The time derivative of the first law of thermodynamics


Oct 27, 2013 (5 years and 2 months ago)


Thermodynamic Energy Equation

The temperature tendency is


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


where q is sensible heating or cooling.

Substitute (2) into (1)


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


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


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



as is given as equation (4.3.4) in Bluestein except with the term



Static Stability Parameter

Setting (5) equal to (6) gives:


Substitution of the relation between omega and vertical velocity gives


Rearranging terms gives t
he static stability parameter


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



Poisson’s Relation is


Taking the natural log of both sides gives


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


Remembering that


and substituting the gas law and the hydrostatic equation gives



Rearranging terms and using the definition for the lapse rates gives


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
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
integration of

the vertical equation
motion and obtaining a solution
for the height, z.


Using the gas law and the definition of potential temperature



Notice that the right hand side of equation (17) conta
ins a factor proportional
to the static stability parameter, asi n (15a).

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).


z’ is the final elevation of oscillation, z’

is the initial elevation of the
air parcel and

is known

as the Brunt
Vaisala frequency,
given by the


Larger values of N occur for highly stable atmosph
eres 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.

Equation (19) conceptually says that
the mo
re stable the atmosphere the smaller (and quicker) the oscillations.