## 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.

[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 *E*(Δ^{3})/*E*(Δ^{2})^{3/2} where Δ = η − *E*(η) is the deviation about its expected value, *E*(η). In this notation *E*(Δ^{2}) 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 η_{e} that occurs with probability *p*_{e}. It is straightforward to show that the skewness is sgn(η_{e})(1 − 2*p*_{e})/ which can be approximated by sgn(η_{e})/ for small *p*_{e}. 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 + σ^{2}_{N}/σ_{S}^{2})^{−3/2} where σ^{2}_{S} and σ_{N}^{2} 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 σ^{2}_{N} ≪ σ^{2}_{S} 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.