An energy consistent summation-by-parts finite difference discretization for the non-hydrostatic atmospheric dynamics equations on the cubed-sphere grid
-
Dmitry A. Markhanov
,
Vladimir V. Shashkin
Abstract
It is expected that atmospheric models with kilometer-scale horizontal resolution will become accessible for use in climate research in the near future. Such high-resolution models require efficient, accurate and conservative numerical solvers for the non-hydrostatic atmospheric dynamics equations on quasi-uniform spherical meshes. In this article, we address this problem by presenting a novel high-order accurate, energy-conserving spatial approximation for non-hydrostatic dynamics equations on the cubed-sphere mesh. The key novel feature of our work is the application of the Summation-By-Parts Finite-Difference (SBP-FD) method for horizontal approximation. SBP-FD enables high-order accurate and stable approximations on logically rectangular multiblock meshes, such as the cubed-sphere. The horizontal gradient and divergence operators constructed using SBP-FD satisfy the discrete analogue of the Ostrogradsky–Gauss theorem, which is essential for energy conservation. As compared to previous works, we explicitly present expressions for discrete vertical mass and entropy fluxes in height-based vertical coordinates, ensuring the compensation potential and thermodynamics energy tendencies with energy contributions from pressure gradient and gravity forces. The proposed spatial approximation is verified using a suite of widely accepted idealized test cases, the results are in good agreement with reference solutions. The robustness of presented spatial approximation is confirmed with the stability in long-term Held–Suarez experiment.
Funding statement: The work was supported by Moscow Center of Fundamental and Applied Mathematics at INM RAS.
References
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A Thin spherical shell mapping
Thin spherical shell domain is the Cartesian product of a sphere S ⊂ ℝ3 and a section [0, H]. The points of SH = S × [0, H] are defined with 4-dimensional vectors
We consider a mapping ℳ from the model coordinates (α, β, η) to a subset of thin spherical shell domain S × [hs(α, β), H], where hs is the orography (atmosphere bottom) height. The mapping is the product (successive application) of horizontal mapping H : (α, β, η) → (S, η) and vertical mapping V : (S, η) → (x, y, z, h). The horizontal mapping is equiangular gnomonic cubed-sphere [22]. The vertical mapping depends on (α, β) or equivalently on the position of the point on the sphere (x, y, z).
The covariant basis vectors of ℳ are defined as
The metric tensor of ℳ is defined as 3×3 matrix
B Auxiliary statements for energy conservation proof
Statement B.1
where xw is an arbitrary w-points quantity column vector.
Proof. The identity follows from the vertical interpolation SBP-property (2.8) and the definition of · (2.13).
Statement B.2
Proof. First, using equation (B.1) and commutation of Hadamard product with the multiplication by diagonal matrix we write:
Then, using Ghor,ξ definition under equations (2.21) and (B.1) we can write
and the same for β-component.
Combining equations (B.3) and (B.4) and recalling the definition of the entropy flux (2.16) we rewrite the last three terms of (B.2) as
that is equal to
Statement B.3
Proof. The proof is exactly the same as for Statement B.2, but using ρ instead of Θ,minstead of q and Φ instead of Π.
Statement B.4
Proof. We first prove for the horizontal part. Given the definition of
Covariant and contravariant horizontal wind components are linked through horizontal mapping con-travariant metric tensor
For the vertical part of Statement B.4 we use definition of [vη , w]w (2.18) and the · property (B.1):
Then, using interpolation SBP-property (2.8) twice:
Finally, we can use discrete product rule (2.10) and commutation property (2.12) to derive
summing this with the horizontal part equation (B.9) and recalling K = (vα ◦ vα + vβ ◦ vβ + Iwpw2)/2 we get (B.7).
Statement B.5
Proof. Using 〈·〉 property (B.1) twice: for ρ from left to right and for 〈mα〉 from right to left we can write:
and the same takes place for β-part of equation (B.13).
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Articles in the same Issue
- Frontmatter
- Analysis of forced response and internal climate variability in the INMCM Earth system model
- Validation of the MGO regional climate model with a new parameterization of land surface processes
- An energy consistent summation-by-parts finite difference discretization for the non-hydrostatic atmospheric dynamics equations on the cubed-sphere grid
- Reproducibility of atmospheric circulation regimes over the winter Northern Hemisphere by the INMCM5 Earth system model
- Modelling magnetic field of a compact electro-magnetic device of experimental setup to study ROT effects in fission of heavy nuclei
Articles in the same Issue
- Frontmatter
- Analysis of forced response and internal climate variability in the INMCM Earth system model
- Validation of the MGO regional climate model with a new parameterization of land surface processes
- An energy consistent summation-by-parts finite difference discretization for the non-hydrostatic atmospheric dynamics equations on the cubed-sphere grid
- Reproducibility of atmospheric circulation regimes over the winter Northern Hemisphere by the INMCM5 Earth system model
- Modelling magnetic field of a compact electro-magnetic device of experimental setup to study ROT effects in fission of heavy nuclei
