We study here if the surface mixed-layer instabilities (SMLI) observed in the regional HYCOM simulation are real feature of the oceans using the ARGO data. One fundamental relationship associated with these SMLI is between the submesoscale horizontal gradient of density and its vertical gradient. The vertical gradient can be observed from, for instance, the Brunt-Vaisala frequency (BVF). There is at present no available product with sufficient resolution to locate the submesoscale horizontal gradient of density (or even temperature) but a study of the regional HYCOM simulation (see ...) shows that there is some correlation between these gradients and the unstable manifold, calculated using the finite-size Lyapunov exponents (FSLE).

In the following, we have extracted ARGO data from a domain (158°W-146°W and 22°N-25°N) northeast of the Hawaiian Archipelago from which we can compute the BVF. The annual cycle of the BVF for that domain is shown in Fig. 1. From February to April, the near-surface BVF is generally low and it is during that period that SMLI occur in the HYCOM simulation.

The histogram of near-surface value of BVF during that period is shown in Fig. 2. The histogram is highly skewed; there is a lot of small values and a few large values. We speculate that the large values are associated with SMLI and thus large values of FSLE.

To see this relationship, for each vertical profile of the values shown in Fig. 2, we keep the maximum value and plot it against the local maximum value of FSLE in Fig. 3 (that value is computed using a radius of 0.25°, about 25 km). There is an indication that large BVF values are associated with large FSLE values and inversely.

To have an idea of the statistical significance of the relationship, we re-compute the regression but, now, with the FSLE drawn randomly inside the domain. We perform that calculation 100 times. An example of a such relationship is shown in Fig. 4. The regression slope from Fig. 3 is +0.0229. In a first calculation with FSLE randomly sampled, we find that the mean regression plus or minus one standard deviation is -0.0017 and +0.0249. In a second calculation, we find -0.0031 and +0.0236. Thus the observed regression is about one standard devation from the mean regression obtained if the null hypothesis is correct. I do not yet know how that translates into a level of statistical significance.

I performed the calculation with 1000 realizations. I obtain -0.0024 and +0.0247 for the mean regression slope plus or minus one standard deviation. The histogram of that slope is shown in Fig. 5, together with the mean plus or minus one standard deviation and the actual value from observations. We see that the distribution is Gaussian with a positive mean.

With a radius of 0.1° instead of 0.25°, I obtain the same qualitative result: the regression slope is 0.0182 and the mean regression plus or minus one standard deviation obtained from a 100 random realizations are -0.0050 and +0.0211. Again, the observed slope is about one standard devation from the mean regression obtained with the null hypothesis. Finally, even when the values for which FSLE is zero are removed, the regression of randomly-sampled realizations stays on average positive.

The reason why even the regression from randomly-sampled realizations is positive is the following. Most of the values have low FSLE and low N; then, for large N, FSLE can be either low (in which the regression is near zero) or large (in which the regression is positive) and thus, on average, the regression is positive.

I used `link_BVF_and_FSLE.m` in `RESEARCH/PROJECTS/MARINE_BIOLOGY/SUBMESOSCALE_PROCESSES/ARGO/analysis` on the central disk.