Geography Reference
In-Depth Information
are negligible following the motion.) In addition, the equation of state must be
replaced by an appropriate measure of the relationship between temperature and
density:
ρ
=
ρ
0
[1
−
ε (T
−
T
0
)]
(10.75)
10
−
4
K
−
1
for water) and ρ
0
where ε is the thermal expansion coefficient (ε
≈
2
×
is the density at the mean temperature T
0
.
Chapter 1 indicated that the character of the motion in a fluid is crucially depen-
dent on the characteristic scales of parameters such as velocity, pressure perturba-
tion, length, and time. In the laboratory the scales of these parameters are generally
many orders of magnitude different from their scales in nature. It is still possible,
however, to produce quasi-geostrophic motions in the laboratory provided that
the motions are slow enough so that the flow is approximately hydrostatic and
geostrophic. As shown in Section 2.4.2, the geostrophy of a flow does not depend
on the absolute value of the scaling parameters, but rather on a nondimensional
ratio of these parameters called the Rossby number. For annulus experiments,
the maximum horizontal scale is set by the dimensions of the tank so that it is
convenient to define the Rossby number as
Ro
≡
U/ [ (b
−
a)]
where is the angular velocity of the tank, U is a typical relative velocity of the
fluid, and b
a is the difference between the radius b of the outer wall and the
radius a of the inner wall of the annular region.
Using the hydrostatic approximation
∂p
∂z
−
=−
ρg
together with the geostrophic relationship
ρ
−
1
0
2
V
g
=
k
×
∇
p
and the equation of state (10.75) we can obtain a
thermal wind
relationship in
the form
∂
V
g
∂z
=
εg
2
k
×
∇
T
(10.76)
Letting U denote the scale of the geostrophic velocity, H the mean depth of fluid,
and δT the radial temperature difference across the width of the annulus, we obtain
from (10.76)
a)] (10.77)
Substituting the value of U in (10.77) into the formula for the Rossby number
yields the
thermal Rossby number
U
∼
εgH δT
/
[2 (b
−
εgH δT
2
2
(b
a)
2
Ro
T
≡
−
