# 04.07.11: Relationship between FSLE and SST gradient¶

Here, I explore how good is the relationship between the unstable manifolds (estimated via the finite-size Lyapunov exponent or FSLE) and the gradient of sea surface temperature (SST) using the regional HYCOM simulation. Especially, when should I expect FSLE and BVF to be correlated in the observations? All the time or only during a particular period of the year? What is the probability that a FSLE filament is associated nearby with a SST filament? To answer these questions, I perform different statistics.

First, I compute the spatial correlation between FSLE and SST gradient for each time step after each quantity has been smoothed spatially (because in about half the cases, a FSLE filament is only nearby a SST filament). The result is shown in Fig. 1, together with his 95% confidence interval (calculated using the null hypothesis that the FSLE and SST gradient are unrelated). We can see that, as expected from visual inspection, FSLE and SST filaments are related. There is some seasonality suggesting that the FSLE and SST filaments are well correlated in Winter-Spring and may be uncorrelated in June-July but there is a large interannual variability and more years would be needed to conclude more decisively.

Figure 1: Spatial correlation between FSLE and SST gradient over the domain 201°E-208°E and 22°E-25°N. The dash red lines show the 95% confidence interval. Each quantity has been smoothed spatially by a 0.1-° running mean before computing the correlation. The result does not change much if the smoothing is performed over 0.05 or 0.75° instead.

Second, I explore what is the chance that a point inside a FSLE filament indicates also a SST filament. A FSLE filament is defined as a point for which FSLE is larger than 0.2 1/day and a SST filament by a point for which the SST gradient is larger than 0.2e-4°C/m. Fig. 2a first shows the percentage of the domain that is within a FSLE and/or a SST filament. We see that both types of filaments increase in late Winter and beginning of Spring and are the rarest in Fall. The fact that both typoes evolve similarly is consistent with the previous result that the two are related.

Figure 2: (a) Percentage of domain area that within a FSLE and/or SST filament. (b) Percentage of points within a FSLE (SST) filament that are also nearby a SST (FSLE) filament. The definition of “nearby” is that at least 10% of the area within 0.25°of that point is within a filament. The horizontal dash lines show the 95% confidence interval. The interval was computed by repeating the same calculating but using maps of FSLE and SST each drawn randomly. The calculation was repeated N=10,000 times but we obtain almost the same result with N=1,000. The non-horizontal dash lines shown another calculation of the 95% confidence interval. This time, to compute the interval at time t, we use the original map of FSLE at time t, rotates it by a random angle and flip it so that the northeast corner becomes the southwest corner and inversely. This assures that the population from which we compute the histogram and the confidence interval has the same density of filaments. The calcul is yet not perfect as it uses filaments from the south of the Hawaiian Archipelago, a region that does not have necessarily the same dynamical regime. Because the calcul is very long, I stopped it for February 2010, enough to see that the temporal variation of the percentage is due mainly to the change in density.

Fig. 2b shows the percentage of points within a FSLE (SST) filament that are also nearby a SST (FSLE) filament. The dash lines show the 95% confidence interval. It is high because of the large density of FSLE and SST filaments –so that it is easy for a point within, say, a FSLE filament to be near a SST filament just by chance. However, during late Winter and Spring, the percentage obtained is unlikely to be due solely to chance, confirming once again that during that period, FSLE and SST filaments are highly correlated.

The calculation of the confidence interval suffers, however, of the change in filament density shown in Fig. 2a. Thus, the confidence is likely to be overestimated for the periods where density is weak and likely to be underestimated for the periods where density is high. Because most of the time, the density is low (Fig. 2a), the correction will be larger for the periods where density is high; this is a problem as it is the period we are interested in.

For this reason, I perform another estimate using a population of maps that have the same density as the one at time t. To construct this population, I take the map of FSLE at time t, rotate it by a random angle and transpose it –so that the northeast corner becomes the southwest corner. Because the calcul is very long, I stopped it for February 2010, enough to see that the temporal variation of the percentage is due mainly to the change in density. This suggests that the proximity of FSLE and SST filaments are mainly controlled by their density, not by their spatial correlation.

The first conclusion we draw is that not all FSLE filaments are associated with a SST filament but the chance that it is is high during end of Winter and beginning of Spring. This means that, in the observations, I should disregard the rest of the year to study the relationship between FSLE and sudden change in stratification. The second conclusion is that the proximity of FSLE and SST filaments is not a priori indicative of spatial correlation; observations of the spatial structure itself is needed to conclude on the spatial correlation.