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A Stability Analysis of Divergence Damping on a Latitude–Longitude Grid

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A Stability Analysis of Divergence Damping on a Latitude–Longitude Grid 2976 MONTHLY WEATHER REVIEW VOLUME 139 A Stability Analysis of Divergence Damping on a Latitude–Longitude Grid JARED P. WHITEHEAD Department of Mathematics, University of Michigan, Ann Arbor, Ann Arbor, Michigan CHRISTIANE JABLONOWSKI AND RICHARD B. ROOD Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, Ann Arbor, Michigan PETER H. LAURITZEN Climate and Global Dynamics Division, National Center for Atmospheric Research,* Boulder, Colorado (Manuscript received 24 August 2010, in final form 25 March 2011) ABSTRACT The dynamical core of an atmospheric general circulation model is engineered to satisfy a delicate balance between numerical stability, computational cost, and an accurate representation of the equations of motion. It generally contains either explicitly added or inherent numerical diffusion mechanisms to control the buildup of energy or enstrophy at the smallest scales. The diffusion fosters computational stability and is sometimes also viewed as a substitute for unresolved subgrid-scale processes. A particular form of explicitly added diffusion is horizontal divergence damping. In this paper a von Neumann stability analysis of horizontal divergence damping on a latitude–longitude grid is performed. Stability restrictions are derived for the damping coefficients of both second- and fourth- order divergence damping. The accuracy of the theoretical analysis is verified through the use of idealized dynamical core test cases that include the simulation of gravity waves and a baroclinic wave. The tests are applied to the finite-volume dynamical core of NCAR’s Community Atmosphere Model version 5 (CAM5). Investigation of the amplification factor for the divergence damping mechanisms explains how small-scale meridional waves found in a baroclinic wave test case are not eliminated by the damping. 1. Introduction are highly dependent on the numerical scheme (Lander and Hoskins 1997). For example, Skamarock (2004) This paper focuses on the characteristics of an explicit estimated with the help of kinetic energy spectra that the diffusion mechanism in the finite-volume (FV) dynam- ‘‘effective resolution’’ of the Weather Research and ical core (Lin 2004) that is part of the Community At- Forecasting (WRF) model is 7Dx where Dx symbolizes mosphere Model version 5 (CAM5) at the National the grid spacing. In any case, subgrid scales smaller than Center for Atmospheric Research (NCAR; Neale et al. 2Dx fall below the resolution of the grid. Slightly larger 2010). A dynamical core is, broadly speaking, that part scales (for WRF these are the scales between 2Dx and of an atmospheric general circulation model (GCM) 7Dx) are represented by the model, but not resolved ef- associated with the fluid dynamics. It includes both the fectively. This paper seeks to quantify the effects of dif- resolved and subgrid-scale flow. The spatial scale of the fusive processes on these unresolved and underresolved resolved flow is determined by the grid spacing of the subgrid scales in GCMs. Of particular interest here is the discrete mesh. The actual ‘‘believable scales’’ of a model impact of a specific, explicitly added diffusive mechanism on the underresolved scales. * The National Center for Atmospheric Research is sponsored The dynamical core of each model is engineered to by the National Science Foundation. satisfy a delicate balance between numerical stability, an accurate representation of the equations of motion, and Corresponding author address: Jared Whitehead, 2074 East Hall, computational cost. In an effort to balance these factors, 530 Church St., Ann Arbor, MI 48109. each model employs some form of diffusion, filtering, or E-mail: [email protected] a posteriori fixers (Williamson et al. 2009). Often these DOI: 10.1175/2011MWR3607.1 Ó 2011 American Meteorological Society Unauthenticated | Downloaded 09/30/21 09:38 PM UTC SEPTEMBER 2011 W H I T E H E A D E T A L . 2977 practices are poorly documented, with the presumption ‘‘manage’’ such numerical issues, and hence avoid or al- that their impact on model performance is small. This leviate the need for an additional explicit diffusive term. presumption is often justified when considering their Considering the treatment of dispersion errors, the man- effect on a fully resolved, isolated wave. However, agement of nonlinear or linear computational instabilities, model performance is determined by a complex spec- the effects of grid staggering, and inherent diffusion, all trum of motions on many scales and the interactions of models have some forms of diffusion mechanisms that are these scales. The decay of waves, ultimately, has im- not fully grounded in the basic physics of the fluid flow. portant implications for the mean circulation of the at- The overarching goal is to avoid the accumulation of en- mosphere (Andrews and McIntyre 1978). Therefore, ergy or enstrophy at the smallest scales near the truncation relegating this ultimate decay to a set of poorly under- limit. stood subgrid-scale processes leaves an undocumented In this paper we explore the linear von Neumann sta- impact on the model circulation and possibly the climate bility characteristics of a second- and fourth-order hori- statistics in long time integrations. Scientific rigor re- zontal divergence damping mechanism applied on a quires the evaluation of the effects of adding diffusion, regular, equal-angle latitude–longitude grid. As a specific filters, and fixers (Jablonowski and Williamson 2011). instance, we consider the divergence damping imple- We assert that this might be a particularly important mentation in the FV dynamical core of CAM5 (Lauritzen aspect of dynamical cores as we push models to higher et al. 2011), which utilizes explicit time stepping (Neale resolution in pursuit of more realistic representations of et al. 2010). The second-order divergence damping both climate and weather. This assertion is based on the mechanism in CAM5 is also implemented in earlier ver- fact that it has been implicit in both model construction sions of CAM [CAM4 and the finite-volume dynamical and dynamical meteorology that the scales of interest core in CAM3.1, see Collins et al. (2004)]. This FV dy- are quasi-nondivergent. This will not be true for models namical core was developed at the National Aeronautics with grid sizes of order 10 km or finer that start to re- and Space Administration (NASA) Goddard Space Flight solve motions in the mesoscale regime. Center, and has some similarities to the Goddard Earth Spurious, dispersive phenomena are a common prob- Observing System version 5 model (GEOS5; Rienecker lem inherent to computational fluid dynamics. Exam- et al. 2008) and the National Oceanic and Atmospheric ples include the Gibbs phenomenon, nonpropagating Administration (NOAA) Geophysical Fluid Dynamics numerical modes, and spectral blocking (e.g., Rood Laboratory’s (GFDL) Atmospheric Model version 2.1 1987). These phenomena may propagate, interact non- (AM2.1; Delworth et al. 2006). linearly, and negatively impact the model solution. An The analysis carried out in the following sections is not economic method to reduce these dispersive modes is to specific to the finite-volume dynamical core and is rele- add an explicit diffusion term to the equations of motion vant to any model that employs divergence damping on a prior to discretization. Most often, a hyperdiffusion regular latitude–longitude grid with an explicit time- technique is used as documented in Jablonowski and stepping scheme. Our particular analysis is carried out on Williamson (2011). This provides a mechanism to dis- a staggered D-grid (Arakawa and Lamb 1977). However, sipate these known spurious modes, presumably, before the analysis technique generalizes to other grid staggering they have a negative impact on the overall dynamics. options and is, in fact, identical for C grids. We explore Generally, the strength of the diffusive process is em- the divergence damping mechanism from both a theoret- pirically tuned so that the kinetic energy spectra imitate ical and practical viewpoint. The latter includes selected observation (Boville 1991; Takahashi et al. 2006). This dynamical core test cases that demonstrate the impact of technique is classically utilized in spectral transform- the diffusive processes directly on the model simulations. based schemes where the Gibbs phenomenon is present This provides a guiding method to analyze the additional when sharp gradients in the flow field arise. Explicit diffusion incorporated in other climate or weather diffusion or filtering processes are also widely used in models. Similar tests and analysis can be performed on finite-difference or spectral element dynamical cores. In other forms of explicit diffusion, providing a systematic contrast, flux-limiting finite-difference and flux- or slope- framework that brings to light the various methods for limiting finite-volume methods typically introduce an in- introducing diffusion to a model. Simulations for two herent nonlinear diffusion via the numerical scheme that different idealized dynamical core test cases indicate that prevents unphysical oscillations from appearing (Durran this linear stability analysis is very accurate, providing 1999). Here, the phrase ‘‘unphysical’’ refers to obvious more impetus to perform similar stability analyses on overshoots and undershoots of numerical estimates. other models’ methods of explicit diffusion or damping. Sometimes the modeler will deliberately choose an in- This paper is organized as follows. In section 2 we
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