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

Abstract: 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 $\ell_{|}$, the growth rate of the columnar eddies and its dependency on the Rossby number $ Ro_{\varepsilon}$ 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 $\varepsilon$ is sought. Comparison with current available scaling laws shows that the relation proposed by Baqui & Davidson (2015), i.e., $\varepsilon\sim u'^3/\ell_{|}$, where $u's$ is the r.m.s. velocity, approximates well part of our data, more specifically the range $0.39\le Ro_{\varepsilon} \le 1.54$. However, relations proposed in the literature fail to model the data for the second and most interesting range, i.e., $0.06\le Ro_{\varepsilon} \le 0.31$, 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.