Geoscience Reference
In-Depth Information
(x
−
X
c
,
−
H
c
)
(
−
x
+
X
,
H
m
)
null field. This procedure gives the gradient of the surface
elevation,
I
=
(x
,
R
=
(
,
η
=
(∂η/∂x
,
∂η/∂y)
.
Integrating
∇
−
X
c
)
2
+
H
c
−
x
+
X)
2
+
H
m
η
over
x
and
y
, we can obtain the field
of surface elevation
η
and hence the barotropic pressure
p
=
ρgη
. It is also important to obtain the velocity
field of the flow. Fortunately, the gradient of the sur-
face elevation can be easily converted into a velocity field
in a rapidly rotating fluid where geostrophic and quasi-
geostrophic approximations can be employed. Consider
the shallow-water equation
∂
V
∂t
+
(
V
∇
0
x/g
,1
)
(
−
N
=
0
x
2
/g
2
+1
,
(5.9)
where
x
,
X
c
,
X
are the
x
coordinates of the point on the
surface, the camera lens, and the point on the slide and
H
m
and
H
c
are the heights of the color mask and the camera
above the surface of water (see Figure 5.1). The equations
resulting from the reflection law give in the first approx-
imation that at certain rotation rate of the tank,
0
,the
rays coming to the camera from all points of the surface
originate from one point of the color mask, namely its
center (for details see
Afanasyev et al.
, [2009]). The rota-
tion rate determines the focal distance of the paraboloid
such that at this particular value of the rotation rate the
camera sees the surface uniformly illuminated by nearly
one color (that of the center of the mask). We call this
rotation rate a null point and all altimetric measurements
are performed at this rate. The value of
0
is determined
by the geometry of the setup:
0
=
g(H
c
+
H
m
)
)
V
+
f
0
−
γ r
2
(
k
·
∇
×
V
)
=
−
g
∇
η
, (5.12)
where
V
is the horizontal velocity vector and
k
is the ver-
tical unit vector. The relative importance of the unsteady
term (the first term) and the nonlinear term (second
term) is determined by the temporal Rossby number
Ro
T
=1
/(T)
and the Rossby number Ro=
U/(L)
.
Here
T
is the time scale of the flow evolution, while
U
and
L
are velocity and length scales of the flow. A third
dimensionless number,
γ
∗
, is given by the ratio of the
γ
-term to the Coriolis parameter,
γ
∗
=
γ L
2
/f
0
. If all three
dimensionless numbers are small, we may neglect the cor-
responding terms in (5.12) and find that the main balance
is between the Coriolis term and the pressure gradient
term, which gives the geostrophic equation
f
0
k
.
(5.10)
2
H
c
H
m
Usually the camera and the mask are at approximately the
same height such that
0
=
g/H
m
. Thus the lower the
mask/camera, the faster the rotation rate must be.
A more accurate solution of the reflection law equa-
tions shows that a color does vary across the surface at the
null point (in the absence of perturbations). Although an
analytical expression of the solution exists, it is more con-
venient to use measured values. In the beginning of each
experiment before the flow is started, one image show-
ing this color distribution is recorded. A color-matching
procedure implemented in a numerical code allows us to
relate the color observed in each point of the surface to
its position on the mask (
X
,
Y
) where the particular color
originates. Thus the null-point image gives the values of
(
X
null
,
Y
null
)
across the surface of water. When the flow
is started, a pressure field created by the flow perturbs the
surface of water. The surface slope changes with respect to
that of the null ield. As a result a ray coming to the camera
originates from a point on the mask (
X
,
Y
) that is differ-
ent from that of the null point (
X
null
,
Y
null
)
. Figure 5.1b
shows the geometry of the reflection from the perturbed
surface. The slope
∂η/∂x
=
α
due to the perturbation can
be found from the geometry in the form
∂η
V
g
=
×
−
g
∇
η
.
(5.13)
The geostrophic velocity
V
g
is then obtained by taking
the cross product with vector
k
of both sides of the above
equation,
V
g
=
g
f
0
(
k
×
∇
η)
.
(5.14)
Substituting (5.14) into the small terms of equation (5.11)
such that the total velocity
V
remains only in the Coriolis
term
f
0
(
k
×
V
)
, we then solve for
V
to obtain
η)
+
γ r
2
f
0
g
f
0
∂
∂t
∇
g
f
0
V
=
V
g
−
η
−
J (η
,
∇
V
g
,
(5.15)
where
∂A
∂y
is the Jacobian operator. Note that we assumed in fact that
V
=
V
g
+
V
a
, where the ageostrophic part of the total
velocity is relatively small,
V
a
<
V
g
. The right-hand side
(RHS) of equation (5.15) contains only the components
of
J(A
,
B)
=
∂A
∂x
∂B
∂y
−
∂B
∂x
η
and their time derivatives. These quantities are mea-
sured by AIV. Note that in order to measure
∇
η
, we need
only one image of the flow, while to measure the time
derivative, two successive images are required. Thus, (5.15)
allows one to calculate the (barotropic) velocity of the
flow in the quasi-geostrophic approximation. Note that
the fact that we use quasi-geostrophic velocity conversion
∇
∂x
=
1
X
null
)H
m
(X
null
−
(X
−
.
(5.11)
x)
2
+
H
m
2
The
y
component of the slope is determined in a simi-
lar manner. The mask coordinates (
X
,
Y
) are determined
by color matching similar to the color matching for the
