The ideal gas constant,
R
, has different values for each particular gas or a mixture of
gases. For dry air parcel that contains no water vapor,
d
R
= 287 J kg
-1
K
-1
. Since density
=
V m
the ideal gas equation can be rewritten as:
In different case, the equation can also be modified by eliminating air density and
replacing it with specific volume,
, it is defined as the ratio of gas volume to its mass
or simply the inverse of air density. The standard unit of specific volume is commonly
expressed in m
3
kg
-1
, depicting the volume occupied by one unit of mass at a given
temperature and pressure. 20
.. ..........
T R
p
In general, the thermodynamics properties
of an air mass determine the particular weather condition in the atmosphere over the
area in which the air mass covers. As an air mass travels from one place to another, it is
being exposed to new environments and its thermodynamics
properties may
change gradually over time. These changes is then
used as a fundamental to understand different atmospheric phenomenon ranging from the
smallest cloud microphysical processes to the general
circulation of
the atmosphere
Wallace and Hobbs 2006.
a. The Hydrostatic Equation
The pressure of air in the atmosphere at any height is determined by the force per unit
area exerted by the weight of air influenced by gravity that acts on its surface. Thus, as the
atmospheric pressure
decreases with
increasing height, there will be an upward motion caused by the pressure gradient force.
Figure 9 Balance of vertical forces of the atmosphere in the state of hydrostatic
balance.
Applied to an atmosphere at rest, the upward pressure gradient force is opposed by
the downward pull of gravity. When there is a balance between these two forces, the
atmosphere is in the state of hydrostatic equilibrium.For atmosphere in hydrostatic
equilibrium, when forces are in balance, there is no net vertical force acting on it. Thus, there
is no vertical acceleration occurs. The hydrostatic balance, as illustrated in Figure 9
is mathematically expressed as
21 ..
.......... dz
g dp
Most of the time, the atmosphere
approximates hydrostatic balance; however, this balance is not achieved for an intense
small-scale system such as tornadoes and thunderstorms
where the
air rapidly
accelerates in a vertical manner Ahrens 2004. If the pressure of a fixed point on the
Earth at height z is pz, then the hydrostatic equation to an infinite height is given by
22 ..
..........
z p
z p
dz g
dp
Since
p = 0, the pressure at height
z
, z
p , is equal to the weight of air and is the
result of gravity force that acts on the air above its level. Hence, the air pressure at the
mean sea level, p
, would be 1013.25 hPa or 1.013 × 10
5
Pa; also known as 1 atmosphere 1 atm.
b. Geopotential
As gravity is a conservative force, the work done by gravity does not depend on its
path and is identically zero Lynch and Cassano 2006. Since the work done by
gravity is equal to zero, it is represented as a gradient of a function; called geopotential.
The geopotential is defined as the required work that must be done against gravity to
raise a unit mass of air from sea level to a given height Salby 1996
.
23 ..
..........
z
dz z
g z
The use of geopotential in momentum equations has the advantage to avoid using gas
density, thus creating simpler equations. It is useful for most atmospheric applications since
direct measurement of air density can be extremely difficult. If gravity were constant, a
surface of geopotential at all places would be
19 ..
.......... T
R p
