A. EQUATIONS RELATED TO DISCUSSIONS ON BUOYANCY:
Generally, involving combinations of (2), (3), and (4).
(i) Acceleration an air parcel experiences due to density differences at a given level can be related to the difference in temperature of the air parcel , T_{p}, with respect to the temperature of the surrounding air, T_{e}, AT A GIVEN LEVEL The temperatures are in Kelvin:
The difference [(T_{ap} – T_{e}) ]is called the LIFTED INDEX, commonly evaluated at the 500 mb level (negative for unstable conditions).
(ii) A true measure of the potential buoyancy is a measure of the "positive" area on a SkewT Ln P diagram. This represents the portion of the parcel ascent curve in which the parcel is warmer and, thus, less dense than the air surrounding it. The positive area represents a potential source of energy for parcels at the ground that are lifted to the elevation (LFC) above which they become warmer than their surroundings. To obtain this, one needs to algebraically add this parameter at every level of the parcel's ascent until it reaches the point at which it becomes the same temperature as its surroundings again (Equilibrium Level). The parameter is known as Convective Available Potential Energy (CAPE) or Positive Buoyancy (B+).
Note that this equation really states that CAPE is
directly proportion to the total acceleration a parcel would experience due to
buoyancy from the LFC to the EL.
(iii) The vertical velocity due to buoyancy at the top of the "positive area" on a sounding can be evaluated from the following:
w = [2 X CAPE]^{1/2}
where w is the vertical velocity in
the Cartesian Cooridinate system
B. HYPSOMETRIC RELATION: Heating the atmosphere causes it to expand (special
application of the gas law, explained below) obtained by substitution of
the hydrostatic equation (4) into the equation of state.(1). This relation
provides the basis of explaining many, many things that synoptic meteorologists
see on weather maps and charts.
Thickness of layer between two pressure surfaces is
directly related to the mean temperature of the layer.
Also, if we consider the thickness of a layer that is often of importance to
synoptic meteorologists, the layer approximately between the ground (1000 mb)
and about 6 km (500 mb), the Hypsometric Relation is
Applications:

C. PRESSURE TENDENCY EQUATION: With respect to the sea level weather map, pressure
changes occur because of net accumulation
or net deficit of air in the air column above.
The PRESSURE TENDENCY EQUATION is obtained by manipulation of (5) and the
substitution into (5) of (1) to eliminate the density. (To be discussed in
detail in JPM's portion of Metr 200/201).
The pressure tendency equation allows one to assess how pressure changes at a
given level developthus accounting for the evolution of the surface pressure
field, for example. It is
the result of an Application of the
Principle of Conservation of Mass.
Since upper tropospheric divergence tends to be
larger than the compensating lower level convergence, generally speaking, areas of divergence in the upper
tropsophere identify regions in which the air columns underneath will be
experiencing a net export of mass (or weight). Thus, surface low pressure areas tend to
develop under regions of upper
tropospheric divergence and vice versa. (Careful, this relates to PRESSURE TENDENCIES, not
necessarily to the position of Highs and Lows on weather maps).
D. POISSON'S EQUATION



The equation above is the First Law of Thermodynamics. Substitution of the expression for density in the Gas Law into the right hand term, and assuming adiabatic conditions (the sensible temperature change = 0), and rewriting the equation, gives Poisson's Relation:
where the subscripts indicate two different levels of the
atmosphere, if diabatic effects are neglected, pressures are in millibars and
temperatures in Kelvin.
If level 1 is the 1000 mb level, then this equation can be
solved for the 1000 mb temperature, which is called the potential
temperature. The equation,
rewritten, is also called Poisson's Equation.
where R_{d}/C_{p} = k = 0.286
The potential temperature is the temperature of an air
parcel brought dry adabatically to a pressure elevation of 1000 mb. Dry adiabats are often labeled with
their respective potential temperature values.
It turns out that the concept of stability really relates to a comparison of an air parcel's potential temperature compared to that of the air around it at a given elevation. Poisson's Relation can be solved at other pressure elevations, for each potential temperature. The family of lines thus generated can be plotted on a diagram that has temperature (in Kelvin) as one coordinate, and p^{k} (increasing downward) as the vertical coordinate.
Poisson's Relation can be used to construct the lines on a thermodynamic diagram. Consider, for example, the 1000 mb temperature of 0C (273K). Insertion of 273K into the equation above, and putting a pressure of, say , 500 mb in the denominator of the right, allows one to determine the 500 mb temperature that corresponds to the dry adiabatic decrease in temperature with height. If one plots that point on a diagram with pressure decreasing upwards, the result is a line corresponding to a potential temperature of 273K. It also corresponds to the dry adabat that extends from the 1000 mb level to 500 mb.
If one does this for all possible temperatures and pressures, one obtains one form of a thermodynamic diagram called a Stuve or Pseudoadiabatic Chart. An example is provided below. For operational purposes, only the portion of the chart that has realistic temperatures for earth (1000 mb tempertures between 220 and 345) is generally reproduced. The lines are nearly parallel on such a chart and are really potential temperature contours (also known as ISENTROPES and DRY ADIABATS)..