Conservation of Angular Momentum–Vorticity

Abstract

This chapter is devoted to concepts that can be deduced from mathematical statements centered around angular momentum of a classical (spin-free) fluid continuum in three-dimensional Euclidian space. Circulation is defined as the integral along a closed simple three-dimensional curve of the scalar product of a smooth velocity field with a vectorial line increment. It leads directly into Kelvin’s circulation theorem, which states that the circulation of an ideal barotropic fluid subject to conservative forces along a material curve is preserved. Its generalization in a non-inertial system to non-barotropic, viscous fluids shows how the growth rate of the circulation around a material curve is contributed among frame rotation, non-barotropicity and dissipative stress contributions. The definitions of ‘vortex lines’, ‘vortex tubes’ and ‘vortex surfaces’ then provide the occasion to prove among other things that vortex tubes in ideal fluids must either be closed within the interior of the domain of the fluid or else end at boundaries. The Helmholtz vorticity theorem, derived from the momentum equation for an ideal, density preserving barotropic fluid by straightforward mathematical transformations states that the material time rate of change of the vorticity per unit mass in three-dimensional space is given by vortex stretching and vortex tilting. This result also holds for such a fluid in a non-inertial frame and leads naturally to the Taylor-Proudman theorem that ‘steady slow flow of a density preserving fluid takes place in a plane perpendicular to the axis of rotation’. Potential vorticity theorems, originally derived by Ertel, are special conservation laws of fluid dynamics in rotating systems for a scalar quantity formed with the vorticity weighted by a flow related quantity. There are several forms of this conservation law, particularly relevant in metrology and oceanography.