L14 Physics of dry air and moist air
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Potential temperature
Pseudo-adiabatic charts
Skew T – ln p charts
Moist air
Saturated adiabatic lapse rate
Normand’s Rule: Cloud base
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:
c p dT   dp  0
Ideal Gas Law (see Lecture 8)
p   RT
so:   1  RT

R is the specific gas constant for air
R = 287 J kg-1 K-1
c p dT   dp  0
substitute in α:
c p dT 
RT
p
dp  0
p
Divide by RT:
c p dT

R T
dp
0
p
Integrate both sides, from the starting (p,T) to
the surface (p0,T0), noting cp/R is a constant:
T
cp
R

dT
T0  

T
p


p0
dp
p
Remember integral of 1/x is the natural log of x:
x2
1
 x dx
x1

ln x 
x2
x1
 x2 

 ln  x 2   ln  x1   ln 

x
 1
 p 
T 

ln    ln 

R  
 p0 
Integrating:
cp
Remember:
a (ln b )  ln b
Hence:
cp
T 
 
 
R

p
p0
a

or:
R
 p  cp

 
  p 0 
T
Rearrange to give potential temperature, θ:
R
 p0  cp

  T 
 p 
R = 287 J kg-1 K-1
cp = 1004 J kg-1 K-1
Hence R/cp = 0.286
Happily, we can look at this graphically:
e.g., the ‘Pseudo-adiabatic’ chart
 p0 

  T 
 p 
0 . 286
Re-arrange: p 0 .286  T
p0
0 . 286

• 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
Pseudo-adiabatic
chart
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
Earth’s
atmosphere
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
• 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
What are
all the
lines on the
skew T-ln p
chart?
isobar
isotherm
dry
adiabat
saturated
adiabat
saturation
mixing
ratio
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
1. Heated
2. Cooled
to be comfortable?
Follow
the dry
adiabat
to 850 hPa
Cabin pressure
850 hPa
Temperature ~43°C
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
evaporation
condensation
Mixing ratio w = mvapour/mdry [normally given units g/kg]
At saturation: evaporation balances condensation
Saturation mixing ratio ws = 0.622 es / p
Relative Humidity = w/ws x 100% = e/es x 100%
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
Saturation mixing ratio
Derived
Constant p: w increases
with T
Constant T: w increases
with decreasing p
Using ideal
gas law,
and def. of
saturation
water
vapour
pressure
(Clausius
Clapeyron,
Dr Essery)
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
As unsaturated air lifts dry
adiabatically, it will eventually
saturate: Normand’s Rule
This level is the
lifting
condensation
level
LCL = Cloud
base
Let’s look at some real data
Albemarle, 00z Monday 17 Oct
Albemarle, 00z Tuesday 18 Oct
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