This paper considers the classic problem of the dynamics of axisymmetric waves
on a rectilinear vortex, in the absence of viscosity. The waves alter the axial pressure
distribution and thus generate axial flows which depend on the radial distribution of
vorticity. To simplify this problem, models have been introduced which average ...
This paper considers the classic problem of the dynamics of axisymmetric waves
on a rectilinear vortex, in the absence of viscosity. The waves alter the axial pressure
distribution and thus generate axial flows which depend on the radial distribution of
vorticity. To simplify this problem, models have been introduced which average over
the cross-section and eliminate the radial dependence. One approach, pioneered by
Lundgren & Ashurst (1989), J. Fluid Mech. 200, 283–307, averages the momentum
equation. Another averaging method, due to Leonard (1994), Phys. Fluids 6, 765–
777, focuses on the vorticity equation. The present paper takes a fresh look at
the derivation of these two distinct models, which we refer to as the momentum
wave model and vorticity wave model respectively, using the tools of differential
geometry to develop a hybrid Eulerian–Lagrangian approach. We compare these
models with area waves in the asymptotic limit of a slender vortex, with radial
structure retained. Numerical calculations are presented to show the differences
between waves in the full slender vortex system and those in the momentum and
vorticity wave models. We also discuss modification of the vorticity wave model to
allow an external irrotational flow, and simulations are presented where a vortex is
subjected to uniform axial stretching. Our approach can also be developed to model
more complicated configurations, such as occur during vortex collisions.