  Rapidly rotating, stably stratified three-dimensional inviscid flows conserve
both energy and potential enstrophy. We show that in such flows, the forward
cascade of potential enstrophy imposes anisotropic constraints on the
wavenumber distribution of kinetic and potential energy. The horizontal kinetic
energy is suppressed in the large, nearly horizontal wave modes, and should
decay with the horizontal wavenumber as $k_h^{-3}$. The potential energy is
suppressed in the large, nearly vertical wave modes, and should decay with the
vertical wavenumber as $k_z^{-3}$. These results augment the only other exact
prediction for the scaling of energy spectra due to constraints by potential
enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that
in the quasi-geostrophic approximation for rotating stratified flows, the
energy spectra must scale isotropically with total wavenumber as $k^{-3}$. We
test our predicted scaling estimates using resolved numerical simulations of
the Boussinesq equations in the relevant parameter regimes, and find reasonable
agreement.
