A quasi-Hamiltonian discretization of the thermal shallow water equations

Abstract

The rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model.

Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models

“…Study of the effective resolution suggests that n " 3 is a sweet spot in terms of maximizing resolvable wavenumber and order of convergence per unit computational cost. Following this idea, a discretization of the thermal shallow water equations using the M GD 3 family can be found in [12] and work is ongoing to extend this to the fully compressible Euler equations using Eulerian vertical coordinates. An extension of this study of 2D shallow water dispersion relationships to incorporate time discretization is also underway.…”

“…More details about the tensor product construction can be found in [12,14,23,29]. In this paper we consider the Qń Λ k family from finite element exterior calculus with A " P C n and B " P DG n´1 [4,5] and the M GD n family (mimetic Galerkin differences) with A " GD n and B " DGD n´1 [12,13]. These are the extension of the element pairs in [13] to quadrilaterals.…”

“…Recall that GD n is the space of order n Galerkin differences in 1D, and that DGD n´1 is the corresponding compatible L 2 space in 1D constructed following [14]. More details about this construction can be found in [12]. A diagram of the degrees of freedom for n " 3 is found in Figure 11, and the basis functions for W 0 , W 1 and W 2 are found in Figure 12.…”

Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

“…In a compatible finite element setting, this has already been achieved for potential vorticity upwinding for the rotating shallow water equations in [1], while upwinding for the velocity field for the incompressible Euler equations has been introduced in [17], using a geometric approach including Lie derivatives. Further, upwinding for buoyancy has been introduced for the thermal rotating shallow water equations in [6]. For the compressible Euler equations, energyconserving upwinding schemes for the density and temperature fields remain to be formulated.…”

A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere

“…The use of mimetic discretizations to represent the solution variables, and the adjoint properties of the differential operators implied by those spaces, allows for the conservation of energy via the exact balance of energetic exchanges, as well as the orthogonality of vorticity evolution to those exchanges [2][3][4]. By satisfying exactly the balance between energetic exchanges, it is hoped that these methods may improve the statistical behaviour of climate simulations over long time scales by mitigating against internal biases in the representation of dynamical processes.…”