A stochastic differential equation framework for gravity wave parametrisation with testing in an idealised setting

Abstract Parametrisations of unresolved gravity waves used in general circulation models can be made more computationally efficient by introducing a stochastic component to the forcing. An additional advantage of introducing stochasticity is that intermittency associated with the scheme could be tuned to resemble the intermittency of observed gravity wave sources, and could therefore act to improve the physical fidelity of the scheme. Here, it is argued that using stochastic differential equations (SDEs) to drive the stochastic component provides a natural general framework to develop such schemes. The Holton–Lindzen–Plumb model of the quasi‐biennial oscillation (QBO) is used to demonstrate the flexibility of the approach. The QBO generated in a (computationally expensive) deterministic broadband multiwave simulation is accurately reproduced using a number of quite different (cheap) stochastic schemes. The method of stochastic averaging is used to prove a matching result that shows that a wide class of such schemes, driven by different SDEs, can each reproduce the deterministic QBO provided that the characteristic time‐scale of the SDEs is sufficiently short. However, each scheme has different intermittency properties: as is increased, their QBOs are shown to diverge, despite the time‐averaged source spectrum in each case remaining unchanged. The SDE framework therefore provides great flexibility to tune a stochastic parametrisation to match observed intermittencies, meaning that future parametrisations can be developed that can account for non‐steady gravity wave forcing in a physically consistent manner.