In this work, we describe a new nonhydrostatic model generalized from Arakawa's potential-enstrophy conserving hydrostatic scheme [Arakawa and Lamb (1981)] and analyze its behavior when applied to a stretched orthogonal curvilinear global-to-urban scale grid having zonally uniform grid spacing near the poles. The model, called PECCAN (Potential Enstrophy Conserving Compressible Atmospheric Nonhydrostatic), solves the governing equations for a fully compressible 3-D nonhydrostatic atmosphere. It is unique because it conserves the domain integrals of all of the following quantities simultaneously: (1) potential enstrophy in the limit of 2-D barotropic frictionless flow, (2) total energy (sum of kinetic, potential, and internal energies) for frictionless flow, (3) mass, and (4) potential enthalpy (the product of specific heat, density, and potential temperature) for adiabatic flow. The governing equations in PECCAN are expressed in general orthogonal curvilinear coordinates. Thus, the model can simulate flows in Cartesian, cylindrical, spherical, and Lambert (polar stereographic and Mercator) coordinate systems, among others, with stretched coordinates in any spatial direction. Stretched coordinates allow the model to cover most of the globe with a coarse horizontal grid while covering one or more regions of interest with a much finer grid. In the vertical, the model uses the altitude coordinate with step topography. To advance the equations forward in time, it can use any one of several explicit time integration schemes. These include the leapfrog, third and fourth order Runge-Kutta, second order Adams-Bashforth, and Heun schemes. To overcome the time step limitation due to sound waves, the model has the option of using a mode-splitting technique in which the velocity divergence terms in the continuity and thermodynamic energy equations and the pressure gradient terms in the momentum equations are advanced using either a small time step explicitly or a large time step implicitly while the remaining meteorologically important terms are advanced using a long time step.

A new feature of the PECCAN model is that it generalizes the potential enstrophy and energy conserving numerical scheme of Arakawa and Lamb (1981) for the 2-D shallow water equations to discretize the nonlinear terms in the governing momentum equations of 3-D atmospheric flow. Various authors have demonstrated that, at least for 2-D shallow water flow, using a numerical scheme that maintains the various global conservation properties of the governing equations, especially that of potential enstrophy (PE), leads to the correct transfer of energy among scales in the model and eliminates spurious numerical sources of momentum and energy. Arakawa and Lamb (1981) have also shown that a PE-conserving numerical scheme for 2-D shallow water flow converges to the correct solution on a significantly coarser grid than does a non-PE-conserving scheme.

To resolve a given region of interest in the flow, one of two approaches has traditionally been used: limited-area boundary conditions or nested grids. However, these approaches each have certain disadvantages. Limited-area boundary conditions can often be mathematically ill-posed, requiring the use of buffer or relaxation zones, or they can be difficult to enforce. Nested grids can violate conservation properties, and they may require the use of buffer zones at grid interfaces to suppress discontinuities in the solution. For these reasons, in this study we use a single stretched grid spanning the globe that adequately resolves the region of interest. If this region is on the urban scale, the grid spacing will generally vary from hundreds of kilometers away from the region down to a kilometer or less in it. However, a stretched spherical coordinate system has its own disadvantage, which is that a small zonal grid spacing in the mid-latitudes leads to an even smaller grid spacing near the poles (because the meridians converge as they near the poles). This can impose a severe limitation on the time step. To avoid this situation, we use a stretched orthogonal coordinate system (ξ,η) in which curves of constant η converge to the poles in such a way that the grid spacing is zonally uniform near the poles. As a result, the grid spacing there will not be smaller than on a regular (i.e. unstretched) spherical grid. We have found that such a grid must have very large grid aspect ratios (> 1000) in some locations, and, for computational efficiency, it might have relatively large (> 10%) changes in grid spacing from one grid box to the next. Traditional schemes used in regional and urban scale models are likely not well-suited for use with such a grid. On the other hand, due to the superior convergence properties of potential enstrophy conserving schemes, it is conceivable that a model using such a scheme can be applied on the global-to-urban scale grid described above more effectively than a model that uses a non-PE-conserving scheme. In this work, we explore this issue by studying the effect of the grid on a synoptic-scale circulation system as it is advected by an otherwise uniform flow through various portions of the grid, e.g. coarsely resolved portions with order one aspect ratios, partially-resolved portions with very large aspect ratios, and finely resolved portions (also with order one aspect ratios).

REFERENCES

Arakawa, A. and Lamb, V. R. (1981). A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Wea. Rev., 109:18-36.