Internal waves are waves which, as the name suggests, occur in the interior of a fluid. You may be asking what sort of restoring force makes waves in the interior of a fluid possible, or you may question how important waves which do not occur on the surface of a fluid are. The answer is that just like surface waves, gravity is the restoring force for internal waves. The density differences in the interior of the fluid are tiny compared to those at the surface (density of water is 1000 times that of air). These tiny density differences, which are almost always present in the oceans and the atmosphere make it possible to have very large internal waves. Large internal waves can have very large currents associated with them and hence they can transport material in the ocean or atmosphere along with them. To start off we will discuss some cases where internal waves play an important role.

Cloud
formations
known
as
**
morning
glories** are
caused
by
a
cloud
being
trapped
in
an
atmospheric
internal
wave.
These
clouds
can
be
several
hundred
kilometers
wide
and
one
to
two
kilometers
in
height.

Another
interesting
phenomena
caused
by
internal
gravity
waves
is **dead-water**.
When
a
boat
is
travelling
over
water
that
strongly
resembles
a
two
layer
fluid,
a
thin
light
layer
of
fresh
water
at
the
top
and
a
thick
dense
layer
of
salt
water
below,
the
boat
is
unexpectedly
slowed
down
by
a
mysterious
force.
The
force
is
due
to
the
ship
generating
a
wake
of
waves
at
the
interface
of
the
two
layers.
The
energy
that
the
ship
exerts
to
propel
itself
forward
is
translated
into
the
generation
of
the
internal
waves.
The
physical
oceanography
department
of
the
Royal
Netherlands
Institute
for
Sea
Research
contains
a
video
of
a
lab
demonstration
of
dead
water.
The
video
is
quite
large,
but
worthwhile
to
look
at.

Near
mountain
ranges
internal
waves
can
form
in
the
atmosphere,
grow
to
their
breaking
point
and
subsequently
cause
very
strong
winds
which
can
cause
a
lot
of
damage.
This
type
of
wind
is
often
referred
to
as **downslope
winds **and
the
internal
waves
responsible
for
it
as **mountain
waves**.

## Two layer fluid

To model internal waves mathematically, we first consider a two layer fluid, that is, a fluid which has two layers each with different density, the denser layer is located on the bottom. We will assume that the layer are immiscible since this allows us to ignore the effects of mixing. The difference in density can be caused by several different things, for example salinity (higher salinity implies higher density) or temperature (lower temperature implies higher density). We will denote the top layer of fluid with a subscript 1 and the bottom layer with a subscript 2. Below is a diagram describing the variables in the problem:

If we assume that the fluid is incompressible and irrotational then the potential function for each layer solves Laplace's equation.

^{2} =gk, surfaces of constant density and surfaces of constant pressure coincide, hence this mode behaves similar to the deep water gravity wave. The other mode is referred to as the baroclinic mode,

in this mode the surface wave and the interface wave are out of phase, and the disturbance amplitude is generally larger at the interface than at the surface. Here is an animation where the depth of the bottom layer is taken to be infinite. On thing to note in this animation is the phase speeds (which can be computed by c = k/w)

## Continuously stratified fluid

In
reality
it
is
very
unlikely
that
you
will
find
a
fluid
which
consists
precisely
of
two
or
more
distinct
layers.
It
is
more
likely
that
the
fluid
is
continuously
stratified
with
a
region
where
the
density
varies
rapidly,
known
as
a **pycnocline.** Think
of
a
lake
in
the
summer,
which
is
quite
warm
for
the
top
meter
or
so,
but
if
you
swim
deeper
you
will
quickly
encounter
very
cold
(and
hence
dense)
water.
Internal
waves
in
a
stratified
fluid
have
peculiar
properties
compared
to
surface
gravity
waves
as
we
shall
see
when
we
discuss
group
velocity.

To start off we will talk about the concept of buoyancy frequency. Assume we have a stabily stratified fluid,

(rho bar is the background density) We have a small fluid parcel of volume. Two forces will be acting on this fluid parcel, a gravitational force and a force due to buoyancy. Assuming the pressure field is unchanged and we displace the fluid parcel by a small amount vertically, z -> z + delta. Then the gravity force acting on the fluid parcel is,

downward and the buoyancy force is,

upward. Using Newton's 2nd law we get the following expression:

Which after some approximations we get,

This is an expression for for simple harmonic motion if,

is constant and we define the buoyancy frequency N by,

With the Boussinesq approximation we get the simpler expression

Applying the Boussinesq approximation (density is constant everywhere except in the gravity, or buoyancy term) and then linearizing the momentum equations, the energy equation and the incompressibility equation, we get the following governing equations:

One thing to note is for the case of the continuously stratified fluid we have not made the irrotationality assumption. Through quite a bit of manipulation of the governing equations we can determine the following expression for the vertical velocity:

From this expression we get the following dispersion relation (assuming the vertical velocity field has the form):

Note
that
the
frequency
only
depends
on
the
direction
of * k*
=
(k,l,n)
and
not
its
magnitude.
There
is
an
upper
bound
on
the
frequency
(namely
the
buoyancy
frequency).
If
we
then
suppose
that
the
wave
motion
has
the
form,

we find, since the motion is incompressible, i.e. the velocity is orthogonal to the wave number vector. Waves for which this is true are called shear waves. We can compute the group velocity to be,

which can easily be shown to be orthogonal to the phase velocity.

Internal waves can be generated by using a horizontal cylinder that oscillates at a frequency smaller then the buoyancy frequency. This will cause waves to be generated, so that energy moves away from the cylinder in four beams, with the crests and troughs (or phase) moving perpendicularly to the direction in which the beams carry energy away from the oscillating cylinder. In the picture below the orange dot represents the cylinder generating the waves and the red and green lines represent the crests and troughs of the wave.