M. Läuter

Semi-Implicit Formulations of the Navier-Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

We present semi-implicit (implicit-explicit) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge-Kutta method for all the tests studied, especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including inertia-gravity waves, density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R^3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O(@Dx^k^+^1) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step @Dt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

A parallel adaptive barotropic model of the atmosphere

The parallel adaptive model PLASMA has been developed for modeling a barotropic atmosphere. This model adapts the computational grid at every time step according to a physical error indicator. Thus, compared to uniform grid experiments the number of grid points is reduced significantly. At the same time, the error increases only slightly, when comparing with uniform grid solutions. For the discretization of the underlying spherical shallow water equations a Lagrange-Galerkin method is used. The unstructured triangular grid is maintained by the grid generator amatos and the large linear systems are solved by the parallel solver interface FoSSI. Experimental convergence is shown by means of steady-state and unsteady analytical solutions. PLASMA yields satisfactory results for quasi standard experiments, that is the Rossby-Haurwitz wave and zonal flows over an isolated mountain.

A Self-Adaptive Finite Element Model of the Atmosphere

An adaptive finite elemente method is introduced ina simplified, nonlinear, global atmospheric circulation model to investigateinternally generated climate variability in the atmosphere.This article focusses on two aspects of the parallel implementation:fast grid partitioning with a space filling curve approach and couplingMPI-parallel solver libraries to the OpenMP-parallel model.