Salby M.L. Fundamentals of Atmospheric Physics
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590
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Answers to Selected Problems
Chapter 1
Vi m ri
2. V. ~"
Vi ~ ni
3. Va -- n-- 7 .
6. (a) NH2o -- 0.0231, (b) M - 28.71 g mo1-1, (c) R- 289.58 J kg-
(d) PH2o- 0.0173 kg m -3, (e) rH2o- 0.0147, (f) ~ -0.0236.
8. En20 -- 1000 ~
(mm).
- fo
rH2oPddZ
9. (a) 47 mm, (b) 9.7 mm.
12.
AT e --
4.4 K.
13. (a) 0.
16.
T(z)- gh2
[1
+ (z) 2]
2-~ a 9
17. 236 K.
Chapter 2
1. (a) 5 km, (b) 0.9 km.
2. (a) 0950 - 30 ~ C.
3. (a) 3.06 x 103 J, (b) 3.10 x 103 J, (c) 4.06 x 104 J.
6. 15.85.
Chapter 3
1. (a) 298 K, (b) -6.3 J
K -1,
(C) 7.05 J
K -1,
(d) 0.75 J
K -1.
2. (a) -20.7 J K -1, (b) 21.5 J K -1.
3. 0.90.
4. (a) 0.967.
5. (a) 277 K, (b) 285 K.
7. 199 J K -1 kg -1.
9. (a) 1.3
x 10 2
J K -1 mo1-1, (b) 5.2
x 10 4
J mo1-1.
595
1 K-l,
596
Answers to Selected Problems
Chapter 4
6. (a) 95 ~ C, (b) 86 ~ C, (c) 71 ~ C.
7. (a) 599 K.
8. (a) 188 K, (b) in the wintertime Antarctic stratosphere.
Chapter 5
1. (a) 33 ~ C, (b) 10.2 g kg -1, (c) 26%.
3. (a)
T(z, t; x) = r~[z - h(x - ct)] -
Fdh(x
- ct),
(b)
r(z, t; x) =
r~[z - h(x - ct)].
7. (a) 75%, (b) 25 ~ C, (c) 325 K, (d) 770 mb, (e) 400 K, (f) 22.5 g kg -1,
(g) 450 mb.
8. 53%.
9. (a) 53%, (b) 100%.
11. (a) 20.1 ~ C, (b) 32.6 ~ C.
15. (e) 328 K.
16. 9~ C.
18. (a) 920 mb, (b) 610 mb.
19. 9%.
20. (a) 5.4%, (b) 0.51 kg, (c) 7.22
x 10 6
J.
22. (a) 13.6 km.
23. 3.4 km.
25. (d) 1.3 km.
Chapter 6
3. w = -R f Td
In p =
-RA.
4.
W-- cp(T2- TI) [I- ( p34
R --
T' ~~
7. (a) 2{(~+ Tcos4~)~- (-~)exp[ -A~+(6-
z- ,2
8. 23K.
9. (a) z= -Rf(~:- 's) [1-- 's[1-e'~('s-O] }
(~_~)
9
12. (a) 158 m.
Answers to Selected Problems
597
Chapter 7
( hI ) N 2 ~ ez/hl--
2. (a) unstable" z <
z c,
stable: z >
Zc; z c
- h 1
In ~ , (b)
--- g ez/h 1
4. 1.24
10 -4
kg s -2, (b) 5.41
10 -4
kg s -2.
5 (a)wo
9 "~".
9
10.
13.
14.
15.
18.
19.
20.
24.
25.
z 9
~2
6. (a)0--T 0 a exp[r'd ( -~)]
-~o az + ,
(b) unstable: z <
Zc,
stable: z >
z c,
Zc /-aTo _ a
=V ra
3Fa
(a) Ttrigger- r 0 -q- --~, (b) At "~ 5
hr.
(a) stable: z < a, unstable: z > a; (b) Az = 2a (exclusive of instability).
27 km.
15kin.
doe dO e
> 0 (stable), = 0 (neutral), a0e
-37 ~ < 0 (unstable).
57 m s -1.
(a)
TI(z)
= 4z-20, (b) 5 km, (e) Wmax = 58 m s -1 at 5 km, Zmax = 9.5 km.
(a) K' > 0 for z < 9.8 km, (b) K' > 0 for z < 5.7 km, (c) K' > 0 for
z < 5.3 km.
(a) 244 K at 13 km.
(a) potentially unstable (stable) for z < 1.6 km (z > 1.6 km), (b) poten-
tially unstable (stable) for z < 4.8 km (z > 4.8 km).
Chapter 8
.
3.
4. (a)
5. (a)
9. (a)
10. (a)
11. (a)
12. (a)
13.
14.
15.
16.
(a) 254 K, (b) 290 K, (c) 270 K, (d) 240 K.
-32.4 K.
4.6 K.
u(z) = porH
exp - --g-
227 K.
95 , (b) 3.4.
393 K, (b) 393 K.
1
:asw-5)
k aLw ,
(b) 278 K.
2]
z
1
T(t)
= (0.12~rt + 1) 3.
\ "0
,~ atm-- atm hood - .
L--aLWWaLWaLW
~hooo It7 _hood
2--aLW '~SW
,~ cos
Os-
uLW ~rTnood; (a) 298 K, (b) 392 K.
(a) 324 K, (b) 212 K.
(a) [asw + (1 - asw)aLw ]
Zn%0 (_a._~)n,
(b) 252 K.
598
Answers
to Selected Problems
17. (a) 291 K, (b) 252 K, (c) 294 K.
18. (a) 5.0 mb, (b) 6.4 mb.
2+(U-1)a
19. (a) F~ - F02_-- ~ (U - 1), F o 2+a (U - 2) Fo 2-a
~ **,
= Fo [I + N2ava] (N).
22. -97 K.
24. (a) o.T4RE(Z)_ Fo{3k
gclz' 11 / ,~
T ~P0 exp[- fo
RTRE(2 t) ] "Jr- , (b) TRE(Z)
_[3kF~ z
, (c)lT-~-poexp(-~) ]_ __ - T~- Fs(0)Zr.
q-d~-po exp (-~)] 1
[ 3kF0
32. (a) 16 days, (b) 0.
33. (a) 4.6 K, (b) 17.6 K.
Chapter 9
3 (a)
as an dM 4
dn
d(loga) --
4"n'a2d(loga)
(b)
- "n'a3p
9 ~ ' d(loga) ~ 3 d(loga)"
8. AG = 16'n'~
3[nwKTln(e-~)] 2
9. (a) 0.60/xm, (b) 0.19/xm.
10. 1.6 g m -3.
11. (a) ~ 100 m.
2 pwga 2
(b) 87/xm, (c) 88/xm.
15. (a)
W t -- -~ Ix '
17. (a) 0.76 years, (b) 76 years.
19. (a) 150 m, (b) 1500 m.
26. cos | = cos 0 cos 0' + sin 0 sin 0' cos(~b - ~b').
27. (a) 0.013, (b) 0.051, (c) 0.301.
28. /3s(~r) - 64zr5 n
a 6 (m2-I
~2
A 4 '0 2 m-----~] "
35. (a)
a e --5.7/xm,
(b) ~l- 785 g m -2 (c)
r c -208.
36.
r c
= 410.
38. (a) Et- 240 g
m -2,
(b)
Pt
= 0.24 g
m -3.
(
)
E_a_e-,'*-E+a+e-~'~-(1-a)a+E_ e-~'~-e-~'*
39. (9.40), but with C =
a2e~,*_a2e-~,,~+(l_a)a+a_(e-,~_e~,~) ,
i~ 9 ,
{
_~, \
E+a_e-~ ,c-Ea+e~'c
+(1-a)a E ~e v~*
_ _ -e Us c)
D~
31 * *
a2+e ,r e-r'~ +(1-a)a+a_(e-r'*-er'c )
41. Csw = -131 W m -2, CLW -- 125 W m -2, C- -6 W m -2.
42. 3.1 mm day -1.
47. Strong forward scattering implies that the reduction of direct transmission
is offset by an enhancement of diffuse transmission.
Answers to Selected Problems
599
Chapter 10
6. (a) two thermodynamic degrees of freedom, (b) two independent families
of surfaces, (c) the families coincide.
10. (c) For a given time interval, the deformation experienced by a material
volume approaches zero with its dimension.
do _ O.
14. (a) 77d~ _ V.v- 0, (b) 17
15. v orthogonal to Vq,.
18. (a) 3.5 ~ , (b) 34.8 ~ (c) 348 ~ (d) 0.019, 0.19, 1.9.
Chapter 11
6. [(u + Or cos q~)r cos 4~] - -r cos q~
prcos~b Oh Jr-
Da 9
dVh . () 11
11. -d7 +fkxv h--V~dO-D h oO__O; a ~-14r + Tr
, a~-- ~ 7r -- Vcr.v h-O;
\ /
dO (t. d _ 3__ +11 h . V,n. + 7r3--7
dt ~ zr' dt ~ 3t
19. (a)
Ug-
fa och' Vg
facos4~ oh, (b)
Ug- fa acb' 1)g
facos4~ Oh"
3Vg
20. alnp = ~k x VpT.
21. (a) w - srh {1-
e -z*/h
},
(b) convergence inside the boundary layer must
be compensated by divergence aloft.
Chapter 12
_1 k
3. (a) Vg of x Vzp, (b) vg ~k x Voq*.
__ 1 3". __ 1 3" (b) Ug -- 1 3.. _ 1 3.
4.
(a)
Ug -- f Oy' Vg f Ox' -- fa Oqb' Vg
facos~ 3--A-'
(C)
Ugch = --~, Ug r = r 36"
cot4~ cot~b
6. (a) k-(V x l/g) -- %V2(I )+
~Hg,
(b) V-13g - a l)g.
7. 105 m s -1.
10. (a) u(r)- const
-7-, (b) F(r) - 27r. const -5/= f(r), (c) vorticity is concentrated
at r = 0 in a point vortex.
(b) Diffusion.
(a) vii = K2+f2, v• -- K2+f2, (b) 6 - tan- , (c) 6.4 ~ (d) 17.1 ~
K
V2(i).
g " l'lh -- K 2 +f2
4.6 ~ .
2 (x2+y 2 )
(a) v - ~ exp ---g:- [-(~Y + Xx)i + (~x - Xy)j],
~--~X4
(b) V. v -
exp
~ -Xe+y2
/
---yz-) [x~ ~2- 1]
4.exp( x +y2)[x2+y ]
(c) k. (v • - --z= - 1.
11.
13.
14.
15.
16.