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Notes from Stirring of the northeast Atlantic spring bloom: A Lagrangian analysis based on multisatellite data by Lehahn et al. (2007, JGR)


The paper studies the formation of meso and submesoscale structure in surface chlorophyll (SC) during the North Atlantic Spring bloom. Calculation of geostrophic flow, Lagrangian trajectories and finite-size Lyapunov exponents (FSLE), together with observations of SC and sea surface temperature (SST) show the two dominant effects of geostrophic flow on SC: (1) a direct effect by advecting SC –isolating SC at the core of an eddy, forming spirals around eddies or filament-like structure–, and (2) an indirect effect by creating new SC via upwelling of nutrients along unstable manifold.

The histogram of the angle between SC fronts and ridges of high FSLE values (Fig. 8) is a convincing statistical evidence that SC fronts do indeed appear along unstable manifolds. I found, however, that the case by case comparisons are not quantitative enough.

Details of the calculation of Lagrangian diagnostics

  • Runge-Kutta scheme of the fourth order used to compute Lagrangian trajectories.
  • The procedure to compute the finite-size Lyapunov exponents (FSLE) is similar to that used by d’Ovidio et al. (2004) with an initial separation of 0.01 deg. and final one of 0.8 deg. “In order to avoid the dependence on the orientation of the pairs, Lyapunov exponents are obtained by diagonalizing the linear transformation of a square whose diagonals are formed by two pairs [Ott, 1993]”.

Difference between finite-size and finite-time Lyapunov exponents

For finite-size Lyapunov exponents, the final distance delta_f is fixed and the time tau is calculated. For finite-time Lyapunov exponents, the time tau is fixed and the final distance delta_f is computed. “From a theoretical viewpoint finite-size and finite-time Lyapunov exponents are quite different”.

Validity of the FSLE calculation

“The above description of hyperbolic structures is rigorous for stationary velocity fields, but can still be applied to the time-dependent case, provided that the evolution of the velocity field is on a slower timescale than tracer advection.” This, again, suggests to introduce the ratio R=tau/T, where tau is the timescale associated with the Lagrangian diagnostic (time scale of the advection within the hyperbolic structure and T is the typical timescale of advection of the hyperbolic structure itself. FSLE calculations are valid if R is much smaller than one (see a similar comment made in the notes on d’Ovidio et al. (2009). T can be calculated from the Lagrangian calculation, using the actual distance followed by the probed parcel and the averaged speed of the parcel during this course. However,

“[f]or the ocean, there is a clear timescale separation, because the propagation of the eddies (and thus of the unstable manifolds) is much slower than the mesoscale velocity field. Midlatitude mesoscale eddies have typical lifetimes of several months and a speed of the order of 10 km/week while mesoscale velocities are 1 order of magnitude larger.”

An interesting discussion of time-independent and time-dependent cases

“Figures 4c–4f compares the cases of stationary (i.e., frozen in time) and time-dependent velocities. In the first case, synthetic tracer trajectories are forced to follow altimetric isolines. Indeed, in geostrophic balance, altimetric isolines are the streamlines and therefore for a stationary velocity field the altimetric isolines coincide with the trajectories. A consequence of this stationarity is that the eddies are characterized by concentric closed circles (‘tori’, as they are properly referred to) that perfectly isolate from the surrounding (Figures 4c and 4e). On the other hand, for the case of time-dependent velocities, the identity between trajectories and streamlines does not hold anymore. In particular, a tracer released inside an eddy does not follow a closed path and after one revolution does not come back exactly to its initial position. In this way, the impermeable barriers formed by the concentric isolines are replaced by spirals, as can be easily seen by detecting transport barriers with the Lyapunov exponent calculation (Figure 4f), or simply by releasing a tracer (Figure 4d). The spirals are tighter and closely resemble impermeable tori where the streamlines are not strongly affected by the time dependency, that is, at the eddy cores. In these regions, passive tracers are trapped for a long time and can escape the eddy only after several revolutions. [...] On the other hand, tori break in looser spirals at the periphery, allowing a relatively stronger exchange. The formation of spirals is a purely dynamical phenomenon, that is controlled by the time variability of the velocity field and not by the spatial scale of the velocity field. For this reason, a passively advected tracer can be distributed with spatial structures that are below the resolution of the velocity field itself. Note however, that spirals can also result from nongeostrophic stationary velocity fields, such as the wellknown Ekman spiral.”


See Lapeyre and Klein (2006) and Legal et al. (2006) for the relationship between unstable manifolds and in situ chlorophyll production via upwelling of nutrients. Look at, also, the several papers referenced in the discussion section: Levy and Klein, 2004; Levy, 2003, Mahadevan and Archer, 2000; Levy et al., 2001; Martin et al., 2002; Lapeyre et al., 2006; Levy et al. (2005a). See also studies that take into account biological dynamics: Lopez et al., 2001; Martin et al., 2002; Srokosz et al., 2003.

Fig. 6 in Lehahn et al. 07

Questions and some answers

  • Again and again, what are the stable and unstable manifolds? Does the unstable manifold correspond to the stretching/divergent lines and the filaments or not? The answer is yes. So why, again, the stable manifold corresponds to large FSLE values computed forward in time while the unstable manifold corresponds to large FSLE values computed backward in time? If you put a patch on the stable manifold initially (the convergent line) and follow it forward in time, it will quickly stretch along the unstable manifold and correspond to a large value of the FSLE. If you then plot this large value at the initial position of the patch, you are locating the stable manifold, not the unstable one. Inversely, if you put the patch on the unstable manifold initially and follow it backward in time, it will stretch rapidly along the stable manifold and correspond to a large value of FLSE. If you plot this value at the initial position, you are locating the unstable manifold. That is why, large values of FSLE computing backward locate unstable manifold. Notice now that if you put a patch on the unstable manifold and follow it forward in time, you will obtain large values of FLSE as well but much smaller than the case where the patch started on the convergent line –because the stretching within the divergent line is less dramatic (needs more time) than the stretching from the convergent to the divergent line.
  • Is it not clear to me why we should use finite-size rather than finite-time Lyapunov exponents and when which type is more appropriate than the other.

Ideas of projects

  • Nobody has addressed the effect of ageostrophic dynamics, such as Ekman dynamics, on the stirring of SC. The success of predicting SC filaments using geostrophic velocities only suggests, however, that ageostrophic effects may be secondary. We could use the output of high-resolution General Circulation Model (GCM) and compare the unstable manifolds obtained using either the geostrophic or full velocity field. Another use of the GCM would be to study the effect of resolution of SLA on the unstable manifolds due to the geostrophic flow. What is the SLA resolution needed to predict reasonably the unstable manifolds? See studies that take into account biological dynamics: Lopez et al., 2001; Martin et al., 2002; Srokosz et al., 2003.
  • The simulation of particles advection in Fig. 9 is not too convincing. Could we not pick up an initial time where a bloom has appeared, initialize a patch of synthetic tracers in the area of the bloom, calculate the trajectories and compare with ulterior observations of SC. The problem is that over the time period of the advection, SC is not passive and kept growing and dying, is not it?
  • “A natural extension of our results is therefore the integration of a Lagrangian analysis with biogeochemical models and with vertical velocity estimation by density anomalies and wind stress. A similar combined approach should be able to attempt a quantitative prediction of chlorophyll gradients and a parameterization of transport on plankton bloom.” I like this one, although it is a complex one.


  • Lapeyre, G., and P. Klein (2006), Impact of the small-scale elongated filaments on the oceanic vertical pump, J. Mar. Res., 64, 835–851.
  • Legal et al. (2006): Diagnosis of the vertical motions in a mesoscale stirring region, J. Phys. Oceanogr., 37, 413–1424.