  We study the motion of an incompressible, inviscid two-dimensional fluid in a
rotating frame of reference. There the fluid experiences a Coriolis force,
which we assume to be linearly dependent on one of the coordinates. This is a
common approximation in geophysical fluid dynamics and is referred to as
beta-plane. In vorticity formulation the model we consider is then given by the
Euler equation with the addition of a linear anisotropic, non-degenerate,
dispersive term. This allows us to treat the problem as a quasilinear
dispersive equation whose linear solutions exhibit decay in time at a critical
rate.
  Our main result is the global stability and decay to equilibrium of
sufficiently small and localized solutions. Key aspects of the proof are the
exploitation of a "double null form" that annihilates interactions between
spatially coherent waves and a lemma for Fourier integral operators which
allows us to control a strong weighted norm.
