Title: Geographical distribution and anisotropy of the inverse kinetic energy cascade, and its role in the eddy equilibrium processes
Abstract: Journal of Geophysical Research: OceansVolume 120, Issue 7 p. 4891-4906 Research ArticleFree Access Geographical distribution and anisotropy of the inverse kinetic energy cascade, and its role in the eddy equilibrium processes Shihong Wang, Shihong Wang Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China College of Earth Science, University of Chinese Academy of Sciences, Beijing, ChinaSearch for more papers by this authorZhiliang Liu, Corresponding Author Zhiliang Liu Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, ChinaCorrespondence to: Z. Liu, [email protected]Search for more papers by this authorChongguang Pang, Chongguang Pang Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, ChinaSearch for more papers by this author Shihong Wang, Shihong Wang Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China College of Earth Science, University of Chinese Academy of Sciences, Beijing, ChinaSearch for more papers by this authorZhiliang Liu, Corresponding Author Zhiliang Liu Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, ChinaCorrespondence to: Z. Liu, [email protected]Search for more papers by this authorChongguang Pang, Chongguang Pang Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, ChinaSearch for more papers by this author First published: 23 June 2015 https://doi.org/10.1002/2014JC010476Citations: 17AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The geographic character of the inverse cascade is analyzed based on the spectral kinetic energy flux calculated in the global ocean, using sea surface height (SSH) data from satellites, reanalysis data, and model outputs. It is shown that the strongest inverse cascade occurs mostly in high-energy eastward-flowing currents, such as the Antarctic Circumpolar Current (ACC), the Kuroshio Extension, and the Gulf Stream, which matches the global distribution pattern of the eddy kinetic energy (EKE). Hence, the eddy scales predicted by the local linear baroclinic instability and from the altimeter observation are mapped out and compared with the energy injection scale and the arrest-start scale of the inverse cascade, respectively. Generally, agrees well with in the midlatitude and high-latitude oceans, especially in the Northern Hemisphere. falls within the arrest ranges of the inverse cascade and is quite close to . Finally, the depth dependence and the anisotropy of the inverse kinetic energy cascade are also diagnosed in the global ocean. We have found that the strength of the inverse cascades decreases with increasing depth, but the global pattern of the strength is nearly invariable. Meanwhile, the variations in depth hardly affect the and . After considering the anisotropy in the spectral flux calculation, a possible inertial range for the zonal spectral kinetic energy flux is expected, where the cascade magnitude will keep a nearly constant negative value associated with the oceanic zonal jets. Key Points: The spectral kinetic energy flux is calculated in the global ocean The inverse cascade may explain the eddy-scale gap Results may explain multiple zonal jets observed in the ocean 1 Introduction Satellite altimeter data indicate that the sea surface height (SSH) variability is dominated by mesoscale eddy signals, with temporal and spatial scales of approximately 50–200 days and 100–500 km, respectively. The eddy kinetic energy (EKE) is about 50 times that of the mean flow [e.g., Robinson, 1983; Stammer, 1997], and the approximate agreement between the linear theory and observations support the view that the baroclinic instability is the main EKE source [Ferrari and Wunsch, 2008]. Additionally, a fully developed geostrophic eddy is not a simple linear process, but rather a turbulent process, involving excitations and interactions on various scales. How these mesoscale eddies interact among themselves and how the energy transfer associated with these interactions takes place are interesting questions. As is well known, the 2-D geostrophic turbulence theory predicts an inverse kinetic energy cascade [Taylor, 1917; Charney, 1971; Salmon, 1978, 1980]. Additionally, the 3-D theory predicts a forward cascade in the baroclinic modes for the total energy, including the kinetic energy and potential energy, at scales larger than the Rossby deformation scale [Charney, 1971; Salmon, 1980; Fu and Flierl, 1980]. While recent studies have employed concurrent satellite altimeter missions to examine the energy pathways in a wave number space at mesoscales, a robust inverse kinetic energy cascade above the deformation scale was found [e.g., Scott and Wang, 2005, hereafter SW05; Qiu et al., 2008; Tulloch et al., 2011; Arbic et al., 2013]. Because the altimeter data predominately reflects the first baroclinic mode [Wunsch, 1997], the inverse kinetic energy cascade observed by the altimeters seemingly contradicts the theoretical predictions. Motivated by the need to address this discrepancy, Scott and Arbic [2007, hereafter SA07] used a two-layer quasi-geostrophic (QG) model to explore the energy pathways in geostrophic turbulence and found a robust inverse cascade of kinetic energy for the baroclinic mode, which makes the most significant contribution to the surface layer inverse cascade. Meanwhile, the well-known forward cascades of the baroclinic potential and total energy were also robust above the deformation scale. It should be noted that Arbic and Flierl [2004] once proposed that turbulence damped by strong bottom friction is equivalent to barotropic and mid-oceanic eddies between the equivalent barotropic and barotropic limits. This may cause the inverse kinetic cascade to be directly analogous to the 2-D geostrophic turbulence cascade. Additionally, in the surface quasi-geostrophic (SQG) model, which is driven by the density anomaly evolution at the boundaries [Held et al., 1995], Capet et al. [2008] also showed that the surface kinetic energy experiences a clear and significant inverse cascade over a wide range of scales. Although the physical mechanism behind the observed inverse cascade is still unclear, the existence of the inverse kinetic energy cascade is robust. Here we are very interested in the geographical distribution characteristics of the inverse cascade in the global ocean. Previous researchers have done valuable work on the spatial distribution pattern. SW05 calculated the spectral fluxes within the unoverlapped subdomains containing 64 × 64 grids and obtained the geographic distribution of the spectral fluxes in South Pacific. Tulloch et al. [2011] calculated the spectral fluxes at some chosen locations in the oceans around the globe. For each location, the SSH were partitioned into overlapping boxes with 32 × 32 grid points, across zonal intervals of 1/3°. Meanwhile, it is shown that both the choice of the window function used to suppress the Gibbs phenomena before taking the discrete Fourier transform (DFT) and the area of the subdomain may induce some differences in the spectral energy flux. For example, Arbic et al. [2013] compared the Tukey and overlapping Hanning window, and noted that the estimated spectral fluxes are sensitive to the spatial window; SW05 investigated the dependence of the flux on the rectangular area and found a small shift toward the smaller scale when smaller grid sizes were used. In this study, we will explore the global spatial distribution of the inverse cascade, focusing particularly on the amplitude, energy injection scale, and scale at which the upscale energy starts being arrested, which may be helpful for further analysis of the physics of inverse cascades. Additionally, the gap between the observed eddy length scales and the spatial scales predicted by linear theory is well known: the observed eddy field has the maximum kinetic energy at wave numbers smaller than the wave numbers of the fastest growth predicted by the linear stability theory [Salmon, 1980; Stammer, 1997; Smith, 2007; Ferrari and Wunsch, 2008]. As shown in Figure 1, the spatial scales predicted by the linear theory (dotted line) are significantly smaller than the observed eddy scales (solid line) (specific calculation details are given in section 3): the difference is about 100 km. This means that the linear theory fails to predict the spatial scales of eddies, although it successfully predicts their temporal scale [Smith, 2007], and a nonlinear interaction must be introduced. Chelton et al. [2011] noted that almost all of the observed mesoscale features, tracked by an automated procedure from the SSH outside the tropical band 20°S–20°N, are nonlinear by the metric , where U is the maximum circum-average geostrophic speed within the eddy interior and c is the translation speed of the eddy. Figure 1Open in figure viewerPowerPoint Zonally averaged observed eddy length scales, (solid line), estimated from the altimeter-observed sea surface height (SSH) anomaly; and zonally averaged eddy scales predicted by linear theory, (dotted line), and estimated by solving the linearized quasi-geostrophic (QG) equation with WOA09. Because the inverse cascade, initially observed by SW05, transfers the kinetic energy from scales close to the deformation radius to larger scales, it might be able to bridge the gap between the linear theory and the observations. The energy injection scale of the inverse cascade defined by SW05 is the smallest scale at which the inverse cascade begins to occur. Therefore, the consistency between the energy injection scale and the most unstable scale predicted by the linear theory is an interesting question. Tulloch et al. [2011] compared the energy injection scales with the eddy scales predicted by the linear theory, and found that both the scales have a general trend and vary less strongly with altitude than the Rossby Deformation Radius. In this study, we will also compare the two scales in the global ocean, similar to Tulloch et al. [2011]. Additionally, we will explore the consistency between the scale at which the inverse cascade starts to be arrested and the observed eddy length scale to show that the observed eddy length scale falls within the arrest ranges of the inverse cascade. There may be two primary possible "sinks" of the upscale kinetic energy. In the first case, the upscale energy is transferred down the water column, analogous to the barotropization in the 3-D geostrophic turbulence, and is finally dissipated in the bottom layer [Vallis, 2006]. A second possibility is that the inverse cascade is anisotropized by the meridional planetary potential vorticity gradient, the β effect, and the flows finally evolve into zonal jets [Rhines, 1975, 1979; Theiss, 2004; Galperin et al., 2010]. In this paper, we will calculate the spectral flux throughout the water column using the model output velocity to diagnose the depth dependence of the inverse cascade. Up to now, most of the studies on the observed inverse kinetic energy cascade focused on the isotropic flux calculated with a total wave number [Scott and Wang, 2005; Schlösser and Eden, 2007; Tulloch et al., 2011]. Therefore, whether an anisotropic inverse cascade in stratified geostrophic turbulence is caused by the planetary potential vorticity gradient, analogous to the effect on the barotropic inverse cascade, remains unanswered. When the effect is introduced in 2-D geostrophic turbulence, inverse cascades become anisotropic, and ultimately results in multiple zonal jets [Rhines, 1975; Holloway and Hendershott, 1977; Galperin et al., 2010]. Recently, both eddy-resolving simulations and observations have revealed robust patterns of multiple zonal jets in the ocean [Maximenko et al., 2005; Richards et al., 2006; Maximenko et al., 2008; van Sebille et al., 2011]. Galperin et al. [2004] argued that the zonal jets are caused by the effect, similar to zonal jets on giant planes because they share similar spectra. However, the direct evidence for the fact that the multiple zonal jets are caused by the effect through an anisotropic inverse cascade is much needed, since the effect can introduce anisotropy even without a cascade [Qiu et al., 2008]. Therefore, another goal of this study is to diagnose the anisotropy of the inverse cascade in the stratified oceans. The paper has been organized as follows. In section 2, the methodologies and data have been outlined. In section 3, we have presented a case study of spectral flux in the Gulf Stream region, followed by the geographical distribution of inverse cascades and its depth dependence, focusing on the amplitude, the energy injection scale, and the scale at which the upscale energy starts to accumulate, assuming that the isotropic conditions prevailed. In the final subsection, the anisotropic characteristics of the inverse cascade have been analyzed. A summary of the results has subsequently been presented in section 4. 2 Methods and Data Spectral Kinetic Energy Flux Based on the assumption of geostrophic balance, an estimation of the time evolution of the EKE can be obtained from the SSH in wave number space. Here we have followed the method of Qiu et al. [2008] to derive the spectral energy flux; this method is an extension of the general method proposed by Frisch [1995]. Consider the horizontal momentum equations on a Cartesian section of the rotating earth: (1) (2)where and are the frictional terms, including any vertical advection of the momentum. First, taking the DFT of equations 1 and 2, and getting the horizontal momentum equations in physical space; then multiplying the equations in the physical space by the respective complex conjugates, and , where the star indicates a complex conjugate and caret indicates DFT. Finally, summing up these two equations, i.e., , we obtain an evolution equation for the spectral kinetic energy density , (3) The forcing term arises from the vortex-stretching term and includes the rate of conversion of the mean available potential energy (MAPE) to EKE. Note the f term dropped out because the Coriolis force is perpendicular to velocity. In addition, the dissipation term, , arises from the frictional terms. The horizontal nonlinear advection terms give rise to the spectral energy transfer: (4)where cpkm (cycles per kilometer), is the grid spacing, and is the number of grid points in each direction ( for this study), derived from the DFT. The horizontal wave number vector is , where . Physically, represents the transmission of EKE between the different spatial modes, and can be estimated from the altimetrically derived SSH data by assuming geostrophy: (5) (6) In order to get the spectral energy flux , we have defined the isotropic wave number as . By summing all the spectral energy transfer at modes with , we get the kinetic energy flux at wave number as (7) The long-term mean of reaches a statistical steady state, so the following approximate balance of terms becomes valid: (8)so (9)where the overbar denotes the long-term mean. Equations 7 and 9 provide keys for interpreting the spectral flux. According to equation 7, the positive (negative) flux indicates that the kinetic energy is transferred from a small (large) wave number to a larger (smaller) wave number, which is a forward (inverse) cascade. Moreover, according to equation 9, when the slope of the spectral flux is positive (negative), the forcing term is stronger (weaker) than the dissipation term , which implies a power source (sink) at this scale in the long-term mean. Eddies at the scale corresponding to the steepest positive slope transfer most of the kinetic energy to the others, and the eddies at the scale corresponding to the steepest negative slope accumulate most of the upscale kinetic energy. Calculation of the Eddy Scale Predicted by Linear Theory Eddy scales predicted by the linear baroclinic instability theory are computed following Tulloch et al. [2009, 2011], which is analogous to Smith [2007]. Using the World Ocean Atlas 2009 (WOA09), a local quasi-geostrophic linear stability calculation is performed on a grid of wave numbers , where is the local Rossby deformation wave number which is defined as the square root of the first nonzero eigenvalue of the vertical-mode equation, (10) We have restricted the instability calculation to K < 3KR because the scale of the mesoscale instabilities typically peaks at Kbci within a factor of 2 of KR, which is not only in accordance with the models of baroclinic instability [Charney, 1947; Green, 1960], but is also shown by Tulloch et al. [2011, Figures 4 and 8]. Linear eddy scales are computed by solving the linearized QG equations about the local climatological stream function and stratification . (11) (12)where is the vortex-stretching operator, is the buoyancy, and is the QG potential vorticity (PV). A wave solution of the form is assumed. At a given , the eigenvalue is the corresponding wave frequency and its imaginary part produces the growth of the wave. The with the largest imaginary part is defined as the baroclinic growth rate, and the reciprocal of the corresponding is defined as the wave number of the maximum linear growth, . Further calculation details can be found in Smith [2007] and Tulloch et al. [2009]. Therefore, the eddy scale predicted by the linear theory is . Observed Eddy Length Scale Various methods exist for the estimation of eddy length scales from the SSH anomalies [Eden, 2007; Tulloch et al., 2011]. The most common one is the peak wave number of the isotropic kinetic energy spectrum. However, isotropic spectra are computed from the rectangular boxes, and the eddies with scales approaching the box size become coarsely quantized. Hence, we use the centroid length scale defined as (13)where is the spectral EKE density: , , and . The magnitudes of the observed eddy length scales vary significantly with different estimates. The biggest difference is nearly 100 km, and the centroid length scale provides the smallest estimate [see Tulloch et al., 2011, Appendix B and Figure B1]. Satellite Altimeter Data The gridded altimeter data product provided by SSALTO/DUACS and AVISO is used. The data set merges all the available satellite altimeter SSH measurements from October 1992 to December 2012, and is interpolated onto a 1/3° Mercator grid with a 7 day temporal resolution. This product enables much better detection of the mesoscale signals than a single satellite [Pascual et al., 2006]. However, it also has lower capture capability than raw along or cross-track data because of the smoothing techniques used during the gridding process [Le Traon et al., 1998]. Model and Reanalysis Data Arbic et al. [2013] noted that the smoothing inherent in the construction of the gridded altimeter data could shift the energy injection scale of toward lower wave numbers (larger scales). In this study, we have also used the eddy-resolving model outputs and reanalysis data to repeat the calculation of to assess the robustness of the observed inverse cascade characteristics. We have used 20 year outputs (1993–2012) from a quasi-global eddy-resolving ocean general circulation model (OGCM) for the Earth Simulation (OFES). The horizontal resolution is 0.1° and the domain is from 75°N to 75°S. The OFES climatological simulation was spun-up for 50 years from the observation data without motion (see details in Masumoto et al. [2004]). Following the climatological simulation, the hindcast simulation was conducted from 1950. The period of simulation extends to the present year. We have also used the 20 year (1993–2012) reanalysis data produced by the Estimating the Circulation and Climate of the Ocean project, Phase II (ECCO2), which is a part of the World Ocean Circulation Experiment (WOCE) and aims to produce the best-possible global time-evolving synthesis of almost all available ocean and sea-ice data, with a resolution that detects ocean eddies. Data constraints include altimetry, gravity, drifter, hydrography, and sea-ice observations (for full data details see Menemenlis et al. [2008]). The spatial resolution is 1/4° × 1/4°. Note that, in this paper, both the model results and reanalysis data were interpolated to an identical grid of the AVISO data. 3 Results and Discussion To set the stage for global discussions, in Figure 2, we have displayed the time-averaged computation over the Gulf Stream region (centered at 34°N, 65°W) from AVISO SSH ( ), OFES SSH ( ), and ECCO2 SSH ( ) data, respectively. Consistent with the results of the previous studies, all have a prominent negative lobe, indicating that the inverse cascade is a robust feature of oceanic general circulation. Figure 2Open in figure viewerPowerPoint Time-averaged spectral kinetic energy flux versus isotropic wave number in the Gulf Stream (rectangle containing 32 × 32 grids centered at 34°N, 65°W). is calculated from AVISO SSH, is calculated from ECCO2 SSH, and is calculated from OFES SSH. Black dots represent energy injection wave numbers. The vertical solid line is , the vertical dash-dotted line is , and the vertical dashed line is . Referring to Figure 2, there are three specific wave numbers on one spectral flux. The first one is the zero point at a large wave number with a positive slope indicating the start of the inverse cascade, which is defined by Scott and Wang [2005] as the energy injection wave number. In this study, we have followed this approximation. Furthermore, the corresponding scale is the energy injection scale, Linj. The second one, where the wave number at which and is minimum, denotes the one at which upscale energy starts being arrested with a corresponding scale of . The third one is the zero point at the small wave number indicating the apparent end of the inverse cascade, the corresponding scale of which is defined as . The amplitude of the inverse cascade is defined as the minimum value of the spectral flux in this study. In Figure 2, we can see that the amplitude of is much smaller than that of and , and displays a significant inverse cascade over a narrower waveband. There may be three reasons for these differences in spectral fluxes form different data sets: (i) the gridded data product provided by AVISO has a lower resolution than using the raw along-track data, and the interpolation procedure used to construct the gridded AVISO data may reduce the amplitudes of , and shift the zero crossings toward lower wave numbers, as already noted by Arbic et al. [2013]. (ii) Mesoscale turbulence near the surface is significantly affected by the eddies that are not resolved by conventional altimetry data, especially at small scales [Sasaki and Klein, 2012]. (iii) The values for lateral turbulent diffusivities used today in OGCMs are still uncertain [Bryan et al., 1999; Eden et al., 2007], which could cause the smaller-scale instabilities in models not present in the real ocean. In the following subsections, we will further diagnose the universalities of these differences. It should be noted that the wave number at which and is minimum (or ) is insensitive to the data source (vertical solid line in Figure 2). In order to study the characteristics of the global distribution of the inverse cascade, we have calculated in the global ocean away from the coast every 1/3°. At each coordinate, is computed over a subdomain containing 32 × 32 grid points. This box size, similar to that chosen in previous studies [Qiu et al., 2008; Tulloch et al., 2011], is large enough to contain the scales of interest. For each box and each snapshot, the SSH field was first detrended by fitting a linear plane via least squares, and this plane was then subtracted from the corresponding snapshot. A Hanning window was applied to the detrended data before DFT. We ignored the small variations in grid spacing within a given box. The spectral flux at each grid point was obtained by averaging the time series of the fluxes, which are very erratic. Note that since the surface are calculated using SSH data under geostrophic balance, we have avoided the regions extremely close to the equator, 10°S–10°N, in this study. Geographic Distribution of the Inverse Cascade's Amplitude Previous researchers prefer the normalized spectral flux to highlight the energy injection scale, and have not paid much attention to the real amplitude. Figure 3 shows the global distributions of the amplitudes estimated from (Figure 3a), (Figure 3c), and (Figure 3d). The global distributions of the amplitudes calculated from the three data sets show quite similar patterns, but those from are the largest, followed by and , in decreasing order. This is mostly because the high resolution of the OFES simulation, which resolves a broader part of the unstable spectrum and leading to more energy flux [Sasaki and Klein, 2012]. Figure 3Open in figure viewerPowerPoint (a) Amplitude of the inverse cascade computed from AVISO SSH, estimated as the minimum value of the flux (indicates the biggest inverse flux). Note that the sign of the flux has been reversed to make the value positive. Units are cm2 s−2 and the field is plotted on a log10 scale. (b) Geostrophic kinetic energy (cm2 s−2) computed from the AVISO SSH anomaly. Data here have been multiplied by where is the latitude, to avoid the equatorial singularity in noisy data. Figure 3b is taken from Stammer [1997]. (c) Amplitude of the inverse cascade computed from OFES SSH. (d) Amplitude of the inverse cascade computed from ECCO SSH. The amplitude of the flux varies by about 2–3 orders of magnitude across the global ocean. In highly energetic predominantly eastward-flowing currents, such as the Antarctic Circumpolar Current (ACC), the Kuroshio Extension, and the Gulf Stream, the amplitudes of are about 2–3 orders of magnitude greater than those in areas of weaker kinetic energy, such as gyre interiors. The amplitudes are also large in subtropical countercurrents (STCCs), but weaker than those in strong eastward current zones. The global distribution patterns of amplitudes are quite similar to the EKE pattern (Figure 3b), which shows that EKE in strong kinetic energy regions is 100 times greater than in other regions [Stammer, 1997]. It should be noted that, in the tropical oceans, amplitudes are unexpectedly small, while the EKE in these regions are very high (which is disguised by the multiplication of amplitudes by ). We know that most mesoscale signals occur as the linear Rossby wave and as nonlinear eddies [Chelton and Schlax 1996; Chelton et al., 2007], and Chelton et al. [2011] observed only a small number of nonlinear eddies throughout the equatorial regions. Therefore, most of the mesoscale signals in the tropics may be in the form of linear Rossby waves, which could not be reflected by the spectral kinetic energy flux, since the spectral flux arises from the nonlinear terms. Comparison Between the Energy Injection Scale and the Eddy Scale Predicted by Linear Theory SW05 demonstrated the distribution characteristic of in South Pacific within each unoverlapped subdomain, and Tulloch et al. [2011] calculated the averaged spectral fluxes at some chosen locations around the global ocean using the overlapped subdomains. Here we have followed Tulloch et al. [2011] with the exception that the spectral fluxes in the global ocean were calculated with every 1/3° of zonal variation and the geography of was depicted, which is helpful to obtain the detailed distribution characteristics. Figure 4a shows the geographic distribution of the computed from the AVISO SSH. We can clearly see that decreases with increasing latitude, from about 200 km at 10° latitude to about 100 km at 60° latitude (Figure 4d). In high-energy predominantly eastward-flowing currents, such as the ACC, the Gulf Stream and its northeast extension, and the Kuroshio and its eastward extension, reaches a local maximum. Figure 4Open in figure viewerPowerPoint Map of the energy injection scale (km), inferred from two decades of data: (a) AVISO SSH, (b) OFES SSH, and (c) ECCO2 SSH. Regions near continents for which no calculation was made are shaded white. Plotted values in each panel are subjected to the condition that the energy injection wave number at each grid. (d) Zonally averaged energy injection scales, alongside eddy scales predicted by the linear theory (blue dashed line). We have also presented the global distributions of energy injection scales computed from OFES SSH (Figure 4b) and ECCO2 SSH (Figure 4c). The patterns agree well with that from AVISO. A notable feature is that AVISO's is noisier than the others, which is probably because the AVISO SSH is more erratic than the other two data sets. Another notable feature is that calculated from OFES and ECCO2 are uniformly smaller than AVISO, by about 50 km in extratropical regions, as shown in