Covering convection with thermal blankets: fluid-structure interactions in thermal convection

TL;DR

The paper addresses how thermal convection interacts with freely moving plates to form large-scale continental structures via a thermal blanket effect. It develops a 2D spectral solver with moving Robin boundary conditions and smooth indicator functions to simulate 1, 2, and multiple floating plates on Rayleigh-Bénard convection, quantifying the two-way coupling between heat transport and plate motion. A central contribution is identifying the covering ratio $Cr = d/\Γ$ as the key control parameter, with a critical value $Cr^*$ that separates passive from translating plate dynamics and governs stable supercontinent formation; higher Rayleigh numbers and aspect ratios modify $Cr^*$, enabling larger stable continents and offering Earth-like scaling insights. The findings provide a quantitative framework for fluid-structure interactions in mantle-like systems and point to physically plausible extensions to three-dimensional geometries relevant to geophysics.

Abstract

The continental plates of Earth are known to drift over a geophysical timescale, and their interactions have lead to some of the most spectacular geoformations of our planet while also causing natural disasters such as earthquakes and volcanic activity. Understanding the dynamics of interacting continental plates is thus significant. In this work, we present a fluid mechanical investigation of the plate motion, interaction, and dynamics. Through numerical experiments, we examine the coupling between a convective fluid and plates floating on top of it. With physical modeling, we show the coupling is both mechanical and thermal, leading to the thermal blanket effect: the floating plate is not only transported by the fluid flow beneath, it also prevents the heat from leaving the fluid, leading to a convective flow that further affects the plate motion. By adding several plates to such a coupled fluid-structure interaction, we also investigate how floating plates interact with each other and show that, under proper conditions, small plates can converge into a supercontinent.

Figures (15)

• Figure 1: Rayleigh-Bénard convection coupled to a free-floating plate leads to different dynamics of plate motion. (a) Schematics of the interaction between Rayleigh-Bénard convection and the floating plate. The fluid is heated from below and has an open free surface, the floating plate on this free surface is transported by the fluid force exerted from below. (b) Different states of plate motion. A small plate with $d/D=0.2$ oscillates between two sidewalls of the convection cell, while a big plate with $d/D = 0.7$ is trapped in the middle of the convection cell. Here $L = (D-d)/2$ is the bound of plate center $x_p$. (c) Flow visualization reveals two counter-rotating large-scale circulations when the plate is located at the center of the convection cell. Image credit: zhong2007bmac2018stochastic.
• Figure 2: Schematics of the floating plate problem. The fluid domain $\Omega$ is heated from the bottom surface $\partial\Omega_0$ and has an open surface on top ($\partial\Omega_1$), floating plates $P_1, P_1, P_2,\dots$ cover part of this open surface and shield the heat from leaving the fluid.
• Figure 3: Smooth step and indicator functions. (a) Smooth step function $W_\epsilon$ that has a transition interval of $[-\epsilon,\epsilon]$. Four values of $\epsilon = 0.01,0.1,0.2,0.4$ are plotted. (b) Smooth indicator function $\hat{a}$ for locating the region of solid plates. The parameters plotted are $x_p^{(1)} = 1$, $x_p^{(2)} = 3$, $d = 1$, and $\epsilon = 0.05$.
• Figure 4: Motion of a small plate ($\hbox{Cr}{} = 0.1$) is random and bidirectional. (a) A snapshot of flow and temperature fields beneath a plate. The small plate is trapped at a cool converging center. (b) Vertically averaged temperature $\bar{\theta}$ and vertical velocity $\bar{v}$ at the same moment of (a). The shaded region indicates the location of the plate. At the converging center, the averaged temperature is low and the flow moves downward. (c)-(d) The displacement $x_p$ and velocity $u_p$ of the plate show behavior of a random walk with jumps.
• Figure 5: Motion of a large plate ($\hbox{Cr}{} = 0.6$) is unidirectional. (a)-(b) Flow and temperature fields beneath the plate. (c)-(d) The displacement $x_p$ and velocity $u_p$ of the moving plate, which shows a unidirectional motion with nonzero mean velocity.