Geoscience Reference
In-Depth Information
11.3 The mixed layer: conserved-scalar fields
In a horizontally homogeneous ABL themean value
C
of a conserved scalar satisfies
∂C
∂t
+
∂cw
∂z
=
0
.
(11.19)
As indicated in
Figure 11.1
,
in general
cw
is nonzero at the bottom and top of
the mixed layer.
In
a horizontally homogeneous situation where
∂C/∂z
does not
depend on time,
cw
varies linearly with
z
(Problem 11.14)
.
11.3.1 “Top-down” and “bottom-up” diffusion
Since a conserved scalar constituent
c
is governed by a linear differential equation,
we can superpose its solutions in the same velocity field. Thus we can consider
separately the
C
profiles that result from the scalar fluxes at the top and bottom
of the mixed layer. We'll call these individual processes
top-down
and
bottom-up
diffusion. These labels refer not to the direction of the scalar flux, but rather to
where the flux is applied.
†
We'll consider dynamically passive scalars - ones that do not affect the velocity
field - in a quasi-steady, horizontally homogeneous CBL. Temperature and water
vapor are not passive, for they induce buoyancy forces. But we'll assume that two
conserved scalars with the same boundary and initial conditions and in the same
turbulence field diffuse identically, so that one can infer the diffusion properties
of temperature and water vapor in a given velocity field from those of passive,
conserved tracers diffusing in that velocity field.
We write the scalar flux profile as the sum of two linear sub-profiles, each with
one nonzero boundary flux:
cw
0
1
cw
1
z
z
i
.
z
z
i
=
z
z
i
cw(z)
=
cw
0
+
(cw
1
−
cw
0
)
−
+
(11.20)
The right panel of
Figure 11.11
sketches an example of top-down diffusion of a
trace constituent
c
. There is a positive flux of constituent at the top of the mixed
layer and zero flux at the surface. Here the top flux is due to the entrainment of
air with lower constituent concentration (
c
negative) in downward (
w
negative)
turbulent motion and high
er-c
oncentration, boundar
y-la
yer air in upward motion.
In a quasi-steady state the
cw
profile is linear, with
∂cw/∂z
positive so that
∂C/∂t
is negative, as sketched.
A bottom-up case is sketched in the left panel of
Figure 11.11
.
Here the flux
divergence is negative so that
C
increases with time. Here the entrainment at the
˜
†
In upper-ocean applications the
top
is the thermocline and the
bottom
is the surface.
