Continuum dislocation dynamics: analogies to fluid turbulence?

The dislocations which mediate plastic flow in crystals are described in the continuum with a nine-component tensor field. We study a nonlinear evolution law for this dislocation density, which shows several intriguing analogies to fully developed turbulence. (a)~As in the infinite Reynolds number limit, where vortex singularities are conjecured to form in finite time, our equation form wall singularites related to those in Burger's equation\footnote{S. Limkumnerd and J. P. Sethna, Phys. Rev. Letters \textbf{96}, 095503 (2006)}. To resolve these walls accurately we apply a central upwind scheme\footnote{A. Kurganov and E. Tadmor, J. of Comp. Phys. \textbf{160}, 1, 241-282 (2000)}. (b)~As in turbulence, we find self-similarity and scaling in the resulting cell wall morphologies when dislocation climb is forbidden. When climb is allowed (i.e., high temperatures) we form non-fractal walls representing grain boundaries. (c)~As in turbulence, where chaos allows only statistical convergence at long times, our two-dimensional simulations appear to have no weak solutions in the continuum limit -- the cascade of structure to short length scales appears to be sensitively dependent on the ultraviolet cutoff.