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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)
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