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

Abstract

A new horizontally explicit/vertically implicit (HEVI) time splitting scheme for atmospheric modelling is introduced, for which the horizontal divergence terms are applied within the implicit vertical substep. The new HEVI scheme is implemented in conjunction with a mixed mimetic spectral element spatial discretisation and semi-implicit vertical time stepping scheme that both preserve the skew-symmetric structure of the non-canonical Hamiltonian form of the equations of motion. Within this context the new HEVI scheme allows for the exact balance of all energetic exchanges in space and time. However since the choice of horizontal fluxes for which this balance is satisfied is not consistent with the horizontal velocity at the end of the time level the scheme still admits a temporal energy conservation error. Linearised eigenvalue analysis shows that similar to a fully implicit method, the new HEVI scheme is neutrally stable for all buoyancy modes, and unlike a second order trapezoidal HEVI scheme is stable for all acoustic modes below a certain horizontal CFL number. The scheme is validated against standard test cases for both planetary and nonhydrostatic regimes. For the planetary scale baroclinic instability test case, the new formulation exhibits a secondary oscillation in the potential to kinetic energy power exchanges, with a temporal frequency approximately four times that exhibited by a horizontally third order, vertically second order trapezoidal scheme. For the nonhydrostatic test case, the vertical upwinding of the potential temperature diagnostic equation is shown to reduce spurious oscillations without altering the energetics of the solution, since this upwinding is performed in an energetically consistent manner. For this test case, which is configured on an affine geometry, the exact balance of energy exchanges allows the model to run stably without any form of dissipation.

References in corpus (10)

• Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms
• Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements
• Mixed Mimetic Spectral Element Method for Stokes Flow: A Pointwise Divergence-Free Solution
• Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
• A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere
• A Mixed Mimetic Spectral Element Model of the 3D Compressible Euler Equations on the Cubed Sphere
• An energetically balanced, quasi-Newton integrator for non-hydrostatic vertical atmospheric dynamics
• Petrov-Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems
• Efficient IMEX Runge-Kutta methods for nonhydrostatic dynamics
• Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction