There has recently been much interest in the so-called quasigeostrophic thermal active scalar equations. In particular, numerical studies of strong front formations in these equations have been done in a attempt to determine whether finite-time singularities can occur. The equations themselves describe the evolution of temperature on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. Hence there is an interest in the geophysical community in their behavior. Constantin, Majda, and Tabak have also shown that there is a compelling analogy between the growth rate of the perpendicular gradient of the scalar field and vortex stretching in the three-dimensional Euler equations. Therefore finite time singularities in the quasigeostrophic equations may shed light on the open question of singularity formation in 3D Euler.
Previously completed numerical studies have used psuedo-spectral methods to integrate the equations mainly because of the need to invert a non-standard elliptic operator (the "square root" of the Laplacian) which diagonalizes in Fourier space. Subsequently these methods use a uniform numerical grid which quickly loses resolution near front formation. To produce an adaptive mesh method for this equation, an adaptive routine for inverting the square root of the Laplacian was created in collaboration with Leslie Greengard of the Courant Institute and Zydrunas Gimbutas (currently at MadMax optics). See the references below for a copy of the paper. The method utilizes integral equations and the Fast Multipole method. This algorithm has been successfully incorported into an AMR code for the quasi-geostrophic scalar and numerical simulations are planned. As can be seen in the figure below, the formation of fronts for this problem (which is one studied by Tabak, etal.) occurs in a relatively small area of the pysical domain, hence adaptive mesh techniques should prove very valuable.