Transcript METO 621
METO 621
Lesson 10
Upper half-range intensity
• For the upper half-range intensity we
use the integrating factor e-t/m
B (t ) t / m
d t / m dI 1 t / m
I e
I e
e
dt
m
dt m
• In this case we are dealing with
upgoing beams and we integrate from
the bottom to the top.
Upper half-range intensity
0
d t '/ m
t */ m
d
t
'
I
e
I
(
0
,
m
,
)
I
(
t
*,
m
,
)
e
t * dt '
t*
dt ' t '/ m
dt ' t '/ m
e
Bt (t ' )
e
Bt (t ' )
0
t*
m
0
m
or
t */ m
I (0, m, ) I (t *,m, )e
t*
0
dt '
m
B (t ')e
t '/ m
Upper half-range intensity
• To find the intensity at an interior point t,
integrate from t* to t and obtain
I (t , m, ) I (t *,m, )e
(t *t )/ m
t*
t
dt '
(t 't )/ m
B (t ')e
m
• What happens when m approaches zero. This
is where the line of sight traverses an infinite
distance parallel to the slab. Here
It (t , m 0, ) B (t )
Formal solution including Scattering and
Emission
• Note that the source is now due to
thermal emission and multiple scattering
a(t )
ˆ
ˆ
ˆ
ˆ
S(t , ) 1 a(t )B(t )
d
'
p(
t
,
',
)I(
t
,
')
4 4
• The independent variable is the extinction
optical depth, the sum of the absorption and
scattering optical depths. We can write
ˆ)
dI (t ,
ˆ ) S (t ,
ˆ)
m
I (t ,
dt
Formal solution including scattering and
emission
• The method of using an integrating factor
can be applied as before
t (P1,P 2)
ˆ
ˆ
I t (P2 ), I t (P1), e
•
t (P P2 )
ˆ
dt S(t,)e
t (P1 )
In slab geometry the solutions become
t / m
I (t , m, ) I (0, m, )e
t (P2 )
t
0
dt '
m
(t t ' )/ m
S(t ', m, )e
Formal solution including scattering and
emission
I (t , m, ) I (t *,m, )e
•
where
•
and
(t *t )/ m
t*
t
dt '
S (t ')e(t 't )/ m
m
I (t,m 0,) S (t ,m 0, )
S(t , m, ) 1 aB(t )
a
4
2
1
0
0
a
4
2
1
0
0
d
'
d
m
'
p(
m
',
';
m
,
)I
(t , m', ')
d
'
d
m
'
p(
m
',
';
m
,
)I
(t , m', ')
Radiative Heating Rate
• The differential change of energy over
the distance ds along a beam is
(d E) dI dAdt d d
4
• If we divide this expression by dsdA,
(the unit volume, dV), and also ddt then
we get the time rate of change in radiative
energy per unit volume per unit frequency,
due to a change in intensity for beams
within the solid angle d.
Radiative Heating Rate
• Since there is (generally) incoming
radiation from all directions, the total change
in energy per unit frequency per unit time
per unit volume is
dI
ˆ I )
d
d
(
4 ds 4
• The spectral heating rate H is
ˆ I )
d (
4
Radiative Heating Rate
• The net radiative heating rate H is
ˆ I
d d (
)
0
4
• In a slab geometry the radiative
heating rate is written
1
F
I
H d
2 d cosd(cos )
z
z
0
0
1
where F F F is the radiative flux in the
z direction
Separation into diffuse and direct(Solar)
components
• Two distinctly different components of the
shortwave radiation field. The solar
component:
S t / m 0
ˆ
ˆ )
(
0
S t / m 0
(m m0 )( 0 )
I (t ,, ) F e
S
F e
• We have two sources to consider, the Sun
and the rest of the medium
I (t,m,) I (t ,m, ) I (t,m,)
d
S
Diffuse and direct components
• Assume (1) the lower surface is black, (2)
no thermal radiation from the surface, the
we can write the half range intensities as
ˆ)
dI (t ,
ˆ ) (1 a)B
m
I (t ,
dt
a
ˆ
ˆ
ˆ
d
'
p(
',
)I
(
t
,
')
4
a
4
ˆ
ˆ
ˆ ')
d ' p(',)I (t ,
Diffuse and direct components
• And for the +ve direction
ˆ)
dI (t ,
ˆ ) (1 a)B
m
I (t ,
dt
a
ˆ
ˆ
ˆ ')
d
'
p(
',
)I
(t ,
4
a
4
ˆ
ˆ
ˆ ')
d
'
p(
',
)I
(t ,
Diffuse and direct components
Now substitute the sum of the direct and diffuse components
ˆ)
ˆ)
dId (t ,
dIS (t ,
ˆ ) I (t ,
ˆ ) (1 a) B
m
m
I d (t ,
S
dt
dt
a
a
ˆ
ˆ
ˆ
ˆ ' ,
ˆ ) I (t ,
ˆ ')
d
'
p
(
'
,
)
I
(
t
,
'
)
d
'
p
(
S
d
4
4
a
ˆ ' ,
ˆ ) I (t ,
ˆ ')
d
'
p
(
d
4
Diffuse and direct components
But I S is thedirect solar beam and dIS I S dt / m
henceonegets
ˆ)
dI (t ,
*
ˆ
ˆ)
m
Id (t , ) (1 a)B S (t ,
dt
a
ˆ
ˆ
ˆ
d
'
p(
',
)I
(
t
,
')
d
4
d
a
4
ˆ
ˆ
ˆ
d
'
p(
'
,
)
I
(
t
,
')
d
Diffuse and direct components
• where
a
ˆ
S (t ,)
4
*
S t / m 0
ˆ
ˆ
ˆ
ˆ
d
'
p(
',
)F
e
(
0)
a
S t / m 0
ˆ
ˆ
p(0 ,)F e
4
• One can repeat this procedure for the
upward component
Diffuse and direct components
ˆ)
dI (t ,
*
ˆ
ˆ)
m
Id (t , ) (1 a)B S (t ,
dt
a
ˆ
ˆ
ˆ
d
'
p(
'
,
)
I
(
t
,
')
d
4
d
a
4
ˆ
ˆ
ˆ
d
'
p(
'
,
)
I
(
t
,
')
d
Diffuse and direct components
a
ˆ
S (t , )
4
*
S t / m 0
ˆ
ˆ
ˆ
ˆ
d
'
p(
',
)F
e
(
0)
a
S t / m 0
ˆ
ˆ
p(0 ,)F e
4
Diffuse and direct components
• If we combine the half-range intensities we
get
dI(t ,u, )
a
u
I(t ,u, )
dt
4
2
1
0
1
d' du', p(u'.';u,)I(t,u',')
(1 a)B S* (t ,u, )
• Where u is cosand not |cos|