  We provide numerical evidence for the existence of a cascade of filament
instabilities in the surface quasigeostrophic system for atmospheric and
oceanic motions 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.
