A compatible finite‐element discretisation for the moist compressible Euler equations

Abstract

A promising development of the last decade in the numerical modelling of geophysical fluids has been the compatible finite‐element framework. Indeed, this will form the basis for the next‐generation dynamical core of the Met Office. For this framework to be useful for numerical weather prediction models, it must be able to handle descriptions of unresolved and diabatic processes. These processes offer a challenging test for any numerical discretisation, and have not yet been described within the compatible finite‐element framework. The main contribution of this article is to extend a discretisation using this new framework to include moist thermodynamics. Our results demonstrate that discretisations within the compatible finite‐element framework can be robust enough also to describe moist atmospheric processes. We describe our discretisation strategy, including treatment of moist processes, and present two configurations of the model using different sets of function spaces with different degrees of finite element. The performance of the model is demonstrated through several test cases. Two of these test cases are new cloudy‐atmosphere variants of existing test cases: inertia–gravity waves in a two‐dimensional vertical slice and a three‐dimensional rising thermal.

A solution to the trilemma of the moist Charney–Phillips staggering

“…$$ The same perturbation of Bryan and Fritsch (2002) is then applied, but to this definition of $$$ {\theta}_{\mathrm{e}} $$$. To find the initial values of the prognostic variables from these requirements, the routine of Bendall et al ., (2020) is used. Condensation and evaporation are applied through a saturation adjustment step at the end of each dynamical core time step; the details of this routine are described by Bendall (2019), although here it is used with constant $$$ {L}_{\mathrm{v}} $$$.…”

“…In the context of GungHo, the natural choices of space for the moisture mixing ratio would be or . To replicate the arrangement used by ENDGame, this work takes , which was also the choice made by Bendall et al ., (2020) in a similar finite‐element discretisation of the moist atmosphere. The trilemma challenge that this article addresses is then how to achieve conservation of total water content in a manner consistent with that for dry density held in .…”

Compatible finite element methods for geophysical fluid dynamics

“…This recovery process also allows the introduction of limiters to prevent overand undershoots. This was done in Bendall, Gibson, Shipton, Cotter and Shipway (2020), applied to compressible Euler solutions with moisture, where limiters are critical for stability. This produced the first atmospheric simulations using compatible finite elements with moist physics.…”

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

“…The discretisation can in principle be applied equally to fully three-dimensional problems and scenarios with a non-constant potential background temperature (by approximately eliminating δθ in the linearised momentum equation). A suitable nonlinear solver strategy for this case can be found in [6].…”

“…The latter is prone to secondary plumes (see e.g. [6]), and to avoid these in our discussion here, we consider the next higher order set of finite element spaces (k=3) for this test case. The domain Ω is given by a horizontally periodic square of 10 km side length.…”