  We prove stability for arbitrarily long times of the zero solution for the
so-called $\beta$-plane equation, which describes the motion of a
two-dimensional inviscid, ideal fluid under the influence of the Coriolis
effect. The Coriolis force introduces a linear dispersive operator into the 2d
incompressible Euler equations, thus making this problem amenable to an
analysis from the point of view of nonlinear dispersive equations. The
dispersive operator, $L_1:=\frac{\partial_1}{|\nabla|^2}$, exhibits good decay,
but has numerous unfavorable properties, chief among which are its anisotropy
and its behavior at small frequencies.
