Geoscience Reference
In-Depth Information
At last for the derivative of the phase function the following relations are
correct:
∂
χ
L
i
(
P
)
∂
χ
λ
x
a
(
P
,
)
x
a
(
P
i
,
,
)
λ
j
)
=
λ
)
L
j
(
)
(5.38)
∂σ
z
,
a
(
P
i
,
∂σ
z
,
a
(
P
i
,
λ
and with accounting for (5.9) after simple transformations we obtain:
⎛
⎞
1
∂
χ
λ
χ
λ
x
a
(
P
i
,
,
)
x
a
(
P
i
,
,
)
)−
1
2
)
=
⎝
D
(
P
i
,
χ
λ
χ
,
λ
χ
,
λ
χ
⎠
,
,
D
(
P
i
,
)
x
a
(
P
i
,
)
d
∂σ
z
,
a
(
P
i
,
λ
σ
z
,
a
(
P
i
,
λ
)
−1
(5.39)
where
χ
λ
=
χ
λ
χ
λ
σ
a
,
z
(
P
i
,
λ
D
(
P
i
,
,
)
b
i
(
,
)+2
c
i
(
,
) ln(
)) .
The derivative with respect to air temperature
. A big quantity of values depends
on temperature. Begin from the photon free path and obtain the following for
it:
∂
∆τ
(
P
1
,
P
2
))
∂
=
∂
∆τ
(
P
1
,
P
2
))
∂α
P
(
P
i
)
∂α
P
(
P
i
)
∂
(
(
(5.40)
T
(
P
i
)
T
(
P
i
)
and for the volume coefficient of the molecular scattering:
∂σ
P
,
m
(
P
)
∂
L
i
(
P
)
∂σ
P
,
m
(
P
i
)
=
.
(5.41)
∂
T
(
P
i
)
T
(
P
i
)
An important feature of calculating the derivatives with respect to temperature
isthenecessityofaccountingforthetemperaturedependenceintheformulaof
the recalculation of the volume extinction coefficients in terms of atmospheric
pressure (5.1). It is obtained as follows:
=
α
P
(
P
i
)
1
.
∂α
P
(
P
i
)
∂
α
z
(
P
i
)
∂α
z
(
P
i
)
1
T
(
P
i
)
T
(
P
i
)
+
(5.42)
∂
T
(
P
i
)
∂σ
P
,
m
(
P
i
)
|∂
The analogous relation is written for derivative
T
(
P
i
), and for the
aerosol scattering volume coefficient the following is obtained:
∂σ
P
,
a
(
P
i
)
∂
=
σ
P
,
a
(
P
i
)
T
(
P
i
)
.
T
(
P
i
)
Now the expression for the extinction coefficient is derived:
∂α
z
(
P
i
)
∂
T
(
P
i
)
=
∂σ
z
,
m
(
P
i
)
+
∂κ
z
,
m
(
P
i
)
.
(5.43)
∂
∂
T
(
P
i
)
T
(
P
i
)
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