The anomalous diffusion of wave disturbances in hydrodynamic-type systems

Abstract

It is shown that solutions of many model non-linear equations of the hydrodynamic type (including the Navier-Stokes equations in the theory of a viscous fluid) in the form of localized wave disturbances undergo anomalous decay and the diffusion-type spreading under the action of Gaussian additive force noise and multiplicative parametric noise that depend randomly on time. The universality of this anomaly (i.e. its independence of the specific form of the equations and, in many respects, of the characteristic properties of the noise) is demonstrated; this reflects the generalized Galilean invariance of the hydrodynamic-type equations and the Gaussian property of random sources. The integrability of certain important hydrodynamic models enables the general conclusions to be supported by particular analytic results. Examples of expressions for the mean flow velocities, developed under the action of random forces on algebraic solitons of the completely integrable equations: equations for internal waves (the Benjamin-Ono equations), the two-dimensional Korteweg-de Vries equation (the Petviashvili-Kadomtseve equation), and the two-dimensional non-linear Schrödinger equation (the Davey-Stewartson equations), are presented in terms of the error function. It is established that for stochastic equations of the hydrodynamic type rapidly decaying multiplicative noise leads to normal Fick diffusion only when there is no additive force noise. Force noise produces anomalous diffusion of wave disturbances, which is described by Richardson's law (with a cubic relation between the temporal and squared spatial scales). The result obtained confirms the universal nature of Richardson's law, which has been demonstrated previously for many examples of turbulent and wave stochastic processes [1, 2].