Spectral intermode coupling in a model of isotropic turbulence

We investigate the nonlinear coupling between the so-called explicit modes, identified with wave numbers k such that 0< or =k< or =k(c), and implicit modes, defined such that k(c)< or =k< or =k(max). Here k(c) is an arbitrarily chosen cutoff wave number and k(max) is the ultraviolet cutoff as determined by viscous damping. The stresses arising from the nonlinearity in the Navier-Stokes equations are categorized as "implicit-implicit" (or "Reynolds") and "explicit-implicit" (or "cross"). These arise from dynamic coupling between different regions of wave number space. Their respective effects on momentum, kinetic energy, and energy flux are assessed. The analysis is based on a model system comprising the Navier-Stokes equations and the Edwards-Fokker-Planck energy equation [S. F. Edwards, J. Fluid Mech. 18, 239 (1964)] which is known to retain all the symmetries of homogeneous, isotropic turbulence. The Reynolds stress is found to be responsible for long-range energy transfers. It can be represented by an effective viscosity and is mainly determined by dynamical friction. The cross term is more complicated, involving both diffusive and frictional effects. For long-range coupling it can be expressed as a modification of the effective viscosity, while for short-range coupling it may be modeled on the assumption that implicit scales are slaved to explicit scales. Thus, both the random and coherent aspects of intermode coupling in turbulent flows are relevant in the cross term. The imposition of a continuity requirement on energy transfer leads to a new parametrization that represents the effect of absent modes in a truncated spectral simulation, and takes into account the phase-coupling (coherent) effects, as well as the usual viscositylike (random) effects.