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

Abstract

A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a nonhydrostatic gravity wave as driven by a temperature perturbation.

A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

“…It can be shown that, by doing so, important conservation properties can be maintained. Since then, much work has been produced and a rich variety of different flavours of structure preserving discretizations have been presented: finite differences/finite volumes [62,63,64,65,66], discrete exterior calculus (DEC) [67], finite element exterior calculus (FEEC) [51,68,69] and the works by the authors [70,71,72,73,74].…”

Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction

“…A Poisson bracket based formulation of the dry compressible Euler equations can be found e.g. in [17,22]. Note that the Poisson bracket and the skew symmetric operator presented in these papers correspond to a different Poisson bracket to the one considered here, relying on the use of a different set of underlying fields (with a mass weighted thermal field Θ).…”

“…For the compressible Euler equations, which form the basic equation set of dynamical cores in NWP, such Hamiltonian based formulations already exist e.g. for hexagonal C-grids [17], the compatible finite element method [22], and for general classes of mimetic discretisations [38].…”

Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere

“…For the purposes of this article the salient feature of this method is that it allows for the preservation of the skewsymmetric structure of the compressible Euler equations, and thus the conservation of energy and energetic exchanges in the discrete form. For a more detailed discussion the reader is referred to previous work on the use of this method for geophysical flow modelling [8,10,11], as well as more foundational works on the subject [12][13][14][15]. In order to begin this discussion we introduce the finite dimensional subspaces…”

“…In the present article a new HEVI scheme is introduced, motivated by a desire to preserve the exact balance of energetic exchanges. The scheme is implemented within the context of a mixed mimetic spectral element spatial discretisation [8] and energy conserving implicit vertical time stepping method [9] that together allow for the exact balance of all energy exchanges in space and time. This is achieved by evaluating the horizontal mass and temperature fluxes within the vertically implicit sub-step, after first computing a provisional horizontal velocity from which to derive a second order temporal representation of the horizontal fluxes.…”