A variational principle based study of KPP minimal front speeds in random shears

Abstract

The variational principle for Kolmogorov–Petrovsky–Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein–Uhlenbeck (O–U) process in the case of two-dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in the case of the O–U process of the shear. The variational principle based computation agrees with these analytical findings, and allows further study of the speed enhancement distributions as well as the dependence of the enhancement on the shear covariance. Direct simulations in the small rms regime suggest a quadratic speed enhancement law for non-KPP nonlinearities.