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Register a new userConnecting wall modes and boundary zonal flows in rotating Rayleigh--Benard convection
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Using direct numerical simulations, we study rotating Rayleigh-Benard convection in a cylindrical cell for a broad range of Rayleigh, Ekman, and Prandtl 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 time scales 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.
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For rapidly rotating turbulent Rayleigh--Benard 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.
We study the stability of steady convection rolls in 2D Rayleigh--Benard convection with free-slip boundaries and horizontal periodicity over twelve orders of magnitude in the Prandtl number $(10^{-6} leq Pr leq 10^6)$ and five orders of magnitude in the Rayleigh number $(8pi^4 < Ra leq 3 times 10^7)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $Gamma = 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$-plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $Pr to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to non-linear relaxation osculations and bursts, which we can explain with a weakly non-linear analysis. In the complementary large-$Pr$ limit, we observe that the stability boundary reaches an asymptotic value $Ra = 2.54 times 10^7$ and that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.
The effect of rotation on the boundary layers (BLs) in a Rayleigh-Benard (RB) system at a relatively low Rayleigh number, i.e. $Ra = 4times10^7$, is studied for different Pr by direct numerical simulations and the results are compared with laminar BL theory. In this regime we find a smooth onset of the heat transfer enhancement as function of increasing rotation rate. We study this regime in detail and introduce a model based on the Grossmann-Lohse theory to describe the heat transfer enhancement as function of the rotation rate for this relatively low Ra number regime and weak background rotation $Rogtrsim 1$. The smooth onset of heat transfer enhancement observed here is in contrast to the sharp onset observed at larger $Ra gtrsim 10^8$ by Stevens {it{et al.}} [Phys. Rev. Lett. {bf{103}}, 024503, 2009], although only a small shift in the Ra-Ro-Pr phase space is involved.
Recently, in Zhang et al. (2020), it was found that in rapidly rotating turbulent Rayleigh-Benard convection (RBC) in slender cylindrical containers (with diameter-to-height aspect ratio $Gamma=1/2$) filled with a small-Prandtl-number fluid ($Pr approx0.8$), the Large Scale Circulation (LSC) is suppressed and a Boundary Zonal Flow (BZF) develops near the sidewall, characterized by a bimodal PDF of the temperature, cyclonic fluid motion, and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount ($>60%$) of the total heat transport for $Pr < 1$ but decreases rather abruptly for larger $Pr$ to about $35%$. In this work, we show that the BZF is robust and appears in rapidly rotating turbulent RBC in containers of different $Gamma$ and in a broad range of $Pr$ and $Ra$. Direct numerical simulations for $0.1 leq Pr leq 12.3$, $10^7 leq Ra leq 5times10^{9}$, $10^{5} leq 1/Ek leq 10^{7}$ and $Gamma$ = 1/3, 1/2, 3/4, 1 and 2 show that the BZF width $delta_0$ scales with the Rayleigh number $Ra$ and Ekman number $Ek$ as $delta_0/H sim Gamma^{0} Pr^{{-1/4, 0}} Ra^{1/4} Ek^{2/3}$ (${Pr<1, Pr>1}$) and the drift frequency as $omega/Omega sim Gamma^{0} Pr^{-4/3} Ra Ek^{5/3}$, where $H$ is the cell height and $Omega$ the angular rotation rate. The mode number of the BZF is 1 for $Gamma lesssim 1$ and $2 Gamma$ for $Gamma$ = {1,2} independent of $Ra$ and $Pr$. The BZF is quite reminiscent of wall mode states in rotating convection.
We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.
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