PNNL

Abstract

Stochastic parameterizations are used in numerical weather prediction and climate modeling to help improve the statistical distributions of the simulated phenomena. Earlier studies by Hodyss et al. (2013, 2014) have shown that convergence issues can arise when time integration methods originally developed for deterministic differential equations are applied naively to stochastic problems. In particular, the numerical solution can cease to converge to the physically relevant solution. In addition, Hodyss et al. demonstrated that introducing a correction term (known in stochastic analysis as the Ito correction) to various deterministic numerical schemes, can help improve their solution accuracy and ensure convergence to the physically relevant solution without substantial computational overhead. The usual formulation of the Ito correction is valid only when the stochasticity is represented by white noise; in contrast, the lack of scale separation between the resolved and parameterized atmospheric phenomena in numerical models motivates the use of colored noise for the unresolved processes. In this study, a generalized formulation of the Ito correction is derived for noises of any color. It is applied to a test problem described by a diffusion-advection equation forced by a spectrum of fast processes. We present numerical results for cases with both constant and spatially varying advection velocities to show that for the same time step sizes, the introduction of the generalized Ito correction helps to substantially reduce time-stepping error and significantly improve the convergence rate of the numerical solutions when forcing in the governing equation is rough (fast varying); alternatively, for the same target accuracy, the generalized Ito correction allows for the use of significantly longer step size and hence helps to reduce the computational cost of the numerical simulation.