Abstract We develop the theory of instabilities in a rotating gaseous disc and in shallow water for the case where there is a break in the surface density and sound velocities, as well as the rotation rate, at a particular radius. Different instabilities of sub-sonic and supersonic flows have been investigated. We also prove the identity of the linearised dynamical equations for the gaseous disc of the Galaxy and for our rotating shallow water experiments.

Introduction

The present paper pursues two aims: (1) to prove that gradient instabilities can lead to spiral structure in galaxies, and (2) to give the theory of gradient instabilities in rotating shallow water, when viscosity effects can be neglected. The behaviour in rotating shallow water has been investigated in an experiment known as “Spiral” at the Plasma Physics Department of the Institute of Atomic Energy.

It is natural to ask why such different subjects as galactic discs and shallow water are combined in this one paper. The reason is that the dynamical behaviour of a gaseous galactic disc and rotating shallow water are described by one and the same set of differential equations. Clearly, shallow water may be considered as a 2-D gaseous dynamical system (Landau & Lifshitz 1986) similar to the gaseous disc of our galaxy. However, viscosity effects near the bottom in the experimental set-up are absent in galaxies and the latter contain forces of self-gravitation that are absent in shallow water.