Geoscience Reference
In-Depth Information
Pr
−
2
we have an inertial/Coriolis balance
in the Ekman layer (i.e., there is no proper Ekman layer),
and the entire flow is characterized by local conserva-
tion of angular momentum [hence
V
=
O(fL)
, which
is proportional to
Thus, for
P
O
(1)
; see Figure 1.19].
(iii) Weak rotation: For
P
Pr
−
2
, the viscous
term in (1.56) becomes dominant in the Ekman layer
(i.e., normal Ekman layers exist), thus rescaling
V
to
O(κ
Ra
1
/
2
P
(Pr
-2
)
(Ra
1/6
)
O
O
(1)
O
O(Ra
-
1/4
)
Very weak
Weak
Moderate
Strong
Very strong
1
/
2
/L)
. This balance extends into the inte-
rior, while the previous balance in the sidewall layer is
unchanged from regime (i). Despite the new scaling for
V
, the dominant balances (and scaling for Pe) in (1.40)
also remain unchanged from (i). The rescaling of
V
does,
however, affect the interior balance in the azimuthal vor-
ticity equation, from a buoyancy/viscous balance to a
buoyancy/Coriolis balance characteristic of the “thermal
wind” balance typical of geostrophic flow. The reason
why the (now geostrophic) scale for
V
does not go as
−
1
typical of a thermal wind scale is because
γ
is now
increasing rapidly with
(
γ
=
O(
P
O(Ra
-
3/4
)
Figure 1.19.
Schematic diagram showing the dependence
of derived parameters on internal parameters in the various
axisymmetric regimes defined in terms of
assuming
=
O(
1
)
.
Quantities represented are
VL/(κ
Ra
1
/
2
)
(solid line);
/(κ
Ra
1
/
4
)
(dashed line);
γ
(dash-dotted line);
/L
(dotted line), and Ro
(dash-crossed line). (Adapted from
Read
[1986] with permis-
sion).
P
3
/
2
)
), which more
P
P
−
1
dependence of
V
for constant
γ
. Note also the zonal Rossby number Ro =
V/fL
=
O(
Pr
−
1
than outweighs the
P
−
1
/
2
and is therefore
1 (see Figure 1.19).
(iv) Moderate rotation: In this regime, the Ekman
layer thickness is comparable with that of the sidewall
buoyancy layer and so is expected to begin to exert a
strong influence on the meridional circulation and trans-
port. Anticipating that
V
will eventually tend toward
the thermal wind scale
O(
(assuming hereafter for simplicity that
= 1) which is pro-
portional to
. Based on a consideration of the full range
of
, we can effectively identify up to six distinct regimes
of axisymmetric flow (see also Figure 1.19):
(i) No rotation,
P
=0.
(ii) Very weak rotation, 0
P
Pr
−
2
.
P
P
−
1
)
, this range of
delin-
eates the regime where
V
reaches a maximum
V
o
=
O(κ
Ra
1
/
2
/L)
. If the Ekman layer exercises dominant
control over the radial mass transport,
will be rescaled
to
O(V
o
L
E
1
/
2
)
=
O(κ
Ra
1
/
4
P
(iii) Weak rotation, Pr
−
2
P
1.
(iv) Moderate rotation,
P
1.
Ra
1
/
6
.
(v) Strong rotation, 1
P
Ra
1
/
6
.
We now briefly outline their characteristics:
(i) No rotation: This has already been discussed above
in Section 1.4.2.1, with consequent scales for
and Pe.
Note that we can obtain an estimate of isotherm slope
γ
=
T
h
/T
(where
T
h
is the horizontal temper-
ature contrast) from a consideration of the balances
in the zonal vorticity equation. Provided Pr
(vi) Very strong rotation,
P
P
−
1
/
2
)
, implying a slow
broadening of the sidewall advective/diffusive boundary
layer from
to
1
/
2
.
(v) Strong rotation: As
P
is increased beyond 1, the
Ekman layers fully dominate the meridional circulation.
By this point, the isotherm slope
γ
has become
O(
1
)
and so cannot increase any further. Then
V
rescales to
the familiar thermal wind scale
V
=
O(κ
Ra
1
/
2
L
−
1
P
1, a
buoyancy/viscous balance holds in the interior, so that
P
−
1
)
.
The expansion of the advective/diffusive sidewall layer
accelerates to
=
O(
Ra
−
1
/
4
L
3
/
2
)
, extending the influ-
ence of thermal diffusion further into the interior. The
heat transport Pe is rescaled to
O(
Ra
1
/
4
P
gα∂T/∂x
=
O(gαTγ/L)
(1.54)
4
χ(
=
O(νL
−
4
)
.
=
ν
∇
(1.55)
P
−
3
/
2
)
, though
it remains
1 (see Figure 1.19).
(vi) Very strong rotation: In this final regime, the dif-
fusive thermal sidewall layer expands to fill the interior
and no separate thermal boundary layer and interior
can be distinguished (though Stewartson
Hence
γ<
Ra
−
3
/
4
1 and isotherms are quasi-
horizontal.
(ii) Very weak rotation: When
f
is no longer zero,
(1.36) is coupled to (1.38) and gyroscopic torques render
v
non-zero. We obtain an estimate for the zonal veloc-
ity scale
V
by scaling (1.38) in the Ekman layer using its
characteristic depth
h
=
1
/
3
layers may
E
exist in this limit). The critical value for
distinguishing
regimes (v) and (vi) simply arises from equating
[see
(v) above] with
L
so that
P
1
/
2
L
. Hence, (1.36) becomes
E
>
Ra
1
/
6
. All other balances
remain unchanged from (v), i.e., the geostrophic interior
and strong Ekman layers. Heat transport in this regime,
P
∂χ
∗
∂z
∗
2
v
∗
=
fL
V
1
/
2
+
J(v
∗
,
χ
∗
)
.
Pr
P
∇
(1.56)