This test is described in Straka et al. (1993). Cold air is placed in a neutrally stratified environment; the cold air falls, and then spreads along the surface as a gravity current. A constant viscosity (75 m^2/s) is used, which allows for a converged solution, provided sufficient resolution is used.

**Test of the pressure solvers:**

This test case is first used to evaluate the pressure solvers in CM1. Results for grid spacing of 50 m are shown in this figure:

Potential temperature perturbation is contoured every 0.5 K (starting at -0.5 K).

The three compessible solvers (panels a-c) yield essentially identical results. There is no perceptible loss in accuracy from using the time-split solver (panels b-c). As expected, the acoustic damping terms (both explicit and implicit) have no substantive impact on the flow of interest.

The simulation using the anelastic solver (panel d) produces a very similar solution, although some minor differences can be seen. Most notably, the gravity current is slightly slower with this option (as compared to results from the compressible solver); at the moment, I have no explanation for this difference.

**Convergence tests:**

Results using sixth-order advection with different resolution are shown in this figure:

Potential temperature perturbation is contoured every 0.5 K (starting at -0.5 K).

Because there is no numerical diffusion with the sixth-order advection scheme, the dispersive errors at coarse resolution are obvious. As resolution improves, the correct solution is obtained.

Results using fifth-order advection with different resolution are shown in this figure:

Potential temperature perturbation is contoured every 0.5 K (starting at -0.5 K).

Because this advection scheme contains high-order flow-dependent diffusion, the results at low resolution are much more acceptable than results from the sixth-order advection scheme. With high resolution (e.g., 50 m grid spacing), results from the fifth-order and sixth-order advection schemes are essentially identical; this suggests that a converged solution is obtained for this grid spacing.

A quantitative evaluation of the differences in resolution are provided by "self-convergence" tests, wherein the grid spacing is changed incrementally with everything else (e.g., timestep) held fixed. See Straka et al. (1993) for more details. Results in terms of the L2 Norm for potential temperature are shown in this figure:

For grid spacing of 200 m or less, both advection schemes produce a second-order convergence rate.

**References:**

Straka, J. M., R. B. Wilhelmson, L. J. Wicker, J. R. Anderson, K. K. Droegemeier, 1993: Numerical solutions of a nonlinear density current: A benchmark solution and comparisons. *Int. J. Numer. Methods Fluids,* **17,** 1-22.