Salby M.L. Fundamentals of Atmospheric Physics
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Appendix D
Vector Identities
A • (B • C) = (A.C)B-(A.B)C
A.(B • C)=(A • B).C =B. (C •
(A x B) . (C x D) = (A . C)(B . D) - (A . D)(B . C)
V(fg) =
fVg + gV f
V. (fA) = Vf .A + fV.A
V x (fA)= Vf xA + fV xA
V. (A x S) =S. (V xA)-A. (V x S)
V.VxA=0
VxVf=O
V. Vf = V2f
V x V x f = V(V. f) - V2f
V(A
.B) =A. VB +B. VA +A x VB +B x V
xA
V(A x B) = B . VA - A . VB + A(V . B) - B(V . A)
1V(A.A)-A x (V xA)
A . VA -
(D.1)
(D.2)
(D.3)
(D.4)
(D.5)
(D.6)
(D.7)
(D.8)
(D.9)
(D.10)
(D.11)
(O.12)
(D.13)
(D.14)
580
Appendix E
Curvilinear Coordinates
Spherical Coordinates (a, 4~, r)
v,/,=
1 8q,
r cos 4) OA
1 Og, Oq,
ea
+- ~--~-e,/, +
r 3r er
(E.1)
1 8A a 1 8
V.A= t
r cos r 8A r cos 4~ 04)
1 8 (r2Ar)
(cos 6A,) + 7Tr
(E.2)
VxA=
1 { E Ar
(r 2cOs$) rcos~b 3q~
O(rA4') ]
3 (rcos
~Aa) - OAr]
+r ~rr --~- e6
E
]}
+ 3A - 3---~(r
cos ~bAx)
e r
1 o~2 I/t
g 2 I// -- r2 C0S2 ~b ~)A2
+
1 3 ( O~b) 1 3 (r2O~b)
r 2 COS 4) 34) COS q~--~ + 7 ~rr -~r
(z.3)
(E.4)
2
OA r
A~ + t
V2A = V2Aa - r2 c0s2 4) r 2 cos q~ 3A
2
sin
~b
OA 6 ]
r2 c0s2 4) o~A e
A6
-a t- V 2A6 - r2 c0s2 4 )
2 3A r
2
sin
~b
OAa ]
I
e ~b
r 2 8~b r 2cos 2~b 3A
2 2 8 (sin ) _ 2
8Ax ]
+ V2Ar - -~ Ar -
rZcosq~ 3~b 4~A6 r2cosq~ 3A
e r
(E.5)
581
582
Appendix E
Cylindrical Coordinates (r, ~b, z)
V~ - O~ 1 ?0 O~
e--~ e r -t- - --~ e 4, + --
r
Oz ez
1 OA 4, 3A z
V .A = __1 O (rAr) 4 f
r Or r Odp @z
V xA- [10A
z
I
r O4,
FOUr
Oz er+L-~z
Az]
Or e4~
1
e(rA4,)
+ -
r Or
1 OA r ]
r ?4) e~
lO
V2O--rO r r-fir- r q
1 021//
020
I
r 2 0t~ 2 o~z 2
V2A- rv2Ar
Ar
L
1.2
r 2 34) er nt-
V2A6
n t- V2Azez
(E.6)
(E.7)
(E.8)
(E.9)
A~ 2 OAr ]
(E.10)
Appendix F
Pseudo-Adiabatic Chart
Figures on the following pages show adiabats (solid lines), which are character-
ized by constant values of potential temperature 0, pseudo-adiabats (dashed lines),
which are characterized by constant values of equivalent potential temperature 0 e, and
isopleths of saturation mixing ratio with respect to water r~ (thin solid lines), all as
functions of temperature and pressure with the ordinate proportional to
(Po/P)".
583
(w~l) ::IClnlI.L7V ::::llV~lXOl:::lddV
I I, I, I, I I I I I, I, I, I, I , I , I , I I
I I
0 0 ~ 0 ~ 0 0 0 0 0 0 0
0 r ~ ~ 0 L'O 0 tO 0 LO 0
0 0 L~3 0 CO 0 0 0 0 0 0 0
(qw) ? nsS? d
o
r
O~
',e'
v
Ill
if-
o I--
W
17
UJ
I--
o
584
(W~l) =Ion1117v 31VIAIIXOEIddV
11,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I .... I,,,,I,,,,I,,,,I,,,,I,,I
f !
o o o o o o o o o o o o
o
U3 0 L'~ 0 L'~ 0 tO 0 U'J 0 U3
tt) U3 ~O CO 1 ~ 1 ~- {30 GO O~ O~ 0
0
i hVl
f -
9 /...: ...............
/: /" i
....~,:.r ...........
...... ! .......... /.
/ /
/
/
0 0 0 0 0 0 0 0 0 0 0 0
~ CO qO 1 ~ 1L ~ OD GO Ob O~
0
0
(qw)
3WRSS3~icl
v
ILl
ILl
ILl
585
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