Directional energy flux in anisotropic turbulence*

In Navier-Stokes turbulence, energy is transferred among three wave numbers via a triad interaction.  The energy flux, which should be equal to the total energy dissipation rate if the energy transfer is local in the wavenumber space, plays a key role in the energy-cascading process that makes the self-similar structures in turbulent flows.  The energy flux for a wave number in isotropic turbulence systems is given by a scalar and is independent of the magnitude of the wavenumbers in the inertial subrange.  On the other hand, the energy flux in anisotropic turbulence systems depends on the directions as well as the magnitudes of the wave numbers, and should be a vector.  The energy flux vector in the anisotropic systems cannot be uniquely determined in a similar way used for the isotropic energy flux, which is obtained from the scalar energy transfer rate according to the continuity equation of energy.  In this work, we propose a way to determine the energy flux vector in anisotropic turbulence by using the pseudo-inverse matrix, which selects the minimal-norm vector, and will discuss the validity of the energy flux vector in rotating turbulence.