## 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 due to 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 allow for locally (and thus globally) conservative transport of total fluid mass and constituent (i.e., tracer) mass, an issue related to conservation remains when they are applied in fluid flow solvers: the lack of consistency between the numerical representation of the total mass continuity and constituent mass conservation equations (Jöckel et al. 2001; Zhang et al. 2008). 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).

*f*plane:

**v**= (

*u*,

*υ*) is the horizontal velocity vector,

*f*is the Coriolis parameter,

*g*′ is the reduced gravity,

*h*is the total fluid depth (a surrogate for total fluid mass), and

*hq*is the depth portion (mass fraction) of an arbitrary constituent, where

*q*is its specific concentration. Numerical consistency is satisfied if, for

*q*

_{0}= 1, the discretization scheme of the constituent equation (4) collapses to that for the continuity equation (3), also known as free-stream preservation.

The difficulty in maintaining consistency, as will be discussed in more detail, can partly be attributed to the conventional linearization around a constant mean reference state in the semi-implicit form of a CISL continuity equation. To eliminate the reference state, Thuburn (2008) developed a fully-implicit CISL-based scheme for the shallow-water equations that requires solving a nonlinear Helmholtz equation at every time step. The solution of the Helmholtz equation is potentially problematic and expensive (Thuburn et al. 2010). To reduce the dependence of their semi-implicit scheme on a reference state, Thuburn et al. (2010) used an alternative iterative approach to solve the nonlinear system, but it requires multiple calls to a Helmholtz solver per time step, again making the scheme potentially expensive.

In addition to consistency and mass conservation, another desirable property is that the new scheme should be shape preserving. A shape-preserving scheme ensures that no new unphysical extrema are generated in a field due to the numerical scheme (e.g., Machenhauer et al. 2009). For example, specific concentrations of a passive constituent should not go outside the range of its initial minimum and maximum values. Nonshape-preserving schemes may generate unphysical specific concentrations, such as negative concentration values due to undershooting.

In this paper, using a shallow-water system, we present a new SLSI formulation that uses a CISL scheme for mass conservation and ensures numerical consistency between the total mass and constituent-mass fields. The new scheme is based on the CISL transport scheme called the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme developed by Lauritzen et al. (2010). Like other typical conservative SLSI solvers, the algorithm requires a single linear Helmholtz equation solution and a single application of CSLAM. To ensure shape preservation, the scheme is further extended to use existing shape-preserving filters.

The paper is organized as follows. In section 2, the conservative semi-Lagrangian scheme CSLAM is described and a discussion of the issue of consistency between total-mass and constituent-mass conservation in its semi-implicit formulation is provided. A new consistent semi-implicit discretization of the CSLAM continuity equation, including the implementation of the shape-preserving schemes, is proposed in section 3. Results from four test cases are presented in section 4, highlighting the stability and accuracy of the new scheme for linear and highly nonlinear flows, as well as showing the shape-preserving ability of the scheme. And finally, in section 5, a summary of the results and a potential extension of the new scheme are given.

## 2. Mass conservation and consistency in SLSI solvers

### a. CSLAM—A CISL transport scheme

*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 (3) is given by

*h*over

^{n}*δA**, which gives departure cell-averaged height values

*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 (dark gray region

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

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

*h*field, and then the integration of the reconstruction function over the departure cell area. The area integration in CSLAM is transformed into a series of line integrals using the Gauss–Green theorem, and involves solving for a set of weights that depends only on the departure cell boundary. The use of line integrals greatly enhances the transport scheme's computational efficiency for multitracer transport as the weights can be reused for all tracer species in the model. For full details on the transport scheme, see Lauritzen et al. (2010).

^{n}### b. A discrete semi-implicit continuity equation in velocity-divergence form using CSLAM

*t*is the model time step,

*H*

_{0}is the constant mean reference height,

**v**

^{n+1}is the velocity field implicitly coupled to the momentum equations,

*n*+ 1, and

**v**

^{n}is the velocity field at time-level

*n*. Their semi-implicit correction term [first term in brackets in (5)] is the correction to the explicit solution

_{eul}and ∇

_{lag}denote discretized divergence operators based on the Eulerian and Lagrangian forms, respectively. Using notations in Fig. 2, the Eulerian divergence operator is given by

Definition of an Eulerian arrival grid cell, and its associated velocities at the cell faces (*u _{l}*,

*u*,

_{r}*υ*,

_{t}*υ*) and cell corners (

_{b}*u*,

_{c}*υ*)

_{c}_{i}for

*i*= 1, 2, 3, 4.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Definition of an Eulerian arrival grid cell, and its associated velocities at the cell faces (*u _{l}*,

*u*,

_{r}*υ*,

_{t}*υ*) and cell corners (

_{b}*u*,

_{c}*υ*)

_{c}_{i}for

*i*= 1, 2, 3, 4.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Definition of an Eulerian arrival grid cell, and its associated velocities at the cell faces (*u _{l}*,

*u*,

_{r}*υ*,

_{t}*υ*) and cell corners (

_{b}*u*,

_{c}*υ*)

_{c}_{i}for

*i*= 1, 2, 3, 4.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

*n*+ ½). As explained in LKM, this approximation stems from their trajectory algorithm, where the trajectory is approximated as two segments: (i) from the departure point to the trajectory midpoint (computed iteratively), and (ii) from the midpoint to the arrival grid point (computed using extrapolated winds; see Fig. 1 in LKM). Since the Lagrangian divergence is calculated based on the change of cell area over time, and departure cell areas are computed using the split-trajectory algorithm, the split approximation can also be applied to the divergence term (LKM).

Ideally, to be consistent, the implicit and the extrapolated divergences would both be solved in a Lagrangian fashion; however, this would lead to a nonlinear elliptic equation instead of a standard Helmholtz equation (Lauritzen 2005). To retain a linear elliptic equation, LKM implemented a predictor-corrector approach to correct for the Eulerian discretization of the implicit divergence term, and found that this step was necessary to maintain stability in their model. In our implementation of the LKM solver using CSLAM, we follow the approach of LKM, where the predictor-corrector term [second term in brackets in (5)] is evaluated by integrating the departure cell-averaged value over *δA**.

### c. Numerical inconsistency in semi-implicit continuity equations in a velocity-divergence form

*hq*is typically solved explicitly using

*h*is the shallow-water height (analogous to total air mass in a full model), and

*q*is the specific concentration of an arbitrary constituent. The cell-integrated transport equation in its flux form helps conserve constituent mass, analogous to the amount of water vapor and other passive tracers in an atmospheric model—an important constraint especially for long simulations. Since the departure cell areas are the same for both total fluid mass and the constituent mass, the weights of the line integrals in CSLAM will need to be computed only once per time step, and represents one of the advantages of this scheme.

If the discrete constituent equation is consistent with the discrete continuity equation, the former should reduce to the latter when *q* = 1, and an initially spatially uniform specific concentration field should remain so. For a divergent flow, however, the semi-implicit correction term in (5) may become large enough such that (7), in its explicit form, is no longer consistent (Lauritzen et al. 2008).

*HQ*

_{0}is a constant mean reference constituent mass, the velocities

**v**

^{n+1}are solutions from the Helmholtz solver, and

**v**

^{n}are the same velocities as in (5).

However, the dependence on a constant mean reference constituent mass *HQ*_{0} may create a source of numerical errors for regions with little constituent mass. For example, in regions where

The issue with an inconsistent constant mean reference state for the total fluid mass and constituent mass fields can be resolved with the formulation we present in the next section.

## 3. A consistent and mass-conserving semi-implicit SW solver

**v**

^{n+1}in

*hq*and

^{n}**v**

^{n}are the constituent mass and velocity at time-level

*n*, respectively. This scheme also resolves the problem of spurious generation of constituent mass for regions with near-zero specific concentration (as described in the previous section). The specific concentration

*q*is diagnosed by decoupling the constituent mass using

*hq*

^{n}^{+1}, we substitute the solutions of

**v**

^{n+1}derived from the Helmholtz solution

*h*

^{n}^{+1}into (10). Prior to diagnosing

*q*using (11), we must correct the solution

*h*

^{n}^{+1}by substituting solutions of

**v**

^{n+1}back into (9); otherwise, the values of

*h*

^{n}^{+1}can become inconsistent with

*hq*

^{n}^{+1}. The consistent

*h*

^{n}^{+1}solution is then used to solve for

*q*using (11) and in the next time step. To compute

*h*and

*q*, and then compute

*hq*(

*x*,

*y*) using

*h*=

*h*(

*x*,

*y*) and

*q*=

*q*(

*x*,

*y*) are the reconstruction functions, and

*x*and Δ

*y*are the grid spacing in the

*x*and

*y*directions, and each of the fluxes are evaluated as

*x*

_{1},

*y*

_{1}), (

*x*

_{2},

*y*

_{2}), (

*x*

_{3},

*y*

_{3}), (

*x*

_{4},

*y*

_{4}), and the departure cell corner points as

*x*- and

*y*-velocity components at the

*i*th vertex, where

*i*= 1, 2, 3, 4.

_{r},

_{l},

_{t}, and

_{b}is identical to the Eulerian velocity divergence and the remaining terms give the geometric correction for a Lagrangian representation (see Fig. 9 in Lauritzen 2005). Using this velocity divergence, we now approximate the Lagrangian flux-divergence term in (9) as

*h*with

*hq*, we can further combine each of the terms in brackets of the constituent equation (10), which becomes

**v**′ is defined as the difference between the velocity field used in the Eulerian flux divergence in (12) and that derived from the Lagrangian flux areas

_{r},

_{l},

_{t}, and

_{b}, divided by the cell face length. The corrective velocity

**v**′

^{n+1}in (15) is computed using

**v**

^{n+1}from the Helmholtz solver and the Lagrangian flux areas based on extrapolated winds divided by the cell face length. The velocity

**v**′

^{n}used in the predictor-corrector term in (15) is computed using the velocity field

**v**

^{n}at time-level

*n*and the Lagrangian flux areas based on

**v**

^{n}, and again divided by the cell face length. Shape-preserving schemes (e.g., the first-order upwind scheme) or higher-order methods such as flux-corrected transport schemes or flux-limiter schemes can then be applied to the fluxes in (15). The first-order upwind scheme is used here, where the upstream values (denoted by the asterisks)

**v**′

^{n+1}and −

**v**′

^{n}, respectively [see e.g., Durran (2010), his Eq. (5.109)]. The first-order upwind scheme is numerically diffusive (Durran 2010), but the damping effect on the correction and predictor-corrector terms should be minimal as the corrective velocities

**v**′

^{n+1}and

**v**′

^{n}are typically very small. To ensure shape preservation in the explicit CSLAM solution, we implement a simple 2D monotonic filter (Barth and Jespersen 1989) that searches for new local minima and maxima in the reconstruction function of

*q*, and scales the function if these values exceed those in the neighboring cell.

*h*, and (A1) and (A2) for the velocity components] revealed an instability related to the averaging of the C-grid velocities to the cell corner points in the continuity equation and its interaction with the rotational modes. Following Randall (1994), we can write a generalized discretized dispersion relation for the linearized shallow-water equations as

*f*and

_{u}*f*are the discrete Coriolis operators,

_{υ}*k*and

_{u}*l*are the discrete height-gradient operators,

_{υ}*k*and

_{h}*l*are the discrete velocity-divergence operators in the continuity equation (the letter subscripts refer to the equations in which they appear), and

_{h}*c*

^{2}=

*gH*. In the linearized shallow-water dispersion relation for C grid, the last two terms on the lhs of (16),

*f*and

_{u}k_{h}l_{υ}*f*, cancel each other and thus there are no numerical frequencies

_{υ}k_{u}l_{h}*ω*with imaginary parts that amplify in time. Although the CSLAM-SW model uses the C grid, we have found that the discretization of the linearized Lagrangian divergence is equivalent to taking an average of the

*u*and

*υ*velocities to the corners of the grid cell followed by an averaging back to the cell faces (i.e., the discretization is equivalent to using a 1–2–1 averaging of the

*u*velocities in the

*y*direction, and of the

*υ*velocities in the

*x*direction, at the Eulerian gridcell faces). This averaging leads to noncancellation of

*f*and

_{u}k_{h}l_{υ}*f*, and growing modes. We have found that using the averaging operators

_{υ}k_{u}l_{h}*x*- and

*y*-momentum equations, respectively, recovers the cancellation and eliminates the unstable mode.

## 4. Test cases

We present four test problems involving divergent flows: a radially propagating gravity wave (with two different initial perturbations), and two highly nonlinear barotropically unstable jets [the Bickley and the Gaussian jets from Poulin and Flierl (2003)]. The gravity wave problem (section 4a) is a simple case to assess the stability and accuracy of the new SLSI solver (CSLAM-SW) with respect to an imposed mean flow speed and the gravity wave propagation speed. We also use this test case to highlight the issue of numerical inconsistency in the constituent transport scheme of LKM. The nonlinearity of the unstable jet in the second problem is particularly useful in testing the stability limits of the new scheme. The Bickley jet (section 4b) has a moderate gradient in the initial height profile, while the steeper profile in the Gaussian jet (section 4c) drives a more unstable jet. These strong gradients provide a severe test for advection schemes. In addition to those from LKM, solutions from a traditional semi-Lagrangian formulation and an Eulerian formulation (see the appendix) are also presented for comparison. We use the highly divergent Gaussian jet case to compare the solutions between the shape-preserving CSLAM-SW solver described by (15) and the LKM with a shape-preserving explicit transport scheme (section 4d).

### a. A radially propagating gravity wave

*f*= 0) 2D radially propagating gravity wave is initiated by a circular height perturbation

*h*′ and advected by a mean background flow:

*h*

_{0}is the initial background height, Δ

*h*is the magnitude of the initial height perturbation,

*x*,

_{c}*y*) is the center of a 200 km × 200 km domain. We perform tests for two different initial height perturbations: a linear case with Δ

_{c}*h*= 10 m and

*h*

_{0}= 990 m; and a nonlinear case with Δ

*h*= 500 m and

*h*

_{0}= 1000 m. A reduced gravitational acceleration of

*g*′ ≈ 0.0204 m s

^{−2}is used, giving an initial gravity wave speed

^{−1}for the two cases, respectively. The mean advection speed

^{−1}) in the atmosphere to the speed of the jet stream (≈100 m s

^{−1}). The background flow velocities

*u*

_{0}≠

*υ*

_{0}are also chosen to ensure that the flow does not align with the mesh.

The model domain consists of 400 × 400 grid cells, with a grid spacing of Δ*x* = Δ*y* = 500 m, and is periodic in both *x* and *y* directions. Since there is no analytical solution to the test problem, to evaluate CSLAM-SW, we produce a fine-resolution Eulerian reference solution with a grid spacing of Δ*x* = Δ*y* = 100 m and a time step of Δ*t* = 10 s. The center of the gravity wave disturbance in the reference solution is stationary (i.e., *u*_{0} = *υ*_{0} = 0 m s^{−1}), and we compare the solutions by translating the gravity wave disturbance in CSLAM-SW to the center of the domain.

*l*

_{2}norm of error as the error measure, which for a uniform mesh is

*i*,

*j*are the grid indices,

*h*(

*i*,

*j*) is the model solution, and

*h*

_{ref}(

*i*,

*j*) is the Eulerian high-resolution reference solution. The

*l*

_{2}norm of error in the height field (at time

*T*= 1 × 10

^{5}s) for different time-step sizes is shown in Fig. 3 for all four models. Results from both the linear and nonlinear initial perturbations are plotted. The time truncation error in CSLAM-SW is very comparable to those in the other two semi-Lagrangian models for both cases. Except for the Eulerian model, all model solutions converge as the time-step size is reduced to less than Δ

*t*= 50 s. At this point, differences between the errors are mainly due to the spatial discretization schemes (more noticeably in the nonlinear case). The Eulerian model and the traditional semi-Lagrangian model have a commonality that they both use a “true” C-grid divergence operator in the continuity equation; whereas as discussed in section 3, the CISL computation of divergence in both CSLAM-SW and LKM consists of an extra averaging operator. For this reason, one may see a smaller spatial discretization error in the traditional semi-Lagrangian model and “coarse” Eulerian model when compared to an Eulerian high-resolution reference solution than those in the CISL models, as is the case in Fig. 3.

Comparison of the height field *L*_{2} error norms for the radially propagating gravity wave solutions. Errors are plotted at time *T* = 1 × 10^{5} s for the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) test cases computed on a 500-m mesh. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Comparison of the height field *L*_{2} error norms for the radially propagating gravity wave solutions. Errors are plotted at time *T* = 1 × 10^{5} s for the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) test cases computed on a 500-m mesh. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Comparison of the height field *L*_{2} error norms for the radially propagating gravity wave solutions. Errors are plotted at time *T* = 1 × 10^{5} s for the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) test cases computed on a 500-m mesh. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

To evaluate the consistency in CSLAM-SW and LKM, a constituent with an initially constant specific concentration distribution (*q*_{0} = 1) is initialized in each model. The CSLAM explicit transport scheme conserves constituent mass in both models; however, as discussed in section 2c, when numerical consistency is violated, constancy of the specific concentration is not guaranteed, and generation or removal of constituent mass is possible. The specific concentration is diagnosed by decoupling the constituent mass variable using (11). A time step of Δ*t* = 100 s is used. Figure 4 shows an example of the specific concentration error in LKM at time *T* = 1 × 10^{5} s for both the linear and nonlinear perturbation cases. The error is largest near the leading edge of the gravity wave, where the flow is most divergent and the semi-implicit correction term is nonzero. Figure 5 shows the variation in error with time step size for both the linear and nonlinear perturbations at the same simulation time as in Fig. 4. The error measures used are the maximum absolute error, the mean absolute error, and the root-mean-squared error. Errors in the solutions from LKM and CSLAM-SW are shown in solid and dashed lines, respectively. Since the inconsistent semi-implicit correction in (5) is proportional to Δ*t*, errors in the scalar field grow with time-step size, which can become a major issue for semi-Lagrangian models that take advantage of larger stable time steps. For the nonlinear test, the maximum absolute error from LKM is in the order of 10^{−2} to 10^{−1}, and is significant for constituents like water vapor, which has a typical mixing ratio of roughly 0.1%–3% in air. On the other hand, CSLAM-SW using a consistent formulation is free-stream preserving (up to machine roundoff) for both cases and all time-step sizes tested.

Specific concentration error (*q* − *q*_{0}) in LKM for a divergent flow initialized with a constant *q*_{0} = 1 in the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) height perturbation cases. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific concentration error (*q* − *q*_{0}) in LKM for a divergent flow initialized with a constant *q*_{0} = 1 in the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) height perturbation cases. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific concentration error (*q* − *q*_{0}) in LKM for a divergent flow initialized with a constant *q*_{0} = 1 in the (a) linear (Δ*h* = 10 m and *h*_{0} = 990 m) and (b) nonlinear (Δ*h* = 500 m and *h*_{0} = 1000 m) height perturbation cases. Note the different scales in the plots.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Variation of specific concentration error (*q* − *q*_{0}) (maximum absolute error, mean absolute error, and root-mean-square error) with time-step size in LKM (solid line) and CSLAM-SW (dashed line) for the (a) linear height perturbation and (b) nonlinear height perturbation cases.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Variation of specific concentration error (*q* − *q*_{0}) (maximum absolute error, mean absolute error, and root-mean-square error) with time-step size in LKM (solid line) and CSLAM-SW (dashed line) for the (a) linear height perturbation and (b) nonlinear height perturbation cases.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Variation of specific concentration error (*q* − *q*_{0}) (maximum absolute error, mean absolute error, and root-mean-square error) with time-step size in LKM (solid line) and CSLAM-SW (dashed line) for the (a) linear height perturbation and (b) nonlinear height perturbation cases.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

### b. Bickley jet—*Ro* = 0.1

The stability of CSLAM-SW is further evaluated with two perturbed jets; we begin with the Bickley jet from Poulin and Flierl (2003). The Bickley jet is simulated at the Rossby number, Ro = *U*/*fL* = 0.1, where *U* is the flow velocity scale, *f* is the Coriolis parameter, and *L* is the length scale of the jet width. We choose the Froude number, Fr = (*fL*)^{2}/*g*′*H* = 0.1. The jet is characterized by greater heights to the left of the channel and dropping off to smaller heights to the right, geostrophically balanced by a mean flow velocity down the channel (Fig. 6). A height perturbation is superimposed at the initial time, causing wave amplification and eventual breaking of the jet into vortices, and formation of a vortex street along the channel. These vortex streets consist of thin filaments of vorticity with strong horizontal velocity shear, making it a good test because it is challenging for all numerical schemes. A more detailed description of the evolution of these jets can be found in Poulin and Flierl (2003).

Initial mean (top) height *h*_{0} and (bottom) velocity *υ*_{0} profiles for the Bickley jet (Δ*h* = 1 m, Δ*υ* = 1 m s^{−1}) and Gaussian jet (Δ*h* = 50 m, Δ*υ* = 56 m s^{−1}).

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Initial mean (top) height *h*_{0} and (bottom) velocity *υ*_{0} profiles for the Bickley jet (Δ*h* = 1 m, Δ*υ* = 1 m s^{−1}) and Gaussian jet (Δ*h* = 50 m, Δ*υ* = 56 m s^{−1}).

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Initial mean (top) height *h*_{0} and (bottom) velocity *υ*_{0} profiles for the Bickley jet (Δ*h* = 1 m, Δ*υ* = 1 m s^{−1}) and Gaussian jet (Δ*h* = 50 m, Δ*υ* = 56 m s^{−1}).

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

*u*

_{0},

*υ*

_{0}, and

*h*

_{0}) and height perturbation

*h*′ of the Bickley jet is given by

*h*is the maximum amplitude of the height perturbation and depends on Ro,

*g*′ is the gravitational acceleration,

*a*is the jet width,

*Y*is the length of the channel, and

*n*is the wavenumber mode of the height perturbation. In our simulations,

*L*=

*a*= 1 × 10

^{5}m,

*X*(width of channel) =

*Y*= 2 × 10

^{6}m,

*f*= 1 × 10

^{−4}s

^{−1}, and

*g*′ = 10 m s

^{−2}. For the specified scale of the jet width and a flow with Fr = 0.1, the mean height of

*h*

_{0}is 100 m. The amplitude of the height perturbation Δ

*h*= 1 m is determined by the scale of the initially geostrophically balanced flow speed (

*U*~ 1 m s

^{−1}) for Ro = 0.1. We choose the most unstable mode of wavenumber

*n*= 3 (Poulin and Flierl 2003) for all of our jet simulations.

Each grid domain has 202 × 202 grid cells and a grid spacing of Δ*x* = Δ*y* = 9950 m, with solid boundary conditions at *x* = −*X*/2 and *x* = *X*/2 and periodic boundary conditions in *y* where *y* ∈ [−*Y*/2, *Y*/2]. A time step of Δ*t* = 2000 s was used in all simulations. Based on the initial gravity wave speed *c* ≈ 32 m s^{−1} and initial flow speed |*υ*| = 1 m s^{−1}, the Courant numbers are Cr_{gw} = 6.4 and Cr_{adv} = 0.2, respectively.

To maintain numerical stability in the Eulerian model, we implemented a second-order explicit diffusion term with a numerical viscosity parameter *β _{x}* =

*β*=

_{y}*ν*Δ

*t*/Δ

*x*

^{2}= 0.02 (where

*ν*is analogous to the physical viscosity). This value corresponds to the numerical Reynolds number, Re =

*UL*/

*ν*= 10

^{2}, a factor of 10 smaller than that used in the forward-in-time Eulerian model of Poulin and Flierl (2003). Explicit diffusion was not applied to any of the semi-Lagrangian models because the schemes have sufficient inherent damping to maintain numerical stability. For the traditional semi-Lagrangian model, however, we found that time off centering in the semi-implicit scheme was needed to maintain stability.

Figure 7 shows the solutions from CSLAM-SW and the three comparison models. Although the exact form of the initial height perturbation was not provided in Poulin and Flierl (2003), we were able to reproduce results very similar to theirs [cf. Fig. 4c of Poulin and Flierl (2003)]. The most noticeable difference among the different model solutions is in the shape and magnitude of the relative vorticity maxima and minima. CSLAM-SW showed very similar vortex shapes to those from LKM and TRAD-SL. The vortices in the Eulerian results are similar to those from the Eulerian model of Poulin and Flierl (2003). The difference between the Eulerian solution and the semi-Lagrangian solutions can be attributed to the inherent damping in the reconstruction step of the CISL schemes and the gridpoint interpolation in the traditional semi-Lagrangian scheme.

Solutions of the Bickley jet at time *T* = 5 × 10^{6} s (after 2500 time steps) for Ro = 0.1, Fr = 0.1 and Cr_{adv} = 0.2. Plotted are positive (solid line) and negative (dashed line) vorticity between −1 × 10^{−5} s^{−1} and 1 × 10^{−5} s^{−1} with a contour interval of 5 × 10^{−7} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Solutions of the Bickley jet at time *T* = 5 × 10^{6} s (after 2500 time steps) for Ro = 0.1, Fr = 0.1 and Cr_{adv} = 0.2. Plotted are positive (solid line) and negative (dashed line) vorticity between −1 × 10^{−5} s^{−1} and 1 × 10^{−5} s^{−1} with a contour interval of 5 × 10^{−7} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Solutions of the Bickley jet at time *T* = 5 × 10^{6} s (after 2500 time steps) for Ro = 0.1, Fr = 0.1 and Cr_{adv} = 0.2. Plotted are positive (solid line) and negative (dashed line) vorticity between −1 × 10^{−5} s^{−1} and 1 × 10^{−5} s^{−1} with a contour interval of 5 × 10^{−7} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

### c. Gaussian jet—*Ro* = 5.0

*h*= 50 m for Ro = 5.0, and Δ

*t*= 100 s is used. With an initial gravity wave speed and maximum flow speed of 38 and 56 m s

^{−1}, respectively, Cr

_{gw}= 0.4 and Cr

_{adv}= 0.56. We note that

*U*>

*c*(i.e., the flow is supercritical). Despite the existence of supersonic waves in the solution, CSLAM-SW is stable even at larger Courant numbers.

As pointed out in Poulin and Flierl (2003), jets in this Rossby regime are highly unstable and of particular interest is the formation of an asymmetric vortex street with triangular cyclones and elliptical anticyclones. As the vortex street is advected toward the deeper water, a strong cutoff cyclone develops due to vortex stretching (adjacent to the main anticyclonic feature). All of our models, including CSLAM-SW, were able to reproduce these features [Fig. 8; cf. Fig. 10e in Poulin and Flierl (2003)]. As in the Bickley jet case, we find that CSLAM-SW produced solutions similar to the other two semi-Lagrangian models (LKM and TRAD-SL).

Solutions of the Gaussian jet for Ro = 5.0 and Cr_{adv} = 0.56 at time *T* = 1.8 × 10^{5} s (after 1800 time steps). Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} s^{−1} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Solutions of the Gaussian jet for Ro = 5.0 and Cr_{adv} = 0.56 at time *T* = 1.8 × 10^{5} s (after 1800 time steps). Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} s^{−1} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Solutions of the Gaussian jet for Ro = 5.0 and Cr_{adv} = 0.56 at time *T* = 1.8 × 10^{5} s (after 1800 time steps). Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} s^{−1} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

In addition to comparing solutions of CSLAM-SW at time steps allowable by the Eulerian scheme, we also tested the stability of CSLAM-SW at a much larger Cr_{adv} = 2.5. Figures 9a–c show solutions at various times from the previous CSLAM-SW simulation (Cr_{adv} = 0.56), and Figs. 9d–f show solutions at each of the corresponding time for Cr_{adv} = 2.5, using the largest time step allowable by the Lipschitz condition for this flow. The solution from the Cr_{adv} = 2.5 simulation is almost identical to the solution using Cr_{adv} = 0.56.

CSLAM-SW solutions of the Gaussian jet for Ro = 5.0 at three different times (from left to right on each row) of the simulation at time *T* = 5 × 10^{4}, 1.0 × 10^{5}, and 1.4 × 10^{5} s. (a)–(c) Solutions using a Cr_{adv} of 0.56 (same simulation as in Fig. 8) (d)–(f) Solutions using a larger Cr_{adv}. of 2.5. Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

CSLAM-SW solutions of the Gaussian jet for Ro = 5.0 at three different times (from left to right on each row) of the simulation at time *T* = 5 × 10^{4}, 1.0 × 10^{5}, and 1.4 × 10^{5} s. (a)–(c) Solutions using a Cr_{adv} of 0.56 (same simulation as in Fig. 8) (d)–(f) Solutions using a larger Cr_{adv}. of 2.5. Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

CSLAM-SW solutions of the Gaussian jet for Ro = 5.0 at three different times (from left to right on each row) of the simulation at time *T* = 5 × 10^{4}, 1.0 × 10^{5}, and 1.4 × 10^{5} s. (a)–(c) Solutions using a Cr_{adv} of 0.56 (same simulation as in Fig. 8) (d)–(f) Solutions using a larger Cr_{adv}. of 2.5. Plotted are positive (solid line) and negative (dashed line) vorticity between −5 × 10^{−4} and 5 × 10^{−4} s^{−1} with a contour interval of 5 × 10^{−5} s^{−1}.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

The CSLAM-SW is numerically stable for the highly nonlinear flow in the Gaussian jet and at Courant numbers much greater than unity. To check that consistency and shape preservation in such a highly divergent flow can be maintained, we repeat the Gaussian jet case using CSLAM-SW and the shape-preserving extensions described in section 3.

### d. Gaussian jet—*Ro* = 5.0 with shape preservation

The shape-preserving CSLAM-SW solver in (15) is tested using the divergent flow of the Gaussian jet as described in section 4c. We also test the LKM solver with the Barth and Jespersen (1989) filter implemented in the explicit scalar transport scheme of *t* = 100 s is used for results in Figs. 10 and 11.

Specific concentration error (*q* − *q*_{0}) in LKM for the Gaussian jet at time *T* = 1.8 × 10^{5} s, initialized with a constant *q*_{0} = 1 field.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific concentration error (*q* − *q*_{0}) in LKM for the Gaussian jet at time *T* = 1.8 × 10^{5} s, initialized with a constant *q*_{0} = 1 field.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific concentration error (*q* − *q*_{0}) in LKM for the Gaussian jet at time *T* = 1.8 × 10^{5} s, initialized with a constant *q*_{0} = 1 field.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific constituent concentration *q* at time *T* = 1.8 × 10^{5} s. Initial minimum and maximum *q* are 0.1 and 1.0, respectively. Regions with unphysical overshooting (red) and undershooting (purple) are highlighted.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific constituent concentration *q* at time *T* = 1.8 × 10^{5} s. Initial minimum and maximum *q* are 0.1 and 1.0, respectively. Regions with unphysical overshooting (red) and undershooting (purple) are highlighted.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

Specific constituent concentration *q* at time *T* = 1.8 × 10^{5} s. Initial minimum and maximum *q* are 0.1 and 1.0, respectively. Regions with unphysical overshooting (red) and undershooting (purple) are highlighted.

Citation: Monthly Weather Review 141, 7; 10.1175/MWR-D-12-00275.1

To test for numerical consistency in the two solvers, we repeat the consistency test described in section 4a by initializing a constant specific concentration field *q*_{0} = 1. The shape-preserving CSLAM-SW solution is able to maintain numerical consistency between *h* and *hq* up to machine roundoff for this highly divergent flow and the result is independent of time-step size. As for LKM, despite the shape-preserving transport scheme in the solver, numerical inconsistency is still an issue with a maximum absolute error (defined as the deviation from *q*_{0} = 1) of 6.79 × 10^{−3}, a mean absolute error of 4.82 × 10^{−4}, and a root-mean-square error of 1.06 × 10^{−3} at time *T* = 1.8 × 10^{5} s (Fig. 10), and as in section 4a, the error is a function of the time-step size (not shown).

*x*direction and has a sharp gradient that coincides with the center of the initial jet:

*q*diagnosed from

*hq*from the nonshape-preserving CSLAM-SW, LKM with shape-preserving transport, and the shape-preserving CSLAM-SW are presented in Figs. 11a–c. The simulation time

*T*= 1.8 × 10

^{5}s in the figure corresponds to the vorticity field shown in Fig. 8.

For the nonshape-preserving CSLAM-SW solver (Fig. 11a), *q* reaches an unphysical peak value of 1.233 and an unphysical minimum value of −0.145 (specific concentrations cannot be negative). The LKM solver with shape-preserving transport (Fig. 11b) has less severe errors than the nonshape-preserving CSLAM-SW, but loses its shape-preserving ability as a result of numerical inconsistency. The minimum and maximum *q* values are 0.099 97 and 1.0063, respectively, at time *T* = 1.8 × 10^{5} s. The overshooting of *q* (which may generate spurious constituent mass) appears to be greater in amplitude than the undershooting for this flow. Overshooting occurs mostly within the strongest anticyclones (negative vorticity centers on the left side of the channel, highlighted in solid black lines in Fig. 11b). Using the shape-preserving CSLAM-SW solver (Fig. 11c), minimum and maximum values of *q* are kept within its physical limits (0.1 and 1.0, respectively, up to machine roundoff) and shape preservation is ensured.

## 5. Conclusions

A conservative and consistent semi-Lagrangian semi-implicit solver is constructed and tested for shallow-water flows (CSLAM-SW). The model uses a new flux-form discretization of the semi-implicit cell-integrated semi-Lagrangian continuity equation that allows a straightforward implementation of a consistent constituent transport scheme. Like typical conservative semi-Lagrangian semi-implicit schemes, the algorithm requires at each time step a single Helmholtz equation solution and a single application of CSLAM.

Specifically, our new discretization uses the flux divergence as opposed to a velocity divergence that requires linearization about a constant mean reference state. For traditional semi-implicit schemes, the dependence on a constant mean reference state makes it difficult to ensure consistency between total fluid mass and constituent mass. When numerical consistency is not maintained, constituent mass conservation can be violated even for solvers that use inherently conservative transport schemes. More unacceptably, constituent fields may no longer preserve their shapes (e.g., losing constancy or positive definiteness).

We have shown an example of a traditional discrete cell-integrated semi-Lagrangian semi-implicit continuity equation (LKM), in which inconsistency can generate significant numerical errors in the specific constituent concentration. The inconsistent semi-implicit correction term in LKM causes errors to grow proportionally with time-step size and with the nonlinearity of the flow. The ideal radially propagating gravity wave tests using the LKM solver showed a maximum absolute error in an initially constant specific concentration (*q*_{0} = 1) field ranging from an order of 10^{−7} to 10^{−3} in the linear case, and an order of 10^{−4} to 10^{−1} in the nonlinear case. The orders of magnitude of these errors are significant relative to the specific concentration of tracers and water vapor in the atmosphere. The consistent formulation in the new CSLAM-SW on the other hand eliminates these errors (up to machine roundoff).

The new flux-form solver (CSLAM-SW) is tested for a range of flows and Courant numbers for the shallow-water system, and is stable and compares well with other existing semi-implicit schemes, including a two-time-level traditional semi-Lagrangian scheme and an Eulerian leapfrog scheme. The Gaussian jet test (the more nonlinear jet of the two presented) showed that CSLAM-SW remains numerically stable when large time steps are used.

We have also identified and eliminated a computational unstable mode in CSLAM-SW and LKM, using the discrete dispersion relation of the linearized shallow-water equations. The numerical instability, associated with the Lagrangian divergence operator on a C grid, can be eliminated by introducing a new averaging operator on the Coriolis terms in the momentum equations.

Shape preservation in CSLAM-SW is ensured by applying a 2D shape-preserving filter in the CSLAM transport scheme and the first-order upwind scheme to compute the predictor-corrector and flux-form correction terms. As shown in the Gaussian jet case, without any shape-preserving filter, unphysical negative and unreasonable positive specific concentrations may develop as a result of undershoots and overshoots. For inconsistent formulations such as that in LKM, the use of a shape-preserving explicit transport scheme cannot guarantee shape preservation either because of numerical consistency errors. CSLAM-SW, on the other hand, allows for straightforward implementation of existing shape-preserving schemes and filters and ensures shape preservation (up to machine roundoff).

The initial testing of the semi-implicit formulation in CSLAM-SW shows promising results. We are currently implementing the extension of CSLAM-SW to a 2D (*x*–*z*) nonhydrostatic, fully compressible atmospheric solver. The desirable properties of mass conservation, consistency, and shape preservation for moisture variables and tracers will likely be important for both short- and long-term meteorological applications.

## Acknowledgments

This work was done as a part of the National Center for Atmospheric Research Graduate Visitor Advanced Study Program. The authors thank Joseph Klemp for his suggestions on the dispersion relation analysis of CSLAM-SW. 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.

## APPENDIX

### Numerical Schemes for Comparison

#### a. A two-time-level traditional semi-Lagrangian semi-implicit model

*β*for numerical stability purposes. The discretized system is given by

*h*′ =

*h*−

*H*

_{0}. The operators are defined as

The *R ^{n}* terms define the known terms that are evaluated at time level

*n*and interpolated to the departure point. The

*n*and

*n*− 1 to time level

*β*is set to 0.1 for all runs.

#### b. An Eulerian leapfrog semi-implicit advective model

*β*= 0.1) to eliminate spurious oscillations. Numerical viscosity is also applied for certain test cases (see section 4b). Using the same notations as for the traditional semi-Lagrangian model, the discretized system is given by

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