On the zero viscosity limit

TL;DR

This paper investigates whether the zero viscosity limit in a simplified shell model matches the no viscosity case, revealing non-analytic behavior of velocities as viscosity approaches zero.

Contribution

It demonstrates that in a three-shell GOY model, two velocity functions are non-analytic at zero viscosity due to oscillatory Bessel functions.

Findings

Two velocities expressed via Bessel functions

Velocities are non-analytic at zero viscosity

Oscillatory behavior prevents a simple limit

Abstract

The question of whether the zero viscosity limit $\nu\to 0$ is identical to the no viscosity $\nu\equiv 0$ case is investigated in a simple shell (GOY) model with only three shells. We find that it is possible to express two velocities in terms of Bessel functions. The third velocity function acts as a background. The relevant Bessel functions are infinitely oscillating as $\nu\to 0$ and do not have a limiting value. Therefore two of the velocity functions of this three-shell model are not analytic functions of $\nu$ at the point $\nu =0$. We also mention a perturbative method which may be used to improve the model.