ON THE GEOMETRY OF SOLUTIONS OF THE QUASI-GEOSTROPHIC AND EULER EQUATIONS

We study solutions of the two-dimensional quasi-geostrophic thermal ac tive scalar equation involving simple hyperbolic saddles. There is a n aturally associated notion of simple hyperbolic saddle breakdown, It i s proved that such breakdown cannot occur in finite time, At large tim e, these solutions may grow at most at a quadruple-exponential rate, A nalogous results hold for the incompressible three-dimensional Euler e quation.