## 1. Introduction

Semi-Lagrangian semi-implicit (SLSI) schemes have been widely used in climate and numerical weather prediction (NWP) models since the pioneering work of Robert (1981) and Robert et al. (1985). The more lenient numerical stability condition in these schemes allows larger time steps and thus increased computational efficiency. Traditional semi-Lagrangian schemes are not inherently mass conserving because of their use of gridpoint interpolation, and the lack of conservation can lead to accumulation of significant solution errors (Rasch and Williamson 1990; Machenhauer and Olk 1997). To address this issue, conservative semi-Lagrangian schemes, also called cell-integrated semi-Lagrangian (CISL) transport schemes (Rancic 1992; Laprise and Plante 1995; Machenhauer and Olk 1997; Zerroukat et al. 2002; Nair and Machenhauer 2002; Lauritzen et al. 2010), have been developed. Although CISL transport schemes, when applied in fluid flow solvers, allow for locally (and thus globally) conservative transport of total fluid mass and constituent (i.e., tracer) mass, a lack of consistency arises between the numerical representation of the total dry air mass conservation, to which we will refer as the continuity equation, and constituent mass conservation equations (Jöckel et al. 2001; Zhang et al. 2008; Wong et al. 2013). Numerical consistency in the flux-form equation for a tracer requires the equation for a constant tracer field to correspond numerically to the mass continuity equation; this consistency ensures that an initially spatially uniform passive tracer field will remain so. The lack of numerical consistency between the two can lead to the unphysical generation or removal of model constituent mass, which can introduce significant errors in applications such as chemical tracer transport (Machenhauer et al. 2009).

Recently, Wong et al. (2013) introduced a new flux-form formulation of the semi-implicit CISL height conservation equation for the shallow-water equations (SWE) solver. They showed that the scheme is accurate and stable even for highly nonlinear barotropically unstable jets and large Courant numbers. They also found that the use of a shape-preserving filter in an inconsistent formulation of the continuity equations is ineffective, highlighting the importance of numerical consistency in these models.

In this paper, the flux-form semi-implicit SWE formulation is extended to the fully compressible two-dimensional (*x*–*z*) moist nonhydrostatic equations for the atmosphere. A nonhydrostatic model permits fast-moving internal gravity and acoustic waves. Here, we integrate the terms responsible for the acoustic waves in a semi-implicit manner to allow large time steps while maintaining stability for these waves. As in Wong et al. (2013), our nonhydrostatic solver is based on CSLAM, a CISL transport scheme developed by Lauritzen et al. (2010) that has been implemented in the National Center for Atmospheric Research (NCAR) High-Order Methods Modeling Environment (HOMME; Erath et al. 2012). We refer to this new conservative and consistent nonhydrostatic solver that uses CSLAM for transport as CSLAM-NH.

The semi-implicit CISL nonhydrostatic solver has six main advantages and desirable properties. As we will show, our nonhydrostatic cell-integrated semi-Lagrangian solver is 1) inherently mass conserving, 2) shape preserving, and, with the new formulation, 3) has numerically consistent transport. 4) The discretization does not depend on a mean reference state, but maintains the same framework as typical semi-implicit CISL solvers, where 5) a single linear Helmholtz equation is solved and 6) a single application of CSLAM is needed per time step.

The paper is organized as follows. The governing equations of the two-dimensional fully compressible nonhydrostatic system are first described in section 2. We then present the proposed discretization of the governing equations, including a consistent formulation of the moisture conservation equations (section 3). The desirable properties of the nonhydrostatic solver are discussed in section 4. We test the nonhydrostatic solver with three idealized test cases and compare results with an Eulerian split-explicit time-stepping scheme (section 5). A fourth test case on numerical consistency is also presented in section 5 to demonstrate the shape-preserving ability of the solver with additional passive tracers. A summary is given in section 6.

## 2. Governing equations

*x*–

*z*) moist Euler equations in Cartesian geometry:

*π*= (

*p*/

*p*

_{0})

^{κ}is the Exner function,

*κ*=

*R*

_{d}/

*c*

_{p},

*γ*=

*c*

_{p}/

*c*

_{υ}= 1.4,

*R*

_{d}= 287 J kg

^{−1}K

^{−1},

*c*

_{p}= 1003 J kg

^{−1}K

^{−1}, and

*g*= 9.81 m s

^{−2}. Perturbation variables from a time-independent hydrostatically balanced background state are used to reduce numerical errors in the calculations of the pressure gradient terms (Klemp et al. 2007). The hydrostatically balanced background state is defined as

*ρ*

_{m}=

*ρ*

_{d}(1 +

*q*

_{υ}+

*q*

_{c}+

*q*

_{r}), where

*q*

_{υ},

*q*

_{c}, and

*q*

_{r}are the mixing ratios for water vapor, cloud, and rainwater, respectively. The

*F*

_{(·)}terms represent diffusion, and any diabatic effects and parameterized physics when moisture is present.

*ρ*

_{d}. The flux variables are given as

*θ*

_{m}is the modified potential temperature

*θ*

_{m}=

*θ*(1 +

*a′q*

_{υ}), where

*a*′ ≡

*R*

_{υ}/

*R*

_{d}≃ 1.61 and

*q*

_{j}= (

*q*

_{υ},

*q*

_{c},

*q*

_{r}).

## 3. A consistent and mass-conserving nonhydrostatic solver

### a. CSLAM—A conservative transport scheme

^{1}where the departure grid cell area

*δA** is found by tracing the regular arrival gridcell area Δ

*A*back in time one time step Δ

*t*(Fig. 1a). The CSLAM discretization scheme for the lhs of Eqs. (3), (4), and (5) is given by

*ϕ*= Θ

_{m},

*ρ*

_{d}, or

*Q*

_{j}. The superscript denotes the time level, and

*ϕ*

^{n}over the departure cell area

*δA**, which gives the cell-averaged departure value

*δA** in CSLAM is found through iterative trajectory computations from the four vertices of an arrival grid cell (unfilled circles in Fig. 1b) to their departure points (filled circles in Fig. 1b). The departure cell area is then approximated using straight lines as cell edges

^{2}(dark gray region

*δA*in Fig. 1b). To integrate the field

*ϕ*

^{n}over

*δA*, CSLAM implements a remapping algorithm that consists of a piecewise quasi-biparabolic subgridcell reconstruction of

*ϕ*

^{n}in the two coordinates as in Nair and Machenhauer (2002) with an additional cross term as described in Jablonowski (2004) that helps smooth subgrid distribution near sharp gradients:

*a*

^{x},

*b*

^{x},

*a*

^{z}, and

*b*

^{z}of the reconstructed parabolic function in the two coordinates are obtained as in Nair and Machenhauer (2002), and the cross-term coefficients

*c*

^{xz}and

*c*

^{zx}are obtained as in Jablonowski (2004). An average of the two coefficients of the cross term,

*c*

^{xz}and

*c*

^{zx}, is taken to avoid a directional bias (Jablonowski 2004). The cell-average value over the Eulerian grid cell is denoted as 〈

*ϕ*

^{n}〉.

*w*

^{(i,j)}that depends only on the departure cell boundary. As described in Lauritzen et al. (2010), the discrete conservative transport scheme for departure cell

*k*is

*x*,

*z*,

*x*

^{2}, and

*z*

^{2}terms, respectively;

*xz*term in Eq. (7); and

*l*is the index for the Eulerian grid cell(s) with which departure cell

*k*overlaps (of a total of

*L*

_{k}overlapping Eulerian grid cells). The partitioning of the areal integration into computation of coefficients and weights greatly enhances the transport scheme’s computational efficiency for multitracer transport, as the weights can be reused for the remapping of all tracer species in the model. For full details on the basic CSLAM, see Lauritzen et al. (2010); for high-resolution spherical implementations of CSLAM, the reader is referred to the modifications to the scheme documented in Erath et al. (2013). A rigorous assessment of the accuracy of linear transport using CSLAM [for the test case in Lauritzen et al. (2012)] and a comparison of CSLAM to a collection of state-of-the-art transport schemes can be found in Lauritzen et al. (2013).

### b. Trajectory algorithm

To find the departure cell area, we trace the vertices of each arrival grid cell back one time step Δ*t* using a trajectory algorithm described in Lauritzen et al. (2006). The trajectory is approximated and split into two segments: departure grid point to trajectory midpoint, and trajectory midpoint to arrival grid point. The split-trajectory approximation facilitates the semi-implicit formulation of the flux-form conservation equation (section 3d).

*n*and velocities extrapolated to time-level

*n*+ 1, respectively. The first segment (from the departure point position

**r**

^{n+1}) is approximated using

*n*+ 1 using a two-time-level extrapolation:

Higher-order approximations to the trajectory can be made by including an acceleration term as described in McGregor (1993). Tests including an acceleration term (not shown) showed that such a higher-order approximation made little difference to the solutions for this suite of tests.

### c. Discretization of the momentum equations

*du*/

*dt*= ∂

*u*/∂

*t*+

*u*∂

*u*/∂

*x*+

*w*∂

*u*/∂

*z*and

*dw*/

*dt*= ∂

*w*/∂

*t*+

*u*∂

*w*/∂

*x*+

*w*∂

*w*/∂

*z*[lhs of Eqs. (1) and (2), respectively] are computed using a gridpoint interpolation to the departure point. The two-time-level discretizations of the momentum equations are

*D*,

*D*/2, and

*A*denote evaluation at the departure, midpoint trajectory, and arrival grid points, respectively, and the superscripts denote the time level. The spatial operators are defined as

*β*≠ 0. Numerical diffusion is represented in

*F*

_{u}and

*F*

_{w}in the form of second-order diffusion with physical viscosity

*ν*:

*n*+ 1/2 using

*π*′, respectively) at every iteration of

*u*

^{n+1}and

*w*

^{n+1}guesses at that iteration, and we have yet to find a feasible way to incorporate this in the Helmholtz solver that ensures convergence at large time steps. The implicit treatment of the buoyancy terms will be the scope of future work.

### d. Discretization of the thermodynamic equation

*ν*times the Prandtl number) and the diabatic tendency from the microphysical scheme are evaluated explicitly and integrated over the departure cell area. Since the predictor-corrector and the forcing terms depend only on values at the previous time level, they can be evaluated along with

_{m}at the new time level except for a final saturation adjustment that takes place at the end of the time step to correct the diabatic tendency using the microphysics scheme. The new tendency is then carried over to the next time step to be used as an estimate of the diabatic term in Eq. (12).

The form of the semi-implicit correction term [square-bracketed terms in Eq. (13)] stems from the split-divergence approximation used in the trajectory computation. The semi-implicit discretization for

*r*,

*l*,

*t*, and

*b*denote the right, left, top, and bottom cell faces of an Eulerian grid cell (Fig. 2), respectively. We use an exact computation of the Lagrangian flux divergence in an Eulerian manner, where Lagrangian flux areas

Geometric representation of the Lagrangian flux divergence, defined as the flux-area difference between the Eulerian arrival grid cell (solid square) and the departure cell (dashed polygon) in one time step. Velocities associated with the Eulerian grid cell at the cell faces (*u*_{l}, *u*_{r}, *w*_{t}, *w*_{b}) and cell vertices (*u*_{c}, *w*_{c})_{i} for *i* = 1, 2, 3, 4 are also shown. White arrows at the four corners indicate the computed trajectories of each departure gridcell vertex.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Geometric representation of the Lagrangian flux divergence, defined as the flux-area difference between the Eulerian arrival grid cell (solid square) and the departure cell (dashed polygon) in one time step. Velocities associated with the Eulerian grid cell at the cell faces (*u*_{l}, *u*_{r}, *w*_{t}, *w*_{b}) and cell vertices (*u*_{c}, *w*_{c})_{i} for *i* = 1, 2, 3, 4 are also shown. White arrows at the four corners indicate the computed trajectories of each departure gridcell vertex.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Geometric representation of the Lagrangian flux divergence, defined as the flux-area difference between the Eulerian arrival grid cell (solid square) and the departure cell (dashed polygon) in one time step. Velocities associated with the Eulerian grid cell at the cell faces (*u*_{l}, *u*_{r}, *w*_{t}, *w*_{b}) and cell vertices (*u*_{c}, *w*_{c})_{i} for *i* = 1, 2, 3, 4 are also shown. White arrows at the four corners indicate the computed trajectories of each departure gridcell vertex.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

The terms proportional to Δ*t*/2 correct for the geometric differences between the Eulerian and Lagrangian flux divergences (shaded areas in Fig. 2). [For full details on the derivation of **∇**_{lag} · (Θ_{m}**v**), see Wong et al. (2013).]

### e. Helmholtz equation

*R*

_{u},

*R*

_{w}, and

*R*

_{Θ}represent the known terms in Eqs. (10), (11), and (17), respectively. The explicit solution

_{m}is performed in a consistent manner. To ensure this, in CSLAM we follow Nair and Lauritzen (2010) in separating the subgridcell reconstructions for

*ρ*

_{d}and

*θ*

_{m}, and compute the second-order reconstruction function Θ

_{m}(

*x*,

*z*) as

*ρ*

_{d}〉 and 〈

*θ*

_{m}〉 are Eulerian gridcell values, and

*ρ*

_{d}(

*x*,

*z*) and

*θ*

_{m}(

*x*,

*z*) are reconstruction functions, respectively. To check for consistency, we substitute a field of constant

*θ*

_{m}[i.e.,

*θ*

_{m}(

*x*,

*z*) = 〈

*θ*

_{m}〉 = 1] in Eq. (19) and see that the rhs of Eq. (19) properly reduces to

*ρ*

_{d}(

*x*,

*z*).

In summary, the solution procedure for obtaining solutions for *u*^{n+1}, and *w*^{n+1}, is as follows: (i) obtain solution for *u*^{n+1} and *w*^{n+1}, respectively; and (iii) recalculate *u*^{n+1} and *w*^{n+1} to eliminate any roundoff errors. This solution procedure is similar to that used in Wong et al. (2013) for the shallow-water equations.

### f. Discretization of the continuity equation

**v**

^{n+1}into

**v**′

^{n+1}.

### g. Discretization of moisture conservation equations

The flux variables of mixing ratios of water vapor *q*_{υ}, cloud water *q*_{c}, and rainwater *q*_{r} are included as prognostic variables in the nonhydrostatic solver. Moist mass conservation equations are integrated using CSLAM. To ensure moisture conservation, numerical consistency between the continuity equation and the moisture conservation equations needs to be ensured. Numerical inconsistency between the continuity equation and other scalar conservation equations can lead to spurious generation or removal of scalar mass, despite using inherently mass-conserving advection schemes.

*Q*

_{j}=

*ρ*

_{d}

*q*

_{j}, where

*q*

_{j}= (

*q*

_{υ},

*q*

_{c},

*q*

_{r}) is

**v**′

^{n},

**v**′

^{n+1}, and the computations for

**∇**

_{eul}· (·) are the same as in Eq. (21). The explicit CSLAM solution

_{m}and any diabatic tendencies from the microphysics.

### h. Diabatic processes

Microphysical processes are modeled using the simple warm-rain Kessler parameterization, as described in Klemp and Wilhelmson (1978). In the evaluation of the thermodynamic and moisture conservation equations, the diabatic forcing is approximated in

### i. Diagnostic equation of state

*p*

_{0}is the reference surface pressure set to 100 kPa.

### j. Consistency and shape preservation

*Q*

_{j}using Eq. (19) described in section 3e to ensure consistency. To ensure shape preservation, we follow the two steps as described in Wong et al. (2013). First, we use the simple 2D filter by Barth and Jespersen (1989) that searches for new local minima and maxima in the reconstruction function of a scalar field such as moisture mixing ratio

*q*

_{j}, and scales the function if these values exceed those in the neighboring cell. For chemistry applications, preservation of linear correlations in tracers is important, and it has been found that the limiter preserves linear correlations between tracers, whereas typically linear correlation is only preserved when the limiter is not applied. Second, to ensure shape preservation in the flux-divergence terms, we compute the upwind moist species mixing ratio

*Q*

_{j}from

*ρ*

_{d}. Then, the flux divergences are computed by centering density to each of the cell faces:

**v**′.

## 4. Desirable properties of CSLAM-NH

The flux-form nonhydrostatic semi-implicit CISL solver CSLAM-NH has six main advantages and desirable properties: (i) is inherently mass conserving using the conservative semi-Lagrangian transport scheme CSLAM, (ii) ensures numerically consistent transport, (iii) is independent of a mean reference state, (iv) is shape preserving, and (v) like typical semi-implicit solvers, CSLAM-NH requires solving a single linear Helmholtz equation and (vi) a single application of CSLAM at each time step.

_{m,ref}:

_{m,ref}is a mean reference state that is often time independent and varies with height. A choice of reference state can be the hydrostatic background state

*q*, are often solved explicitly using the CISL transport scheme:

*ϕ*=

*ρ*

_{d}

*q*is the scalar mass and

*n*over the departure cell. Such explicit schemes would lead to numerical inconsistency between the discrete CISL continuity equation in Eq. (25) and the discrete constituent mass conservation equations such as in Eq. (26). If the discrete conservation equation is consistent with the discrete continuity equation, the former should reduce to the latter when

*q*is a constant, and an initially spatially uniform passive tracer field should remain so. The inconsistent flux-divergence correction term in Eq. (25) can spuriously generate or remove moisture or tracer mass in the model.

*ϕ*

_{ref}is difficult, making a numerically consistent formulation such as in Eq. (27) hard to implement.

The formulations we present for the thermodynamic, density, and moisture conservation equations [Eqs. (17), (21), and (23), respectively] are all numerically consistent with one another. These consistent formulations are made possible by avoiding the use of a mean reference state. In our formulation, we use the explicit CSLAM solution instead of a mean reference state in the flux-divergence correction terms. This approach eliminates the difficult choice of a mean reference state *ϕ*_{ref} for moisture or tracer mass.

Even if an appropriate choice of *ϕ*_{ref} can be found, using a time-independent mean reference state in Eq. (27) can be problematic for regions with little moisture or tracer mass *ϕ*_{ref}, the flux divergences are likely nonzero for a divergent flow and can, therefore, generate or remove unphysical mass (Lauritzen et al. 2008). In the nonhydrostatic solver presented here, by computing the flux divergences of the explicit solution

As scalar mass conservation is not guaranteed in an inconsistent solver, these solvers also generally do not preserve the shape of scalar fields such as mixing ratios, even when shape-preserving filters are applied to the transport scheme. The implications are that the scalar field may no longer be positive-definite, and new unphysical minima and maxima can occur because of under- and overshooting, respectively. The consistent and shape-preserving transport in the proposed solver ensures that no new (unphysical) minimum and maximum (within machine roundoff) will occur.

## 5. Idealized test cases

Two of the three idealized test cases presented—namely, a density current simulation and a gravity wave simulation—are commonly used as benchmarks for testing nonhydrostatic solvers. The third idealized test case is a 2D squall-line simulation, where the stability of the model is tested with latent heating modeled by a simple warm-rain microphysics scheme. In addition to comparing with available solutions in the literature, comparisons with an Eulerian split-explicit model with second-order advection are also presented.

### a. Density current

The nonlinear density current test case, described in Straka et al. (1992), is widely used as a benchmark test for nonhydrostatic solvers (e.g., Klemp et al. 2007; Xue et al. 2000). An initial cold temperature perturbation is centered in the domain, and the negatively buoyant cold air descends to the surface and forms symmetric density currents propagating in opposite directions. Straka et al. (1992) provides a well-documented comparison among various compressible and quasi-compressible solvers as well as a high-resolution benchmark solution.

*x*= 0.0 km, with −25.6 ≤

*x*≤ 25.6 km and 0 ≤

*z*≤ 6.4 km. As described in Straka et al. (1992), the initial condition is given by a temperature perturbation Δ

*T*:

*x*

_{c},

*z*

_{c}) = (0.0, 3.0) km is the center of the perturbation, and its width and depth are given by

*x*

_{r}= 4.0 km and

*z*

_{r}= 2.0 km. The surface temperature

*θ*

_{0}is at 300 K in a horizontally homogeneous and neutral environment. A constant physical viscosity of 75 m

^{2}s

^{−1}is used. The domain is an

*x*-periodic channel with reflective boundary conditions along the upper and lower boundaries as specified by Straka et al. (1992) that require ∂

*u*/∂

*z*=

*w*= ∂

*ρ*/∂

*z*= ∂Θ/∂

*z*= 0.

Following Straka et al. (1992), we simulate the density current test case using grid spacings Δ*x* = Δ*z* = 400, 200, 100, 50, and 25 m, with Eulerian time step sizes of Δ*t* = 4, 2, 1, 0.5, and 0.25 s, respectively. Figure 3 shows the potential temperature perturbation (*θ*′) from its mean state from CSLAM-NH and the Eulerian split-explicit scheme with second-order advection at the simulation end time of 15 min using different model resolutions.

Potential temperature perturbation (K) after 15 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 400, 200, 100, 50, and 25 m. Contour intervals are every 1 K, starting at 0.5 K. Mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) after 15 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 400, 200, 100, 50, and 25 m. Contour intervals are every 1 K, starting at 0.5 K. Mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) after 15 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 400, 200, 100, 50, and 25 m. Contour intervals are every 1 K, starting at 0.5 K. Mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

The density current is clearly underresolved using a 400-m grid spacing, with only the main rotor marginally resolved (7 ≤ *x* ≤ 9 km). A grid spacing of 200 m gives a better resolution of the main rotor as well as a second rotor (11 ≤ *x* ≤ 12 km); however, the leading third rotor is still underresolved. For resolutions finer than Δ*x* = Δ*z* = 100 m, all three rotors are well resolved with the solutions converging and indistinguishable by eye between 50- and 25-m grid spacings. The differences among the model resolutions agree well with those documented in other nonhydrostatic solvers such as in Straka et al. (1992), Xue et al. (2000), Skamarock and Klemp (2008), and Melvin et al. (2010).

Positions of the density current front (specified to be at *θ*′ = −1 K), the minimum and maximum *θ*′ values in the domain, and **Σ****Σ***θ*′ values are colder than those in SLICE; therefore, the front locations also traveled farther out from the center line. Analytically, the maximum *θ*′ should remain zero throughout the simulation, as is the case in the higher-resolution runs (50 and 25 m). The contribution of positive *θ*′ values in **Σ**^{−5} K and 1 × 10^{−8} K, respectively), increasing up to the order of 1 × 10^{−1} K at 200 m. [Straka et al. (1992) only reported values up to four decimal points.]

Statistics for the density current simulations at time 15 min using CSLAM-NH at various grid resolutions and time steps. Comparison values from the nonhydrostatic *x*–*z* solver using SLICE in Melvin et al. (2010) are also presented. REFC25 are values taken from Straka et al. (1992). The

For the next simulation, a mean background wind of *x* = Δ*z* = 200, 100, and 50 m are shown in Fig. 4. Time step sizes are the same as in Fig. 3. The solutions from CSLAM-NH in general show proper symmetry about the translating centerline, although very subtle differences between the secondary rotors in the left- and right-moving currents are noticeable at 200- and 100-m grid spacing. As a comparison, the Eulerian split-explicit second-order advection scheme shows noticeably larger errors in the right-moving current as expected because of the right-moving current moving at a greater speed than the other (causing larger advective phase errors).

As in Fig. 3, but for (from top to bottom) Δ*x* = Δ*z* = 200, 100, and 50 m and a mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

As in Fig. 3, but for (from top to bottom) Δ*x* = Δ*z* = 200, 100, and 50 m and a mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

As in Fig. 3, but for (from top to bottom) Δ*x* = Δ*z* = 200, 100, and 50 m and a mean wind

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

For this test case, we found that the maximum stable time step size in CSLAM-NH is double of that of the Eulerian scheme. Figure 5 shows solutions for tests where *x* = Δ*z* = 100 m using a time step size of 3 and 4 s, whereas the maximum stable Eulerian time step size is Δ*t* = 2 s. The solution using a large time step of 4 s is almost indistinguishable by eye from the 25-m high-resolution solutions (Fig. 3). With mean advection

Potential temperature perturbation (K) from CSLAM-NH after 15 min for grid spacing Δ*x* = Δ*z* = 100 m using time step sizes Δ*t* = (top) 3 and (bottom) 4 s. Mean wind *t* ≤ 2 s for numerical stability of this gravity current. Contoured as in Fig. 3.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) from CSLAM-NH after 15 min for grid spacing Δ*x* = Δ*z* = 100 m using time step sizes Δ*t* = (top) 3 and (bottom) 4 s. Mean wind *t* ≤ 2 s for numerical stability of this gravity current. Contoured as in Fig. 3.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) from CSLAM-NH after 15 min for grid spacing Δ*x* = Δ*z* = 100 m using time step sizes Δ*t* = (top) 3 and (bottom) 4 s. Mean wind *t* ≤ 2 s for numerical stability of this gravity current. Contoured as in Fig. 3.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

### b. Gravity wave

*θ*

_{0}:

*θ*

_{0}= 10

^{−2}K,

*a*= 5 km is the half-width of the initial perturbation,

*H*= 10 km is the total depth of the domain, and

*x*

_{c}= 0.25

*L*, where

*L*=300 km is the length of the domain. The background atmospheric stratification has a constant Brunt–Väisäla frequency

*N*= 10

^{−2}s

^{−1}. For one simulation, no mean wind

*u*, Θ, and

*ρ*values into the boundary, consistent with the free-slip boundary conditions, and setting

*w*= 0.

We run the gravity wave test case at grid spacings Δ*x* = Δ*z* = 1 km, 500 m, and 250 m using Eulerian time step sizes Δ*t* = 12, 6, and 3 s, respectively. Solutions from CSLAM-NH at the three resolutions for *x* and *z* is maintained. The density variation in the full Euler equations results in solutions that are asymmetric in *z*, as observed in the CSLAM-NH solutions, the second-order Eulerian solutions, the fifth-order Eulerian solutions, as well as the SLICE nonhydrostatic vertical model solutions.

Potential temperature perturbation (K) after 50 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 1000, 500, and 250 m. Contour interval is 5 × 10^{−4} K (zero contour line not plotted). Solid lines indicate positive contours and dashed lines indicate negative contours. Solution is translated using a mean wind *x* = 0.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) after 50 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 1000, 500, and 250 m. Contour interval is 5 × 10^{−4} K (zero contour line not plotted). Solid lines indicate positive contours and dashed lines indicate negative contours. Solution is translated using a mean wind *x* = 0.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) after 50 min for (left) CSLAM-NH and (right) the Eulerian split-explicit scheme for (from top to bottom) Δ*x* = Δ*z* = 1000, 500, and 250 m. Contour interval is 5 × 10^{−4} K (zero contour line not plotted). Solid lines indicate positive contours and dashed lines indicate negative contours. Solution is translated using a mean wind *x* = 0.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Like in the density current test, we impose a mean advection wind

Testing of CSLAM-NH using larger time steps in this gravity wave test case reveals a numerical stability condition that is sensitive to the stratification *N*. (We note that CSLAM-NH is unconditionally stable for *N* = 0, i.e., for a near-pure advection case of the initial warm perturbation.) Figure 7 shows the instability for *N* = 0.01 s^{−1} with and without a mean wind imposed. We further evaluate the maximum stable CSLAM-NH time step size for the gravity wave case with the mean advection wind speed of *x* = Δ*z* = 1 km) over a range of *N* (0.01, 0.015, and 0.02 s^{−1}). Since the gravity wave phase speed varies with *N*, we increase/decrease the simulation time length as appropriate such that the gravity wave solutions are similar to those shown in Fig. 6; for example, for *N* = 0.015 s^{−1}, the simulation time is reduced to 2000 s. Test results showed that the maximum stable CSLAM-NH time step sizes are Δ*t*_{max} = 38, 35, and 32 s for *N* = 0.01, 0.015, and 0.02 s^{−1}, respectively, whereas in the case of the Eulerian split-explicit scheme, the maximum stable large time steps are found to be Δ*t* = 60, 55, and 50 s (with small time step size of 2.4 s), respectively, limited by the stability condition of the advection scheme. The buoyancy terms in the vertical momentum equation are integrated explicitly in CSLAM-NH, and handled implicitly in the Eulerian scheme. When we remove the buoyancy terms from the implicit step and solve them explicitly in the Eulerian model, the time step sizes required to obtain solutions of similar accuracy as those from the vertically implicit model are reduced by 20%–35%, and are closer to those found in CSLAM-NH. The devising of an integration scheme that handles the buoyancy terms implicitly in CSLAM-NH will require a robust and stable way of updating the density perturbation in the Helmholtz solver, and this will be addressed in future work.

Potential temperature perturbation (K) solutions of the gravity wave case using increasingly large CSLAM-NH time steps (Δ*x* = Δ*z* = 1 km) where (a)–(c) *t* = 60–100 s and (d)–(f) *t* = 24–40 s. Contoured as in Fig. 6.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) solutions of the gravity wave case using increasingly large CSLAM-NH time steps (Δ*x* = Δ*z* = 1 km) where (a)–(c) *t* = 60–100 s and (d)–(f) *t* = 24–40 s. Contoured as in Fig. 6.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Potential temperature perturbation (K) solutions of the gravity wave case using increasingly large CSLAM-NH time steps (Δ*x* = Δ*z* = 1 km) where (a)–(c) *t* = 60–100 s and (d)–(f) *t* = 24–40 s. Contoured as in Fig. 6.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

### c. 2D (x–z) squall line

We perform a test case of a 2D squall line as described in Weisman and Klemp (1982) to evaluate mass conservation, consistency, and shape-preservation in the nonhydrostatic solver, in addition to testing for any small-scale computational instability in the model due to latent heating.

*x*= 0.0 km, with −100 ≤

*x*≤ 100 km and 0 ≤

*z*≤ 20 km. As in Weisman and Klemp (1982), a conditionally unstable thermodynamic profile is used to initialize the horizontally homogeneous environment. Constant physical horizontal and vertical eddy viscosities of 250 m

^{2}s

^{−1}are used. A warm thermal perturbation near the surface is prescribed to initiate convection (Weisman et al. 1988). The initial thermal perturbation has a maximum of Δ

*θ*

_{0}= 3 K, centered at

*z*

_{c}= 1.5 km and along the center line (

*x*

_{c}= 0) of the domain, with a horizontal radius

*x*

_{r}of 10 km and a vertical radius

*z*

_{r}of 1.5 km. The shape of the perturbation is a cosine hill given as

^{−1},

*u*

_{s}= 10 m s

^{−1}, and

*z*

_{ts}= 2.5 km. The environmental potential temperature and relative humidity profiles at the initial time are

*θ*

_{tr}= 343 K,

*z*

_{tr}= 12.0 km, and

*T*

_{tr}= 213 K are the potential temperature, geometric height, and actual temperature at the tropopause, respectively. The maximum water mixing ratio is capped at 14 g kg

^{−1}. The surface potential temperature

*θ*

_{0}= 300 K. The skew

*T*–log

*p*diagram for this sounding can be found in Fig. 1 of Weisman and Klemp (1982). Numerical simulations (unless otherwise stated) use a grid spacing Δ

*x*= Δ

*z*= 500 m, a time step Δ

*t*= 5 s, and a time-off-centering parameter of

*β*= 0.1 to maintain numerical stability. Like the gravity case, the boundary condition is implemented by linear extrapolating

*u*, Θ, and

*ρ*values into the boundary and setting

*w*= 0, consistent with the free-slip boundary conditions.

A comparison of the squall-line development among CSLAM-NH (with shape preservation), the fifth-order split-explicit, and the second-order split-explicit Eulerian models is presented in Fig. 8. Instantaneous and accumulated surface precipitation integrated across the model domain are presented in Fig. 9; also shown is the rate of condensation over the entire domain. Maximum updraft velocity is shown in Fig. 10. The series of updraft velocity peaks highlight the continuous triggering of new convective systems along the advancing front.

Vertical cross sections of vertical velocity (color shading; m s^{−1}) and solid contours of the convective cloud structure (*q*_{c} = 0.1 g kg^{−1}) at times (from top to bottom) *T* = 0.6, 0.8, 1.3, and 1.7 h of the simulation for the 500-m grid-spacing runs with a Δ*t* = 5.0 s from (left) CSLAM-NH, (middle) fifth-order split-explicit Eulerian model, and (right) second-order split-explicit Eulerian model.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Vertical cross sections of vertical velocity (color shading; m s^{−1}) and solid contours of the convective cloud structure (*q*_{c} = 0.1 g kg^{−1}) at times (from top to bottom) *T* = 0.6, 0.8, 1.3, and 1.7 h of the simulation for the 500-m grid-spacing runs with a Δ*t* = 5.0 s from (left) CSLAM-NH, (middle) fifth-order split-explicit Eulerian model, and (right) second-order split-explicit Eulerian model.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Vertical cross sections of vertical velocity (color shading; m s^{−1}) and solid contours of the convective cloud structure (*q*_{c} = 0.1 g kg^{−1}) at times (from top to bottom) *T* = 0.6, 0.8, 1.3, and 1.7 h of the simulation for the 500-m grid-spacing runs with a Δ*t* = 5.0 s from (left) CSLAM-NH, (middle) fifth-order split-explicit Eulerian model, and (right) second-order split-explicit Eulerian model.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Moisture statistics for (top) surface precipitation rate (kg s^{−1}), (middle) accumulated surface precipitation (kg), and (bottom) condensation rate (kg s^{−1}) from the microphysics using CSLAM-NH, Eulerian fifth-order horizontal advection, and Eulerian second-order horizontal advection at Δ*x* = Δ*z* = 500 m.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Moisture statistics for (top) surface precipitation rate (kg s^{−1}), (middle) accumulated surface precipitation (kg), and (bottom) condensation rate (kg s^{−1}) from the microphysics using CSLAM-NH, Eulerian fifth-order horizontal advection, and Eulerian second-order horizontal advection at Δ*x* = Δ*z* = 500 m.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Moisture statistics for (top) surface precipitation rate (kg s^{−1}), (middle) accumulated surface precipitation (kg), and (bottom) condensation rate (kg s^{−1}) from the microphysics using CSLAM-NH, Eulerian fifth-order horizontal advection, and Eulerian second-order horizontal advection at Δ*x* = Δ*z* = 500 m.

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Updraft intensities using CSLAM-NH (red) and Eulerian fifth-order horizontal advection (black) at Δ*x* = Δ*z* = 500 m and Δ*t* = 5 s (solid), and Δ*x* = Δ*z* = 250 m and Δ*t* = 2.5 s (dashed).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Updraft intensities using CSLAM-NH (red) and Eulerian fifth-order horizontal advection (black) at Δ*x* = Δ*z* = 500 m and Δ*t* = 5 s (solid), and Δ*x* = Δ*z* = 250 m and Δ*t* = 2.5 s (dashed).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Updraft intensities using CSLAM-NH (red) and Eulerian fifth-order horizontal advection (black) at Δ*x* = Δ*z* = 500 m and Δ*t* = 5 s (solid), and Δ*x* = Δ*z* = 250 m and Δ*t* = 2.5 s (dashed).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

All three models (CSLAM-NH, Eulerian fifth-order advection, and Eulerian second-order advection) show similar development of the convective system (Fig. 8). At 0.6 h, all three models show an initial downshear orientation of the system due to the ambient wind shear. As the storm continues to develop with the cold pool strengthening behind the system (not shown), convergence and enhanced uplift lead to the storm tilting in a near-upright position (*T* = 0.8 h). At 1.3 h, a new cell is triggered near the edge of the cold pool, where uplift of the warm moist air in the boundary layer is enhanced. At 1.7 h, the cold pool is strong enough to generate a circulation such that the system develops an upshear orientation, as described in Rotunno et al. (1988). Comparing to the simulations from the Eulerian second-order model, those from CSLAM-NH show closer resemblance to those from the Eulerian fifth-order model. The better agreement is also evident in the moisture statistics (Fig. 9), especially in the accumulated surface precipitation amounts and condensation rate in the domain.

Focusing on the two models that show more comparable results, the first maximum updraft velocities from CSLAM-NH (34.1 m s^{−1}) is slightly greater than that from Eulerian fifth-order advection (31.6 m s^{−1}) (Fig. 10). CSLAM-NH appears to show a weaker second peak updraft velocity (21.9 m s^{−1}) than the Eulerian fifth-order model (28.3 m s^{−1}); however, the stronger first peak (~34 m s^{−1}) and weaker second peak (~25 m s^{−1}) are also observed in a higher-resolution simulation using the Eulerian fifth-order model at a grid spacing of 250 m and large time step size of 2.5 s (dashed black line in Fig. 10). For comparison, maximum updraft from CSLAM-NH at Δ*x* = 250 m and Δ*t* = 2.5 s (red dashed line in Fig. 10) is also shown, and at the higher resolution, the two models agree very well with each other.

The maximum stable time step in the Eulerian split-explicit fifth-order advection scheme is a large time step of 20 s and acoustic time step size of 1.25 s. The maximum CSLAM-NH stable time step is limited to 15 s because of the violation of the Lipschitz stability condition in the vicinity of the updraft when a larger time step is used (the instability occurs when the storm reaches its first maximum vertical updraft, which generates a strong horizontal wind shear between the updraft and the neighboring downdraft). In Fig. 11, we see at larger time step sizes, maximum updraft velocities remain close to the small time-step solutions.

Timing and intensity of the maximum vertical updraft using Δ*x* = Δ*z* = 500 m at different CSLAM-NH time step sizes (solid lines), as compared to the Eulerian fifth-order horizontal advection vertical velocity (dashed lines). (Only the first hour is plotted.)

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Timing and intensity of the maximum vertical updraft using Δ*x* = Δ*z* = 500 m at different CSLAM-NH time step sizes (solid lines), as compared to the Eulerian fifth-order horizontal advection vertical velocity (dashed lines). (Only the first hour is plotted.)

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Timing and intensity of the maximum vertical updraft using Δ*x* = Δ*z* = 500 m at different CSLAM-NH time step sizes (solid lines), as compared to the Eulerian fifth-order horizontal advection vertical velocity (dashed lines). (Only the first hour is plotted.)

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

With the 2D squall-line test case, we examine the shape-preservation properties of CSLAM-NH using the shape-preserving scheme by Barth and Jespersen (1989) in the CSLAM transport scheme and the upwind scheme for the flux-correction terms in the transport equations. An analogous implementation of these schemes for a shallow-water model is described in Wong et al. (2013).

To verify that consistency is achieved, an additional passive tracer with mixing ratio *r* is introduced into the model. The passive tracer initially has a constant mixing ratio of *r*_{0} = 1.0 g kg^{−1} and we form the discretized conservation equation as in Eq. (23). The minimum and maximum values of *r* are maintained at 1.0 g kg^{−1} (up to machine round-off of order 10^{−14}) throughout the simulation using the consistent formulation in CSLAM-NH.

For a passive tracer that uses an inconsistent discrete conservation equation such as in Eq. (26), unphysical minima and maxima of the passive tracer mixing ratio are generated (Fig. 12). At the end of the squall line simulation at 2 h, the minimum and maximum mixing ratios *r* are 0.986 and 1.021 g kg^{−1}, respectively (i.e., the error is on the order of 1 part in 100). We note that the shape-preserving limiter described in Barth and Jespersen (1989) was also applied in CSLAM in this test. Because of numerical inconsistency, however, the limiter becomes ineffective agreeing with the results in Wong et al. (2013). This discrepancy from constancy highlights the importance of ensuring numerical consistency to properly maintain conservation of moisture and tracer mass in a semi-implicit CISL solver.

Mixing ratio errors (g kg^{−1}) due to numerical inconsistency associated with Eq. (26) for times *T* = 0.6, 0.8, 1.3, and 1.7 h. The passive tracer is initialized with a uniform mixing ratio field of 1.0 g kg^{−1}. The consistent formulation in CSLAM-NH [which does not use Eq. (26)] ensures mixing ratio constancy of the same passive tracer up to machine round-off of order 10^{−14} (not shown).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Mixing ratio errors (g kg^{−1}) due to numerical inconsistency associated with Eq. (26) for times *T* = 0.6, 0.8, 1.3, and 1.7 h. The passive tracer is initialized with a uniform mixing ratio field of 1.0 g kg^{−1}. The consistent formulation in CSLAM-NH [which does not use Eq. (26)] ensures mixing ratio constancy of the same passive tracer up to machine round-off of order 10^{−14} (not shown).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

Mixing ratio errors (g kg^{−1}) due to numerical inconsistency associated with Eq. (26) for times *T* = 0.6, 0.8, 1.3, and 1.7 h. The passive tracer is initialized with a uniform mixing ratio field of 1.0 g kg^{−1}. The consistent formulation in CSLAM-NH [which does not use Eq. (26)] ensures mixing ratio constancy of the same passive tracer up to machine round-off of order 10^{−14} (not shown).

Citation: Monthly Weather Review 142, 4; 10.1175/MWR-D-13-00210.1

## 6. Summary

A new cell-integrated semi-Lagrangian (CISL) nonhydrostatic atmospheric solver, CSLAM-NH, for the moist Euler equations is introduced in this paper. The two-dimensional (*x*–*z*) Cartesian nonhydrostatic solver uses a CISL transport scheme, CSLAM, for conservative transport. It also incorporates a new approach to ensure numerical consistency among the CISL continuity equation and the conservation equations for potential temperature, moisture species, and passive tracers. A semi-implicit time integration scheme is used to stably handle the fast-moving acoustic waves in the compressible system.

Based on a recently tested shallow-water equations solver, the extended nonhydrostatic atmospheric solver presented here, CSLAM-NH, possesses a number of features ideal for weather and climate modeling purposes. The solver

is inherently mass conserving through the use of a conservative transport scheme CSLAM,

ensures numerical consistency between the continuity equation and other scalar mass conservation equations (the lack of which may lead to violation of scalar mass conservation),

does not depend on a mean reference state,

can be easily implemented with existing shape-preserving filters to ensure shape preservation of scalar fields,

requires a single linear Helmholtz equation solution (as in typical semi-implicit solvers) per time step, and

requires a single application of CSLAM per time step.

Here, we tested the nonhydrostatic extension for three idealized test cases: a density current, a gravity wave, and a squall line. To represent microphysical processes in the squall-line test, the Kessler warm-rain microphysics parameterization scheme is coupled to the dynamics. The 2D solver currently does not admit flow in the *y* direction, and therefore, Coriolis terms are neglected; however, the tests we present allow for sufficient testing of typical meteorological flows. Results compare well with other existing Eulerian (such as ARW-WRF) and nonhydrostatic CISL solvers (such as the nonhydrostatic SLICE model). In the density current and gravity wave tests, we see that CSLAM-NH allows for stable time steps 2 times larger than that in an Eulerian model. In the highly nonhydrostatic flow of the squall-line test case, the maximum stable time step size is of similar magnitude as the Eulerian split-explicit model. The strong wind shear across the storm updraft imposes a time step limit in CSLAM-NH due to the Lipschitz stability condition (violation of which leads to the crossing of trajectories).

Plans to extend the nonhydrostatic solver to include orographic influences are also under way. This work involves transformation of the nonhydrostatic equations into a terrain-following height coordinate. In traditional semi-Lagrangian semi-implicit solvers, flow over topography has been found to trigger spurious resonances and time off-centering in the implicit scheme has been found to eliminate these noises. Thus far, without orography, we have found that our nonhydrostatic solver only requires time off-centering (*β* = 0.1) in the squall-line case to maintain numerical stability. The nonhydrostatic solver with orography will allow us to test the conservative and consistent transport and stability of the new semi-implicit CISL discretization under the influence of a terrain-following coordinate.

## Acknowledgments

This work was done as a part of the National Center for Atmospheric Research Graduate Visitor Advanced Study Program. The first author would also like to acknowledge the Canadian Natural Science and Engineering Research Council for their financial support via the Discovery Grant to the last author.

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^{1}

Note that CSLAM may also be cast in flux form (Harris et al. 2011).

^{2}

Higher-order edge approximations have been explored in Ullrich et al. (2012).