A vertical slice frontogenesis test case for compressible nonhydrostatic dynamical cores of atmospheric models

TL;DR

The paper addresses the need for standardized, fast benchmarks for compressible nonhydrostatic atmospheric cores that exhibit frontogenesis. It introduces a compressible vertical slice Eady test grounded in a variational framework that conserves energy $E$ and potential vorticity $q$, solved with a compatible finite element discretisation and an implicit midpoint time-stepping scheme to produce a reference solution. The results show quasi-periodic lifecycles of fronts with front formation around days 4–7, and demonstrate sensitivity to advection form (vector-invariant vs advective) via a Hollingsworth-like instability, while higher resolution enhances peak velocities and reveals discretisation effects. The test case offers a practical benchmark for intercomparison and testing of subgrid schemes, with potential extensions to moisture and boundary-layer processes and a goal of rigorous semigeostrophic convergence analysis in future work.

Abstract

A new test case is presented for evaluating the compressible dynamical cores of the atmospheric models. The test case is based on a compressible vertical slice model that can be obtained by simple modification of a standard three dimensional compressible dynamical core. On the one hand, an advantage of the test case is that is quasi-2D, so it can be run quickly on a standard workstation, enabling rapid experimentation with numerical schemes and discretisation choices. On the other hand, the test case exhibits frontogenesis, a challenging regime for numerical discretisations which usually only arises in 3D model configurations for the compressible case. Numerical results of the test case using an implicit time-stepping method with a compatible finite element discretisation are presented as a reference solution. An example comparison between advective and vector-invariant forms for the advective nonlinearity in the velocity equation demonstrates one possible use of the scheme. The comparison shows a Hollingsworth-like instability when the vector invariant form is used.

Figures (5)

• Figure 1: Snapshots of out-of-slice velocity fields at days 2, 4, 7, and 11 in (a) the control run and (b) the high-resolution run.
• Figure 2: Snapshots of in-slice temperature perturbation field at days 2, 4, 7, and 11 in (a) the control run and (b) the high-resolution run.
• Figure 3: The root mean square of the out-of-slice velocity in the control run (black) and in the high-resolution run (red).
• Figure 4: Time evolution of energy in the control run (black) and in the high-resolution run (red). Thick line represents the evolution of total energy. Dotted and solid lines represent the evolutions of in-slice and out-of-slice components of the kinetic energy. Dot-dashed line represents the evolution of potential energy.
• Figure 5: Snapshots of out-of-slice velocity and in-slice temperature perturbation at day 6. top: advective form, bottom: vector invariant form.