Документ взят из кэша поисковой машины. Адрес оригинального документа : http://geonew.phys.msu.ru/junior_students/course_quot_fundamentals_of_geophysics_and_ecology_quot/2011%20Lecture%2006.pdf Дата изменения: Tue Mar 20 21:33:40 2012 Дата индексирования: Sat Feb 2 21:42:46 2013 Кодировка:




z

r v


r v = (u x , u y , u r v (u, v, w )

z

)

y

rr v = v( x , y, z, t )

p = p( x, y, z, t )


0

x

= ( x, y, z, t )




z



y

dz dy dx



0

x




F



~ dm = dx dy dz

q (, , , ...) q (, )




n z = {0,0,1}
z y

n y = {0,1,0} n x = { ,0,0} 1

0

x


z

F = [( x + dx ) - ( x )] dydz n x = {- 1,0,0} F = dxdydz x n x = { ,0,0} 1
y



0

x


z

xx yx zx



xy z

xz z zz



xz





xy

y




xx
j i



0

u u i+ ij = - pij + x x x j




r ma =



r F



+




r F



dx dy dz
r dv

~ dx dy dz x

dxdydz

r rr 1 = g + 2[v в ] + " " x dt




p( x , y, z) dy dz

x

n

x

n

p( x + dx, y, z) dy dz
!


p( x , y + dy, z) dx dz
n
y

p( x , y, z) dx dz

n


y


p( x , y, z + dz) dx dy

n

z

nz p( x , y, z) dx dy






F

grad p x

=- =- =-

1 p x 1 p y 1 p z

r F

grad p

= - grad p r 1 = - p

1



F

grad p y

r F

grad p

F

grad p z

r , , x y z


z y
,

0

x

r v = (u (z ), 0, 0

)


u xz = z


u z

dx dy
z + dz



u dx dy z z


z y

0

x

r v = (u ( y), 0, 0

,

)


u xy = y

u y

dx dz
y + dy

u dx dz y y


z y

0

x

r v = (u ( x ), v, w

,

)


u xx = x

u x

dy dz
x + dx

u dy dz x x



F
. x

1 u = x x

1 u 1 u + + y z z y



F

. x

2u 2u 2u = + + x 2 y 2 z 2

= u = u


r F

.

r = v




, / -3 10 2 10
-5

, / -6 10
2

1510
-6 2

-6





680 10 / c
10 / c
17 2






r ma =
r dv dt



r F



+



r F



r rr = g + 2[v в ]-

r p

r + v


rr v = v ( x , y, z, t ) r r rr r dv v = + v, v
dt t

()






u= du dt du dt

u ( x , y , z , t ) = u (x ( t u u x u = + + t x t y u u u = + u+ v t x y

), y( t ), z( t ), t y u z + t z t u +w z

)

du i u i = + dt t


j=1

3

u i u x j

j


r v t

rrr + v, v = r rr = g + 2[v в ] -

()

-

r p

r + v

3 5


( )

( x )u ( x )dydz

z y x

( x + dx )u ( x + dx )dydz


m t

= dxdydz

t

=

= -[( x + dx ) u ( x + dx ) - ( x )u ( x )]dydz t t =- [u x

]

-

[v y

]

-

[w z

]

r + div[v ] = 0




r v t

rrr + v, v = r rr = g + 2[v в ]-

()

-

r p

r + v



t = (p)

r + div ( v ) = 0




()

r v t

rrr r rr + v, v = g + 2[v в ] - r + div ( v ) = 0

()

r p

r + v

t = ( p)

!!!


+ r
r v r r r p r rr r + (v, )v = - + g + 2[v в ] + v; t : r T r + (v, )T = T; t s r r + (v, ) s = s; t r + div ( v ) = 0; t = (p, T, s).


+ r

r v r r r p r rr r + (v, )v = - + g + 2[v в ] + v; t T T T T r r +v +w u + (v, )T = T; x y z t s r r s s s + (v, ) s = s; u +v +w t x y z r u u u + div ( v ) = 0; u +v +w t x y z = (p, T, s).


,

r v=0



rr v=v



0

( )

- C

T z

=Q

u = z p = p0



T = T0


,
, ..

: q q (-, - ) q q etc.


rr v = v 0 ( x , y, z) p = p 0 ( x , y, z )

( t=0)

T = T0 ( x , y, z) s = s 0 ( x , y, z )

r v r = a 0 ( x , y, z) t K «»










1: « ()»

= 0 = const r r rrr rr r v p r + v, v = - + g + 2[v в ]+ v t r r div v = 0 + div ( v ) = 0 t = (p) 0

()


r v t

1: « ()»

rrr + v, v = -

()
~ u

r p
0

r rr r + g + 2[v в ]+ v

r div v = 0
u x y L w w ~ z H + v

H << L w << u

u


x

+

v y

=-

w z

w



~

H L

u




2: « »

r v t

rrr + v, v = -

()

r p

r rr r + g + 2[v в ]+ v

t = (p)

r + div ( v ) = 0


3: « () »

rrr rr r p r + g + 2[v в ]+ v + v, v = - t r + div ( v ) = 0 : t " " " " = (p)
- 0xy {u = v = w = 0} {w = 0}

r v

()

r


4: « , » r

rr rrr p r + v, v = - + g + 2[v в ] t 0 r div v = 0

r v

()

r v1 , p1 r - , v 2 , p 2 r r Av1 + Bv 2 , Ap1 + Bp2 - A, B -


«» :
1. 2.
strophe (.) ­ ,
( ) ( )

z:-

r p

r +g =0



x y

rr :- + 2[v в ] = 0

r p


«» : 1.

dw

0 onst w = ~ = cu -g =- L dt z g = const H << L
= -0 g p(z ) = p 0 - 0 g z

1 p

H

dp dz


g

dp



dz z dp dz

= -g ( z) = -g p RaT

( z ) =

p( z )

R aT dp dz =- p H
H= R aT g

{T, g, R a } f (z)
0




dp dz g =- p H z ln p(z) - ln p 0 = -

ln p z H

p(z) p
0

p( z ) = p 0 e
0

-z / H

H p( z ) z ln =- p0 H R aT H= g

=-

z

z 0

H ­




Ra =

R µ

=

8.31 [ / ] 0.029 [ / ]



287 [ / ]

H=

R aT g

=

287 [ / ] 288 [K ] 9 .8 [ / c ]
2



8434 []







p = p 0e H=

-z / H

R aT g

8




(40N, )

p = p 0e

-z / H

H = 6.8
232 K


«» : 2.

r v t

rrr + v, v = -

()

r p
0

r rr + g + 2[v в ]

r div (v ) = 0

!


Gaspard-Gustave de Coriolis French, Mathematics, Physics 1792-1843

F



rr = 2m[v в ]


x = - cos

r

Z



z = sin
X

y = 0


r = (- cos , 0, sin r v = ( u , v, w )

)

i (v z - w y ) rr [v в ] = u v w = j(w x - u z ) = k (u - v ) x y z y x i(v sin ) i(v sin ) 1. w << {u, v} = j(- w cos - u sin ) j(- u sin ) 2. F = 0 k (v cos ) 0 i j k
z


2v sin rr 2[v в ] - 2u sin 0 fv rr 2[v в ] = - f u 0 f = 2 sin






t=

L

U

?
Carl-Gustaf Rossby

T


Swedish-US meteorologist 1898-1957

T t

=

TU L

R

o


TU Ro = L

3[]в 0.01[ / ] Ro = = 0.1 0.3[ ]

24 60 60[]в 0.01[ / ] Ro = 3000 0.3[]


r v t U ~

rrr + v, v = -

()

r p

r rr + g + 2[v в ] U f f = 2 sin


UU

T L T ~ L/U r v

R o << 1
rrr + v, v

t Ro = rr 2[v в ]

()

UU L ~ U f =

U Lf




TU Ro = L

U Ro = , f = 2 sin Lf


24 60 60[c]в 10[ / ] Ro = 0.86 6 10 [ ]

24 60 60[c]в1[ / ] Ro = 0.086 6 10 []




2.

-

r p
0

rr + 2[v в ] = 0

r div v = 0


2.

- -

1 p 0 x 1 p 0 y

+ 2v sin = 0 - 2u sin = 0


g p( x , y, z ) = p
z y



+ 0 g[( x , y) - z]
(x)

?
0 x

dp dz

= -g

0

-

1 p 0 x

= -g

x




- -

1 p 0 x 1 p 0 y

+ 2v sin = 0 - - 2u sin = 0

1 p 0 x p y =0

= -g

x

u=0 v= g 2 sin x


g
z y

(x)

0

x


g
z


y F
grad

(x)
F


0

x




- ( XVIII ): « , , ­ »


() 500


«» :
3.

v 1 p - =0 r r
2


«» :
4......