Stochastic ray theory for long-range sound propagation in deep ocean environments

Abstract

The motion of sound ray trajectories in deep ocean environments, including internal wave induced scattering, is considered. Using the empirical Garrett–Munk internal wave spectrum and results from the study of stochastic differential equations, a framework for studying and modeling stochastic ray motion is developed. It is argued that terms in the ray equations involving internal wave induced sound speed perturbations δc can be neglected, but those involving ∂δc/∂z cannot. It is then shown in that the (Markov) approximation that spatial variations of ∂δc/∂z are delta correlated is remarkably good. These results lead naturally to an extremely simple system of coupled stochastic ray equations (in ray depth z, ray slowness p and travel time T) in which stochasticity enters the system only through the equation for p. Solutions to the stochastic ray equations—or the corresponding Fokker–Planck equation—describe approximately the density of acoustic energy in range, depth, angle and time. Two dimensionless parameters are introduced: (1) an acoustic Péclet number which is a measure of the ratio of the strength of deterministic ray refraction to that of stochastic scattering induced ray diffusion; and (2) a measure of the ratio of the strength of scattering induced ray diffusion to that of wave diffraction. Numerical solutions to the stochastic ray equations are compared to full wave simulations. These results show that, even in the weak scattering regime (large acoustic Péclet number), the inclusion of internal wave induced scattering may lead to important qualitative corrections to predictions of distributions of acoustic energy.

Ocean acoustic wave propagation and ray method correspondence: Internal wave fine structure

“…A numerical scheme has been introduced by Colosi and Brown, 17 which allows the efficient computation of a random ensemble of individual realizations of the typical sound speed fluctuations. This scheme conforms to the Garrett-Munk spectral and statistical phenomenological description of the internal waves 18,26 and has the form…”

Travel time stability in weakly range-dependent sound channels

“…1 correspond to an ensemble of 200 rays with a fixed launch angle, each in the same background sound speed structure but with an independent realization of the internalwave-induced perturbation superimposed. (Independent realizations were generated using the same internal wave field by staggering the initial range at which rays were launched with ∆r = 5 km; the vertical derivative of internal-wave-induced sound speed perturbations has a horizontal correlation length shorter than 5 km 13 , so this simple procedure ensures statistical independence.) Figure 1 shows two examples of a portion of what is commonly referred to as a timefront, consisting of many smooth branches that meet at cusps.…”

Wave chaos and mode–medium resonances at long-range sound propagation in the ocean

“…8,16,32 Although our perturbation ␦c takes into account only one internal wave mode, it causes a chaotic ray behavior whose properties closely resemble that observed in more realistic numerical models. 10,33 …”

“…In a more realistic environmental model where the sound speed perturbation is caused by random internal waves with statistics determined by the Garrett-Munk spectrum, the situation is different: steep rays are less chaotic than flat ones. 6,33 In Refs. 11 and 18, it has been shown that in our environmental model the coexistence of chaotic and regular rays causes an appearance of the gap in the timefront.…”