Geoscience Reference
In-Depth Information
Note that the minus sign appears in Equation (19.9) because the fluctuation w
,
which is positive away from the surface, results in a fluctuation in horizontal wind
speed at the level z which is negative relative to the mean wind speed at that level.
The turbulent kinematic flux of water vapor is by definition the time average
value of the product of fluctuations in specific humidity and vertical wind speed,
and it can be obtained by substituting for q
from Equation (19.7) and w
from
(19.9) and averaging, i.e.
⎛⎞
⎛⎞
q
u
⎪ ⎪
(19.10)
qw
¢¢ ¢
.
=−
z
.
− −
c z
¢
.
⎬ ⎨
⎜⎟
⎜⎟
z
z
⎝⎠
⎝⎠
⎪ ⎪
which simplifies to:
⎛⎞⎛⎞
q
u
(19.11)
2
qw cz
¢¢
.
=−
.(
¢
) .
⎜⎟⎜⎟
z
z
⎝⎠⎝⎠
The basis of mixing length theory is the postulate that, providing turbulence is
solely frictional in origin, it can be characterized by a hypothetical mixing length,
l
, whose value is assumed to represent the effective average vertical size of the
turbulent eddies in the turbulent field that transports atmospheric entities. In the
surface layer of the ABL the mixing length is defined such as to simplify
Equation (19.11) by:
2
2
l
=
cz
.(
¢
)
(19.12)
and Equation (19.11) becomes:
⎛⎞⎛⎞
q
u
2
(19.13)
qw
¢¢
.
=−
l
.
⎜⎟⎜⎟
z
z
⎝⎠⎝⎠
In a similar way, it can be shown that the momentum flux is given by:
⎛⎞⎛⎞
u
u
(19.14)
2 .
uw
¢¢
=−
l
⎜⎟⎜⎟
z
z
⎝⎠⎝⎠
By comparing Equation (19.13) with Equation (19.6) and Equation (19.14) with
Equation (19.4) it follows that:
KK ⎛⎞
== ⎜⎟
u
2 .
l
M
V
⎝⎠
(19.15)
But it still remains necessary to assign a value to the mixing length,
l
.
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