On stochasticity of zonostrophic instability

Abstract

This paper concerns the behaviour of reduced models for turbulence–mean-flow interactions, in particular for the modelling of zonostrophic instability, which accounts for the spontaneous generation of zonal flows observed in many planetary and geophysical fluid systems. In modelling, an idealised sea of random waves or turbulent eddies is usually driven by a stochastic body force. However, the stochasticity of the zonal mean flow is often neglected, following an ergodic assumption that the zonal mean is equal to the ensemble mean. In this paper, we examine the stochasticity of the zonal mean flow and its consequences for zonostrophic instability. From large numbers of realisations of an idealised fluid system, we show that the zonal mean flow typically exhibits considerable stochasticity. The actual assumption needed for the theory of zonostrophic instability is a weaker assumption, that the waves and mean flows are statistically decorrelated, and we examine its validity. From the evolution equation for mean flows in stochastic waves, we derive an analytical expression for the weak correlation between waves and mean flow. Results indicate that this correlation becomes stronger and has an impact on growth rates as the instability becomes prominent. We further show that neglecting the correlation will result in an under-estimation of the growth rate, and we derive an improved dispersion relation based on this correlation. Theoretical predictions are verified by numerical simulations.