Nonlinear energy in a wave turbulence system

Single-wavenumber representations of nonlinear energies are required to investigate energy budget due to nonlinear interactions among Fourier modes in wave turbulence. While we have reported in a previous paper that the single-wavenumber representations successfully works for the Foppl-von Karman equation, we will show here that for the Majda-McLaughlin-Tabak model the single-wavenumber representations of nonlinear energies is not necessarily unique. Introducing auxiliary variables composed differently from complex amplitudes, two natural representations of the nonlinear energy are obtained. It is numerically observed that the two kinds of the nonlinear-energy spectra, based on these two representations, are qualitatively similar, but the energy budgets are clearly different. To select the appropriate single-wavenumber representation of the nonlinear energy, the properties which an eligible single-wavenumber representation should have are discussed.