Scaling of Huygens-front speedup in weakly random media

Phenomena from combustion to seismic waves to population spreading can be modelled by Huygens' principle of front propagation, first stated as a law of geometrical optics. In each application, the boundary of the affected region (idealized as a sharp front) advances normal to itself at a locally specified speed. In a uniform medium, where this speed is constant, an initially wrinkled front flattens out over time; but in a spatially varying medium, a competition occurs as wrinkling is continually reintroduced. A central problem for the latter case is to determine the overall statistically steady propagation rate of the wrinkled front, which exceeds the average local speed because the front is defined by the fastest paths through the medium (first passage). Here we show mathematically that for a medium with weak isotropic random fluctuations, under mild conditions on its spatial structure, the speedup of the front passage rate scales with the 4/3 power of the fluctuation amplitude. This 4/3 result agrees with previous heuristic analysis and numerical simulations, and eliminates ambiguities concerning the physical relevance of the scaling. In the important case of premixed fluid combustion, the result describes the weak-advection scaling of the turbulent flame speed. Our analysis relates the front dynamics to the evolution of a pressure-free fluid obeying the white-noise-driven Burgers equation, itself a widely studied model of turbulence. The results reported here enable adaptation of existing treatments of Burgers turbulence to estimate the prefactor of the speedup scaling and its dependence on medium structure, with implications even for the opposite, practical limit of strongly advected flames