Options and extensions for the stochastic shallow convection scheme in ICON

<p>The Icosahedral Model (ICON) of the German Weather Service (Deutscher Wetterdienst, DWD) is used for numerical weather prediction at global and regional scales. In the limited area mode, resolution is typically on the order of a few kilometers horizontal grid spacing. Deep convective transport is partially resolved at these scales, but shallow convection remains poorly represented without a parameterization.</p><p>A stochastic shallow convection scheme was developed in collaboration with the Max Planck Institute for Meteorology, and is now being implemented in ICON with a view towards operational use. The scheme is scale-adaptive and renders resolution-dependent tuning of the convection parameterization unnecessary. Mass flux limiters essential for the stable operation of the unaltered convection scheme can be removed when the stochastic perturbations are introduced.</p><p>Alongside the original, explicit stochastic scheme an approximation using stochastic differential equations (SDE) has been developed. The advantage of the SDE version is a lower computational and memory cost, and the ability to save and restart the model‘s stochastic cloud state easily.</p><p>Equivalence of the two versions can be demonstrated by running one version interactively, the other passively (“piggy-backing”). While the SDE approximation is computationally more efficient, the explicit version of the scheme can be easily extended to keep track of additional properties of the shallow cloud ensemble. For example, the convective updraft core fraction can be calculated for use in the diagnostic subgrid cloud scheme. Or knowledge of individual clouds’ depth can be used to derive a more realistic lateral detrainment profile than is currently in use.</p><p>We demonstrate the performance of the scheme and illustrate options and applications in single column mode, case studies and month-long hindcasts.</p>

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

A New TVD Scheme for Gradient Smoothing Method Using Unstructured Grids

An improved [Formula: see text]-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.