# First calculation of FSLE using OFES¶

This note presents the first calculation of finite-size Lyapunov exponents (FSLE). Two different methods are used. Both are based on the computation of Lagrangian trajectories using a 4th-order Runge-Kutta algorithm and both use neighboring particles. They differ in the last step to compute the exponents: one uses the distance between parcels, the other computes the axes of an ellipse from the deformation of a hypothetical circle. The second one can also provide the direction of the stable/unstable manifold but this information has not yet been implemented.

The first calculation has been performed using the geostrophic flow during January and February 2000 deduced from the sea surface height (SSH) of the QUICKSCAT 0.1° 3-day OFES output.

First, the two methods give similar qualitative and quantitative results with little discrepancy between the two (Fig. 1).

Figure 1: Stable manifold for Feb. 1, 2000 with initial distance of 0.1° and final distance of 1° using (a) trajectory separations and (b) ellipse deformation.

Fig. 2 show the stable and unstable manifolds for Feb. 1, 2000 with the former computed from onward-in-time trajectories and the latter from backward-in-time trajectories. Fig. 3 shows a composite of the two with stable manifold shown with positive values and unstable one with negative values. It is a little bit messy so it is hard to recognize where hyperbolic points are located.

Figure 2: (a) Stable and (b) unstable manifold for Feb. 1, 2000 with initial distance of 0.1° and final distance of 1° using ellipse deformation.

Figure 3: Stable (positive) and unstable (negative) manifold for Feb. 1, 2000 with initial distance of 0.1° and final distance of 1° using ellipse deformation.

The calculation with either method is also not too sensitive to the value of the final distance chosen (Fig. 4). We also see that a period of integration of 30 says is adequate for this type of regime (Fig. 5).

Figure 4: Stable manifold for Feb. 1, 2000 with initial distance of 0.1° and various final distances, all using ellipse deformation.

Figure 5: Stable manifold for Feb. 1, 2000 with initial distance of 0.1° and final distance of 1° using ellipse deformation and for various periods of integration.

Next step would be to compute the average FSLE for this region for one or more years and compare it to Fig. 6 of Calil and Richards (2009).