Geoscience Reference
In-Depth Information
Dimensionless gradients
The theory of surface layer scaling is assumed to apply in the surface layer of the
ABL where it is also assumed that all vertical fluxes (of momentum, sensible heat
and water vapor, for example) are constant with height. The theory also applies
above uniform horizontal surfaces where there are no changes in the mean values
of atmospheric variables and fluxes of atmospheric entities in horizontal direc-
tions, and no subsidence. In these conditions, the prognostic equation for turbu-
lent kinetic energy is simplified to the form given earlier as Equation (18.17).
Remembering that because the momentum flux is assumed independent of
height in the surface layer, Equation (19.19) requires that
u
*
is also independent of
height. Equation (18.17) can be re-written in dimensionless form by multiplying
by the factor [
k
(
z
-
d
)/(
u
*
)
3
], as follows:
e
(
w
′′
q
)
∂′
(
wp
′
)
( ) ( ) 1 ( )
() () ()
* * *
I II III
(
kz d
−∂
kz d
−
kz d
−
v
=
g
−
3
3
3
u
∂
t
u
r
u
∂
z
q
v
a
(20.1)
)
()
*
IV V VII
kz d
−
)
∂
u kz d we
kz
(
−
)
∂
′
(
−
3
d
(
)
−
uw
′
′
−
−
e
3
3
()
u
∂
z
()
u
∂
z
u
*
*
From the definition of
u
*
in Equation (19.19), it follows that
u
2
=−
(
u
¢¢
and
)
*
term IV in Equation (20.1) can be simplified to:
kz d
u
(
−∂
)
u
f
=
(20.2)
M
z
∂
*
The function on the right hand side of Equation (20.2) is called
the dimensionless
gradient of wind speed
. When written in this dimensionless form, the wind speed
gradient in the surface layer has been normalized to allow for the local surface-
related dependency on
u
*
and for height above the zero plane displacement. The
dimensionless function
f
M
on the left hand side of Equation (20.2) is as yet unspec-
ified, both in terms of functional form and purpose. However, multiplying
Equation (20.2) by (-
u
*
2
/
f
M
) gives:
∂
u
−=−
u
2
ku
*
()(
f
−
1
z
−
∂
d
)
(20.3)
*
M
z
Comparing this equation with Equation (19.27) suggests that the function
f
M
is
a dimensionless function whose reciprocal acts to change the effective average
vertical size (i.e. the mixing length) of the turbulent eddies operating in the
surface layer. In neutral stability conditions, comparison between Equations
(19.27) and (20.3) requires that
f
M
=
1, but in other conditions of thermal
