Geoscience Reference
In-Depth Information
4 Scaling Analysis
4.1 Rotating Case
The main goal of this section is to define the radial velocity of the jet propagation,
U
(m/s), in dimensionless form using the main external dimensional parameters:
buoyancy flux (initiated by line source),
B
(m
3
/s
3
); Coriolis parameter,
f
(s
1
);
depth of the tank,
H
(m); time,
t
(s); thermal diffusivity,
k
T
(m
2
/s); and kinematical
viscosity,
(m
2
/s).
Results of laboratory experiments have shown that the radial propagation velocity
of the jet is proportional to the buoyancy flux,
B
, and Coriolis parameter,
f
,as
U
~
B
1/2
,
U
~
f
1
, and that it does not depend on time. As a working fluid in all
the experimental runs is fresh water, we can assume that the
k
T
and
n
n
are nearly
constant.
Thus, the main dimensionless parameters governing the process are the (flux)
Rayleigh number, Ra
F
¼
g
a
FH
3
=r
0
c
p
k
T
n
, and the Ekman number,
n
fH
2
Ek
¼
(1)
Here,
g
is the acceleration due to gravity (m/s
2
);
the thermal expansion
coefficient (
C
1
);
F
the heat flux (Wt/m);
H
the vertical length scale (m);
r
0
the
reference density (kg/m
3
);
c
p
the water heat capacity (Wt/m kg
C);
k
T
the thermal
diffusivity (m
2
/s);
a
the viscosity (m
2
/s). The heat flux,
F
, causes the buoyancy
n
c
p
(m
3
/s
3
) into the surface layer. Thus, it is more convenient to
use for the Rayleigh number the expression:
fluxes,
B
¼
F
a
g
=r
BH
3
k
Ra
B
¼
(2)
T
n
Therefore, we may parameterize the velocity of the buoyant jet as follows:
U
B
1
=
3
Ra
1
=
2
Ek
(3)
In the final form, the formula for the radial velocity of the jet is defined as:
B
1
=
2
1
=
2
k
T
H
1
=
2
f
;
U
B
1
=
3
n
1
=
2
B
1
=
3
n
k
T
H
1
=
2
f
U
;
or, using the Pr number,
