Heisenberg's Statistical Theory of Turbulence and the Equations of Motion for a Turbulent Flow

It is shown that consistent with Heisenberg's statistical theory of turbulence, equations of motion for a turbulent flow can be derived. These equations are linear integro-differential equations expressing the non-local interaction of eddies with different wave numbers on the basis of Heisenberg's statistical theory. The nonlocal terms in these equations of motion for turbulent flow have to be determined from the energy spectrum of the turbulent motion, which is obtained from a nonlinear integro-differential equation. A general solution of the linear equations of motion is obtained by an arbitrary superposition of solutions. However, only those linear superpositions are permitted which are selfconsistent with the energy spectrum of turbulent motion. In contrast to the Navier-Stokes equations, the non-linearity occurs here only in the equation for the energy spectrum and not in the equations of motion itself. This fact greatly facilitates the integration of these equations. Our analysis is extended to include turbulent convection. In the spirit of Heisenberg's hypothesis, equations of motion and energy are formulated which are consistent with the equations of the energy spectrum for free turbulent convection derived by Ledeoux, Schwarzschild and Spiegel. With this method, one may treat turbulent convection problems which arise in stellar and planetary atmospheres where the classical solution of laminar free convection cannot be applied.