Geoscience Reference
In-Depth Information
Figure 11.11 Sketches of profiles of themean value and vertical flux of a conserved
scalar in bottom-up (left) and top-down (right) diffusion. From
Wyngaard
(
1987
).
mixed-layer top, in conjunction with the change in
C
that develops across the
interfacial layer, will generate top-down diffusion as well. Thus the bottom-up
process, unlike the top-down one, is unlikely to exist alone.
A natural question is whether top-down and bottom-up diffusion have the same
properties. This would be the case in a flow with statistical symmetry about its
horizontal midplane - for example, convection between parallel plates, the bottom
plate heated and the top plate cooled, the two surface heat fluxes being equal in
magnitude. Here the profiles of the eddy diffusivities
K
b
and
K
t
are symmetric
about the midplane:
K
t
(
1
K
b
(z/z
i
)
. But since the ABL is not symmetric
about its midplane, top-down and bottom-up diffusion in the ABL should also lack
that symmetry.
†
To examine this asymmetry we use the linearity of its conservation equation
to represent a conserved scalar field as the sum of top-down and bottom-up parts
coexisting in the velocity field:
−
z/z
i
)
=
c
˜
=˜
c
b
+˜
c
t
,C
=
C
t
+
C
b
,c
=
c
t
+
c
b
.
(11.21)
The simplest mixed-layer similarity hypothesis is that the mean scalar gradient in
each process depends only on the boundary flux, the mixed-layer scales
w
∗
and
z
i
,
and
z
:
∂C
t
∂z
cw
1
w
∗
z
i
g
t
(z/z
i
),
∂C
b
∂z
cw
0
w
∗
z
i
g
b
(z/z
i
),
=−
=−
(11.22)
where
g
t
and
g
b
are dimensionless functions. Then the symmetry question iswhether
g
b
(z/z
i
)
=
g
t
(
1
−
z/z
i
)
.
†
In fact, one expects that turbulent boundary layers in general lack that symmetry.
