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! v
"
"
! v =! k " #$ + #%
"# (! k $ "%) = 0
" #"$ = 0
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Illustration showing the relationship among the rotational wind, divergent wind, and the velocity potential at 300 hPa for January (sample data from the NCAR Community Climate Model, CCM2).
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d! v dt
= "1#$p +
! g
d d t
dr d t
d! r dt
=dr! r
dt+! " #! r
$! v = ! v r +
! " #! r
d! v dt
=dr! v
dt+! " #! v
! " # (
! " #! r ) = $"2 ! R
d! v dt
=dr! v r
dt+ 2! " #! v r $"
2 ! R
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dr! v r
dt= "2! # $! v r "
1%&p +
! g +#2 ! R
! g
"2 ! R
! g
"! v "t
= #! v $ %! v # 2
! & '! v # 1
(%p +
! g
! v =! i u +! j v +! k w
d! v dt
=! i du
dt+! j dv
dt+! k dw
dt+ u d! i
dt+ v d! j
dt+ w d
! k
dt
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d! i
dt= u
tan"r! j #! k r
$
% &
'
( )
d! j
dt= #u tan"
r! i # v
r! k
d! k
dt=
ur! i + v
r! j
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dudt
= 2"sin# +utan#a
$
% &
'
( ) v *
uwa* 2"wcos# * 1
+,p,x
dvdt
= * 2"sin# +utan#a
$
% &
'
( ) u *
vwa*1+,p,y
dwdt
=u2 + v 2
a* 2"ucos# * 1
+,p,z
* g
"p"z
= #$g
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v!
"! v "t
= #! v $ %! v # w "
! v "z
# fk & ! v # 1'%p
"w"t
= #! v $ %w # w "w
"z#1&"p"z
# g
"#"t
= $%& (#! v )
p = "RT
"#"t
= $! v % &# $ w
"#"z
+1cp
pS
p'
( )
*
+ ,
R /C p
˙ H
f = 2"sin#
˙ H
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dudt
= 2"sin# +utan#a
$
% &
'
( ) v *
uwa* 2"wcos# * 1
+,p,x
dvdt
= * 2"sin# +utan#a
$
% &
'
( ) u *
vwa*1+,p,y
dwdt
=u2 + v 2
a* 2"ucos# * 1
+,p,z
* g
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w v
f u + 1"#p #y
= 0
uw v
w v !
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( , 2005)
27 Vallis, 2006
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" # dp dt!
d" = gdz
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" # dp dt ~ w /$g
"! v "t
= #! v $ %! v #& "
! v "p
# fk ' ! v #%(
"#"p
= $RTp
"#! v + $
%$p
= 0
"#"t
= $! v % &# $'
"#"p
+1cp
pS
p(
) *
+
, -
R /C p
˙ H
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"! v "t
= #! v $ %! v #& "
! v "p
# fk ' ! v #%(
Ro "UfL
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fk " ! v g = #$%
! v g =1f! k " #$
ug = "1f#$#y, vg =
1f#$#x
"#! v g = 0
$ = 0
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"! v g"p
= #Rfp! k $ %T
"ug"p
=Rfp"T"y
"vg"p
= #Rfp"T"x
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"u g"p
=Rfp"T "y
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