1. Introduction
The vastest systematic propagation in the upper ∼1000 m ocean interior is perhaps from nonlinear eddies. The patterns and dynamics of eddy motion have been a subject of active research in modern oceanography. As a clue of eddy kinematics, their trajectories have been individually and statistically analyzed in recent decades (Morrow et al. 2004; Fu 2009; Chelton et al. 2011b), thanks to the life-long tracking ability provided by satellite altimetry since the 1990s (e.g., Le Traon and Dibarboure 1999). The importance of such near-real-time tracking is linked to the prominent role of the eddy in transporting heat and substances (through a parcel of trapped fluid) across the global ocean (e.g., Zhang et al. 2014) given its numerous amount and ubiquitous nature. Over the past half a century, significant progress has been made in characterizing and understanding the intrinsic drift and forced shift of global eddy motions as briefly summarized below.
Theoretically, the analytical solution of oceanic eddy motion can be traced back to that of its atmospheric counterpart, vortex. Two pioneering works with fundamental conclusions are worth mentioning. Bjerknes and Holmboe (1944) noted that the planetary gradient of the Coriolis parameter (the so-called β effect) produces a mass imbalance in a circular vortex, causing it to move westward irrespective of its polarity. Meanwhile, Rossby (1948) pointed out that the β effect exerts a net meridional force on cyclones and anticyclones, directing poleward and equatorward, respectively. A few decades later, McWilliams and Flierl (1979) and Nof (1981) were among the first physical oceanographers to follow their meteorological pioneers and determine the intrinsically westward drift of isolated baroclinic eddies under planetary effect. As far as the propagation speed is concerned, Nof’s formula is extended by Killworth (1983), and generalized by Cushman-Roisin et al. (1990) (see section 2c for further discussion).
Observationally, the patterns and dynamics of eddy motion are much more complex since none of the true eddies can be ideally isolated to a free drift mode. This can be immediately understood by going through relevant literature of the past two decades. Instead of focusing on isolated eddies, Fu (2009) proposed a maximum cross correlation method to derive the propagation features of global oceanic variability. This approach assumes that the localized overall variability is dominated by mesoscale eddies, but also with contributions from fronts, current meanders, and planetary waves. Outside the equatorial zone, the direction of zonal propagation is westward and the actual velocity of propagation is influenced by the mean flow. The westward eddy propagation can be enhanced by westward mean currents such as the tropical current systems, or reversed in the presence of strong eastward mean currents such as the Gulf Stream and the Antarctic Circumpolar Current (ACC). At tropical latitudes, the meridional propagation shows convergence toward (divergence away from) the equator at the western (eastern) part of the basin. At mid-to-high latitudes, the pattern of eddy propagation is highly correlated with the path of mean flows steered by the bottom topography which is most pronounced in the Southern Ocean.
In contrast to the work by Fu (2009), other researchers carried out similar investigations by adopting a different approach of automatic tracking of numerous individual eddies (e.g., Morrow et al. 2004; Chelton et al. 2011b), which largely excludes noneddy mesoscale variances. Considering eddies with a life span longer than three months from different regions, Morrow et al. (2004) found different pathways in the propagation of cyclonic and anticyclonic eddies: Consistently westward for all in zonal movement, but divergently equatorward and poleward in meridional shift for the two polarities, respectively. For example, it is shown that about 2 (3) of the cyclones (anticyclones) propagate poleward (equatorward) in the South Atlantic region west of the Agulhas Retroflection with a speed of ∼0.3 (0.2) km day−1. A comprehensive study on global nonlinear eddy activities has been conducted by Chelton et al. (2011b). They observed that some 75% of the eddies with a lifetime longer than 16 weeks propagate nearly due west with small (about 1.5°) opposing meridional deflections of cyclones and anticyclones (poleward and equatorward, respectively), and with propagation speeds that are nearly equal to the phase speed of the baroclinic Rossby waves. These characteristics are predominantly consistent with the theory of large nonlinear vortices on a β plane (e.g., Nof 1981; Cushman-Roisin et al. 1990).
In addition to the well-known westward propagation and slight polarity-based meridional deflections, Ni et al. (2020) recently showed that mesoscale eddies also move randomly in all directions oceanwide as a result of eddy–eddy interaction. The speed of this movement decreases with latitude and equals the baroclinic Rossby wave speed at about ±25°, separating the global ocean into a high-latitude isotropic turbulence regime and a low-latitude anisotropic wavelike regime.
The compatibility of mesoscale eddy and Rossby wave propagations has been a puzzling issue for decades (e.g., Robinson 1983; Chelton and Schlax 1996; Chelton et al. 2011b; Oliveira and Polito 2013; Polito and Sato 2015). According to Polito and Sato (2015), the streamline of an isolated vortex is approximately circular. The Rossby wave, which normally consists of crests and troughs perpendicular to the direction of propagation, is neither isolated nor circular. In the real ocean, the distinction between mesoscale eddies and Rossby waves is not always clear due to their coexistence. Chelton et al. (2011b) argued that the westward propagating variability consists mostly of nonlinear eddies (given the “blobby” nature of sea surface height, SSH, and the fact that the rotational speed of the SSH feature is mostly greater than its propagation speed) rather than linear Rossby waves. They admitted that the Rossby wave interpretation of propagation features in earlier studies (e.g., Chelton and Schlax 1996) was a consequence of the coarse resolution of SSH fields constructed from TOPEX/Poseidon. In contrast, Oliveira and Polito (2013) found that the use of cross correlation between the sea level anomaly (SLA) and the sea surface temperature (SST) anomaly has the capability to distinguish between signals from baroclinic Rossby waves and mesoscale eddies which are blended in mid latitudes. Polito and Sato (2015) observed that there is a significant portion of eddies coinciding with Rossby wave extrema, the majority of which often stay there for the entire lifetime. Therefore, they concluded that an eddy often rides on a Rossby wave. Based on a series of laboratory experiments, Espa et al. (2020) confirmed that nonlinear eddy–wave interactions facilitate flow self-organization into zonal patterns in which Rossby waves and westward propagating eddies coexist, and are often inseparable as the former maintains the instability that sustains the latter.
Although remarkable progress has been made in characterizing and understanding the pattern and velocity of global eddy propagation thanks to the availability of nearly three decades of altimeter data, a number of critical questions and problems remain open which include but not limited to the following: 1) Besides the primarily westward and secondarily poleward/equatorward classic description, a detailed global phase map of eddy propagation direction is still absent. 2) A hidden cancellation effect in meridional eddy deflection might be overlooked given the ∼1/3 violations of the generic polarity-dependent rules. 3) The confusion between the two intrinsically westward propagations (mesoscale eddies and Rossby waves, described here as “duality”) is far from being resolved. In the present work, substantial efforts have been made to explore these unresolved issues. The rest of the paper is organized as follows: the data and theory for deriving global eddy propagation is described in section 2. The results on divergence of eddy direction and dispersion of eddy speed are provided in section 3. The effects of currents, wind, and bottom topography on eddy propagation are analyzed and discussed in section 4, and some concluding remarks are presented in section 5.
2. Data and theory
a. Merged altimeter data
The altimeter data used in this study are the delayed-time products (available at https://marine.copernicus.eu/) generated by Copernicus Marine Environment Monitoring Service (CMEMS) from a combination of multiple altimeter missions (CMEMS 2020). Specifically, the CMEMS “all-satellite” delayed product of daily merged mean SLA with a grid size of 1/4° longitude/latitude is employed to ensure consistent eddy tracking. Based on a series of eddy identification and tracking schemes (Liu et al. 2016; Sun et al. 2017; Tian et al. 2019), a global eddy dataset of more than 28 years spanning from January 1993 through March 2020 was created.
b. Current, wind, and topographic data
1) Current data
Eleven years (1 January 2006–31 December 2016) of gridded current data used in this analysis are the Mercator Ocean GLORYS2V4 reanalysis product generated by the CMEMS Global Monitoring and Forecasting Centre (available at https://resources.marine.copernicus.eu). The GLORYS2V4 data provide an eddy-permitting (1/4° resolution) global ocean simulation for 75 vertical levels (from 0 to 5500 m). These reanalysis daily fields are built to be as close as possible to the observations while maintaining agreement with the model physics as described in detail in Amanda et al. (2019).
2) Wind data
More than 20 years of daily gridded wind products (available at http://www.remss.com/measurements/wind/) constructed from QuikScat (20 July 1999–31 December 2009, version 4) and WindSat (1 January 2010–31 December 2019, version 7.0.1) measurements are used in this study. These data were produced by the Remote Sensing Systems and came in files with orbital data mapped to a 1/4° longitude–latitude Earth grid. Detailed information can be found in Meissner and Wentz (2009), and Ricciardulli and Wentz (2011).
3) Topographic data
Topographic data (available at https://www.ngdc.noaa.gov/mgg/global/) are obtained from the ETOPO1 1-Arc-Minute Global Relief Model at the National Geophysical Data Center of NOAA. The data processing, evaluation, assembly, and assessment of Etopo1 are described in Amante and Eakins (2009). In addition, the spatial resolution of 1 arc min is resampled to 0.1° for better adaptation to the spatial resolution of other datasets.
c. Theoretically predicted eddy propagation
According to the international thermodynamic equation of seawater 2010 (TEOS-10), the potential density (ρθ) is calculated from the temperature and salinity data using the Gibbs Seawater Oceanographic toolbox (ICO et al. 2010):
(a) Theoretically predicted westward eddy propagation speed using climatological data of temperature and salinity from World Ocean Atlas 2018. (b) Altimeter-derived mean speed of westward eddy propagation during 1998–2020 (same for subsequent figures unless otherwise specified). (c) The difference between (b) and (a). The spatial resolution is 1° × 1° for all panels (same for subsequent figures unless otherwise specified). “Westward” (similar for eastward, northward, southward in subsequent figures) refers to a 90° sector with respect to due west.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
In practice, the eddy propagation direction at each 1° × 1° cell for a given day is defined as a 7-day average (centered at the day of concern) of the daily eddy directions. In addition, eddies with lifetimes less than 10 days are excluded, and “infant” (born within three days) as well as “dying” eddies (within last three days of lifetime) are also discarded due to their instabilities. To create a map of annual climatological directions of eddy motion, a vector composition of eddy propagation velocity within each 1° × 1° cell is conducted. The speed of zonal and meridional eddy movement is defined as the ratio of the distance traveled by the eddy over the time elapsed (i.e., 7 days), on the basis of which the annual mean of the two velocity components in each cell can be calculated. The two orthogonal components are then vectorized to obtain an annual climatological direction. Such a vectorization scheme is expected to minimize the effect of high-frequency perturbations induced by instantaneous shift of individual eddies.
3. Results
a. Divergence of eddy direction
The hemispherically divided directional probability of eddy propagation is shown for both anticyclonic eddies (AEs) and cyclonic eddies (CEs) in Fig. 2a. The four curves share a common feature of a nearly symmetric leaf-like pattern with a west dominance, particularly for the Northern Hemisphere. A secondary tip toward due east with a distinct Southern Hemisphere preference is also evident. The discrepancy between the two polarities appears to be quite small. Such an overall pattern confirms a previous finding that global eddy motions are basically westward but spread in all directions with a slight east peak (Fu 2009; Chelton et al. 2011b; Ni et al. 2020).
(a) Accumulated directional probability map of global eddy propagation for the two hemispheres and two polarities (see legend for details). The azimuthal resolution is 1°. (b) As in (a), but for three selected 1° × 1° cells as indicated in Fig. 3a.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
Figure 3 shows the climatological phase maps of global eddy propagation for the two polarities which are, to our knowledge, the first of their kind. Instead of being straightly westward, the details of the mean eddy propagation direction and the divergence of CEs and AEs varies geographically in complex ways, apparently because of the effects of mean currents on the potential vorticity gradient (PVG; see the appendix). Systematic westward propagation which accounts for approximately 65% of the entire eddy population (mostly within the ±40° region) is apparently the primary mode except for the equatorial zone. In contrast, an eastward secondary mode for nearly 15% of the eddy trajectories can be identified over the ACC and the equatorial currents, as well as the main flows of the Kuroshio and Gulf Stream. As revealed in Fig. 2a, the directional spectrum of eddy motion exhibits a continuum structure. The remaining ∼20% of the eddies move randomly in various meridional directions between the two principal modes, forming an essentially westward pattern with a 360° divergence including a few irregular bands of 180° reversals. Since mesoscale eddies play an important role in maintaining midlatitude oceanic fronts (Jing et al. 2020), the sharp reversals in the phase map may serve as a practical criterion to identify these fronts, notably the southern and northern ACC fronts along the two boundaries of this circumpolar flow (see the sharp red–blue contrast in Figs. 3a,b) as discussed in Chapman et al. (2020).
Global phase maps of eddy propagation direction for (a) AEs and (b) CEs. Site A (20°S, 270°E), B (50°S, 95°E), and C (45°S, 315°E), marked by triangles in (a), are selected for further analysis (see Fig. 2b). (c) A similar map but incorporating both AEs and CEs for an enlarged area surrounding site C as indicated in (a). Only eddies with a lifetime longer than 10 days are used in the calculation (same for subsequent figures unless otherwise specified). The probability values are labeled on corresponding sectors of the color pie with an azimuthal interval of 30°.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
At local scale, we select three sites (see labels A, B, and C in Fig. 3a) to examine their cell-wise features of eddy propagation direction. Site A is located in the tropical southeast Pacific where consistent westward eddy propagations prevail. It is not surprising that its directional radar map has an absolute west dominance (∼95% for the west half plane, see the light purple line in Fig. 2b). On the contrary, site B falls into the Indian Ocean sector of the ACC zone whose radar map mirrors site A with eastward eddy motions accounting for ∼86% of the total population (see the light blue line in Fig. 2b). In contrast, a striking amphidromic structure is observed near site C around which numerous eddies circulate (Fig. 3c, see also the dark purple line in Fig. 2b). This is a novel example of nontidal amphidrome (Chen et al. 2014), which displays a systematic counterclockwise rotation of eddy trajectories with two sharp “U turns” at the west and east ends, respectively. This unique eddy loop is believed to be associated with the Zapiola Anticyclone of the mean circulation in the Argentine Basin (Fu 2006, 2009).
As far as the meridional deflection of eddy motion is concerned, the AE equatorward/CE poleward general description might be oversimplified. This argument is supported by the polarity-divided distributions of eddy deflecting angles as illustrated in the middle column of Fig. 4. The corresponding eddy propagation speeds and origin-normalized eddy trajectories are also presented for reference (see the left and right columns in Fig. 4). Figures 4e and 4f clearly reveal that the movements of 38.4% (41.1%) AEs (CEs) are actually poleward (equatorward), contradicting the existing common understanding. Appropriate statements ought to be such that 1) the meridional deflection of eddies is geographically dependent rather than purely polarity determined. A striking pattern of eastern-and-midlatitude dominance (see the tongue-shaped reddish zones in Figs. 4e,f) is observed for equatorward deflection regardless of eddy polarity, and the opposite is true for a tropical ocean enhancement of poleward deflection. 2) Approximately 60% of the AEs (CEs) deflect equatorward (poleward) irrespective of their meridional speeds (see Figs. 4b,c). The net effect of deflection is AE equatorward/CE poleward for westward propagating eddies, but otherwise for eastward propagating ones (see Figs. 4h,i). In addition, it is noted that there is a slight overall preference of equatorward motion of all eddies (51.7%, see Fig. 4g), as also evidenced in Chelton et al. (2011b, see their Fig. 19). The eastern boundary current systems might be one of the major sources of this net equatorial deflection (e.g., Morrow et al. 2004), as a result of the similarities of the mean structures of these systems and their effects on the mean total potential vorticity gradient in these regions. It may also be related to the spatial phasing of the wind stress curl residuals acting to displace the mesoscale eddies equatorward while suppressing their amplitude (White and Annis 2003).
Geographical distributions of eddy propagation properties. (top) All eddies, (middle) AEs, and (bottom) CEs. (a)–(c) The meridional eddy propagation speed, (d)–(f) the azimuth angle of equatorward (positive) and poleward (negative) eddy deflections, and (g)–(i) the density maps of eddy trajectory (positive/negative for eastward/westward in longitude, and positive/negative for equatorward/poleward of due west) relative to the initial location of each eddy (the percentages of eddy trajectories falling into each quadrant are also marked on the three panels).
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
It should be pointed out that the propagation of mesoscale eddies is actually controlled by the total PVG, as referred to “effective β vector” by Samelson (2010). In the case of a zero mean flow, the total PVG reduces to the β effect. But in the presence of a nonzero mean flow, the PVG vector can deviate considerably from meridional. In most places of the ocean, the deviations are controlled by the vortex stretching terms that arise from vertical shear of the mean currents. The reader is referred to the appendix for a quantitative estimate of the total PVG in the global ocean with further discussions.
It is also worth to check the correspondence between the geometric/dynamic properties and meridional deflection of mesoscale eddies. In doing so, the globally averaged angles of deflection with respect to eddy radius, amplitude, and maximum geostrophic velocity are shown in Fig. 5. For westward propagating eddies of both polarities, the poleward deflection increases when they become bigger, stronger, and faster; CEs are found to be consistently larger than AEs for a given property value, but this appears to be reversed for the eastward propagating larger eddies as also evidenced in Figs. 4h and i.
Variation of eddy meridional deflection against eddy properties: (a) radius, (b) amplitude, and (c) maximum geostrophic velocity.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
b. Dispersion of eddy speed
We now return to the observed full speed of eddy propagation (Fig. 1b) for a comparison with the theoretical estimation (Fig. 1a). As shown in the difference map (Fig. 1c), the altimeter-derived eddy velocity is tremendously lower than the predicted values in the tropical oceans. The opposite is true for the extratropical regions but in a more disorganized manner. This result suggests that the theory of free eddy drift under geostrophic approximation produces comparable results only in limited midlatitude zones of the global ocean. A similar conclusion is also obtained by Oliveira and Polito (2013) who compared the phase speed of Rossby waves and propagation speed of eddies within the zonal band of 10.5°–35.5°S in the South Atlantic. They found that in 70% of the cases, the absolute difference between the two speeds is less than 11%. They further pointed out that a positive bias of 32% ± 11% was observed when comparing their scale-based eddy speed with those presented by Chelton et al. (2011b). In fact, Chelton et al. (2011b) noted that the eddy speeds relative to the local long Rossby wave phase speeds are about 20% slower in the Northern Hemisphere than in the Southern Hemisphere, which serves as additional evidence of the dispersion effect.
The dispersion of eddy propagation speed can be revealed straightforwardly by superimposing the theoretical (thick dashed line) and climatological (thick solid line) speeds of the baroclinic mode of annual Rossby waves on the zonal distribution of data density of westward eddy speed (in color) derived from satellite altimetry as illustrated in Fig. 6a. It becomes immediately evident that a varying degree of dispersion takes place asymmetrically around the theoretically predicted curve at all latitudes whose standard deviation is shown in Fig. 6b. Combining these two panels one recognizes that the numerous eddy propagations largely follow the theory in their westward speeds, particularly in the zonal bands between 20° and 50° of the two hemispheres (see the position of the thin solid line in Fig. 6a). These latitudes coincide with the belts of most concentrated westward eddy propagations. Eddy motions in other three aspects (east, south, and north) also exhibit considerable scatterings with a common equatorial peak in the mean values (Figs. 6c–e), although the maxima of data density are shifted in latitude compared to their west counterpart (Fig. 6a).
(a) Zonal distribution of data density of westward eddy propagation speed derived from altimeter data. Also overlaid are the climatological (thick solid line) and theoretical (thick dashed line) speeds of the annual baroclinic Rossby wave. The thin solid line depicts the zonal mean of data density. (b) Zonal distribution of standard deviation of westward eddy propagation speed with respect to the theoretical speed (thick dashed line) of the annual baroclinic Rossby wave. (c)–(e) As in (a), but for the eastward, southward, and northward eddy propagation speeds, respectively.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
As previously reported, the surface manifestations of westward propagating mesoscale eddies and annual Rossby waves are similar in terms of ocean topography and translation speed, which are therefore heavily mixed in altimeter measurements (e.g., Chelton et al. 2011b). Figure 7a shows the geographical pattern of most inseparable zones of strong eddy–wave coupling in the global ocean, which are identified by eddy translation speed falling into ±10% of the Rossby wave phase speed. Five eddy–wave duality zones (EWDZs) are found around two subtropical belts centered at 27°N and 26°S, which are also evidenced in existing case studies (e.g., Chelton and Schlax 1996; Barron et al. 2009; Chelton et al. 2011a; Oliveira and Polito 2013; Polito and Sato 2015). Coincidentally, for example, Oliveira and Polito (2013) considered 25.5°S as the transition latitude between the dominance of wave and eddy signals (with variability dominated by eddies poleward of it, see their Figs. 3d–f). Globally, the identified transition from wave dominance to eddy dominance in previous studies varies from ±25° (e.g., Chelton et al. 2011b) to ±30° (e.g., Tulloch et al. 2009). Interestingly, Polito and Sato (2015) used a vivid term of “Rossby wave riding by eddies” to describe their main finding of the coincidence of wave extrema (crests and troughs) with a large number of vortices throughout their lifetimes at preferred latitudes, which falls also into our subtropical EWDZs. Note that the color map in Fig. 6a is derived from individual eddy speeds rather than integral slopes of SLA in the time–longitude diagram as performed in the aforementioned studies. It could be expected that with the upcoming launch of next generation interferometric altimeter (such as SWOT with submesoscale resolving capability, see, e.g., Fu and Ubelmann 2014), the practical difficulty of distinguishing waves from eddies may be significantly eased.
(a) Geographical distribution of the percentage of westward eddy propagation speed falling into ±10% of the climatological phase speed of annual Rossby wave corresponding to Fig. 1a. (b),(c) As in (a), but for eddy speed 10% lower or 10% higher than the phase speed of annual Rossby wave, respectively. (d)–(f) The zonal distributions corresponding to (a)–(c).
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
As for the mechanism of eddy–wave duality, a plausible interpretation is that the potential energy of baroclinic waves and kinetic energy of turbulent eddies are convertible at scales on or close to the Rossby deformation scale via nonlinear interactions (Tulloch et al. 2009). Such an energy cascade is unlikely to occur when the eddy and wave scales are unmatched. Furthermore, we try to understand the influence of horizontal scale on Rossby wave speed. Following a comparison between the generalized Rhines scale (which takes into account the wavelength adjustment of the Rossby wave; Rhines 1975) and the Rossby deformation scale (which is assumed to be a lower limit for the wavelength of the observed waves), it is found that the two scales are consistent in the tropical oceans within ±30° (Eden 2007) where the EWDZ is located. That is, eddies scale is the smaller of the deformation radius and the Rhines scale with a critical latitude near 30°, where the deformation scale is similar to the Rhines scale. This argument can also be confirmed by the fact that the Rhines scale is roughly constant in the tropics with a wavelength around 600 km but drops quickly to 200 km in midlatitudes (see Fig. 6d in Tulloch et al. 2009). Moreover, Tulloch et al. (2009) reveal that a transition from a field dominated by waves to one dominated by eddies also occurs at about ±30°, broadly consistent with the transition that is required to fit linear theory to altimetric observations. Therefore, the horizontal scale of the wave will have limited impact on the zonal Rossby wave speed (shown in Fig. 6a) as far as the identified EWDZ (see Fig. 7a) is concerned (given the 27°N/26°S versus ±30° small shifts between our result and their finding).
Outside the EWDZs, the eddy–wave speeds are dispersed, and are divided into a tropical wave-faster zone (WFZ, ∼20°S–20°N, as also claimed by Klocker and Marshall 2014) and two extratropical eddy-faster zones (EFZs, poleward from ∼±40°, see Figs. 7b,c). An implication of such a dispersion is that wave dominance and eddy dominance can be respectively expected within the WFZ and EFZs in terms of dynamic energy. As observed by Chelton and Schlax (1996), the latitudinal structure of sea level associated with tropical Rossby waves has two symmetric maxima at 4°N and 4°S with much more apparent westward propagations. On the other hand, according to Chelton et al. (2011b) the percentage of eddy kinetic energy explained from 30% to 40% in the quiescent oceans to 60%–70% in most of the eddy-abundant regions typically around 49°N and 59°S of the two hemispheres (see Fig. 6e of Chen and Han 2019), which belong to our EFZs. Besides, it can also be anticipated that relatively large eddies prevail in the WFZ while the opposite is true in the EFZs (see Fig. 6 in Chen et al. 2019).
The latitude-dependent duality/dispersion relationships between eddy and Rossby wave propagations can be further revealed by superimposing theoretical annual baroclinic Rossby waves on individual tracks (instead of gridded SLA) of AEs and CEs in time–longitude diagrams as shown in Fig. 8. As far as the South Pacific sector of 200°–260°E is concerned, the wave–eddy propagation relation can be divided into four zones: 1) WFZ (Figs. 8a,b); 2) EWDZ (Fig. 8c); 3) EFZ (Figs. 8d,e); 4) reversal zone (Fig. 8f) in which eddies propagate eastward against Rossby waves. Also evidenced in these panels are the decreasing westward eddy propagation speed down to 45°S (Figs. 8a–e), short versus long eddy lifetimes in low/high and middle latitudes (Figs. 8a,d,f), as well as two highly productive eddy sources around (15°S, 230°E) and (55°S, 230°E) (Figs. 8b,e).
Time–longitude diagram of eddy propagation at six selected latitudes in the southeast Pacific Ocean. The red and blue lines correspond to individual tracks (within ±1° latitude) of AEs and CEs, respectively. The solid and dashed black lines indicate the crests and troughs of theoretical annual Rossby wave propagations.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
4. Discussion
In addition to the intrinsically westward propagation of oceanic eddies, external forcings such as current, wind, and topographic effects may also strongly modulate their paths and velocities (Fu 2006; Wilson 2016).
a. Current effect
The global distribution of depth-mean ocean current variability derived from GLORYS2V4 is shown in Fig. 9. The left column displays (from top to bottom) the west, east, south, and north components of the flow magnitude in which zonal dominance is apparent, especially for the east component associated with the western boundary currents, the equatorial currents, and the ACC (Fig. 9b). Strikingly, the advection of eddies by the depth-mean velocity is able to explain, through the Doppler effect, the observed eastward (reversed) propagation over the ACC (Fu 2009; Klocker and Marshall 2014). In fact, the real impact on eddy propagation is, to a large extent, determined by the phase pattern (rather than magnitude) of the global circulation system (Fig. 9g). Comparing Fig. 9g and Figs. 3a,b one finds that a significant phase coherency exists between currents and eddies with several common features: equatorial east, tropical west, and extratropical east. Such a similarity suggests that the ocean current is the primary external dynamic factor for reshaping eddy motion, as is largely confirmed in the probability maps of west- and east-going currents in association with eddy propagations of the same direction (Figs. 9e,f). In addition, a secondary effect of the currents in determining the mean eddy propagation direction is through the total vorticity gradient, which modifies the β effect as discussed above.
Global distribution of depth-mean ocean current (0–2000 m) derived from GLORYS2V4 for the period 2006–16. (a)–(d) The west, east, south, and north components of the depth-mean flow. (e),(f) Probability maps of westward and eastward currents in association with eddy propagations of the same direction. (g) A global map of depth-mean current direction with probability values labeled on corresponding sectors of the color pie.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
To further reveal the eddy–current correspondence, radar maps of current and eddy velocities are plotted in Fig. 10 (top panels). The overall match between eddy (contours) and current (colors) data density in the speed-direction domain implies that the ocean current plays a critical role in modulating eddy activities. This argument is particularly true when comparing the westward and eastward subsets (left and right panels in Fig. 10), although the nonlinear relation between current and eddy motions becomes more evident for the latter. Note that the net velocity of eddy motion is introduced by a combination of its intrinsic westward propagation and the advection of mean flow over the depths of its vertical extent (Fu 2006). In particular, it is believed that the eddy motion is mostly affected by the current at the depth of eddy core within 1000 m of the global upper ocean (see Fig. 4 in G. Chen et al. 2021), which is polarity sensitive with CEs being much deeper than AEs at the same location.
Radar maps of depth-mean ocean current and eddy propagation velocities for (a) westward and (b) eastward propagating eddies. Colors and contours correspond to data density of current and eddy, respectively. The northeast- and northwest-oriented scales correspond to current velocity and eddy speed, respectively. (c),(d) As in (a) and (b), but for combined wind stress and eddy propagation velocity. The northeast- and northwest-oriented scales correspond to wind stress and eddy speed, respectively.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
b. Wind effect
Previous studies have noted that wind stress acts systematically (energize or damp) on ocean eddies (e.g., Dewar and Flierl 1987; McGillicuddy et al. 2008; Gaube et al. 2015; Xu et al. 2016; Wilson 2016). On the one hand, the eddy–wind interaction provides an energy flux into or out of the eddy field, which could have a direct impact on eddy energetics (Wilson 2016; Eden and Dietze 2009; Zhai and Greatbatch 2007). Ocean eddies are proved to be systematically energized when the wind stress shear is in the same direction as their rotation, while are damped if the wind stress shear is in the opposite direction (Xu et al. 2016). This variation of eddy energy will inevitably affect its dynamics, lifetime, and propagation conditions. On the other hand, the SST signatures in eddies alter the wind stress that is capable of modifying eddy dynamics. It is found that the eddy SST-induced wind stress residuals produce residual Ekman pumping that results in anomalous vertical motions in the interior of the eddies. The vertical Ekman pumping alters local potential vorticity gradient, leading to the equatorward trends for warm eddies, while poleward trends for the cold eddies (White and Annis 2003; Seo et al. 2016). Although these mechanisms have been effectively proposed, the global pattern of eddy–wind coupling remains unclear at least from an observational perspective. The purpose of Fig. 11 is therefore to reveal the geographical characteristics of the eddy–wind correlation on a global scale based on satellite observations. Going through Figs. 11a–d and Fig. 11g, the components of wind stress in four directions as well as their combined phase map are clearly evident, among which the pronounced westerlies in the midlatitudes of the two hemispheres are particularly noteworthy. For westward-propagating eddies, the strong relationship with wind stress exerts a strengthening effect in the tropical oceans via the mechanisms discussed above, while the reversed direction of the wind stress in the extratropical latitudes may considerably hinder their movement in an opposite way (Fig. 11e). For eastward propagating eddies, however, a remarkable eddy–wind interaction can be expected in the ACC region, which in turn supports the argument that wind stress is one of the major driving forces for the eastward propagation of eddies in the Southern Ocean (Fig. 11f). As a result, the spatial characteristics of wind stress modulation on eddy motion in terms of strength and consistency are intuitively demonstrated, which provide an observational confirmation for both theoretical analysis and model simulation.
As in Fig. 9, but for sea surface wind stress variability derived from scatterometers (see section 2b for specific data periods).
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
The influence of wind stress on eddy motion is visually similar to that of current in a climatological sense. This can be seen by comparing corresponding panels between Figs. 9 and 11, as a result of the close coupling between the upper ocean and the lower atmosphere. The only exception appears in the equatorial zone where the doldrums play little role in eddy dynamics. It has to be pointed out, however, wind impact on eddy motion is obviously indirect and thus much weaker compared to current effect. This statement can be somehow verified by comparing the corresponding top and bottom panels in Fig. 10 where the pattern of data density of wind speed stretches slightly eastward in contrast to the considerable westward preference of eddy speed.
c. Topographic effect
Apparently, the propagation of eddies is affected by the local topography, especially when interacting with a continental slope, an island, or a seamount. Numerical studies have shown that the barotropically dominated eddy propagation strongly depends on the orientation and magnitude of the topographic slope and the barotropicity of the eddy (e.g., Smith and O’Brien 1983; Grimshaw et al. 1994; Jacob et al. 2002). Although barotropical structure is not typical of mesoscale eddies, the simulations may give theoretical insight into vertical interactions as influenced by topography. However, in cases in which eddies are initially surface-intensified whose main core is not in direct contact with the topography, vertical baroclinic instability of these eddies becomes important. It has been demonstrated that the depth of eddy core is ∼226 m on global average [see Fig. 4 in G. Chen et al. (2021) for the geographical distributions], with pycnocline displacements of more than ±40 m due to the upward/downward trapping of the interior water parcels for cyclonic/anticyclonic vortex (X. Chen et al. 2021). Interactions between surface-intensified baroclinic eddies and topography have been considered in several laboratory, numerical and theoretical studies (e.g., Thierry and Morel 1999; Sutyrin and Grimshaw 2010). In theory, these eddies are mainly affected by topography in two aspects. First, topography gradients would affect the evolution of eddies’ vertical structure by the baroclinic stability of the surrounding fluid. In the baroclinically unstable ocean, the eddy is regarded as a potential vorticity anomaly whose motion is influenced by the variations of the vertical shear. When encountering a bottom slope, the topographically induced flow dispersion leads to an asymmetric eddy structure which then gives the eddy a nonlinear self-advective tendency, and further alters its kinetics and propagation trajectory. Meanwhile, the estimated radius of deformation over a steep bottom (corresponding to topography-induced baroclinic modes) is 20%–50% larger than over a flat bottom. Having a larger radius will in return impact eddy propagation since the phase speed of long planetary wave varies with the square of the radius (LaCasce and Groeskamp 2020). Second, topography with steep gradients can scatter disturbances created by the upper-layer eddy motions by producing smaller-scale eddies or lower-layer feedbacks which would evolve quickly to the upper layer under the influence of topographic Rossby waves. This generated lower-layer flow plays an important role in controlling eddy propagation and dispersion over both the abyss and near the slope and shelf (e.g., Sutyrin et al. 2003; Vic et al. 2015).
To examine the effect of sea floor topography on eddy motion, the variations of its propagation speed anomaly (positive for acceleration between two successive days) against topographic gradient (positive for uphill) for different depth ranges are shown in Fig. 12. Going through the five panels, three characteristics become clearly visible irrespective of topographic aspect. First, the translation speed slows down when eddies climb upward, and vice versa. Second, a higher acceleration rate (both positive and negative) is observed in waters shallower than 3000-m depth, especially in the upper 1000 m ocean. As the eddy perturbation decays with depth, its vertical imprint almost disappears below 1000 m, causing a gradual diminishing of the topography effect on eddy motion. Third, the bottom relief has the least impact on eastward propagating eddies compared to those toward other three aspects. Or more generally, the impact of topography on eddy’s meridional motions is greater than that of zonal motions. It is consistent with theory that the steep-induced baroclinic instability leads to the local change of the potential vorticity gradient, which makes eddy produce the trend of meridional deflection, along with a more pronounced variation of meridional velocity. Additionally, the shadow effect of island and shelf on eddy propagation is also recognizable in the phase maps (see Figs. 3a,b). To summarize, the topographically steered current modifies the potential-vorticity-gradient-induced eddy propagation, and the topographic influence is determined by the orientation and magnitude of the bottom slope, as can be theoretically expected from potential vorticity conservation (Jacob et al. 2002).
Variations of eddy propagation speed anomaly (positive for acceleration between two successive days) against ocean topographic gradient (positive for uphill) for different depth ranges for (a) all, (b) eastward, (c) southward, (d) westward, and (e) northward propagating eddies and their corresponding topographic aspects. The red, yellow, green, and blue lines represent ocean depths of 500, 1000, 2000, and 3000 m, respectively. Note the two opposite vertical scales for uphill and downhill in each panel.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
5. Concluding remarks
Using 28 years (1993–2020) of satellite altimeter data along with simultaneous current, wind, and topographic data, the directional divergence of global eddy propagation has been climatically investigated, and the speed dispersion of it has been systematically explored.
In terms of the divergence of eddy propagation, existing understanding favors a primary zonal divergence with a west dominance, and a secondary meridional divergence with polarity dependency. Based on the eddy phase map generated in this study, however, we reveal that the global eddy motion actually has a full-angle divergence with a leaf-like radar pattern. The traditional view of polarity dependency of eddy deflection (i.e., poleward for CEs and equatorward for AEs) is found to be theoretically true but practically misleading. In fact, geographically correlated discrepancy is equally important to polarity induced deviation given the ∼40% of the violations, although the net effect still follows the theoretical rule.
As far as the westward eddy propagation speed is concerned, there has been a long-standing puzzle between the “compatible” and therefore confusing mesoscale eddies and the annual Rossby waves. We approach the problem by introducing two key concepts of “duality” and “dispersion.” It is argued that the eddy–Rossby wave relationship has a duality nature (tight eddy–wave coupling, or waves riding by eddies) in five subtropical zones centered around two critical latitudes (27°N and 26°S) in the two hemispheres, outside which their relationship has a dispersive nature with waves propagating faster in the tropical oceans and otherwise in the extratropical oceans. Another perspective to conceive the eddy–wave duality is that the observed systematic westward propagation exhibits more large-scale wave characteristics in the tropical oceans while more mesoscale eddy characteristics in the extratropical oceans. Such a conceptualization is expected to better unify existing theories and observations.
Finally, it is worth pointing out that the results obtained in this analysis are believed to be statistically significant in a climatological sense, and hopefully allow a more accurate tracking of eddy trajectory, which may lead to improved weather forecasts and climate predictions for oceanic eddies.
Acknowledgments.
This research was jointly supported by the National Natural Science Foundation of China (Grant 42030406), the Wenhai Program of the S & T Fund of Shandong Province for Pilot National Laboratory for Marine Science and Technology (Qingdao) (Grant 2021WHZZB1500), and the Ministry of Science and Technology of China (Grant 2019YFD0901001). We thank the two anonymous reviewers for valuable comments and suggestions that greatly improved the manuscript. We also thank Dr. Graham Quartly at PML U.K. for helpful discussions during the writing of this manuscript.
APPENDIX
Eddy Meridional Deflection Determined by the Total Potential Vorticity Gradient
It has been revealed decades ago that the meridional deflection of eddies from due westward propagation is controlled by the beta effect (e.g., McWilliams and Flierl 1979; McDonald and Straub 1995; Early et al. 2011). However, in most places of the ocean, the potential vorticity gradient (PVG) forced by the vertical baroclinic instability of the large-scale flow would deviate significantly from meridional, which is regarded as an “effective beta vector” (Walker and Pedlosky 2002; Samelson 2010). This will inevitably exert a driving force on the deflection of eddies and thus affect their trajectory. Here we examine the essential role of PVG deflection in eddy propagation based on the full background shear and stratification in a global hydrographic dataset, and the 1° × 1° gridded Argo product (available at http://apdrc.soest.hawaii.edu/projects/Argo/data/gridded/On_standard_levels/) which provides temperature (T) and salinity (S) data of the upper 2000-m ocean with 27 depth levels. The total PVG is estimated according to Tulloch et al. (2009) with the following procedures.
The global distributions of the zonal/meridional components of the total PVG are shown in Figs. A1a and A1b, from which the total PVG vector at each location can be composed (Fig. A1c). The overall direction of the PVG is quite consistent with various current systems, such as the equatorial currents, the western boundary currents, and the circumpolar currents. We further estimate the meridional deflection of the PVG in the global ocean (Fig. A1d), and find that it has a general similarity to Figs. 4d–f, which show the pattern of eddy meridional deflection under the β effect. In typical areas where eddies are seen to be significantly deflected (e.g., the North Pacific, the southeast Pacific, and the North Atlantic), the total PVG bears a similarly enhanced deflection behavior with a stronger magnitude, implying that the vertical shear of the current field is one of the main controlling factors of eddy meridional deflection. It is evident that in the eastern boundary of the subtropical Pacific, PVG exhibits a significant equatorial deflection, which leads to the equatorial deflection of eddies (either AE or CE). In the North Atlantic, the poleward deflection of the PVG drives the eddies to produce a similar shift. This geographical correlation further indicates that the meridional deflection of eddies is geographically correlated rather than polarity determined. However, limited by the insufficient accuracy of the hydrographic data and the inconsistent spatial–temporal sampling between altimeters and Argo floats, there are still some regions where the two patterns are uncorrelated or even anticorrelated, although the deflection of eddies generally follows that of the PVG from a global perspective. Given the above results and discussion, the effective impact of the PVG deflection on eddies’ meridional motion is at least qualitatively confirmed by current hydrological data, but can be quantitatively improved when better hydrological measurements become available in the future.
Global distributions of the (a) zonal and (b) meridional components of the total PVG computed using Argo T/S data. (c) Global phase map of the PVG direction. (d) Azimuth angle of the equatorward (positive) and poleward (negative) deflections of the total PVG.
Citation: Journal of Physical Oceanography 52, 4; 10.1175/JPO-D-21-0122.1
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