Meridional transports from OBP integration
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 ().
Reconstructing north- and southward transports
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.
Left panel: time mean of AMOC from ECCO2, with maximum at
32∘ N, 909 m depth, minimum at 47∘ N, 2990 m depth, both
indicated by marker X. Right panel: variability of AMOC from ECCO2, maximum at
34∘ N, 1200 m depth, indicated by marker X.
Root mean square error and correlation coefficients for reconstructed net
transport in three different depth layers and from different OPB resolutions
(native ECCO2 grid vs. GRACE-like resolutions). Left: rms error for the
computed time series and rms signal time series (red dashed); center:
selection of two cases of rms error and different scales on the plots; right:
correlation coefficients of the derived time series versus the model truth.
Red dashed line indicates the 95 % significance level.
Transport time series anomalies at 30∘ N, reconstructed
from different OBP data sets versus the model baseline. Left: including land
hydrology signal; right: without land hydrology signal.
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. ).