Parameterisation of small-scale random forcing in β-plane turbulence

Abstract

Zonal jet formation in $\beta$ -plane turbulence is investigated with the focus on whether an accurate closure can be developed for the eddy momentum fluxes due to small-scale random forcing. The approach of Srinivasan and Young (J. Atmos. Sci., vol. 71, 2014, pp. 2169–2185) is developed to give a relatively simple expression for the local Reynolds stress, due to a white-in-time random forcing with characteristic length scale much less than the jet spacing. In typical jet flows, however, it is demonstrated that the Srinivasan–Young flux is not the full story because momentum fluxes due to jet-scale waves, present as a result of distinct barotropic instabilities of the eastward and westward jets, respectively, also play a key role in the momentum balance. Numerical simulations that explicitly include the random forcing are then compared with those in which the Srinivasan–Young closure is applied. For typical jet flows, good agreement of the equilibrium zonal flow is found provided that the closure simulation is not truncated to be purely zonal, i.e. jet-scale secondary barotropic instabilities are allowed to develop. Flows in which the geometry or external forcing acts to suppress the development of secondary instabilities are also simulated, and for these flows the Srinivasan–Young closure is shown to be successful as a purely zonal closure. It is argued that vortex condensates in isotropic forced-dissipative 2D turbulence are an example of this latter situation.