Transport by Negative Eddy Viscosity in Soliton Turbulence

Abstract

The forced Schrödinger equation is used to describe the micro-hydrodynamical state of strong soliton turbulence. It is transformed into a master equation, and is decomposed into a macro-group, a micro-group, and a submicro-group, representative of the three transport processes of spectral evolution, transport property, and relaxation. The loss of memory in the relaxation gives the closure for the transport property to approach its equilibrium. The kinetic equation for the macro-distribution is derived and is reverted to the continuum by the method of moments to find the equation of spectral evolution. The spectral flow is found to be governed by three types of transport. At the large-scale side of the spectrum, the fluctuating driving field forms a cascade transport with negative viscosity called the reverse cascade. The nonlinear modulation forms another cascade transport. Its function is to drain the solitons toward the high wave number side of the spectrum. This process is regulated by a coupling function as the result of the coupling between the driving field and the soliton field in the coupling subrange.The equations of spectral densities for the two sub-ranges are solved in terms of the driving field asFE(k) ∼ k-m, FN(k) ∼ k-nfor the soliton field fluctuations and the density fluctuations, respectively. If the driving field is an emission due to the acoustic turbulence, having an intensity X'k2 ∼ k-2the spectral densities of soliton turbulence have the power lawsm = 1, n = 1 in the inertia subrangem = 3, n = 5 in the coupling subrangewith increasing wave numbers.Since the Schrödinger equation does not possess a molecular dissipation, or another equivalent dissipation, the transport by the direct cascade, as originated from the nonlinear modulation, can not find a sink to form an inertia subrange, as would be expected from the Navier-Stokes turbulence. This means that the nonlinear Schrödinger equation in its normal form, i.e. without the driving term, can not lead to chaos and yield a fully developed turbulence.