Valentin Resseguier

Geophysical flows under location uncertainty, Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction, an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.

Geophysical flows under location uncertainty, Part III SQG and frontal dynamics under strong turbulence conditions

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. This change can be consistently applied to all fluid dynamics evolution laws. This paper continues to explore benefits of this framework and consequences of specific scaling assumptions. Starting from a Boussinesq model under location uncertainty, a model is developed to describe a mesoscale flow subject to a strong underlying submesoscale activity. Specifically, turbulent diffusion and rotation effects have similar orders of magnitude. As obtained, the geostrophic balance is modified and the Quasi-Geostrophic assumptions remarkably lead to a zero Potential Vorticity. The ensuing Surface Quasi-Geostrophic model provides a simple diagnosis of warm frontolysis and cold frontogenesis.

Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. In this paper, simplified geophysical dynamics are derived from a Boussinesq model under location uncertainty. Invoking usual scaling approximations and a moderate influence of the subgrid terms, stochastic formulations are obtained for the stratified Quasi-Geostrophy and the Surface Quasi-Geostrophy models. Based on numerical simulations, benefits of the proposed stochastic formalism are demonstrated. A single realization of models under location uncertainty can restore small-scale structures. An ensemble of realizations further helps to assess model error prediction and outperforms perturbed deterministic models by one order of magnitude. Such a high uncertainty quantification skill is of primary interests for assimilation ensemble methods. MATLAB® code examples are available online.

Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

We present here a new stochastic modelling in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating, component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unre-solved small-scale velocity component. They bring to the reduced system an explicit subgrid term enabling to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or nonstationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flows data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers Re = 300 and Re = 3900, respectively. The finite dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects as well as structured energy dissipation effects take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.