Pulses in the zero-spacing limit of the GOY model

We study the propagation of localised disturbances in a turbulent, but mome ntarily quiescent and unforced shell model (an approximation of the Navier- Stokes equations on a set of exponentially spaced momentum shells). These d isturbances represent bursts of turbulence travelling down the inertial ran ge, which is thought to be responsible for the intermittency observed in tu rbulence. Starting from the GOY shell model, we go to the limit where the d istance between succeeding shells approaches zero ("the zero spacing limit" ) and helicity conservation is retained. We obtain a discrete field theory which is numerically shown to have pulse solutions travelling with constant speed and with unchanged form. We give numerical evidence that the model m ight even be exactly integrable, although the continuum limit seems to be s ingular and the pulses show an unusual super exponential decay to zero as e xp(-constant sigma") when n --> infinity, where a is the golden mean. For f inite momentum shell spacing, we argue that the pulses should accelerate, m oving to infinity in a finite time. Finally, we show that the maximal Lyapu nov exponent of the GOY model approaches zero in this limit. (C) 2000 Elsev ier Science B.V. All rights reserved.