Mohamed Zerroukat

Application of the parabolic spline method (PSM) to a multi-dimensional conservative semi-Lagrangian transport scheme (SLICE)

The recently devised one-dimensional parabolic spline method (PSM) for efficient, conservative, and monotonic remapping is introduced into the semi-Lagrangian inherently-conserving and efficient (SLICE) scheme for transport problems in multi-dimensions. To ensure mass conservation, an integral form of the transport equation is used rather than the differential form of classical semi-Lagrangian schemes. Integrals within the SLICE scheme are computed using multiple sweeps of PSM along flow-dependent cascade directions to avoid the large timestep-dependent splitting errors associated with traditional fixed-direction splitting. Accuracy of the overall scheme, including at large timestep, is demonstrated using two-dimensional test problems in both Cartesian and spherical geometries and compared with that of the piecewise parabolic method (PPM) applied within the same SLICE framework.

Short Note: A simple mass conserving semi-Lagrangian scheme for transport problems

A mass conserving semi-Lagrangian (SL) scheme is achieved with a combination of a simple explicit smoothness-based mass correction and a standard non-conservative interpolating SL scheme. The resulting mass correction can be incorporated into any existing SL scheme with negligible extra cost. A more selective and less damping monotonicity filter by comparison to traditional filters is also presented. Results from various tests from the literature show that, in addition to mass conservation, the proposed scheme has negligible impact on the overall accuracy of the standard non-conservative SL scheme.

The monotonic Quartic Spline Method (QSM) for conservative transport problems

A quartic spline based remapping algorithm is developed and illustrative tests of it are presented herein. To ensure mass conservation, the scheme solves an integral form of the transport equation rather than the differential form. The integrals are computed from reconstructed quartic splines with mass conservation constraints. For higher dimensions, this remapping can be used within a standard directional splitting methodology or within the flow-dependent cascade splitting approach. A high-order grid and sub-grid based monotonic filter is also incorporated into the overall scheme. This filter is independent of the underlying spline representation adopted here, and is of more general application.

A deep non-hydrostatic compressible atmospheric model on a Yin-Yang grid

The singularity in the traditional spherical polar coordinate system at the poles is a major factor in the lack of scalability of atmospheric models on massively parallel machines. Overset grids such as the Yin-Yang grid introduced by Kageyama and Sato 1 offer a potential solution to this problem. In this paper a three-dimensional, compressible, non-hydrostatic atmospheric model is developed and tested on the Yin-Yang grid building on ideas previously developed by the authors on the solution of Elliptic boundary value problems and conservation on overset grids. Using several tests from the literature, it is shown that this model is highly stable (even with little off-centering), accurate, and highly efficient in terms of computational cost. The model also incorporates highly efficient and accurate approaches to achieve positivity, monotonicity and conservative transport, which are paramount requirements for any atmospheric model. The parallel scalability of this model, using in excess of 212 million unknowns and more than 6000 processors, is also discussed and shown to compare favourably with a highly optimised latitude-longitude model in terms of scalability and actual run times.

A moist Boussinesq shallow water equations set for testing atmospheric models

The shallow water equations have long been used as an initial test for numerical methods applied to atmospheric models with the test suite of Williamson et al. 1] being used extensively for validating new schemes and assessing their accuracy. However the lack of physics forcing within this simplified framework often requires numerical techniques to be reworked when applied to fully three dimensional models. In this paper a novel two-dimensional shallow water equations system that retains moist processes is derived. This system is derived from three-dimensional Boussinesq approximation of the hydrostatic Euler equations where, unlike the classical shallow water set, we allow the density to vary slightly with temperature. This results in extra (or buoyancy) terms for the momentum equations, through which a two-way moist-physics dynamics feedback is achieved. The temperature and moisture variables are advected as separate tracers with sources that interact with the mean-flow through a simplified yet realistic bulk moist-thermodynamic phase-change model. This moist shallow water system provides a unique tool to assess the usually complex and highly non-linear dynamics-physics interactions in atmospheric models in a simple yet realistic way. The full non-linear shallow water equations are solved numerically on several case studies and the results suggest quite realistic interaction between the dynamics and physics and in particular the generation of cloud and rain. Novel shallow water equations which retains moist processes are derived from the three-dimensional hydrostatic Boussinesq equations.The new shallow water set can be seen as a more general one, where the classical equations are a special case of these equations.This moist shallow water system naturally allows a feedback mechanism from the moist physics increments to the momentum via buoyancy.Like full models, temperature and moistures are advected as tracers that interact through a simplified yet realistic phase-change model.This model is a unique tool to test numerical methods for atmospheric models, and physics-dynamics coupling, in a very realistic and simple way.

On the monotonic and conservative transport on overset/Yin-Yang grids

In this paper, we outline a simple and a general methodology to achieve positivity, monotonicity and mass conservation with transport schemes on general overset grids. The main feature of the approach is its reduced complexity, which simplifies the use of higher-order schemes and higher dimensions on general grids and in particular for overset grids. The method also does not degrade substantially the order of the overall scheme despite the extra constraints of monotonicity and conservation. The approach is applied to achieve mass conservation with semi-Lagrangian schemes and its performance is analyzed using simple one-dimensional overlapping grids and a two-dimensional spherical Yin-Yang grid. The Yin-Yang grid is a special overset grid for the sphere and it is of a special interest in the atmospheric modeling community, as it is one of the grids that may resolve the scaling issue of existing longitude-latitude-grid based atmospheric models on massively parallel machines.