Jets and large-scale vortices in rotating Rayleigh-Bénard convection

One of the most prominent dynamical features of turbulent, rapidly rotating convection is the formation of large-scale coherent structures, driven by Reynolds stresses resulting from the small-scale convective flows. In spherical geometry, such structures consist of intense zonal flows that are invariant along the rotation axis. In planar geometry, long-lived, depth-invariant structures also form at large scales, but, in the absence of horizontal anisotropy, they consist of vortices that grow to the domain size. In this work, through the introduction of horizontal anisotropy into a numerical model of planar rotating convection by the adoption of unequal horizontal box sizes (i.e., $L_x\le L_y$, where the $xy$ plane is horizontal), we investigate whether unidirectional flows and large-scale vortices can coexist. We find that only a small degree of anisotropy is required to bring about a transition from dynamics dominated by persistent large-scale vortices to dynamics dominated by persistent unidirectional flows parallel to the shortest horizontal direction. When the anisotropy is sufficiently large, the unidirectional flow consists of multiple jets, generated on a time scale smaller than a global viscous time scale, thus signifying that the upscale energy transfer does not spontaneously feed the largest available mode in the system. That said, the multiple jets merge on much longer time scales. Large-scale vortices of size comparable with $L_x$ systematically form in the flanks of the jets and can be persistent or intermittent. This indicates that large-scale vortices, either coexisting with jets or not, are a robust dynamical feature of planar rotating convection.

Figure 1

Time series of the r.m.s. values of the three velocity components for $L_y=8$. The gray dashed lines indicate the times of the snapshots in Fig. 2.

Figure 2

Snapshots of the depth-averaged axial vorticity taken at the times indicated by the gray dashed lines in Fig. 1.

Figure 3

Snapshots of the horizontal and vertical cross sections of (a) the axial vorticity and (b) the vertical velocity for $L_y=1$.

Figure 4

Snapshots of the horizontal and vertical cross sections of (a) the axial vorticity and (b) the vertical velocity for $L_y=8$.

Figure 5

Snapshots of the depth-averaged axial vorticity (bottom panel) and profile of the mean $u_x$ as a function of $y$ (top panel) for (a) $L_y=2$, (b) $L_y=4$, and (c) $L_y=8$.

Figure 6

Flow anisotropy coefficient $\alpha$, defined by (4), as a function of $L_y$.

Figure 7

Space-time diagram of the depth-averaged axial vorticity for $\mathrm{Ek}={10}^{-5},\mathrm{Ra}=3\times {10}^8,L_x=1$, and $L_y=4$: (a) $x$-averaged and (b) the r.m.s. of the fluctuating component.

Figure 8

(a) Time series of the r.m.s. values of the three velocity components and (b) space-time diagram of the depth-averaged axial vorticity for $\mathrm{Ek}={10}^{-4},\mathrm{Ra}/{\mathrm{Ra}}_c=4.2$, and $(L_x,L_y)=(1,4)$.

Figure 9

${\text{Re}}_x$ as a function of $L_y/n$ (for cases with $n>0)$ for $\mathrm{Ek}={10}^{-5},\mathrm{Ra}=3\times {10}^8$, and $L_x=1$.

Figure 10

Time series of the r.m.s. values of the three velocity components for the simulation with $(L_x,L_y)=(1,1)$ whose initial condition is constructed from $(L_x,L_y)=(0.5,1)$ $\left(\mathrm{Ek}={10}^{-5},\mathrm{Ra}=3\times {10}^8\right)$.

Figure 11

(a) Time series of the r.m.s. values of the three velocity components for the simulation with $\mathrm{Ek}={10}^{-4},\mathrm{Ra}/{\mathrm{Ra}}_c=4.2$, and $(L_x,L_y)=(2,2.15)$. (b), (c) Snapshots of the $z$-averaged axial vorticity at the times indicated by the dashed vertical lines in (a).

Figure 12

Time series of the r.m.s. values of the three velocity components for the simulation with $\mathrm{Ek}={10}^{-4},\mathrm{Ra}/{\mathrm{Ra}}_c=4.2$, and $(L_x,L_y)=(2,2.2)$.

Figure 13

$\mathrm{\Gamma }$ as a function of (a) the normalized Rayleigh number and (b) the local Rossby number for $\mathrm{Ek}={10}^{-5}$.

Figure 14

$\text{Nu}$ as a function of the Reynolds number based on $u_x$ for the simulations of Table 1 $(L_x=1$ and varying $L_y)$ and for a reference case without a large-scale flow $\left(L_x=L_y=0.3\right)$.

Figure 15

Profiles of (a) the time-averaged mean flow, ${\langle u_x\rangle }_{xz}$, (b) the mean axial vorticity, ${\langle {\omega }_z\rangle }_{xz}$, and (c) the mean heat flux at the upper boundary, $q$; $\mathrm{Ek}={10}^{-5},\mathrm{Ra}=3\times {10}^8,L_x=1$, and $L_y=4$. The red bands correspond to the regions of positive mean vorticity.

Figure 16

Snapshot of the depth-averaged axial vorticity in the saturated regime for the case of $L_x=2,L_y=4$.