Journal of Geophysical Research: Oceans

Skewness of sea level variability of the world's oceans



[1] Skewness of sea level variability for the world's oceans is calculated using gridded altimeter data for the period 1993–2001. Many well-known ocean features can be identified in the skewness map, including the Gulf Stream, Kuroshio Extension, Brazil-Malvinas Confluence, and the Agulhas Retroflection. It is shown, through an idealized example and results from a quasi-geostrophic model, that sea level skewness can be used to identify the mean path of unstable ocean jets and also regions dominated by eddies with a preferred sense of rotation. These ideas are confirmed with a more detailed analysis of the skewness fields for the northwest Atlantic and Agulhas Retroflection region. Finally, it is argued that sea level skewness, like variance, is a potentially powerful diagnostic for testing the realism of high-resolution ocean circulation models.

1. Introduction

[2] Satellite remote sensing has resulted in a much improved global description of the surface properties of the ocean, and a better understanding of many important phenomena and processes [e.g., Le Traon, 2002]. Remote sensing is also providing critical data streams for assimilation into operational models of basin- and global-scale circulation [e.g., Pinardi and Woods, 2002]. The altimeter, which can measure temporal variations in sea surface height with an accuracy of several cm, is arguably the most important sensor for physical oceanographic applications [e.g., Le Traon, 2002].

[3] The altimeter measures the distance between the sensor and the sea surface. To convert this measurement into a dynamically meaningful quantity, the position of the sensor with respect to the geoid is required. To date it has not been possible to define the geoid with sufficient accuracy to allow useful dynamical calculations at wavelengths shorter than about 2000 km [e.g., Le Traon, 2002]. The result is that most studies of altimeter data have focused on variability about the time mean. The most common statistic used to describe such variability is the variance. Maps of variance for the world's oceans [e.g., Ducet et al., 2000] clearly identify regions of strong mesoscale variability (e.g., the Gulf Stream, Kuroshio Extension, Agulhas Retroflection and the Brazil-Malvinas Confluence). Maps have also been produced of the variance of slopes of the sea surface. Such maps are usually interpreted in terms of the eddy kinetic energy of the surface flow and have provided new insights into aspects of regional oceanography [e.g., Ducet and Le Traon, 2001]. Observed sea level variances and eddy kinetic energies based on altimeter data are also used routinely to assess the realism of eddy-resolving ocean models [e.g., Treguier et al., 2003].

[4] One cause of elevated sea level variance is a meandering mid-ocean jet. The effect is illustrated in Figure 1. The top plot shows the sea surface topography across an idealized jet that can translate horizontally and thereby cause variations in sea level at a fixed horizontal position. The middle plot shows the probability density function (pdf) of sea surface height at three locations assuming the horizontal translations of the jet have a Gaussian distribution with zero mean and a typical magnitude that is comparable to the jet width (see legend for details). As expected, the standard deviation is a function of position (see bottom plot of Figure 1) with maximum variability at the mean position of the jet where the sea surface slope is greatest.

Figure 1.

Sea level variability caused by a horizontally translating jet. (top) Sea level profile as a function of cross-jet position x. The three curves show the sea level profile when the horizontal displacement perturbation x′ is 0 (middle curve) and ±L/2 (bounding curves, where L is a measure of jet width). The equation for the sea level profile is η = tanh[(x + x′)/L]. (middle) Probability density functions of sea surface height at x = 0 and ±3L/2 under the assumption that x′ has a zero mean Gaussian distribution with σ = L/2. (bottom) Cross-jet structure of the standard deviation and skewness of sea level (after normalization by their maximum values of 0.42 and 2.1, respectively). To mimic noisy observations and also to ensure that skewness is well behaved away from the center of the jet where the sea level variance drops to zero, the sea level used for the bottom plot is assumed to be the sum of contributions from the translating jet and a zero mean Gaussian noise process with a standard deviation of 0.05.

[5] It is also clear from Figure 1 that the pdf of sea level height is, in general, not symmetric. For example the pdf at the location with the highest mean sea level has an extended tail pointing toward low values of sea level; that is, the pdf is skewed to the left. The most common way of quantifying the skewness of a random quantity, η say, is by E3)/E2)3/2 where Δ = η − E(η) is the deviation about its expected value, E(η). In this notation E2) is the variance and so skewness is just the normalized third moment about the mean. The bottom plot of Figure 1 shows that skewness, like variance, is also a strong function of cross jet position: it is zero at the mean position of the jet, where the changes in surface height induced by the translating jet are symmetric, and it decreases(increases) toward higher (lower) mean sea levels. This means that for the simple jet profile shown in Figure 1, it is possible to infer the mean position of the jet from the skewness. It is also possible to infer the sign of the sea surface slope and thus the direction of the associated geostropic flow. In subsequent sections we will show that maps of observed sea level skewness can also be used to make inferences about the position and direction of mean flows in the real ocean.

[6] Another situation in which one could expect to find a skewed sea level distribution is a region populated by intense eddies with the same sense of rotation. The reason is that when one of the eddies sits over a fixed location it will cause an anomalously large change in sea level. The net result will be a sea level distribution with negative skewness for cyclonic eddies and positive skewness for anticyclonic eddies. To quantify the effect, assume that sea level at a fixed location is zero apart from an eddy contribution ηe that occurs with probability pe. It is straightforward to show that the skewness is sgn(ηe)(1 − 2pe)/equation image which can be approximated by sgn(ηe)/equation image for small pe. Thus the skewness depends only on the proportion of time eddies appear at the location of interest; the shorter this time, the greater the skewness.

[7] One attractive feature of skewness is that it has some robustness against additive noise with zero skewness (e.g., Gaussian noise). To see this, let S denote a random variable corresponding to a signal of interest, and let N denote an independent random variable corresponding to noise. If the noise has zero skewness, it is straightforward to show that the skewness of S + N is simply the skewness of S scaled by (1 + σ2NS2)−3/2 where σ2S and σN2 are the variances of S and N respectively. Thus adding symmetric noise to a signal will not change the sign of the skewness. Further, if σ2N ≪ σ2S the noise will have little effect on the magnitude of the skewness. As a physical illustration of the effect, consider a field of nonlinear eddies (the signal) to which is added random fields of linear, freely propagating Rossby waves (the noise). To first order, the linear Rossby waves will generate a symmetric sea level distribution and will therefore not change the sign of the skewness which is determined by the nonlinear eddies.

[8] In this paper we present maps of skewness of sea level variability for the world's oceans calculated from gridded altimeter data for the period 1993–2001. The maps indicate spatially coherent structures, similar in character to the maps of sea level variance that have already been published and interpreted. We also show that skewness, like variance, can be used to identify physical features such as mean currents and fields of eddies with a preferred sense of rotation. We illustrate these points with examples that include the Gulf Stream and Agulhas Retroflection regions. We also argue, in part on the basis of results from an idealized quasi-geostrophic model, that sea level skewness is a potentially useful diagnostic for assessing the realism of eddy resolving models of the deep ocean.

[9] The present study builds on the earlier study of Niiler et al. [2003] which provides a comprehensive description of the near surface mean flow and mesoscale variability in the Kuroshio Extension. Niiler et al. [2003] provide a map of the skewness of the surface geostrophic vorticity (their Figure 10b) which is then used to delineate regions dominated by cyclonic or anticyclonic eddies. The main differences between the present study and that of Niiler et al. [2003] are (1) we focus on sea level, rather than surface geostrophic vorticity (which can be thought of as high–wave number filtered sea level), (2) we provide a global description of skewness and discuss in detail the North Atlantic and Agulhas Retroflection region, (3) we show that skewness can be used not only to identify regions dominated by eddies of specified rotation but also the mean path of unsteady jets, and (4) we argue, on the basis of our quasi-geostrophic model integrations and regional analyses, that sea level skewness is a potentially powerful diagnostic for model validation.

[10] The structure of the paper is as follows. A quasi-geostrophic model, configured for the well-known double-gyre problem, is described in section 2 and its sea level fields are used to test the above predictions about skewness in the presence of a meandering, unstable jet. The altimeter data used in this study are described in section 3 and maps of standard deviation and skewness for the world's oceans, and several subregions, are presented. The results are summarized, and suggestions are made for future work, in the final section of the paper.

2. Skewness of Sea Level Variability of a Quasi-Geostrophic Model

[11] To help interpret the maps of observed skewness presented in the next section we now consider a rectangular, midlatitude ocean forced by a steady wind stress curl. The ocean consists of two layers, with the lower layer infinitely deep and at rest. Following McCalpin and Haidvogel [1996] we assume the dynamics are quasi-geostrophic [e.g., Pedlosky, 1992] and the interface height (h) evolves according to

equation image

Subscripts denote differentiation with respect to time (t), and the zonal (x) and meridional (y) coordinates of a β plane. ∇2 denotes the horizontal Laplacian and J(a,b) = axbyaybx. The parameter γ2 = f02/gH where f0 is the Coriolis parameter defined in the middle of the domain, g′ = Δρ/ρ0 is reduced gravity, and H is the reference thickness of the upper layer. The symbol equation image denotes dissipation and is assumed to be of the form

equation image

where r denotes an interfacial friction coefficient, and Ah and Ab are scale-selective horizontal friction coefficients. The symbol equation image denotes forcing by the wind stress curl and, following McCalpin and Haidvogel [1996], it is assumed to be of the form

equation image

where y* = y/Ly is the nondimensional meridional coordinate ranging from 0 (southern boundary) to 1 (northern boundary). The parameter αs controls the asymmetry of the wind stress curl.

[12] The initial condition is a flat interface. The boundary conditions are derived from a zero normal flow condition, and partial slip conditions appropriate for the terms in equation image. The numerical scheme is based on first-difference approximations and the code was written, and made available, by John McCalpin. For further details on the boundary conditions and numerics see McCalpin and Haidvogel [1996] and references cited therein.

[13] The model domain is 3600 km in the zonal direction, 2800 km in the meridional direction and centered on 30°N. The horizontal grid spacing is uniform in x and y and equal to 20 km. The time step is 2 hours. The mean depth of the upper layer is 600 m and the internal Rossby radius is taken to be γ−1/2 = 47.6 km. The friction coefficients are r = 10−7 s−1, Ah = 100 m2 s−1 and Ab = 8 × 1010 m4 s−1. The wind stress scale, τ0, is set equal to 0.1 Pa and the wind stress asymmetry factor, αs, is set equal to 0.05. The wind-forcing ramps up to its full strength over the first 100 days of model integration. The above set of parameters corresponds closely to the reference case discussed in detail by McCalpin and Haidvogel [1996].

[14] A typical sequence of snapshots of sea surface height is shown in the top row of Figure 2. The snapshots clearly show two gyres separated by a meandering, unstable jet that generates eddies in the central part of the domain. The anticyclonic(cyclonic) eddies subsequently propagate westward to the north(south) of the jet.

Figure 2.

Sea surface height variability of the quasi-geostrophic model. (top) Snapshots of sea surface height (in m). (bottom left) Mean, (bottom middle) standard deviation, and (bottom right) skewness of sea level based on output for 6.84 to 87.61 years following the start of the integration. The mean and standard deviation are in meters. The skewness has been calculated assuming that the sea level is the sum of contributions from the quasi-geostrophic model and a zero mean Gaussian noise process with a standard deviation of 0.15 m. This makes the skewness approach zero in regions with low variance.

[15] The bottom row of Figure 2 shows the mean, standard deviation and skewness of the model's sea level fields. The mean field clearly shows the expected double gyre and a jet that reaches about halfway across the domain. Note the oscillation in the mean path of the jet. As expected, the standard deviation is largest in the vicinity of the meandering jet and in the adjacent regions occupied by westward propagating eddies. The map of skewness is in accord with the discussion given in the Introduction. In particular, the line of zero skewness coincides closely with the mean position of the jet and skewness is positive (negative) in the region with low (high) mean sea level. Clearly given just the map of skewness one could make reasonable inferences about the mean path and direction of the jet, and also the regions dominated by cyclonic and anticyclonic eddies. This encourages us to examine the skewness of observed sea level variability for the world's oceans as described in the next section.

3. Maps of Variability for the World's Oceans

[16] The sea level observations used in this study are in the form of weekly gridded fields at 1/3° resolution for the period 1993–2001, inclusive. The gridded fields are based on delayed-mode data from the TOPEX/Poseidon, Jason 1, ERS1/2 and ENVISAT satellite altimeter missions processed by the Space Oceanography Division of CLS (Collection Localisation Satellites) located in Toulouse, France. (The work was carried out as part of the Environment and Climate EU ENACT project, EVK2-CT2001-00117, with support from CNES, the French Space agency.) Corrections were made to the along-track data by CLS to suppress residual orbit errors, tides and the inverse barometer effect [Le Traon et al., 1998].

[17] The spatial mapping was performed by CLS on the along-track data using an optimal interpolation procedure [Ducet et al., 2000]. The covariance function includes the effect of signal propagation velocities that can vary with geographical location. The spatial and temporal scales, and the horizontal propagation velocities, were computed from five years of TOPEX/Poseidon and ERS sea level data. In this study we focus on the time variability of sea level and so do not discuss the problem of defining a mean sea surface topography.

3.1. Global Maps

[18] The standard deviation of the sea level time series for each grid point are mapped in Figure 3. This type of figure, and similar ones for the eddy kinetic energy of surface flow, have been published and discussed extensively in the literature [e.g., Ducet et al., 2000]. We simply note here that the map indicates known regions of intense mesoscale variability (e.g., the Gulf Stream, Kuroshio Extension, Brazil-Malvinas Confluence, and the Agulhas Current system).

Figure 3.

Standard deviation (in cm) of sea level for the world's oceans based on weekly gridded altimeter data for the period 1993–2001, inclusive. The data are described in section 3.

[19] To illustrate the different character of sea level variability in selected regions we have chosen 8 locations (Figure 4) and plotted their time series in Figure 5. There are clearly striking differences in the character of sea level variability at nearby locations. For example, the two locations in the Gulf Stream region have very different skewness values (see right plots) even though they are separated by less than 150 km. The times series from the more southerly Gulf Stream location shows the occasional occurrence of very low values resulting in a negative skewness. Given the results from the quasi-geostrophic model discussed in section 2 it is possible that these negative sea level anomalies are due to meanders of the Gulf Stream or the passage of cold core rings. Similarly striking contrasts in skewness are evident in the Kuroshio Extension and Agulhas Retroflection region. The time series for the eastern Tropical Pacific is dominated by the large El Nino of 1997–1998 which caused high sea levels in the eastern Pacific in late 1997. Another interesting time series is from the northern rim of the North Pacific (marked Aleutian Chain on Figure 5). It is dominated by three positive sea level events that contributed significantly to the skewness of 2.19 for this location.

Figure 4.

Locations (circles) for which time series of sea level are plotted in Figure 5 and regions (rectangles) for which maps of skewness are presented in Figures 8 and 12.

Figure 5.

Time series of sea level for the eight locations shown in Figure 4. Each series has been standardized prior to plotting by removing its mean and dividing by its standard deviation. The plots to the right of each time series show the histogram and sample skewness of each series.

[20] To provide a global view of skewness it has been calculated for each grid point and mapped in Figure 6. The most dominant, large-scale feature in this map is found in the tropical Pacific. On the basis of the time series shown in Figure 5 this feature is attributed to a single low-frequency event: the El Nino of 1997–1998. Another large-scale feature is the extended region of positive skewness in the western equatorial Indian Ocean which may be related to the tropical Indian Ocean dipole mode [e.g., Saji et al., 1999]. Interesting regions outside of the tropics include the Gulf Stream, Agulhas Current and Retroflection off South Africa, the Kuril-Kamchatka Trench along the western boundary of the North Pacific, the Gulf of Alaska and the Alaskan Stream, and the Leeuwin Current off western Australia. Outside of the tropics, the standard error of the mean skewness estimates is typically 0.2 (see Appendix A). This gives us confidence that the extratropical, large-scale patterns of skewness are statistically significant. They are briefly discussed in the following subsections.

Figure 6.

Skewness of sea level for the world's oceans based on weekly gridded altimeter data for the period 1993–2001, inclusive. The data are described in section 3.

3.2. Gulf Stream Region

[21] Reverdin et al. [2003] recently provided a comprehensive description of the surface circulation of the North Atlantic on the basis of the trajectories of about 1800 surface drifters. The following description of the circulation, and the referenced figure, are taken from Reverdin et al. [2003]. The mean surface circulation shown in Figure 7 was calculated by first binning the drifter velocities into 0.5° latitude by 1° longitude bins and then averaging. The main boundary currents are clearly evident including the Florida Current, Gulf Steam and Labrador Current. Of particular interest here is the Gulf Steam. As it leaves Cape Hatteras, the Gulf Stream is well defined and flows to the northeast. It soon takes a more eastward path and when it reaches the New England Seamounts (63°W, 38°N) there is evidence of a northward shift of the mean flow. At 55°W there is a significant weakening and branching of the mean flow. Further downstream (50°W, 41°N) the northern branch merges with eastward flowing waters of the Labrador Current and together they flow around the northern edge of an anticyclonic feature in the Newfoundland Basin (sometimes referred to as the Mann Eddy).

Figure 7.

Map of the mean surface circulation of the northwest Atlantic based on an analysis of close to 1800 surface drifters by Reverdin et al. [2003] and published as their Figure 2b. The colored background shows the eddy kinetic energy based on the sum of variances of the binned meridional and zonal drifter velocities. Red corresponds to high energy.

[22] An enlarged version of the skewness map for the northwest Atlantic is shown in Figure 8. The zero skewness line agrees well with the observed mean path of the Gulf Stream calculated by Reverdin et al. [2003] including its northeast path after leaving Cape Hatteras, subsequent turn to the east and northward shift at the New England Seamounts (compare Figures 7 and 8). There is also the suggestion in the zero skewness line of the western edge of the Mann Eddy. An important point to note is that skewness can only detect a mean flow if the current varies. This is presumably the reason why the Florida Current is not evident in the skewness. The regions of positive and negative skewness defined either side of the mean Gulf Stream are broader than the range of Gulf Stream meanders. Skewness beyond the meander range is presumably due to warm and cold core eddies on the north and south side of the stream respectively. Note that the region of positive skewness (anticyclonic vorticity) in the Slope Water is bounded to the north by the shelf slope. This is to be expected because such eddies are generally confined to deep water.

Figure 8.

Enlarged version of the global skewness map (Figure 6) for the northwest Atlantic. The line shows the smoothed 1000 m depth contour.

[23] Figure 9 shows the standard deviation and skewness of sea level along 68°W, across the Gulf Stream. This plot is similar to the corresponding figure based on the idealized model (Figure 1) and is consistent with results from the quasi-geostropic model (Figure 2). Such agreement supports the idea that skewness can be used to infer the mean path and direction of unsteady jets in the real ocean.

Figure 9.

Standard deviation and skewness of sea level across the Gulf Stream. The longitude of the meridional section is 68°W, and it runs from 33°N to 42°N. The standard deviation and skewness have been scaled by their maximum absolute values of 37.7 cm and 2.15, respectively.

[24] Another subregion in Figure 8 with a coherent skewness feature is the Gulf of Mexico. The line of zero skewness agrees with the mean path of the Loop Current that enters the Gulf through the Yucatan Strain before exiting through the Florida Strait. The region of positive skewness to the west of the Loop Current is consistent with the westward propagation of anticyclonic Loop Current eddies [e.g., Sturges and Leben, 2000] which are expected to generate positive sea level skewness.

3.3. Agulhas Current System

[25] Another interesting region in the global skewness plot is just south of South Africa where the warm, saline waters of the Indian Ocean meet the relatively cold, fresh waters of the South Atlantic. This region is of general interest to physical oceanographers for two reasons. First, interocean exchange in this region is believed to be an important component of the global thermohaline circulation [e.g., Gordon, 1986]. Second, this region is one of the most energetic parts of the world's oceans (see Figure 3) and warm core rings generated at the Agulhas Retroflection are some of the strongest ever observed.

[26] The Agulhas Current runs southwestward along the east coast of Africa. (See Figure 10 for a schematic of the circulation in this region. The figure was presented by de Ruijter et al. [1999a]. The same authors also provide a good review of the observational, theoretical and modeling studies of this region.) Cyclonic eddies, called Natal Pulses, have been observed to propagate to the southwest in the Agulhas Current [e.g., Lutjeharms and Roberts, 1988; de Ruijter et al., 1999b]. Schouten et al. [2002] provide evidence that the Natal Pulses, and hence the meanders of the Agulhas Current, are triggered by offshore anticyclones that are generated in either the Mozambique Channel or just south of Madagascar. As the meanders move south, they grow in amplitude and can be accompanied by smaller-scale cyclonic instabilities along the continental slope [e.g., Penven et al., 2000].

Figure 10.

Schematic of the circulation of the Agulhas Current system presented by de Ruijter et al. [1999a].

[27] At about 37°S, the Agulhas Current leaves the continental slope and enters the South Atlantic as a free jet. It quickly retroflects and returns to the Indian Ocean along a meandering path that coincides approximately with the Subtropical Convergence. As indicated in Figure 10, the Agulhas Retroflection can occasionally shed energetic, warm rings that move into the South Atlantic. Transport by such eddies is believed to be an important contribution to the interocean exchange of heat and salt. Quartly and Srokosz [1993] plotted the instantaneous frontal position of the Agulhas Current system on the basis of advanced very high resolution radiometer (AVHRR) sea surface temperature images for a 3 year period starting March 1, 1985. A plot of frontal positions (redrawn in Figure 11) clearly shows the path of the Agulhas Return Current, its northward deflection around the Agulhas Plateau and subsequent eastward, quasi-steady meandering path with a zonal wavelength of about 5 degrees of longitude.

Figure 11.

Positions of fronts and rings in the Agulhas Current system. The fronts were estimated from gradients in AVHRR SST images for the months of September–November 1985–1987, inclusive. The figure is a slightly modified form of Figure A1 of Quartly and Srokosz [1993], and both are based on data generated by Eric Chassignet. The areas shaded light gray are shallower than 3000 m.

[28] An enlarged view of skewness in the Agulhas Current region is shown in Figure 12. One striking feature of this plot is the close agreement between the line of zero skewness and the path of the Agulhas Return Current estimated by Quartly and Srokosz [1993], including its deflection by the Agulhas Plateau and subsequent meandering. Note the sign of skewness is also consistent with the direction of the Agulhas Return Current: in the Southern Hemisphere, geostrophy requires that mean sea level is higher on the north side of the Agulhas Return Current and so any jet meandering, or simple eddy generation, will lead to positive skewness on the south side of the current.

Figure 12.

Enlarged version of the global skewness map (Figure 6) for the Agulhas Current system. The lines show the smoothed 1000 and 3000 m depth contours.

[29] Another significant feature in the skewness map is the band of negative skewness along the continental slope of the east coast of Africa, south of about 30°S. We suggest this is the expression of meanders in the Agulhas Current. Note also the tongue of positive skewness on the outer edge of the Agulhas Current that can be traced back to the southern tip of Madagascar; we suggest this reflects the path of the anticyclones that have been proposed as one of the triggers for Natal Pulses [Schouten et al., 2002].

[30] Figure 12 also indicates a tongue of positive skewness to the west of the retroflection (i.e., west of about 20°E). At first sight this may appear to be the signature of propagating warm core rings shed from the retroflection. However, observational and modeling studies suggest that warm core rings exist north of this tongue of positive skewness. A possible explanation is that skewness depends on the relative contributions of cyclones and anticyclones to the sea level variability and if they are equal, the skewness will be zero. Observational and modeling studies suggest that cyclones originate close to the African coast and move to the west-southwest, slowly decaying, while the stronger anticyclonic rings move from the retroflection to the west-northwest [Richardson and Garzoli, 2003; Matano and Beier, 2003; Treguier et al., 2003]. Thus it is possible that the contributions of cyclones and anticyclones cancel closer to the west coast of Africa, resulting in the southward displacement of the positive skewness tongue in the Agulhas ring area shown in Figure 12.

3.4. Northwest Pacific

[31] Anticyclonic eddies are often observed in the Kuril-Kamchatka Trench which runs along the western boundary of the North Pacific, from the northern tip of mainland Japan to Kamchatka. For example, Thomson et al. [1997] released a near-surface satellite-tracked drifter off the east coast of the Kuril Islands, close to the center of the Kuril-Kamchatka Trench, in September, 1993. The drifter immediately started to move around a large anticyclonic eddy which Thomson et al. [1997] speculate may be a quasi-permanent feature of the regional circulation. After about 40 days the drifter drifted inshore and started to move around another smaller anticyclonic eddy before entering the Sea of Okhotsk. Isoguchi and Kawamura [2003] recently used ten years of altimeter data to track seven anticyclonic eddies in the Kuril-Kamchatka Trench. They found that some of the eddies could be tracked for almost two years.

[32] The global map of skewness (Figure 6) clearly shows a band of high skewness coincident with the Kuril-Kamchtka Trench, consistent with the study of Isoguchi and Kawamura [2003]. The skewness map also hints at a reduction in skewness in the vicinity of the Bussol Strait at 46°N, the widest and deepest of the channels connecting the Sea of Okhotsk to the North Pacific. It is not clear at this time if this break in skewness reflects a disruption of the propagation of anticyclonic eddies along the trench.

3.5. Gulf of Alaska

[33] The large-scale, cyclonic circulation of the northeast corner of the North Pacific Ocean, north of about 50°N, is called the Alaskan Gyre. The coastal boundary current that runs poleward along the coast of southeast Alaska is called the Alaska Current. At about 155°W, close to Kodiak Island, it merges with the Alaskan Stream and flows westward along the Alaskan Peninsula and Aleutian Island Chain. There is evidence for strong interannual variability in the transport of these boundary currents that may be linked to ENSO events. (See Okkonen et al. [2001] for more details on the large-scale circulation and additional references.)

[34] In comparison to the mean circulation, relatively little is known about the mesoscale variability in this region. Thomson and Gower [1998] analyzed a sequence of thermal images for March 1995 and identified a train of six anticyclonic eddies along the shelf slope, stretching from about 51°N to almost Kodiak Island. They suggest the simultaneous appearance of these eddies was the result of a basin-scale instability of the baroclinic boundary current system, triggered by an abrupt reversal in the large-scale wind field. Meyers and Basu [1999] examined six years of altimeter data along a track that paralleled the shelf break between 45°N and 60°N. They found most of the eddy activity north of 50°N, with the anticyclonic eddies more numerous and stronger.

[35] On the basis of an analysis of six years of altimeter data, Crawford et al. [2000] identified anticyclonic eddies in the Alaskan Stream with surface height anomalies and lifetimes approaching 0.7 m and 3 years respectively. As the eddies propagated westward along the northern slope of the Aleutian Trench, they were associated with seaward meanders of the Alaskan Stream. Okkonen et al. [2001] used over seven years of altimeter data to define eddy corridors for the anticyclonic eddies in the vicinity of the Alaskan Stream, between 140°W and 180°W. In accord with Crawford et al. [2000], they observed long-lived anticyclones propagating westward at speeds approaching 4 cm s−1. On the basis of their observational analysis and additional model studies, Okkonen et al. [2001] conclude that the dynamics of the Alaskan Stream eddies are uncertain at this time. They speculate that coastal Kelvin waves and wind-forced coastal downwelling may play a role in eddy generation, and that planetary solitary waves (Rossby solitons) may be part of the propagation mechanism.

[36] The global map of skewness (Figure 6) clearly shows a band of high skewness coincident with the Aleutian Trench, consistent with the above description of anticyclonic eddies and meanders in the Alaskan Stream. The maps also shows generally positive skewness in eastern Gulf of Alaska, consistent with the observational studies of Thomson and Gower [1998] and Meyers and Basu [1999].

3.6. Southeast Indian Ocean

[37] The Leeuwin Current flows along the shelf break off the west coast of Australia. It transports relatively warm and fresh water of tropical origin southward to Cape Leeuwin and then eastward into the Great Australian Bight. Ridgway and Condie [2004] recently presented observational evidence that the Leeuwin Current is part of an extended, shelf break current system that runs from the northern tip of the west coast of Australia (North West Cape) to the southern tip of Tasmania. Both warm and cold eddies have been observed on the seaward side of the Leeuwin Current. Morrow et al. [2004] recently used 5 years of altimeter data from the TOPEX/Poseidon and ERS satellite missions to track a large number of cyclonic and anticyclonic eddies between 90–120°E, 20–40°S. They generally found the anticyclonic eddies close to shore and north of about 32°S. Most of the cyclonic eddies were in deep water and south of about 32°S. Both types of eddy moved westward but the cyclonic eddies had a significant poleward component to their translational velocity.

[38] The skewness map is in general agreement with above description of the eddy field between 90–120°E, 20–40°S (compare Figure 6 of this paper with Figure 1 of Morrow et al. [2004]). In particular the skewness is positive close to shore and north of about 32°S, consistent with an eddy field dominated by anticyclones; in deep water, south of about 32°S, the negative skewness is consistent with more cyclones.

4. Summary and Discussion

[39] The main conclusion of this study is that the skewness of observed sea level, like variance, is a useful statistic for mapping the physical oceanographic state of a region, and also testing the realism of high-resolution ocean models. The variance of sea level has been used for many years to identify regions of intense mesoscale variability. Skewness is sensitive to the mean path and direction of unstable jets, and also the dominant sense of rotation of eddy fields [Niiler et al., 2003]. Thus variance and skewness provide complementary information about the state of the upper ocean. One attraction of using skewness, as opposed to more subjective measures like the frequency and intensity of cyclones in a region, is that it is defined unambiguously and is very straightforward to calculate from both model output and observations. It is also moderately insensitive to the addition of independent noise with a symmetric probability density function.

[40] It is important to recognize that there is not, in general, a one-to-one correspondence between skewness and physical features in the real ocean. For example it is only possible to detect the mean path of a jet if it meanders. Thus one would not expect a topographically controlled boundary flow like the Florida Current to have a strong signature in skewness and this was observed to be the case as noted in section 3. Similarly, skewness cannot distinguish between an eddy field with an equal mix of cyclonic and anticyclonic eddies of the same intensity, and a region with no eddies. The conclusion is that, in general, the relationship between skewness and mean flows and eddy field structure will vary from region to region. For regions with a clear, interpretable relationship between skewness and, for example, the mean path of an unstable flow (e.g., Gulf Stream, Agulhas Retroflection) or the extent of an eddy rich region (e.g., western Gulf of Mexico), observed skewness could be useful in terms of defining the mean state, and also quantifying seasonal and possibly interannual variability.

[41] The skewness of the quasi-geostrophic model output described in section 2 is sensitive to the choice of model parameters, particularly the friction coefficients. This suggests that skewness of observed sea level variability will be useful in testing the realism of high-resolution ocean circulation models of the real ocean. This could be important, for example, in the Agulhas Retroflection region where the role of eddies in the exchange between the Indian Ocean and South Atlantic is uncertain [Matano and Beier, 2003; de Ruijter et al., 1999a]; sea level skewness could be used to test the realism of regional models and indirectly their estimates of interocean exchange. In general we would argue that if a model cannot reproduce a statistically significant, large-scale feature in the observed skewness field then the model is wrong in some fundamental way, at least as far as its representation of mesoscale variability in the upper ocean is concerned. Exactly the same situation arises when a model's sea level variance disagrees with the map of observed sea level variance. Our experience with an eddy permitting, 1/3° model of the North Atlantic is that it is fairly easy to reproduce the observed distribution of sea level variance in the Gulf Stream region, but much more difficult to reproduce the observed skewness. The reason is probably that the intensity, size and frequency of the warm and cold core rings are not realistic in this relatively coarse model and the discrepancy is reflected in the maps of skewness (but less so in the maps of variance). We are presently working on a higher resolution model and will report on the effect on skewness in a future publication.

[42] We conclude by noting that there are interesting features in the global map of skewness that are not easily explained at the present time. For example it is not immediately obvious what causes the unusual skewness distribution in the vicinity of the Brazil-Malvinas Confluence or the near zonal band of negative skewness to the south of Australia. The explanation of such features will probably have to come from realistic, high-resolution models.

Appendix A:: Statistical Significance of the Skewness Estimates

[43] The standard deviation of sample skewness based on a random sample of size N drawn from a normal distribution is approximately equation image for large N. One of the reasons we cannot use this result to assess the statistical significance of our skewness estimates is that the sea level observations are serially and seasonally dependent. Both of these effects will reduce the effective degrees of freedom of the estimator to a value less than N.

[44] To assess the statistical significance of the skewness estimates we have taken a simple, pragmatic approach. We first calculated skewness for each of the N = 9 calendar years for which we had altimeter data. We then averaged these 9 annual skewness maps to find the mean skewness at each grid point. Over most of the world's oceans the mean skewness thus calculated closely resembles the map shown in Figure 6. The major differences were found in the tropical Pacific and Indian Ocean. These difference are readily explained by the strong interannual variability in these tropical regions which is removed by the annual averaging approach. (The yearly mean is removed from each annual series for each grid point as part of the annual skewness calculation). Over the rest of the ocean, including the Gulf Stream and Agulhas Retroflection regions, the maps of skewness calculated by the two methods are quite similar, indicating that skewness is generally the result of variability with timescales less than one year.

[45] From the N = 9 annual skewness maps it is also possible to calculate the standard deviation of the annual skewness values at each grid point (s say), and thus the standard error of the mean skewness using the well-known formula s/equation image. This standard error calculation assumes the annual skewness estimates are uncorrelated (a reasonable assumption) but does allow for seasonality and serial dependence within each calendar year. A typical standard error of the mean skewness over most of the ocean is s/equation image = 0.2; apart from a few isolated spots, all of the standard errors are less than 0.4. Overall this calculation gives us confidence that, outside of the tropics, the large-scale patterns of skewness, which have typical skewness magnitudes exceeding unity, are significantly different from zero.


[46] We thank David Griffin, Frank Shillington, and Deidre Byrne for useful discussions and Kassiem Jacobs for help with the QG modeling. The three reviewers of the manuscript also provided very insightful and constructive comments. We also thank Eric Chassignet, Will de Ruijter, Graham Quartly, and Gilles Reverdin for providing data and permission to use their figures. This work was funded by Natural Sciences and Engineering Research Council of Canada and the Canadian Foundation for Climate and Atmospheric Sciences.