A Curious Phenomenon in a Model Problem,Suggestive of the Hydrodynamic Inertial Range and Smallest Scale of Motion

We consider a certain infinite system of ordinary differential equations, regarded as a highly simplified model of how energy might be passed up the spectrum in the Navier-Stokes equations, into the smaller scales of motion. Numerical experiments with this system of equations reveal a very striking inertial range and smallest scale phenomenon. In the case of steady data, the solution tends to a steady state in which the decay, as a function of mode number, is nearly linear until it reaches a very small value, beyond which it decays at a doubly exponential rate. This change in the character of the decay occurs in a sharply defined range of one or two mode numbers, effectively defining a largest significant mode number, which would translate in the spectral analogy to a smallest significant length scale. The first objective of this paper is a formulation and proof of what is observed in this experiment, especially concerning the decay of steady solutions with respect to mode number. Although similar numerical experiments with nonsteady data give convincing evidence of the same smallest scale phenomenon, some of our methods of proof for steady solutions do not generalize to nonstationary solutions. Consequently, our results for nonstationary solutions are less complete than for steady solutions. But, at the same time, their proofs seem more relevant to the Navier-Stokes equations. We conclude by describing and conjecturing about the results of further experiments with related equations, in which the coefficients are varied or the viscosity is set equal to zero. The ultimate objective of this paper is to begin a rigorous investigation of smallest scale phenomena in simple model problems, hoping for insights and generalizations that might be applied to the Navier-Stokes equations.