@misc{OgajaWill, author = {Ogaja, Jack and Will, Andreas}, title = {Fourth order, conservative discretization of horizontal Euler equations in the COSMO model and regional climate simulations}, series = {Meteorologische Zeitschrift}, volume = {25}, journal = {Meteorologische Zeitschrift}, number = {5}, issn = {1610-1227}, doi = {10.1127/metz/2016/0645}, pages = {577 -- 605}, 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 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 and thus using the symmetric 4th order scheme without explicit horizontal diffusion increases the effective resolution by a factor of two to approximately 5Δx. 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 2m air temperature.}, language = {en} } @misc{WillOgaja, author = {Will, Andreas and Ogaja, Jack}, title = {Higher order spatial discretisation methods for non-hydrostatic models of the atmosphere on regular grids}, series = {Mathematical Theory and Modelling in Atmosphere-Ocean-Science, Report No. 34/2010}, volume = {34}, journal = {Mathematical Theory and Modelling in Atmosphere-Ocean-Science, Report No. 34/2010}, pages = {2089 -- 2090}, abstract = {In turbulence modeling small stencils, conservation of the integrals of motion and high order of approximation of the mathematical operators for filtering and derivatives could be realised ([2]). Nowadays such methods are going to be developed for state of the art atmospheric LAMs. In the following the behaviour in idealised test cases of different numerical approximations of an incompressible model and the current status of the development of a state of the art model ( COSMO) for RCM and NWP is presented.}, language = {en} } @misc{WillOgaja, author = {Will, Andreas and Ogaja, Jack}, title = {Higher order horizontal schemes in COSMO 5 . 0 at different resolutions}, series = {CLM-Community Newsletter}, volume = {8}, journal = {CLM-Community Newsletter}, pages = {7 -- 8}, language = {en} } @phdthesis{Ogaja, author = {Ogaja, Jack}, title = {Higher order horizontal discretization of Euler-equations in a non-hydrostatic NWP and RCM model}, publisher = {Cuvillier}, address = {G{\"o}ttingen}, isbn = {978-3-7369-9294-8}, pages = {133}, abstract = {For the first time, efficient fourth order accurate horizontal discretization of the Euler equations has been implemented in an operational NWP and RCM model implicitly conserving the kinetic energy of the rotational flow, which allowed simulation of regional climate over a European domain with COSMO model without horizontal numerical filter. The first spatial scheme implemented is a central difference fourth order scheme which is a natural extension of the COSMO spatial schemes by consistent discretization of linear and nonlinear terms of the model's Euler equations to fourth order accuracy. The second, is a symmetric fourth order accurate scheme previously used for turbulence studies. It has been shown analytically that none of the previously implemented numerical schemes (COSMO schemes) is higher order(higher than second order) accurate. In contrast, the new schemes have been shown to be fourth order accurate. It is further shown that the symmetric scheme conserves rotational part of the momentum and kinetic energy a priori. The ageostrophic divergent part is discretized fourth order accurate. Analysis of the numerical errors of the schemes show that the new schemes exhibit significantly improved accuracy in terms of amplitude error, and slight improvement in terms of phase errors in comparison with the COSMO schemes. Linear stability analysis of the new schemes reveal improved linear stability which allows significantly larger time steps compared to the COSMO schemes. Theoretical analysis further reveal significant decrease in alias error of the symmetric schemes, which reduces significantly the non-linear instability of the schemes compared to other schemes. Together with conservation of the kinetic energy, it allows stable simulations without numerical diffusion. A two-dimensional numerical idealized test has been used to reveal the second order and fourth order convergence properties of the schemes. Instability attributed to inconsistent discretization of non-linear and linear terms of the model's Euler equations has been demonstrated using the same idealized test case and a real case study. The model's numerical accuracy, stability and simulation quality using different schemes have been further assessed by analysis of a series of regional climate simulations over European domain using ERA-interim re-analysis boundary data. The results show a significant improvement of the models stability and effective resolution. The results also reveal significant effects of the new schemes on the coupling between the model's resolved dynamics and the sub-grid scale physical parameterizations. This has particularly enhanced the bias in the model's simulated convective precipitation.}, language = {en} }