SCALING AND LINEAR-RESPONSE IN THE GOY TURBULENCE MODEL

The GOY model is a model for turbulence in which two conserved quantit ies cascade up and down a linear array of shells. When the viscosity p arameter, v, is small the model has a qualitative behavior which is si milar to the Kolmogorov theories of turbulence. Here a static solution to the model is examined, and a linear stability analysis is performe d to obtain response eigenvalues and eigenfunctions. Both the static b ehavior and the linear response show an inertial range with a relative ly simple scaling structure. Our main results are: (i) The response fr equencies cover a wide range of scales, with ratios which can be under stood in terms of the frequency scaling properties of the model. (ii) Even small viscosities play a crucial role in determining the model's eigenvalue spectrum. (iii) As a parameter within the model is varied, it shows a ''phase transition'' in which there is an abrupt change in many eigenvalues from stable to unstable values. (iv) The abrupt chang e is determined by the model's conservation laws and symmetries. This work is thus intended to add to our knowledge of the linear response o f a stiff dynamical system and at the same time to help illuminate sca ling within a class of turbulence models.