Geography Reference
In-Depth Information
5.3.3
The Mixing Length Hypothesis
The simplest approach to determining a suitable model for the eddy diffusion coef-
ficient in the boundary layer is based on the mixing length hypothesis introduced
by the famous fluid dynamicist L. Prandtl. This hypothesis assumes that a parcel
of fluid displaced vertically will carry the mean properties of its original level for
a characteristic distance ξ
and then will mix with its surroundings just as an aver-
age molecule travels a mean free path before colliding and exchanging momentum
with another molecule. By further analogy to the molecular mechanism, this dis-
placement is postulated to create a turbulent fluctuation whose magnitude depends
on ξ
and the gradient of the mean property. For example,
∂ θ
∂z
;
∂
u
∂z
; v
=−
¯
∂
v
∂z
¯
θ
=−
ξ
u
=−
ξ
ξ
where it must be understood that ξ
> 0 for upward parcel displacement and ξ
< 0
for downward parcel displacement. For a conservative property, such as potential
temperature, this hypothesis is reasonable provided that the eddy scales are small
compared to the mean flow scale or that the mean gradient is constant with height.
However, the hypothesis is less justified in the case of velocity, as pressure gradient
forces may cause substantial changes in the velocity during an eddy displacement.
Nevertheless, if we use the mixing length hypothesis, the vertical turbulent flux
of zonal momentum can be written as
∂
u
∂z
¯
−
u
w
=
w
ξ
(5.24)
with analogous expressions for the momentum flux in the meridional direction
and the potential temperature flux. In order to estimate w
in terms of mean fields,
we assume that the vertical stability of the atmosphere is nearly neutral so that
buoyancy effects are small. The horizontal scale of the eddies should then be
comparable to the vertical scale so that
w
|∼|
V
|
|
and we can set
∂
V
∂z
w
≈
ξ
where
V
and
V
designate the turbulent and mean parts of the horizontal velocity
field, respectively. Here the absolute value of the mean velocity gradient is needed
because if ξ
> 0, then w>0 (i.e., upward parcel displacements are associated
with upward eddy velocities). Thus the momentum flux can be written
ξ
2
∂
V
∂z
∂
u
∂z
=
¯
K
m
∂
u
∂z
¯
u
w
=
−
(5.25)
