Lecture 10 part 2
Slice method
1. Particle method (Lect. 8-9):
assumption of particle motion within motionless environment.
2. Basic approaches of slice method: 1938-1939 (Bjerknes J.,Petersen S., Shishkin N. )
Lecture 10. Slice method |
1 |
Let’s pick out a sufficiently large
layer of air
S ' cross sec tion(inner)
w' 0
z2 is upper layer
z1 is lower layer
S '' cross sec tion(outer)
w'' 0
Lecture 10. Slice method |
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Slice method assumptions:
•all variations of the quantities within the picked out layer come out adiabatically;
•no advection (horizontal mass motion) is observed;
•the mass of air above any layer remains unchanged.
Lecture 10. Slice method |
3 |
The latter means that fluxes of air mass through section S’ and S’’ are equal:
'S 'w' ''S ''w '' |
(10.8) |
is density
' ''
S ' w' S '' w '' |
(10.9) |
Lecture 10. Slice method |
4 |
To derive the stability criterion:
T is temperature difference, T ' is updraft flux temp.,
T '' is the downdraft one at z level
T T ' T ''
T 0 unstable state( positive acceleration)
T 0 stable state( negative acceleration)
Lecture 10. Slice method |
5 |
Suppose, particles come from lower level to upper one:
T ' T |
' (z z ) |
(10.10) |
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T '' T |
'' (z |
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z ) |
(10.11) |
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' is adiabatic lapse rate in updraft flux
'' is adiabatic lapse rate in downdraft flux
z z |
w' t |
(10.12) |
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z2 z |
w'' t |
(10.13) |
Lecture 10. Slice method |
6 |
From (10.10)-(10.13) we obtain:
T T |
T |
( ' w ' '' w '' ) t |
(10.14) |
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T1 T2 (z2 z1 ) (z2 z) (z z1 ) t(w' w'' )
T ' (w ' w '' ) ( 'w ' ''w '' ) t |
(10.15) |
Lecture 10. Slice method |
7 |
Let’s use (10.9)
S ' w' S '' w '' |
(10.9) |
T ( ' (w ' w '' ) ( ' w ' '' w '' ) t |
(10.15) |
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S ' |
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T ( ' |
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( |
'' ) w |
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t |
(10.16) |
S |
'' |
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Lecture 10. Slice method |
8 |
Let’s consider 3 cases:
1.' '' a
•Updraft and downdraft air isn’t saturated
T ( a |
)(1 |
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S ' |
)w |
' |
t |
(10.17) |
S '' |
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•Criteria of stability of both methods (parcel and slice ones) give the same result.
Lecture 10. Slice method |
9 |
2nd case:
2.' '' ' a
•Updraft and downdraft air fluxes are saturated
T ( a ' )(1 SS''' )w ' t
•Criteria of stability of both methods (parcel and slice ones) give the same result.
Lecture 10. Slice method |
10 |