Transcript Lecture 9, Slice Method

The Slice Method
Chapt 4 page 51.
A conceptual model accounting for compensating motion
by ambient air as a parcel or column rises.
Does inclusion of subsidence in the environmental (non-cloud)
air make it easier or harder to form a Cb?
Introduced by
Bjerknes, J. 1938: Saturated ascent of air through a
dry adiabatically descending environment.
Quart. J. Royal Meteorological Society, 65
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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Conditions to which the method applies:
• Initially horizontal layer of saturated air.
• There may be several regions in which the
air is ascending and cooling moist
adiabatically.
• Within the remainder of the layer there must
be regions of descent with warming at the
dry adiabatic rate.
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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Let
* Ascending air have total horizontal
area A and upward speed w.
* Descending air have area A’ and vertical
speed w’.
Assume
Rate at which mass descends through a fixed
reference level in the slice of originally saturated
air is equal to the rate at which mass ascends
through the reference level.
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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In time dt, the mass dm transported upward and
the mass dm` transported downwards may be
written as:
dm = rAwdt = rAdz = - AdP/g
dm` = r`A`w`dt = r`A`dz` = - A`dP`/g
dz/dz` = vertical distance traveled by ascending/descending air.
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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At the initial moment, the slice is horizontally
homogenous: r = r`. Since dm = dm`, we can
divide the ascending equation by the descending:
Aw Adz Adp
1=
=
=
Aw
Adz
Adp
A w dz dp
or
= =
=
A w dz dp
Assume also that advection is negligible.
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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Consider the layer as conditionally unstable: (s    d )
Let
zo – reference level in the layer
T – initial temperature of ascending air
T’ – initial temperature of descending air
Tf, Tf’ – final temperatures of a/de-scending air
 – lapse rate of ambient air
When ascending air reaches zo : Tf = T – sdz
When descending air reaches zo : Tf’ = T’ + ddz
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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T’
po-dp’
po
zo+dz’
Tf’ = T’+ddz’
To
Tf = T-sdz
po+dp
zo
zo-dz
T
So for the unstable case Tf > Tf’
or T - sdz > T’+ddz’
But for the initial conditions
T = To+ dz
T’ = To- dz’
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Z.Q. Li
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The two preceding expressions can be combined:
( – s)dz > (d – )dz’ for instability to occur.
A dz
But
=
A dz 
( – s)A’ > (d – )A : unstable
( – s)A’ = (d – )A : neutral
( – s)A’ < (d – )A : stable
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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Define the lapse rate for neutral equilibrium
Ad  As
n 
A  A
Then the stability criteria reduce to
 > n
 = n
 > n
- unstable
- neutral
- stable
Note: n is a weighted average of d and s with
the weighting factors the areas of ascent and descent.
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Z.Q. Li
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Thus, for the conditionally unstable case we are considering,
accounting for compensating vertical motions by the ambient
air requires the lapse rate to be steeper for instability to occur.
The slice method is not easy to apply in practice
because it requires knowledge of the relative areas of ascent
(A) and descent (A’).
The stability criteria indicate that the chances for
development of slice instability are greatest when A’ is large
and A is small for then n is small and more easily exceeded.
Severe storms tend to have small updrafts and large are areas
of subsidence. Cloud models must include this and it is a
useful concept for understanding that the free trop. is
generally subsiding where it is not rising convectively
(Anderson et al., Nature Communications 2015).
Copyright © 2010 R.R. Dickerson &
Z.Q. Li
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