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Notes on Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents by d’Ovidio et al. (2004, GRL)

In this paper, the authors compute the finite-size Lyapunov exponents (FLSE) from the velocity field output of a high-resolution numerical model. The model has 1/8 deg. resolution in the horizontal and 30 levels in the vertical. The analysis is performed on the second layer, thus avoiding the wind-driven Ekman dynamics that dominate mostly in the first layer.

They argue that the horizontal velocity field in that layer is close to be horizontally non-divergent because of the weakness of the vertical velocity compared to the horizontal one. Furthermore, because we are interested in the transport due to mesoscale, the FLSE are calculated over 1-10 days corresponding to horizontal displacement of 100 km. Over this spatial scale, the averaged vertical velocity (from the model output?) is less than 1 m/day so that over that period, particles have travelled less than 10 m in the vertical, less than the thickness of the layer studied. Thus, it is reasonable to assume that the flow stays horizontal within that layer. [Question: does the flow has to be horizontally non-divergent to calculate the FLSE?].

The Lyapunov exponents “are defined as the exponential rate of separation, averaged over infinite time, of fluid parcels initially separated infinitesimally. In realistic situations (such as the case of the Mediterranean Sea where boundaries at finite distance strongly influence the circulation) the infinite-time limit in the definition makes the Lyapunov exponent a quantity of limited practical use. Recently, the Finite Size Lyapunov Exponent (FSLE) has been introduced [Aurell et al., 1997; Artale et al., 1997] in order to study non-asymptotic dispersion processes, which is particularly appropriate to analyze transport in closed areas.”

Given one particle at time t and position x and another particle at time t and at a distance d0 from the first particle, the trajectories of each are computed and the time tau necessary for the two particles to be separated by a distance df is calculated. The FLSE is inversely proportional to tau and is a function of x, t, d0 and df:

“the FSLE represents the inverse time scale for mixing up fluid parcels between length scales d0 and df.”

Four values of FLSE are computed, depending on the position of the second particle to the first one (to the west, the north, the east or the south) and the final FLSE kept is the largest of the four. [Question: Are we not forgetting larger FLSE values by chosing only four relative positions?]

The authors chose d0 as the horizontal grid spacing Dx. I do not understand perfectly the reason why:

“If one chooses d0 much smaller than Dx, all the points of a stretching manifold laying further than d0 from any grid point are not tested, and thus the method gives only a rather discontinuous sampling of the structure. On the other hand, if d0 is much larger than Dx, the same stretching manifold is detected (‘smeared’) on several grid points, with a loss in spatial resolution.”

The final distance df is chosen as the mesoscale (about 1 deg.) at which we want to study the mixing properties.

Calculating FLSE for each day and taking the time average gives the spatial distribution of low and high mixing areas. High FLSE values also correspond to a good approximation to repelling material lines and stable manifolds. The largest values of FLSE, and so the strongest areas of mixing, are seen at the edges of eddies while the center of eddies correspond usually to low level of mixing.

By calculating FLSE from an integration backward in time, FLSE maxima indicate areas of maximum compression, attracting lines or unstable manifolds. [Question: Why the low values of FLSE when integrating onward with time are not the same than the high values of FLSE when integrating backward with time?]. Regions of strong mixing are those where compressing and stretching lines cross each other. The authors introduce an index to isolate specifically these areas of strong mixing. [Question: Why the maximum values of FLSE when integrated onward with time alone do not correspond to strong mixing? In other words, why do we have to combine the FLSE calculated onward with time with those calculated backward in time to know regions of strong mixing? Here is a partial answer to these questions:

“Because of the approximate incompressible character of the horizontal flow, the temporal variations of forward and backward FSLEs are strongly correlated, and one expects that the same information can be obtained from just one of the FSLEs.”

Thus, in the case of horizontally nondivergent flow, it seems that indeed one type of FLSE, and one index only, is enough but because it is not perfectly the case, the authors checked by comparing the two indeces of strong mixing. Question: The authors show temporal coherence between the two indeces? Do we have also spatial coherence? We should if I understand correctly.] The advantage of using the index based only on the FLSE computed onward is that it can be calculated from floater experiments. [Question: Why and how?]

[Question: Should we apply the same technique with Pierre to the features of Tropical Instability Vortices and see if the results concur with Pierre’s detailed Lagrangian study?]


  • Artale, V., G. Boffetta, A. Celani, M. Cencini, and A. Vulpiani (1997), Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient, Phys. Fluids, 9, 3162–3171.
  • Aurell, E., G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani (1997), Predictability in the large: An extension of the concept of Lyapunov exponent, J. Phys. A Math. Gen., 30, 1–26.