Geoscience Reference
In-Depth Information
The ingredients listed under (2) and (3) are combined into another dimensionless
quantity, z
L
(in the literature sometimes denoted as ζ (zeta)). The deinition of z
L
is:
z
L
g
H
c
1
3
zg
u
κ θ
θ
≡−
z
κ θ ρ
v
=
v
*
,
(3.19)
u
2
v
v
p
*
*
where θ v * is the virtual potential temperature scale. This is comparable to the temper-
ature scale θ * , but with the surface sensible heat lux H replaced by the surface virtual
heat lux H v 16 : θ
≡− H
cu
p
v
. The inclusion of κ in the Obukhov length is an arbitrary
v
*
ρ
*
choice made in the past and the inclusion of the minus sign ensures that z / L has the
same sign a s the Richardson number. L is a length scale, called the Obukhov length.
Recall that θ v is the absolute virtual potential temperature, that is, in Kelvin, so that
the Obukhov length is not very sensitive to the temperature of the air. We come back
to the physical interpretation of z
L
in Section 3.5.2 .
Question 3.12: Verify that indeed the Obukhov length L has units of length.
Now the central assumption of MOST is that any of the dimensionless gradients
given in Eq. ( 3.18 ) are a universal function (called φ ) of
z
L :
u
κ
z
z
L
=
φ
m
z
u
*
θκ
θ
z
z
L
=
φ
h
z
*
(3.20)
q
κ
z
z
=
φ
e
z
q
L
*
q
κ
z
z
L
=
x
φ
x
z
q
x
*
Because the functions φ z
L
are not necessarily the same for each of the quantities,
they are identiied with a subscript that indicates the variable. The shape of the
φ -functions cannot be known beforehand and needs to be determined from experi-
ments (step 3 and 4 of dimensional analysis, see page 98).
16 Based on the approximation of the kinematic virtual heat lux, the surface virtual heat lux becomes:
c
L
θ β .
q
p
v
HH
v = +
q
+
061
.
cEH
= +
1
061061
.
+
.
1
1061
.
θ
p
 
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