The Lapse Rate of Special
Atmospheres and their
application.
Atms 4310 / 7310
Lab 5
Anthony R. Lupo
The Lapse Rate of Special
Atmospheres and their application.
Take (Start with) the hypsometric equation
for example:
RT  pu 
 pu 
z  zu  zl  
ln    H ln 
g  pl 
 pl 
H = RT/g
are two!!!
(physical interpretation), there
The Lapse Rate of Special
Atmospheres and their application.

(The Big) “H”: This is scale height, the height
of all the gas in the atmosphere if the gas had
same pressure and temperature.
 Another interpretation is that this is the height
of the atmosphere at 1/e of the Surface
pressure (e- folding depth)
 Does it work?
The Lapse Rate of Special
Atmospheres and their application.
 Well, if 360 hPa = 7 km, H, assuming T = 250 K, is 7175
m.
 Thus, it does work!
 Ok, let’s take a look at a “special atmosphere”

-(zu – zl) / H = ln(pu/pl)
 Take the antilog of both sides:

Exp (-(zu – zl) / H) = pu/pl
The Lapse Rate of Special
Atmospheres and their application.
 Or…….

pu = pl exp (-(zu – zl) / H)
 We get a relationship for pressure and
height if we assume a constant
temperature atmosphere (e.g., T = 250 K).
The Lapse Rate of Special
Atmospheres and their application.
  This is what we call an isothermal
atmosphere. Another name for it is “barotropic”
(density is a function of pressure) but we’ll talk
about this later (atms 4320). Thus, we can
substitute r for p.
 Uses of this “special atmosphere”:
 Assume zl = 0 pl = sea level pressure.

P = Pslp exp(-z/H)
The Lapse Rate of Special
Atmospheres and their application.
  We can then assume T = Tsurface. If we know
the surface pressure, and our elevation, we can
reduce our pressure to sea level!!! This is how
it’s done to produce the sea level pressure
map!!!
 Give an example:
 p = 1000 mb
 Tsurf = 68o F
z station elevation= 200m
The Lapse Rate of Special
Atmospheres and their application.
But keep in mind: standard US practice is
to average the previous 12 h with the
current for Tsurf)
We get:
Pslp = 1023.6 hPa
The Lapse Rate of Special
Atmospheres and their application.
 Now, let’s find a relationship that uses a constant lapse
rate given a surface pressure (Po) and Temperature (To).
 Start w/ hydro. Balance and Eqn of state (Like
Hypsometric equation)

We can assume
 T = (To – Ge z)
 where Ge is the “environmental” lapse rate
The Lapse Rate of Special
Atmospheres and their application.
 If surface is zo = 0 the integral is (we
also need to “strategically” multiply by 1 to
integrate as well, can you spot it?):
1 p g
G z
pl p t  RG zl To  Gz t
Then solve for z:
pu
zu

To   p 
z  1   
G   po 

RG
g





The Lapse Rate of Special
Atmospheres and their application.
  Use of this relationship:
 Typically use with “US standard atmosphere” where,
 To = 288 K
 Po = 1013.25
 lapse = 6.5 C/km
 Then in potential vorticity (PV) thinking, this standard
atmosphere can be used as the base, where PV
anomalies are calculated relative to the base. (We will
not touch this).
The Lapse Rate of Special
Atmospheres and their application.
Calculation of Altimeter settings.
Convert to inches 1 mb = 0.02953 inches
The Lapse Rate of Special
Atmospheres and their application.
The End!
The Lapse Rate of Special
Atmospheres and their application.
Questions?
Comments?
Criticisms?