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Notes on Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin by d’Ovidio et al. (2009, Deep-Sea Research I)


In this paper, the authors use the sea surface anomaly (SLA) from altimetric data over the western Mediterranean Sea to calculate and compare Eulerian diagnostics–mainly Okubo-Weiss (OW) criterion–, with the Lagrangian diagnostic of finite-size Lyapunov exponent (FSLE). Eulerian diagnostics are making the assumption that the flow is time-independent; in consequence, they cannot capture the submesocale filament-like features that arise from horizontal advection of the time-dependent flow. FLSE diagnostics, however, make the assumption that lines of divergence/divergence stay the same over the time of integration and they do capture the submesoscale features. For instance, the paper shows the good correlation between FLSE and the structure of a patch of synthetic tracers advected by the altimetry-deduced geostrophic flow.

“We find that at (sub-)mesoscale resolution the information provided by Eulerian diagnostics and Lyapunov exponents coincides only for very stationary eddies, while providing two distinct and complementary pictures of the circulation in all the other cases: the Eulerian analysis provides the eddies that populate the mesoscale, while the Lagrangian analysis yields the tracer filaments generated by the spatio-temporal variability of these eddies.”

The comparison with a real tracer, such as sea surface temperature (SST), is less convincing however but still more informative than using Eulerian diagnostics. The authors speculate that the discrepancies between FLSE and SST are due mainly to the altimetry data, their coarse resolution and their exclusion of ageostrophic motions. But they also note that the discrepancies are not dramatic and could be limited easily with higher resolution altimetry data expected in the near future.

Sparse notes

  • This paper has a nice summary (p. 16) of previous studies that used finite-size and finite-time Lyapunov exponents (FSLE and FTLE respectively). Especially, Abraham and Bowen (2002) and LaCasce and Ohlmann (2003).
  • In this paper, three probing tracers, forming an equilateral triangle initially, are used and the calculation is stopped once one of them is at the distance df from the probed tracer (see d’Ovidio et al. (2004) for a slightly different scheme).
  • Convergent lines correspond to the stable manifold while divergent lines correspond to the unstable manifold. Filaments appear in general along the divergent lines/unstable manifold:
Fig. 1 in d'Ovidio et al. 09
  • I would like to understand this part:

    “Due to the hyperbolic structure of which the manifold is a part, the tracer front approaches the manifold exponentially fast. The distance dr between the tracer front and the manifold depends on the initial front-to-manifold distance di, the [Local Lyapunov] exponent l of the manifold, and the time of integration t:

    dr = di exp(-l*t).”

    • Is there not a problem of sign? Compare with (2). Is it because they look at FSLEs computed backward in time?
    • Maxima in FSLE calculated forward in time should corresponds to the divergent lines, the unstable manifold and the front, is not it? So why do they use the FLSE calculated backward in time?
  • This quote

    “Note that for a time-dependent velocity field, the sketch of Fig. 1 holds only if the hyperbolic structures evolve in time on a time scales lower than the tracer advection, so that the tracer can actually relax over the manifold.”

    suggests that we could introduce the ratio R of t (the advective time scale within the hyperbolic structure) and T, the advective time scale of advection of the hyperbolic structure. FSLE diagnostics would thus be valid only when R is much smaller than one. The problem is that it seems difficult to define t and T. A separation of meso and submesoscale fetaures would be needed. Question: By ignoring diagnostics when R is too large, could we filter potentially spurious fronts?

  • As in d’Ovidio et al. (2004), di is set as the resolution of the data –0.01 deg. using the SST data–, and df as 1 deg., which is approximatively the radius of the eddies.

    “Values of df smaller or larger up to 50% do not change significatively the calculation.”

    Typical values of periods of integration are 25 to 45 days. The altimetric data used are a combination of ERS-ENVISAT and TOPEX/Poseidon-JASON data. Geostrophic velocities are deduced from the Sea Surface Anomaly (SLA) and a Runge-Kutta algorithm is used to compute trajectories backward and forward in time (see p. 19 for details).

  • Issue of spurious eddies found with OW but not in SSH or FSLE:

    “Across the Algeriancurrent, the OW field identifies some possibly spurious eddies that do not appear in either the FSLE or SSH. This is due to the fact that both SSH and FSLE are not invariant under a transformation of coordinates to a frame of reference moving at a constant velocity with respect to the original (Galilean transformation) and therefore, eddies are hidden or partially hidden by the presence of the Algerian current. Since OW is Galilean invariant it is able to detect these eddies but itfails in the detection of the Algerian current which appears as a coherent structure characterized by manifolds (barriers) parallel to the altimetric streamlines in the FSLE picture.”

    Question: Could OW and SSH/FLSE be equivalent in regions where there is no strong mean flow, such as in the North Pacific subtropical gyre away from the Hawaiian Islands?

  • The correlation between lines of maximum FLSE and synthetic tracer trajectories are high, suggesting that we could use FLSE diagnostic to have an idea of water advection –ignoring, however, vertical advection.

    • Has a similar study be done for the North Pacific subtropical gyre and observations at station ALOHA?
    • Is this actually surprising, given that FLSE calculation are based on the same trajectories over a much longer period?

    Note: I am not sure I understand the issue on the release dates in Fig. 4. Maybe this discussion contains a partial answer to this question, as well reasons why FSLEs are a better tool than simply following water parcels:

    “In fact, the advection of a synthetic tracer can be considered itself as a direct, Lagrangian diagnostic of transport barriers. This approach, however, has some limitations. Lagrangian structures are sampled unevenly, since the tracer tends to spend longer times in regions with low velocities. For this reason, even strong barriers may require a carefully choice of the initial conditions and of the integration time. An example of this appears in Fig. 4, where part of the Almeria–Oran front is not entirely visible and would have required to probe the velocity field with a larger number of tracer blobs. When doing the tracer experiment of Fig. 4 we also observed that, not surprisingly, the tracer has a long residence time over the Balearic Abyssal Plain and therefore for long integration times mostof the time the tracer shades the barrier of this region only,independently of its initial condition. On the other hand, a reduction of the integration time increases the gap between the tracer front and the actual barrier position, since the tracer front has a shorter time to relax to it (see the discussion of Eq.(3)). In fact, an attempt to overcome these limitations requires the use of backward trajectories (for which barriers are regions of maximal separation) and of variable integration times, basically yielding a FSLE-type calculation. For this reason, the FSLE can be considered an optimized way of advecting a passive tracer for the detection of transport structures. Note also that, besides guaranteeing a uniform sampling of the velocity field and the precise localization of transport barriers, the FSLE also provides the information of the intensity of thebarrier at the same cost of tracer advection.”

  • A good idea for a project:

    “The recent availability of realistic, sub-mesoscale resolving biogeochemical models (e.g. Resplandy etal., 2008) should give the possibility of exploring the role of unresolved ageostrophic components and of tracer activity on filament detection.”

References to read

  • See Lehahn et al. (2007) for more details on the FSLE computation from altimetric data:

    Lehahn et al. (2007): Stirring of the Northeast Atlantic spring bloom: a Lagrangian analysis based on multi-satellite data, Journal of Geophysical Research, 112.