Hasselmann's Paradigm for Stochastic Climate Modelling based on Stochastic Lie Transport

TL;DR

This paper develops a stochastic climate model based on Hasselmann's paradigm, incorporating stochastic Lie transport into the atmospheric component while keeping the ocean deterministic, and introduces variants with desirable mathematical properties.

Contribution

It introduces SALT and LA-SALT models that incorporate stochastic transport into climate modeling, preserving circulation and enabling linear moment equations.

Findings

SALT preserves circulation in the stochastic model.

LA-SALT's higher moments follow linear deterministic equations.

Local existence results are established for the models.

Abstract

A generic approach to stochastic climate modelling is developed for the example of an idealized Atmosphere-Ocean model that rests upon Hasselmann's paradigm for stochastic climate models. Namely, stochasticity is incorporated into the fast moving atmospheric component of an idealized coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. More specifically the stochastic model SALT (stochastic advection by Lie transport) is constructed by introducing stochastic transport into the material loop in Kelvin's circulation theorem. The resulting stochastic model preserves circulation, as does the underlying deterministic climate model. A variant of SALT called LA-SALT (Lagrangian-Averaged SALT) is introduced in this paper. In LA-SALT, we replace the drift velocity of the stochastic vector field by its expected value. The remarkable property of LA-SALT is that the evolution of its higher moments are governed by linear deterministic equations. Our modelling approach is substantiated by establishing local existence results, first, for the deterministic climate model that couples compressible atmospheric equations to incompressible ocean equation, and second, for the two stochastic SALT and LA-SALT models.