Geoscience Reference
In-Depth Information
10.2.4.5 Quasi-steadiness, local homogeneity
We'll now apply to the surface layer the general criteria of
Chapter 9
for the neg-
ligibility of the effects of unsteadiness and inhomogeneity. We can write for the
Reynolds stress
equation (8.69)
, for example,
u
2
τ
u
∼
u
2
u
3
u
3
∗
z
,
∼
∼
∼
time change
τ
u
,
rate of production
(10.25)
with
τ
u
the time scale of the unsteadiness. The time-change term will be
negligible if
u
2
u
3
z
u
∗
τ
u
∗
z
,
τ
u
or
1
.
(10.26)
10 m,
u
∗
=0.3ms
−
1
,and
If
z
means a factor of 10 less than, this requires that
τ
u
be greater than about 5 minutes in order that the surface layer be quasi-steady.
This is not difficult to meet in practice.
With horizontal
=
inhomogeneity the Reynolds stress equation has a mean
advection term:
U
u
2
U
∂uw
L
x
,
mean advection
∼
∂x
∼
(10.27)
where
L
x
is a horizontal scale of the inhomogeneity. For this to be negligible
we need
U
u
2
u
3
L
x
U
u
∗
L
x
z
,
or
z
.
(10.28)
The value of
U/u
∗
depends on height, stability, and the roughness of the surface,
but typically ranges from about 10 to 30; if
means a factor of 10 greater than,
this indicates that
L
x
must be greater than 100 to 300 times the height. If
z
10m,
then
L
x
must be greater than 1 to 3 km. Sites this uniform can be difficult to find.
=
10.2.4.6 The Richardson number as an alternative stability variable
The M-O stability variable
z/L
involves surface fluxes, but in practice the turbulent
fluxes at some height above the surface are used. The first such flux measurements
were made in the 1950s, and well into the 1970s they were made only in research
applications. Today, relatively inexpensive, reliable turbulent-flux instrumentation
is commercially available, but stress measurements in particular are still plagued
by scatter caused by the slow convergence of the time average.
The turbulence Richardson number of
Chapter 9
is
buoyancy force
/
mass
inertia force
/
mass
gθ
θ
0
u
2
Ri
t
=
∼
(
9
.
6
)
