An order-parameter field theory for turbulent fluctuations

With the use of the Herivel-Lin variational principle to describe incompressible inviscid fluids, a fluctuation Hamiltonian is derived in terms of a complex field $\Psi (\overrightarrow{\mathrm{r}}, t)$ representing the fluid. The source-free part of the field current describes the velocity field. The field interaction is found to be of current-current type, i.e., ${({\Psi }^{*} \mathrm{grad} \Psi -\mathrm{c}.\mathrm{c}.\mathrm{})}^2$ instead of the commonly used ${\left|\Psi \right|}^4$ type. The pressure, together with a complete energy-momentum tensor of the $\Psi$ field, is introduced, depending on the boundary conditions. The latter imply a long-range interaction near the onset of turbulence. Viscosity is taken into account by proper extension of the equation of motion of $\Psi (\overrightarrow{\mathrm{r}}, t)$, which turns out to be of the Landau type. This equation is solved exactly for laminar plane shear flow. The eddy interaction in $k$ space for fully developed turbulence is given together with a model Hamiltonian for the effects of viscosity. Finally the transient behavior of the $\Psi$-field amplitude of a fixed spatial mode near the onset of turbulence is compared with experiments.