Projet de thèse en Mathématiques Appliquées
Sous la direction de Clémentine Prieur et de Eric Blayo.
Thèses en préparation à Grenoble Alpes , dans le cadre de École doctorale mathématiques, sciences et technologies de l'information, informatique (Grenoble) , en partenariat avec LJK - Laboratoire Jean Kuntzmann (laboratoire) et de MOISE (equipe de recherche) depuis le 01-10-2015 .
Pas de résumé en français disponible.
Parameter estimation for subgrid modeling, application to ocean and atmosphere
Atmospheric and oceanic simulation models are widely used for weather, general circulation and climate. In this context, the simulated flows are spectrally broad band across global to micro scales. Since a full representation in all different scales is not computable, numerical models therefore contain parameterizations to account for the interactions with key processes too small to be resolved by the computational grid. The absence of such parameterizations would lead to unacceptable model errors. The primary goal of these subgrid-scale (SGS) models is to obtain correct statistics of the energy-containing scales of motion well-resolved on a given computational grid. The most important feature of a SGS model is thus to provide adequate transport of energy from the resolved grid scales to the unresolved grid scales. The mathematical formulation of SGS models is often devised empirically as a function of a set of unknown parameters through the use of the eddy-viscosity and diffusivity concept [e.g. Stull, 1988]. Conventional parameterizations are deterministic. However, in recent years weather and climate models have also begun experimenting with stochastic SGS models [Frederiksen et al., 2012; Mana & Zanna, 2014]. This approach proved to be promising because there are many possible realizations of the subgrid state for a given large-scale state. In this general context, the objective of the PhD is to propose efficient, physically consistent, and innovative ways to calibrate the unknown parameters of SGS models using parametric and/or nonparametric estimation techniques. The tuning of those parameter values is generally either done using observations and/or high-resolution numerical simulations (a.k.a. Large Eddy Simulation, LES). The latter source of information is considered in the context of this work. All parameters and relationships will be calibrated by using data from high-resolution models to guide the building of a parametrization for a coarser resolution model. A first difficulty is to properly define the relationships between the coarse scales and the small scales to upscale high-resolution information to the resolution of the model of interest. A corollary is to carefully extract independent realizations of the relevant quantities from the high-resolution data that may result from one single long-term deterministic simulation. The mathematical theory and the practical tools for the estimations of parameters of SGS models will be first experienced on a simplified set of equations, using for instance the one-dimensional Burgers' equation, before envisioning more complex flows. The objectives are: 1) Choose among a set of existing subgrid scale models (either stochastic or deterministic), relevant SGS schemes for the turbulent flows of interest. 2) Calibrate the unknown parameters associated to those schemes by: a) defining a criterion of "good recovery", such as e.g. the consistency of low and high resolution energy spectra, at least in a predetermined range of the spectrum. b) proposing an efficient optimization procedure with respect to the criterion chosen in a). It could happen that several criteria are of importance, leading to a multi-objectives optimization issue. In case the calibration procedure involves a snapshot (or sample), the choice of the snapshot is also a point to discuss. 3) Validate the model, once it has been calibrated. One may focus in a first time on the subgrid modeling of Burgers' equation [e.g. Love, 1980], which will serve as a "test case" for the "real" equations modeling geophysical fluids we are interested in. 4) Study the relative role of numerical errors (i.e. discretization errors) vs physical errors (i.e. subgrid terms) to ensure that the calibration procedure does not compensate for nonphysical errors.