Recently,

Mesoscale eddies (MEs) at spatial scales from approximately 50 up to 500 km are energetic patterns of ocean surface flow fields. The birth of MEs often occurs along shorelines triggered by shear-driven barotropic instabilities or at the edges of surface currents by density-anomaly-driven baroclinic instabilities

Kinetic energy (KE) is a quantitative characteristic of ocean flow fields

In this work we extend a previous analysis by

Snapshot of the global geostrophic flow field on a randomly chosen day (23 September 2018) from the AVISO data bank

Our primary data source is the AVISO data bank

For the validation, we exploited the vortex data bank
assembled by

We are aware of the fact that there is a continuously growing set of data repositories, most of them based on AVISO altimetry. We recently learned about the recent development by

Several studies of the shape of ocean MEs revealed that they are close to Gaussian humps or troughs

We do not repeat all the details described in

We note here that a Gaussian vortex represents a specific case of a general class of solutions of the vorticity equation. The nondimensional form of the tangential velocity field is

In order to characterize correlations for the two integrals

Pearson correlation is the most common metric for the evaluation of a linear association between two time series. An often-used alternative metric to test arbitrary but monotonous association between two time series is provided by the Spearman rank correlation coefficient, which is actually Pearson's correlation coefficient applied to the ranks of the observations. According to our tests, the two coefficients are essentially identical for the integrals, and therefore
we use Pearson's

Special considerations are required to determine the area of integration. The reason is that practically no correlations exist between

A finite area of integration at any tessellation is not closed: eddies come and go, emerge and decay. Still, when the Gaussian hypothesis holds, then a strong correlation is expected between integrated kinetic energy and enstrophy.
If the correlations between

We noted before that the integrated kinetic energy over an infinite domain as in Eq. (

For the reason explained above, we reverse the consideration of Eq. (

A well-known and widely analyzed feature of eddy trajectories is the general tendency for westward propagation in the absences of strong countercurrents

The beta-plane approximation allows an analytical solution with the result that all eddies (cyclonic and anticyclonic) propagate westward, and the speed does not depend on the size and height (or depth) of a vortex obeying geostrophic equilibrium. The simple formula for the drift speed is the same as for long nondispersive Rossby waves

We will compare our results with the linear estimate. We used the gridded dataset for Rossby radii compiled by

The appealing aspect of the beta-plane approach is that the drift speed does not depend on the characteristics of individual eddies, although they transport kinetic energy and vorticity westward. We exploit this fact to estimate westward propagation velocities by evaluating the cross-correlation

For the validation, we again explored the eddy census by

Pearson correlation coefficients

The geographic distribution of the Pearson correlation coefficient

The strong correlations permit giving an estimate for an effective proxy vortex radius

Geographic distribution of the effective radius

For the validation of the results, Fig.

The agreement between the five estimates of zonal mean effective radii

Zonal and temporal mean values of the characteristic vortex sizes

Note that averaging should be performed with care. This is due to a finite size effect: it often occurs that no eddies (more precisely, no eddy centers) are identified in a given day and in a given tile by the eddy census. Actually, from the aggregated dataset of size 7792

The dashed line in Fig.

Temporal mean eddy kinetic energy

In order to compare amplitudes of proxy vortices and amplitudes (stored height parameters) in the eddy census database, one can explore Eq. (

Temporal mean eddy kinetic energy ratio

Time-dependent cross-correlation analysis of the EKE field by Eq. (

Figure

These statistics also suffer from an additional finite size effect, besides the occasional lack of identified vortices. When the location of an ME is close to the boundary of the given tile, its contribution to

An essential and well-known source of input data errors is the difficulty of measuring eddy amplitude from altimeter data. Most of the recorded amplitudes are small in the range of a few centimeters, and the reference level is usually the approximate height of the identified close contour (depending on the method). Small eddies (in both extent and height) are poorly resolvable at the available spatial resolutions, and therefore estimates are certainly loaded with errors. In order to check the sensitivity of the method to amplitude errors, we repeated all the calculations whereby the amplitude values are systematically shifted up by

Zonal mean values of the characteristic zonal drift speed

Hovmöller plots (time–longitude diagrams) of eddy kinetic energy propagation for three particular regions. Latitudes are indicated in the title of panels. Black lines guide the eye for characteristic slopes. Color scales are logarithmic.

Snapshots of the geostrophic flow field and integrated kinetic energy (color scale is linear) for two time instances in the northern Pacific region analyzed in Fig.

The results for the analysis of (mostly) westward drift (see Sect.

Figure

The observed discrepancies can be explained by the fact that mesoscale eddies are not the only form of kinetic energy transport on the ocean surfaces. We have already noted that the eddy kinetic energy field estimated from the geostrophic velocity anomalies (see Fig.

In order to further elaborate the possible reasons for the discrepancies between the drift speed values obtained by the two methods, we studied the Hovmöller diagrams (time–longitude plots) for several latitudes. Three examples are illustrated in Fig.

Since we already discussed the results in the subsections above, here we list the main findings of this work.

The eddy kinetic energy (EKE) obtained from the 2D geostrophic velocity anomalies [

The time-dependent integrated eddy kinetic energy

The effective zonal mean radii of proxy vortices

Integrated eddy kinetic energies are obtained by two estimates. Firstly,

As for the propagation of kinetic energy and drift of eddies, the estimates from time-lagged cross-correlation analysis of EKE fields are in general agreement with vortex tracking statistics considering order of magnitude and sign (see Fig.

We can shortly conclude that the original proposal by

Global geostrophic velocity fields are openly available after registration at the EU Copernicus Marine Service (

IMJ designed the research. IMJ and MV performed the research. HK and JACG contributed new numerical and analytical tools. IMJ and HK analyzed data. IMJ, HK, JACG, and MV wrote the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Hungarian National Research, Development and Innovation Office under grant numbers FK-125024 and K-125171, as well as by the Visitor Programme of Max Planck Institute for the Physics of Complex Systems. Jason A. C. Gallas was supported by CNPq, Brazil, under grant PQ-305305/2020-4.

This research has been supported by the Max-Planck-Institut für Physik Komplexer Systeme (Visitor Programme), the Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, National Research, Development and Innovation Office (grant nos. FK-125024 and K-125171), and the Conselho Nacional de Desenvolvimento Científico e Tecnológico, Ciência sem Fronteiras (grant no. PQ-305305/2020-4).The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Ilker Fer and reviewed by Takaya Uchida and one anonymous referee.