The Atlantic Meridional Overturning Circulation consists of a northward flow in the upper layer of the ocean (mostly between the surface and 1000 m depth; ; ) and a return flow to the south in the deeper layer of the ocean (between approximately 1000 and 5000 m depth). The meridional stream function ψ(y,x) is derived from meridional velocities (vy) by integration over longitudes (x) and from the surface (η) to depth (z) (): ψ(y,z)=∫WE∫zηvydzdx. As the large-scale flows are dominated by a geostrophic balance, the meridional transport per unit depth T(z), at a particular latitude (y) and depth (z), can be derived from the zonal bottom pressure differences pE and pW at the eastern and western basin boundaries by taking T(z)=pE(z)-pW(z)ρ0f, where the constants are the Coriolis parameter f and the mean sea water density ρ0 (; ). Acceleration and stress terms are neglected, as they only play a role in the Ekman layer and in the deep bottom layers. For a more rigorous derivation and justification for Eq. () we defer to and , and references therein. Using the geostrophic approximation, the depth-integrated meridional transports T(z) at a particular latitude (y) can then be used to give the meridional stream function ψ at that latitude: ψ(y,Δz)=1ρ0f∫z1z2(pE-pW)dz. Equation () provides a method to derive the geostrophic component of the AMOC stream function (or volume transport between two layers) from ocean bottom pressure data along the boundaries of the ocean basin. Possibly intervening topography (i.e., mid-ocean ridges) should in theory be considered when evaluating Eq. (), but demonstrated in an ocean model that, for interannual time-variable transports in the North Atlantic, bottom pressure variability is concentrated along the western boundary, and it is sufficient to use only the outermost eastern and western points across the basin section (if a basin-mean or depth-averaged boundary pressure is removed from pW in Eq. (), it is also possible to use only bottom pressure on the western boundary (). Furthermore, the dominance of the western boundary variations was recently confirmed from hydrographic in situ data (). We reconfirmed this with the ocean model ECCO2 (see below for details), and we thus use pE and pW only for our analyses in the North Atlantic. While knowledge of pE and pW along the boundaries is in principle sufficient to infer ψ(y,z), the actual measurement of these terms is challenging. In situ BPRs suffer from notorious drift problems and thus require drift corrections that usually inhibit any inferences about longer-than-annual variations (). An alternative measurement of OBP variations can be obtained from time-variable gravity observations from space as currently acquired by the GRACE satellites. The main challenge for OBP inferred from time-variable gravity is the limited horizontal resolution, as well as the required signal sensitivity. Due to the altitude (about 450 km) and orbit configuration of the two GRACE satellites, the horizontal spatial resolution is limited to approximately 300 km (e.g., ; ). Much of the AMOC-related OBP signals occur along the narrow and steep western boundary slope and are thus difficult to resolve (which is also a limitation to in situ pressure observations). In Sect. , we therefore quantify these resolution issues using synthetic data at GRACE-like spatial resolutions to quantify the feasibility of the OBP AMOC approach. Also note that GRACE can only resolve OBP variations relative to a (arbitrary) time mean. Therefore, all terms in Eqs. ()–() are taken to be anomalies and only AMOC variations can be inferred, but not its long-term average. The mean flow in the North Atlantic and the resulting OBP anomalies are illustrated in Fig. .

We use the ECCO2 () to reconstruct the AMOC variability from simulated OBP observations in the North Atlantic as in Eq. (). ECCO2 is an ocean state estimate which optimally fits ocean observations using a Green's function approach. The OBP-derived AMOC reconstructions are compared against the model baseline AMOC, which represents the model truth and is based on the meridional velocities according to Eq. (). Similar to , we extract zonal OBP profiles following the model's bathymetry and then interpolate these values to regularly spaced 100 m depth intervals. Monthly ECCO2 OBP data at a horizontal resolution of 0.25∘ are computed for the time period January 1993 through December 2012, for the area of the North Atlantic, 80 to 0∘ W and 20 to 60∘ N. We remove a global mean OBP term to enforce mass conservation (). For the subsequent analysis, we remove a trend and the mean annual climatology signal from all time series (OBP, velocities), and we smooth with a 15-month running mean to focus on interannual signals only.

Conventional GRACE gravity field solutions are given in global spherical harmonic basis functions without any type of constraints (e.g., ). In contrast, the GRACE mascon solutions employ geophysical constraints and provide an improved spatial localization. A best-fitting gravity value is estimated for each mascon cell (here: 3∘ × 3∘ equal-area cells). Importantly, the mascon solution makes the application of empirical post-processing filters (i.e., destriping) unnecessary and thus features a better signal-to-noise ratio at smaller spatial scales. We evaluate OBP output from the ECCO2 ocean state estimate as provided at a 0.25∘ degree resolution. To create GRACE-like synthetic observations that match actual GRACE resolution, we bin-average the OBP fields to a 3∘ equal-area grid. This grid is identical to the JPL mascon RL05M grid (). Second, the OBP data are smoothed to resemble the resolution of the standard GRACE solutions, which are represented in spherical harmonics truncated at degree and order 60, and smoothed with a Gaussian filter with 300 km radius. This would be necessary for real GRACE data in order to reduce noise and correlated errors. This processing provides approximately the resolution that is currently achieved with the GRACE satellites. However, we do not consider instrument and resulting measurement errors in the gravity field retrieval from GRACE measurements in order to focus on the issues of spatial resolution and signal leakage. The spatial smoothing and averaging of the 0.25∘ OBP fields lead to significant resolution reduction in particular in highly energetic regions like the Gulf Stream, as well as in regions of steep bathymetry (Figs.  and ). Additional post-processing filters employed to reduce correlated errors in GRACE would further dampen geophysical signals (e.g., ).

For each of the synthetic OBP data sets, meridional transport time series are computed in 1∘ latitude increments and over 100 m depth intervals, and the rms differences between reconstructed and model reference time series are computed for each depth and latitude (Fig. ). The results for OBP without a hydrology signal (Fig. , top row) at the 0.25∘ native ECCO2 resolution lead to errors smaller than 0.5 Sv km-1 for depths between 1000 and 5000 m. At latitudes lower than 50∘ N, errors above 1000 and below 5000 m vary with latitude and depend on the bathymetry gradients. At GRACE-like resolutions (panels b and c in Fig. ), the errors are slightly higher across all depths, and at specific latitudes, e.g., at 25–30∘ N, there are significant signal leakage errors that introduce significant transport retrieval errors. The steep topography (Fig. ) at these latitudes causes one 3∘ mascon to cover depth layers from above 1000 m to below 3000 m. Very high errors (> 1.5 Sv km-1) occur in the upper 100 m depth for all latitudes due to the non-geostrophic, wind-driven transport in the Ekman layer, which cannot be recovered from east–west OBP difference observations. In all following computations of the geostrophic volume transports, we therefore exclude the upper 100 m (in the OBP-derived as well as in the reference transport time series).

While the GLDAS hydrology signal does not affect the results on the ECCO2 model grid (Fig. , panel d), significant leakage errors from land hydrology are introduced when the OBP and hydrology signals are spatially smoothed to GRACE-like resolutions (Fig. e and f). Without hydrology leakage, errors of 1.5 Sv km-1 and larger only occur in the uppermost 100 m when Ekman transports are not accounted for, and at 25∘ N for the mascons. With hydrology leakage effects, the GRACE-like OBP resolutions lead to high errors that extend into deeper layers, down to 3000 to 5000 m depth for latitudes 32 to 40∘ N. This effect is highly latitude-dependent, since the lower resolution only degrades the results if smoothing occurs over too many depth layers and/or the coastline. In this way, the results are very dependent on the bathymetry and the proximity of depth contours to land points, as well as signal amplitudes over land. In addition, pressure variations over steep bathymetry cannot be adequately resolved in the spatially smoothed data. For the mascon resolution, the leakage effect changes with mascon latitudes. Significant hydrology signal leakage occurs especially between 35 and 40∘ N, down to a depth of 2000 m. For spherical harmonics, the leakage effects are more smeared out over depths and latitudes. Between 20 and 40∘ N down to 3000 m depth, errors are between 1 and 2 Sv km-1. For the mascon results, the CRI filter reduces much of the leakage artifacts; the major leakage effect between 35 and 40∘ N is reduced from errors exceeding 2 Sv km-1 to less than 1 Sv km-1 by 2000 m depth.

Another strategy to reduce leakage is to optimize the placement of the individual mascons in longitude direction (for each mascon latitude). When mascon boundaries align with the coastline, hydrology leakage is reduced; when an individual mascon does not cover too many depth layers, leakage between depths in the ocean is reduced. The optimal mascon position (in longitude) is found by minimizing both types of errors simultaneously. While there are latitudes where land leakage is not reduced as much by optimal positioning as by the CRI filter (e.g., 22, 33∘ N), errors in the deeper layers between 2000 and 3000 m depths are smaller than for the CRI results. For 30 to 50∘ N and between 1500 and 5000 m depth most errors are below 0.5 Sv km-1 with the position-optimized mascons, while they tend to be between 0.5 and 1 Sv km-1 in the results with CRI. Note that CRI only treats and reduces land leakage; it does not mitigate leakage between different ocean depths layers ().

The maximum of the mean model AMOC in ECCO2 lies at 32∘ N and 909 m depth (Fig. ). Thus, net transports from the surface to about 909 m are northward, and net transports below about 909 m to a depth of about 5000 m are southward. However, the depth of maximum overturning varies with latitude and in time. The circulation below about 5000 m is linked to the Atlantic Bottom Water and is not considered in the following. As mentioned before, the uppermost 100 m of the ocean is also excluded, because the Ekman circulation and related transports cannot be recovered from OBP gradients. Over interannual periods, the net water volume transported northward should equal the water volume transported back south (e.g., or for 10-day timescales). Thus, it should be sufficient to observe either the northward or the southward transport in order to reconstruct the interannual AMOC transport variations, as long as the depth of maximum overturning circulation is known. Since we do not know the correct depth of maximum overturning for each latitude and time, we make an assumption of a constant depth, which introduces only a small error.

In what follows, three different depth layers are considered in more detail: 100 to 909, 909 to 3000, and 3000 to 5000 m depth; detectability of the AMOC signal in each of these three layers from GRACE-like OBP resolutions is assessed. The first layer covers the northward transport (down to the maximum of the mean AMOC, Fig. ); the second layer covers steep ocean basin slopes for most latitudes (Fig. ); and the third layer covers deeper transport, where the bathymetry is less steep (Fig. ) and therefore can be expected to be more favorable for GRACE-like resolutions.

Figure  shows rms errors and correlation coefficients for the reconstructed transport versus the model baseline for the three layers. The center plot is a selection of the plots on the left, with an adjusted axis to enhance details for the solutions with smaller errors (ECCO2 native resolution and mascons with CRI). The reconstructed transport from OBP at the ECCO2 native 0.25∘ resolution (black curves Fig. ) matches the model baseline transport best; it shows smallest rms errors (about 0.5 Sv and below, with a maximum of 1 Sv) for all the three layers and the highest correlation coefficients. The average rms error and correlation level are similar for all the three layers under consideration. However, when smoothing to GRACE-like resolutions, rms differences become larger and correlation coefficients smaller, due to the much coarser resolution of 3∘. For these resolutions, overall and maximum rms errors (Fig. , left) are larger for the medium depth layer (909 to 3000 m) than for the upper and the deep layers. The larger errors between 20 and 45∘ N in the medium layer for data at GRACE-like resolutions are caused by the steep slopes between about 1000 and 3000 m depth (Figs.  and ). When the data are smoothed, the OBP values cannot be attributed to the correct depth as the depth interval for one 3∘ smoothing interval becomes very large. Between 45 and 60∘ N, the depth gradient for a 3∘ longitude interval becomes much smaller; i.e., more than one 3∘ pixel is needed to cover the depth gradient from 909 to 3000 m. Thus, OBP at individual depth layers can be better resolved and the transport reconstruction is more accurate, leading to smaller rms error. For the upper transport, rms errors are high (0.5 to 2 Sv) for spherical harmonics and mascons, and especially high between 30 and 40∘ N. These errors are attributed to leakage effects from land hydrology signals (Fig. ). In the upper layer, the coastline resolution improvement correction makes a big difference: for mascons with CRI (red curve in Fig. ) the rms errors are at a level similar to the ECCO2 native 0.25∘ resolution and below 1 Sv. In the deep layer (3000 to 5000 m depth), there are still high rms errors of about 3 Sv between 30 and 40∘ N for spherical harmonics (black dashed curve), because land hydrology leakage extends to depths below 3000 m for these latitudes (Fig. ). The CRI algorithm and position optimizing of mascons corrects for these errors; therefore, rms errors for mascons with CRI and position-optimized mascons (red and blue curves) are about and below 1 Sv in the deep layer for 20 to 45∘ N. Beyond 45∘ N, the GRACE resolution is well capable of capturing all the OBP signal, since the bathymetry is less steep. Therefore, rms errors decrease and drop below 0.5 Sv for 50 to 60∘ N. In order to show more detail with respect to the signal rms, the two solutions with smaller rms error, i.e., the original ECCO2 grid and mascons with CRI, are plotted again in the center of Fig. . Even for these better-performing solutions, the signal rms is of the same order of magnitude as the rms error, with the rms error from the mascon solutions exceeding the signal rms by far in the intermediate layer. The rms errors for the original ECCO2 resolution are mostly just below the signal rms. Even though rms errors for mascons are higher, the results in the upper and deeper layers achieve smaller rms error than signal rms for selected latitudes. As mentioned before, the overturning transport signal is on the edge of detectability in GRACE gravity data, but we show in this study that it is possible with CRI improved mascons for selected latitudes.

Correlation coefficients vary a lot with latitude. While correlation coefficients are highest for the 0.25∘ ECCO2 resolution, the difference to the GRACE-like resolutions is the largest in the medium layer, due to steep basin boundary in this layer, as explained above. Even though there are a few latitudes with poor correlation in the deep layer for the GRACE-like resolutions (e.g., between 25 and 30∘ N, and 40 to 50∘ N), the correlation coefficients are overall higher than in the upper two layers, where most correlation coefficients are below 0.5. Most correlation coefficients with the time series from OBP at the original ECCO2 resolution are significant (Fig.  black curve above red dashed 95 % significance level), while significance of the correlation coefficients varies a lot with latitude for GRACE-like resolutions (all other curves). Especially in the deep layer, there are several latitudes where correlation coefficients for mascons with CRI are well above the 95 % significance level, e.g., 20–25, 30–40, and 55–60∘ N. Again, this indicates that the less steep bathymetry in the deep layer is more favorable for GRACE-like resolutions.

In conclusion, Fig.  shows that the upper and the deep layer transport can be reconstructed from GRACE-like OBP resolutions with rms error of 0.5 Sv and correlation coefficients of about 0.7, as long as leakage from land hydrology is accounted and corrected for. The medium layer (909 to 3000 m depth) is much less suitable for transport reconstruction from GRACE-like OBP resolutions, because the steep bathymetry in this layer cannot be resolved well by GRACE.

Figure  shows one example for reconstructed transport time series at 30∘ N. The left-hand side of the figure shows the results for OBP time series including continental hydrology, while the right-hand side shows the corresponding time series, but for the OBP signal only, without hydrology. The magnitude of the model reference signal which we are trying to recover is about the same for the upper and the intermediate layer (well below 2 Sv), but it reaches and exceeds 2 Sv in some months for the deep layer. In the upper and intermediate layer, there is a very large signal magnitude in the time series derived from spherical harmonics and position-optimized mascons including land hydrology. This large signal magnitude is caused by leakage of the continental hydrology signal (larger magnitude than OBP signal). It is not present in the solution without hydrology. Also note that leakage affects even the intermediate depth layer at this latitude, i.e., below 909 m depth. The original ECCO2 grid is not affected by hydrology; therefore the solid black curves are the same in the plots on the right and on the left. Leakage from continental hydrology does not affect the very deep layers; thus, the results on the right and on the left for the deep layer are the same. Without any leakage, reconstruction of the transport signal works well for all different OBP time series for the upper layer. However, this scenario is not very realistic. Even though a good portion of the signal can be recovered, the solution from spherical harmonics shows the largest discrepancies from the model reference for the scenario without hydrology. From the mascon resolution, the signal can be recovered well in the deep layer; however, there are some discrepancies in the intermediate layer which are due to signal leakage across different depth due to steep bathymetry. In conclusion, first and foremost, continental hydrology has to be taken into account, for example with the CRI filter for the mascons. Second, leakage across steep bathymetry contaminates the transport signal derived from mascon-resolution OBP. Favorable latitudes and depth layers for less steep bathymetry gradients have to be chosen.

Finally, Fig.  shows our AMOC reconstruction for 30∘ N, derived by summing up the time series for the intermediate and the deep transport layer, i.e., showing the total southward transport (which we assume to be compensating for the entire AMOC northward transport). The model reference time series is matched closely by the time series derived from OBP at the original ECCO2 grid. There are some larger discrepancies between the model reference and the time series derived from mascons (with CRI) (blue curve in Fig. ). Nevertheless, the model reference can be recovered with an rms error of 0.90 Sv and a correlation coefficient of 0.63. While in the deep layer transport (bottom panel, left, in Fig. ) the time series derived from mascons with CRI and the original ECCO2 grid are very similar (black and blue solid curves), they differ for the intermediate layer, while the black curve (ECCO2 grid) is closer to the model reference. This is what introduces errors in the mascon time series in Fig. . While the CRI takes care of continental hydrology leakage, there is leakage across the steep bathymetry at depths between 909 and 3000 m (compare solid blue curves on the left- and right- hand side, intermediate panel in Fig. ).