Geoscience Reference
In-Depth Information
1. Fornoshear stress on the upper and lower boundaries, the upper boundary held at a
constant temperature and all heating from below (
A
=
0),
Ra
c
=
27
π
4
/
4
=
658. At this
Rayleigh number the horizontal dimension of a cell is 2.8
d
.
2. Fornoslip on the boundaries, the upper boundary held at a constant temperature and
all heating from below (
A
=
0),
Ra
c
=
1708. At this Rayleigh number the horizontal
dimension of a cell is 2.0
d
.
3. Fornoslip on the boundaries, a constant heat flux across the upper boundary and
all heating from within the fluid (
Q
=
0),
Ra
c
=
2772. At this Rayleigh number the
horizontal dimension of a cell is 2.4
d
.
4. Fornoshear stress on the boundaries, a constant heat flux across the upper boundary
and all heating from within the fluid (
Q
=
0),
Ra
c
=
868. At this Rayleigh number the
horizontal dimension of a cell is 3.5
d
.
Thus, although the exact value of the critical Rayleigh number
Ra
c
depends on
the shape of the fluid system, the boundary conditions and the details of heating,
it is clear in all cases that
Ra
c
is of the order of 10
3
and that the horizontal cell
dimension at this critical Rayleigh number is two-to-three times the thickness of
the convecting layer. For convection to be vigorous with little heat transported
by conduction, the Rayleigh number must be about 10
5
.Ifthe Rayleigh number
exceeds 10
6
, then convection is likely to become more irregular.
A second dimensionless number
6
used in fluid dynamics, the
Reynolds number
(
Re
), measures the ratio of the inertial to viscous forces,
ρ
ud
η
Re
=
ud
υ
=
(8.33)
where
u
is the velocity of the flow,
d
the depth of the fluid layer and
the kinematic
viscosity.
Re
indicates whether fluid flow is laminar or turbulent. A flow with
Re
υ
1isturbulent.
Re
for the mantle is about 10
−
19
-10
−
21
,sothe flow is certainly laminar.
A third dimensionless number, the
Nusselt number
(
Nu
), provides a measure
of the importance of convection to the heat transport:
1islaminar, since viscous forces dominate. A flow with
Re
heat transported by the convective flow
heat that would be transported by conduction alone
Nu
=
Qd
k
T
Nu
=
(8.34)
where
Q
is the heat flow,
d
the thickness of the layer,
k
the thermal conductivity
and
T
the difference in temperature between the top and bottom of the layer. The
Nusselt number is approximately proportional to the third root of the Rayleigh
number:
Nu
≈
(
Ra
/
Ra
c
)
1
/
3
(8.35)
6
Fluid dynamics utilizes several dimensionless numbers, all named after prominent physicists.
