Advances in Meteorology

Volume 2016 (2016), Article ID 4513823, 12 pages

http://dx.doi.org/10.1155/2016/4513823

## An MCV Nonhydrostatic Atmospheric Model with Height-Based Terrain following Coordinate: Tests of Waves over Steep Mountains

^{1}National Meteorological Center, China Meteorological Administration, Beijing 10086, China^{2}Center of Numerical Weather Prediction, China Meteorological Administration, Beijing 10086, China^{3}State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100086, China^{4}Department of Energy Sciences, Tokyo Institute of Technology, Yokohama 226-8502, Japan^{5}School of Human Settlement and Civil Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Received 20 November 2015; Accepted 4 February 2016

Academic Editor: Xiao-Ming Hu

Copyright © 2016 Xingliang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A nonhydrostatic atmospheric model was tested with the mountain waves over various bell-shaped mountains. The model is recently proposed by using the MCV (multimoment constrained finite volume) schemes with the height-based terrain following coordinate representing the topography. As discussed in our previous work, the model has some appealing features for atmospheric modeling and can be expected as a practical framework of the dynamic cores, which well balances the numerical accuracy and algorithmic complexity. The flows over the mountains of various half widths and heights were simulated with the model. The semianalytic solutions to the mountain waves through the linear theory are used to check the performance of the MCV model. It is revealed that the present model can accurately reproduce various mountain waves including those generated by the mountains with very steep inclination and is very promising for numerically simulating atmospheric flows over complex terrains.

#### 1. Introduction

Mountain weather processes, such as lee waves, rotors, and downslope windstorms, which have great influence on the air quality over complex mountainous terrains [1, 2], involve a wide range scales of air motions and present a challenge to numerical modeling. With the rapid development of computer hardware, it is now possible for the atmospheric models to represent the complexity in topography with the increasing horizontal resolutions and thus provide more accurate predictions for the mountain weather processes. The ability to simulate the atmospheric flows over complex mountainous areas becomes highly demanded for the nonhydrostatic atmospheric numerical models, such as MM5 [3], MC2 [4], COAMPS [5], LM [6], ARPS [7], WRF [8], and GRAPES [9]. Though the significant advancements have been achieved during the past several decades, adequate simulations and predictions of the complex terrain-forced weather processes, for example, mountain waves, still remain an issue unsatisfactorily resolved. Further efforts are still required to develop more reliable dynamic cores with accurate representations of the topographic effects to improve the simulation of atmospheric flows over complex terrains.

Recently, a new nonhydrostatic model was developed by using the multimoment constrained finite volume (MCV) method [10]. Different from the conventional finite volume method, the unknowns (or the degree of freedom (DOFs)) are defined as the values at the solution points distributed within each mesh cell. In contrast to the direct multimoment method [11–18] where the moments are directly used as the predicted unknowns, the MCV formulation [19] updates the DOFs as the point values at the solution points whose time evolution equations are derived by applying a set of constrained conditions imposing on different types of moments, such as volume-integrated average (VIA), point value (PV), and spatial derivative values (DV) and is thus simple, efficient, and easy to implement. Being a high order scheme, more accurate numerical results can be obtained in terms of the equivalent DOF resolution in comparison with the traditional finite volume method even with relatively coarse grid spacing. The rigorous numerical conservation in MCV model is exactly guaranteed by a constraint on the VIA through a finite volume formulation of flux form. Being a new nodal-type high order conservative method, it is much beneficial to compute the metric and source terms in an MCV model, which are always involved in the treatments of spherical geometry in the horizontal direction and coordinate transformation in the vertical direction for topographic effect. The MCV model has some appealing features for atmospheric modeling, such as the rigorous numerical conservation, good computational efficiency, and flexible configuration for solution points, and thus can be expected as a practical framework of the dynamic cores, which well balances the numerical accuracy and algorithmic complexity [10, 20]. The competitive results of the widely adopted benchmark tests can be referred to [10, 20–24].

To deal with the bottom topography, the height-based terrain following coordinate is adopted in our MCV model. Since Phillips’ pioneering work [25], the terrain following coordinates, mainly classified into the pressure-based coordinate and the height-based coordinate, have been widely adopted to represent the underlying mountainous surface in atmospheric models. During the past several decades, the height-based coordinate [26] has got an increasing popularity mainly due to its applicability for both hydrostatic and nonhydrostatic models and computational simplicity. A variational grid generation technique [27] was adapted to mountain wave simulation as well. Modified version of terrain following coordinate has been also proposed [28] to circumvent to some extent the drawbacks of the coordinate in representing steep topography. In this study, a set of benchmark tests of mountain waves generated by a constant background flow over mountains with different steepness, which essentially represent the complex mechanisms involved in mountain weather processes, are examined by the MCV nonhydrostatic model using the height-based terrain following coordinate in order to verify the performance of the MCV model in simulating the mountain weather processes and its potential for the further numerical investigations on the flows in the atmosphere boundary layer and the air quality over complex mountainous terrain. The numerical results are evaluated in comparison with the semianalytical solutions obtained from the linear theory [29], as well as the numerical solutions from other representative models.

The remainder of this paper is organized as follows. In Section 2 the compressible nonhydrostatic atmospheric model using the MCV scheme and the height-based terrain following coordinate is briefly introduced. Section 3 describes the mountain tests and the semianalytical solutions through the linear theory. Section 4 discusses numerical results of various mountain waves by the MCV model. Finally, a short conclusion is given in Section 5.

#### 2. 2D MCV Nonhydrostatic Atmospheric Model

In order to consider complex topography as the bottom boundary of the atmospheric model, a height-based terrain following coordinate is used in this study to map the physical domain to the computational domain through the transformationwhere is the elevation of topography, the altitude of the model top, and .

Using a height-based terrain following coordinate, 2D compressible and nonhydrostatic governing equations for atmospheric dynamics are written in flux form aswherewhere is density, are velocity vector in the physical domain, is the vertical velocity in the transformed coordinates, , and is the Jacobian of transformation.

The thermodynamic variables are split into a reference state and the deviations to improve the accuracy of the numerical model aswhere the reference pressure and the density satisfy the local hydrostatic balance, is the position vector, , and .

The constants used in the simulations are specified as follows. Gravitational acceleration , ideal gas constant for dry air , specific heat at constant pressure , specific heat at constant volume , , reference pressure at the surface , and constant .

The MCV scheme is adopted in this model to solve the governing equations (2). The MCV scheme is a general numerical framework for developing high order numerical models to solve the hyperbolic systems. A major feature, which distinguishes MCV scheme from other conventional numerical schemes, is the local high order spatial reconstruction. For the sake of brevity, we omit the details of the numerical formulation of MCV nonhydrostatic model in the present paper. The fourth-order MCV scheme and the 3rd TVD Runge-Kutta time scheme [30] are adopted in this study. A local Lax-Friedrich approximate Riemann solver [31] is used for computational efficiency. The interested readers are referred to [10] for details.

The boundary conditions on the bottom surface are of crucial importance in atmospheric models, especially for simulations of the waves generated by complex topography. For the test cases studied in this paper, no-flux condition is imposed along the bottom boundary and nonreflecting condition is used for the lateral and the top boundaries.

The no-flux boundary condition requires the velocity field to satisfy the relationwhere is the outward unit normal vector of the bottom surface and is velocity vector on the bottom boundary.

The nonreflecting boundary conditions are realized by a sponge layer along the lateral and top boundaries that relaxes the numerical solution to the prescribed reference. The damping terms are added to the momentum and potential temperature equations aswhere is the relaxation coefficients and is the specified reference state. More details about the strength of the Rayleigh damping and the formulations of semidiscrete no-flux condition are described in [10].

#### 3. Mountain Wave Test Cases

Satomura et al. [32] suggested a set of mountain wave tests to evaluate the capability of atmospheric models in reproducing topographic effects. The isolated bell-shaped bottom mountain to trigger the waves is specified aswhere is the maximum height of the mountain, is the center of physical domain, and is the half width of the mountain.

Six test cases with different mountain height and horizontal width adopted in this study are given in Table 1. In these cases, the mountain slopes (measured by averaged inclination angles ) vary from 0.006 to 45 degrees. The cases of large inclination angles indicate steep mountains and are very challenging for the height-based terrain following coordinate. The initial hydrostatic conditions in these tests are specified in terms of Exner pressure and potential temperature via the hydrostatic relation . Setup of each case will be described in detail in Section 4.