Numerical Simulation of a Self-Similar Cascade of Filament Instabilities in the Surface Quasigeostrophic System

We provide numerical evidence for the existence of a cascade of filament instabilities in the surface quasigeostrophic system for rotating, stratified flow near a horizontal boundary. The cascade involves geometrically shrinking spatial and temporal scales and implies the singular collapse of the filament width to zero in a finite time. The numerical method is both spatially and temporally adaptive, permitting the accurate simulation of the evolution over an unprecedented range of spatial scales spanning over ten orders of magnitude. It provides the first convincing demonstration of the cascade, in which the large separation of scales between subsequent instabilities has made previous numerical simulation difficult.

Figure 1

Snapshots of the patch boundary at selected times prior to the onset of the first filament instability; $\theta =2\pi$ inside the patch boundary and $\theta =0$ outside. The final frame shows an $8\times$ magnification of a portion of the central filament.

Figure 2

Maximum curvature and inverse filament width as a function of (a) time $t$ and (b) rescaled time $\tau =-\log (t_s-t)$, with $t_s=1.7917134235333878$. Corresponding times in each panel are indicated by the diamond symbols; the dashed lines indicate the function $1/(t_s-t)$. Note the double-log scale in (a) and the single-log scale in (b).

Figure 3

Contour shape at time $t=1.79171342351654$, or $\tau =24.8068$, at successive magnifications of $20\times$ (linear dimension). The central shaded square in each panel is shown magnified in the adjacent panel of the same color.