Geoscience Reference
In-Depth Information
does not mean of course that the flows in the tank are
necessarily quasi-geostrophic. In the experiments we mea-
sure a “true” field of the gradient of the surface elevation
(in fact the pressure gradient), but the calculated velocity
is more accurate when the dimensionless numbers RoT
T
,
Ro, and
γ
∗
are small.
Yet another important application of AIV is to study
baroclinic flows. In a baroclinic flow AIV gives only the
surface (barotropic) velocity. In some cases this may be
enough for an adequate description of the flow, especially
if the baroclinic component of velocity is relatively weak.
However, both barotropic and baroclinic components can
be measured in the special case of a two-layer fluid. In
this case AIV is used in combination with the optical
density method. Here we describe this method, referring
to
Afanasyev et al.
[2009] for more details. In order to
measure the thickness of one of the layers, we dye this
layer with a red food dye. The layer does not have to cover
theentireareaof thedomain.Insomeexperimentsthedyed
fluid is supplied from a source to form boundary currents,
lenses, or plumes that have a distinct boundary separating
the dyed fluid from the clear ambient water. The thickness
h
1
of the dyed layer can be measured by relating the value
of the chromaticity
u
or
v
to the thickness of the dyed
fluid. This approach differs from that used by others [e.g.,
Linden and Simpson
, 1994;
Cenedese and Dalziel
, 1998],
whousedthebrightness
L
forthispurpose.Chromaticityis
chosen because it is less affected by background variations
of brightness. For calibration purposes a wedge-shaped
cuvette filled with the fluid with the same concentration
of dye as that used in the experiment is placed above the
tank. The profile of the chromaticity
a
across the cuvette
gives the relation between the depth of the fluid in the
cuvette and its color intensity. This relation is used to
calculate
h
1
in every pixel of the image of the flow. The
baroclinic component of the flow can then be calculated
using (5.15), where the layer thickness
h
1
is used instead of
thesurfaceelevation
η
andthereducedgravity
g
=
g ρ /ρ
,
where
ρ
is the density difference between the layers, is
used instead of
g
. The optical density method requires a
uniform background light. The method can be used either
in a separate experiment or in combination with AIV.
Almost simultaneous measurements can be performed by
switching back and forth from the background light to the
color mask light. The time difference between the velocity
fields measured by the optical density method and AIV
will be determined by the frame rate of the camera.
a rotating layer occur due to variation of the Coriolis
parameter or due to variation of the depth of the layer
h(r)
, respectively. Here we call both planetary and topo-
graphic waves “Rossby waves,” assuming their dynamic
equivalence via the conservation of PV. These waves are
of relatively low frequency, with the frequency being pro-
portional to the parameter
β
or
γ
, depending on which
approximation is used. In fact, they constitute a subset of
inertial waves.
Phillips
[1965] showed that Rossby waves in
a rotating annulus with quadratically varying depth can be
obtained from a set of inertial wave modes when appropri-
ate boundary conditions are applied. The theory of both
inertial waves and Rossby waves is well known, and the
details can be found in textbooks. Here we describe only
briefly the surface elevation and velocity fields of both
types of waves for the purpose of showing their surface
signature. AIV allows one to observe and reconstruct the
structure of the waves by their surface signature.
Consider first inertial waves in a rotating layer of water.
For simplicity, we assume that the Coriolis parameter and
the depth of the layer are constant. Let us start with
linearized equations of motion for a layer of depth
H
0
∂u
∂t
−
1
ρ
∂p
∂x
,
∂v
∂t
+
f
0
u
=
1
ρ
∂p
∂y
,
f
0
v
=
−
−
(5.16)
∂w
∂t
=
1
ρ
∂p
∂z
,
∂u
∂x
+
∂v
∂y
+
∂w
−
∂z
=0
with boundary conditions
w(
0
)
=
∂η
H
0
)
=0.
(5.17)
∂t
,
p(
0
)
=
ρgη
,
w(
−
We look for a solution in the form of horizontally prop-
agating modes with vertical structure determined by their
z
-dependent amplitudes,
(u
,
v
,
w
,
p
,
η)
=
(u
0
(z)
,
v
0
(z)
,
w
0
(z)
,
p
0
(z)
,
η
0
)
×
exp
[
i(ωt
−
kx
−
ly)
]
.
(5.18)
The system of equations (5.16) can then be reduced to
a single equation for pressure, which yields a solution of
the form
p
=
ρgη
cos
γ
n
(z
+
H
0
)
cos
γ
n
H
0
(5.19)
with dispersion relation
γ
n
tan
γ
n
H
0
=
ω
2
−
g
,
(5.20)
5.4. INERTIAL AND PLANETARY/
TOPOGRAPHIC WAVES
where
γ
n
=
ω
2
k
2
+
l
2
A rotating fluid system sustains inertial waves with fre-
quencies below the value of the Coriolis parameter
ω
f
0
[e.g.,
Batchelor
, 1967]. Planetary or topographic waves in
≤
f
0
−
ω
2
.
(5.21)
