Geoscience Reference
In-Depth Information
4.2 Temperature flux in a boundary layer
With the decomposition
T
=
+
∂
2
∂x
i
∂x
i
.
∂
∂t
+
U
i
∂
∂u
i
θ
∂x
i
=
∂x
i
+
α
(4.2)
The molecular-diffusion term is important only in the diffusive sublayer on the
surface. If that surface is very large in extent and horizontally homogeneous, the flow
approaches horizontal homogeneity inwhich
U
i
=[
(z, t)
.
This is a common model of the atmospheric boundary layer. Then the time rate of
change of the mean fluid temperature above the surface is due to the divergence of
the turbulent flux,
U(z,t),
0
,
0
]
and
=
∂
∂t
=−
∂u
i
θ
∂wθ
∂z
∂x
i
=−
.
(4.3)
If the surface is warmer than the fluid above, rising (positive
w
) fluid tends to
be warmer than its local environment (positive
θ
), since it came from the warmer
region nearer the surface; likewise, sinking (negative
w
) fluid tends to be cooler
than its environm
ent
(negative
θ
), since it came from t
he c
ooler region farther away.
Thus, w
e ex
pect
wθ
to be positive. Correspondingly,
wθ
is negative over a cooler
surface.
wθ
vanishes at the boundary-layer top because the turbulence vanishes
there. Thus the divergence of the turbulent temperature flux is nonzero within the
boundary layer, which through
Eq. (4.3)
causes
to change with time.
The familiar near-surface warming of the atmosphere on a sunny morning is
well described by
Eq. (4.3)
(interpreted for potential temperature, defined in
Part II
).
Warming typically occurs at nearly the same rate overmuch ormost of the deepening
boundary layer. If so,
Eq. (4.3)
says that the temperature flux divergence is nearly
constant with height.
Typical daytime, clear-weather
wθ
and
profiles are sketched in
Figure 4.1
.
The
re is a pronounced negative mean temperature gradient near the surface, where
wθ
is positive. This is consistent with the existence of an eddy diffusivity
K
such that
wθ
K∂/∂z
.
Abo
ve the surface themagnitude of
∂/∂z
decreases with height
more sharply than
wθ
does, implying that
K
increases away from the surface. The
top of the daytime boundary layer is typically determined by a “capping” inversion -
a stably stratified layer (
increasing with height) that damps turbulence and acts
as a “lid” for risi
ng
convective elements. The entrainment of this inversion by
turbulence makes
wθ
negative in this interfacial region, as sketched in
Figure 4.1
.
As
Figure 4.1
indicates, this region of positive
∂/∂z
aloft can extend down into
the boundary layer. As a result the eddy diffusivity for temperature can be poorly
=−