  In the presence of inertia-gravity waves, the geostrophic and hydrostatic
balance that characterises the slow dynamics of rapidly rotating, strongly
stratified flows holds in a time-averaged sense and applies to the
Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this
wave-averaged balance and illustrate its accuracy in numerical solutions of the
three-dimensional Boussinesq equations, using a simple configuration in which
vertically planar near-inertial waves interact with a barotropic anticylonic
vortex. We further use the conservation of the wave-averaged potential
vorticity to predict the change in the barotropic vortex induced by the waves.
