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! v

"

"

! v =! k " #$ + #%

"# (! k $ "%) = 0

" #"$ = 0

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7

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).

8

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

9

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

10

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

11

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

12

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

13

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

14

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

34

"! 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

36

"! 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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