A scenario for finite time singularity in the quasigeostrophic model

A possible route to finite-time singularity in the quasigeostrophic system, via a cascade of filament instabilities of geometrically decreasing spatial and temporal scales, as suggested by Held et al., 1995 (J. Fluid Mech., vol. 282, 1--20), is investigated using high-resolution numerical integrations. A number of initial temperature distributions are considered, which can be classified into two distinct types according to their spatial continuity. For the case of continuous initial temperature distributions, a hybrid contour-dynamical--pseudo-spectral algorithm is used to minimize dissipation in the direction of maximum temperature gradient and to ensure that maximum temperature levels are conserved exactly. In all cases, primary and secondary instabilities appear to be well-resolved, but the large difference in spatial scale between successive instabilities places severe limitations on the inferences that may be made about the behavior of the exact solutions at extremely small scales. Filament instability is also shown to be potentially important in the closing saddle scenario investigated in many previous studies.

The second type of initial temperature distribution considered is that of a patch of uniform temperature with a single discontinuity along a closed contour at the patch edge. In this case, a pure contour dynamical method is used to solve the equations in an infinite domain, without the need for an underlying numerical grid. The contour is represented by a set of points connected by global splines to ensure continuity of the tangent vector and its first derivative; the points are advected by the velocity field obtained by an integral around the closed contour. Increasing the point density around the contour in regions of high curvature and strain allows accuracy to be retained even as scales shrink over several orders of magnitude. Evidence is presented of singularity formation in finite time, which may involve either the formation of a self-similar corner region or the repeated instability of filaments of ever shrinking spatial and temporal scales, depending on details of the initial patch shape.