Physics–Dynamics Coupling with Element-Based High-Order Galerkin Methods: Quasi-Equal-Area Physics Grid

Abstract

Atmospheric modeling with element-based high-order Galerkin methods presents a unique challenge to the conventional physics–dynamics coupling paradigm, due to the highly irregular distribution of nodes within an element and the distinct numerical characteristics of the Galerkin method. The conventional coupling procedure is to evaluate the physical parameterizations ( physics) on the dynamical core grid. Evaluating the physics at the nodal points exacerbates numerical noise from the Galerkin method, enabling and amplifying local extrema at element boundaries. Grid imprinting may be substantially reduced through the introduction of an entirely separate, approximately isotropic finite-volume grid for evaluating the physics forcing. Integration of the spectral basis over the control volumes provides an area-average state to the physics, which is more representative of the state in the vicinity of the nodal points rather than the nodal point itself and is more consistent with the notion of a “large-scale state” required by conventional physics packages. This study documents the implementation of a quasi-equal-area physics grid into NCAR’s Community Atmosphere Model Spectral Element and is shown to be effective at mitigating grid imprinting in the solution. The physics grid is also appropriate for coupling to other components within the Community Earth System Model, since the coupler requires component fluxes to be defined on a finite-volume grid, and one can be certain that the fluxes on the physics grid are, indeed, volume averaged.

A Total Energy Error Analysis of Dynamical Cores and Physics‐Dynamics Coupling in the Community Atmosphere Model (CAM)

“…It clearly exhibits unphysical oscillations coinciding with element boundaries. Details of the spectral‐element method, its coupling to physics, and associated noise issues are discussed in detail in Herrington et al (). The noise in the solutions is even visible in the 500‐hPa pressure velocity annual average (Figure a).…”

“…One‐year average of mass (kg/m 2 ) “clipped” in physics‐dynamics coupling (so that state is not driven negative) when using ftype=0 (‘dribbling’) physics‐dynamics coupling for (a) water vapor, (b) cloud liquid, and (c) cloud ice, respectively. Interestingly, the element boundaries systematically show in the plots which is likely related to the anisotropy of the quadrature grid (Herrington et al, ).…”

“…The element noise in CAM‐SE with combination (ftype=2) seen in both and 500‐hPa pressure velocity can be “removed” by using CAM‐SE‐CSLAM (Figure d), which uses a quasi equal‐area physics grid and CSLAM (Lauritzen et al, ) consistently coupled to the SE method (Lauritzen et al, ). The noise patterns in vertical velocity off the western coast of South America are present in all CAM‐SE simulations (and hence not related to PDC algorithm) are also removed by using CAM‐SE‐CSLAM (Herrington et al, ).…”

“…The “clipping” changes the water mass budget without accounting for it in water fluxes or in the thermodynamics and hence lead to TE conservation errors (both kinetic and thermal energy). Change of vertical grid/coordinate errors : If the vertical coordinates in physics and in the dynamical core are different, then there can be spurious PDC energy errors even when using the state‐update method for adding tendencies to the dynamical core state. For example, many nonhydrostatic dynamical cores (e.g., Skamarock et al, ) use a terrain‐following height coordinate whereas physics uses pressure. Change of horizontal grid errors: If the physics tendencies are computed on a different horizontal grid than the dynamical core, then there can be spurious energy errors from mapping tendencies and/or variables between horizontal grids (e.g., Herrington et al, ). Compensating energy fixers: To avoid TE conservation errors which could accumulate and ultimately lead to a climate drift, it is customary to apply an arbitrary energy fixer to restore TE conservation. Since the spatial distribution of many energy errors, in general, is not known, global fixers are used.…”

Exploring a Lower‐Resolution Physics Grid in CAM‐SE‐CSLAM

“…Element‐based Galerkin methods are susceptible to grid‐imprinting and may need to be considered when contemplating a particular physics grid (Herrington et al, , hereafter referred to as H18). Grid imprinting manifests at the element boundaries, since the global basis is least smooth ( C 0 ; all derivatives are discontinuous) for quadrature nodes lying on the element boundaries, and the gradients (e.g., pressure gradients) are systematically tighter producing local extremes.…”