Transition to three-dimensional flow in thermal convection with spanwise rotation

We investigate by direct numerical simulation Rayleigh-Bénard convection in a rotating rectangular cell with rotation vector and gravity perpendicular to each other. The flow is two-dimensional near the onset of convection with convection rolls aligned parallel to the rotation axis of the boundaries. At a sufficiently large Rayleigh number, the flow becomes unstable to three-dimensional disturbances which changes the scaling of heat transport and kinetic energy with Rayleigh number. The mechanism leading to the instability is identified as an elliptical instability. At the transition, the Reynolds and Rossby numbers $\mathrm{Re}$ and $\mathrm{Ro}$ based on the kinetic energy of the flow are related by $\mathrm{Re}\propto {\mathrm{Ro}}^{-2}$ at small $\mathrm{Ro}$ with a geometry-dependent prefactor.

Figure 1

$\mathrm{Nu}$ as a function of $\mathrm{Ra}$ for 2D (black dots) and 3D (empty red symbols) simulations for nonrotating convection (circles), $\mathrm{Ek}={10}^{-2}$ (squares), $7\times {10}^{-3}$ (diamonds), $3\times {10}^{-3}$ (triangles up), ${10}^{-3}$ (triangles down), $4\times {10}^{-4}$ (plus), ${10}^{-4}$ (x), and $4\times {10}^{-5}$ (stars). The solid and dashed lines in the left panel indicate the power laws $\mathrm{Nu}=0.18{\mathrm{Ra}}^{0.35}$ and $\mathrm{Nu}=0.54{\mathrm{Ra}}^{0.26}$, respectively, whereas the right panel shows the compensated Nusselt number $\mathrm{Nu}/{\mathrm{Ra}}^{1/3}$.

Figure 2

Reynolds number $\mathrm{Re}$ (left panel) and compensated Reynolds number $\mathrm{Re}/{\mathrm{Ra}}^{2/3}$ (right panel) as a function of $\mathrm{Ra}$ with the same symbols as in Fig. 1. The solid and dashed lines in the left panel indicate the power laws $\mathrm{Re}=0.09{\mathrm{Ra}}^{0.67}$ and $\mathrm{Re}=0.99{\mathrm{Ra}}^{0.44}$, respectively.

Figure 3

The anisotropy Eq. (7) as a function of $\mathrm{Ra}$ for $\mathrm{Ek}=4\times {10}^{-5}$.

Figure 4

The vertical bars indicate intervals of $\mathrm{Ra}$ in which the transition from 2D to 3D flows occurs as a function of $\mathrm{Ek}$. The solid line is given by $982{\mathrm{Ek}}^{-0.9}$.

Figure 5

Line segments crossing the stability limit of 2D flows in the ($\mathrm{Re},\mathrm{Ro}$) plane (left panel). The continuous curve results from the stability analysis for an elliptical vortex with $\beta =0.78$ and $k_{\parallel }=2\pi$, $k_{\perp }=\pi$. The right panel shows the same data and the same curve in the ($\mathrm{Ek},\mathrm{Ro}$) plane.

Figure 6

Ellipticities of the streamlines of 2D convection flows at the center of the cell deduced from the eigenvalues $1/a$ and $1/b$ of the Jacobian of the velocity field. Ellipticity is defined as the parameter $\beta$ in Eq. (10).

Figure 7

Streamlines of the two-dimensional flows at $\mathrm{Ek}=4\times {10}^{-5}$ and $\mathrm{Ra}=8\times {10}^6$ corresponding to $\mathrm{Ro}=6.6\times {10}^{-2}$ (left panel), $\mathrm{Ek}=7\times {10}^{-3}$, and $\mathrm{Ra}={10}^5$ corresponding to $\mathrm{Ro}=1.24$ (middle panel) and $\mathrm{Ek}=9\times {10}^{-2}$ and $\mathrm{Ra}=8\times {10}^4$ corresponding to $\mathrm{Ro}=13.3$ (right panel).

Figure 8

The stability limit of an elliptical vortex computed for $k_{\parallel }=5\pi /3$, $k_{\perp }=2\pi$ and $\beta =0.8$ (solid line, intended as a fit at small $\mathrm{Ro}$) or $\beta =0.76$ (dashed line, intended as a fit at large $\mathrm{Ro}$) together with the same numerical data as in Fig. 5 delimiting the stability limit of 2D convection flow.