Theoretical aspects of upscale error growth through the mesoscales: an analytical model

• Published in: Quarterly journal of the Royal Meteorological Society 2017-10, Vol.143 (709), p.3048-3059
• Main Authors: Bierdel, L., Selz, T., Craig, G.C.
• Format: Article
• Language: English
• Subjects: Boussinesq approximation; Convection; Convective clouds; Convective instability; Errors; Frameworks; geostrophic adjustment; Green's function; Heat; Heat sources; Heat transfer; Instability; Latent heat; Latent heat release; Mathematical models; mesoscale dynamics; Numerical simulations; predictability; upscale error growth; Weather forecasting
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3160

Recent numerical studies suggest that convective instability and latent heat release quickly amplify errors in numerical weather predictions and lead to a complete loss of predictability on scales below 100 km within a few hours. These errors then move further upscale, eventually contaminating the balanced flow and projecting on to synoptic‐scale instabilities. According to this picture, the errors have to transition from geostrophically unbalanced to balanced motion while propagating through the mesoscale. Geostrophic adjustment was suggested as the dynamical process of this transition, but so far has not been clearly identified. In the current study, an analytical framework for the geostrophic adjustment of an initial point‐like pulse of heat is developed on the basis of the linearized, hydrostatic Boussinesq equations. The heat pulse is thought to model a convective cloud or an error within the prediction of a cloud. A time‐dependent solution for both transient and balanced flow components is derived from the analytical model. The solution includes the Green's function of the problem, which enables the extension of the model to arbitrary heat sources by linear superposition. Spatial and temporal scales of the geostrophic adjustment mechanism are deduced and diagnostics are proposed that could be used to demonstrate the geostrophic adjustment process in complex numerical simulations of midlatitude convection and upscale error growth. An analytical solution is found that describes the transient and balanced response of a horizontally unbounded and rotating atmosphere to a buoyancy source. Characteristic spatial and temporal adjustment scales are derived. Three diagnostics that allow for an identification of the geostrophic adjustment process in numerical simulations of the atmosphere are suggested. The image shows (a) the full buoyancy solution at z=H/2 as a function of distance from the source and time (Hovmöller diagram) and (b) the spatial dependence of the solution for 5, 20 and 40 h and infinity. Grey solid, dashed and dotted lines in (a) are the times of the cross‐sections displayed in (b). Vertical grey lines in (b) denote the position of the discrete front at different times.