Geoscience Reference
In-Depth Information
Similarly,
∂wθ/∂z
is of order
Q
0
/h
, so when nondimensionalized with surface-
layer scales it is also very small:
z
Q
0
∂wθ
∂z
∼
z
h
1
.
(10.5)
Because vertical gradients of these turbulent fluxes are negligible when expressed
in surface-layer scales, the surface layer has been called the
constant-flux
layer
.
Horst
(
1999
) has suggested the more appropriate name
surface-flux
layer.
10.2 Monin-Obukhov similarity
10.2.1 Dimensional analysis
The Monin-Obukhov (M-O) similarity hypothesis has provided much of the
foundation for our understanding of the atmospheric surface layer. It rests on dimen-
sional analysis and the
Buckingham Pi Theorem
, which prescribes the optimal
approach to determining a dependent variable in a physical problem. That theo-
rem says that if one can identify the
m
1 parameters governing that dependent
variable, and if
n
is the number of dimensions represented by it and the governing
parameters, then:
−
•
m
n
independent dimensionless quantities can be formed, the nondimensionalization
being done with the governing parameters. An
independent
quantity is one that cannot be
made from the others. The choice of the dimensionless quantities is arbitrary.
• These
m
−
n
independent dimensionless quantities are functionally related, so the
dependent variable can be taken to be a function of the governing parameters.
−
Identifying the governing parameters requires an overall grasp of the physics of
the problem and good intuition. We'll illustrate this process with examples from
Chapter 1
.
Kolmogorov
(
1941
) hypothesized that the length and velocity scales
η
and
υ
of
dissipative-range turbulence depend only on the viscous dissipation rate per unit
mass,
, and the fluid kinematic viscosity,
ν
:
η
=
η(, ν),
υ
=
υ(,ν).
(10.6)
The dimensional analyses for the unknowns
η
and
υ
are similar, so we'll do
η
.
By hypothesis the dependent variable
η
is governed by
and
ν
,so
m
−
=
1
2,
3. The units of
η
are L; of
,
L
2
T
−
3
;of
ν,
L
2
T
−
1
.Thereare
n
m
=
=
2
dimensions so
m
−
n
=
1 independent dimensionless quantity can be formed from
