An intrusion layer in stationary incompressible fluids: Part 1: Periodic waves

Abstract

Waves on a neutrally buoyant intrusion layer moving into otherwise stationary fluid are studied. There are two interfacial free surfaces, above and below the moving layer, and a train of waves is present. A small amplitude linearized theory shows that there are two different flow types, in which the two interfaces are either in phase or else move oppositely. The former flow type occurs at high phase speed and the latter is a low-speed solution. Nonlinear solutions are computed for large amplitude waves, using a spectral type numerical method. They extend the results of the linearized analysis, and reveal the presence of limiting flow types in some circumstances.

An intrusion layer in stationary incompressible fluids Part 2: A solitary wave

“…There it was assumed that all three fluid layers had the same density. This linearized solution was generalized in Paper 1 [8] to allow all three fluid layers to have different densities, and it was confirmed there also that two different propagation modes are possible. There is a high-speed type in which both interfaces move in phase, and a low-speed branch for which the two interfaces move oppositely.…”

“…Similar kinds of flows may be produced pyroclastically due to volcanic eruptions, as discussed by Nield & Woods [12]. Three-layer atmospheric intrusion flows can also occur above rivers, and the 'Bridgewater Gerry' in Hobart in Tasmania is a striking example of this effect; see Figure 13 in Paper 1 [8], for example.…”

“…As in Paper 1 [8], both the upper and lower fluids are at rest and are of infinite depth. The theory of solitary waves has a long and distinguished history, and an overall review of the classical work in this field has been given by Miles [13].…”

“…as in Paper 1 [8]. The first of these parameters F is the Froude number for the moving middle intrusion layer, and represents a ratio of its speed to a characteristic speed dependent on its depth.…”

“…That approach corresponds reasonably closely to the viewpoint of the present paper. The governing equations for this problem are reviewed in § 2, and the results for the (linearized) small amplitude solution for waves of finite wavelength are summarized from Paper 1 [8]. In § 3, the weakly nonlinear theory is developed, leading to a Korteweg-de Vries equation for the interface shapes.…”