Elsevier

Journal of Marine Systems

Volume 126, October 2013, Pages 33-42
Journal of Marine Systems

On the inversion of submesoscale tracer fields to estimate the surface ocean circulation

https://doi.org/10.1016/j.jmarsys.2012.02.014Get rights and content

Abstract

In this paper, we demonstrate the feasibility of inverting the information contained in oceanic submesoscales, such as the ones evidenced in tracer observations of sea surface temperature (SST), to improve the description of mesoscale dynamics provided by altimetric observations. A small region of the Western Mediterranean Sea is chosen as a test case. From a SST snapshot of the region in July 2004, information is extracted to improve the velocity field as computed by geostrophy from the AVISO altimetric data at the same location and time. Image information is extracted from SST using a binarization of the SST gradients. Similarly, image information is extracted from the dynamic topography using finite size Lyapunov exponents (FSLE). The inverse problem is formulated in a Bayesian framework and expressed in terms of a cost function measuring the misfits between the two images. The large amount of information which is already available from ocean color satellites or which will be available from high-resolution altimetric satellites such as SWOT, is a strong motivation for this work. Moreover, the image data assimilation approach which is explored here, is a possible strategy for handling the huge amount of satellite data imprinted by small scale information.

Introduction

Mesoscale dynamics have been shown to be a key ingredient of the ocean circulation. The ubiquitous presence of mesoscale eddies in the world ocean has been blatantly revealed by satellite altimetry (LeTraon and Morrow, 2011). Today, there is an increasing interest for submesoscale activity at scales of O(1 km). The energetics and dynamical role of such submesoscale processes have been largely underestimated until recently (Capet et al., 2008b). Submesoscale dynamics can result from stirring actions in the mesoscale flow, in which frontogenesis is one of the dominant processes (Thomas et al., 2008). These dynamics are also evidenced in various recent field observations and in numerical models (Capet et al., 2008b, Klein et al., 2010, Marchesiello et al., 2003, Marchesiello et al., 2011).

Among the observations, high-resolution satellite images of tracer fields – such as sea surface chlorophyll or sea surface temperature (SST) – clearly evidence filaments and frontal structures at these submesoscales. In the mid-term, submesoscale dynamics should be observable at the surface by high-resolution 2D altimetry (thanks to the SWOT satellite mission, for example). However it is still unclear as to how much (and with what accuracy) of the submesoscale signal would be seen by altimetry (Chavanne and Klein, 2010).

Accounting for the ocean submesoscales is a challenge for future oceanographic research and liable to bring much to an improved understanding and modeling of the ocean dynamics. Interestingly, submesoscale dynamics are also viewed as a key element of the biogeochemical behavior of the ocean and the filament structures might be indicative of enhanced primary production (e.g. Lévy et al., 2001). The observational prospects especially from space offer wide possibilities for actually observing those features. On the modeling side, present ocean models do not truly resolve submesoscale activities at global scales but progress is expected in the near future as regional models are already able to do so. In any case, the presence of high-resolution information will raise specific issues. In particular we will be confronted to a considerable amount of data from satellites which we may not be prepared to handle and use. Today, we are not yet able to optimally exploit the detailed high-resolution information that is contained in ocean color or SST satellite images for example.

The present work stands within this general oceanographic context. But it stands also within the specific methodological context of data assimilation. Data assimilation is an ensemble of techniques that aim at performing an optimal combination of information of various types (and with various errors) to obtain an improved description of a dynamical system. What is of interest for us here is how data assimilation can be adapted to handle huge masses of information and more specifically to find a way of assimilating structured information as for example the one that is observed in submesoscale tracers. A possible answer is image data assimilation as was discussed in particular by Titaud et al. (2010). Following these authors, we adopt the “image” terminology to refer to the spatial structures evidenced by the high-resolution original data. These image structures are simplified representations of these data and obtained through image processing techniques of different complexities. Then, data are substituted by this structure information and no longer by the original pointwise data. This is a way to reduce the size of the dataset while keeping the dynamical information content of the data. A lot of work in the literature is concerned with image treatment but this is not per se our interest here. The use of images leads to specific formulations of the data assimilation problem since contrary to the classical approach, this is not information on the system per se that is used but a proxy which characterizes the information structure (and its time evolution). Some recent studies have shown how Lagrangian tools can be used to link turbulence properties and tracer dynamics by deriving relevant proxies (Abraham and Bowen, 2002, d'Ovidio et al., 2009, Lehahn et al., 2007). Several types of proxies could be thought of such as the finite-size Lyapunov exponents (FSLE, Aurell et al., 1997), the finite-time Lyapunov exponents (FTLE, (Titaud et al., 2011)) or multiscale analysis (Yahia et al., 2010).

In this data assimilation context, the general objective of this paper is to explore how submesoscale information that is contained in satellite tracer fields can be used to better describe ocean dynamics at meso and larger scales. The overall problem is rather complex and this work has a clear preliminary nature. To tackle it, we made several choices:

  • Nowadays, submesoscales are synoptically observed only by tracers and in particular through SST observations. In parallel, altimetry provides a relatively faithful access to mesoscale dynamics and in particular velocity field through the geostrophic derivation of the SSH. As a first step, we restrict ourselves to explore the capability of SST information at the submesoscales to correct for the mesoscale (and larger scale) dynamics as described by altimetry.

  • To perform image data assimilation, the finite-size Lyapunov exponents field (FSLE) is used as the proxy variable for the velocity field. Images were thus created from FSLE fields. Images from the SST high-resolution controlling data are also required: they were simply obtained by computing the normalized gradients.

  • In this work, we consider a simpler data assimilation problem in the sense that the date is fixed and therefore there is no time evolution. In other words, an inversion is performed.

Then, the goal of this work is to demonstrate that the submesoscales from the SST field can be inverted to improve the description of the altimetric velocity. The inversion from the SST to the SSH-derived velocity field is made by considering the misfits between two images from SST and SSH observations. The main challenges are: (i) from an oceanographic point of view, it concerns the capability to make use of submesoscale information to control large scale ocean circulation, (ii) from a data assimilation perspective, the capability of using images as a go-between for different types of data. Note that, in a recent work, Titaud et al. (2011) have addressed a partly similar sensitivity study in the context of numerical simulations (all information used were provided by model simulations). Therefore, one aim of the present work is to extend this study to real data and prove in a real context the feasibility of the approach.

The paper is organized as follows. In Section 2 the observations used in this study are presented as well as the FSLE definition and computations. The inverse method is detailed in Section 3. And the results are presented and discussed in Section 4. Conclusions are finally given in Section 5.

Section snippets

Observations and related quantities

The founding breakthrough for the present study lies in the observational evidence that mesoscale flow stirring can be characterized quite faithfully by the distribution of Lyapunov exponents (Abraham and Bowen, 2002, d'Ovidio et al., 2004). This was further developed, for instance by Lehahn et al. (2007), who demonstrated a direct link between observed high-resolution chlorophyll images and mesoscale velocity fields derived from altimetric observations. Lyapunov exponents were also shown to

Formulation of the inverse problem

To formulate the inverse problem, we consider the AVISO velocity map (Fig. 3a) as a background velocity ub and the observation λ^o of the tracer pattern (Fig. 2c) as an additional information about velocity. The problem can then be solved using the Bayes theorem:pa(u)pb(u)pλ^o|uwhere pb(u) = p(u|ub) is the prior probability distribution for the velocity u, pλ^o|u is the conditional probability of the observed structure λ^o given the velocity u, and pa(u)=pu|ub,λ^o is the posterior probability

Results of the estimation process

The prerequisite for a successful estimation process lies in the good conditioning of the cost function as expressed in Eq. (8). In this section, we present the investigations that have been made on the cost function behavior in two successive stages: first by looking at the simple inversion of the intermediate FSLE field, second at the inversion of the actual data. Let us recall that the feasibility of the FSLE inversion has already been demonstrated when data are simulated by a numerical

Conclusion

In this paper, a case study has been presented that demonstrates the feasibility to correct an observed velocity field (derived from SSH data by geostrophy) using a satellite tracer image (here a SST field). Using maximum probability and minimum variance estimators, the velocity field has been corrected so that a better match to the submesoscale SST filament structures is reached. The corrections are consistent with the submesoscale tracer observations based on the rationale that has been

Acknowledgments

This research has been conducted with the support of the CNES (French Space Agency) and the CNRS (French National Research center). We also thank Francesco d'Ovidio (from LOCEAN-IPSL) and Konstantin Koshel (from V.I.Il'ichev Pacific Oceanological Institute) for providing codes to calculate the FSLE.

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