SYMMETRIES IN HYDRODYNAMIC TURBULENCE AND MHD DYNAMO THEORY

The three-dimensional equations of ideal hydrodynamics and ideal MHD a re expanded in eigenfunctions of the curl, and the resulting basic int eractions of these nonlinear systems are analysed. As the equations ar e invariant under time and amplitude reversal, a criterion defining th e arrow of time is introduced. A new parameter, the center of energy, serves to characterize a basic interaction. In the 3D Euler equations we find four different interactions and their mirror images, two of wh ich can transport energy to smaller wavenumbers. This can lead to the appearance of structures in turbulent flow and throws doubt on a deriv ation of Kolmogorov's law based on a cascading of energy to higher wav enumbers. In the corresponding two-dimensional equations, which are is omorphic to the guiding centre model in plasma physics, only one inter action exists, with a strong inverse cascade, which can lead to accumu lation of energy in the spatially largest accessible modes. In MHD the ory it is possible to separate magnetic from kinetic interactions. The former give again four basic interactions, two being regular and two being inverse cascades. One of these is quite strong, and can lead to the MHD dynamo effect. Kinetic energy can be transferred into magnetic energy. The dynamo effect is is accompanied by alignment of velocity and magnetic fields. We show that stationary velocity fields may lead to exponentially growing magnetic fields and we give an explicit crite rion for this instability.