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

Brief communication: Multiscaled solitary waves

It is analytically shown how competing nonlinearities yield multiscaled structures for internal solitary waves in stratified shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the Korteweg–de Vries (KdV) equation or its usual extensions. The multiscaling phenomenon exists or does not exist for almost identical density profiles. The trapped core inside the wave prevents the appearance of such multiple scales within the core area. The structural stability of waves of large amplitudes is briefly discussed. Waves of large amplitudes displaying quadratic, cubic and higher-order nonlinear terms have stable and unstable branches. Multiscaled waves without a vortex core are shown to be structurally unstable. It is anticipated that multiscaling phenomena will exist for solitary waves in various physical contexts

Brief communication "On one mechanism of low frequency variability of the Antarctic Circumpolar Current"

In this paper we present a simple analytical model for low frequency and large scale variability of the Antarctic Circumpolar Current (ACC). The physical mechanism of the variability is related to temporal and spatial variations of the cyclonic mean flow (ACC) due to circularly propagating nonlinear barotropic Rossby wave trains. It is shown that the Rossby wave train is a fundamental mode, trapped between the major fronts in the ACC. The Rossby waves are predicted to rotate with a particular angular velocity that depends on the magnitude and width of the mean current. The spatial structure of the rotating pattern, including its zonal wave number, is defined by the specific form of the stream function-vorticity relation. The similarity between the simulated patterns and the Antarctic Circumpolar Wave (ACW) is highlighted. The model can predict the observed sequence of warm and cold patches in the ACW as well as its zonal number