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 the
-plane
approximation. In vorticity formulation, the model we consider is then given
by the Euler equation with the addition of a linear anisotropic, nondegenerate,
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.
