The Slice Method Chapt 4 page 51.

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The Slice MethodChapt 4 page 51.
A conceptual model accounting for compensating
motion by ambient air as a parcel or column
rises.
Introduced by Bjerknes, J. 1938 Saturated
ascent of air through a dry adiabatically
descending environment. Quart. J. Royal
Meteorological Society, 65
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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.

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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.
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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
rAwdt rAdz - AdP/g
dz/dz vertical distance traveled by
ascending/descending air.
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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
Assume also that advection is negligible.
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Consider the layer as conditionally unstable
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 g
lapse rate of ambient air When ascending air
reaches zo Tf T Gsdz When descending air
reaches zo Tf T Gddz
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The two preceding expressions can be combined
(g Gs)dz gt (Gd g)dz for instability to
occur.
(g Gs)A gt (Gd g)A unstable (g
Gs)A (Gd g)A neutral (g Gs)A lt
(Gd g)A stable
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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
gn is small and more easily exceeded. Severe
storms tend to have small updrafts and large are
areas of subsidence.