Helicity; Equations of Gas Dynamics; The Ertel Invariant

Abstract

The notion of helicity

$$ \chi\doteq\mathbf{u}\cdot\boldsymbol{\Omega}\quad ( \boldsymbol{\Omega} \doteq\operatorname{rot}\mathbf{u} ), $$

(3.1)

although being less widely known, is rather important for the description of such phenomena as tornadoes and typhoons. Unlike stream lines, the vortex lines are frozen into the fluid, according to the Kelvin theorem. Hence for non-stationary processes, the mutual location of vortex and stream lines, i.e., the structure and topology of the flow, change in time. The value of χ serves as a measure of this local structure change. On the other hand, intuition suggests that if the vortex lines are knotted or linked, the total number of such linkings should not change during the evolution, at least for an unbounded volume of fluid, since according to the Kelvin theorem the vortex lines are never born and never disappear. This is why it is interesting to derive the evolution equation for helicity to resolve the question on the existence of an integral topological invariant.