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Register a new userThe onset of zonal modes in two-dimensional Rayleigh-Benard convection
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Added by
Philip Winchester Mr
Publication date
2021
fields
Physics
and research's language is
English
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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.
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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.
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.
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 perform a bifurcation analysis of the steady state solutions of Rayleigh--Benard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialisation strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this non-linear problem, including disconnected branches of the bifurcation diagram, without the need of any prior knowledge of the dynamics. One of the disconnected branches we find contains a s-shape bifurcation with hysteresis, which is the origin of the flow pattern that may be related to the dynamics of flow reversals in the turbulent regime. Linear stability analysis is also performed to analyse the steady and unsteady regimes of the solutions in the parameter space and to characterise the type of instabilities.
Rayleigh-Benard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimension (2D) RB convection and the other one three-dimension (3D) RB convection with a rotating axis parallel to the plate. We explore the parameter range of Rayleigh numbers Ra from $10^7 to $10^9$ and Prandtl numbers $Pr$ from $1$ to $100$. We show that zonal flow, which was observed, for example, by Goluskin emph{et al}. emph{J. Fluid. Mech.} 759, 360-385 (2014) for $Gamma=2$, is only stable when $Gamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. With increasing $Gamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $Gamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. For the 3D simulations, we fix $Ra=10^7$ and $Pr=0.71$, and compare the flow for $Gamma=8$ and $Gamma = 16$. We demonstrate that with increasing aspect ratio $Gamma$, zonal flow, which was observed for small $Gamma=2pi$ by von Hardenberg emph{et al}. emph{Phys. Rev. Lett.} 15, 134501 (2015), completely disappears for $Gamma=16$. For such large $Gamma$ only convection roll states are statistically stable. In between, here for medium aspect ratio $Gamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2D case.
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