  We consider the propagation of linear gravity waves on the free surface of
steady, axisymmetric flows with purely azimuthal velocity. We propose a
two-dimensional set of governing equations for surface waves valid in the
deep-water limit. These equations come from a closure condition at the free
surface that reduces the three-dimensional Euler equations in the bulk of the
fluid to a set of two-dimensional equations applied only at the free surface.
Since the closure condition is not obtained rigorously, it is validated
numerically through comparisons with full three-dimensional calculations for
vortex flows, including for a Lamb--Oseen vortex. The model presented here
overcomes three limitations of existing models, namely: it is not restricted to
potential base flows; it does not assume the base flow to have a flat free
surface; and it does not require the use of infinite-order differential
operators (such as $\tanh(\nabla)$) in the governing equations. The model can
be applied in the case of rapid swirl (large Froude number) where the base free
surface is substantially deformed. Since the model contains only derivatives of
finite order, it is readily amenable to standard numerical study.
