Rossby Number Effects on Columnar Eddy Formation and the Energy Dissipation Law in Homogeneous Rotating Turbulence

Two aspects of homogeneous rotating turbulence are quantified through forced Direct Numerical Simulations in an elongated domain, which is in the direction of rotation about $340$ times larger than the typical initial eddy size. First, by following the time evolution of the integral length-scale along the axis of rotation ${\mathrm{\ell }}_{\mathrm{\parallel }}$, the growth rate of the columnar eddies and its dependency on the Rossby number $R{o}_{\epsilon }$ is determined as $\gamma =4 \exp(-17 Ro_{\varepsilon})$,where$\gamma$ is the non-dimensional growth rate. Second, a scaling law for the energy dissipation rate $\epsilon$ is sought. Comparison with current available scaling laws shows that the relation proposed by Baqui & Davidson (2015), i.e., $\epsilon \sim u^{\mathrm{\prime }3}\mathrm{/}{\mathrm{\ell }}_{\mathrm{\parallel }}$, where $u^{\mathrm{\prime }}s$ is the r.m.s. velocity, approximates well part of our data, more specifically the range $0.39\le R{o}_{\epsilon }\le 1.54$. However, relations proposed in the literature fail to model the data for the second and most interesting range, i.e., $0.06\le R{o}_{\epsilon }\le 0.31$, which is marked by the formation of columnar eddies. To find a similarity relation for the latter, we exploit the concept of a spectral transfer time introduced by Kraichnan (1965). Within this framework, the energy dissipation rate is considered to depend on both the non-linear time-scale and the relaxation time-scale. Thus, by analyzing our data, expressions for these different time-scales are obtained that results in~$\varepsilon\sim u’^4/(\ell_{\bot}^2 Ro_{\varepsilon}^{0.62} \tau_{nl})$,where$\ell_{\bot}$ is the integral length-scale in the direction normal to the axis of rotation and ${\tau }_{nl}$ is the non-liner time-scale of the initial homogeneous isotropic field.