Fourth order, conservative discretization of horizontal Euler equations in the COSMO model and regional climate simulations

Abstract

Horizontal spatial schemes of third order and above used for discretization of COSMO (Consortium for Small Scale Modeling) Euler equations can be described as quasi-higher order schemes since interpolation of the advecting velocities and differencing of the pressure gradient term remain second order accurate. For NWP and Regional Climate modeling, upwind schemes of either third or fifth order have been recommended combined with an explicit numerical diffusion. We have implemented fully fourth order central difference horizontal schemes for the model's Euler equations with two types of discretization of the advection terms: the first is a natural extension of the COSMO fourth order scheme by introducing fourth order interpolation of the advecting velocity, and the second is a symmetric type discretization which is shown to conserve the rotational part of kinetic energy. We combine both advection schemes with fourth order discretization of the pressure gradient term. To make the schemes completely fourth order, we consider all metric terms resulting from coordinated transformations. Theoretical analysis of the new schemes compared to the model's existing third order upwind scheme exhibits: a slightly increased group velocity error due to a wider stencil, a similar dispersive error, a significant reduction of the amplitude error, a significantly minimized aliasing error due to symmetric advection-discretization, and a significant increase in effective Courant number which potentially allows longer time steps. Using 20-year climate simulations, we show that the new symmetric fourth order scheme is more stable than the extended COSMO fourth order scheme and third order upwind scheme, and that an explicit numerical diffusion can be avoided when using the symmetric scheme. We show that a 20 % dispersive (phase) and diffusive (amplitude) errors limit result to the model's effective resolution of approximately 5Δx$5\Delta x$ for all 4th and 3rd order schemes. Considering the same error limit for simulated kinetic energy spectra show that the horizontal numerical diffusion is reducing the model's effective resolution to more than 10Δx$10\Delta x$ and thus using the symmetric 4th order scheme without explicit horizontal diffusion increases the effective resolution by a factor of two to approximately 5Δx$5\Delta x$. We further show that both implicit diffusion in upwind schemes and explicit numerical diffusion necessary for the current model's stable runs have effects of equal magnitude on the model's predicted climatologies. Climatologies show that fourth order schemes enhance vertical turbulence mixing in the planetary boundary layer which reduces parameterized convection. This consequently results to approximately 20 % peak reduction of summer precipitation and an increase of approximately 0.5 degrees Kelvin in summer 2 m air temperature.

Keywords

• fourth order
• conservative scheme
• Euler equations
• COSMO
• effective resolution
• climate modeling