Abstract: The quasi-geostrophic equation is considered in this paper.A criterion is given for the finite time blow-up of a classical solution in terms of its small time behaviors along the trajectories.The result is deduced from a differential inequality for the reciprocal of the length of the gradient of the solution along a particle trajectories.The method and the result are similar to those of Chae on the 3 dimensional incompressible Euler system.The result gives some inspiration for the further research of quasi-geostrophic equation.

Key words: Quasi-Geostrophic equation, classical solution, blow-up criterion