Stochastic Galerkin method for cloud simulation

M Lukáčová-Medvid’Ová,A Chertock,B Wiebe,A Kurganov,P Spichtinger

doi:10.1515/mcwf-2019-0005

Abstract

AbstractWe develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to 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 examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.

Highlights

We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds
The Navier-Stokes equations are solved by an implicit-explicit (IMEX) nite-volume method, while for the cloud equations we develop a stochastic Galerkin method based on the generalized polynomial chaos (gPC)
We study a mathematical model of cloud dynamics, which is based on the compressible nonhydrostatic Navier-Stokes equations for moist atmosphere, ρt + ∇ · =,t + ∇ · ρu ⊗ u + p Id − μm ρ ∇u + (∇u) = −ρge, (2.1)

Summary

Introduction

Clouds constitute one of the most important components in the Earth-atmosphere system. They in uence the hydrological cycle and by interacting with radiation they control the energy budget of the system. A common way of obtaining such an ensemble is by using a mass or size distribution, which would lead to a Boltzmann-type evolution equation. There are some approaches available in literature to formulate cloud models in such a way [4, 18, 19], a complete and consistent description is missing. Since measurements of size distributions of cloud particles are di cult, we are often restricted to averaged quantities such as, for example, mass of water per dry air (mass concentrations). Models are often formulated in terms of so-called bulk quantities, that is, mass and number

Chertock

Conclusion

A Appendix

B Appendix