Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 142

non-physical, because there no longer is a density current: the cold, dense air mass
takes up the entire depth of the atmosphere and there is no place for advancing
air to flow left of the edge of the cold pool or above the domain at the top, so
that c ¼ 0. This case has been referred to as the ''lock exchange'' problem. Klemp
et al. in 1994 showed that it requires time-dependent solutions (the cold, dense air
does not remain locked up, but is released, like water held up by a dam which is
suddenly released, and its structure evolves with time). In any event, it is also a
case that is not relevant meteorologically because cold pools in convective storms
do not extend up to the troposphere. The solution for the very restrictive case
when h ¼ H = 2is

c 2

1

¼

2 g h ½ð 1 0 Þ= 0

ð 3 : 30 Þ

Klemp et al. in 1994 showed that this steady-state solution is also non-physical
because energy is not conserved unless turbulent friction is included. The more
relevant solution is for the case when H !1 (i.e., or when h H and the cold
pool is relatively shallow) so that

c 2

¼ 2 g h ½ð 1 0 Þ= 0 ð 3 : 31 Þ

This formula appears frequently in the literature and is used very often. The speed
c of the shallow cold pool, in the absence of surface drag is therefore

1 = 2

c ¼f 2 g h ½ð 1 0 Þ= 0 g ð 3 : 32 Þ

If surface drag is included and its effect is assumed to slow down the movement of
the cold pool, then

1 = 2

c ¼ K fg h ½ð 1 0 Þ= 0 g ð 3 : 33 Þ

where K is an empirical constant 1-1.5. The formula (3.32) is similar to that for
the phase speed of a shallow-water gravity wave, even though in the case of the
latter net mass is not transported along with the wave, while in the case of the
former mass is transported by the gust front. If a density current enters a region
where there is a strong stable layer, then gravity waves (a bore) may be generated,
but one can still distinguish between the density current and gravity waves above
it. Formulas (3.32) and (3.33) may be expressed more conveniently for use with
meteorological data by using the expression for buoyancy in terms of potential
temperature (2.148) (instead of in terms of density) as follows:

1 = 2

c ¼f 2 g h ½ð 0 1 Þ= 0 g ð 3 : 34 Þ

and

1 = 2

c ¼ K fg h ½ð 0 1 Þ= 0 g ð 3 : 35 Þ

where 0 is the average potential temperature of the environmental air; and 1 is

the cooler, average potential temperature of the cold pool.
In the reference frame of the moving density current, ambient air slows down
as it approaches the leading edge and rises up and over the leading edge ( Figure
3.38 ). When the ambient air is unsaturated and stable for the displacements it
undergoes as it rises over the cold pool, a laminar shelf or arcus cloud may form