The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 31

where l is the density of all liquid substances (including cloud droplets, drizzle

drops, and raindrops); i is the density of all ice substances (includes all ice

crystals, graupel, and hailstones); T v is the virtual temperature; and T cv is the
cloud virtual temperature. It follows from (2.20) and the observation that the
water vapor mixing ratio r v 1 that
T v ¼ T ½ 1 þð R v = R d Þ r v =ð 1 þ r v Þð 1 þ 0

:

609r v Þ T

ð 2 :

21 Þ

From (2.20) and (2.21) it is seen that the cloud virtual temperature
T cv ¼ð 1 þ 0 : 609r v r l r i Þ T ð 2 : 22 Þ

where r l is the liquid water mixing ratio; and r i is the ice mixing ratio. The
buoyancy for moist air containing both liquid and ice condensate is therefore
B ¼ g T 0 ð 1 þ 0

609r v r l r i Þ=

: T ð 2 : 23 Þ

An upward (downward) buoyancy force is thus diminished (increased) by the
loading of water substance and increased (decreased) by the presence of water
vapor.
It will be demonstrated later that the total vertical acceleration following an
air parcel is given not only by Archimedean buoyancy, but also by the vertical
perturbation pressure gradient force. This force is in part associated with air that
is moved out of the way ahead of the air parcel and air moving in behind the air
parcel as it is accelerated vertically by Archimedean buoyancy. This air movement
is accomplished by a perturbation high-pressure area ahead of the air parcel to
force air out of the way and by a perturbation low-pressure area behind the air
parcel to suck air in behind it. When the pressure gradient force is directed in the
opposite direction to air motion, as it is in this case, the pressure gradient force is
said to be ''adverse''. If the air parcel is moving vertically at a constant speed and
the adverse vertical perturbation pressure force exactly counteracts the buoyancy
force, the flow is hydrostatic (as noted earlier). If the air parcel accelerates verti-
cally, then the vertical perturbation pressure gradient does not exactly counteract
the buoyancy force. It will be shown shortly how one can compute the vertical
pressure gradient force and thereby determine net vertical acceleration. There is
also a component to the vertical perturbation pressure gradient force associated
with the wind field in the environment; this component will also be discussed later.

2.2 THERMODYNAMICS

The first law of thermodynamics for air is as follows:
dQ ¼ C v dT þ pd ð 2 : 24 Þ

where Q is heat energy; C v is the specific heat at constant volume of a volume of
air containing a mixture of dry air, water vapor, and other water substances