地球物理流体力学第四章

第四章层结流体的准地转运动Quasigeostrophic

Motion

of

a

Stratified

Fuid主要讲述：层结流体中准地转运动的位涡方程；层结流体中的Rossby波；层结流体中的Rossby波的垂直模态。§4.1

球坐标下的方程组：尺度分析The

Equations

of

Motion

in

Spherical

Coordinates:

Scalingdt

rdt

r

r-dv

+

wvdt

r¶p

+

Ff1

¶p-

2W

cos

fu

=

- -

g

+

Frr

¶r1

¶p+

tan

f

+

2W

sin

fu

=

- +

Fr

rr

¶f

qrr

cos

f

¶l1du

+

uw

-

uv

tan

f

-

2W

sin

fv

+

2W

cos

fw

=

-dw u

2

+

v

2u

2r

+¶r1

¶u

+

1

¶(v

cos

f

)

¶w

+

2w

=

0dt

r

cos

f

¶l

r

cos

f

¶fdr

+

r

(大气)2T

+

Q

dt

cpT

rdq

=

q

k热力学方程(海洋)•cp2ar0=

k

r

-

Qdr

dt状态方程p

=

rRTr

=

r0

1

-

a

(T

-

T0

)]z*

=

r

-

r0y*

=

(f

-

f0

)

r0x*

=

lr0

cos

f0以下记：尺度分析：

p

ppsssgDrsp

=

ps

1

+

eFp

1

+~)

+

p(

x

,

y

,

z

,

t

)

=rsUf

0

L*

*

*

*p*

=

ps

(

z**

s

*

*

*

*

*r

=

r

(

z

)

+

r~(

x

,

y

,

z

,

t

)由静力近似：¶p*

/

¶z*

=

-r*

g¶~p

/

¶z*

=

-r~gfi

¶ps

/

¶z*

=

-rs

g

=

o

(r

s

eF

)而¶~p

/¶z*

~

o(~p

/D

)=o(rsUf

0

L

/D

)fi

rsUf

0

L

r

~

o~准地转近似：e

=

o(L

/

r)<<1

限制下的情形gD

£

o(e)-2F

=

L2

/

R

2

=

0

海洋=10f

2

L2

大气=10-1

且d

=D

/L

£

epFps

¶zgD-

¶psp

=

p

1+

e*

s\

r*

=

rs

(1+

eFr)

¶x

¶y

¶z

rsdt

r

dt分析连续方程：eF

dr

+

drs

+

(1

+

eFr

)

¶u

+

¶v

+

¶w

+

D

2w

-

Lv

tan

q

=

0L=

D

Uw

*已取w零级近似的水平运动方程：¶y¶x¶pv

=u

=

-

¶p0000一级近似：d

vdtd

u¶y¶p¶x¶p101

0

0

dt1010

0+

u

+

bu y

=

--

v

-

bv y

=

-

¶x

¶y

¶u

¶vdtd

0

(z0

+

by)=

-

1

+

1

=

0¶x

¶yrs

¶z¶u0

¶v0

¶

w0

rs

)得：

+

+=

0+

+¶x

¶yrs

¶z¶u1

¶v1

¶

w1rs

)0

w

=

0即w

=

ew

+

e2

w

+...1

2为求w1需用热力学方程cccRpcvpp

R

ln

p*

=

1-ln

p*

-

ln

r*

+

Cln

p*

-

ln

r*

+

C

=大气：lnq*

=ln

T*

-pFssspcvp

¶z-

¶ps=c-

ln(1

+

eFr

)+

C-

ln

rln

p

+

ln 1

+

ecsvs

sp

pc p

¶zc

-

¶ps=

cvp

-eFr

+

Cln

p

-

ln

r

+

eF\

q*

=

qs

(z*

)(1

+

eFq

(x,

y,

z,

t

)000\

qp

-

rc p

¶zp

s=

cv

-

¶ps=

-eFrs

rg¶

rs

p)=

-rg

=

eFgD~¶z*

¶~prs

¶z¶rsrs

¶z

¶z=

+

p0¶

rs

p0

)

¶p0\

-r0

=sp¶z=

cv--

¶zqs

¶z¶qsc

ps

¶z

r

¶zD¶z¶ps

¶rs-

p0¶p0

p0¶p0\

q0

=dts¶x

¶y\r

¶z(z

+

by

)=

-

+

=¶(w1

rs

)¶v1

¶u10d

0¶

+

v¶¶x

¶y= +

udt00

d0

¶¶t100

0.65

300

100

104

1

=+==-

=

o(10-1

)TDc

g

D

¶Td*

p

s

*sss(g

-g)=

oq

¶z

q

¶z T

¶z¶q

D¶q¶z=

¶p00\

q=dtT

+

Q

+

s

=

QeF

qs

¶zdq

q

kdq

(1+eFq)w¶q2*

dt*

cpT

r**-

\+

¶x

¶z

¶y+

S

¶zSdtz

S

ssssd0

1

¶u0¶q0

¶v0

¶q0

¶

rsq0

r

¶z

rs

Q

dt

=

r

¶z

Sd0q0

¶¶

rs

=

Q

-r

¶z

r

¶¶(w1

rs

)¶y

¶y¶z

¶z¶x

¶x¶z

¶z 0

= 0

=-

0¶q

¶2

p

¶u¶q

¶2

p

¶v但 0

= 0

= 0

,\201Lf

2

L2N

2

D2

L2+

w

S

=

Q0

0dt=

D

s

*

0s

S

=

s

=

s

=q

¶z f

2

L2g¶q

D2Fq

¶z¶qd

q

-

=+\Sdt¶x

¶ydt

ssssd

0

¶v1

¶u10d

0¶

rsq0

r

¶z¶

rs

Q

dt

=

r

¶z

Sd

0q0

¶

rs

=

Q

-r

¶z

Sr

¶z¶(w1

rs

)(z

+

by

)=

-若绝热，并令：y

=p0

=

\dt

¶

rs

Q

rs

¶z

Srs

¶z

S¶z¶

rs

¶p0

z

0

+

by

+d

0

(1)-¶y

¶x

¶x2

+

¶t

¶x

¶y

s

=

0s¶

r

¶y

r

¶z

S¶z

+ +

by

+

¶ ¶y

¶ ¶y

¶

¶2y

¶2y¶y2边界条件，对于大气只有下边界条件：2DhE1/

2

B

=

ew1w=

v

z0

+V0E1/

2\

w1

=

v

z0

+V0 hB2e++ =

-00

01d

q

Q

d

¶y

/

¶z

QSdt

S

Sdt

S而w

=

-Sdt

B

+

B

¶x

¶y

¶y

¶x

-+=

-

v

¶y

¶h¶y

¶h¶x

¶y2¶2y

2E1/

2

¶2yz

=02e

d

¶y

/

¶z

绝热：

0

恢复到有量纲的位涡方程为：2*

=

0*

*

*¶

¶

¶ss

*

s¶x

¶y

¶tr¶

rs

f¶

p*

-

ps

r

¶z

N

¶z

+u*

+v*

z*

+

b0

y*

+•cp2dt*ar0r*

-

Qdr*对海洋：

=

k1dt

r

¶z

eF

dr

+

w

¶rs

(1+

eFr)=

-Qs*

s

r

=

r

(1+

eFr)sdt

F

r

¶zd

r

w

¶r

0 0

+

1

s

=

-Qdt¶z

¶z¶z

1

=

0

0

+¶w

d

¶

r

/

S

)

¶

Q

/

S

)ssr

¶z=r

¶z

¶z0¶

r

p

)

¶p0

+

p

¶rss

00-r

=s¶rs但对海洋

<<1

fir

¶z¶z¶p00r

=

-

\¶z

S

¶z

S

dt

00¶

r

=

¶

Q

+

by

-d

0

z

¶

Q

=¶\¶z

S

¶p¶z

S¶z

dt

d

000z

+

by

+00f

2

L2

L2N

2

D2

L2f

2

L2D2s

*s=

s

=

D

S

=

s

=

s

Fr

¶z

r

¶z-

¶r

-

g¶r2-

+¶y

¶y

¶x¶t

¶x+

+

+

by

=

0

(2)¶x

¶y2

¶z

S¶z¶2y

¶

¶y

¶

¶2y

¶

¶y

¶

¶y若绝热，并令：y

=p0上边界条件：curltdt

er

f

UDdhs

0t001+fi

w

(x,

y,1,

t

)=

FS

dt Sdt

¶z

Sdt

¶z

11

d0

r0

=

d0

¶p0

=

d0

¶y

但绝热时w

=

-=\Sdt

¶z

s

0z

=1(

F

<<1)curlter

f

UDt0d0

¶y

§4.2

层结流体中的Rossby波Rossby

Wave

in

a

Stratified

Ocean2++

-

¶t

¶x

¶y

¶y

¶x

s

=

0s¶

r

¶y

r

¶z

S¶z

+

by

+¶

¶2y

¶2y

¶ ¶y

¶

¶y大气：22++-¶y

¶z

S¶z

¶t

¶x

¶y

¶y

¶x

¶x

+

by

=

0¶

¶y

¶x

¶y2¶2y¶

¶2y

¶ ¶y

¶

¶y海洋：

+若远离边界，=

1

=

constS

=

const,rs

¶z

H-

¶rsH

很大。大气中

H

=

o

(1)，海洋中设解：y

=Az

/2

H

cos(kx

+ly

+mz

-st

)(

)/

Sk

2

+

l

2

+

m2

+1/

4H

2-

bk得：s

=假定波动的垂直尺度很小（远离边界）则：1/m2

<<D2或H

2k

2

+

l

2

+

m2

/

S\

s

=

-

bk

2

m

D/

m2

L=

=L2

S

m2

L2

L2

m是两波节点之间的垂直距离。

L

是等密度面的变形半径，L大气LD

=D

/

f0

,

Lm

=

LD

/

m

D

/f0

,

海洋LD

=rs

¶z*g¶rsqs

¶z*g¶qs

D===22

2

2(k

+

l

+

m

/

S

)c

=cgz

=cgy

=gx

2bkl

(k

2

+

l

2

+

m2

/

S

)2

2bkm

/

S

¶s

b(k

2

-

(l

2

+

m2

/

S

)¶k

(k

2

+

l

2

+

m2

/

S

)2¶s¶l¶s¶m<

0=-2m2

/

Scgzs

/

m

(k2

+l2

+m2

/

S)2注意所以垂直方向相速与群速总是反向§4.3

Rossby波垂直模态Rossby

Wave

Vertical

Modes小扰动、水平无界、无地形和摩擦、刚盖、无风应力强迫近似下的海洋：+

+=

0=

02z

=1s

¶t¶z

z

=0¶x¶x

¶y2¶t

¶2y

s

+

br

¶z

S¶z