Geoscience Reference
In-Depth Information
The above equation can be written in terms of
√
z
as follows:
dz
z
05
dz
(
)
=
2
dz
=
2
zd
( .
z
Therefore, substituting in Eq. (1.36)
z
z
r
w
π
2
zd z
r
w
π
dz
*
*
∫
*
*
∫
Γ=
=
12
/
2
2
1
2
z
ln
z
ln
z
2
2
x
2
x
2
r
w
π
dz
z
()
ln
r
w
π
*
*
∫
*
*
∫
Γ=
=
li
()
z
2
2
(1.37)
x
1
x
1
x
=
z
and
x
=
z
.
1
1
2
2
In the above equation,
z
1
and
z
2
refer, respectively, to the lower and upper limits
of integration and li is Soldner's integral or the logarithm integral. The large eddy
growth time Γ can be computed from Eq. (1.37).
1.7
General Systems Theory and Classical Statistical
Physics
A summary of Lebowitz's (Lebowitz
1999
) discussion on the essential role of clas-
sical statistical mechanical concepts underlying the formulation of precise physical
laws for observed macroscale phenomena in nature is given below. Nature has a
hierarchical structure, with time, length, and energy scales ranging from the submi-
croscopic to the supergalactic. Surprisingly it is possible and in many cases essen-
tial to discuss these levels independently—quarks are irrelevant for understanding
protein folding and atoms are a distraction when studying ocean currents. Neverthe-
less, it is a central lesson of science, very successful in the past 300 years, that there
are no new fundamental laws, only new phenomena, as one goes up the hierarchy.
Thus, arrows of explanations between different levels always point from smaller to
larger scales, although the origin of higher level phenomena in the more fundamen-
tal lower level laws is often very far from transparent. SM provides a framework for
describing how well-defined higher level patterns or behavior may result from the
nondirected activity of a multitude of interacting lower level individual entities. The
subject was developed for, and has had its greatest success so far in, relating meso-
scopic and macroscopic thermal phenomena to the microscopic world of atoms and
molecules. SM explains how macroscopic phenomena originate in the cooperative
behavior of these microscopic particles (Lebowitz
1999
).
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