Stochastic Galerkin method for cloud simulation

In this talk, we consider a mathematical model of cloud physics that consists of the Navier-Stokes equations coupled with the cloud evolution equations for water vapor, cloud water, and rain. In this model, the Navier-Stokes equations describe weakly compressible flows with viscous and heat conductivity effects, while microscale cloud physics is modeled by the system of advection-diffusion-reaction equations. We aim to explicitly describe the evolution of uncertainties arising from unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for example, the ones leading to changes in macroscopic cloud patterns as a shift from hexagonal to rectangular structures.