Validation of compressible numerical models is limited by the availability of analytic solutions for non-Boussinesq fluids. One common Boussinesq solution used for model validation is that of a Beltrami flow (Shapiro, 1993). While Boussinesq solutions are valuable and have been used to approximately validate compressible models using very shallow domains, they are not able to provide validation for flow extending over a large depth. In this case, an anelastic solution is better suited. For a Beltrami flow, this requires replacing the constant abnormality with one that is a function of density, allowing the vorticity to remain non-divergent while maintaining the Beltrami relationship. Also the viscous term is not the standard one for compressible flow although it does reduce to the standard one for incompressible flow in the Boussinesq limit. Separating variables results in a Sturm-Liouville equation for the amplitude of vertical velocity as a function of height. This is solved using a Galerkin method, and the velocity and pressure fields are computed. The solutions apply to an anelastic, adiabatic atmosphere with a rigid lid and rigid bottom and a no-slip (free-slip) condition on the rotational (divergent) wind in these surfaces. Periodic conditions apply at the lateral boundaries. The Beltrami flow solution can be used to test advection, pressure gradients and viscous dissipation.

This solution will be applied to the Advanced Regional Prediction System (ARPS) to show its utility in model validation.