General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 10

1.5.2 Logarithmic Wind Profile in the ABL

The height interval in which this incremental change d W in the vertical velocity oc-
curs is d R which is equal to r . The height up to which the large eddy has grown is
equal to R (see Fig. 1.3 ).
Using the above expressions Eq. (1.2) can be written as follows:

w
k

dR
R

w
k dR

* *

dW = ln . = (1.3)

Integrating Eq. (1.3) between the height intervals r and R , the following relation for
W can be obtained as follows:

w

k dR w

R

* *

∫

W = ln =

k Rr

(lnln) −

r

(1.4)

w
k

R
r

w
k

=

* *

W = ln ln . z

In Eq. (1.4), it is assumed that w ∗ and r * are constant for the height interval of inte-
gration. The length scale ratio R / r is denoted by z . A normalized height with refer-
ence to the turbulence scale r can be defined as z = R / r .
Equation (1.4) is the well-known logarithmic velocity profiles in turbulent shear
flows discussed originally by von Karman ( 1956 ) and Prandtl ( 1932 ) (for a recent
review, see Marusic et al. 2010 ), and to the recently discovered logarithmic varia-
tion of turbulent fluctuations in pipe flow (Hultmark et al. 2012 ). Observations and
the existing theory of eddy diffusion (Holton 2004 ) indicate that the vertical wind
profile in the ABL follows the logarithmic law which is identical to the expression
shown in Eq. (1.4). The constant k (Von Karman's constant) as determined from
observations is equal to 0.4 and has not been assigned any physical meaning in the
literature. The new theory proposed in this study enables prediction of observed
logarithmic wind profile without involving any assumptions as in the case of ex-
isting theories of atmospheric diffusion processes such as molecular momentum
transfer (Holton 2004 ). Also, the constant k now has a physical meaning, namely, it
is the fractional volume dilution rate of the large eddy by the turbulent scale eddies
for dominant large eddy growth.

1.5.3 Fractional Upward Mass Flux of Surface Air

Vertical mixing due to turbulent eddy fluctuations progressively dilutes the rising
large eddy and a fraction f equal to Wr / w * R of surface air reaches the normalized
height z as shown in the following. The turbulent eddy fluctuations carry upward
surface air of frictional origin. The ratio of upward mass flux of air/unit time/unit