Biological activity and mesoscale horizontal stirring in the surface  ocean of the 4 Eastern Boundary Upwelling Systems: a comparative study. 

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Generalities and basic principles:

Altimetry satellites basically determine the distance from the satellite to a target surface by measuring the satellite-to-surface round-trip time of a radar pulse. However,  this is not the only measurement made in the process, and a lot of other information can be extracted from altimetry. The magnitude and shape of the echoes (or waveforms) also contain information about the characteristics of the surface which caused the reflection. The best results are obtained over the ocean, which is spatially homogeneous, and has a surface which conforms with known statistics. Surfaces which are not homogeneous and contain discontinuities or significant slopes, such as ice, rivers or land surfaces, make accurate interpretation more difficult. Several different frequencies are used for radar altimeters. Each frequency band has its advantages and disadvantages : sensitivity to atmospheric perturbations for Ku-band, better observation of ice, rain, coastal zones, land masses… for Ka-band. Radar altimeters permanently transmit signals to Earth, and receive the echo from the reflecting surface. The satellite orbit has to be accurately tracked (Doris system allows a very precise location of the satellite on its orbit) and its position is determined relative to an arbitrary reference surface, an ellipsoid. The sea surface height (SSH) is the deviation from the sea surface to a reference ellipsoid or a mean sea surface (see Fig. 2.6).
It is used today for a large panel of applications, going from large scale circulation, tides, mean sea level, continental water monitoring, and last but not least, the study of mesoscale circulation.

Sea Surface Height, Quikscat wind stress and derived geostrophic currents:

In this section, multi-satellite data are used to infer mesoscale velocity fields of the surface ocean in two dimensions.
The satellite surface currents we used, Ut(ut, vt) in m.s−1, are the sum of the gridded geostrophic velocities, Ug(ug, vg) and the Ekman currents at 15 m depth, Ue(ue, ve) (Sudre and Morrow [2008]).
The geostrophic currents are calculated from the a Sea Surface Height composite field. The Mapped Sea Level Anomaly (MSLA) from the Data Unification and Altimeter Combination System (DUACS) are combined with the mean dynamic topography RIO05 (Rio et al. [2005]) to obtain a time-variable sea surface height Materials and Methods: a set of complementary tools to study the influence of physical processes on ecosystem dynamics at mesoscale.
(SSH) data product. The MSLA product for the 1999-2006 period merges altimetric measurements from five altimeter missions (Topex/Poseidon, ERS1 and 2, Geosat Follow-on, Envisat and Jason-1). The mean dynamic topography (MDT) is the mean sea surface height that is due to the permanent ocean circulation, with the marine geoid removed. The MDT product used hereafter is derived from the RIO05 product, which is based on multiple in situ and satellite data sets, including GRACE gravity data. We use the weekly DUACS SSH data that has a spatial resolution of 1/3◦ projected onto a Mercator grid. We convert this spatial resolution onto a 1/4◦ regular grid using a standard bilinear interpolation algorithm, which is more easily compared to other satellite data products (e.g. 1/4◦ resolution SST or Ocean Colour).
Firstly, we calculate the total SSH as the sum of the altimetric MSLA and the RIO05 MDT. The SSH gradients ( @h @y , @h @x ) are calculated linearly from the surrounding grid points using a finite difference formulation. Outside the equatorial band (5◦S to 5◦N), the surface currents are calculated from these SSH gradients, assuming a geostrophic balance: ug = − g f @ h @ y vg = g f @ h @ x (2.2).
Where f = 2 sin ‘ the Coriolis parameter depending on latitude, g is the acceleration due to gravity and h is the height of the sea surface above a level surface (SSH).

Eulerian / Lagrangian description.

The description of fluid motion can be addressed following two different ways: one can evaluate the velocity, pressure and density fields at fixed spatial locations in the fluid, or either follow the trajectory of each fluid particle. The first approach is called Eulerian and the second one Lagrangian. In principle both are equivalent, and if we denote by v(x, t) the Eulerian velocity field, providing us the value of the fluid velocity at any space-time point (x, t), then the motion of a fluid particle with initial localization x(0) is given by: dx dt = v(x, t). (2.5).
This expression establishes the physical connection between the Eulerian and Lagrangian description. It clearly says that when a particular fluid particle is known to be at a specific space-time point, its Lagrangian velocity must be equal to the Eulerian field value at that space-time point.

Dynamical systems and manifolds.

A dynamical system of general form is often expressed by dx dt = v(x(t), t) (2.6)  (t0) = x0 (2.7). In the differential Eqs. (2.6), (2.7), t represents time and it is the independent variable, and the dependent variable, x(t), represents the state of the system at time t. The vector function v(x, t) typically satisfies some level of continuity.
Materials and Methods: a set of complementary tools to study the influence of physical processes on ecosystem dynamics at mesoscale.
As time evolves, solutions of Eqs. (2.6), (2.7) trace out curves. In dynamical systems terminology, solutions flow along their trajectory. Numerical solutions of Eqs. (2.6), (2.7) can almost always be found by numerical integration of v, however such solutions are not convenient for general analysis. While the exact solution of Eqs. (2.6), (2.7) would be ideal, the analytic solution of Eqs. (2.6), (2.7) can not be calculated in general.
If v is independent of time t, the system is known as time-independent, or autonomous, and there are a number of standard techniques for analyzing such systems. The global flow geometry can be understood by studying invariant manifolds of the fixed points of Eqs. (2.6), (2.7), in particular stable and unstable manifolds often play a central role.
A fixed point of v is a point xc such that v(xc) = 0. The stable manifolds of a fixed point xc are all trajectories which asymptote to xc when t → ∞. Similarly, the unstable manifolds of xc are all trajectories which asymptote to xc when t → −∞ . Often, stable and unstable manifolds separate distinct regions with different flow geometry.
The stable and unstable manifolds can help uncover the global flow geometry of a dynamical system (see Fig. 2.8). The notion of stable and unstable manifolds becomes ambiguous for time-dependent systems, which are the most relevant for us. Such systems rarely even have fixed points in the traditional sense. Many dynamical systems of practical importance are time-dependent, especially in cases where the dynamical system represents the motion of a geophysical fluid. These time-dependent dynamical systems typically have regions of dynamically distinct behavior which can be thought of as being divided by separatrices. However, for such systems these regions change over time, and hence so do the separatrices. We consider a generic hyperbolic point and its associated stable and unstable manifolds (see Fig. 2.8).
If we integrate two points that are initially on either side of a stable manifold forward in time, then these points will eventually diverge from each other. Likewise, if we started with two points on either side of an unstable manifold, then these points would quickly diverge from each other if integrated backward in time. This is why these manifolds are often called separatrices, since they separate trajectories.
We would like to define such structures by looking at the divergence or stretching between trajectories. To find separatrices that are analogous to stable manifolds, we measure stretching forward in time and to find separatrices that are analogous to unstable manifolds, we measure stretching backward in time (Fig. 2.8). However, the analogy between these separatrices and traditional definitions of stable and vertical separatrice is equivalent to the unstable manifold, whereas the quasi horizontal separatrice represents the stable manifold. The intersection between the two black lines is the hyperbolic point.
unstable manifolds is not straightforward. For time-dependent flows, we refer to theses separatrices as Lagrangian Coherent Structures (LCS), a common name in fluid mechanics that will be defined in the following. While there are numerous ways to measure stretching, we have found that the Local Lyapunov Exponent (Finite- Time Lyapunov Exponent and Finite-Size Lyapunov Exponent) provides the best measure when trying to understand the flow geometry of general time-dependent systems.

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The non asymptotic Finite-Size Lyapunov Exponents.

The existence of chaotic behavior systems was first introduced by the French mathematician Henri Poincaré in the 1890s in a paper on the stability of the Solar System. Some time later, other scientists found additional chaotic systems and they developed new mathematics and theories (Kovalevska, Hopf, Kolmogorov, Lorentz among others). Chaos is a motion irregular in time, unpredictable in the long term, hyper-sensitive to initial conditions and complex, but ordered, in the phase space: it is associated with a fractal structure. In present day literature, a system is said to be chaotic if small (i.e. infinitesimal) perturbations grow exponentially with time, which is connected to a positive Lyapunov exponent.
The classical Lyapunov exponent is defined as the exponential rate of separation, averaged over infinite time, of particle trajectories initially separated infinitesimally.
Consider x(t0) and x(t) = x(t0) + x(t) the position of two particle separated Materials and Methods: a set of complementary tools to study the influence of physical processes on ecosystem dynamics at mesoscale. initially by a distance x(t0). The global Lyapunov exponent is defined by = lim t→∞ lim x(t0)→0 1 t ln |x(t)| |x(t0)| , (2.8).
The Lyapunov exponent is quite useful in the study of time-independent dynamical systems. The seminal work of Lyapunov [1992] was very important in laying the theory of Lyapunov exponent for time-independent systems. Then the manuscript by Barreira and Pesin [2002] contains a good modern and comprehensive treatment of the subject. However, many dynamical systems of practical importance, especially in the realm of fluids, are time-dependent and only known over a finite interval of time and space. Because of its asymptotic nature, the classical Lyapunov exponent is not suited for analyzing these dynamical systems. The infinitetime limit in Eq.(2.8) makes the Lyapunov exponent of limited practical use when dealing with experimental data. A generalization of the Lyapunov exponent, called the Local Lyapunov exponent (LLE), has been proposed to study the growth of noninfinitesimal perturbations (distance between trajectories) in dynamical systems. Recently the concept of a LLE has been applied to study dispersion in turbulent flow fields. The LLE is a scalar value which characterizes the amount of stretching about the trajectory of point x over a time interval. The LLE varies as a function of space and time. The LLE is not an instantaneous separation rate, but rather measures the average, or integrated, separation between trajectories. This distinction is important because in time-dependent flows, the instantaneous velocity field is often not revealing much about actual trajectories, that is, instantaneous streamlines can quickly diverge from actual particle trajectories. However the LLE accounts for the integrated effect of the flow because it is derived from particle trajectories, and is thus more indicative of the actual transport behavior. Depending on which asymptotic character is eliminated, there are two non-asymptotic Lyapunov exponents: finite-time (FTLE) and finite-size (FSLE) Lyapunov exponents, that are very similar.
Here we will detail only the Finite-Size Lyapunov Exponent (FSLE) which is a measure for the growth rate of a perturbation.

Table of contents :

1 General Introduction 
1.1 Global Climate change and Biogeochemistry.
1.2 Spatial and temporal scales in the Ocean.
1.3 Mesoscale physical processes influence marine ecosystems.
1.4 The Eastern Boundary Upwelling systems.
1.5 Thesis objectives and plan.
1.6 Résumé Introduction (français).
2 Materials and Methods: a set of complementary tools to study the influence of physical processes on ecosystem dynamics at mesoscale. 
2.1 In-situ data from oceanographic surveys.
2.1.1 Data from CTD sensors.
2.1.2 Data from water sample measurements and zooplankton net.
2.1.3 Other data in marine sciences.
2.2 Satellite data
2.2.1 Ocean Color.
2.2.2 Ocean altimetry.
a – Generalities and basic principles:
b – Sea Surface Height, Quikscat wind stress and derived geostrophic currents:
2.3 The Finite-Size Lyapunov Exponents: a lagrangian powerful tool.
2.3.1 Eulerian / Lagrangian description.
2.3.2 Dynamical systems and manifolds.
2.3.3 The non asymptotic Finite-Size Lyapunov Exponents.
2.3.4 Lagrangian Coherent Structures (LCS) as ridges in the FSLE field
2.4 Academic and realistic numerical modelling.
2.4.1 Interests and principles.
2.4.2 Hydrodynamical and biological models.
3 A mesoscale survey of the northern and central Iberian Peninsula Upwelling System: spatial variability and bio-physical interactions. 
3.1 Article 1: A mesoscale survey of the northern and central Iberian
Peninsula Upwelling System: spatial variability and bio-physical interactions, Rossi et al., Progr. Oceanogr.
3.2 Résumé de l’article 1 (français).
3.3 Perspectives and other study derived from the survey.
3.3.1 Distribution of Volatile Halogenated Organic Compounds in the Iberian Peninsula Upwelling System.
3.3.2 Zooplankton communities and size spectra in the Iberian Peninsula Upwelling System.
4 Punctual small scale physical processes observed during MOUTON 2007 and their academic studies. 
4.1 Article 2: Effect of the wind on the shelf dynamics: formation of a secondary upwelling along the continental margin, Rossi et al., Ocean Modelling, .
4.2 Résumé de l’article 2 (français).
4.3 Article 3: Influence of a bottom topography on an upwelling current:  generation of long trapped filaments, Meunier, Rossi et al., in revision, Ocean Modelling.
4.4 Résumé de l’article 3 (français).
5 Biological activity and mesoscale horizontal stirring in the surface  ocean of the 4 Eastern Boundary Upwelling Systems: a comparative study. 
5.1 Article 4: Comparative study of mixing and biological activity of the Benguela and Canary upwelling systems, Rossi et al., 2008 Geophysical Research Letters
5.2 Résumé de l’article 4 (français).
5.3 Article 5: Horizontal stirring and biological activity in the surface ocean of the four Eastern Boundary Upwelling Systems, Rossi et al., 2009 Nonlinear Processes in Geophysics
5.4 Résumé de l’article 5 (français).
6 Conclusions and perspectives. 
6.1 Conclusions
6.2 Perspectives
6.2.1 Mesoscale variability of the Iberian Peninsula Upwelling System.
6.2.2 Inhibiting effect of mesoscale turbulence from FSLE on the surface chlorophyll in the EBUS: toward an identification of effective processes.
a – An academic modelling of the Benguela Upwelling System.
b – Toward a 3D realistic coupled modelling of the IPUS using HYCOM.
c – Extension of the Finite Size Lyapunov exponents theory.
6.2.3 General perspectives.
6.3 Conclusions et perspectives (français)
6.3.1 Conclusions (français).
6.3.2 Perspectives (français).
References

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