Quasilinear approach to ray tracing in weakly turbulent, randomly fluctuating media

Abstract

Ray propagation in weakly turbulent media is described by means of a quasilinear (QL) approach in which the dispersion relation and the ray equations are expanded up to, and including, second-order terms in the medium and ray fluctuations, leading to equations for the ensemble-averaged ray and its root-mean-square (rms) spreading. An important feature of the QL formalism is that the average ray does not coincide with the zero-order, unperturbed ray but may exhibit a drift with respect to the latter that is governed by the mean squared fluctuations. The theory is complete in that equations can be set for all quantities necessary to compute the ray trajectory and the rms spreading along its path, yet they obey an infinite downward recurrence in which equations involving lower-order derivatives of the medium fluctuations are recursively generated by the subsequent higher-order derivative, and which must thus be truncated for practical purposes. Using as examples the propagation of rays in homogeneous media with fluctuations arising from the presence of either a single random mode or a multimode isotropic turbulent spectrum, the QL formalism is validated against Monte Carlo (MC) calculations and, whenever possible, its numerical implementation is verified by comparison with analytical predictions. Choosing $4%$ both for the level of fluctuations and for the maximum ratio between the wavelengths of the propagating ray and of the turbulent modes, so as to remain within the validity of the second-order expansion in the random perturbations and of the eikonal approximation, the overall agreement between QL and MC results is fairly good, particularly for quantities such as the distance traveled by the average ray, its perpendicular rms spread, and the averages of the wave-vector components.