Geoscience Reference
In-Depth Information
virtual potential temperature and gradient of specific humidity in the surface layer
are normalized to allow for the local surface-related dependency on
u
*
and height
above the zero plane displacement.
At this stage, the mixing length hypothesis has been rewritten such that the
effective mixing length which controls turbulent transport in the surface layer
can be modified by the value of the dimensionless functions
f
M
,
f
H
and
f
V
.
These functions are all equal to unity in neutral conditions, but their value may
alter with the extent of buoyant production or destruction in the turbulent
field. It is next necessary to assume that the functional forms of dimensionless
functions
f
M
,
f
H
and
f
V
are a universal feature of turbulence in the surface layer
and independent of the actual surface itself. Although these dimensionless
functions are not known, if they are indeed universally applicable they can
presumably be defined by calculation from measured gradients and fluxes at
one place using Equations (20.2), (20.5) and (20.8), and the functional forms
so defined may then be applied everywhere. But the next step is to define a
dimensionless measure of the rate of buoyant production/destruction in
terms of which the empirical functional form of
f
M
,
f
H
and
f
V
can be defined
by experiment.
Obukhov length
Equation (20.1), i.e., the dimensionless form of the prognostic equation for
turbulent kinetic energy, can be used to define the required dimensionless
measure of buoyant production/destruction in terms of which
f
M
,
f
H
and
f
V
can
be parameterized. In this equation, Term II describes the contribution of buoyancy
to the production or destruction of turbulence. This term can be re-written in the
form (
z
-
d
)/
L
where
L
is called the
Obukhov length
and defined by:
3
()
*
−
q
u
v
L
=
(20.10)
gk
(
q¢ ¢
w
)
v
Because all the other factors on the right hand side of Equation (20.10) are positive,
the Obukhov length, and therefore (
z
-
d
)
/
L
, ta
kes its sign as being opposite to the
sign of the kinematic sensible heat flux,
q¢ ¢
. Consequently,
L
is negative when
the heat flux is positive (often in daytime conditions), and
L
is positive when the
heat flux is negative (often in nighttime conditions). One physical interpretation
of the Obukhov Length is that the value (-
L
) corresponds to the height at which
buoyant production of turbulent kinetic energy begins to dominate over shear
(mechanical) production when the mean profiles are assumed to be logarithmic
through the whole surface layer.
Not surprisingly, the factor (
z
-
d
)/
L
is formally related to the flux form of the
Richardson number, at least in neutral conditions, as can be shown by rearranging
terms in (
z
-
d
)/
L
as follows:
v
w
