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Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
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Discussion papers
https://doi.org/10.5194/gmd-2019-122
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-2019-122
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Development and technical paper 21 Jun 2019

Development and technical paper | 21 Jun 2019

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This discussion paper is a preprint. It is a manuscript under review for the journal Geoscientific Model Development (GMD).

Extending Square Conservation to Arbitrarily Structured C-grids with Shallow Water Equations

Lilong Zhou1,3, Jinming Feng2, and Lijuan Hua1 Lilong Zhou et al.
  • 1College of Earth and Planetary Sciences, University of Chinese Academy of Sciences
  • 2Key Laboratory of Regional Climate-Environment for Temperate East Asia, Institute of Atmospheric Physics, Chinese Academy of Sciences
  • 3Numerical Weather Prediction Center of China Meteorological Administration

Abstract. The square conservation theory is widely used on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass, total absolute vorticity and instantaneous total energy. The first 2 integral invariants are entirely conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge-Kutta. The method of strictly conserving the total energy simultaneously uses both an anti-symmetrical spatial discrete operator and square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK, attempting to extend the square conservation theory to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations by Institute of Atmospheric Physics (IAP) transformation, expressing the temporal trend of the evolution variable by using the original operators of TRiSK. Using the square conservative Runge-Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.

Lilong Zhou et al.
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Lilong Zhou et al.
Model code and software

TMCORE L. Zhou and L. Dong https://doi.org/10.5281/zenodo.3241647

Idealized Global Spectral Atmospheric Models (GFDL) with standard shallow water test cases L. Zhou and GFDL https://doi.org/10.5281/zenodo.3249878

Lilong Zhou et al.
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Short summary
Atmospheric simulation should obey the physical conservation law, in a closed atmospheric system, some of the invariants should be conserved, i.e. the total mass, total energy, total absolute vorticity and so on. In this paper, we've improved the conservative properties of the atmospheric model, the tests show that better conservative properties bring us less simulation error, and the stability of model is also improved.
Atmospheric simulation should obey the physical conservation law, in a closed atmospheric...
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