Free-slip Boundaries Research Articles

Three-dimensional treatment of convective flow in the earth's mantle

A three-dimensional finite-element method is used to investigate thermal convection in the earth's mantle. The equations of motion are solved implicitly by means of a fast multigrid technique. The computational mesh for the spherical problem is derived from the regular icosahedron. The calculations described use a mesh with 43,554 nodes and 81,920 elements and were run on a Cray X. The earth's mantly is modeled as a thick spherical shell with isothermal, free-slip boundaries. The infinite Prandtl number problem is formulated in terms of pressure, density, absolute temperature, and velocity and assumes an isotropic Newtonian rheology. Solutions are obtained for Rayleigh numbers up to approximately 106 for a variety of modes of heating. Cases initialized with a temperature distribution with warmer temperatures beneath spreading ridges and cooler temperatures beneath present subduction zones yield whole-mantle convection solutions with surface velocities that correlate well with currently observed plate velocities.

Stability of finite-amplitude convection

Internally generated vertical vorticity enters the lowest-order amplitude equations for free-slip boundaries in an essential way when the Prandtl number P is finite and the flow three dimensional. For parallel rolls the band of stable wavenumbers is substantially modified from what was previously believed to be correct. In particular there are no stable states for P<0.301. Numerical simulations for free boundaries and larger P suggest a mechanism through which the box size determines the critical Rayleigh number for noisy time dependent convection. A stability analysis of model amplitude equations for rigid boundaries agrees qualitatively with the numerical results of Cleaver or Busse for P≲O(1). There is now considerable continuity between the stability diagrams for rigid and free boundaries.

Disappearance of stable convection between free-slip boundaries

A new secondary instability has been found for nearly parallel rolls that eliminates all wavelengths previously thought to be stable for Prandtl numbers less than 0.301 or 0.782, depending on the circumstances. There is currently no known counterpart to this instability between rigid boundaries.

Galerkin Approximations to Flows within Slabs, Spheres, and Cylinders

This Letter introduces infinite-order accurate and efficiently implementable Galerkin (spectral) approximations to time-dependent incompressible flows within slabs, spheres, and cylinders with either rigid no-slip or free-slip boundaries. The unusual choice of Chebyshev polynomials as Galerkin expansion functions is crucial for the efficiency of the method.