Scaling Laws for Stagnant-Lid Convection with Buoyant and Negatively Buoyant Crustal Layers

Abstract:

The formation of a buoyant crust can influence mantle convection on the Earth and other planets. In particular, in a stagnant-lid regime where the crust is not readily recycled, a thick crust can potentially form; if thick enough, it can influence the thickness of the lithosphere and hence heat flux due to convection. We perform numerical models of stagnant-lid convection with buoyant crustal layers of varying thickness and density, to determine how they influence convective heat flux; Rayleigh number and Frank-Kamenetskii parameter for viscosity are also systematically varied. We find two end-member behaviors: a thin crust limit where convection is largely unaffected by the presence of the crust, and the thickness of the lithosphere is approximately the same as it would be if the crust were absent; and a thick crust limit where the crustal thickness itself determines lithospheric thickness and heat flux. We find that the transition between these two regimes occurs when the crustal thickness plus the thickness of a thermal boundary layer beneath the crust is approximately equal to the lithosphere thickness expected if no crust were present. We determine scaling laws for the internal temperature and convective heat flux of the mantle, or Nusselt number, as a function of Rayleigh number, crustal thickness, buoyancy number, and Frank-Kamenetskii parameter for viscosity. Heat flux and internal temperature of the mantle are independent of crust thickness, until it is thick enough for convection to enter the thick crust limit. Then, increasing crust thickness causes mantle heat flux to drop, due to the thicker lithosphere that develops. Likewise, mantle internal temperature increases as crust thickness increases as mantle heat loss is less efficient. We determine the range of mantle temperatures, as a function of planet size, where convection will be in the thick crust limit, assuming that crust thickness will be limited by the transition to eclogite. We find that the thick crust limit can prevail over a wide range of mantle temperatures, and therefore have a major influence on convection, for small planets, similar to Mars. Finally, we show that convection is unlikely to develop in the crust, independent of convection in the underlying mantle, due to the low temperatures and high viscosities in this region.