A Phenomenological Theory of Rotating Turbulence

We present direct numerical simulations of statistically-homogeneous, freely-decaying, rotating turbulence in which the Rossby number, $\mbox{Ro}=u_{\bot } /2\Omega \ell_{\bot } $, is of order unity. The initial condition consists of fully-developed turbulence in which Ro is sufficiently high for rotational effects to be weak. However, as the kinetic energy falls, so also does Ro, and quite quickly we enter a regime in which the Coriolis force is relatively strong and anisotropy grows rapidly, with $\ell_{\bot } <<\ell_{//} $. This regime occurs when $\mbox{Ro}\sim \mbox{0.4}$ and is characterised by an almost constant perpendicular integral scale, $\ell_{\bot } \sim \mbox{constant}$, a rapid linear growth in the integral scale parallel to the rotation axis, $\ell_{//} \sim \ell _{\bot } \Omega t$, and a slow decline in the value of Ro. We observe that the rate of dissipation of energy scales as $\varepsilon \sim {u^{3}} \mathord{\left/ {\vphantom {{u^{3}} {\ell_{//} }}} \right. \kern-\nulldelimiterspace} {\ell_{//} }$ and that both the perpendicular and parallel energy spectra exhibit an $k^{-5/3}$ inertial range; $E(k_{\bot } )\sim \varepsilon^{2/3}k_{\bot }^{-5/3} $ and $E(k_{//} )\sim \varepsilon ^{2/3}k_{//}^{-5/3} $. We show that these power-law spectra have nothing to do with Kolmogorov's theory and are not a manifestation of traditional critical balance theory, as this requires $\varepsilon \sim u^{3}/\ell _{\bot } $ and $E(k_{//} )\sim (\varepsilon^{4/5}/\Omega ^{2/5})k_{//}^{-7/5} $. Finally, we develop a spectral theory of the inertial range that assumes that the observed linear growth in anisotropy, $\ell_{//} /\ell_{\bot } \sim \Omega t$, also occurs on a scale-by-scale basis all the way down to the Zeman scale.