Theoretical Aspects of Upscale Error Growth through the Mesoscales

Recent numerical studies suggest that error growth in the atmosphere is an initially localized, highly intermittent phenomenon that expands upscale and plays a significant role in contaminating the larger scale flow at longer forecast lead times. In particular it has been found that latent heat release associated with deep moist convection is a primary mechanism for small-scale error growth that leads to a complete loss of predictability on scales below 100 km within a few hours. The errors introduced by slight changes in the amplitude and position of the heating induced by condensation within the convective clouds then expand upscale and propagate through the mesoscales, a process whereby the errors transition from geostrophically unbalanced to balanced. Geostrophic adjustment following convective heating was suggested as a possible dynamical process underlying this transition but appears difficult to extract from numerical simulations.

In the current study an analytical formulation for the geostrophic adjustment of an initial pointlike pulse of heat is developed on the basis of the linearized, hydrostatic Boussinesq-equations. The heatpulse is thought to model a convective cloud or an error within the prediction of a cloud. A time-dependent solution for both the transient and the balanced flow components is derived from the analytical model. The solution is the Green's function for the mathematical problem, which allows for the simple construction of a solution for arbitrary forcings. From the solution, the temporal and spatial adjustment scales are identified, and diagnostics are developed that allow an identification of the geostrophic adjustment process in numerical simulations. The predictions are then tested within highly idealized numerical simulations. The high level of agreement between error growth characteristics in the numerical perturbation experiments and the analytical predictions suggests that geostrophic adjustment following convective heating plays a major role in upscale error growth through the mesoscales.