Chapter 5
Soundings
• There are four basic
types of sounding
observations.
– (1) Radiosondes
• An instrument
package lifted by a
balloon with sensors
for pressure,
temperature, humidity.
– (2) Pibals (Pilot balloons)
 Carry no instruments. Are usually
tracked with theodolite. The
balloon is assumed to rise at a
constant rate once the correct
amount of gas is placed in the
balloon.
 By knowing the time of flight and
the elevation and azimuth angle to
the balloon, the position of the
balloon, thus the wind speed and
direction at various heights can be
obtained.
– (3) Rawinsondes
Combines radiosonde
(instrument package) and
method of tracking:
Tracked by either a radio
direction finder antenna, a
radar, or by GPS
– (4) Dropsondes
- dropped from an aircraft
or from a constant
pressure balloon
(2) Upper-air Maps
• Remember the format for
plotting data on an upper-air
map.
• Error on pg. 5
–The height tendency is plotted to the lower-right
of the station circle.
–This is not the position of the pressure tendency
on surface maps.
•That goes to the right of the station circle.
Radiosonde/rawinsonde - a circle.
Aircraft observation
- a square.
Satellite derived wind - a star.
(3) Sounding Diagrams
• Sounding diagrams are used to
represent the character of the air by
profiles of temperature, moisture, wind
as measured vertically through the
atmosphere above a location.
• Common ones used are:
– Stuve diagram
– Emagram
– Tephigram
– Skew-T log-P diagram
• Pressure may be plotted on a linear vertical scale, or
on a logrithmic vertical scale.
• Temperature is plotted on the horizontal axis opposite to the convention of plotting data which puts
the independent parameter on the horizontal axis.
A Log-pressure diagram
A linear pressure diagram
A Stuve diagram
A Skew-T log-P diagram
• There are several sets of lines on these diagrams
and a point
plotted on the
diagram has a
different
meaning
depending on
which set of
lines you are
considering.
• The
hodogram in
the upperleft shows
how the
balloon
moved.
• Other lines on the Skew-T diagram
– Potential Temperature (also called dry
adiabats).
• Potential temperature is the temperature a
parcel of air would have if it descended to a
particular pressure level (no exchange of heat
with the environment).
• The standard reference pressure is 1000 mb.
• Can by calculated by Poisson’s Equation.
Rd
Po = reference pressure
Po c p
Rd=gas constant for dry air 287J/kg
  T  
P 
cp=specific heat of dry air at constant
pressure, 1004J/kg


• Saturation Mixing Ratio lines: the value of the mixing
ratio of saturated air at the given temperature and
pressure with respect to a flat water surface.
• Determined using the Clausius-Clapeyron Equation:
rs  ro  e
 L  1 1 
   

Rv To T 

ro = 0.611kPa
To=273.15oK
Rv=461J/kg oK
Lv=2.5 x 106 J/oK
Ld=2.83 x 106 J/oK
I am using “r” to represent mixing ratio.
Sometimes “e” is used.
• Saturation Equivalent Potential
Temperature: (Sometimes just called
equivalent temperature.) The
temperature a parcel of air at a given
temperature and pressure would have if
it were saturated, and if all that water
were condensed and removed and the
parcel brought down to some reference
level, usually 1000 mb.
• Converting water vapor to liquid water
releases latent heat which is then
absorbed by the gas molecules of the
air, so the air temperature increases.
So, an air temperature’s saturated
equivalent potential temperature is
always warmer than its potential
temperature.
(5) Vertical Derivatives and the
Hydrostatic Equation
• Consider the vertical derivative of
temperature with respect to pressure.
This is simply the instantaneous change
of temperature with pressure at some
T
level; or
P
• One could also determine


T
z
• Procedure:
– Pick the level (pressure) at which you wish to
compute the derivative.
– Draw a line tangent to the temperature profile at
that level.
– Going from higher pressure to lower pressure,
pick a point about 50mb lower along the tangent
line and determine the pressure and temperature.
– Pick a point about 50mb above along the tangent
line and determine the pressure and temperature.
– Determine (P1-P2) and (T1-T2) from these values,
making certain that P1 and T1 are the values lower
in the atmosphere.
• We usually would like to have these
derivatives (of some element); such as
temperature, with respect to height. So,
it is useful to be able to convert from
pressure derivatives to height
derivatives.
• We can get that using the Hydrostatic
Equation.
• The Hydrostatic Equation results from
considering the vertical forces acting on
an air parcel which is not moving.
– These are gravity (directed down) and
– the vertical pressure gradient force
(directed up) which results from pressure
being higher near the Earth’s surface and
less as height increases.
• If the air is not moving vertically, the
magnitude of these forces are the
m P
same.

 mg
 z
• If we consider unit mass (mass=1), then
we can
 cancel it on both sides and we
are essentially dealing with
accelerations.
1 P

g
 z
P
  g
z
•
or
or P  gz
• The Hydrostatic equation.
this has density
in
it
which
is
• However, 

difficult to measure and most
thermodynamic diagrams don’t have
scales for density.
• We can get rid of density using the Ideal
Gas Law equation.
•Considering the Ideal Gas Law equation: PV  nR
–V = volume
–n = number of molecules (moles) of the gas.
*
T
–R* = universal gas constant = 8.3169 J/moleoK
 volume and, if we multiply
•If we divide both msides by V,
d
both sides by:
m
d
Where md is* the molecular mass of dry air/mole,
we get: P  n  md R T

md V
But, (n × md) is simply the mass, so we can replace
with density, ρ.
n  m d 


 V 
•
R*
And, m d is simply the gas constant for dry air, Rd which equals
287 J/kgoK. (note error, pg. 17).
P  Rd T
• This results in:
• Then, we can combine the hydrostatic and ideal gas law

P
equation.
z
  g
• Writing the Ideal Gas Law equation for an expression for density
P
gives:

Rd T
• and substituting
 into the Hydrostatic Equation gives:
P
Pg

z
Rd T

• Consider this equation for dry air, we can write it as:
1 dP
g

P dz
Rd T
d ln P

g
Rd T
• Which is:
dz
• We
can see that pressure decreases in a natural

logarithmic manner from the equation showing the
derivative of pressure
with height.

• And, that change of pressure with height is
dependent on temperature.
• If a layer of the atmosphere is isothermal, changes in
height of pressure surfaces are directly proportional
to the logarithm of pressure.
• The heights, as related to pressure, on a sounding
diagram are from the standard atmosphere, and
would not be correct for the actual atmosphere.
However, we can make a height scale on the
sounding diagram which fits the actual atmosphere.
(6) Lapse Rate

T
z
• Lapse rate is the rate at which
temperature decreases with height.
• If all we have is temperature
and

pressure data, then using the
Hydrostatic equation we can get an
expression for the change of
temperature with height related to the
change of temperature with pressure.
• The Hydrostatic equation can be written
as: P  gz
• Then we can write the lapse rate
T
T
T T

or

g



as:
P  gz
P z
P

Rd T

Also,
remember that
So you could substitute in for density.

(7) The Buoyancy Equation
• Suppose the atmosphere is not “in balance” as
expressed by the Hydrostatic equation. Then, there
is vertical motion.
• The Vertical Momentum Equation expresses this
situation.
• This is the acceleration produced because of a
difference between the gravitational force/unit mass
and the pressure gradient force/unit mass.
• “D” is called the total derivative - the rate of change of
the value of a quantity associated with a particular air
Dw
1 P
parcel.
 g 
Dt

 z
• This expresses what is happening to an
air parcel that is moving vertically.
 1 P 
• The term   z  is normally negative,
since pressure decreases upward and

positive
“z-direction” is upward.
• (Note: error, first line, paragraph 4, pg.
21. Drop the word force.
• The air around the parcel - assuming it is not
moving vertically - is expressed by the
hydrostatic equation which, if we move all
terms to the right of the equal sign, can be
written as: 0  g  1 P
 o z
• Where o is the density of the environmental
air.

• The density
of the parcel can be written as a
perturbation of the environmental air density;
or:
o + ’
• The vertical momentum equation
then becomes:
• Subtracting the environmental
hydrostatic equation from this one
(change sign and add) gives:
• Combining terms and rearranging
Dw
1 P
'
gives:

Dt
 o z  o  '

• From the second term above, we
can see that:
1 P

• So we can substitute
- g for
 o z
• And get:
 '
Dw
 g
Dt
 o  '

Dw
1 P
 g 
Dt
 o  ' z
1 P
0  g 
 o z
Dw
1 P
1 P


Dt
 o z  o   ' z
g 
1 P
 o z
Dw
'
 g
Dt
 o  '
• So, if ’ is negative, (density is less than

the environment
- which occurs when
the temperature of the parcel is warmer
than the environment) then the right
side of the equation is positive and the
parcel accelerates upward.
• However, as an air parcel rises of sinks
through other air, air molecules drag
against the parcel creating a drag force.
This is usually small enough that for
most purposes it can be ignored.
• The vertical momentum equation can also be
written using potential temperature.
Dw
'
g
Dt
o
• The right side does not have a negative sign,
because awarm parcel would have a higher
potential temperature, just as it would have a
lower density. (The density term is not in the
equation.)
(8) The Thermodynamic Equation
• This equation expresses how the potential
temperature of an air parcel changes over time due
to various processes (primarily adding or removing
heat energy by phase changes or with the
D  Q
environment).

Dt
T cp
• If there is no phase change occurring, and assuming
no exchange of heat with the environment, the right
side is zero
 and there is no change of potential
temperature.
• On a thermodynamic diagram, you are following
parallel to the dry adiabats.
Stability
Remember:
(1) Is Absolutely
Unstable Air.
(2) Is Absolutely
Stable Air.
(3) Is Conditionally
Unstable Air.
• If we take a
parcel of air and
lift it, (assuming it
is not saturated),
we follow parallel
to the dry adibats
(potential
temperature
lines).
(9) Latent Heat Release
• What happens if the
air becomes saturated
and continues to rise.
• Remember: It
becomes saturated at
the Lifting
Condensation Level.
Questions
• Do: 2, 4, 5, 6, 7.