1. Chapter 5
Soundings

2. There are four basic types of sounding observations.
(1) Radiosondes
An instrument package lifted by a balloon with sensors for pressure, temperature, humidity.

3. (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.

4. (3) Rawinsondes
Combines radiosonde (instrument package) and method of tracking:
Tracked by either a radio direction finder antenna, a radar, or by GPS

5. (4) Dropsondes
- dropped from an aircraft or from a constant pressure balloon

6. (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.

7. (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

8. 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

9. A Stuve diagram
A Skew-T log-P diagram

10. 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.

11. The hodogram in the upper-left shows how the balloon moved.

12. 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.
Po = reference pressure
Rd=gas constant for dry air 287J/kg
cp=specific heat of dry air at constant pressure, 1004J/kg

15. 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:
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.

17. 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.

18. 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.

20. (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 level; or
One could also determine

21. 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.

23. 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.

24. 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.

25. If the air is not moving vertically, the magnitude of these forces are the same.
If we consider unit mass (mass=1), then we can cancel it on both sides and we are essentially dealing with accelerations.

26. or or
The Hydrostatic equation.
However, this has density in it which is 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.

27. Where md is the molecular mass of dry air/mole, we get:
Considering the Ideal Gas Law equation:
V = volume
n = number of molecules (moles) of the gas.
R* = universal gas constant = J/moleoK
If we divide both sides by V, volume and, if we multiply both sides by:
Where md is the molecular mass of dry air/mole, we get:
But, (n × md) is simply the mass, so we can replace
with density, ρ.

28. And, is simply the gas constant for dry air, Rd which equals 287 J/kgoK. (note error, pg. 17).
This results in:
Then, we can combine the hydrostatic and ideal gas law equation.
Writing the Ideal Gas Law equation for an expression for density gives:
and substituting into the Hydrostatic Equation gives:

29. Consider this equation for dry air, we can write it as:
Which is:
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.

30. 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.

31. (6) Lapse Rate
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.

32. The Hydrostatic equation can be written as:
Then we can write the lapse rate
as:
Also, remember that
So you could substitute in for density.

33. (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 parcel.

34. This expresses what is happening to an air parcel that is moving vertically.
The term is normally negative, since pressure decreases upward and positive “z-direction” is upward.
(Note: error, first line, paragraph 4, pg Drop the word force.

35. 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:
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 + ’

36. The vertical momentum equation then becomes:
Subtracting the environmental hydrostatic equation from this one (change sign and add) gives:
Combining terms and rearranging gives:
From the second term above, we can see that:
So we can substitute - g for
And get:

37. 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.

38. 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.

39. The vertical momentum equation can also be written using potential temperature.
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.)

40. (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 environment).
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.

41. Stability Remember: Is Absolutely Unstable Air.
Is Absolutely Stable Air.
Is Conditionally Unstable Air.

42. 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).

43. (9) Latent Heat Release
What happens if the air becomes saturated and continues to rise.
Remember: It becomes saturated at the Lifting Condensation Level.

44. Questions
Do: 2, 4, 5, 6, 7.