1. Introduction
The quasi-bidimensional nature of ocean motions makes two-dimensional turbulence a paradigm for understanding the dynamical properties of ocean flows (Rhines 1979; McWilliams 1984). One of the most remarkable aspects of two-dimensional (2D) turbulent flows is the presence of coherent vortices that emerge under a wide range of initial conditions if forcing and dissipation are sufficiently small. This has led to some widely studied approaches in the past in which such flows are considered as an ensemble of localized vortices that dominate the statistical properties of the flow. The idea of structures with concentrated vorticity as fundamental building blocks of the turbulent flow has been largely exploited in the past to analyze numerical simulations of 2D turbulence, to build conceptual models, and to determine statistical and mixing properties of turbulent flows, scaling laws, and so on (Jiménez 1996; Babiano et al. 1987; McWilliams 1990; Weiss and McWilliams 1993).
A question that recently has received much interest is related to the role of such coherent structures in the dispersion of tracers. Indeed, the presence of vortices induces the anomalous dispersion of passive tracers associated with non-Gaussian statistics of velocity fields (Provenzale 1999; Pasquero et al. 2001). Numerical experiments of decaying turbulence show that the variation of the Reynolds number (Re) modifies the size and separation of the vortices. For small values of Re, the flow is characterized by the presence of large vortices and Gaussian probability density functions (PDFs) of velocity. However, as the Reynolds number increases (and the number of vortices increases, with the vortices becoming smaller), the tails of the velocity PDF smoothly increase. Then, the observed velocity PDFs for large enough Re are non-Gaussian (Bracco et al. 2000a, 2005, manuscript submitted to J. Geophys. Res.). Furthermore, a detailed analysis of the velocity field showed that, although the velocity PDF inside vortices is Gaussian, the velocity field generated by the vorticity of these vortices is not (Bracco et al. 2000a). This suggests that the existence of long tails in the velocity PDF of barotropic turbulence is due to the nonlocal effect of vortices, as was proposed by Jiménez (1996). However, experiments with forced 2D turbulence have shown that large-scale forcing diminishes the role of coherent vortices in the pertinent statistics (Schorghofer and Gille 2002).
Ocean fields also exhibit non-Gaussian velocity PDFs. This has been obtained from the analysis of global satellite altimetry (Llewellyn Smith and Gille 1998; Gille and Llewellyn Smith 2000), drifting buoys (Swenson and Niiler 1996; Bracco et al. 2000b; Maurizi et al. 2004), current meters (LaCasce 2005), and high-resolution numerical simulations of the ocean circulation (Bracco et al. 2003). Such observational evidences lead to the interpretation that the non-Gaussian property may be associated with the presence of coherent vortices that populate the ocean. Nevertheless, it can also be related, as proposed by Llewellyn Smith and Gille (1998), to the large-scale spatial inhomogeneity of the eddy kinetic energy (EKE), which is expected to be higher in regions dominated by the presence of unstable jets. From the analysis of altimeter data it is found that velocity PDFs are predominantly Gaussian for small ocean regions, but in regions dominated by organized flows such as western boundary currents PDFs are closer to an exponential distribution (Llewellyn Smith and Gille 1998). However, in a recent paper, LaCasce (2005) found that Eulerian velocity PDFs computed from current meters in the North Atlantic Ocean were significantly non-Gaussian and statistically indistinguishable from Lagrangian measurements so long as the float data are averaged in bins of appropriate size.
The aim of this work is to analyze to what extent non-Gaussian deviations of velocity PDFs are due to coherent vortices in a semienclosed basin such as the Mediterranean basin. The Mediterranean Sea is dominated by the entrance of freshwater incoming from the Atlantic Ocean through the Strait of Gibraltar. The instability of this inflow and local wind action often generate coherent vortices in several parts of the basin that enhance the mixing of these incoming light waters with the saltier resident waters (e.g., Millot 1999, 2005). Analysis of altimetric maps shows that PDFs of the velocity field at basin scale also appear non-Gaussian, in agreement with previous cited works. To discern to what extent this is due to the presence of mesoscale vortices in the basin we propose to make a partition of the flow between coherent vortices and a background field based on the Okubo–Weiss parameter, following the ideas of Bracco et al. (2000b), and then reanalyze the velocity PDF of both fields.
The paper is structured as follows: section 2 describes the sea level anomaly maps used in this study, the criterion of vortex identification, and the methodology followed to estimate PDFs. Section 3 studies velocity PDFs in different topological domains and presents an explicit separation of the contribution of coherent vortices from other contributions. Section 4 discusses the results and presents conclusions.
2. Data and methods
a. Sea level anomalies
In this study we used sea level anomaly (SLA) maps produced by the Collecte Localisation Satellites (CLS) in Toulouse, France, which combine the signal of the European Remote Sensing Satellite (ERS) and the Ocean Topography Experiment (TOPEX)/Poseidon altimeters. These maps are processed including usual corrections (sea state bias, tides, inverse barometer, etc.) and with improved ERS orbits using TOPEX/Poseidon as a reference (CLS 1996; Le Traon and Ogor 1998). SLA is regularly produced by subtracting a 4-yr mean value (1993–96) and, prior to the analysis, data are low-pass filtered using a 35-km median filter and a Lanczos filter with a cutoff wavelength of 42 km in order to reduce altimetric noise (Larnicol et al. 1995). The data used here actually span 7 years from October 1992 to October 1999, but with a gap between December 1993 and March 1995 due to the 168-day ERS-1 orbit. SLA maps are finally built every 10 days using an improved space–time objective analysis method, which takes into account long wavelength errors, on a regular grid of 0.2° × 0.2° (Le Traon et al. 1998; Larnicol et al. 2002). This leaves a total of 213 SLA maps. Data in the Aegean Sea were not considered because of the high density of islands. Velocities are estimated assuming a geostrophic relationship.
b. Vortex extraction
c. Velocity PDF
Probability density functions of the velocity at basin scale have been estimated by computing the histograms of both components (zonal and meridional) of the geostrophic velocity calculated from the whole time series of SLA maps. First, the mean value of the set has been removed and, then, the velocities have been normalized by the number of samples and divided by the bin width. Usually, a reasonably interval of the bin width is between 0.2σ and (0.4–0.5)σ to have a normalized bias error less than 1% (Bendat and Piersol 1985; Emery and Thomson 1998). This bin width is a compromise between the reduction of the random error and the suppression of the bias error. Table 1 lists the values of σ for all velocity fields considered here. From the whole range of values in the table a bin width of 2 cm s−1 has been taken, which is between the lower bound of 0.7 cm s−1 (≈0.2σ) and a very conservative value of 4.68 cm s−1 (≈0.4σ).
The above estimation of the results’ significance has been done by considering that the observed values, or at least a randomly picked subset of observations, were independent. To improve the reliability of significance tests, we have also tried to give a rough estimation of the degrees of freedom of the system. The total number of observations is 106 for the whole time series, distributed onto 213 maps separated every 10 days and with about 5 × 103 points per map. Previous altimetric studies suggested that the spatial correlation scale lies between 80 and 100 km and the temporal correlation scale between 10 and 20 days (Ayoub et al. 1998). Therefore, to construct SLA maps, the temporal scale was chosen to be 15 days and the spatial scale 150 km, which also considers the need to build an isotropic estimation of the SLA field (Larnicol et al. 2002). The number of grid points within a circle of radius equal to 150 km is approximately 187. Then, we consider that the number of independent points within a single SLA map is around 5 × 103/187 ≃ 29. On the other hand, if a temporal correlation scale of 10 were selected, which is consistent with not only Ayoub et al.’s (1998) estimation but also Salas et al. (2001), each map would be considered as an independent realization. However, since this scale has been taken as 15 days for the SLA maps, we consider that there are only 213/1.5 = 142 independent maps. Therefore, we estimate the number of degrees of freedom as 142 × 29 = 4118. Returning to the above significance tests, if the values of skewness and kurtosis, reported in Table 1, are considered along with our estimation of the degrees of freedom, the JB test indicates that the null hypothesis can be rejected in all the situations considered at the 95% level of confidence (not shown).
3. Results
a. Observed PDF
Figure 2 shows the resulting PDFs of both velocity components, and their moments are listed in the first row of Table 1. These PDFs are characterized by long tails and a sharp core indicating that its departure from the Gaussian distribution is due to not only energetic events but also a larger contribution of small values. Therefore, the distribution resembles an exponential distribution. This departure from Gaussianity is also evident from the values of kurtosis, which are greater than 0 [the definition of kurtosis includes a −3 term that makes it zero for a Gaussian distribution; Press et al. (1994)].
b. Vortex separation
The velocity PDFs associated with the fields ubg and uυ are represented in Fig. 6 and their first moments in Table 1. PDFs of the velocity field induced by the background distribution of vorticity (ubg) are characterized by a wide Gaussian-like core with small tails when compared with the other distributions. Its kurtosis is about a factor of 4 smaller than uυ but its skewness is the highest. The kurtosis of the background field is reduced significantly when compared with the initial velocity field (from 2.37–2.69 to 0.74–0.72), while the kurtosis of the vortex-induced velocity field is rather high for both components (2.97–3.21), which is clearly non-Gaussian, being closer to an exponential shape even at the core.
Although the PDF of the background field is closer to a Gaussian distribution, it is still significantly different from it (see section 2c). To explore this point further, the spatial distribution of the eddy kinetic energy associated with the background field, Ebg, has been computed (Fig. 7). As can be seen, the Ebg is not homogeneously distributed through the basin and the highest values of Ebg are well correlated with regions characterized by a high density of coherent structures (cf. with Fig. 1). A possible explanation for this is that the definition of the background field contains both the vorticity of the background, characterized by small values of W (usually taken as |W| ≤ 0.2σW), and the vorticity of cells that surround vortex cores (approximately taken as W > 0.2σW). If this last contribution is subtracted, the kurtosis of the resulting background velocity field is still reduced (from 0.74–0.72 to 0.64–0.55) but not completely eliminated. Although the vortex identification based on the Okubo–Weiss parameter is a robust criterion for the identification of vortex cores in altimetric data (see Isern-Fontanet et al. 2003; Morrow et al. 2004; Isern-Fontanet et al. 2006), it is very difficult to properly identify the surrounding cell of the vortices and therefore eliminate its contribution.
c. Coherent vortex separation
The separation of the contribution of each type of vortex leads to two distributions with different shapes and different kurtosis (see Table 1 and Fig. 8). The distributions of the velocity field generated by weak vortices has a more or less Gaussian core while their tails are relatively small in comparison with the PDF of the velocities induced by intense vortices, which looks like an exponential distribution. This suggests that the main contribution to non-Gaussianity is mainly due to intense vortices. Indeed, uiυ has the highest kurtosis of all velocity fields, while the velocity field obtained by the elimination of the contribution of intense vortices (v − uiυ) is almost Gaussian, having the smallest kurtosis of the fields obtained by integrating the vorticity (see Fig. 9).
4. Discussion and conclusions
Results indicate that geostrophic velocity PDFs derived from SLA maps for the Mediterranean basin are significantly non-Gaussian. After the application of a flow partition based on the Okubo–Weiss parameter, this non-Gaussianity appears to be mostly due to the presence of coherent vortices, while the velocity PDF associated with the background is quite close to a Gaussian distribution. The extraction of the velocity field associated with the vortices has significantly reduced kurtosis. Indeed, similar results are obtained if only the contribution of intense vortices, which represent only about 20% of the vortices identified in the dataset, is removed from velocities. These vortices are those with amplitude values such that a ≥ 2σW and correspond to what are generically called mesoscale vortices (although they are larger than the mesoscale) in the common literature on the Mediterranean.
Nevertheless, the separation between a background field with a Gaussian velocity PDF and a non-Gaussian velocity field induced by vortices is not complete. PDFs of the background field still contain small deviations from the Gaussian in their tails. As previously outlined, one reason may be due to the way the Okubo–Weiss criterion separates the field, which does not allow for extraction of the surrounding cells of vortex cores. Indeed, several studies have underlined its limitations when the flow is rapidly evolving (Basdevant and Philipovitch 1994; Hua and Klein 1998). Another reason, as proposed by Llewellyn Smith and Gille (1998), may be large-scale inhomogeneities of the EKE. The EKE in the Mediterranean was found to be inhomogeneously distributed (see Fig. 11 in Isern-Fontanet et al. 2006). Although the spatial distribution of EKE associated with the background field (Fig. 7) is much more uniform and in some regions the vortex contribution is effectively removed (i.e., the area between the Balearic Islands and the Spanish coast around 40°N, 2°E), a certain degree of inhomogeneity is still apparent. In much of the domain the inhomogeneity of the EKE associated with the background field is highly correlated with regions where there is a great concentration of coherent vortices. For example, high values of EKE in the Algerian Basin, the central Ionian Sea or the Crete Passage could be explained by the contribution of cells surrounding the vortex. But a closer look into the EKE distribution for the background also reveals some patterns or traces that are probably associated with the presence of ocean currents. Beside that main currents do not appear on SLA maps, some traces of their locations and paths are not totally absent because of their associated variability, although they are not comparable with the strong signals of coherent vortices. This leads one to conclude that the origin of the non-Gaussianity of the velocity field in the basin is mostly due to vortices, although some less important contributions from a different origin cannot be discounted.
A question that arises from these results is to what extent the vortices influence the dynamics of the flow in the entire basin. Our results seem to confirm that intense vortices play a major role in the dynamics of the basin. However, more refined analysis needs to be done, in particular to improve the field separation between vortices and the rest of the flow. As argued by Farge et al. (1999), methods as the one applied here do not preserve the smoothness of the vorticity field. Therefore, spurious discontinuities may contaminate the components in which the flow is split. Turiel et al. (2005, manuscript submitted to J. Atmos. Oceanic Technol., hereinafter TIG) have recently analyzed the vorticity field for the same dataset but following a completely different technique to separate the flow based on wavelets, introduced by Farge et al.: coherent vortices are extracted by wavelets projecting the vorticity, and then the field is reconstructed, discarding part of the wavelet coefficients. The velocity field associated with this coherent part, which is built from a relatively small number of wavelet coefficients, accounts for most of the enstrophy and energy of the original field (TIG). This reinforces the idea that this small percentage of intense vortices responsible for the non-Gaussian character of the velocity PDF dominate the dynamics in the Mediterranean Sea.
Another question to be studied is the role of the vertical structure in the nonlocal contribution of vortices. A 2D vortex, or an equivalent barotropic vortex, has a larger range of influence than a baroclinic one (Bracco et al. 2004). Thus, its nonlocal effect could be different. Unfortunately, the data used in this study do not allow us to recover this information, nor is the vertical structure of coherent vortices presently known with enough detail. However, some field measurements have shown that at least some vortices in the Algerian Basin, here labeled as intense, have a very deep signature and extend down to the bottom of the Mediterranean (Millot 1985; Ruiz et al. 2002). Such vortices follow preferential paths that closely map the large-scale barotropic circulation at deep and intermediate levels, clearly constrained by f /H isocontours, with f being the planetary vorticity and H the water depth (Testor et al. 2005; Isern-Fontanet et al. 2006). Thus, it is expected that such marked barotropic character leads to a stronger nonlocal contribution than the one from a baroclinic vortex.
Last, a direct consequence of our work concerns the quantification of mixing and dispersion processes in the Mediterranean Sea. Non-Gaussian velocity PDFs induce anomalous dispersion due to the presence of these coherent vortices, and classical constant eddy diffusivity approaches should be discarded in favor of more complex ways of parameterizing the dispersion. To account for the anomalous dispersion due to vortices, Lagrangian models for particle dispersion may be formulated through a two-component stochastic process that separates the dynamical component associated with the background-induced motion and the vortex-induced dynamics (Thompson 1987; Pasquero et al. 2001). In such cases, the vortex-induced contribution part can be linked to the characteristics of the non-Gaussian Eulerian velocity PDF through a Fokker–Planck equation. However, this point is beyond the scope of present work and is left for future studies.
Acknowledgments
This is a contribution to the IMAGEN project funded by the Spanish R+D Plan (REN2001-0802-C02-02) and MERSEA project funded by the European Union (AIP3-CT-2003-502885). Altimetric maps for the period analyzed were elaborated upon and provided by CLS (Toulouse, France) under contract to the MATER project funded by the European Commission (MAS3-CT96-0051).
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Distribution of the centers of (top) all vortices and (bottom) intense vortices (a0 > 2) identified in the dataset. Adapted from Isern-Fontanet et al. (2006).
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDF for the Mediterranean Sea. The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDFs outside of vortices (vbg) and inside vortices (vυ) for the Mediterranean Sea. The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDFs inside weak vortices (vwυ) and inside intense vortices (viυ) for the Mediterranean Sea. The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Vorticity PDFs of the original field (ω), outside vortices (ωbg), and inside vortices (ωυ) for the Mediterranean Sea. The dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDFs of the field induced by the vorticity outside vortices (ubg) and inside vortices (uυ). The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Eddy kinetic energy associated with the field ubg (Ebg).
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDFs of the field induced by the vorticity inside weak vortices (uwυ) and inside intense vortices (uiυ) for the Mediterranean Sea. The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Velocity PDF of the field v − uiυ. The solid line is for the zonal component, the dashed line is for the meridional component, and the dotted line is a Gaussian distribution with σ = 1.
Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2971.1
Number of points (N), variance (σ2), skewness (s), and kurtosis (k; its definition includes a −3 term that makes it zero for a Gaussian distribution) of the zonal (u) and meridional (υ) components of the velocities defined in the text (see the rightmost column). Velocities labeled v are the observed velocities or subsets of the observed velocities; velocities labeled u are velocities obtained by integration of the vorticity (cm s−1), and α is the Kolmogorov–Smirnov statistics.
