Solitary waves with recirculation zones in axisymmetric rotating flows

In this paper, we describe a theoretical asymptotic model for large-amplitude travelling solitary waves in an axially symmetric rotating flow of an inviscid incompressible fluid confined in an infinitely long circular tube. By considering the special, but important, case when the upstream flow is close to that of uniform axial flow and uniform rotation, we are able to construct analytical solutions which describe solitary waves with `bubbles', that is, recirculation zones with reversed flow, located on the axis of the tube. Such waves have amplitudes which slightly exceed the critical amplitude, where there is incipient flow reversal. The effect of the recirculation zone is to introduce into the governing amplitude equation an extra nonlinear term, which is proportional to the square of the difference between the wave amplitude and the critical amplitude. We consider in detail a special, but representative, class of upstream inflow conditions. We find that although the structure of the recirculation zone is universal, the presence of such solitary waves is quite sensitive to the actual upstream axial and rotational velocity shear configurations. Our results are compared with previous theories and observations, and related to the well-known phenomenon of vortex breakdown

Large-amplitude solitary waves with vortex cores in stratified and rotating flows

Most theoretical studies of solitary waves are for the weakly nonlinear regime, where models such as the Korteweg-de Vries equation are commonly obtained. However, observations of solitary waves often show that these waves can have large amplitudes, to the extent that they may contain vortex cores, that is, regions of recirculating flow. In this work, we report on theoretical asymptotic models, which describe explicitly the structure of solitary waves with recirculation zones, for certain special but important upstream flow configurations. The key feature which enables this construction is that, both for stratified shear flows and for axisymmetric swirling flows, the steady state vorticity equation is almost linear when the upstream flow is almost uniform. That is, for stratified shear flows the upstream flow and the upstream stratification are almost constant, while for rotating flows the upstream axial flow and angular velocity are almost constant. This feature enables the asymptotic construction of solitary waves described by a steady-state generalised Korteweg-de Vries equation in an outer zone, matched to an inner zone containing a recirculation zone. These recirculation zones exist for wave amplitudes just greater than a certain critical amplitude for which there is incipient flow reversal. The recirculation zones have a universal structure such that their width increases without limit as the wave amplitude increases from the critical amplitude to a certain maximum amplitude, but their existence can be sensitive to the actual upstream flow configuration. Applications are made to observations and numerical simulations of large amplitude internal solitary waves, and to the phenomenon of vortexbreakdown

On vorticity waves propagating in a waveguide formed by two critical layers

A theoretical model for long vorticity waves propagating on a background shear flow is developed. The basic flow is assumed to be confined between two critical layers, respectively, located near the lower and upper rigid boundaries. In these critical layers even small disturbances will break, and eventually a thin zone of mixed fluid will appear. We derive a nonlinear evolution equation for the amplitude of a wave-like disturbance in this configuration, based on the assumption that the critical layers are replaced by thin recirculation zones attached to the lower and upper rigid boundaries, where the flow is very weak. The dispersive and time-evolution terms in this equation are typical for Korteweg - de Vries theory, but the nonlinear term is more complicated. It comprises nonlinearity associated with the shear across the waveguide, and the nonlinearity due to the flow over the recirculation zones. The coefficient of the quadratic nonlinear term may change sign, depending on the presence or otherwise of recirculation zones at the upper or lower boundary of the waveguide. We then seek steady travelling wave solutions, and show that there are no such steady solutions if the waveguide contains no density stratification. However, steady solutions including solitary waves and bores can exist if the fluid between the critical layers is weakly density stratified

Asymmetric internal solitary waves with a trapped core in deep fluids

We describe an asymptotic model for long large-amplitude internal solitary waves with a trapped core, propagating in a narrow layer of nearly uniformly stratified fluid embedded in an infinitely deep homogeneous fluid. We consider the case of a mode one asymmetric wave with an amplitude slightly greater than the critical amplitude, for which there is incipient over-turning, that is wave-breaking. We then incorporate a vortex core located near the point at which this incipient breaking occurs. The effect of the vortex core is to introduce into the governing equation for the wave amplitude an extra nonlinear term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. Thus the derived new equation for the wave amplitude incorporates both the nonlinearity arising due to the flow over the recirculation core, and the nonlinearity associated with the ambient stratification; the dispersion term however remains of the Benjamin-Ono type. We find that as the wave amplitude increases above the critical amplitude, the wave broadens, which is in marked contrast to the case of small amplitude waves where a sharpening of the wave crest normally occurs. The limiting form of the broadening wave is “a deep fluid bore”. The wave speed is found to depend nonlinearly on the wave amplitude and the traditional linear dependence underestimates this speed.