Geography Reference
In-Depth Information
Fig. 5.3
Balance of forces in the well-mixed planetary boundary layer:
P
designates the pressure
gradient force, Co the Coriolis force, and
F
T
the turbulent drag.
means is needed to determine the vertical dependence of the turbulent momentum
flux divergence in terms of mean variables in order to obtain closed equations for
the boundary layer variables. The traditional approach to this closure problem is
to assume that turbulent eddies act in a manner analogous to molecular diffusion
so that the flux of a given field is proportional to the local gradient of the mean. In
this case the turbulent flux terms in (5.18) and (5.19) are written as
K
m
∂
K
m
∂
¯
¯
u
∂z
v
∂z
u
w
=−
;
v
w
=−
and the potential temperature flux can be written as
K
h
∂ θ
∂z
θ
w
=−
where K
m
(m
2
s
−
1
)isthe
eddy viscosity
coefficient and K
h
is the eddy diffusivity
of heat. This closure scheme is often referred to as
K theory
.
The K theory has many limitations. Unlike the molecular viscosity coefficient,
eddy viscosities depend on the flow rather than the physical properties of the
fluid and must be determined empirically for each situation. The simplest models
have assumed that the eddy exchange coefficient is constant throughout the flow.
This approximation may be adequate for estimating the small-scale diffusion of
passive tracers in the free atmosphere. However, it is a very poor approximation
in the boundary layer where the scales and intensities of typical turbulent eddies
are strongly dependent on the distance to the surface as well as the static stability.
Furthermore, in many cases the most energetic eddies have dimensions comparable
to the boundary layer depth, and neither the momentum flux nor the heat flux is
proportional to the local gradient of the mean. For example, in much of the mixed
layer, heat fluxes are positive even though the mean stratification may be very close
to neutral.
