Geoscience Reference
In-Depth Information
4.2 Temperature flux in a boundary layer
With the decomposition T
θ , ensemble averaging the temperature equation
(1.32) in a turbulent boundary layer yields
=
+
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 to be positive. Correspondingly, is negative over a cooler
surface. 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 and profiles are sketched in Figure 4.1 .
The re is a pronounced negative mean temperature gradient near the surface, where
is positive. This is consistent with the existence of an eddy diffusivity K such that
K∂/∂z . Abo ve the surface themagnitude of ∂/∂z decreases with height
more sharply than 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 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
=−
 
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