Connecting wall modes and boundary zonal flows in rotating Rayleigh-Bénard convection

Using direct numerical simulations, we study rotating Rayleigh-Bénard convection in a cylindrical cell with aspect ratio $\mathrm{\Gamma }=1/2$, for Prandtl number 0.8, Ekman number ${10}^{-6}$, and Rayleigh numbers from the onset of wall modes to the geostrophic regime, an extremely important one in geophysical and astrophysical contexts. We connect linear wall-mode states that occur prior to the onset of bulk convection with the boundary zonal flow that coexists with turbulent bulk convection in the geostrophic regime through the continuity of length and timescales and of convective heat transport. We quantitatively collapse drift frequency, boundary length, and heat transport data from numerous sources over many orders of magnitude in Rayleigh and Ekman numbers. Elucidating the heat transport contributions of wall modes and of the boundary zonal flow are critical for characterizing the properties of the geostrophic regime of rotating convection in finite, physical containers and is crucial for connecting the geostrophic regime of laboratory convection with geophysical and astrophysical systems.

Figure 1

Phase diagram of states of rotating Rayleigh–Bénard convection: $\text{Ra}/{\text{Ra}}_w$ vs $\text{Ek}$. Boundaries are for ${\text{Ra}}_w, {\text{Ra}}_c, {\text{Ra}}_g$, and ${\text{Ra}}_t$ defined in the text.

Figure 2

Instantaneous temperature fields (left—horizontal at $z=H/2$ with streamlines; right—vertical at $r=0.98R$) for $\text{Ek}={10}^{-6}$. Corresponding $\text{Ra}$ and $\epsilon$: (a) $3\times {10}^7$, 0.071, (b) $5\times {10}^8$, 17, (c) $1\times {10}^9$, 35. Directions of rotation and wall mode precession are shown.

Figure 3

$\text{Nu}$ vs $\text{Ra}$ for $\text{Pr}=0.8, \text{Ek}={10}^{-6}$ and $\mathrm{\Gamma }=1/2$. (a) Time-dependent wall modes $\text{Ra}<5\times {10}^8$ and onset of bulk convection for $\text{Ra}\gtrsim 9\times {10}^8$. Vertical bars are standard deviations of $\text{Nu}$ fluctuations. Short-dashed line indicates linear fit to the data near onset (inset $\text{Nu}-1$ vs $\epsilon$ near onset). ${\text{Nu}}_{\text{off}}$ is the amount contributed by wall modes at bulk convection onset. [(b), (c)] Larger range of $\text{Ra}$ with total $\text{Nu}$ (solid, black) vs $\text{Ra}/{\text{Ra}}_c$ and contributions averaged over regions defined by $r/R\le r_0$ (bulk modes) and $r/R>r_0$ (wall modes/BZF). Black squares are the data from Ref. [20] for fixed $\text{Ra}={10}^9$. Solid lines are linear fits for $\text{Ra}/{\text{Ra}}_c⪅5$.

Figure 4

(a) Nusselt number vs $\epsilon$ , (b) $({\omega }_d-{\omega }_{d_c}){\text{Ek}}^{-5/3}$ (inset linear dependence on $\epsilon$ near onset), and (c) $({\delta }_0/H){\text{Ek}}^{-2/3}$ vs $\epsilon$ [31].

Figure 5

Scaled (a) boundary mode drift frequency $({\omega }_d-{\omega }_{d_c}){\text{Ek}}^{-5/3}{\text{Pr}}^{4/3}\approx 0.022(\text{Ra}-{\text{Ra}}_w)$ and (b) sidewall boundary length scale $({\delta }_0/H){(\text{Ra}-{\text{Ra}}_w)}^{-1/6}\approx 1.0{\text{Ek}}^{2/3}$.