Geoscience Reference
In-Depth Information
and neglected centrifugal accelerations. It is convenient to
define a meridional stream function
χ
such that
u
=
∂χ
dynamical balances in the interior and principal bound-
ary layers and obtain the dependence of
and the zonal
velocity scale on external parameters over as wide a range
as possible. Initial assumptions are restricted as follows:
(i) Aspect ratio
is not too different from unity.
(ii) Single thickness scales are assumed,
for the side
and
h
for the horizontal boundary layers.
(iii) Outside the boundary layers there is a distinct inte-
rior flow with length scales
L
and
H
such that
(
,
h)
N
∂χ
∂x
.
∂z
,
v
=
−
(1.35)
The steady-state equations for momentum, continuity,
and heat then reduce to a zonal momentum equation
2
v
=
f
∂χ
ν
∇
∂z
+
J(v
,
χ)
,
(1.36)
(L
,
H)
.
(iv) Prandtl number Pr
where
f
=2
and the Jacobian is defined as
J(c
,
d)
=
∂c
∂x
1.
∂d
∂z
−
∂c
∂z
∂d
∂x
;
(1.37)
1.4.2.1. Nonrotating Problem.
We assume the flow to
comprise an advective interior and thin sidewall boundary
layers and nondimensionalize in the thin sidewall layer of
thickness
using
x
=
x
∗
,
z
=
Hz
∗
,
T
the azimuthal vorticity equation is
4
χ
=
gα
∂T
f
∂v
2
χ)
,
∇
∂x
−
∂z
−
∇
ν
J(χ
,
(1.38)
T
o
=
TT
∗
,
−
where
T
is the temperature and
α
the volumetric expan-
sion coefficient and vorticity
ζ
is defined as
ζ
=
∂u
χ
=
χ
∗
.
(1.47)
∂v
∂x
=
Thus equation (1.40) becomes
2
χ
;
∂z
−
∇
(1.39)
2
T
∗
+
κH
J(χ
∗
,
T
∗
)
.
∇
(1.48)
and the temperature equation is
For advective/diffusive balance, we require
=
κH
2
T
+
J(χ
,
T)
= 0.
κ
∇
(1.40)
We consider a container of aspect ratio
defined by
=
H
L
.
(1.49)
For this case, (1.38) becomes
(1.41)
4
χ
∗
=Ra
L
4
∂T
∗
∂x
∗
−
1
Pr
J(χ
∗
,
(where
H
and
L
are the vertical and horizontal length
scales, respectively, of the domain) and apply boundary
conditions
χ
=
∂χ
2
χ
∗
)
.
∇
∇
(1.50)
1, we obtain a buoyancy/viscous balance in the
sidewall boundary layer, implying that
=Ra
−
1
/
4
1
/
4
L
,
If Pr
∂z
=
v
=
∂T
∂z
=0,
z
=0,
H
,
(1.42)
(1.51)
χ
=
∂χ
∂x
=
v
=
T
−
T
0
=0,
x
=
−
L/
2,+
L/
2. (1.43)
which was the result obtained by
Read
[1992] [see also
Fein
, 1978;
Friedlander
, 1980] for the principal boundary
layer scale when
N
2
Pr
/f
2
We make use of dimensionless parameters such as the
Ekman number
2
/
3
. In this case, the Nusselt
or Péclet number is obtained from equation (1.49) as
E
E
defined by
ν
H
2
,
κ
=
O
Ra
1
/
4
3
/
4
.
E
=
(1.44)
1=Pe=
N
−
(1.52)
the Prandtl number Pr
(
=
σ/κ)
, and the Rayleigh number
Ra =
gαTL
3
κν
1.4.2.2. Effects of Rotation.
In considering the relative
impact of rotation on the circulation, it seems intuitive
that the Ekman layer will be of importance. It is there-
fore convenient to follow an approach due to
Hignettetal.
[1981] and recently applied to good effect in the context
of convection in rotating systems by
King et al.
[2009] and
King et al.
[2012] in defining a parameter
.
(1.45)
) and Péclet
(Pe) numbers as measuring the ratios of total heat trans-
port and advective heat transport, respectively, to that due
to conduction, which we take to be
It is also convenient to define Nusselt (
N
measuring the
(square of the) ratio of the thickness of the Ekman layer
and sidewall buoyancy layer. Thus
P
=
N
κ
+ 1 = Pe + 1
(1.46)
(where
is a characteristic scale for
χ
). We then carry
out a scale analysis with the aim of deriving the dominant
=Ra
−
1
/
2
E
−
1
,
P
(1.53)
