Abstract
The Piecewise Parabolic Method (PPM), a numerical technique developed in astrophysics for modeling fluid flows with strong shocks and discontinuities is adapted for treating sharp gradients in small-scale meteorological flows. PPM differs substantially from conventional gridpoint techniques in three ways. First, PPM is a finite volume scheme, and thus represents physical variables as averages over a grid zone rather than single values at discrete points. Second, a unique, monotonic parabola is fit to the zone average of each dependent variable using information from neighboring zone averages. As shown in a series of one- and two-dimensional linear advection experiments, the use of parabolas provides for extremely accurate advection, particularly of sharp gradients. Furthermore, the monotonicity constraint renders PPM's solutions free from Gibbs oscillations. PPM's third attribute is that each zone boundary is treated as a discontinuity. Using the method of characteristic the nonlinear flux of quantities between zones is obtained by solving a Riemann problem at each zone boundary in alternating one-dimensional sweeps through the grid. This methodology provides a highly accurate, physically based solution both in the vicinity of sharp gradients and in regions where the flow is smooth.
The ability of PPM to accurately depict the evolution of sharp gradients in small-scale, nonlinear flows is examined by simulating a buoyant thermal and a density current in two dimensions. Comparisons made against Midpoint cloud models reveal that PPM provides superior solutions at equivalent spatial resolution, particularly with regard to resolving shear lines that subsequently become unstable. The PPM model has excellent mass and energy conservation properties, and exhibits virtually no numerical dissipation of resolvable modes. Although PPM is not yet as economical as a conventional gridpoint model, we anticipate that its efficiency can be greatly improved by modifying the treatment of acoustic modes.
