Symmetry Properties of Cameron-Martin-Wiener Kernels

Abstract

The symmetry properties of the Cameron-Martin-Wiener (Wiener-Hermite) kernels are investigated in the isotropic velocity field as required for the invariance of the velocity distribution of the field under rigid rotation and reflection. As a sufficient condition for the invariance, it is found that the even kernels must be isotropic true tensors, and the odd kernels (i) isotropic true tensors or (ii) isotropic pseudotensors. In incompressible turbulent flow, it is shown that the odd kernels must be isotropic pseudotensors. Assuming that the nonlinear term of Burgers' equation can be treated as a perturbation in the very late decay stage, it is shown that the energy spectrum density must vanish at zero wavenumber in order to get a valid physical model of turbulence. Applying the above conclusion, it is seen that the odd kernels in the Burgers' model must be odd functions, which suggests that the model thus obtained is the one-dimensional counterpart of turbulent flow which is incompressible.