Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyancy frequency approximately modeling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component that can contaminate the wave envelope and has a vertical wavelength that decreases as the wave packet interacts with the interface.

Earth's tropopause often has a vertical structure with a very sudden change
in the lapse rate with increasing altitude, and a corresponding sudden
increase in the buoyancy frequency

Recently

A related configuration is a layer with constant

Packets of internal waves that propagate at a steep angle to the horizontal
will experience a modulational instability, as discussed by

The stability of plane monochromatic internal waves propagating at an angle
to the horizontal was treated by

The incident waves treated here are partially reflected at the interface,
resulting in incident, reflected, and transmitted wave packets that are
governed by coupled nonlinear Schrodinger (NLS) equations. Similar coupled
NLS equations have been treated previously by

The reflection of nonlinear internal waves by a sloping bottom has been
treated by several authors; for example

The results given below show that the incident and reflected waves combine
for a short period to create a strong localized mean flow under the interface
that is discontinuous at the interface, as in

The flow is treated as incompressible and inviscid, and attention is
restricted to two dimensions. The stratification is due to the presence of a
non-diffusing quantity, and the flow is assumed to be Boussinesq. The flow is
then governed by

The kinematic condition at a sharp interface between two layers is

The dynamic condition is continuity of total pressure

The waves are horizontally periodic but modulated vertically. The modulation
is assumed to be slow as measured by the small parameter

The linear solution for a wave with upward group velocity is

When an interface is included, the solution in the lower layer requires the
addition of reflected waves:

A peculiar fact of the Boussinesq equations is that when

Further higher harmonics are generated by nonlinear effects at the interface,
as shown previously by

Combining all leading order contributions results in

Separate all dynamic fields into a

A discussion of the general equations governing the wave-induced mean flow is
given by

In the upper layer,

The leading order contributions to the primary harmonic in
Eqs. (

A similar development for the reflected and transmitted waves leads to

The interfacial conditions must be treated in the same manner. The nonlinear
terms in both the kinematic Eq. (

Keeping only cubic terms, the kinematic interfacial condition is

The dynamic condition is

The Schrodinger equation for the incident waves written in terms of slow
variables, Eq. (

However, the results given below show that the vertical integral of

Note that these linear conditions imply continuity of velocity of the wave
components. Importantly the

The amplitude Eqs. (

Transmission coefficient

All variables are rescaled with the horizontal wavenumber

A wave envelope is created at the bottom boundary by imposing the value of
the real part of

The value of

The behavior of the waves with increasing

The behavior with increasing

Behavior of the coefficients of the dispersion and nonlinear terms
in the amplitude equations as

The coefficient of the nonlinear term and the dispersion term in the
amplitude equation governing the incident waves Eq. (

The dispersion coefficient is negative for small values of

Some insight is revealed by separating the wave amplitude into a magnitude
and phase,

Multiply Eq. (

Vertical profiles of the wave magnitudes

Equation (

For the numerical results, first consider

Vertical profiles of the wave magnitudes

Figure

Figure

The mean flow in Fig.

The coefficient of the dispersion term is negative when

The two-layer case with

Contours of vertical velocity in a single layer of constant

Vertical profiles of the wave magnitudes

Figure

Vertical profiles of the wave magnitude in two layers for a sequence
of times. Each profile is shifted by a value of 1.5 for display. The
parameter values are

Figures

Vertical profiles of the mean flow in two layers for a sequence of
times. Each profile is shifted by a value of 1.5 for display. The parameter
values are

Figure

Contours of vertical velocity at

Figures

Vertical profiles of the wave magnitude (left panel), phase (center
panel), and mean flow (right panel) at three times in a single layer of
constant

The amplitude and phase obey Eqs. (

Figures

Figure

The overall mean-flow strength is shown in Figs.

Vertical profiles of the wave magnitudes

Figure

Vertical profiles of the wave magnitude in two layers for a sequence
of times. Each profile is shifted by a value of 1.5 for display. The
parameter values are

Also in Figs.

Vertical profiles of the mean flow in two layers for a sequence of
times. Each profile is shifted by a value of 1.5 for display. The parameter
values are

Vertical profiles of the wave magnitude

Figure

Time history of the maximum of the mean flow. The dashed line is the
velocity jump at the interface while the thick solid line that is the maximum
the interference mean

If the interface is replaced with a rigid lid, then the waves are completely
reflected but otherwise behave in the same manner as above. The interfacial
boundary conditions are replaced with

Time history of the maximum of the mean flow. The dashed line is the
velocity jump at the interface while the thick solid line that is the maximum
the interference mean

An example case with

Time history of the maximum of the mean flow. The dashed line is the
velocity jump at the interface while the thick solid line that is the maximum
the interference mean

Temporal evolution of the vertical integral of the mean velocity for

Atmospheric observations indicate that the tropopause altitude is more likely to experience turbulence and large amplitude waves than other altitudes. The abrupt change in the buoyancy frequency suggests that such observations are related to the dynamics of internal waves near the tropopause. Previous numerical simulations conclude that internal waves will create a wave-induced jet-like mean flow in the tropopause vicinity that is likely responsible for at least some of the observations. An idealized low-dimensional model of such waves is treated here. The model consists of three coupled nonlinear Schrodinger equations along with linear interfacial conditions.

Numerical solutions with weak dispersion (

Vertical profiles of the wave magnitude (left panel), phase (center
panel), and mean flow (right panel) at three time values with a rigid lid,
with

However, the results here also show that dispersion may dominate the motion
and can act to greatly enhance the jet-like flow or weaken it, depending on
the value of

The mean flow found here also has the oscillatory interference component. The
results show that these mean-flow oscillations are transferred to the wave
envelopes and can become exaggerated in the tail of the incident wave packet
when

The observations of

Funding provided by the Physical Mathematics Program at the Air Force Office of Scientific Research. Edited by: R. Grimshaw Reviewed by: three anonymous referees