Stochastic aspects of geophysical equations
Final Report Abstract
This project was concerned with the study of stochastic aspects of equations arising in the field of geophysical fluid dynamics and carried out at the Scuola Normale Superiore di Pisa. Focus were laid on two models, the surface quasi-geostrophic equation and the primitive equations of atmosphere and ocean. The surface quasi-geostrophic equation describes the temperature at the surface of a rapidly rotating fluid, and it is used in connection with the formation of temperature fronts. To take into account uncertainties, the equation driven by space-time white noise, i.e., with a very rough stochastic forcing, was considered on the torus. Using the theory of regularity structures solutions for sufficiently smooth data were constructed. Besides this existence result for this specific equation, also a general way to include non-smoothing integral operators into the theory of regularity structures was demonstrated. The primitive equations are a fundamental system for describing atmospheric and oceanic dynamics, and they are used in the field of weather forecasting. The system with transport noise is of particular interest, because it is closely related to the study of turbulences. Transport noise means in this context, roughly speaking, that a noise term, which depends on the first derivatives of the solutions in a particular way, is present in the equation. Known results on stochastic primitive equations have the drawback that they either do not include transport noise or they need strong structural assumptions on the stochastic term to handle the pressure. By using the approach of stochastic maximal regularity, in this project a global well-posedness with both, noise acting directly on the pressure and full transport noise was achieved in the L2-setting. This is a step on the way to rigorously justify the stochastic model for the primitive equation by deriving it from the stochastic Navier-Stokes system. Additionally, only a parabolicity assumption had to be assumed and no smallness of the coefficients was required. Also results for the additive noise in the Lp-setting and for a stochastic term on the boundary were deduced in the course of the project. The primitive equations with only horizontal viscosity are of particular interest in the field of numerical weather prediction. On the one hand, in the atmosphere the horizontal viscosity is much larger than the vertical one and the limiting case is considered to be a good approximation. On the other hand, numerical (hyper-)viscosity acting only in the horizontal directions is often used in the computer simulations. In this project, the first results for stochastic primitive equations with only horizontal viscosity were proven. Considering the system with physical boundary conditions and a transport noise, that did not depend on the vertical derivative, by a finite dimensional approximation the existence of local solutions was deduced under Lipschitz conditions on the noise and a smallness assumption for the Lipschitz constant for the transport part. Global solutions were obtained for noise with additional structure, namely that the transport part is linear in the first derivatives. Compared to the deterministic situation, for the stochastic system higher regularity of the data is needed in the result, a point which asks for further investigation.
Publications
- Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity. 2021
M. Saal and J. Slavík
(See online at https://doi.org/10.48550/arXiv.2109.14568) - Surface Quasi-Geostrophic Equation driven by Space-Time White Noise. 2021
P. Forstner and M. Saal
(See online at https://doi.org/10.48550/arXiv.2111.04644) - The stochastic primitive equations with transport noise and turbulent pressure. 2021
A. Agresti, M. Hieber, A. Hussein and M. Saal
(See online at https://doi.org/10.48550/arXiv.2109.09561)
