Geoscience Reference
In-Depth Information
(b)
(a)
Pr < 1
Pr>1
Ekman layer
E 1/3 Stewartson layer
Thermal boundary layer
Temperature
profile
with waves
With waves
Without waves
Without waves
Figure 3.9. Schematic diagram of the effect of baroclinic waves on the mean radial temperature profile for (a) Pr < 1 and (b)
Pr > 1. Shown are sections in the r
z plane with cooling at the right wall, heating at the left, and insulated rigid upper and lower
boundaries.
Homogenous Fluid, Pr <E 1 / 2 The fluid interior is
homogeneous and constrained by the Taylor-
Proudman theorem. Dissipation is through Ekman
suction, and the sidewall boundary layers are the
two Stewartson layers of thickness E 1 / 3 and E 1 / 4 .
Weakly Stratified, E 1 / 2 < Pr <E 2 / 3 The Taylor-Pro-
udman theorem is still strong but thermal stratifica-
tion is increasingly noticeable. The E 1 / 4 Stewartson
layer is largely unaffected but buoyancy very close
to the wall affects the E 1 / 3 Stewartson layer and two
new boundary layers develop: one very thin layer of
thickness E 1 / 2 /( Pr ) 1 / 4 in which the viscous stresses
balance buoyancy and an outer, hydrostatic layer of
thickness ( Pr ) 1 / 2 .
Strongly Stratified, Pr >E 2 / 3 The fluid is strongly
stratified, the hydrostatic ( Pr ) 1 / 2 layer merges with
the interior, and only the E 1 / 2 /( Pr ) 1 / 4 buoyancy
layer remains. Ekman suction is very weak and the
interior is controlled by viscous diffusion.
3.5.3. Low-Order Model of Wave-Boundary Layer
Interaction
Based on the scaling by Barcilon and Pedlosky [1967]
and the observations by Read [2003] that the thinner of
the two vertical boundary layers in a way determines the
behavior of the heat transfer in the steady wave through
the Ekman layers and fluid interior, respectively, we can
propose a simple conceptual model of the interaction
between baroclinic waves and the thermal forcing of the
Ekman circulation and the thermal wind in the interior
using the following argument, which is also illustrated
schematically in Figure 3.9.
Since the analysis by Barcilon and Pedlosky [1967]
suggests a change from the E 1 / 3 Stewartson layer to
thermal boundary layers scaling with Pr ,weusethe
E 1 / 3 Stewartson layer as our reference layer in the follow-
ing argument, which is based on the relative thickness of
the vertical velocity and thermal boundary layers.
To determine the relative thicknesses and put these in
the context of the nondimensional parameters, a set of
axisymmetric solutions of the MORALS code for a range
of Prandtl numbers but otherwise fixed parameters was
generated (see Appendix Appendix A: for model speci-
fications). Applying the classification from Barcilon and
Pedlosky [1967], all solutions presented here are nominally
strongly stratified, with the transition from weak stratifi-
cation to strong stratification at Pr
Read [2003] analyzed the heat transfer calculated using
the MORALS model (introduced above in Section 3.3.1)
as a function of a boundary layer ratio, defined as the
squared ratio of the thermal sidewall boundary layer
thickness to the Ekman layer thickness. This demon-
strated that the Ekman layer was the limiting factor when
the thermal boundary layer was wider than the Ekman
layer but that the heat transport by the axisymmetric
flow became constant when the thermal boundary layer
became thinner than the Ekman layer.
0.1. Since the Ekman
number was a constant in all computations, the E 1 / 3 Stew-
artson layer thickness was used as our reference layer.
Figure 3.10 shows the distances of the theoretical and
 
Search WWH ::




Custom Search