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A New Dynamical Core of the Global Environmental Multiscale (GEM) Model with a Height-Based Terrain-Following Vertical Coordinate

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A New Dynamical Core of the Global Environmental Multiscale (GEM) Model with a Height-Based Terrain-Following Vertical Coordinate JULY 2019 H U S A I N E T A L . 2555 A New Dynamical Core of the Global Environmental Multiscale (GEM) Model with a Height-Based Terrain-Following Vertical Coordinate SYED ZAHID HUSAIN,CLAUDE GIRARD, AND ABDESSAMAD QADDOURI Atmospheric Numerical Prediction Research Section, Meteorological Research Division, Environment and Climate Change Canada, Dorval, Quebec, Canada ANDRÉ PLANTE Canadian Meteorological Centre, Environment and Climate Change Canada, Dorval, Quebec, Canada (Manuscript received 20 December 2018, in final form 7 May 2019) ABSTRACT A new dynamical core of Environment and Climate Change Canada’s Global Environmental Multiscale (GEM) atmospheric model is presented. Unlike the existing log-hydrostatic-pressure-type terrain- following vertical coordinate, the proposed core adopts a height-based approach. The move to a height- based vertical coordinate is motivated by its potential for improving model stability over steep terrain, which is expected to become more prevalent with the increasing demand for very high-resolution forecasting systems. A dynamical core with height-based vertical coordinate generally requires an it- erative solution approach. In addition to a three-dimensional iterative solver, a simplified approach has been devised allowing the use of a direct solver for the new dynamical core that separates a three- dimensional elliptic boundary value problem into a set of two-dimensional independent Helmholtz problems. The issue of dynamics–physics coupling has also been studied, and incorporating the physics tendencies within the discretized dynamical equations is found to be the most acceptable approach for the height-based vertical coordinate. The new dynamical core is evaluated using numerical experiments that include two-dimensional nonhydrostatic theoretical cases as well as 25-km resolution global fore- casts. For a wide range of horizontal grid resolutions—from a few meters to up to 25 km—the results from the direct solution approach are found to be equivalent to the iterative approach for the new dynamical core. Furthermore, results from the different numerical experiments confirm that the new height-based dynamical core is equivalent to the existing pressure-based core in terms of solution accuracy. 1. Introduction 2014). The existing pressure-type coordinate system then permits the use of a direct solver for the discretized The dynamical core of the Global Environmental Mul- EBV problem to resolve the dynamical component of tiscale (GEM) model, used operationally by Environ- the flow. The direct solver starts by separating the EBV ment and Climate Change Canada (ECCC) for numerical problem vertically in terms of the vertical eigenvec- weather prediction (NWP), employs a log-hydrostatic- tors of the part of the coefficient matrix that only in- pressure-type terrain-following vertical coordinate. The cludes the discretized difference and average operators system of nonlinear model equations is linearized in the vertical direction (Qaddouri and Lee 2010). For N around a basic state and is then reduced to an elliptic number of model vertical levels, the resulting N verti- boundary value (EBV) problem through numerical cally decoupled two-dimensional Helmholtz problems discretization and elimination of variables (Girard et al. are then separated along the longitude, leading to a system of tridiagonal problems depending only on the Denotes content that is immediately available upon publica- latitude. The tridiagonal problems are finally solved tion as open access. using LU decomposition without pivoting. Such an ap- proach is computationally more efficient than most it- Corresponding author: Syed Zahid Husain, syed.husain@ erative methods, particularly for the spatial resolutions canada.ca of the current operational NWP systems at ECCC. DOI: 10.1175/MWR-D-18-0438.1 For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/ PUBSReuseLicenses). Unauthenticated | Downloaded 09/25/21 04:19 PM UTC 2556 MONTHLY WEATHER REVIEW VOLUME 147 One of the principal incentives for the adoption of has been found to lead to any meaningful stability im- the existing pressure-type vertical coordinate in GEM provement for steep orography. is the computational advantage of the direct solution Although the instability induced by steep orogra- approach that is permissible with such a coordinate. phy is often characterized as a limitation of the terrain- However, numerical experiments carried out at ECCC following nature of the vertical coordinate itself, have revealed that vertical separability, which is an Smolarkiewicz et al. (2007) were successful in re- imperative for the direct solution approach, can be- solving flow around the Pentagon with a model in- come quite restrictive for very high spatial resolutions volving height-based TFC where the maximum slope (e.g., for subkilometer horizontal grid spacing). Fur- was well above the widely acknowledged 458 threshold. thermore, the reduction of the 2D Helmholtz problems Previous experience with the Mesoscale Compress- into the 1D tridiagonal problems requires projection ible Community model at ECCC also suggests that a of right hand side (rhs) of the 2D problems along dynamical core with height-based TFC does not suf- the pertinent eigenvectors which is based on Fourier fer from similar severe orography-induced instability transformation. This necessitates transposing the (Girard et al. 2005). Apparently, a dynamical core coefficient matrix that involves global communica- with a height-based TFC can lead to improved nu- tions, and therefore, becomes inefficient for very large merical stability through better implicit treatment of number of processor cores. As a result, the direct solver the metric terms arising from the vertical coordinate is found to lose scalability with increasing number transformation through iterative solvers. More impor- of processor cores. Initial research at ECCC as well tantly, conventional numerical approximation of the as published research literature (Müller and Scheichl horizontal gradients in a TFC becomes less accurate 2014) reveal that optimized three-dimensional iterative with increasing terrain slope as well as with increasing solvers may possess better scalability in these circum- vertical resolution close to the model surface (Mahrer stances. The different limitations of the existing di- 1984). In this context, Zängl (2012) argues that the rect solver, particularly its potential lack of scalability pressure gradient term, in particular, becomes suscep- for future generations of massively parallel supercom- tible to triggering numerical instability when the height puters, therefore warrants the development of more difference between two adjacent grid points along a scalable iterative solvers at ECCC. A height-based terrain-following vertical level is much larger than the vertical coordinate is considered to be more amenable vertical grid resolution adjacent to the level. Numeri- to such iterative solvers as the metric terms originating cal approximation of the horizontal gradient terms in from the vertical coordinate transformation appear ex- theTFC,however,canbesignificantly improved fol- plicitly in the discretized EBV. lowing the approach proposed by Mahrer (1984). This Another, but more challenging, problem pertaining approach requires determination of the modified hor- to the existing GEM dynamical core is the fact that the izontal differencing stencils associated with each grid- current model exhibits strong numerical instability over point location that minimize the error in the metric steep orography. Tests conducted at ECCC have de- corrections for the terrain-following nature of the co- termined the maximum permissible terrain slope for ordinate. The existing pressure-type TFC varies with maintaining stability to be approximately 458 (Vionnet time and, therefore, would require determination of et al. 2015). This has been considered to be a limitation the pertinent gridpoint locations in the vertical for the inherent to the terrain-following coordinate (TFC) sys- modified differencing stencils at every time step. On tems (Zängl 2012). With a growing demand for very the contrary, the height-based TFC is time-invariant high-resolution operational NWP systems, steep oro- and thus would require the determination of these graphic slopes are expected to become more preva- gridpoint locations only once at the beginning of the lent in the near future. Improving model stability over time integration. Therefore, from a computational ef- steep mountains is therefore of critical importance for ficiency standpoint, a height-based TFC is more suitable the future development of subkilometer NWP systems. for implementing improved numerical approximation of A number of approaches have been investigated to im- horizontal gradients to address instabilities induced by prove numerical stability with the existing pressure-type steep orography. vertical coordinate in GEM. These include increased The aforementioned challenges associated with the off-centering in the discretized vertical momentum existing log-hydrostatic-pressure-type TFC motivated equation, a vertically variable basic state temperature the development of a new dynamical core for the GEM profile, and modifications to the nonhydrostatic contri- model that utilizes a height-based TFC. The primary butions in the linear system arising from the discretized objective of the present study is to demonstrate that, GEM formulation.
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