Stochastic aspects of geophysical equations

• Applicant: Dr. Martin Saal
• Subject Area: Mathematics
• Term: from 2019 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 429483464

In this project, stochastic variants of the surface quasi-geostrophic equation and of the primitive equations, both important equations from the field of geophysical fluid dynamics, will be investigated. 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, while the primitive equations are a fundamental system for describing atmospheric and oceanic dynamics, and they are used especially in the field of weather forecasting. These equations are considered with stochastic data, that is, with initial values and external forces given as noise, and for the surface quasi-geostrophic equation the case of stochastic rather than deterministic diffusion is additionally considered. The focus lies on the question of the existence and uniqueness of solutions, but also the behaviour of solutions in the case of vanishing viscosity will be investigated. Besides the mathematical interest in stochastic partial differential equation, the essential role of these equations in geophysical applications is the main motivation to study them intensively in this project. In the modelling of geophysical systems, stochastic terms are used to account for numerical and empirical uncertainties, and there are important examples of intrinsic stochasticity. One purpose of the project is a better understanding of the influence of these stochastic terms and to study the robustness of the models against such perturbations. Moreover, this kind of stochastic differential equations is mathematically challenging, since the treatment of the nonlinearities is very involved due to the expected low regularity of the solutions. In recent years, new methods have been introduced to handle such problems, and they were successfully applied in the context of parabolic equations and to the wave equation. Making use of those modern methods will be necessary within this project to obtain the desired results and it is a further objective of this project to extend their scope to transport equations by considering the special case of the (inviscid) surface quasi-geostrophic equation. As host institution the Scuola Normale Superiore (SNS) in Pisa, Italy, participates in this project. With several research groups for stochastic analysis it is a centre in this field. The host at the SNS for this project, Franco Flandoli, is a world-renowned expert on stochastic fluid dynamics and transport equations, who has also recently turned to geophysical problems. Thus, the SNS offers ideal conditions to successfully carry out the research project there.

• DFG Programme: Research Fellowships
• International Connection: Italy
• Host: Professor Dr. Franco Flandoli