On the interaction between oncoming internal waves and a dense gravity current in a two-layer stratification

A series of laboratory experiments was conducted to investigate the dynamics of a dense gravity current flowing down an inclined slope into a two-layer stratification in the presence of oncoming internal interfacial waves. The experiment is set up such that the gravity current propagates towards a wave maker emitting interfacial waves such that the current and waves propagate in opposite directions. The results were compared with the case of gravity current without oncoming waves. The gravity current splits into a portion that inserts itself into the pycnocline as an interflow and another that propagates down the slope as an underflow, with the proportionality depending on the characteristics of the gravity current and the oncoming waves when they are present. The interflow is shown to arise from a combination of detrainment and the preferential insertion of fluid with density greater than the upper layer and less than lower layer along the pycnocline. The mass flux of the interflow is observed to be reduced by the oncoming waves, as waves act to decrease the interflow velocity. The internal waves also increase the path length that the interflow must travel. A combination of reduced velocities and increased path length explains the observed reduction in cumulative flux. The trend of the final cumulative flux is consistent with the mass change observed by comparing density profiles obtained before and after the experiment.

Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.

Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive the modal equations from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (group speed, wave action conservation law) and to constructing more general ‘hybrid solutions’ consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. Detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions (i.e. eigenfunctions of the relevant spectral problems) for the surface waves in homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a large family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. Global and local measures of the deformation of wavefronts are introduced and evaluated.