Comparison of Lorenz and Charney–Phillips vertical discretisations for dynamics–boundary layer coupling. Part I: Steady states

Abstract

Accurate coupling between the resolved-scale dynamics and the parametrised physics is essential for accurate modelling of the atmosphere. Previous emphasis has been on the temporal aspects of this so-called physics-dynamics coupling problem, with little attention on the spatial aspects. When designing a model for numerical weather prediction there is a choice for how to vertically arrange the predicted variables, namely the Lorenz and Charney-Phillips grids, and there is ongoing debate as to which is the optimal. The Charney-Phillips grid is considered good for capturing the potential vorticity dynamics and wave propagation, whereas the Lorenz grid is more suitable for conservation. However the Lorenz grid supports a computational mode. It is argued here that the Lorenz grid is preferred for modelling the stably stratified boundary layer. This presents the question: which grid will produce more accurate results when coupling the large-scale dynamics to the stably stratified planetary boundary layer? The question is addressed by examining the ability of both the Lorenz and Charney-Phillips grids to capture the steady state of a set of equations that simultaneously represents both large-scale dynamics and the planetary boundary layer. The results show that the Charney-Phillips grid is able to capture accurately the steady boundary-layer solution provided the Richardson number is calculated without vertically averaging the shear. Averaging the shear suppresses the negative feedback of the shear on the diffusion coefficient; the positive feedback, via the vertical gradient of potential temperature, then leads to the formation of unrealistic step-like features.

Stability analysis of split-explicit free surface ocean models: Implication of the depth-independent barotropic mode approximation

“…Indeed, for N = 10 −3 s −1 and H = 4000 m, we get ≈ 4.10 −4 . The normal mode definition (15)(16)(17) can be expressed in terms of (considering…”

“…This approximation corresponds to the one done in rigid lid ocean models where ∂η ∂t = 0. In this approximation, 0 is an eigenvalue of the Sturm Liouville system (15)- (17) and the approximated modes (denoted here by M ) satisfy the following properties…”

“…If h 0 is computed inside the barotropic integration, then at the end of this integration the correction is naturally applied to the barotropic component of the pressure field. However when discretized on a vertical Lorenz grid, the presence of a computational mode ([16],[17]) prevents the determination of a unique correction on the density field (which is what is needed at the end) from a correction on the pressure field.…”

Choice of function spaces for thermodynamic variables in mixed finite‐element methods

“…Due to the computational mode of the Lorenz staggering, and notwithstanding its better conservation properties, the Charney-Phillips staggering has emerged as the preferred arrangement by many groups for dynamical cores in comparisons with the Lorenz staggering, e.g. Arakawa and Moorthi (1988), Fox-Rabinovitz (1994), Arakawa and Konor (1996) and Holdaway et al (2013a). Examples of operational forecast models running with vertical Charney-Phillips staggering are the Global Environmental Multiscale model (Girard et al, 2014) and the Met Office's Unified Model, which also uses C-grid staggering in the horizontal direction (Cullen et al, 1997;Davies et al, 2005;Wood et al, 2014).…”