Accuracy and Energy Conservation in a Three-Dimensional Anelastic Model

Abstract

Accuracy and energy conservation are examined in a three-dimensional (3D) anelastic model. For both dry and moist (noncondensing) atmospheres, we prescribe analytic solutions of momentum, potential temperature and mixing ratio for both periodic and closed boundaries. Accuracy is assessed by comparing amplitudes and phase speeds from both the numerical and analytic solutions. Kinetic and potential energies and enthalpy (including air, vapor, liquid and latent) are calculated for both the mean and perturbation states. To assess the energetics involved in phase changes, we examine a separate cloud simulation. Two-dimensional (2D) and hydrostatic experiments are also conducted using the cloud simulation. For the linear analytic wave solutions, phase speeds as a function of time step for our semi-implicit model are compared to both implicit and explicit linear stability generated speeds. We show that an explicit scheme enhances the phase speed up to the CFL cutoff while an implicit scheme retards the phase speed. For the quasi-Lagrangian method of moisture advection, we find that a water conservation algorithm is necessary to maintain conservation of total perturbation energy. Similarly. the correct inclusion of moisture in the computation of density is most critical to energy conservation. In comparing a 2D forced cloud to the 3D simulation, only 17% of the perturbation energy which changes form in the 3D case does so in the 2D experiment-in direct relation to the larger cloud in the 3D simulation. And finally, comparing experiments both with and without the hydrostatic assumption, we verify earlier 2D findings that the magnitude of the vertical motion is larger in a hydrostatic model.