## 1. Introduction

The nonhydrostatic effects are found to be very important in the meso- and synoptic-scale atmospheric dynamics and have been included in regional models for several decades. Among others, some examples of regional nonhydrostatic dynamical models are the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5; Dudhia 1993), Environment Canada’s mesoscale compressible model, version 2 (MC2; Benoit et al. 1997), the U.S. Navy Coupled Ocean–Atmosphere Mesoscale Prediction System (Hodur 1997), the German Weather Service’s local model (LM; Doms and Schattler 1999), the University of Oklahoma’s Advanced Regional Prediction System (ARPS; Xue et al. 2000), the NCAR Weather Research and Forecasting Model (WRF; Skamarock and Klemp 2008), and the China Meteorological Administration’s Global/Regional Assimilation and Prediction Systems (GRAPES; Chen et al. 2008). It is observed as a noticeable trend that finer grid resolutions are available for global models due to the rapid progress in computer hardware, and mesoscale dynamics can be solved on a global base, which requires the global models to be built in the nonhydrostatic framework as well (Putman and Lin 2007; Satoh et al. 2008; Walko and Avissar 2008; Gassmann 2011; Ullrich and Jablonowski 2012a).

Another requirement for an atmospheric model to be able to efficiently use next-generation large-scale supercomputers, which usually consist of tens to hundreds of thousands of processors with distributed-memory nodes, is that the dynamical core should be highly scalable for large-scale parallel processing. Some recent studies demonstrate that the dynamical cores based on high-order schemes with local spectral reconstructions are superior in parallel scalability and overcome the barrier that prevents the spherical-harmonic spectral-transform-based method and finite-volume method from efficient implementations on supercomputers toward exaflops computation (Dennis et al. 2005, 2012). Researches have been so far carried out in such a direction by using the spectral element (SE) method (Thomas and Loft 2000; Iskandarani et al. 2002; Giraldo and Rosmond 2004; Fournier et al. 2004; Taylor and Fournier 2010) or the discontinuous Galerkin (DG) method (Levy et al. 2007; Giraldo and Restelli 2008; Restelli and Giraldo 2009; Nair et al. 2009; Blaise and St-Cyr 2012). The latter can be then further divided into modal type and nodal type (Hesthaven and Warburton 2008), and the nodal type is getting an increase in popularity because of its computational efficiency. As pointed by these authors, the high-order accuracy, geometric flexibility, and the scalability on massively parallel computer systems make this class of local reconstruction-based methods quite worthy of further exploration as the potential numerical frameworks of new-generation atmospheric and oceanic models. Meanwhile, it is also noted that the existing high-order spectral convergence methods are more computationally expensive due to for example the numerical quadrature in the Galerkin formulation.

Another type of high-order schemes can be devised by using the multimoment concept (Yabe and Aoki 1991; Yabe et al. 2001; Xiao 2004; Xiao et al. 2006; Ii et al. 2005; Xiao and Ii 2007; Ii and Xiao 2007; Chen and Xiao 2008), where we make use of different kinds of discrete quantities, so-called *moments* in out context, to describe the physical field, such as the pointwise value, derivatives, and volume-integrated value. These moments are locally defined over each mesh cell, which allows us to build high-order local reconstructions. Different moments can be then updated by different formulations that may have different forms but should be consistent with the original conservation law. For example, the point value can be updated by a pointwise Riemann solver or semi-implicit semi-Lagrangian (SISL) solver, while the volume-integrated average value is computed by a finite-volume formulation to ensure the rigorous numerical conservation. In our previous studies, two global multimoment finite-volume shallow-water models have been reported using either a pointwise Riemann solver (Chen and Xiao 2008) or the SISL method (Li et al. 2008). In these schemes, the moments (i.e., cell-integrated average), which point to value and derivative, are directly treated as the prognostic variables.

A more efficient alternative to the multimoment finite-volume formulation is to define the unknowns [or the degrees of freedom (DOFs)] as the values at the points located within each grid cell and to use the time evolution equations of different moments as the constraint conditions to derive the governing equations for updating the unknown point values. The resulting scheme is so-called multimoment constrained finite-volume (MCV) method (Ii and Xiao 2009) where the moments are not directly used as the predicted unknowns (DOFs), but the constraint conditions. The numerical conservation is exactly guaranteed through a constraint on the cell-integrated average, which is cast in a finite-volume formulation of flux form. In the present multimoment constrained method, all predicted unknowns are the nodal values at the solution points and the volume integration is not involved for a conservation law, which makes the numerical formulation very efficient, especially when the physical source term and metric term are included in the governing equations. Our experiences also show that an MCV scheme can use a larger Courant–Friedrichs–Lewy (CFL) number for computational stability compared to other high-order schemes of the same order, and the location of the solution points can be determined in a more flexible manner. We have recently implemented the MCV method to develop global shallow-water models on an icosahedral grid (Ii and Xiao 2010), hexagonal geodesic grid (Chen et al. 2012), and the Yin–Yang grid (Li et al. 2012), and demonstrated its potential as the fundamental numerics for a new type of dynamical cores.

In this paper, we implement for first time the third- and fourth-order MCV method to construct the nonhydrostatic compressible atmospheric dynamical core. The basic numerical formulations are straightforwardly derived by applying the numerical schemes presented in Ii and Xiao (2009) to the dynamic equations of the compressible and nonhydrostatic atmosphere. The remaining paper consists of the following parts. Section 2 describes the 2D compressible governing equations and their alternative form related to the hyperbolicity. The dynamical core based on the third- and fourth-order MCV schemes is presented in section 3. In section 4, the numerical results of some standard benchmark tests are reported to evaluate the presented model in both accuracy and efficiency. Finally, we end the paper with some conclusions and future goals in section 5.

## 2. The governing equations for compressible nonhydrostatic dynamical core

*ρ*is the density,

**u**= (

*u*,

*w*)

^{T}the vector wind in the Cartesian coordinate,

*p*is the pressure, and

*θ*is the potential temperature. Since the potential temperature is related to the air temperature

*T*and pressure

*p*by

*p*=

*C*

_{0}(

*ρθ*)

^{γ}, which exactly closes the equation set where

*C*

_{0}is constant given by

*γ*=

*c*/

_{p}*c*= 1.4,

_{υ}*R*= 287 J kg

_{d}^{−1}K

^{−1},

*c*= 1004.5 J kg

_{p}^{−1}K

^{−1},

*c*= 717.5 J kg

_{υ}^{−1}K

^{−1}, and

*p*

_{0}= 10

^{5}Pa.

### a. Splitting of reference state

*z*)]. The thermodynamic variables are then written aswhere the reference pressure

*p*′ =

*ε*

_{0}(

*ρθ*)′ and

The horizontal mean fields of the thermodynamic variables in the reference state are in hydrostatic balance and make no contribution to the dynamical processes. The deviations of the thermodynamic variables from their mean fields play roles in the dynamic processes. Subtracting the reference state from the governing equations and using the deviations of thermodynamic variables as the computational variables make the system more suitable for numerical procedures, and thus can be interpreted as a preconditioner that effectively enhances the computational stability and efficiency.

### b. Governing equations with the effects of topography

In the presence of topography, the height-based terrain-following coordinate introduced by Gal-Chen and Somerville (1975) is utilized to map the physical space (*x*, *z*) into the computational domain (*x*, *ζ*) via the transformation relationship *ζ* = *ζ*(*x*, *z*).

*φ*denotes an arbitrary field variable. The relation between the vertical components of velocity in

*z*coordinate and

*ζ*coordinate isIt is noted that the governing Eqs. (13)–(16) in the absence of topography will reduce to Eqs. (9)–(12).

### c. Flux Jacobian matrix

*x*and

*ζ*directions, and

**f**and

**g**, and which are computed, respectively, byand

_{x}

**Λ**

_{x}

_{x}and

_{ζ}

**Λ**

_{ζ}

_{ζ}). Some algebraic manipulations yield

**Λ**

_{x}= diag(

*u*,

*u*,

*u*−

*a*,

_{u}*u*+

*a*) and

_{u}**Λ**

_{ζ}= diag(

*w*,

*w*,

*w*−

*a*,

_{w}*w*+

*a*), which are the characteristic speeds of the system. The transformation matrices

_{w}_{x}and

_{ζ}are constructed from the right eigenvectors of

*x*and

*ζ*directions, respectively. In the following numerical formulations, we make use of the approximate Riemann solver, which requires the eigenvalues obtained above.

## 3. Multimoment constrained formulations

### a. Spatial discretization

*x*,

*z*) to the computational domain (

*x*,

*ζ*) through a transformation (Schär et al. 2002):where

*z*(

_{s}*x*) is the elevation of topography,

*z*is the altitude of the model top level, and

_{T}*s*is the scale height.

_{ij}) as a structured mesh where the mesh elements are numbered by

*i*and

*j*, respectively, in

*x*and

*ζ*directions, such thatwhere the mesh cell is defined as

_{ij}= [

*x*

_{i}_{−1/2},

*x*

_{i}_{+1/2}] ⊗ [

*ζ*

_{j}_{−1/2},

*ζ*

_{j}_{+1/2}], and

*I*and

*J*are the total numbers of mesh cells in the

*x*and

*ζ*directions.

*L*th-order scheme, the DOFs in

_{ij}are denoted by

*q*and defined at points (

_{ijlm}*x*,

_{ijl}*ζ*) with

_{ijm}*l*and

*m*= 1, …,

*L*, where

*l*denotes the index of solution points in the

*x*direction and

*m*in the

*ζ*direction. The total number of DOFs used in an

*L*th-order MCV scheme is [

*I*× (

*L*− 1) + 1] × [

*J*× (

*L*− 1) + 1]. Our experience shows that in an MCV scheme the numerical result is not sensitive to the location of the solution points. No significant difference is observed in the solutions between the Gauss-point and the equidistance-point configurations. We use the equidistance-point configuration including the cell boundary points in the present model, which is simpler and easy to implement for real applications. In this case, the point value at a boundary point is shared by the adjacent cells and the physical field is

*C*

^{0}continuous. For simplicity, we drop the cell indices

*i*and

*j*from now on and focus our discussions on the local control volume where the solution points are equally spaced over [

*x*

_{i}_{−1/2},

*x*

_{i}_{+1/2}] and [

*ζ*

_{j}_{−1/2},

*ζ*

_{j}_{+1/2}], respectively, by

*x*direction along the

*m*th line segment

*q*and

*f*represent any component of the conservative variables

**q**and the corresponding component of the flux function

**f**, respectively. The same procedure applies in the

*ζ*direction along the

*l*th line segment

*x*with

*ζ*and flux component

*f*with

*g*for the

*ζ*direction. The numerical details presented below can be straightforwardly extended to the system conservation laws in Eq. (20).

*q*as the following:where Δ

*x*=

_{i}*x*

_{i}_{+1/2}−

*x*

_{i}_{−1/2}and

*x*

_{cpm}represents a constraint point within or at the two ends of line segment

#### 1) Local reconstruction

*φ*,

_{lm}*l*= 1, 2, … ,

*L*, at the solution points

*x*

_{1}to

*x*, a one-dimensional Lagrange interpolation polynomial of degree (

_{L}*L*− 1) is constructed aswhere the Lagrange basis function isThe

*φ*denotes either the conservative state variables

*q*or the flux component of

**f**and

**g**.

#### 2) Multimoment constraints

As addressed by Ii and Xiao (2009), high-order schemes can be constructed by properly choosing multimoment constraints derived from the governing equation in Eq. (28). We summarize the constraint conditions used in this paper as follows:

- C1)The LIA moment over line segment
, which is cast in a finite-volume formulation,where and are the numerical flux at the two ends of segment . Because in the present schemes the segment ends are also the solution points where the solution is readily updated, numerical fluxes can be immediately computed. For example, the flux component *ρu*of the continuity equation at the left boundary point can be directly computed by, where *ρ*_{1m}and*u*_{1m}are the solutions at the left boundary point of the cell, which are updated every time step and shared by the two neighboring cells. The finite-volume formulation in Eq. (34) assures the rigorous conservation in numerical solution. - C2)Being other constraints, the PV-moment values at the two ends of segment
are predicted by a collocation formulation of Eq. (28),where and are the derivatives of the flux function, which need to be computed from the approximate Riemann solver discussed later. - C3)In the fourth-order scheme, an extra constraint condition is imposed on the first-order DV moment at the segment center. (Note that we denote
*x*_{cpm}as the constraint point at the line segment center by*x*, and the first-order derivative_{cm}by *q*hereafter.) Its evolution equation is obtained by differentiating the governing equation in Eq. (28) with respect to_{x}*x*and collocate the resultant equation at cell center, which yieldswhere the second-order derivative value of the flux functionat the segment center is computed from the forth-order approximation:and the line segment center value *f*is obtained from the Lagrange interpolation in Eq. (32)._{cm}

#### 3) Third-order MCV scheme

*L*= 3), three solution points over line segment

*m*here), where the solutions

*q*,

_{lm}*l*= 1, 2, 3, are computed. From the constraints C1 and C2 as well as the Lagrange interpolation in Eq. (32), we get the following semidiscrete equations to update the unknowns for

*q*

_{2m}is derived by replacing the LIA moment in Eq. (34) with the nodal solutions

*q*

_{1m},

*q*

_{2m}, and

*q*

_{3m}, using the LIA moment definition in Eq. (29) and the Lagrange interpolation in Eq. (32).

_{3βm}, we rewrite Eq. (38) into a component form:

*ζ*direction:and

#### 4) Fourth-order MCV scheme

*L*= 4), the unknowns

*q*,

_{lm}*l*= 1, 2, 3, 4, over line segment

_{4βm}, we get the componentwise expression as

_{4lβ}are the entries of the following matrix and vector generated from the spatial discretization in

*ζ*direction:

### b. Approximate Riemann solver

In the above discussions we are left with the need to find the numerical flux and its derivative at the cell boundaries. As part of the solution points are at the cell boundaries, the numerical fluxes

*x*direction over

*x*

_{bpm}for either

*x*

_{1}or

*x*, where the piecewise reconstructed polynomial in Eq. (32) may lead to two different values of the derivatives of the state variable on the left and right sides of

_{L}*x*

_{bpm}denoted by

*x*

_{bpm}. For computational efficiency, a local Lax–Friedrich (LLF) approximate Riemann solver (Shu and Osher 1988) is adopted in this paper, which reads aswhere

*x*

_{bpm}.

^{−}(

*φ*:

*x*) denotes the interpolation function over the left-hand-side cell and Ψ

^{+}(

*φ*:

*x*) the right-hand-side cell of boundary point

*x*

_{bpm}. The numerical procedure described above applies to the

*ζ*direction as well.

### c. Numerical formulation for dynamic viscosity

Although we start with an inviscid form of the governing equations in Eq. (1), the computation of dynamical viscosity term, **∇** · (*μρ***∇***φ _{lm}*), will be involved when the physical viscosity, as in the Navier–Stokes equations, has an effect. It is always the case that some numerical viscosity might be necessary as a filter for high-order schemes where the numerical dissipation is suppressed and noises of small scales do not diminish. In such a case, viscous or dissipative terms are added to the right-hand side of the momentum equations and the energy equation for a given viscosity coefficient

*μ*. Here

*φ*denotes the components of the wind (

_{lm}*u*,

*w*) or the potential temperature

*θ*.

*x*direction, for example,for MCV3 andfor MCV4. The same applies to the

*ζ*direction.

It should be noted that in the formulation of the MCV schemes (at least those used in the present study) the cell boundary values are shared by the neighboring cells, which are uniquely updated by the derivative Riemann solver. The cell boundary values connect the neighboring cells and maintain at least the *C*^{0} continuity for the physical field, which allows the above formula for second-order derivative to work properly without other special treatment, like the local discontinuous Galerkin (LDG) method (Cockburn and Shu 1998).

As mentioned before, the LLF Riemann solver in terms of the derivative of the flux function is utilized in MCV method, which stabilizes the computation through an effective numerical viscosity. However, as observed in other high-order schemes, extra dissipation is required in some cases as will be reported in section 4.

### d. Boundary conditions

We used different boundary conditions for the test cases presented in this paper. The no-flux condition is used along the bottom boundary. According to different test cases, periodical, no-flux and nonreflecting conditions are used, respectively, for the lateral boundaries. We use either no-flux or nonreflecting condition for the top boundary. Next, we describe how the no-flux and nonreflecting boundary conditions are implemented in the MCV schemes. Some representative procedures under other frameworks can be also found in the literature (Durran and Klemp 1983; Giraldo and Restelli 2008; Ullrich and Jablonowski 2012b).

#### 1) No-flux boundary conditions

**n**is the normal direction of the boundary and

**u**= (

*u*,

*w*)

^{T}is the wind field. It can be also interpreted as the slip boundary condition, and is used in all test cases in this paper.

*j*= 1 are boundary cells

*C*

_{i}_{1}and their bottoms coincide with grid line

*ζ*

_{1/2}. One layer of ghost cells

*C*

_{i}_{0}is placed beneath

*ζ*

_{1/2}. Given the values at the solution points of boundary cell

*C*

_{i}_{1},

*q*

_{i}_{1lm},

*m*= 1, 2, … ,

*M*with

*M*= 3 for the third-order scheme and

*M*= 4 for the fourth-order scheme, the above no-flux boundary condition can be realized by mapping the boundary cell values to the corresponding points in the ghost cell as follows:The above mapping ensures that there is not fluid motion across the bottom boundary. The values in the ghost cells are updated at every time step.

#### 2) Nonreflecting boundary conditions

*τ*is the relaxation coefficients and

**q**

_{b}represent some specified background fields such as the mean velocity or the reference state of the thermal-dynamic variables. The strength of the Rayleigh damping

*τ*varies over a finite interval in the interior of computational domain. Following the existing works (Giraldo and Restelli 2008; Ullrich and Jablonowski 2012b), the relaxation coefficients is defined aswhere

*s*∈ (

*x*,

*ζ*) indicates a location in computational domain, and

*s*

_{0}means the location of the boundary. The term

*s*denotes the thickness of the absorbing layer and

_{T}*τ*

_{0}is the inverse time scale of the damping. Different values for the maximum coefficients are adopted when two absorbing boundary layers, such as the upper boundary and the lateral boundary, overlap (Ullrich and Jablonowski 2012b).

### e. Time integration

*q*), we summarize the semidiscrete formulations as

_{lm}*n*, we use the following multistep updating to obtain the values

*n*+ 1:

It should be noted that a multistep time integration scheme might not be the final choice in the context of parallel processing. Although the local reconstruction used in an MCV scheme effectively suppresses data communication, using a time marching procedure with fewer substeps will definitely improve the parallel efficiency.

As a matter of fact, we have implemented other time integration approaches under the framework of the multimoment method, such as the semi-Lagrangian method based on characteristics (Ii and Xiao 2007) and the conservative semi-implicit semi-Lagrangian method (Li et al. 2008), which can be implemented to the MCV schemes in a straightforward manner. For further exploration, we do not see substantial difficulties to use other alternatives for time integration (e.g., Klemp et al. 2007; Restelli and Giraldo 2009; Ullrich and Jablonowski 2012b; Dumbser et al. 2008), for the MCV formulations.

### f. Some remarks

Before ending this section, we want to make some brief remarks on the important features of the MCV schemes.

- In the current MCV schemes, the solutions at the cell boundaries are shared by the neighboring cells. So, the DOFs of solutions required in the third-order MCV scheme is 2
*I*+ 1 for a 1D computational domain of*I*cells, and 3*I*+ 1 are required in the fourth-order schemes. One derivative Riemann solver is required to compute the solution (point value) at each cell boundary for both third- and fourth-order schemes. Thus, in the present formulation we need to solve the Riemann problem for*I*− 1 times in 1D. Given the solution at each cell boundary updated every time step, the numerical flux needed in the constraint on the VIA (or LIA in 1D) moment can be directly computed without the use of Riemann solver. It should be also noted that in some higher-order MCV schemes, as shown in Ii and Xiao (2009) for example, higher-order derivative Riemann problem might be involved, which increases the use of Riemann solvers. - We have experimented to replace the equidistance solution points by Gauss–Chebyshev–(Lobatto) points for both MCV3 and MCV4 schemes and examined the spectra of the semidiscrete equations, which eventually determine the stable CFL numbers of the corresponding schemes. It is found that the constraint conditions in an MCV scheme substantially affect the property of the numerical schemes, such as accuracy and stable CFL number, while the location of the solution points does not play an important role. It reveals that the solution points can be located within each cell in a more flexible manner.
- As mentioned above, the solution points can be more flexibly chosen in an MCV scheme, not limited to the Gauss points or Gauss–Lobatto points as used in the spectral element method or the nodal DG method. We use equally spaced solution points in the present MCV formulations, which simplifies the formula for updating the unknown DOFs (see matrices
_{3}and_{4}given before) and is more attractive for real applications. - All DOFs to be updated are nodal values, which particularly benefits the computations of source terms and metric terms. Moreover, the equations to update the nodal values are derived from a set of constraint conditions without the explicit involvement of the inner product in a weak-form formulation like the Galerkin method. It should also be noted that the constraint conditions can be chosen in view of not only numerical accuracy and efficiency but also underlying physics.
- The largest allowable CFL number for computational stability for 1D advection test was estimated in Ii and Xiao (2009), which reveals that the MCV schemes can use larger CFL number in comparison with other high-order schemes of the same order reported in Zhang and Shu (2005). For example, the largest stable CFL number for the third-order MCV scheme with a third-order Runge–Kutta method is 0.4 in terms of the cell width, while those are 0.145 for the discontinuous Galerkin method (Cockburn and Shu 1989) and 0.209 for the spectral volume method (Wang 2002).

We also measured the stable CFL conditions of the atmospheric core in this paper by examining the largest stable CFL number defined by max_{Ω}[(*U* + *a*)_{ij}/Δ*d _{ij}*] with

*d*= Δ

_{ij}*x*/(

_{i}*L*− 1) = Δ

*ζ*/(

_{j}*L*− 1). The third-order TVD Runge–Kutta time integration scheme is used. The largest stable CFL number for the core using the third-order scheme is 0.52, while that for the fourth-order scheme is 0.5. It shows that the stable CFL number remains nearly unchanged (i.e., the time stepping is linearly in proportion to the interval of the neighboring solution points), which to some extent is appealing for the use of uniformly located solution points.

## 4. Numerical results

We present in this section some standard and widely used benchmark tests including the rising convective thermal bubble, density current, internal gravity waves, and Schär mountain test in the nonhydrostatic atmospheric scope to validate the present compressible nonhydrostatic atmospheric dynamical core in comparison with other existing ones.

*θ*,where

*θ*

_{0}is constant. Thus, the conversion from

### a. A rising convective thermal bubble

*R*is the radius of the bubble. In this test case, Δ

*θ*= 2 K,

*R*= 2000 m, and (

*x*

_{0},

*z*

_{0}) = (10 000 m, 2000 m). The simulation is run for 1000 s on a domain of [0, 20] × [0, 10] km. The boundary conditions are no-flux along all boundaries.

Figure 2 shows the contours of potential temperature perturbations, horizontal wind, and vertical wind computed by the third-order MCV scheme with an equidistance grid spacing of 125 m. As shown in Fig. 2a, sharp gradient of potential temperature perturbations are created in the upper part of the thermal bubble since the center of convective thermal bubble rises faster than anywhere else inside the bubble because of the distribution of the potential temperature perturbation. An explicit dissipation filter in Eq. (54) is used with a viscosity coefficient of *μ* = 10 m^{2} s^{−1}. The dissipation filter effectively eliminates the noises (oscillations) of small scale in both potential temperature and velocity fields. The numerical dissipation in the present model is adaptively controllable, which is different from other methods where numerical dissipation is embedded in the reconstruction procedure, such as Ahmad and Lindeman (2007) and Norman et al. (2011). A proper limiting projection to enforce the monotonicity when implementing an MCV scheme for atmospheric modeling should be another interesting topic worthy of further exploration.

The fine structures in the numerical solutions are resolved. It is observed that the symmetric structures of both potential temperature and wind field as in Fig. 2 are perfectly reproduced similar to previous studies such as Ahmad and Lindeman (2007) and Norman et al. (2011), although different contour intervals are used here.

Figure 3 shows the numerical results of the fourth-order MCV scheme on a coarser grid spacing of 200 m, which corresponds to the results given in Fig. 2 with regard of the total number of DOFs. The numerical dissipation filter in Eq. (55) was used in this example with *μ* = 10 m^{2} s^{−1}. Again, the symmetric structures of the physical fields are perfectly simulated. As shown in Fig. 3, the structures in both potential temperature and velocity fields look more intensive compared to those from the third-order MCV scheme.

We have also run simulations without any dissipation filter using different grid resolutions. We doubly refined the grid spacing for the third-order MCV scheme to 62.5 m and the fourth-order MCV scheme to 100 m, respectively, which requires approximately the same number of DOFs for both schemes. As shown in Figs. 4 and 5, refining the grid resolution effectively reduces the numerical oscillations of small scales. This observation agrees well with the common understanding that a refined grid can resolve better structures of smaller scale, which are more likely to remain in a less dissipative model of higher-order convergence. It reveals that the simulation results on a low-resolution grid adequately reproduce the converged solutions in the present model.

In a qualitative manner, the time histories of the maximum of potential temperature perturbations and vertical velocity in a series of grid refinement experiments without the explicit dissipation filter are plotted in Fig. 6. Simulations were run with gradually refined resolutions for the third-order and fourth-order MCV schemes where the finest grid spacings are 62.5 m for MCV3 and 100 m for MCV4 schemes. In the results of high-resolution simulations, the convexities of vertical velocity during [600, 900] s are observed, and the maximum vertical velocity appears around 800 s. It agrees well with other existing studies (e.g., Ahmad and Lindeman 2007; Norman et al. 2011).

### b. Density current

*θ*= −15 K, (

*x*

_{0},

*z*

_{0}) = (0, 3000 m), and (

*x*,

_{r}*z*) = (4000, 2000 m). The simulation is run for 900 s on a domain of [−26.5, 26.5] km × [0, 6.4] km. Note that the potential temperature perturbation is adopted similar to Giraldo and Restelli (2008), which is a little different from the original temperature perturbation defined by Straka et al. (1993).

_{r}The boundary conditions for all four boundaries are no flux. Being a requirement of the physical process, dynamic viscosity and diffusion are added to the momentum and potential temperature equations with the dissipation coefficient of *μ* = 75 m^{2} s^{−1} in this test case. The second-order derivative terms are computed by the numerical formulation discussed in section 3c.

Figure 7 indicates the potential temperature perturbation contours after 900 s for 400-, 200-, 100-, 50-, and 25-m grid spacings using the third-order MCV scheme. It is observed that two of the three Kelvin–Helmholtz rotors are reproduced at the very coarse resolution of 400 m. The second rotor comes forth when the grid resolution becomes finer than 200 m. The solutions on grids finer than 100 m converge with the vortical structures adequately simulated.

Figure 8 shows the potential temperature perturbation contours after 900 s for 600-, 300-, 150-, 75-, and 37.5-m grid spacings computed by the fourth-order MCV scheme. With the same number in the DOFs of third-order MCV scheme on a 400-m grid, the fourth-order MCV method on a coarser grid of 600 m reproduced the three rotors. The numerical solutions converge when the grid spacing is finer than 300 m, and the solutions on 150- and 75-m grids are visually identical to the converged results shown as those of the 25-m grid for MCV3 and the 37.5-m grid for MCV4. It reveals that a higher-order scheme converges with the numerical solution more rapidly with the same DOFs.

The profiles of the potential temperature perturbation along the horizontal direction at the height of *z* = 1200 m are plotted in Fig. 9 for both third-order and fourth-order MCV schemes. Figure 9a shows the results from the highest grid resolutions for two schemes: 25 and 37.5 m for the MCV3 and MCV4 schemes, respectively. Both solutions are well converged and no noticeable difference is seen between them. It is observed that three valleys exist, which correspond to the three distinct rotors shown in Figs. 7 and 8. Compared with the results of other high-order numerical methods such as the spectral element and discontinuous Galerkin, our results agree well with that of DG3 (see Fig. 8a in Giraldo and Restelli 2008) with competitive performance. Figure 9b shows the numerical results from different grid resolutions computed by the third-order MCV scheme. It is indicated that at coarse resolution such as 400 and 200 m the potential temperature profiles vary significantly. However, there are not much noticeable changes for the potential temperature profiles on high-resolution grids finer than 100 m. With the equivalent numbers of DOFs for different grid resolutions, the numerical results by the fourth-order MCV scheme are shown in Fig. 9c, which reveals an adequate convergence when the grid resolution is finer than 300 m. Again it indicates a better convergence of the fourth-order method, which is competitive with the existing high-order schemes, for example, the DG method as shown in Fig. 8b of Giraldo and Restelli (2008).

Table 1 shows the comparisons of the front location, maximum–minimum of potential temperature, and density perturbations between the numerical results from the third- and fourth-order schemes using the same fine resolution in terms of equivalent number of DOFs. It is evident that the two MCV schemes with high-resolution grids give the converged solution. We have obtained −9.06-K minimum potential temperature perturbation in comparison with that of −9.08 K in DG3 and SE3 [see Table 5 in Giraldo and Restelli (2008)] where the same initial conditions of potential temperature perturbation are utilized.

Comparison of the numerical results between the third- and fourth-order MCV schemes for the density current test. The total number of DOFs is equivalent to that of a conventional finite-volume or finite-difference scheme with a 12.5-m grid spacing. Front location is defined by the location of −1-K contour value of potential temperature perturbation.

### c. Internal gravity waves

*N*

_{0}= 10

^{−2}s

^{−1}to admit IGWs. A potential temperature perturbation is added to the basic potential temperature field as follows:where

*H*= 10 km, Δ

*θ*= 0.01 K,

*x*

_{0}= 100 km,

*a*= 5 km, and

*θ*

_{0}= 300 K. The initial state of the atmosphere is assumed to ride on a constant mean flow of

Figure 10 shows the contour plots of the potential temperature perturbation. The same contouring interval such as Skamarock and Klemp (1994) and Giraldo and Restelli (2008) is adopted for the convenience of comparison. The results of MCV schemes look quite similar with those in Skamarock and Klemp (1994) and Giraldo and Restelli (2008), as well as other existing studies (e.g., Ahmad and Lindeman 2007; Norman et al. 2011). It is noted that the error norms measured by Skamarock and Klemp (1994) cannot be adopted here since the analytic solution is available only for the Boussinesq equations but not for the fully compressible equations.

Table 2 shows the maximum and minimum of vertical velocities and potential temperature perturbations for third- and fourth-order MCV schemes after 3000 s. It can be seen that the third- and fourth-order schemes obtained almost the same numerical outputs though different grid spacings are used here. Specifically, as mentioned by Giraldo and Restelli (2008), the ranges of potential temperature perturbations are *θ*′ ∈ [−1.49 × 10^{−3}, 2.82 × 10^{−3}] in the numerical results of the model based on the technique of flux-based wave decomposition in Ahmad and Lindeman (2007) and *θ*′ ∈ [−1.51 × 10^{−3}, 2.78 × 10^{−3}] from models based on the spectral element and discontinuous Galerkin in Giraldo and Restelli (2008), while our results are *θ*′ ∈ [−1.52 × 10^{−3}, 2.80 × 10^{−3}] as shown in Table 2. It is observed that these numerical results agree well with each other.

Comparison of the numerical results between the third-order scheme on a 100-m grid and the fourth-order scheme on a 125-m grid for the internal-gravity wave test.

Figure 11 gives the profiles of potential temperature perturbations along the horizontal cross section *z* = 5 km after 3000 using the three-order and fourth-order MCV schemes. It is observed that the profiles are perfectly symmetric about the position *x* = 16 km. Compared with the results of the model using fifth-order weighted essentially nonoscillatory (WENO; Norman et al. 2011), the numerical solution of competitive quality are obtained by utilizing the equivalent DOF resolution with the third-order MCV scheme, as shown in Fig. 11a. The fourth-order MCV scheme, with a little increase in the equivalent DOF resolution, results in significantly improved numerical outputs as shown in Fig. 11b. In addition, the fourth-order MCV scheme also has competitive numerical results with the same grid spacing of 250 m compared to those using the schemes of spectral element and discontinuous Galerkin methods [see Fig. 2b in Giraldo and Restelli (2008)].

### d. Schär mountain

*h*

_{0}= 250 m,

*a*

_{0}= 5000 m, and

*λ*

_{0}= 4000 m. A plot of the mountain profile is shown in Fig. 12. As mentioned above, the hybrid vertical coordinate (Schär et al. 2002) is adopted. Following the configuration setup of Schär et al. (2002),

*s*= 3 km is specified in Eq. (26).

The basic state of the atmosphere has a constant mean flow of ^{−1} and a uniform stratification with a constant Brunt–Väisälä frequency of *N*_{0} = 0.01 s^{−1}. The reference potential temperature is given by Eq. (65) with *θ*_{0} = 280 K.

As described in Schär et al. (2002), gravity waves of different scales are generated by the mountain in the lower atmosphere. The larger-scale hydrostatic waves propagate vertically through the whole domain, while the smaller-scale waves decay rapidly with height due to the nonhydrostatic effects. The mountain wave consists of different scales and has more complex structures in low atmosphere. For the Schär mountain test, the semi-analytic solutions based on linear theory can be found by a Fourier transformation (Smith 1979, 1980).

The computational domain is [−25 000, 25 000] m × [0, 21 000] m with the grid spacings of Δ*x* = 250 m in the *x* direction and Δ*ζ* = 210 m in the *ζ* direction. No-flux boundary conditions are used for the bottom boundary, and nonreflecting boundary conditions are imposed by placing sponge layers in the regions of *ζ* ≥ 9000 m for the top boundary and

Figures 13 and 14 show the simulated results of horizontal and vertical wind speeds after 10 h by MCV3 and MCV4 schemes, respectively, against the semi-analytical solution. In comparison with the results of other numerical models (Klemp et al. 2003; Giraldo and Restelli 2008; Ullrich and Jablonowski 2012b; Simmaro and Hortal 2012), the results of the present model look very competitive.

For further comparison, the root-mean-square (RMS) errors against the semi-analytical solution are examined and compared with the results reported in Giraldo and Restelli (2008). Table 3 gives the RMS errors for different physical variables (Π, *u*, *w*, *θ*) in our simulations. Compared with the outputs of high-order numerical methods of local reconstructions, such as the SE method and the DG method reported in Giraldo and Restelli (2008, see Table 6 therein), the RMS errors of our results are smaller in this test case. Moreover, other smoother vertical coordinates such as Schär et al. (2002) and Klemp (2011) can be adopted in future to improve the accuracy. It is also observed that although the MCV4 scheme gives less diffusive results compared to the MCV3 scheme, but it does not show a superiority in numerical accuracy in this particular case. Leaving the lucid explanation to further investigation, we think that part of the reason may be due to the boundary conditions and the vertical coordinate transformation currently used, which might not be necessarily favorable for the high-order schemes.

Root-mean-square errors of the Schär mountain for different physical fields after 10 h for 250 m (*x*) and 210 m (*ζ*) resolution using the MCV3 and MCV4 schemes. The units for variables Π, *u*, *w*, and *θ* are dimensionless, m s^{−1}, m s^{−1}, and K, respectively.

## 5. Conclusions

We have presented a 2D nonhydrostatic compressible atmospheric core by using the third- and fourth-order multimoment constrained finite-volume (MCV) schemes. The unknown DOFs are the point values defined at equally spaced solution points within each mesh cell. High-order reconstructions are conducted at the cell level through the point values by 1D Lagrange interpolations in structured grids. The time evolution equations for updating the unknowns are derived from a set of constraint conditions in terms of multimoments such as volume-integrated average (VIA), point value (PV), and spatial derivative values (DV). There is no any numerical quadrature explicitly involved in the present formulations, and the solution points can be flexibly located within each mesh element, which makes the resulting models very efficient and particularly attractive when dealing with the source terms of physical processes in atmospheric models.

The numerical results of widely used standard benchmark tests including the topographic effects show that the present dynamical core can produce numerical solutions of good quality comparable to other high-order schemes as expected. The numerical formulations based on multimoment constraints, as presented in this paper, are exactly conservative and have significant advantages in algorithmic simplicity, flexibility, and computational efficiency.

As the first implementation of MCV schemes in developing nonhydrostatic compressible dynamical cores, we have very promising results from the present study. Further research will be continued to develop dynamical cores using the same methodology for the global atmosphere where the spherical geometry is another key issue, which has been extensively investigated in our previous researches with promising outputs as well.

## Acknowledgments

This work is supported by the National Key Technology R&D Program of China (Grant 2012BAC22B01), the National Natural Science Foundation of China (Grants 10902116 and 40805045), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant 41221064), and special research of public service sectors (meteorological) of China (Grant GYHY201006013). We gratefully acknowledge Professor F. X. Giraldo for providing the semi-analytic solution of the Schär mountain test for comparison. We also thank the anonymous reviewers for their constructive suggestions.

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