Dynamic scaling in rotating turbulence: A shell model study

We investigate the scaling form of appropriate timescales extracted from time-dependent correlation functions in rotating turbulent flows. In particular, we obtain precise estimates of the dynamic exponents $z_p$, associated with the timescales, and their relation with the more commonly measured equal-time exponents ${\zeta }_p$. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.

Figure 1

Representative plots for the evolution of the normalized time-dependent fourth-order structure function $F_4(k_n,t)$ versus time, which is normalized by the Kolmogorov timescale ${\tau }_{\eta }$ for (a) $\text{Ro}=0.161$ and (b) $\text{Ro}=0.043$. The black, red, blue, and magenta lines correspond to $n=11$, 12, 14, and 16, respectively. (c) Plot of the relative spectral energy density $E_4(k_n,\omega )$ of $F_4(k_n,t)$ for $\text{Ro}=0.043$ versus $k_n$ for different harmonics $\omega$; the vertical dashed line represents the Zeman wave number.

Figure 2

A log-log (base 10) plot of the integral time $T_p^I\left(k_n\right)$ versus $k_n$ for $p=2$ (red circles) and $p=4$ (blue squares) for $\text{Ro}=0.043$. The dashed horizontal lines are best fits ($z_p=0$) for wave numbers lower than the Zeman wave number and the thick black lines are the best fits in the inertial range not dominated by rotation ($z_p\ne 0$; see Table 1). The lower inset shows a plot of $T_p^I$ ($p=2$–6, from the uppermost to the lowermost plot), for $k_n\lesssim k_{\mathrm{\Omega }}$, vs $1/\mathrm{\Omega }$ (except for $\mathrm{\Omega }=0.1$, where the plateau extends for just a few shells) for different values of $p$. The dashed lines are linear fits which show $T_p^I\propto 1/\mathrm{\Omega }$, consistent with the theoretical prediction. The upper inset shows a plot of the fourth-order vs second-order integral timescales, which shows a convincing scaling with slope $z_4/z_2\approx 1.09$, consistent with the exponent ratio obtained from the bridge relation.