overall decrease with increasing latitude at fixed frequency (especially at the highest latitudes and frequencies) than is shown by ECCO; unlike ECCO, the theory shows no sign of a peak in the spectral ratio at the annual band (Fig. 5). Such disagreements necessarily imply either that there are changes in stratification or length scale or that there are important forcing and dynamics that are ignored by Gill and Niiler (1973).

Gill and Niiler (1973) assume a transient baroclinic response. However, at periods longer than it would take baroclinic Rossby waves to transit an ocean basin, baroclinic signals will be in equilibrium with the atmosphere. For instance, first baroclinic mode Rossby waves traveling along () latitude will spend about 1 year (one decade) traversing a basin 5000 km across (Fig. 5a). Therefore, while they can be appropriately applied to studying the transient case of seasonal variability in the midlatitude ocean, the simple scaling arguments of Gill and Niiler (1973) cannot be appropriately applied to investigating the equilibrium case of decadal variability in the tropical ocean. The fact that the spectral ratio in Fig. 5a seems to asymptote to a constant value at the lowest latitudes and lowest frequencies is consistent with the simple model of decadal ocean response to stochastic winds due to Frankignoul et al. (1997), who hypothesize that, at low enough frequency, the quotient of baroclinic and barotropic pressure variance at a given depth will depend only on stratification.

Gill and Niiler (1973) also assume that changes are driven by wind stress curl. So, the annual peak in the spectral ratio might reflect the oceanic response to distinct forcing that projects more favorably onto the baroclinic modes, such as surface buoyancy flux or water mass exchange. For example, Peralta-Ferriz and Morison (2010) interpret the annual cycle in Arctic Ocean in terms of an adjustment to runoff plus precipitation minus evaporation involving quadratic bottom drag and boundary layer turbulence due to background currents.

Finally, Gill and Niiler (1973) ignore the effects of relative vorticity. However, for space scales comparable to, or smaller than, the barotropic Rossby radius of deformation (which is on the order of 2000 km at midlatitudes), these effects might play an important role. For instance, Fu (2003) reasons that intraseasonal sea level variability in the extratropical oceans reflects the damped barotropic response to wind stress curl mediated by relative vorticity generation. As relative vorticity effects become important, Eq. (8) can no longer be interpreted as the quotient of baroclinic and barotropic variabilities; that the decrease in spectral ratio value with increasing latitude at fixed frequency (especially at high latitudes and high frequencies) appears to be stronger in theory than in ECCO might reflect this fact.

5. Summary and discussion

We investigated the nature of variability over the global ocean using GRACE and ECCO, focusing on space scales ( km) and time scales ( month and decade) that are accessible to GRACE (Figs. 12). Estimates of from GRACE and ECCO are generally in good agreement, with both products showing similar areas of enhanced variability (Fig. 1). Model and data time series are significantly correlated over most of the global ocean, and residual differences between the two are mainly smaller than the observed variability (Fig. 2). Disagreements between GRACE and ECCO in some regions (e.g., in the Arctic Ocean) are probably due to a mix of model error and data noise (Fig. 2).

Diagnostic variables furnished by the ocean state estimate allowed us to investigate the vertical structure of variability (Figs. 35). Whereas stronger variations at higher latitudes and shorter periods are affected by barotropic processes, weaker fluctuations at lower latitudes and longer periods can be influenced by baroclinic effects (Fig. 3). Decadal simulations of monthly behavior based on purely barotropic frameworks are expected to be in error by about 30% on average ( at lower latitudes and at higher latitudes; Figs. 4a and 4c). Rossby waves driven by wind stress curl and coupling of the baroclinic and barotropic modes due to interactions between flow and topography could be important influences on baroclinic variability (Fig. 4b).

Scaling arguments due to Gill and Niiler (1973) can be a helpful lens through which to view the vertical structure of variability, as they generally capture the increasingly baroclinic nature of variability as frequency and latitude decrease (Fig. 5). There are, however, important discrepancies between predictions from this simple theory and output from the ocean state estimate: the theory due to Gill and Niiler (1973) does not anticipate whitening of the quotient of baroclinic and barotropic power spectral densities manifested by the state estimate at the lowest latitudes and frequencies, probably because of their assumption of a transient baroclinic response, which is only valid on time scales less than it would take a Rossby wave to cross a basin; also, it does not predict an annual peak in the spectral ratio, possibly implicating important driving mechanisms not considered by Gill and Niiler (1973).

These findings have implications for using GRACE to calculate ocean transports. Several studies employ GRACE to estimate Antarctic Circumpolar Current (ACC) volume transport variability (Zlotnicki et al. 2007; Böning et al. 2010; Bergmann and Dobslaw 2012; Makowski 2013). These papers assume that GRACE records are barotropic in nature. However, a failure to distinguish between barotropic and baroclinic components could contaminate such estimates, since the baroclinic component carries negligible net volume transport (Gill 1982). Our results suggest that, at space and time scales accessible to GRACE, variability at high latitudes is overwhelmingly barotropic, so the assumption of a purely barotropic regime will not be a major source of error in computations of variable ACC flows based on GRACE.

Relatedly, while our use of a model product rather than observational data for the purpose of studying the vertical structure of variability in a continuously stratified ocean was borne out of necessity, if we assume that the ocean can be adequately described by a two-layer system, the barotropic and baroclinic components can be determined solely on the basis of and sea level information—namely, by solving a linear system of two equations with two unknowns (see section 2.2 of Jayne et al. 2003). This observation suggests that barotropic and baroclinic terms could be diagnosed and separated based on data from satellite gravimetry and radar altimetry, for example, which would allow for observation-based computation of barotropic transports in regions where baroclinic effects are important (e.g., low latitudes). Exploration of this interesting possibility is left for future investigation.

Acknowledgments

This work was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA), and was further supported by NASA GRACE Grant NNX12AJ93G and Grant OCE-0961507 from the National Science Foundation. The GRACE ocean data were processed by Don P. Chambers, supported by the NASA MEaSUREs Program, and are available online (at http://grace.jpl.nasa.gov). A listing of available ECCO products can be found on the group website (http://www.ecco-group.org). The comments from two anonymous reviewers were appreciated.

REFERENCES

REFERENCES
A
,
G.
,
J.
Wahr
, and
S.
Zhong
,
2013
: Computations of the viscoelastic response of a 3-D compressible Earth to surface loading: An application to Glacial Isostatic Adjustment in Antarctica and Canada. Geophys. J. Int.,192, 557–572, doi:.
Adcroft
,
A.
, and
J.-M.
Campin
,
2004
:
Rescaled height coordinates for accurate representation of free-surface flows in ocean circulation models
.
Ocean Modell.
,
7
,
269
284
, doi:.
Adcroft
,
A.
,
C.
Hill
, and
J.
Marshall
,
1997
:
Representation of topography by shaved cells in a height coordinate ocean model
.
Mon. Wea. Rev.
,
125
,
2293
2315
, doi:.
Bergmann
,
I.
, and
H.
Dobslaw
,
2012
:
Short-term transport variability of the Antarctic Circumpolar Current from satellite gravity observations
.
J. Geophys. Res.
,
117
,
C05044
, doi:.
Bettadpur
,
S.
,
2012
: UTCSR level-2 processing standards document (for level-2 product Release 0005). GRACE 327-742, CSR Publ. GR-12-xx, Revision 4.0, The University of Texas at Austin, 16 pp.
Boening
,
C.
,
T.
Lee
, and
V.
Zlotnicki
,
2011
:
A record-high ocean bottom pressure in the South Pacific observed by GRACE
.
Geophys. Res. Lett.
,
38
,
L04602
, doi:.
Böning
,
C.
,
R.
Timmermann
,
S.
Danilov
, and
J.
Schröter
,
2010
:
Antarctic Circumpolar Current transport variability in GRACE gravity solutions and numerical ocean model simulations
. System Earth via Geodetic-Geophysical Space Techniques, F. Flechtner et al., Eds., Advanced Technologies in Earth Sciences, Springer, 187–199.
Campin
,
J.-M.
,
A.
Adcroft
,
C.
Hill
, and
J.
Marshall
,
2004
:
Conservation of properties in a free-surface model
.
Ocean Modell.
,
6
,
221
244
, doi:.
Campin
,
J.-M.
,
J.
Marshall
, and
D.
Ferreira
,
2008
:
Sea ice–ocean coupling using a rescaled vertical coordinate z*
.
Ocean Modell.
,
24
,
1
14
, doi:.
Chambers
,
D. P.
,
2011
:
ENSO-correlated fluctuations in ocean bottom pressure and wind-stress curl in the North Pacific
.
Ocean Sci.
,
7
,
685
692
, doi:.
Chambers
,
D. P.
, and
J. A.
Bonin
,
2012
:
Evaluation of Release-05 GRACE time-variable gravity coefficients over the ocean
.
Ocean Sci.
,
8
,
859
868
, doi:.
Chambers
,
D. P.
, and
J.
Schröter
,
2011
:
Measuring ocean mass variability from satellite gravimetry
.
J. Geodyn.
,
52
,
333
343
, doi:.
Cheng
,
M.
,
J. C.
Ries
, and
B. D.
Tapley
,
2011
:
Variations of the Earth’s figure axis from satellite laser ranging and GRACE
.
J. Geophys. Res.
,
116
,
B01409
, doi:.
Cheng
,
X.
,
L.
Li
,
Y.
Du
,
J.
Wang
, and
R.-X.
Huang
,
2013
:
Mass-induced sea level change in the northwestern North Pacific and its contribution to total sea level change
.
Geophys. Res. Lett.
,
40
,
3975
3980
, doi:.
Dee
,
D. P.
, and Coauthors
,
2011
:
The ERA-Interim reanalysis: Configuration and performance of the data assimilation system
.
Quart. J. Roy. Meteor. Soc.
,
137
,
553
597
, doi:.
Duffy
,
P. B.
,
M.
Eby
, and
A. J.
Weaver
,
1999
:
Effects of sinking of salt rejected during formation of sea ice on results of an ocean-atmosphere-sea ice climate model
.
Geophys. Res. Lett.
,
26
,
1739
1742
, doi:.
Frankignoul
,
C.
, and
P.
Müller
,
1979
:
Quasi-geostrophic response of an infinite β-plane ocean to stochastic forcing by the atmosphere
.
J. Phys. Oceanogr.
,
9
,
104
127
, doi:.
Frankignoul
,
C.
,
P.
Müller
, and
E.
Zorita
,
1997
:
A simple model of the decadal response of the ocean to stochastic wind forcing
.
J. Phys. Oceanogr.
,
27
,
1533
1546
, doi:.
Fu
,
L.-L.
,
2003
:
Wind-forced intraseasonal sea level variability of the extratropical oceans
.
J. Phys. Oceanogr.
,
33
,
436
449
, doi:.
Fu
,
L.-L.
, and
R. D.
Smith
,
1996
:
Global ocean circulation from satellite altimetry and high-resolution computer simulation
.
Bull. Amer. Meteor. Soc.
, 77,
2625
2636
, doi:.
Fukumori
,
I.
,
R.
Raghunath
, and
L.-L.
Fu
,
1998
:
Nature of global large-scale sea level variability in relation to atmospheric forcing: A modeling study
.
J. Geophys. Res.
,
103
,
5493
5512
, doi:.
Gaspar
,
P.
,
Y.
Grégoris
, and
J.-M.
Lefevre
,
1990
:
A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: Tests at station Papa and long-term upper ocean study site
.
J. Geophys. Res.
,
95
,
16 179
16 193
, doi:.
Gent
,
P. R.
, and
J. C.
McWilliams
,
1990
:
Isopycnal mixing in ocean circulation models
.
J. Phys. Oceanogr.
,
20
,
150
155
, doi:.
Gill
,
A. E.
,
1982
: Atmosphere-Ocean Dynamics. International Geophysics Series, Vol. 30, Academic Press, 662 pp.
Gill
,
A. E.
, and
P. P.
Niiler
,
1973
:
The theory of the seasonal variability in the ocean
.
Deep-Sea Res. Oceanogr. Abstr.
,
20
,
141
177
, doi:.
Griffies
,
S. M.
,
1998
:
The Gent-McWilliams skew flux
.
J. Phys. Oceanogr.
,
28
,
831
841
, doi:.
Heimbach
,
P.
,
2008
: The MITgcm/ECCO adjoint modelling infrastructure. CLIVAR Exchanges, No. 44, International CLIVAR Project Office, Southampton, United Kingdom, 13–17.
Heimbach
,
P.
,
C.
Hill
, and
R.
Giering
,
2005
:
An efficient exact adjoint of the parallel MIT General Circulation Model, generated via automatic differentiation
.
Future Gener. Comput. Syst.
,
21
,
1356
1371
, doi:.
Hibler
,
W. D.
,
1980
:
Modeling a variable thickness sea ice cover
.
Mon. Wea. Rev.
,
108
,
1943
1973
, doi:.
Jayne
,
S. R.
,
J. M.
Wahr
, and
F. O.
Bryan
,
2003
:
Observing ocean heat content using satellite gravity and altimetry
.
J. Geophys. Res.
,
108
,
3031
, doi:.
Johnson
,
G. C.
, and
D. P.
Chambers
,
2013
:
Ocean bottom pressure seasonal cycles and decadal trends from GRACE Release-05: Ocean circulation implications
.
J. Geophys. Res. Oceans
,
118
,
4228
4240
, doi:.
Landerer
,
F. W.
, and
D. L.
Volkov
,
2013
:
The anatomy of recent large sea level fluctuations in the Mediterranean Sea
.
Geophys. Res. Lett.
,
40
,
553
557
, doi:.
Large
,
W.
, and
S.
Yeager
,
2004
: Diurnal to decadal global forcing for ocean and sea-ice models: The data sets and flux climatologies. NCAR Tech. Note NCAR/TN-460+STR, 112 pp. [Available online at http://nldr.library.ucar.edu/repository/collections/TECH-NOTE-000-000-000-601.]
Makowski
,
J. K.
,
2013
: Understanding transport variability of the Antarctic Circumpolar Current using ocean bottom pressure. M.S. thesis, Dept. of Marine Science, University of South Florida, 60 pp.
Marshall
,
J.
,
A.
Adcroft
,
C.
Hill
,
L.
Perelman
, and
C.
Heisey
,
1997
:
A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers
.
J. Geophys. Res.
,
102
,
5753
5766
, doi:.
Peixoto
,
J. P.
, and
A. H.
Oort
,
1992
: Physics of Climate. American Institute of Physics, 520 pp.
Peralta-Ferriz
,
C.
, and
J.
Morison
,
2010
:
Understanding the annual cycle of the Arctic Ocean bottom pressure
.
Geophys. Res. Lett.
,
37
,
L10603
, doi:.
Philander
,
S. G. H.
,
1978
:
Forced oceanic waves
.
Rev. Geophys.
,
16
,
15
46
, doi:.
Piecuch
,
C. G.
,
2013
:
Dynamics of satellite-derived interannual ocean bottom pressure variability in the western tropical North Pacific
.
J. Geophys. Res. Oceans
,
118
,
5117
5128
, doi:.
Piecuch
,
C. G.
, and
R. M.
Ponte
,
2014
:
Annual cycle in southern tropical Indian Ocean bottom pressure
.
J. Phys. Oceanogr.
,
44
,
1605
1613
, doi:.
Piecuch
,
C. G.
,
K. J.
Quinn
, and
R. M.
Ponte
,
2013
:
Satellite-derived interannual ocean bottom pressure variability and its relation to sea level
.
Geophys. Res. Lett.
,
40
,
3106
3110
, doi:.
Ponte
,
R. M.
,
1999
:
A preliminary model study of the large-scale seasonal cycle in bottom pressure over the global ocean
.
J. Geophys. Res.
,
104
,
1289
1300
, doi:.
Ponte
,
R. M.
, and
C. G.
Piecuch
,
2014
:
Interannual bottom pressure signals in the Australian–Antarctic and Bellingshausen basins
.
J. Phys. Oceanogr.
,
44
,
1456
1465
, doi:.
Quinn
,
K. J.
, and
R. M.
Ponte
,
2012
:
High frequency barotropic ocean variability observed by GRACE and satellite altimetry
.
Geophys. Res. Lett.
,
39
,
L07603
, doi:.
Redi
,
M. H.
,
1982
:
Oceanic isopycnal mixing by coordinate rotation
.
J. Phys. Oceanogr.
,
12
,
1154
1158
, doi:.
Song
,
Y. T.
, and
V.
Zlotnicki
,
2004
:
Ocean bottom pressure waves predicted in the tropical Pacific
.
Geophys. Res. Lett.
,
31
,
L05306
, doi:.
Swenson
,
S.
,
D.
Chambers
, and
J.
Wahr
,
2008
:
Estimating geocenter variations from a combination of GRACE and ocean model output
.
J. Geophys. Res.
,
113
,
B8410
, doi:.
Tapley
,
B. D.
,
S.
Bettadpur
,
M.
Watkins
, and
C.
Reigber
,
2004
:
The Gravity Recovery and Climate Experiment: Mission overview and early results
.
Geophys. Res. Lett.
,
31
,
L09607
, doi:.
Vivier
,
F.
,
K. A.
Kelly
, and
M.
Harismendy
,
2005
:
Causes of large-scale sea level variations in the Southern Ocean: Analyses of sea level and a barotropic model
.
J. Geophys. Res.
,
110
,
C09014
, doi:.
Volkov
,
D. L.
,
2014
:
Do the North Atlantic winds drive the nonseasonal variability of the Arctic Ocean sea level?
Geophys. Res. Lett.
,
41
, 2041–2047, doi:.
Wahr
,
J.
,
M.
Molenaar
, and
F.
Bryan
,
1998
:
Time variability of the Earth’s gravity field: Hydrological and oceanic effects and their possible detection using GRACE
.
J. Geophys. Res.
,
103
,
30 205
30 229
, doi:.
Welch
,
P. D.
,
1967
:
The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms
.
IEEE Trans. Audio Electroacoust.
,
15
,
70
73
, doi:.
Wouters
,
B.
, and
D.
Chambers
,
2010
:
Analysis of seasonal ocean bottom pressure variability in the Gulf of Thailand from GRACE
.
Global Planet. Change
,
74
,
76
81
, doi:.
Wunsch
,
C.
, and
P.
Heimbach
,
2007
:
Practical global oceanic state estimation
.
Physica D
,
230
,
197
208
, doi:.
Wunsch
,
C.
, and
P.
Heimbach
,
2013
:
Two decades of the Atlantic meridional overturning circulation: Anatomy, variations, extremes, prediction, and overcoming its limitations
.
J. Climate
,
26
,
7167
7186
, doi:.
Wunsch
,
C.
,
P.
Heimbach
,
R. M.
Ponte
,
I.
Fukumori
, and the
ECCO-GODAE Consortium Members
,
2009
:
The global general circulation of the ocean estimated by the ECCO-Consortium
.
Oceanography
,
22
,
88
103
, doi:.
Zhang
,
J.
, and
D.
Rothrock
,
2000
:
Modeling Arctic sea ice with an efficient plastic solution
.
J. Geophys. Res.
,
105
,
3325
3338
, doi:.
Zhang
,
J.
,
W. D.
Hibler
III
,
M.
Steele
, and
D. A.
Rothrock
,
1998
:
Arctic ice–ocean modeling with and without climate restoring
.
J. Phys. Oceanogr.
,
28
,
191
217
, doi:.
Zlotnicki
,
V.
,
J.
Wahr
,
I.
Fukumori
, and
Y. T.
Song
,
2007
:
Antarctic Circumpolar Current transport variability during 2003–05 from GRACE
.
J. Phys. Oceanogr.
,
37
,
230
244
, doi:.

Footnotes

1

For clarity, in figures has thickness units, whereas in equations has standard pressure units (Pa).

2

We choose to use the angled-bracket and double-prime notation to denote barotropic and baroclinic components, respectively, following the general usage of Peixoto and Oort [1992, see their section (4.1.2)].

3

For example, using Eqs. (1) and (6) and integrating by parts, it is readily shown that P is equivalently defined as and so, in the case of a flat-bottomed ocean, in Eq. (5) is equivalent to .