Geoscience Reference
In-Depth Information
in density between different parts of the fluid. When the density disturbance is
of thermal origin,
ρ
=
ρ
−
ρ
0
=−
ρ
0
α
(
T
−
T
0
)
(8.30)
where
ρ
0
is the density at a reference temperature
T
0
, and
α
is the volumetric
coefficient of thermal expansion.
In order to use Eqs. (8.26)-(8.30)toevaluate the form of convective flow, it is
usual to present the equations in a parametric form, which means that the values
of density, viscosity, length, time etc. are all scaled to a dimensionless form (e.g.,
see Hewitt
et al
. 1980). For doing this, several dimensionless numbers that com-
pletely describe the flow are routinely used in fluid dynamics. The dimensionless
Rayleigh number Ra
is given by
α
gd
3
T
κυ
Ra
=
(8.31)
where
is the volume coefficient of thermal expansion,
g
the acceleration due to
gravity,
d
the thickness of the layer,
α
T
the temperature difference in excess of the
adiabatic gradient across the layer,
κ
the thermal diffusivity and
υ
the kinematic
viscosity (kinematic viscosity
). The
Rayleigh number measures the ratio of the heat carried by the convecting fluid
to that carried by conduction. Flow at a particular Rayleigh number always has
the same form regardless of the size of the system. Thus it is straightforward for
laboratory experiments to use thin layers of oils or syrups over short times and
then to apply the results directly to flow with the same Rayleigh number in the
Earth (with viscosity, length and time scaled up appropriately).
To evaluate thermal convection occurring in a layer of thickness
d
, heated from
below, the four differential equations (8.26)-(8.30)havetobesolved with appro-
priate boundary conditions. Usually, these boundary conditions are a combination
of the following:
=
dynamic viscosity
/
density, i.e.,
υ
=
η/ρ
(i)
z
=
0or
z
=
d
is at a constant specified temperature (i.e., they are isotherms), or the
heat flux is specified across
z
=
0or
z
=
d;
(ii) no flow of fluid occurs across
z
=
0 and
z
=
d
; and
(iii)
z
=
0or
z
=
d
is a solid surface, in which case there is no horizontal flow (no slip)
along these boundaries, or
z
=
0or
z
=
d
is a free surface, in which case the shear
stress is zero at these boundaries.
Solution of the equations with appropriate boundary conditions indicates that
convection does not occur until the dimensionless Rayleigh number,
Ra
,exceeds
some critical value
Ra
c
.For this layer the Rayleigh number can be written
α
gd
4
(
Q
+
Ad
)
k
κυ
Ra
=
(8.32)
where
Q
is the heat flow through the lower boundary,
A
the internal heat generation
and
k
the thermal conductivity. The critical value of the Rayleigh number further
depends on the boundary conditions.
