This test is described in Skamarock and Klemp (1994). A small-amplitude temperature perturbation is placed into a stably stratified environment, thus triggering propagating inertia-gravity waves. This test is used here to evaluate the accuracy of the pressure solvers in CM1.

**Nonhydrostatic-scale simulations:**

The initial conditions, domain size, and grid spacing are the same as in Skamarock and Klemp (1994). For the truly compressible simulation (psolver=1), the timestep is 1 s. For the time-split compressible simulations (psolver=2,3), the large timestep is 12 s and the small timestep is 1 s, and potential temperature is integrated on the small timesteps. For the anelastic (psolver=4) and incompressible (psolver=5) simulations, the timestep is 12 s. Single precision is used for all simulations.

Results at t=3000 s are shown in this figure:

Contours are potential temperature perturbation with a contour interval of 0.0005 K (negative contours are dashed).

The analytic solution for incompressible flow is shown in Fig. 1b of Skamarock and Klemp (1994); the results from the incompressible simulation [panel (e) here] compare well with the analytic soluton. For the other results, the variation in density with height cause the results to depart from the incompressible solution.

All of the non-incompressible simulations (psolvers=1,2,3,4) compare well with each other, which verifies the accuracy of these solvers. Most importantly, the time-split compressible solvers (psolvers=2,3), which are used for most applications of CM1, produce essentially identical result as the truly compressible result (psolver=1); there are no indications of any loss of accuracy from the use of time-splitting, or by the use of acoustic dampers (which are used with psolver=2,3, but not with psolver=1).

**Hydrostatic-scale simulations:**

The initial conditions, domain size, and grid spacing are the same as in Skamarock and Klemp (1994). For this test, the vertically propagating acoustic waves make the truly compressible solver (psolver=1) and the fully explicit time-split solver (psolver=2) prohibitively expensive; these solvers would never be used for this application and, thus, results are not shown here. For the time-split compressible simulation (psolver=3), the large timestep is 200 s and the small timestep is 25 s, and potential temperature is integrated on the small timesteps. For the anelastic (psolver=4) and incompressible (psolver=5) simulations, the timestep is 200 s. Single precision is used for all simulations.

Results at t=60,000 s are shown in this figure:

Contours are potential temperature perturbation with a contour interval of 0.0005 K (negative contours are dashed).

The analytic solution for incompressible flow is shown in Fig. 3b of Skamarock and Klemp (1994); the results from the incompressible simulation [panel (c) here] compare well with the analytic soluton.

Results from the compressible solver (psolver=3) and the anelastic solver (psolver=4) are essentially identical, which verifies the accuracy of these solvers. This test also shows that this nonhydrostatic model can simulate hydrostatic flows accurately.

**References:**

Skamarock, W. C., and J. B. Klemp, 1994: Efficiency and accuracy of the Klemp-Wilhelmson time-splitting technique. *Mon. Wea. Rev.,* **122,** 2623-2630.