Using a classical example, the Lorenz‐63 model, an original stochastic framework is applied to represent large‐scale geophysical flow dynamics. Rigorously derived from a reformulated material derivative, the proposed framework encompasses several meaningful mechanisms to model geophysical flows. The slightly compressible set‐up, as treated in the Boussinesq approximation, yields a stochastic transport equation for the density and other related thermodynamical variables. Coupled to the momentum equation through a forcing term, the resulting stochastic Lorenz‐63 model is derived consistently. Based on such a reformulated model, the pertinence of this large‐scale stochastic approach is demonstrated over classical eddy‐viscosity based large‐scale representations.

Expanding on a wavelet basis the solution of an inverse problem provides several advantages. First of all, wavelet bases yield a natural and efficient multiresolution analysis which allows defining clear optimization strategies on nested subspaces of the solution space. Besides, the continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain. By choosing differentiable wavelets, accurate high-order derivative regularizers can be efficiently designed via the basis's mass and stiffness matrices. More importantly, differential constraints on vector solutions, such as the divergencefree constraint in physics, can be nicely handled with biorthogonal wavelet bases. This paper illustrates these advantages in the particular case of fluid flow motion estimation. Numerical results on synthetic and real images of incompressible turbulence show that divergencefree wavelets and high-order regularizers are particularly relevant in this context.

Numerical simulations of industrial and geophysical fluid flows cannot usually solve the exact Navier-Stokes equations. Accordingly, they encompass strong local errors. For some applications-like coupling models and measurements-these errors need to be accurately quantified, and ensemble forecast is a way to achieve this goal. This paper reviews the different approaches that have been proposed in this direction. A particular attention is given to the models under location uncertainty and stochastic advection by Lie transport. Besides, this paper introduces a new energy-budget-based stochastic subgrid scheme and a new way of parameterizing models under location uncertainty. Finally, new ensemble forecast simulations are presented. The skills of that new stochastic parameterization are compared to that of the dynamics under location uncertainty and of randomized-initial-condition methods.

Abstract. Based on a wavelet expansion of the velocity field, we present a novel optical flow algorithm dedicated to the estimation of continuous motion fields such as fluid flows. This scale-space representation, associated to a simple gradient-based optimization algorithm, naturally sets up a well-defined multi-resolution analysis framework for the optical flow estimation problem, thus avoiding the common drawbacks of standard multi-resolution schemes. Moreover, wavelet properties enable the design of simple yet efficient high-order regularizers or polynomial approximations associated to a low computational complexity. Accuracy of proposed methods is assessed on challenging sequences of turbulent fluids flows.