1. Copyright © 2010 R.R. Dickerson & Z.Q. Li 1 The Slice Method Chapt 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

2. Copyright © 2010 R.R. Dickerson & Z.Q. Li 2 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.

3. Copyright © 2010 R.R. Dickerson & Z.Q. Li 3 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.

4. Copyright © 2010 R.R. Dickerson & Z.Q. Li 4 In time dt, the mass dm transported upward and the mass dm` transported downwards may be written as: dm =  Awdt =  Adz = - AdP/g dm` =  `A`w`dt =  `  `dz` = - A`dP`/g dz/dz` = vertical distance traveled by ascending/descending air.

5. Copyright © 2010 R.R. Dickerson & Z.Q. Li 5 At the initial moment, the slice is horizontally homogenous:  `. Since dm = dm`, we can divide the ascending equation by the descending: Assume also that advection is negligible.

6. Copyright © 2010 R.R. Dickerson & Z.Q. Li 6 Consider the layer as conditionally unstable: Let z o – reference level in the layer T – initial temperature of ascending air T ’ – initial temperature of descending air T f, T f ’ – final temperatures of a/de-scending air  – lapse rate of ambient air When ascending air reaches z o : T f = T –  s dz When descending air reaches z o : T f ’ = T ’ +  d dz

7. Copyright © 2010 R.R. Dickerson & Z.Q. Li 7 popo zozo p o -dp ’ p o +dp z o -dz z o +dz ’ T ToTo T’T’ T f ’ = T ’ +  d dz ’ T f = T-  s dz So for the unstable case T f > T f ’ or T -  s dz > T ’ +  d dz ’ But for the initial conditions T = T o +  dz T ’ = T o -  dz ’

8. Copyright © 2010 R.R. Dickerson & Z.Q. Li 8 The two preceding expressions can be combined: (  –  s )dz > (  d –  dz ’ for instability to occur. (  –  s )A ’ > (  d –  A : unstable (  –  s )A ’ = (  d –  A : neutral (  –  s )A ’ < (  d –  A : stable

9. Copyright © 2010 R.R. Dickerson & Z.Q. Li 9 Define the lapse rate for neutral equilibrium Then the stability criteria reduce to  n - unstable  n - neutral  n - stable Note:  n is a weighted average of  d and  s  with the weighting factors the areas of ascent and descent.

10. Copyright © 2010 R.R. Dickerson & Z.Q. Li 10 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.