Oleg G. Derzho

Solitary waves with a vortex core in a shallow layer of stratified fluid

This paper is concerned with a theoretical model for large amplitude solitary waves with a vortex core in a shallow layer of stratified fluid with a nearly uniform stratification. Previous work has shown that solitary waves can be calculated up to a critical amplitude for which the horizontal velocity, in a frame for which the wave is at rest, approaches zero at the boundary point beneath the wave crest for a wave of elevation (or above the wave crest for a wave of depression). Here we calculate waves with amplitudes slightly greater than the critical amplitude by incorporating a vortex core located near the aforementioned boundary point. 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. 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 amplit...

Solitary waves of permanent form in a deep fluid with weak shear

The Benjamin–Davis–Acrivos–Ono equation is generalized to account for finite, large amplitude solitary waves in a sheared deep fluid. It is shown how fine structure of stratification and weak noncritical shear in such geophysical flows do affect length (shape), wave (phase) velocity, and even stability of finite amplitude solitary waves.

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 overturning, 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 amplit...

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.

Rotating dipolar gyres on a γ-plane

Nonlinear dipolar vortices/gyres on a γ-plane are investigated both experimentally and theoretically. The solutions describe a fundamental dipolar mode of large scale barotropic motion of the polar ocean or atmosphere on the rotating planet. The entire dipolar gyre is predicted to rotate anticyclonically with a specific angular velocity. The existence and stability of the theoretically predicted flow are confirmed in a laboratory experiment on a rotating platform. The laboratory flows are induced by an electromagnetic method and are observed using the nonintrusive optical method of altimetric imaging velocimetry. The rotation rate of the experimental flow is in good agreement with that predicted theoretically. Detailed measurements of the velocity field and surface elevation demonstrate that an assumption of linearity of the relation between the relative vorticity and the stream function is valid.

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.

Trapped Vortex Cores in Internal Solitary Waves Propagating in a Thin Stratified Layer Embedded in a Deep Homogeneous Fluid

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 is presented. The case of a mode one asymmetric wave with an amplitude slightly greater than the critical amplitude, for which there is incipient overturning, is considered. A vortex core located near the point at which this incipient breaking occurs is then incorporated. 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. The result is 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.