Augmenting KZ finite flux solutions and nonlocal resonant transfer

This short paper is dedicated to Volodja Zakharov, a unique and original scientist, leader, mentor, poet and great friend for more than forty years. We will speak more of the man and his work in the body of the paper. One of his many and singular contributions to science was the development of a special and extremely relevant class of solutions for wave turbulence theory, the study of the statistical evolution of a sea of weakly nonlinear, dispersive waves [1]. The closed kinetic equation describing the evolution of the spectral energy or, equivalently, number density, had been known for many years, and had been explicitly derived in the context of surface gravity waves by Hasselmann in 1962 [2]. Many works addressing the questions of natural closure [3–5], other examples such as Rossby waves [6], surface tension dominated waves [7], plasma waves [8] soon followed. Before Zakharov, there was not much discussion of the statistical steady states, other than the equipartition spectra, to which the solutions of the kinetic equation might relax. The equipartition of conserved density solutions were obvious, readily seen by inspection. But, despite the fact that many of the western authors were familiar with the ideas of Kolmogorov in fully developed hydrodynamic turbulence, Zakharov [9,10] was the only one who realized that there should also be statistical steady states in the wave turbulence context corresponding to the finite fluxes of the conserved densities such as energy and wave action from scales at which they were introduced to scales at which they were dissipated or absorbed. The kinetic equation has hidden symmetries that Zakharov understood should be there and he found them. They led him to solutions that are now called Kolmogorov-Zakharov (or KZ) spectra. For those insights, and in particular for the discovery of inverse fluxes, Zakharov, along with Kraichnan, was awarded the 2003 Dirac Medal. However, as recognized by the present authors [11], these solutions have limited validity in two respects. First, they are rarely universally valid throughout the whole spectrum. Second, in some cases certain integrals, associated with physically important functionals, may not converge. The term nonlocal is often used to describe such situations but quantitative definitions of local and nonlocal remain open challenges. Colloquially, local connotes that the dominant transfer is between neighboring (in scale) wavenumbers; nonlocal connotes that there is significant and direct transfer between widely separated scales. KZ solutions connote local, a cascade, a la Richardson (big whirls make little whirls that feed on their velocity.), of some conserved density. In this short paper, we discuss remedies for these two challenges. First, we addresses how one may repair and augment the KZ spectrum in regions of wavenumbers where the KZ solution no longer obtains because the premises on which the closure is achieved are violated. The resolution may have broad applications. It continues and in fact builds on much of the work begun by one of the authors and Zakharov [12]. The second challenge addresses the possibilities that, and situations where, significant resonant transfer occurs between members of the resonant triads and quartets which have very different scales. The notions of local and nonlocal transfer have been investigated in the three-wave context by the other author [13,14] and, recently in the four-wave context, begun in [15,16]. In this work, we discuss whether a long wave, short wave, pairing in a resonant quartet may explain how the energy in the sea, inserted by a Miles’ type mechanism at small scales, can initiate longer waves directly via resonant wavepackets or whether the longer waves arise from a transient dual cascade in which both energy and waveaction, for a time, flow from small to large scales. In what we present, we do not claim to have made any great breakthroughs but hope that we have begun several necessary conversations.