Energy‐conserving finite‐difference schemes for quasi‐hydrostatic equations

Abstract

†The contributions of Thomas Melvin were written in the course of his employment at the Met Office, UK and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.The pressure-based hydrostatic primitive equations model LMD-Z is extended to solve the quasi-hydrostatic deep-atmosphere as well as the non-traditional shallow-atmosphere equations (with a complete Coriolis force representation).The continuous equations are first derived in their curl form using Eulerian horizontal and non-Eulerian vertical coordinates. The equations are then interpreted as a Hamiltonian system, as they are expressed in terms of functional derivatives of the Hamiltonian. Using a finite-difference scheme on a longitude/latitude grid and based either on a Lagrangian or mass-based vertical coordinate, the discrete scheme is obtained by imitating the Hamiltonian formulation at the discrete level. It is shown how this form leads straightforwardly to the conservation of discrete total energy. The relation between the discrete equations and the discrete antisymmetry property of the Poisson bracket is discussed.The computing infrastructure of the dynamical core is kept essentially unchanged but the modification of the hydrostatic balance requires a mass-based vertical coordinate. Also, absolute angular momentum is used as a prognostic variable instead of relative velocity, which allows time-dependent metric terms and the non-traditional Coriolis force to be absorbed into it.The prototype implementation is applied to idealized circulations of an Earth-like small planet and validates the stability and accuracy of the new dynamical core.

The Limits of the Primitive Equations of Dynamics for Warm, Slowly Rotating Small Neptunes and Super Earths

“…As discussed in Mayne et al (2014b) the physical justification for the traditional approximation is relatively weak, and concerns have been raised over its validity for thick atmospheres (Tokano 2013;Mayne et al 2014b), and terrestrial planets (Gerkema et al 2008;Tort et al 2015). Also as discussed the condition w v tan φ indicates whether the terms omitted by this approximation are negligible or not.…”

“…In addition Tort et al (2015) have shown that the traditional approximation becomes increasingly less valid as, in their case, Ω is increased, yet ΩR p remains constant, for the terrestrial regime. If ΩR p is constant, Eqn (22) is clearly constant for a given φ. Eqn (17) (22) and (17) also make interesting predictions for the atmospheric interaction with condensates.…”

“…Focusing our work on GJ 1214b we find significant differences in the resulting dynamical structure of the simulated atmosphere between the "primitive" and more complete equations. This is caused by the breakdown of the 'shallow-fluid' and traditional approximations, similar to that found for a terrestrial planet by Tort et al (2015). The rest of this paper is structured as follows: in Section 2 we introduce the model we are using, summarise its evolution and describe the setups and parameters adopted.…”

“…Concerns over the validity of the "primitive" equations for "thick" atmospheres, where the atmospheric scale height becomes significant compared to the assumed total planetary radius have been raised for Titan and Venus (see Tokano 2013). Additionally, work has been done assessing the limits of the primitive equations for Earth (see Tort et al 2015, and references therein), and the traditional approximation specifically (Gerkema et al 2008). Several 3D models have been applied to exoplanets which solve the full Navier-Stokes equations, such as the 3D radiation-hydrodynamics model of Dobbs-Dixon & Agol (2013) 1 , the recently developed THOR dynamical core (Mendonça et al 2016), and of course our own work (e.g., Mayne et al 2014b).…”

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties

“…However, a similar Z grid scheme based on a Helmholtz decomposition of the momentum instead of the wind, which does conserve both total energy and potential enstrophy, was developed by Salmon (2005Salmon ( , 2007 using techniques from Hamiltonian mechanics (specifically, Nambu brackets). The idea of using Hamiltonian mechanics to derive conservative models for atmospheric dynamical cores has seen a great deal of interest and progress in the past 10 years (see Gassmann and Herzog, 2008;Gassmann, 2013;Sommer and Névir, 2009;Nevir and Sommer, 2009;Dubos and Tort, 2014;Dubos et al, 2015;Tort et al, 2015;Salmon, 1988;Shepherd, 2003). With the recent development of Hamiltonian formulations for essentially all of the equation sets and vertical coordinates used in atmospheric dynamics, it seems likely that this approach will continue to be employed in the future.…”