Geoscience Reference
In-Depth Information
The rightmost empirical approximation assumes relatively
weak concentrations of sodium chloride solutions [
Weast
,
1981].
Combining equations (10.2) and (10.5), we find that
light follows a parabolic path when passing through a uni-
form stratification at a scant angle from the horizontal
such that
path is 0.14 mm, which is much more easily discernible.
If one pixel of the camera has a vertical resolution of
0.5 mm, then the disturbance in the tank will shift the
image by about a third of a pixel, which can easily be
observed by the change of intensity of light emanating
from the edge of a line.
Through (10.10), we have found the forward equation
in which known changes to the stratification enables us to
predict the vertical displacement of a point in an object
as seen by a camera looking through the stratified fluid.
In the derivation of (10.7), because we have assumed the
disturbance is spanwise uniform, it is a trivial matter to
invert (10.10) so that an observed vertical displacement in
an image can predict the change in the stratification:
1
2
γ N
T
2
y
2
,
z(y)
=
z
i
+
y
tan
φ
i
−
(10.7)
in which
z
i
is the height and
φ
i
is the angle at which
the light ray enters the tank, In deriving (10.7), we have
assumed that
N
T
is independent of
y
, which is the case for
spanwise uniform disturbances in a tank. This assumption
will be relaxed below in the consideration of axisymmetric
and fully three-dimensional disturbances.
Synthetic schlieren is usually employed to measure per-
turbations to the ambient. Directly, it measures how the
squared buoyancy frequency changes as a result of the
compression and stretching of isopycnal surfaces. Over a
distance
y
=
L
T
, light is deflected vertically by
1
2
L
T
2
+
L
o
L
T
−
1
z
1
γ
n
0
n
a
N
2
−
.
(10.11)
If part of an image appears to deflect downward, it means
thatthestratiication betweenitand thecamera has locally
become stronger.
This result is straightforwardly applied to the circum-
stance in which a camera looks through a tank at an
image of horizontal black-and-white lines, as is the case in
Figure 10.3. Even if density perturbations result in image
distortions that shift the lines by a fraction of their width
(which, in fact, is ideal if you wish easily to compute the
line displacement), then (10.11) immediately predicts the
change to the squared buoyancy frequency.
More processing is required if the line is displaced
significantly. The calculation of displacements is even
more difficult if the lines become magnified or contracted
because second derivatives of the refractive index become
significant (in which case a shadowgraph becomes the
more useful, if qualitative, tool). The method breaks
down entirely, as within the valleys of the model hills in
Figure 10.3b, when the lines blur due to three-dimensional
mixing.
The examination of an image of lines provides a
relatively easy method to calculate small vertical dis-
placements in an image and hence
N
2
.Butafar
more informative, though computationally more inten-
sive, application uses an image of dots [
Dalziel et al.
,
2000]. For example, Figure 10.4 shows a qualitative syn-
thetic schlieren image produced with a beam of internal
waves from an oscillating cylinder that passes in front of
an image of regularly spaced black circles on a white back-
ground. The difference image is shown, analogous to that
in Figure 10.3.
The cat-eye-like patterns have a black portion near the
center of the dot surrounded on two sides by white, indi-
cating how the dot shifted both horizontally and verti-
cally as a result of internal waves passing between the
camera and image. The angle and eccentricity of the
1
2
γN
2
L
T
2
,
z
=
−
(10.8)
in which
g
ρ
0
∂ρ
∂z
N
2
N
T
2
N
2
=
≡
−
−
(10.9)
is the change in the squared buoyancy due to the density
perturbation
ρ
.
For example, if internal waves compress isopycnals so
that the ambient
N
2
locally increases by 10% from 1.0 to
1.1s
−
2
, then the light deflects by 0.04 mm crossing a 20 cm
wide tank. This is a small but discernible displacement
that can be captured by a digital camera with sufficiently
high resolution.
The apparent deflection is larger if the image is placed
some distance behind the tank. Not only is the light
deflected downward if the stratification increases, but also
the angle of the light ray at the tank wall changes. So
the apparent image displacement magnifies linearly as the
image is moved further away.
Assuming the tank walls are negligibly thin, one can
predict the total displacement of light from an object a
distance
L
o
from one side of the tank to a camera on the
other side of the tank to be
1
2
γN
2
L
T
2
n
0
n
a
γN
2
L
o
L
T
, (10.10)
in which
n
a
is the refractive index of air and
φ
0
is the angle
from the horizontal at which light enters the camera from
the object.
In the example above, if we now suppose the image is
20 cm behind the tank, then the displacement of the light
z(N
2
)
−
−
