Asymmetric velocity boundary conditions lead to zonal flow in centrifugal convection

TL;DR

This work investigates centrifugal convection in a rapidly rotating annulus under mixed velocity boundary conditions using two-dimensional direct numerical simulations. It shows that asymmetric boundary conditions, especially inner stress-free outer no-slip, promote a robust zonal flow aligned with rotation, which markedly suppresses heat transfer and yields a weak $Nu$–$Ra$ scaling ($Nu\sim Ra^{0.1}$) with strong azimuthal–radial Reynolds-number anisotropy ($Re_φ\gg Re_r$). The study links this behavior to a redistribution of kinetic-energy dissipation from boundary layers to the bulk during zonal-flow formation, consistent with plume suppression, and demonstrates that curvature asymmetry ($\eta$ small) biases the system toward zonal flow, while approaching the planar limit ($\eta\rightarrow1$) recovers convection rolls. A steady free-convective boundary-layer model explains bulk-temperature asymmetry in terms of the heat-flux balance between inner and outer walls, highlighting the role of geometry in these flows. The results offer insight into zonal-flow formation in geophysical and astrophysical rotating systems and suggest directions for extending the analysis to fully three-dimensional settings and broader parameter regimes.

Abstract

We perform direct numerical simulations of rapidly rotating annular centrifugal convection to investigate how mixed (asymmetric) velocity boundary conditions and geometric curvature shape the flow organisation and heat transfer. Motivated by the quasi-two-dimensionalisation under strong rotation and the long spin-up required for large-scale states, we employ two-dimensional simulations and consider four boundary-condition sets: no-slip/no-slip (INON), no-slip/stress-free (INOS), stress-free/no-slip (ISON) and stress-free/stress-free (ISOS). For fixed geometry with the radius ratio $η=0.5$ and over the Rayleigh number $Ra\in[10^6,10^9]$, the heat transfer is strongest for ISOS, followed by INOS and INON, while ISON exhibits a pronounced suppression as a strong zonal flow aligned with the rotation develops. In the three cases dominated by large-scale circulation, the Nusselt number $Nu$ follows an effective classical-type scaling close to $Nu\sim Ra^{0.27}$, whereas the zonal-flow branch displays a much weaker scaling $Nu\sim Ra^{0.1}$ and strong flow anisotropy with the large difference between the radial and azimuthal Reynolds numbers $Re_r\ll Re_\varphi$. A dissipation analysis shows that zonal-flow formation is accompanied by a transition from boundary-layer-dominated dissipation to a relatively low and uniform bulk dissipation, consistent with shear-induced plume suppression. By varying the radius ratio $η$, we demonstrate that increasing $η$ weakens curvature asymmetry and destabilises the zonal-flow state, leading to roll-dominated convection in the planar limit, and we relate the accompanying bulk-temperature asymmetry to the boundary heat flux asymmetry using a free-convective boundary-layer model.

Figures (11)

• Figure 1: Left: schematic diagram of the flow configuration in the stationary reference frame with angular velocity $\Omega$. $R_{i,o}$ and $\Theta_{i,o}$ are the radius and temperature of the inner and outer cylinders, respectively. Hot and cold plumes are generated in the corresponding boundaries and form the roll state, while the strong azimuthal zonal flow leads to the zonal flow state. Right: two typical instantaneous temperature snapshots with velocity vectors are displayed to demonstrate the different flow characteristics between the roll state (upper figure) and the zonal flow state (lower figure).
• Figure 2: Typical instantaneous temperature snapshots and corresponding Nusselt number of different boundary condition sets: (a) INON, (b) INOS, (c) ISON, (d) ISOS. $Ra=10^8, \eta=0.5.$
• Figure 3: The variation of $Nu$ with $Ra$ at various boundary condition sets under fixed system geometry $\eta=0.5.$
• Figure 4: The variations of (a) the radial Reynolds number $Re_r$ and (b) the azimuthal Reynolds number $Re_\varphi$ with $Ra$ at various boundary condition sets under fixed system geometry $\eta=0.5.$
• Figure 5: Time series of the (a) Nusselt number and (b) Reynolds number under the boundary condition set ISON, $Ra=10^8, \eta=0.5$. The dashed black lines in (a) indicate the sampling time for the snapshots shown in the figure \ref{['fig: snapT']}.