Effect of confinement on the transition from two- to three-dimensional fast-rotating turbulent flows

We study the effect of confinement on the three-dimensional linear instability of fast-rotating two-dimensional turbulent flows. Using large-scale friction to model the effect of rigid boundaries at the top and bottom, we study the onset of three-dimensional perturbations on a rapidly rotating flow. The friction term is taken to affect both the evolution of the two-dimensional turbulent flow and the perturbations that evolve on top of it. Using direct numerical simulations, the threshold for the onset of three-dimensional perturbations is traced out as a function of the control parameters. As reported in the earlier work of Seshasayanan et al. [J. Fluid Mech. 901, R5 (2020)], we find that the two different mechanisms, namely the centrifugal and parametric-type instabilities, are responsible for the destabilization across the wide range of parameters explored in this study. In the turbulent regime, we find that the large-scale friction term does not affect the threshold in the case of centrifugal instability, while in the case of parametric instability, the large-scale friction makes the flow stable for a wider range of parameters. For the parametric instability, the length scale of the unstable mode is found to scale as the inverse square root of the rotation rate and the growth rate of the unstable mode is found to be correlated with the minimum of the determinant of the strain rate tensor of the underlying two-dimensional turbulent flow, showing resemblance with elliptical type instabilities. Results from the turbulent flow are then compared with the oscillatory Kolmogorov flow, which also undergoes parametric instability resulting into inertial waves. The dependence of the threshold on the aspect ratio of the system is discussed for both the turbulent and the oscillating Kolmogorov flows.

Figure 1

Panel (a) shows a representation of the instability threshold ${\text{Ro}}_c$ as a function of both $\text{Rh}$ and $\text{Re}$. Above this curve the two-dimensional flow is unstable to infinitesimal 3D perturbations, while below this curve it is stable. Panel (b) shows the time series of the perturbation energy in a log-linear scale for different parameters: the red and magenta curves correspond to $\text{Rh}={10}^5$, $\text{Ro}=2.1\times {10}^{-4}$ and $\text{Rh}=1$, $\text{Ro}=2.9\times {10}^{-3}$, respectively, showing the growth of perturbation while the blue and black curve corresponds to $\text{Rh}=1$, $\text{Ro}=1.6\times {10}^{-3}$ and $\text{Rh}=10$, $\text{Ro}=1.3\times {10}^{-3}$, respectively, showing decay of perturbations. The Reynolds number is fixed at $\text{Re}={10}^5$ and domain size at $qL=2\pi /10$ for all the curves.

Figure 2

Panel (a) shows the instability threshold ${\text{Ro}}_c$ in the $\text{Re}\text{−}\text{Ro}$ plane for different aspect ratios $qL$. The large-scale Reynolds number is fixed at $\text{Rh}=9.4\times {10}^4$ for all the curves. (b) shows the instability threshold ${\text{Ro}}_c$ in the $\text{Rh}\text{−}\text{Ro}$ plane for different aspect ratios $qL$. Here the Reynolds number is fixed at $\text{Re}=1.0\times {10}^5$ for all the curves. The inset in panel (b) shows the threshold with a rescaled $x$ axis given by ${\left(qL\right)}^{-1}\text{Ro}$. The dashed lines in panel (a) corresponds to the scaling law ${\text{Ro}}_c\sim {\text{Re}}^{-1}$ shown for comparison.

Figure 3

The figures show the dependence of the threshold ${\text{Ro}}_c$ curves as a function of the large-scale friction $\text{Rh}$ with panel (a) corresponding to an aspect ratio $qL=2\pi$ and panel (b) to an aspect ratio $qL=2\pi /10$. The different curves correspond to different $\text{Re}$. Filled symbols correspond to centrifugal instability mechanism, while nonfilled symbols denote the parametric instability points.

Figure 4

The real part of $x$ component of 3D-perturbation vorticity field is shown in panel (a) for moderate domain ($qL=2\pi$) corresponding to $\text{Re}=1.1\times {10}^5,\text{Rh}=1.2\times {10}^5$, in panel (b) for thinner domain ($qL=20\pi$) corresponding to $\text{Re}=1.0\times {10}^5,\text{Rh}=1.1\times {10}^5$. Panel (c) shows the variation of the real part of the $x$ component of the vorticity at the mid-$x$ plane ($x/L=0.5$) as a function of $y$. The blue curve shows the profile corresponding to the parametric instability and the red curve shows the profile for the centrifugal instability.

Figure 5

Shows the variation of the length scale of the unstable mode ${\ell }_m$ as a function of the critical Rossby number ${\text{Ro}}_c$ for the points associated with parametric instability. The data is obtained for the parameters corresponding to $qL=2\pi /10$, $\text{Rh}=9.4\times {10}^4$, and varying $\text{Re}$. The dashed line with a scaling ${\text{Ro}}^1$ is shown for comparison.

Figure 6

Panel (a) shows the time series of the logarithm of the perturbation energy for the parameters $\text{Re}=1.1\times {10}^5$, $\text{Rh}=1.2\times {10}^5$, $\text{Ro}=1.6\times {10}^{-3}$, and $qL=2\pi$. The cross symbol on the figure denotes the time instant at which panels (b) and (c) are taken, panel (b) shows the real part of the $x$ component of vorticity ${\widehat{\omega }}_x^r$ of 3D perturbations and panel (c) shows the determinant of the strain rate tensor field for the underlying two-dimensional turbulent flow $S_{\text{det}}$. The black marker in panel (c) denotes the position of the minimum of the determinant of the strain rate tensor.

Figure 7

Shows the scatter plot of the minimum of the determinant of the strain rate tensor $\left[S_{\text{det}}^{\text{min}}(x,y)\right]$ plotted against the growth rate of 3D perturbation for $\text{Re}=1.1\times {10}^5,\text{Rh}=1.2\times {10}^5\text{and}qL=2\pi$. Inset shows the same plot with a rescaled strain rate given by $S_{\text{det}}^{\text{min}}(x,y)\times [L/\left(U\mathrm{\Omega }\right)]$.

Figure 8

Figures show the threshold ${\text{Ro}}_c$ as a function of the Reynolds numbers $\text{Re},\text{Rh}$, for the oscillatory Kolmogorov flow for different aspect ratios $qL$. Parameter corresponding to panel (a) $\text{Re}$ vs $\text{Ro}$ is $\text{Rh}=1.0\times {10}^5$ and panel (b) $\text{Rh}$ vs $\text{Ro}$ is $\text{Re}=1.0\times {10}^5$.

Figure 9

Figures show the dependence of the threshold ${\text{Ro}}_c$ on the aspect ratio $qL$ in panel (a) and the oscillation frequency $\chi$ in panel (b) for the oscillatory Kolmogorov flow. The dashed lines indicate a scaling ${\text{Ro}}^1$ in panel (a) and ${\text{Ro}}^{-1}$ in panel (b), are shown for comparison. In panel (b) the threshold ${\text{Ro}}_c$ is obtained for the parameters $qL=2\pi /10$ and $\text{Rh}=1.0\times {10}^5$.

Figure 10

Schematic showing the thinner (left) and elongated (right) domains and axis of rotation.