The problem on internal waves in a weakly stratified two-layer fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of a scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary-wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.

In this paper, we consider an analytical model of internal solitary waves in
a two-layer fluid with the density continuously increasing with depth in both
layers. This model is a development of non-linear two-layer models previously
suggested by Ovsyannikov (1985), Miyata (1985) and Choi and Camassa (1999),
as well as the latest 2.5-layer models considered by Voronovich (2003) and
Makarenko and Maltseva (2008, 2009a, b). Two-layer approximation is a
standard model of a sharp pycnocline in a stratified fluid with constant
densities in each layer, but which is discontinuous at the interface.
Correspondingly, the 2.5-layer model takes into account a slight density
gradient in stratified layers which is comparable with the density jump at
the interface. In all these cases, internal solitary waves can be described
in closed form by the solutions resulting from the quadrature

We apply the method of derivation involving asymptotic analysis of the non-linear Dubreil-Jacotin–Long equation that results from fully non-linear Euler equations of stratified fluid. The long-wave scaling procedure uses a small Boussinesq parameter which characterizes slightly increasing density in the layers and a small density jump at their interface. This method combines the approaches applied formerly to a pure two-fluid system with the perturbation technique discussed for the first time by Long (1965) and developed by Benney and Ko (1978) for a continuous stratification. The parametric range of a solitary wave is considered in the framework of the constructed mathematical model. It is demonstrated that these wave regimes can approach the parametric domain of the Kelvin–Helmholtz instability. The stability of solitary travelling-wave solutions of the Euler equations for continuously stratified, near two-layer fluids was studied numerically and analytically by Almgren et al. (2012). They demonstrated that the wave-induced shear can locally reach unstable configurations and give rise to local convective instability. This is in good qualitative agreement with the laboratory experiments performed by Grue et al. (2000). It seems that such a marginal stability of long internal waves could explain the formation mechanism of very long billow trains in abyssal flows observed by Van Haren et al. (2014).

We consider a 2-D motion of inviscid two-layer fluid which is weakly
stratified due to gravity in both layers. The fully
non-linear Euler equations describing the flow are

Scheme of the flow.

The layers are separated by the interface

Further we consider a steady non-uniform flow; hence, we have

Now we introduce scaled independent variables

Following Turner (1973), we introduce the densimetric (or internal) Froude
number

The model of fully non-linear travelling waves in a two-layer irrotational
flow, with the interface

In many cases, a parametric range of solitary waves can be determined a
priori as the domain being supercritical with respect to the spectrum of
small-amplitude sinusoidal waves. It is helpful while the critical phase
speed can be simply defined from the dispersion relation of infinitesimal
waves. In our case, linearizing of
Eqs. (

A spectrum of stationary harmonic waves, defined on the

Spectrum of linear waves (coloured modes 1–3)

We emphasize that parameters

The derivation procedure of the non-linear long-wave 2.5-layer model should
involve, in accordance with hypothesis (

Now we substitute power expansion (

Small-amplitude waves can be modelled by a simplified weakly non-linear
version of Eq. (

Solitary-wave regimes are obtained depending on the multiplicity of the roots

Large-amplitude internal waves are generated in deep ocean layers due to the
interaction of internal tides with irregular bottom topography near
underwater ridges (Morozov, 1995, 2018; Morozov et al., 2010). These waves
play a significant role in the energy transformation and mass transport in
the oceanic stratified flows, while they intensify mixing of the abyssal
waters. Note that internal Froude numbers

Internal front in abyssal stratified flow.

Train of interfacial solitary waves affected by the Kelvin–Helmholtz instability.

Profiles of the density, salinity and temperature.

We present in this section a comparison of solutions of suggested
mathematical models with the field data measured for internal solitary waves
in weakly stratified abyssal currents. Figures 3 and 4 demonstrate fragments
of temperature distribution in a quasi-steady shear bottom flow recorded from
a mooring station with a 350 m line of thermistors located over a depth of
4720 m at the entrance to the Romanche Fracture Zone in the equatorial
Atlantic (Van Haren et al., 2014). Trains of short-period (

The upstream parameters used in the calculation were chosen from
conductivity, temperature and pressure (CTD) and lowered acoustic Doppler
current profiler (LADCP) data of density and currents measured immediately at
the fronts of selected waves. Undisturbed depth of lower-layer

Upstream velocities

In this paper we have considered the problem of internal stationary waves at the interface between exponentially stratified fluid layers. We demonstrated that the non-linear DJL model of weakly stratified 2.5-layer fluid flow can be reduced explicitly to an approximate non-linear ordinary differential equation describing large-amplitude internal solitary waves. The parametric range of solitary waves is described, including regimes of broad plateau-shaped solitary waves and internal fronts. These wave regimes can be affected by the Kelvin–Helmholtz instability induced by the velocity shear at the interface; hence, the marginal stability of internal waves could explain the formation mechanism of very long billow trains observed in the Romanche Fracture Zone.

The data can be requested from author Eugene Morozov by the e-mail egmorozov@mail.ru.

Eigenfunction

The coefficient

Denominator

All authors made the same contribution to this work.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Extreme internal wave events”. It is a result of the EGU, Vienna, Austria, 23–28 April 2017.

This work was supported by the Russian Foundation for Basic Research (grant nos. 15-01-03942, 17-08-00085 and 18-01-00648) and Interdisciplinary Program II.1 of SB RAS (project no. 2). Roman Tarakanov was supported by the Russian Science Foundation (grant no. 16-17-10149). Edited by: Kateryna Terletska Reviewed by: two anonymous referees