Higher Order Spatial Discretisation of Euler Equations and Effective Resolution in an Operational Nonhydrostatic NWP and RCM Model
Abstract:Several studies have shown that Limited Area Models exhibit a substantial difference between the model’s horizontal grid and effective resolution. At the same time the effect of increased model resolution is small in comparison to the increased computing costs (approx. one order of magnitude for half grid size). In the recent years the computing power increased faster than the storage space, and higher order spatial discretisation schemes have the potential to increase the effective model resolution while the storage space needs remain constant and increasing the computational costs slightly.
We have implemented fully fourth order horizontal discretisation of the Euler equations in the NWP and RCM COSMO model with two variants of the advection term discretisations. The first is an extension of the model’s higher order discretisation by introducing fourth order interpolations of the advecting velocity in a staggered grid system denoted by N4. The second is a symmetric type discretisation of the advection term which can be shown to conserve the 1st and 2nd moments of the advected quantity if the continuity equation is satisfied, here referred to as S4. Both convective schemes can be combined with fourth order discretisation of the pressure gradient term, referred to as p4. The new fourth order discretisations are referred to as N4p4 and S4p4 and complement the already existing schemes A3p2, A4p2, A5p2 and A6p2. Here AXdenotes the central difference (2,4,6) and upwind (3,5) discretisation of the gradient of the advected velocity.
Using a 5-years long climate simulations over a European domain covering the complex terrain in the Alps region, we show that the new scheme S4p4 is more stable than the existing third order upwind scheme, and that the explicit numerical diffusion can be entirely avoided when using the new S4p4 scheme. By comparing the model's predicted with observed kinetic energy spectra, we show that the new fourth order schemes increases the model's effective resolution by at least a factor of two, and that by applying S4p4 scheme we achieve an effective resolution of 2Δx (where Δx is the model horizontal grid spacing). It is further shown that both implicit diffusion in upwind schemes and explicit numerical diffusion have diffusion effects of equal magnitude on kinetic energy at small scales.