Identification of Marine Eddies from Altimetric Maps (original) (raw)

1. Introduction

The detection and sampling of marine eddies from field measurements has been the objective of several research programs in the past 30 yr (Robinson 1983; Kamenkovich et al. 1986). Since the launching of satellites for ocean observation, remote sensing imagery has provided large-area synoptic views of flow structures that were unavailable with previous in situ sampling strategies. Because of their sampling rate and resolution, infrared images have provided valuable information to survey the dynamics of such structures [see numerous examples in Nihoul and Jamart (1989)]. Thus, marine eddies have been regularly observed, and their properties and dynamics have been studied in great detail (i.e., Hooker et al. 1995, 1997).

However, two major drawbacks limit the use of infrared data for general quantitative purposes: 1) the inherent problems associated with the observation system, mainly related with data hiding during cloud cover periods, and 2) the dynamic restrictions on the use of the sea surface temperature as a passive tracer (Kelly and Strub 1992). Instead, sea surface height from altimetric data is directly a measure of a dynamic property of the ocean. The combination of several altimeters still provides the adequate spatial and temporal resolutions to sample marine eddies (Wilkin and Morrow 1994; Stammer 1997; Goni and Johns 2001; Holland and Mitchum 2001).

When attempting to analyze the individual behavior of marine eddies, a first problem to be addressed is how to systematically define and identify mesoscale eddies from sea level maps. One could be tempted to detect eddies by means of an algorithm to search some geometrical properties of the sea level height field, for example, extrema points and spiral or closed isolines. Under the usual assumption of geostrophic balance the sea level height corresponds to the streamfunction, and this algorithm would select points of zero geostrophic velocity (vg = 0). But this indicator is not invariant under a transformation of coordinates to a frame of reference moving at a constant velocity with respect to the original (Galilean transformation), and thus for some flow regimes, such as a vortex embedded in a background flow, the streamfunction does not exhibit a clear extreme. Vorticity magnitude | ω | should be a more natural and better candidate to be used as a criterion of eddy identification, because eddies should be areas of net vorticity, and it is invariant under a Galilean transformation. However, the vorticity criterion alone can be masked near boundary layers where great values of background vorticity not necessarily associated with coherent vortex can be found. Jeong and Hussain (1995) proposed a general identification criterion that satisfies both requirements: invariance under Galilean transformations and selection of regions with net vorticity.

In this note this criterion is applied to systematically identify marine eddies from altimetry. The Algerian Basin in the Western Mediterranean Sea has been selected to show the performance of the method. The mesoscale variability in the area is mainly due to the presence of long-lived mesoscale anticyclonic eddies generated from instabilities of the Algerian Current (Fuda et al. 2000; Obaton et al. 2000; Salas et al. 2002a). The characteristics of the Algerian eddies are large enough (L ∼ 100 km) to be resolved by the separation of altimetric tracks, and their propagation velocity (C ∼ 2–5 km day–1) is slow enough for their evolution to be adequately sampled (Millot 1991). Several studies have shown the reliability of the altimetric signal to analyze the dynamics of the Mediterranean Sea, particularly within the Algerian Basin (e.g., Vignudelli 1997; Ayoub et al. 1998; Iudicone et al. 1998; Larnicol et al. 2002). However, in these and others studies the analysis of altimetric data was more focused on the statistical properties and interannual variability of the inferred circulation than on the detailed analysis of the dynamics of the Algerian eddies. Ayoub et al. (1998) identified some anticyclonic structures from closed streamline structures. Bouzinac et al. (1998) tried to extract the behavior of the Algerian eddies as complex empirical orthogonal functions (EOFs). In addition to the complex decomposition of EOFs that allows study of time-evolving modes, each mode is spread over the whole domain, so their interpretation in terms of spatially localized and finite-amplitude structures is questionable. Finally, more recent works have provided some interesting questions on such structures by a joint analysis of different datasets (e.g., Fuda et al. 2000; Puillat et al. 2002; Salas et al. 2002b). Of special interest is the work by Puillat et al. (2002), in which several such structures are tracked by an analysis based on the Advanced Very High Resolution Radiometer (AVHRR) and complemented with altimetry when the infrared images were unavailable or confusing. The main problem they found is how to unambiguously identify Algerian eddies, especially when thermal gradients are weak or eddies are embedded in the Algerian Current. Our purpose is to show that this general problem can be addressed by the criterion presented in this paper.

First, some theoretical definitions of vortex identification developed within the frame of turbulence studies are presented briefly, together with the specific dataset used. Then, some results and examples of applying the criterion to sea level maps for the Algerian Basin are presented and summarized.

2. Methods and data

a. Identification of vortices

Hunt et al. (1988) defined eddy cores as regions in which the second invariant of the velocity gradient tensor ∇v is positive and the pressure tends to a minimum. The second invariant of ∇v is defined as

i1520-0426-20-5-772-e1

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where Ω and 𝗦 are the symmetric and antisymmetric components of ∇v. Thus, Q represents the local balance between shear strain rate and vorticity magnitude.

Later, Chong et al. (1990) provided a general classification of flow fields defined by instantaneous streamline patterns from the study of the eigenvalues of the velocity gradient tensor ∇v. They proposed that a vortex core is a domain characterized by having complex eigenvalues of ∇v and that the criterion simply concerns the positiveness of the discriminant of the characteristic equation. Jeong and Hussain (1995) proposed a new definition in which, imposing the existence of a local pressure minimum, a vortex core is the connected region with two negative eigenvalues of 𝗦2 + Ω2. As they discuss, their criterion satisfies the requirements of Galilean invariance and net vorticity and, furthermore, includes similar dynamical properties of the criteria proposed by Hunt et al. (1988) and Chong et al. (1990). Through several analytic and numerical examples, Jeong and Hussain (1995) show that in general the three criteria do not capture exactly the same regions for 3D flows.

However, an interesting point concerning the application of these three criteria to altimetric maps is that they are strictly equivalent for pure planar flows. In this case eddies are defined as the simply connected region where

i1520-0426-20-5-772-e2

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where (u, υ) is the two-dimensional velocity field. The equivalence can be understood because Q is the source term of the Poisson equation for pressure, ∇2_p_ = 2_ρ_ Q, and also it is the discriminant of the characteristic equation for finding the eigenvalues of ∇v. This quantity, except for a factor of 4 and a global change of sign, corresponds to the Okubo–Weiss parameter (Okubo 1970; Weiss 1991), the magnitude of which has direct implications on the dynamics of the vorticity field. Under the assumption that vorticity and strain are slowly varying with respect to the vorticity gradient along a particle path, the sign of Q allows partitioning of the fluid flow into two regions: elliptic and hyperbolic (McWilliams 1984). Elliptic domains are those wherein rotation dominates deformation (Q > 0), and hyperbolic domains (Q < 0) are those in which deformation dominates rotation. Numerical simulations of two-dimensional turbulence have shown that elliptic domains correspond to eddy cores but also to other regions corresponding to the background field (Elhmaïdi et al. 1993). Basdevant and Philipovich (1994) provided numerical evidence that the partition given by the positiveness of the Okubo–Weiss parameter (or, equivalently, Q) is strictly valid in vortex core regions or in the immediate vicinity of saddle points.

In summary, theoretical results presented above show that eddy cores can be identified by selecting the simply connected regions with high positive values of Q. In order to distinguish and isolate the coherent structures from the background field, a threshold for Q must be defined (Elhmaïdi et al. 1993). This threshold is usually based on the statistical properties of the field and depends on the Reynolds number. For two-dimensional turbulence it is typically taken as Q_0 = 0.2_σ Q, where σ Q is the standard deviation for Q (Pasquero et al. 2001). Alternatively, in other works (e.g., Benzi et al. 1986; Bracco et al. 2000), a similar level of threshold, but for the vorticity, has been used to partition the fluid flow.

The application of expression (2) to identify marine eddies within altimetric maps is straightforward. Assuming an almost divergent free flow and geostrophic balance, the streamfunction is proportional to the sea level height. In terms of the streamfunction, condition (2) is written as

i1520-0426-20-5-772-e3

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which is, in fact, the necessary condition for the existence of an extreme (Courant and John 1989). In other words, an eddy core region is a region wherein a streamfunction extreme is possible.

b. Data

In this study, the criterion is applied to sea level anomaly (SLA) maps produced by Collecte Localisation Satellites (CLS), Toulouse, France, from the combination of European Remote Sensing Satellite (ERS) and TOPEX/Poseidon altimeters. These maps are processed including the usual corrections (sea-state bias, tides, inverse barometer, etc.) and with improved ERS orbits, using TOPEX/Poseidon as a reference (AVISO 1997; Le Traon and Ogor 1998). SLAs are regularly produced by subtracting a 4-yr mean value 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). 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).

3. Results

By applying centered differences, maps of geostrophic velocities, Q, and vorticity fields have been computed. Figure 1a shows a typical map of SLA covering the southern region of the Western Mediterranean along the African coast. Regions of Q > 0 are easily detected for each map and separated according to the sign of vorticity (Figs. 1b,c). By looking at these maps several kinds of regions can be appreciated. In general a close correspondence between regions of maximum and minimum values of SLA and regions where Q > 0 is found, mostly coinciding with marine eddies. However, some detected regions or patterns are not directly associated with closed streamline domains, or it may happen that a big structure is split into two separated areas of Q > 0. See, for example, the regions near 0° or 8°E in Fig. 1b. In the first case, these regions are effectively dominated by rotation in front of deformation but are characterized by low levels of Q. The values of Q near the center of the anticyclonic structure centered at 37°N, 2.5°E are of the order of 7 10–11 s–2, while for the cyclonic structure centered at 37°N, 4°E they are of the order of 1 10–11 s–2. In fact, a high enough threshold value for Q filters out most of these areas (Fig. 2).

As pointed out, another situation arises when a large closed streamline pattern splits into two regions with Q > 0. This is the case in which this criterion performs better than finding the extreme values of the closed streamlines areas. Figure 3 shows three consecutive SLA zooming maps and the corresponding regions detected by Q > 0. In the initial maps the streamlines and the Q maps show a similar scenario: a large anticyclonic area that contains two structures enclosing two streamfunction extremes and surrounded by two cyclonic areas. In the middle figures, the streamline pattern appears as a single elongated structure, while the Q field still detects two distinct core regions. Finally, at the end of the sequence, there only appears a strong eddy as denoted by both fields. A remarkable property is that the Q field can be used to filter out the weakest structures; the only difference is that the core regions are wider as the threshold used is lower.

An interesting consequence of the combination of Q > 0 regions with an appropriate threshold value is that intense vortex core regions can be easily extracted and therefore their main kinematic properties computed. To test this methodology (see Fig. 4) it has been applied to the 96-1 and 97-1 eddies described in Puillat et al. (2002, see their Fig. 1). Observed trajectories show very similar patterns. The most remarkable difference is the length of the trajectories, which are shorter using altimetry alone. Also, the use of the Q criterion allowed tracking of the 96-1 eddy for 20 more days than did visual inspection of SLA contours. Notice that trajectories are smoother when derived from altimetry; this is because the use of SLA maps and the identification of eddies allows objective definition of the eddy center. In particular we took it as the mean position of the simply connected region with Q > 0. Alternatively, it can be defined as the point of maximum (or minimum) SLA, but this definition is not invariant under Galilean transformations and has potential problems when an eddy is embedded in a background flow.

4. Conclusions

This paper shows that a criterion based on the sign of Q, which corresponds to an invariant of the velocity gradient tensor, can be used as a way to systematically identify marine eddies from SLA maps. The criterion partitions the field depending on the sign of Q, and eddy cores are recognized as the simply connected regions characterized by positive values of Q. Unfortunately, it captures not only eddy cores but also all regions in which rotation dominates over deformation. This implies that complementary information is needed to isolate the eddy core regions from other structures. As suggested from other published works, a suitable procedure should be to use a threshold value _Q_0 based on the statistical analysis of the field. Here values of _Q_0 have been subjectively selected to show the filtering effect of the choice. Future work should be done in order to better determine which is the most adequate threshold value _Q_0 needed to filter out undesired structures.

The criterion is an objective detector in the sense of being independent of the reference system. Thus it is supposed to be unaffected by the seasonal variations of mean sea level, when they have wavelengths much greater than the size of the eddy regions, and also by the presence of a linear trend in SLA maps. Both effects are equivalent to a uniform Galilean transformation that should modify the streamline field but not the Q map.

Another question is related to the hypothesis underlying the applicability of the Q criterion. Notice that the equivalence between some vortex identification criteria is based on the assumption that the flow is horizontally nondivergent. This assumption is not true for the geostrophic flow, but its horizontal divergence is very small, so in general this assumption should not introduce big errors. Here the comparison with the output of numerical models of ocean circulation would give more information on the limits of applicability.

Finally, the precise identification of eddies allows development of an algorithm for automatically tracking eddies and monitoring of their kinematic properties in a routine way. The comparison with some published results has been straightforward, but in our case the information content has been reduced and the eddies have been almost automatically tracked. The limitations are mostly imposed by the time interval of the SLA maps in comparison with the time evolution of these vortices.

Acknowledgments

This is a contribution to the GPS Radar Altimeter Calibration (GRAC) project funded by the Spanish R + D Plan and the European Union (2FD97-0588). Jordi Isern-Fontanet has been partially supported by contracts from the GRAC and IMAGE (REN2001-0802-C02-02) projects. Altimetric maps for the period October 1992–97 were elaborated and provided by CLS (Toulouse, France) under contract (MAS3-CT96-0051) of the Mass Transfer and Ecosystem Response (MATER) project funded by the European Commission. Altimetric maps for the period October 1997–September 1999, also produced by CLS, were kindly provided by Giles Larnicol. We would like to thank two anonymous reviewers for comments that helped to improve the manuscript.

REFERENCES

Fig. 4.

Fig. 4.

Fig. 4.

Observed positions and propagation velocities derived with the Q criterion of the same Algerian eddies described in Puillat et al. (2002): (top) 96-1 eddy from 19 Feb 1996 to 21 Jul 1998 and (bottom) 97-1 eddy between 20 May 1997 and 28 Sep 1998

Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<772:IOMEFA>2.0.CO;2