QUASI-MONOTONE ADVECTION SCHEMES BASED ON EXPLICIT LOCALLY ADAPTIVE DISSIPATION

The authors develop and test computational methods for advection of a scalar field that also include a minimal dissipation of its variance i n order to preclude the formation of false extrema. Both of these prop erties are desirable for advectively dominated geophysical flows, wher e the relevant scalars are both potential vorticity and material conce ntrations. These methods are based upon the sequential application of two types of operators: 1) a conservative and nondissipative (i.e., pr eserving first and second spatial moments of the scalar field), direct ionally symmetric advection operator with a relatively high order of s patial accuracy; and 2) a locally adaptive correction operator of lowe r spatial accuracy that eliminates false extrema and causes dissipatio n. During this correction phase the provisional distribution of the ad vected quantity is checked against the previous distribution, in order to detect places where the previous values were overshot, and thus to compute the excess. Then an iterative diffusion procedure is applied to the excess field in order to achieve approximate monotone behavior of the solution. In addition to the traditional simple flow tests, we have made long-term simulations of freely evolving two-dimensional tur bulent flow in order to compare the performance of the proposed techni que with that of previously known algorithms, such as UTOPIA and FCT. This is done for both advection of vorticity and passive scalar. Unlik e the simple test flows, the turbulent flow provides nonlinear cascade s of quadratic moments of the advected quantities toward small scales, which eventually cannot be resolved on the fixed grid and therefore m ust be dissipated. Thus, not only the ability of the schemes to produc e accurate shape-preserving advection, but also their ability to simul ate subgrid-scale dissipation are being compared. It is demonstrated t hat locally adaptive algorithms designed to avoid oscillatory behavior in the vicinity of steep gradients of the advected scalars may result in overall less dissipation, yet give a locally accurate and physical ly meaningful solution, whereas algorithms with built-in hyperdiffusio n (i.e., those traditionally used for direct simulation of turbulent f lows) tend to produce a locally unsufficient and, at the same time, gl obally excessive amount of dissipation. Finally, the authors assess th e practial trade-offs required for large models among the competing at tributes of accuracy, extrema preservation, minimal dissipation (e.g., appropriate to large Reynolds numbers), and computational cost.