On the numerical stability of surface-atmosphere coupling in weather and climate models

Coupling the atmosphere with the underlying surface presents numerical stability challenges in cost-effective model integrations used for operational weather prediction or climate simulations. These are due to the choice of large integration time steps compared to the physical timescale of the problem, aiming at reducing computational burden, and to an explicit flux coupling formulation, often preferred for its simplicity and modularity. Atmospheric models therefore use the surface-layer temperatures (representative of the uppermost soil, snow, ice, water, etc.) at the previous integration time step in all surface-atmosphere heat-flux calculations and prescribe fluxes to be used in the surface model integrations. Although both models may use implicit formulations for the time steps, the explicit flux coupling can still lead to instabilities. In this study, idealized simulations with a fully coupled implicit system are performed to derive an empirical relation between surface heat flux and surface temperature at the new time level. Such a relation mimics the fully implicit formulation by allowing one to estimate the surface temperature at the new time level without solving the surface heat diffusion problem. It is based on similarity reasoning and applies to any medium with constant heat diffusion and heat capacity parameters. The advantage is that modularity of the code is maintained and that the heat flux can be computed in the atmospheric model in such a way that instabilities in the snow or ice code are avoided. Applicability to snow-icesoil models with variable density is discussed, and the loss of accuracy turns out to be small. A formal stability analysis confirms that the parametrized implicit-flux coupling is unconditionally stable.