Effect of spatio-temporal noise in the Eddington factor on the scalar flux

Spatial and temporal noise in the Eddington factor, simulating noise arising in hybrid numerical schemes, is modeled as a Gaussian stochastic process and its effect on the scalar flux investigated theoretically. In the small correlation time limit, a nonstandard closed equation for the mean scalar flux is obtained that contains a fourth order derivative of the scalar flux. In an infinite medium setting, this term is shown to have a destabilizing effect on the solution. Specifically, any spatial Fourier mode with wavelength smaller than a critical value, which depends on the noise characteristics, amplifies in time without bound, in contrast to the corresponding nonrandom case which is dissipative for all modes. An asymptotic solution is obtained which shows that the noise effect disappears at late times and the scalar flux limits to the deterministic solution.