In this note, I present the first results concerning the use of the “effective” surface quasi-geostrophy (eSQG) theory (Klein *et al*. 2009) to deduce the relative vorticity field (ζ) and the vertical velocity (w) from the sea surface height (SSH). To test the routine that I wrote, I use Paulo’s output model over the upper 60 m. For each snapshot, SSH is detrended by fitting a 2D plane to SSH and removing it from the original SSH field. The SSH is further tapered at the edges of the domain by a 5%-cosinus taper. No other filtering is applied as significant noise has been obtained in the relative vorticity and vertical velocity field when the SSH is high-pass filtered to remove wavelengths smaller than 20 km. Thus more work is needed to include such additional filtering. The values of N0 and c are those used in Klein *et al*. (2009) –the results are weakly dependent of these values, at least over the upper 60-m of the model ocean.

First ζ from model output and from eSQG theory are compared for day 1 (Fig. 1) and their scattered plot is shown in Fig. 2. Overall, model ζ is relatively well predicted from eSQG theory. Notice, however, that the values of the model zeta inside the eddy (southwest corner of the domain) are much larger than those predicted by eSQG theory which results in a lack of correlation (Fig. 2; blue dots).

Second, w from model output and from eSQG theory are compared for day 1 (Fig. 3). w from the model output is relatively well predicted by eSQG theory for scales larger than 0.25-0.5° but not smaller, consistent with the results of Klein *et al*. (2009): the vertical velocity at small scales is dominated by the dynamics of the wind-driven mixed layer which is excluded in the eSQG theory. The correlation is shown in Fig. 4 and appears satisfying, although eSQG tends to underestimate the vertical velocity field. It is expected that the correlation should be better at a deeper level outside the mixed layer.

In this animation, w at 60 m depth from the model and from the eSQG theory is plotted for every 2-day average of the three months of the model, together with their correlation and the detrended SSH. The correlation between the two quantities for all three months is shown in Fig. 5. It seems that there are two types of extreme vertical velocity events from the eSQG theory, one too small and one too large compared to the vertical velocity from the model output. More work is needed to know which flows are associated with these extreme events and why the eSQG theory fails to predict their correct amplitude.

- It would be nice to compare the model output to the output from eSQG theory for deeper depths (100, 200, 500 m and 1000 m for instance), below the mixed layer where the correlation should be much better (Klein
*et al*. 2009; their Fig. 3c). - Study a larger region.
- Include the horizontal filtering (between 20 and 400 km) used in Klein
*et al*. (2009). - What are the two types of extreme vertical velocities obtained by eSQG theory as shown in Fig. 5?
- Results when the SSH is subsampled in time and space as in T/P SSH.