We calculate static solutions of the `GOY¿ shell model of turbulence and do a linear stability analysis. The asymptotic limit of large Reynolds numbers is analyzed. A phase diagram is presented which shows the range of stability of the static solution. We see an unexpected oscillatory dependence of the stability range upon lg small nu, Greek, where small nu, Greek is the viscosity. This effect depends upon the discrete structure of the shell model and goes to zero as the separation between the shells is brought to zero. These findings show how viscous effects play a role in determining inertial properties of shell models and give some hints for understanding the effects of viscous dissipation upon real turbulence.
Schoerghofer, N., Kadanoff, L., & Lohse, D. (1995). How the viscous subrange determines inertial range properties in turbulence shell models. Physica D, 88, 40-54. https://doi.org/10.1016/0167-2789(95)00186-8