Implementation of the Optical Flow to Estimate the Propagation of Eddies in the South Atlantic Ocean

Abstract

The ocean is filled with mesoscale eddies that account for most of the oceanic kinetic energy. The importance of eddies in transporting properties and energy across the ocean basins has led to numerous efforts to track their motion. Here, we implement a computer vision technique—the optical flow—to map the pathways of mesoscale eddies in the South Atlantic Ocean. The optical flow is applied to the pairs of consecutive sea surface height maps produced from a nearly 30-year-long satellite altimetry record. In contrast to other methods to estimate the eddy propagation velocity, the optical flow can reveal the temporal evolution of eddy motion, which is particularly useful in the regions of strong currents. We present the time-dependent estimates of the speed and direction of eddy propagation in the Eulerian frame of reference. In an excellent agreement with earlier studies, the obtained pattern of eddy propagation reveals the interaction of eddies with the background flow and the bottom topography. We show that in the Antarctic Circumpolar Current, the variability of the eddy propagation velocity is correlated with the variability of the surface geostrophic velocity, demonstrating the robustness of the optical flow to detect the time-variable part of eddy motion.

1. Introduction

The ocean is always in turbulent, highly variable motion. Ocean circulation redistributes mass, heat, salt, nutrients, carbon, and other properties, and plays a crucial role in regulating the global and regional climate, weather, and ecosystems. The advent of satellite observations of sea surface temperature (SST) in the 1970s and sea surface height (SSH) in the 1980s has revolutionized our view on ocean variability. The nearly global coverage of satellite measurements has revealed the distribution and spectral characteristics of ocean variability at various time scales. It has been discovered that ocean variability is dominated by processes with spatial scales ranging from a few tens to a few hundreds of kilometers and a time scale on the order of 100 days [1,2,3]. This type of ocean variability is usually termed as transient mesoscale eddies, including individual coherent eddies (vortices characterized by a clear boundary), meanders, fronts, filaments, and planetary Rossby waves [4]. Transient mesoscale eddies can be found virtually everywhere in the ocean, but the largest and most energetic eddies emerge from the instabilities of the strongest currents, in particular the Antarctic Circumpolar Current (ACC) and western boundary currents, such as the Gulf Stream, Kuroshio, Agulhas Current, etc. Away from strong currents that can significantly influence their motion, eddies tend to propagate westward as Rossby waves. They carry properties horizontally and redistribute them vertically, contribute to the climatically meaningful meridional heat transport, drive regional sea level changes, influence shipping routes and fisheries, etc., e.g., refs. [5,6,7,8].

The importance of mesoscale eddies in ocean dynamics has stimulated numerous efforts to track the movement of individual eddies and to estimate their propagation velocities. The tracking of individual eddies consists of following their trajectories in a Lagrangian manner. Existing techniques include the Okubo–Weiss method [9,10,11,12], algorithms based on the geometry of velocity vectors and streamlines [13,14,15,16,17], a hybrid geometric Okubo–Weiss method [18], a wavelet-based approach [19,20], and an objective method based on geodesic transport theory [21]. Most procedures of eddy detection rely on using either surface geostrophic velocities derived from SSH or closed SSH contours. In altimetry applications, however, the use of geostrophic velocities and moreover the relative vorticity is not optimal, because it requires computation of the first and the second derivatives of SSH, respectively, which amplifies any noise present in satellite measurements and interpolation errors [15]. Different from individual eddy detection and tracking, the regional and global patterns of transient eddy propagation velocities in the Eulerian frame of reference have also been obtained by carrying out a space–time-lagged correlation analysis of SSH fields [22,23].

The objective of this study is to demonstrate the use of the optical flow (OF) technique to derive the time-dependent propagation velocities of transient mesoscale eddies in the Eulerian framework. The OF is a computer vision algorithm that estimates the apparent motion of brightness patterns in a sequence of images, e.g., [24]. Some examples of the earlier implementation of the OF in geosciences include the assessment of the quality of forecasts of meteorological fields [25,26,27], and the estimates of sea-ice motion [28,29] and surface currents from SST images [30,31]. The study reported in this paper is, to our knowledge, the first implementation of the OF method to satellite altimetry SSH data. Like the space–time-lagged correlation analysis of [22,23], the OF method cannot distinguish various forms of eddy variability, e.g., isolated cyclones and anticyclones, planetary waves, meanders, fronts, etc. On the other hand, while the space–time-lagged correlation analysis yields only the time–mean propagation velocities, the OF computes “instantaneous” velocities between the pairs of consecutive satellite images, so given the large number of images, it is well suited for estimating the temporal evolution of motion. In oceanographic applications, this might be particularly useful in regions where the propagation velocities of mesoscale eddies are strongly affected by the variable background ocean circulation.

In this work, we focus on the South Atlantic Ocean and the adjacent regions of the Southern Ocean (Figure 1). This region is characterized by complex dynamics and, therefore, it is particularly representative for testing the OF method. The same region was used by [22] to estimate eddy propagation velocities using a space–time-lagged correlation method, which we also use for validation. The mean dynamic topography contours displayed in Figure 1 indicate the streamlines of the surface flow (with the high SSH to the left of the flow direction in the Southern Hemisphere). The ocean circulation here is dominated by a system of western boundary currents (the Brazil and Malvinas Currents), the eddy-rich eastward-flowing ACC, and the Agulhas Retroflection, which produces prominent eddies that propagate into the South Atlantic. The ACC transports 130–140 Sv (1 Sv = 10⁶ m³ s⁻¹) of water through the Drake Passage [32] and supplies the Malvinas Current that carries 30–40 Sv northward along the Argentine coast [33,34,35]. The southward branch of the South Atlantic subtropical gyre along the Brazil coast is comprised of the Brazil Current with a mean strength of 14 Sv and a standard deviation of about 9 Sv at 34.5°S [36]. The Brazil and the Malvinas currents collide at approximately 39°S forming one of the most energetic regions of the World Ocean—the Brazil–Malvinas Confluence (Figure 1; ref. [37]). A very strong anticyclonic (anticlockwise) barotropic flow transporting approximately 100 Sv exists around a topographic anomaly known as the Zapiola Rise at about 45°S and 45°W [38,39]. This flow is also associated with the anticyclonic propagation of mesoscale eddies [22]. By applying the OF technique to SSH data in the South Atlantic Ocean, we provide a realistic description of eddy propagation velocity validated by several other methods and previous studies.

2. Materials and Methods

2.1. Satellite Altimetry Data

Satellite altimetry has provided accurate, nearly global, sustained observations of SSH since the launch of the TOPEX/Poseidon mission in August 1992. Here, we use the daily SSH maps above the geoid (absolute dynamic topography) from January 1993 to June 2022 processed and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS; http://marine.copernicus.eu, accessed on 2 May 2023). The SSH above the geoid is the sum of the MDT-CNES-CLS22 mean dynamic topography [40] over the 1993–2012 period and the sea-level anomaly with respect to the mean sea surface. Prior to mapping, the along-track altimetry records are routinely corrected for instrumental noise, orbit determination error, atmospheric refraction, sea state bias, tides, as well as static and dynamic atmospheric-pressure effects [41].

The SSH maps are then produced on a global 1/4° × 1/4° (longitude × latitude) grid by the optimal interpolation of measurements from all the altimeter missions available at a given time. Due to the filtering properties of the optimal interpolation, the SSH maps have an effective spatial resolution ranging from about 400 km at 10°S to 100 km at 60°S and an effective temporal resolution varying between 14–45 days [42]. Although the number of simultaneously flying altimeter satellites has varied with time, it has been reported that the impact of satellite constellation on the effective spatial resolution is modest, with a globally averaged gain of resolution of ∼5% in maps constructed with three altimeters compared with two altimeters [42].

Because the smallest repeat period of altimeter satellites (excluding crossover points) is about 10 days and the effective temporal resolution of SSH maps is even longer, using every single daily SSH map for the OF computation is redundant. Therefore, to reduce the computational cost, we use every 10th SSH map (1076 maps in total). The SSH anomalies are computed with respect to the record-long mean SSH above the geoid (absolute dynamic topography). The global mean SSH time series is subtracted from each grid point to obtain the local dynamic SSH anomalies not related to the global mean sea-level rise. The seasonal cycle, which is mostly characterized by standing oscillations, is estimated by fitting both the annual and semi-annual harmonics in a least-squares sense and subtracted from the SSH fields. Furthermore, to focus on the mesoscale variability only, we high-pass filter the time series at each grid point by removing the signals with periods greater than 1.5 years.

2.2. Optical Flow

Here, we provide a brief description of the OF method as it is applied to calculate the perceived eddy propagation velocities between the pairs of consecutive SSH anomaly maps (images). Let η₁(x, y, t) denote the SSH anomaly signal at a grid point or pixel (x, y) on the first map at time t. Suppose that during the time interval ∆t, the water parcel with the signal η₁(x, y, t) has moved to a new location (x + ∆x, y + ∆y) on the second map and the SSH anomaly signal has been transformed to η₂(x + ∆x, y + ∆y, t + ∆t) (Figure 2a). It is assumed that the SSH change over ∆t, i.e., between consecutive maps, is small, so that

$${\eta }_2\left(x+\Delta x,y+\Delta y,t+\Delta t\right)\approx {\eta }_1\left(x,y,t\right)$$

(1)

This is known as the brightness-constancy assumption in computer vision. Considering the effective resolution of SSH maps, it is reasonable to assume that the displacements ∆x and ∆y are relatively small over ∆t = 10 days. Thus, the left-hand side of (1) can be approximated by a Taylor series expansion up to the first-order terms:

$${\eta }_2\left(x+\Delta x,y+\Delta y,t+\Delta t\right)\approx {\eta }_1\left(x,y,t\right)+{\eta }_x\Delta x+{\eta }_y\Delta y+{\eta }_t\Delta t$$

(2)

where (η_x, η_y, η_t) is a vector of the zonal, meridional, and temporal derivatives of η. Substituting (2) into (1) and dividing by ∆t yields the OF constraint equation [24]:

$${\eta }_{x}u+{\eta }_{y}v+{\eta }_t=0$$

(3)

where u = ∆x/∆t and v = ∆y/∆t are the zonal and meridional propagation velocities (optical flow) of the SSH signal, respectively. While the vector derivative (η_x, η_y, η_t) can be estimated by spatio-temporal finite differencing at each pixel, the two unknowns (u and v) cannot be uniquely determined from (3) alone. Known as the aperture problem, this ambiguity translates to the unique computation of motion along the direction of the gradient (η_x, η_y), but not perpendicular to it. Additional constraints are required to arrive at a unique solution. Among many approaches to address this problem, the Lucas–Kanade method assumes that the OF within a small neighborhood W (e.g., an n × m rectangular patch) of each pixel (x, y) is constant [43]. This yields a linear system of equations for the n × m pixels within W with only two unknowns:

$$\left[\begin{array}{c}\begin{array}{cc}{\eta }_x\left(1\right)& {\eta }_y\left(1\right)\\ {\eta }_x\left(2\right)& {\eta }_y\left(2\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\eta }_x\left(nm\right)& {\eta }_y\left(nm\right)\end{array}\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}-{\eta }_t\left(1\right)\\ -{\eta }_t\left(2\right)\\ ⋮\end{array}\\ -{\eta }_t\left(nm\right)\end{array}\right]$$

(4)

The least-squares optimization applied to the above over-constrained linear system of the form $A_{nm\times 2}u=B_{nm}$, where u(u, v) is the optical flow vector, yields the following solution:

$$u={\left(A^{T}A\right)}^{-1}{A}^{T}B$$

(5)

Mathematically, we require a near-unity condition number of $A^{T}A$ for a robust solution. In our application, overcoming the aperture problem, this translates to having sufficient variation of η within W. Generally, SSH maps have sufficiently rich texture within a patch size of 5 × 5 pixels, which we assume in this work (Figure 2a). Any land and out-of-bound pixels falling within W are discarded, reducing the number of equations in (4). For the grid spacing of 1/4° and the patch size of 5 × 5 pixels, the spacing between the estimates of u(u, v) based on non-overlapping W is 1.25°. This is less than the effective spatial resolution of SSH maps everywhere within the domain.

Recall that the derivation of (3) assumes small displacements ∆x and ∆y, on the order of one pixel. For larger displacements, which may happen in dynamically active regions such as the ACC and western boundary currents, the Taylor series approximation (2) and the OF constraint Equation (3) are no longer valid. This is resolved by the recursive implementation of the Lucas–Kanade method applied to multi-resolution Gaussian pyramidal representation of SSH maps [44]. The OF is then computed recursively, starting from the lowest-resolution level of the pyramid, and propagating the resultant flow to and updating it at the next higher-resolution level until the original map (with the highest resolution) is reached (Figure 2b). The number of levels in the pyramid is set to achieve a motion of roughly one-pixel at the lowest resolution. In this work, we use three levels, where the original SSH map resolution is reduced by a factor of two relative to the previous level.

Unlike the space–time-lagged correlation analysis, the OF technique does not require spatial and temporal constraints to isolate the required scales of ocean variability. Instead, it estimates the motion of individual pixels that may belong to SSH patterns of any spatial scale. The scale separation can be performed at the stage of data preparation by applying spatial and temporal filtering, as was done in this work by removing the global mean SSH, the seasonal cycle, and the low-frequency signals with periods greater than 1.5 years. The residual variability is dominated by transient mesoscale eddies. As mentioned above, because u and v are computed between two consecutive images, applying the OF to a series of SSH maps yields the time series of the propagation velocities of SSH patterns.

3. Results

3.1. Time–Mean Eddy Propagation

The intensity of eddy variability in the South Atlantic is demonstrated by the standard deviation of SSH (Figure 1). The most energetic regions are associated with the Agulhas Retroflection, the Agulhas Return Current, and the Brazil–Malvinas Confluence. Elevated eddy energy is also observed along the ACC, especially near the Drake Passage and between 30–40°E. The time–mean speed and direction of the horizontal movement of SSH patterns associated with transient eddy variability estimated with the OF are displayed in Figure 3. Like the space–time correlation analysis, the OF provides a description of the transient eddy propagation velocity in the Eulerian frame of reference. The method reveals the relatively swift propagation of eddies along the Agulhas Current flowing along the southeastern coast of Africa with velocities of 5–7 km/day. Upon reaching the Agulhas Retroflection, the eddy motion becomes weak, and no eddies tend to follow the eastward Agulhas Return Current. Once new eddies are formed by the Agulhas Retroflection (around 20°E, 40°S), they start propagating westward-northwestward with a speed of 4–5 km/day. There is no significant northward eddy propagation along the relatively weak Benguela Current that flows along the southwestern coast of Africa.

Nevertheless, eddies are generated near the eastern boundary and propagate westward as Rossby waves in a vast region north of 35°S. This region represents the northern part of the South Atlantic subtropical gyre. The mesoscale variability towards the equator becomes weak (Figure 1), and westward-propagating signals have smaller amplitudes compared to signals at higher latitudes. The speeds of westward propagation increase from 2–3 km/day at 33–35°S to 5–7 km/day at 10–15°S (Figure 3). There is a remarkable transition between the coherent basin-wide westward propagation north of 33°S and the region south of this latitude and to the east of the Argentine Basin with a very weak westward flow. This meridional contrast is apparently the result of the mean-flow effect on eddy propagation. As illustrated by the contours of the mean dynamic topography (Figure 1), the latitude of 33°S divides the subtropical gyre in two parts: the northern part with the general northwestward direction of the surface flow and the southern part with the eastward direction of surface currents. This means that the mean flow reinforces the westward propagation north of 33°S and compensates for the westward propagation south of this latitude.

Near the western boundary, there is a well-defined band of southwestward velocities of up to 5 km/day between 25°S and 40°S associated with the Brazil Current. Further south, along the Argentine coast, the northward-flowing Malvinas Current is also a pathway of eddy movement, with eddy propagation velocities reaching 7 km/day. The bottom topography plays an important role in constraining the eddy motion, especially in the Southern Ocean where stratification is weak and the vertical extent of eddies can reach the bottom (e.g., [23]). The Argentine Basin represents another good example of eddy propagation steered by the bottom topography (Figure 4). The Southern Ocean eddies enter the basin from the Scotia Sea, mainly through channels in the North Scotia Ridge, and follow the continental shelf break. At about 40°S, eddies carried by the northward Malvinas Current collide with those carried by the southward Brazil Current. This process forms the highly energetic Brazil–Malvinas Confluence—the region with the high standard deviation of SSH (Figure 1) but very slow eddy propagation (Figure 3 and Figure 4).

The most remarkable feature of eddy dynamics in the Argentine Basin is the anticyclonic (anticlockwise) propagation of eddies around the Zapiola Rise (ZR). The ZR is a seamount that rises about 1500 m above the Argentine Abyssal Plain (Figure 4). A very strong anticyclonic recirculation gyre extending from surface to bottom and transporting about 100 Sv exists around this topographic anomaly [38]. The eddy propagation around the ZR mimics the recirculation gyre with velocities comparable to the mean flow. The eddy propagation velocities on the northern flank of the seamount are the strongest in the South Atlantic Ocean and often exceed 10 km/day. On the southern flank of the seamount, eddy propagation is slower with velocities of 1–3 km/day. It is interesting to note that the mean SSH slope is steeper to the south of the seamount crest than to the north (see contours in Figure 1), meaning that the eastward surface geostrophic velocity is greater south of the ZR than the westward surface geostrophic velocity north of the ZR. This suggests that eddy propagation velocities around the ZR are mostly affected by currents at depth rather than by the near-surface flow.

The ACC in the Southern Ocean has a sufficiently strong eastward component that reverses the inherent westward direction of eddy propagation and forces them to move in the direction of the mean flow. As illustrated in Figure 3, the zonal eddy propagation velocities within the ACC are much more variable in space than those within the South Atlantic subtropical gyre. Some regions of the ACC show very slow either eastward or westward eddy propagation, but the Drake Passage, the Scotia Sea, and the region east of 20°W are characterized by relatively swift eastward eddy propagation with speeds reaching 4–5 km/day. The latter segment of the ACC is zoomed in Figure 5, showing the contours of the time-mean SSH overlaid on the mean eastward eddy propagation velocity.

It appears that the intensification of the eastward eddy propagation happens in the locations where the ACC is also stronger, associated with steeper SSH gradients and, consequently, stronger surface geostrophic flows. There are two bands of steep SSH gradients or fronts within this segment of the ACC: (i) the Sub-Antarctic Front (SAF) centered approximately along −0.2 m mean SSH contour (red contour in Figure 5) and (ii) the Polar Front (PF) centered at about –0.6 m mean SSH contour (blue contour in Figure 5) [45,46].

3.2. Time-Variable Eddy Propagation

As noted earlier, the OF method estimates apparent motion between two consecutive images. By computing the OF between each pair of SSH maps (1075 pairs in total), we can investigate the temporal evolution of the eddy propagation velocity. Changes in the eddy propagation velocity can occur due to changes in the background flow and/or changes in the stratification. Chelton et al. [47] concluded that the effects of temporal variations of the stratification can be neglected for the purposes of estimating the phase speed of Rossby waves. They noted, however, that the lack of hydrographic time series at fixed locations make the quantitative estimates of the importance of temporal variability of the stratification difficult to obtain. In the following, we evaluate the temporal evolution of the eddy propagation velocity estimated with the OF method in the ACC region considered above (Figure 5). The region was selected because of the significant influence of the background flow that reverses the intrinsic westward eddy propagation. Therefore, by comparing the variability of the eddy propagation velocity with the variability of the SSH gradient or the geostrophic flow in the region, it is possible to analyze how robust the time-dependent part of the eddy propagation velocity is.

It appears that “instantaneous” velocities are rather noisy and some spatial and temporal filtering is required. The noise can be due to the measurement and optimal interpolation errors of SSH maps, as well as due to the local violation of the brightness-constancy assumption in highly energetic regions. Displayed in Figure 6a is the time–latitude diagram of the zonal component of the eddy propagation velocity averaged between 10°W and 10°E and low-pass filtered with a Lowess filter with a cutoff period of 3 months. Overlaid are the contours of the largest meridional SSH gradients. In the zonal segment considered, both the eastward eddy propagation velocity and the SSH gradient exhibit two distinct bands between about 44–46°S and 48–50°S associated with the SAF and the PF, respectively. Here, the eastward eddy propagation velocities are 2–4 km/day and the meridional SSH gradients often exceed 0.2 m per 100 km. Further downstream, these two fronts converge at 30–40°E (Figure 5).

The meridional position of the two fronts has remained rather stationary during the period of observations, but both the SAF and the PF have intensified over the observational period, as suggested by the SSH gradients. What is interesting is that in some periods the intensity of the fronts is correlated with the eastward eddy propagation velocity. The time series of the eddy propagation velocity and the meridional SSH gradient averaged zonally between 10°W–10°E and meridionally between 44–46°S for the SAF and 48–50°S for the PF are shown in Figure 6b and Figure 6c, respectively. The correlations between the eddy propagation velocity and the SSH gradient are 0.27 for the SAF and 0.46 for the PF, both significant at 95% confidence. Particularly good agreement between the eddy propagation velocity and the SAF intensity is observed in 2001–2009 and in 2015–2021 (Figure 6b). The agreement between the eddy propagation velocity and the PF intensity is more pronounced at interannual time scales, showing the general tendency for the intensification of both the SSH gradient and the eastward-propagation velocity (Figure 6c). Although there is no perfect match between the time series of the collocated eddy propagation velocity and the SSH gradient, the reasonable agreement between the two independent parameters suggests that the optical flow can detect the temporal evolution of the eddy propagation velocity, at least in the regions with strong background flow where the signal-to-noise ratio is sufficiently large.

4. Validation of the Optical Flow Estimates

The detection of westward-propagating eddies and Rossby waves in satellite altimetry measurements has been based on the analysis of time–longitude or Hovmöller diagrams of SSH anomalies. A time–longitude diagram is a two-dimensional plot that represents a time series of zonal sections. Westward-propagating signals (waves) are clearly seen as tilted crests and troughs of SSH anomalies moving linearly from right to left across the diagram. The speed of these signals can be easily calculated from the slope of the diagonal alignments of crests and troughs. Displayed in Figure 7 are the time–longitude diagrams of SSH anomalies along the zonal sections of 10°W–10°E at several latitudes that we use to validate the results obtained with the OF method. The diagrams reveal the westward propagation at 15°S, 25°S, and 33°S, i.e., within the South Atlantic subtropical gyre, and the eastward propagation at 45°S, where the Sub-Antarctic Front of the ACC is located.

We use the two-dimensional Radon transform (2D-RT; e.g., [48]) to objectively estimate the speed of propagating signals in the selected time–longitude plots (Figure 7). The 2D-RT has been widely used in satellite remote sensing for the global analysis of Rossby waves (e.g., [49,50,51,52]). The method consists of projecting the Hovmöller diagram onto a line at a range of angles from 0 to 179° (the time axis is at 90°). When the line is perpendicular to the alignment of crests and troughs of the Rossby waves in the diagram, the projected SSH anomalies have the maximum variance (energy). The angle of the perpendicular to the line with the maximum variance yields the speed of propagation. The 2D-RT is best applied to the regions where the speed is reasonably constant within the selected subdomain. This is why the longitudinal extent of the subdomain used in this study was limited to 20°. The zonal eddy propagation velocities between 10°W–10°E estimated with the 2D-RT are −5.3 km/day at 15°S, −3.6 km/day at 3.6°S, −2.8 km/day at 33°S, and 2.2 km/day at 45°S (positive eastward).

In addition, we estimate the eddy propagation velocities using the space–time-lagged cross-correlation analysis similar to [22,23]. In essence, this method is based on computing correlations between the SSH anomalies at each grid point with the SSH anomalies at all the neighboring locations and at various time lags. At each time lag, the location of the maximum correlation is determined, and a velocity is estimated from the time lag and the distance of the location from the origin. An average velocity vector weighted by the correlation coefficients is ultimately computed from the estimates at various time lags. To focus on the mesoscale eddies, the time lags are limited to less than 100 days. The zonal and meridional dimensions of the box, within which the correlations are computed, are set to latitude-dependent correlation scales from [53], ranging from 100 km at 60°S to about 270 km at 10°S.

Displayed in Figure 8 is a comparison of the eddy propagation velocities between 10°W–10°E estimated with the OF method (blue curve), the 2D-RT (red stars), and the space–time-lagged cross-correlation analysis (red curve). The 10°W–10°E segment clearly captures the westward propagation of eddies in the South Atlantic subtropical gyre and the eastward propagation of eddies within the ACC. There is an excellent agreement between the OF estimates and the other two methods, especially south of 20°S. The largest discrepancy is observed in the tropics, where the OF estimates have the strongest variability (standard deviation of the zonal velocity is about 3 km/day). The zonal phase speed of the first-mode nondispersive long baroclinic Rossby waves between 10°W–10°E in the South Atlantic Ocean, estimated from the global atlas of the first baroclinic Rossby radius of deformation [46], is also shown in Figure 8. There is a reasonable agreement between the observed propagation velocities and the ones predicted by the classical theory at mid-latitudes. At high latitudes, the eddy propagation velocity is governed by the ACC and nearly all eddies are advected eastward, which is not accounted for in the theoretical estimate. Equatorward of 25°S, the westward eddy propagation is much slower than Rossby waves, which is consistent with earlier studies [15,23,49].

5. Summary and Conclusions

The pathways of mesoscale eddies are an important informative characteristic of ocean dynamics. The objective of this paper is to introduce an application of the optical flow (OF) technique to satellite altimetry data to estimate the direction and speed of transient eddy propagation in the South Atlantic Ocean. The South Atlantic Ocean and the adjacent segment of the Southern Ocean feature several regions with specific eddy variability, such as the Brazil–Malvinas Confluence, the Zapiola Anticyclone, the Agulhas Retroflection, and the ACC and, therefore, represent an ideal polygon for testing the OF method. The method does not detect/track individual eddies or other SSH patterns. Instead, it provides a description of eddy motion in the Eulerian frame of reference, similar to the space–time-lagged correlation analysis of SSH fields implemented earlier in [22,23].

The obtained time–mean pattern of eddy propagation velocities (Figure 3) is in an excellent agreement with the results of [22], obtained for the same region (see Figure 1 in [22]). This pattern illustrates the latitude-dependent westward propagation of Rossby waves, as well as the interaction of eddies with ocean currents and with the bottom topography. For example, the westward propagation of eddies is faster in the northern part of the subtropical gyre, where it is reinforced by the northwestward background flow, and it is much slower in the southern part of the gyre, where the mean flow opposes and, thus, slows down the westward propagation of eddies. The eddy propagation in the Argentine Basin, including the Zapiola Anticyclone, provides a great illustration of the steering effect of bottom topography on eddy motion. Another example of the eddy–mean flow interaction is the generally eastward propagation of eddies in the ACC, where the strong background flow reverses the intrinsic westward propagation and forces eddies to move in the direction of the current. The eddy propagation velocities estimated with the OF method are validated by the 2D-RT of time–longitude diagrams at several latitudes and by carrying out the space–time-lagged correlation analysis of SSH fields (similar to [22]). The obtained estimates agree with each other very well, providing strong evidence for the utility of the OF algorithm in oceanographic applications to study the motion of eddies and other sea-surface patterns.

An advantage of the OF over the methods used in earlier studies, such as the Lagrangian tracking of eddies and the space–time-lagged correlation analysis, is that it allows the computation of the time-variable part of the eddy propagation velocity. While it has been reported that changes in the stratification probably have a negligible effect on the phase speed of westward propagating signals [47], the variability of strong currents, such as the ACC, does influence the eddy propagation velocity. Therefore, in the regions of strong background flow, the use of the OF to estimate the time-variable eddy motion might be particularly useful. Note that while changes in the surface geostrophic flow are effectively measured by satellite altimetry, changes in the eddy propagation velocity mainly reflect changes in the deep flow, which are not readily available from existing observations. We find that the “instantaneous” velocities calculated by the OF between two consecutive satellite images can be rather noisy, particularly in the regions of strong variability. Therefore, the removal of outliers and some low-pass spatial and temporal filtering of the resulting (u, v) fields might be required.

The validation of the time-variable part of the eddy propagation velocity is problematic because there are no relevant observations or estimates that can be used as the ground truth. In this work, to validate the result obtained with the OF, we use the meridional SSH gradient in the ACC region, which is proportional to the surface geostrophic flow. We show that although the match between the time series of the eddy propagation velocity and the SSH gradient cannot be perfect, a reasonable agreement between the time series indicates that the time-dependent part of the eddy propagation velocity is dominated by the physical signal rather than by noise due to data errors and/or the violation of the brightness-constancy assumption. Therefore, we conclude that the OF method can be used for the determination of time-variable eddy propagation patterns. The OF provides the potential to study changes in the mid-depth and deep circulation in the regions where strong background flow significantly affects the propagation of eddies. Further research is needed to characterize the sources of errors in the OF computation in more detail and to find an optimal method for noise reduction.