Thermodynamic Energy
Equation

The
temperature tendency is

(1)

where
dT/dt is the individual derivative of temperature. This temperature change experienced by
the air parcel itself, dT/dt can be viewed as a "source/sink" term. The first law of thermodynamics allows us
to quantify this "source" term.

The
time derivative of the first law of thermodynamics is

(2)

where
q is sensible heating or cooling.

Substitute
(2) into (1)

(3)

The
definition of the adiabatic and environmental lapse rates are as follows:

(4)

Substitution
of (4) into (3) gives the Temperature Tendency Equation

(5)

Alternatively,
of the hydrostatic equation into (3) gives the Thermodynamic Energy Equation

(6)

as
is given as equation (4.3.4) in Bluestein except with the term a/c_{p} for dT/dp.

__Static
Stability Parameter__

Setting
(5) equal to (6) gives:

(7a)

Substitution
of the relation between omega and vertical velocity gives

(7b)

Rearranging
terms gives the static stability parameter

(8)

For
a stable atmosphere, the dry adiabatic lapse rate always exceeds the
environmental lapse rate, and the static stability parameter is > 0.

Equation
(6) may now be rewritten

(9)

PoissonÕs
Relation is

(10)

Taking
the natural log of both sides gives

(11a,b)

The
partial derivative with respect to height of (11b) is

(12)

Remembering
that

(13)

and
substituting the gas law and the hydrostatic equation gives

(14a,b)

Rearranging
terms and using the definition for the lapse rates gives

(15a,b)

Equation
(15b) states that the static stability is greatest in situations in which the
vertical gradient of isentropes is the greatest, that is to say, situations in
which there is a large change in potential temperature with height. Thus, isentropes are packed in frontal
zones, inversions and in the stratosphere, whereas, they are not in regions in
which the atmosphere tends toward low static stability.

For
a stable atmosphere, the static stability parameter is always positive. In the restrictive and rare case of
absolutely unstable conditions, the parameter is negative. For a positive static stability
parameter, parcels displaced from an initial elevation will be colder and
denser than their surroundings at a given elevation. If the parcel is displaced and
released, it will oscillate around its initial elevation until it comes to
rest. The period (or frequency) of
these oscillations can be appears in the double integration of the vertical
equation of motion and obtaining a solution for the height, z.

The
solution is in the form of an exponential function, with a power that contains
the vertical derivative of potential temperature, as in equations (14a and
15a). This is known as the
Brunt-Vaisala frequency and is given by the expression:

(16)

Larger
values of N occur for highly stable atmospheres and vice versa. N often appears in equations
involving instabilities in the atmosphere.
The student should keep in mind that the Brunt-Vaisala Frequency is
simply another measure of the static stability.