Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection

For rapidly rotating turbulent Rayleigh–Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like ${Ra}^{1/4}{Ek}^{2/3}$ where the Ekman number $Ek$ decreases with increasing rotation rate.

Figure 1

Sidewall temperature PDFs, $r/R=1$, for $z/H=1/4$ (diamonds), $z/H=1/2$ (circles), and $z/H=3/4$ (squares), with $1/Ro=0$ (a), (c) and 10 (b), (d). Experimental measurements with $Ra=8\times {10}^{12}$ (a), (b) and DNS with $Ra=10^9$ (c), (d), both with $Pr=0.8$. Bimodal Gaussian distributions (solid lines), the sum of two normal distributions (dashed lines), are observed for rapid rotation (b), (d). $⟨·{⟩}_{z_s}$ denotes average in time and over all sensor positions at distance $z$ from the hot plate.

Figure 2

Horizontal cross sections of time-averaged flow fields (DNS), visualized with streamlines (arrows) and azimuthal velocity $⟨u_{\varphi }{⟩}_{\mathrm{t}}$ (colors) (a), (b) at height of thermal BL, $z={\delta }_{\theta }\equiv H/(2Nu)$ and (c), (d) at the midplane, $z=H/2$, with $Ra=10^9$ and $1/Ro=0.5$ (a), (c) and 10 (b), (d). Blue (pink) color indicates anticyclonic (cyclonic) motion. In (d), locations $r=r_0$ of $⟨u_{\varphi }{⟩}_{\mathrm{t}}=0$ (solid line) and $r=r_{u_{\varphi }^{\max }}$ of the maximum of $⟨u_{\varphi }{⟩}_{\mathrm{t}}$ (dashed line) are shown.

Figure 3

For $Ra=10^9$, $1/Ro=10$: (a) $⟨u_{\varphi }{⟩}_{t,\varphi }$ vs $z$ at $r=0.95R$. The inset shows the same data for $0\le z\le H/2$ in a log plot. (b) $⟨u_{\varphi }{⟩}_{t,\varphi }$ vs $r$ at $z=H/2$; radial zero crossing $r=r_0$ (solid line) and radial maximum $r=r_{u_{\varphi }^{\max }}$ (dashed line). (c) Instantaneous thermal field at $r=r_{u_{\varphi }^{\max }}$ vs $z$ and $\varphi$.

Figure 4

For $Ra=10^9$ and $1/Ro=10$: (a), (b) angle-time plots at $r=r_{u_{\varphi }^{\max }}$, $z=H/2$ of (a) $T$ and (b) $u_z$; (c) normalized time-averaged vertical heat flux $⟨{\mathcal{F}}_z{⟩}_{\mathrm{t}}$ at $z=H/2$. In (c), location of $r$ where $⟨{{\mathcal{F}}_z|}_{z=H/2}{⟩}_{\mathrm{t}}=Nu$ (dashed-dotted line) and locations $r=r_0$ of $⟨u_{\varphi }{⟩}_{\mathrm{t}}=0$ (solid line) and $r=r_{u_{\varphi }^{\max }}$ of the maximum of $⟨u_{\varphi }{⟩}_{\mathrm{t}}$ (dashed line) are shown. Color scale from blue (min values) to pink (max values) ranges (a) between the top and bottom temperatures, (b) in $[-u_{\mathrm{ff}}/2,u_{\mathrm{ff}}/2]$, (c) from 0 to midplane magnitude of $⟨{\mathcal{F}}_z{⟩}_{\mathrm{t}}$, which is $\approx 3.4Nu$.

Figure 5

(a) Scaling of BZF widths ${\delta }_0$, ${\delta }_{u_{\varphi }^{\max }}$, ${\delta }_{u_z^{\mathrm{rms}}}$, and ${\delta }_{{\mathcal{F}}_z^{\max }}$ with $Ek$ (DNS for $Ra=10^9$); (b) Fitted peak values of bimodal PDF distributions (normalized by $\sigma$, standard deviation of $T$) at $z/H=1/2$ vs $r/R$: DNS ($Ra=10^9$) and measurements ($Ra=8\times {10}^{12}$), both for $1/Ro=10$; (c) diagram of the bimodal and unimodal temperature distributions at $r=R$, according to our DNS ($Ra=10^9$) and experiments (larger $Ra$) of rotating RBC for $Pr\approx 0.8$ and $\mathrm{\Gamma }=1/2$. Critical inverse Rossby number equals $1/{Ro}_c=2\pm 1$ (shown with a dashed line).