1. L14 Physics of dry air and moist air
Potential temperature
Pseudo-adiabatic charts
Skew T – ln p charts
Moist air
Saturated adiabatic lapse rate
Normand’s Rule: Cloud base

2. Potential Temperature (θ)
The potential temperature of an air parcel is its temperature when compressed (or expanded) adiabatically to surface pressure (p0) (defined as a standard pressure of 1000 hPa).
Again, start from the 1st Law of Thermodynamics, and make dq=0:

3. Ideal Gas Law (see Lecture 8)
so:
R is the specific gas constant for air
R = 287 J kg-1 K-1
substitute in α:

4. Divide by RT:
Integrate both sides, from the starting (p,T) to the surface (p0,T0), noting cp/R is a constant:
Remember integral of 1/x is the natural log of x:

5. Rearrange to give potential temperature, θ:
Integrating:
Remember:
Hence:
or:
Rearrange to give potential temperature, θ:
R = 287 J kg-1 K-1
cp = 1004 J kg-1 K-1
Hence R/cp = 0.286

6. Happily, we can look at this graphically: e. g
Happily, we can look at this graphically: e.g., the ‘Pseudo-adiabatic’ chart
Re-arrange:
So if you plot: p0.286 on y-axis,T on x-axis
For a constant θ, (p00.286/θ) is also a constant, so the graph yields a straight line with gradient given by (p00.286/θ), and passing through T=0 and p=0

7. Pseudo-adiabatic chart
Earth’s atmosphere
y-axis is linear for p0.286
also linear for ln(p)
Useful as now we can follow each line and determine graphically temperature at any pressure, assuming adiabatic expansion/compression

8. Pseudo-adiabatic chart
Earth’s atmosphere
Solves Poisson’s equation graphically!
Disadvantage:
Everything happens in small region of the chart…
This can be overcome by skewing the temperature lines rather than plotting them straight up → The Skew T-ln p chart
Earth’s atmosphere is never here
Potential Temperature: temperature at 1000HPa

9. Examples: Kuching in Malaysia Valentia in Ireland.
Vertical T
Skew T-ln p chart difference to pseudo-adiabatic: ln(p) rather than p0.286 T skewed
Skew T

10. What are all the lines on the skew T-ln p chart?
isobar
isotherm
dry adiabat
saturated adiabat
saturation mixing ratio

11. Example: airplane air If an airplane at 250hPa takes air in at
-51oC and adjusts it to cabin pressure (850 hPa), does the air have to be
Heated
Cooled
to be comfortable?

12. Follow the dry adiabat to 850 hPa
Cabin pressure 850 hPa
Temperature ~43°C

13. Moist air See: L6 Humidity
Air contains some H2O molecules (water vapour)
Vapour pressure (e): partial pressure exerted by the gaseous water (hPa)
Mixing ratio (w): mass of water vapour / mass of dry air
Warmer air can accommodate more water molecules; the maximum for a given temperature is when the air is ‘saturated’
For a given temperature, there is a: saturation vapour pressure (es) saturation mixing ratio (ws)
An air parcel can become saturated, e.g. by ascent and cooling
Once saturated, further cooling will result in condensation of liquid water: i.e. cloud droplets

14. evaporation
condensation
Mixing ratio w = mvapour/mdry [normally given units g/kg]
At saturation: evaporation balances condensation
Saturation mixing ratio ws = es / p
Relative Humidity = w/ws x 100% = e/es x 100%

15. Thermodynamics of saturated air
As long as air remains unsaturated, it will behave like ‘dry’ air
However, once saturated, the condensation of liquid water releases latent heat
This means that an ascending air parcel that becomes saturated will cool less than one that remains unsaturated
We can theoretically derive how much the cooling is modified (not done here, see Wallace & Hobbs p79-87 if interested), and define the ‘Saturated Adiabatic Lapse Rate’ (SALR)
The difference between the DALR and a SALR is largest for warmer air, as the water vapour content, and hence latent heat release are larger
Saturated adiabats are solid green lines on the skew T-ln p chart

16. Saturation mixing ratio
Derived
Using ideal gas law, and def. of saturation water vapour pressure
(Clausius Clapeyron, Dr Essery)‏
Constant p: w increases with T
Constant T: w increases with decreasing p

17. Relative humidity: RH = 100*e/es ≈ 100*w/ws
Dewpoint (Td): Temperature to which air must cool at constant pressure to be saturated
Q: Air at 1000 hPa and 18oC has a mixing ratio of 6 g/kg. What is its relative humidity and dewpoint?
w = 6 g/kg ws
RH=6/13*100=46%
Dewpoint ~6.5oC
13g/kg saturation mixing ratio. => relative humidity 46%
Notice: if we know one moisture variable (mixing ratio) we know them all!

18. As unsaturated air lifts dry adiabatically, it will eventually saturate: Normand’s Rule

19. This level is the lifting condensation level

20. LCL = Cloud base

21. Let’s look at some real data

22. Albemarle, 00z Monday 17 Oct

23. Albemarle, 00z Tuesday 18 Oct

24. Summary
Potential temperature – the temperature of air compressed/expanded to 1000 hPa along a dry adiabat
Pseudo-adiabatic charts – graphically solve equations
Skew T – ln p charts – will use in labs
Moist air – releases latent heat at saturation point
Saturated adiabatic lapse rate – less than DALR – typically 6 K/km
Normand’s Rule: Can estimate cloud base height using surface temperature and moisture