MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Lecture T4
Moist air on a tephigram and stability
Reading: FWC, sections 5.10, 5.12, 5.13
T4 1 - Tephigram for moist air (this is the theory behind some of the concepts
you used in last weeks practical):
1st law of thermodynamics….
Q. What is the theoretical basis for the tephigram you used in last weeks
practical?
Written as:
But R and cP are constants for DRY air. Air in the real atmosphere is rarely (if ever!) dry.
So how do we represent a moist or saturated air and the processes involving it?
Moist air never holds more than 2-3% by mass of water vapour, and often a lot less, so
the constants c p and R for moist air are not too different from their values for dry air. This
means we can use the same tephigram for a sample of moist air as for a sample of dry
air. This makes the tephigram really useful. However, for saturated air (e.g. air parcels
within a cloud) it’s behaviour is very different from that we’ve already described.
T4 2 - Saturation:
Saturation: the parcel can hold no more water vapour.
Q. What will happen to a saturated parcel if we cool it? Or add more water
vapour?
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
How do we measure saturation?
Saturated vapour pressure es: water vapour content of the air at saturation.
: Only depends on temperature.
: Determined from the Clausius-Clapeyron equation
Appendix 6.1 of FWC is a good
place to read more about this
des
Les
.

dT Rv T 2
If we assume that latent heat of evaporation/condensation, L, is taken as a constant ( an
OK assumption although in fact, it decreases slightly with temperature), the ClausiusClapeyron equation can be integrated to give:
 L  1 1
e s  e s 0 exp      .
 Rv  T0 T  
where es0 is a constant of integration. If T0 is 273 K, then es0 is 611 Pa.
Remember that the humidity mixing ratio, r, is related to the vapour pressure?
r
Rd e
.
Rv ( p  e)
If we put es instead of e in this formula we get an expression for the saturated humidity
mixing ratio,rs, that tells us that rs only depends on pressure and temperature. This
means we can plot lines of constant rs on the tephigram (Operationally these are dashed
lines
making
a
small
angle
with
the
isotherms.
For the more mathematically
minded….
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
T4 3 - Saturated adiabatic lapse rate:
What happens when we lift a parcel of saturated air adiabatically?
1.
2.
3.
But how can we represent this process on a tephigram? (Remembering that for dry air a
parcel will follow a dry adiabat.)
The thermodynamic state of the air parcel at the end of the step may be calculated using
the first law of thermodynamics.
Heat released per unit mass = latent heat x change of saturated humidity mixing ratio
So putting this into the first law of thermodynamics (as dQ)
 L drs  c p dT 
RT
dp
p
(The minus sign is needed so that in descent when dp > 0, the air parcel must supply
heat to evaporate some of the liquid water droplets).
We want to find the temperature that the parcel ends up with so rearrange the equation
dT 

1  RT
dp  L drs  .


cp  p
From this equation, we deduce that the temperature fall when the saturated parcel is
lifted is smaller than when an unsaturated or dry parcel is lifted.
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
On a tephigram the parcel ascends along a line which is between the lines of constant T
and lines of constant .
Q. What happens as the parcel is lifted higher and becomes colder?
If we did this calculation at each level we would build up a continuous curve on the
tephigram showing the path of an ascending saturated air parcel. Since the only heat
sources or sinks are intrinsic to the air parcel, we call such a curve a saturated adiabat.
Operationally, the saturated adiabats are labelled by the temperature on them at
pressure pR  105 Pa . This temperature is denoted w and is called the wet bulb potential
temperature. Lines of constant w are the curves on the tephigram, making a large angle
with the  lines near the bottom of the diagram but virtually parallel to them near the
top.
T4 4 - Normand's construction:
It turns out that there is a quick way to estimate when a parcel will become saturate (or
cloudy) from a tephigram if we know the temperature at a level and something about the
water vapour content of the parcel at that level. The method (which you used last week)
is based on the theory we’ve just discussed.
Imagine an unsaturated air parcel at some pressure with humidity mixing ratio r.
We know that its dewpoint temperature, Td can be found using the relationship:
rs (Td )  r (T ) ,
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
(i.e. read off the temperature where the line rs  r intersects the pressure line).
Now suppose the parcel is lifted adiabatically.
1. The parcel will follow the unsaturated adiabat and its temperature will fall.
lifting condensation
level
2. At the same time, the parcel will conserve its moisture content, and so the dewpoint
at any pressure will be given by the intersection of the pressure line with the rs  r
line.
3. The dewpoint will drop less rapidly with height than the temperature, and so at some
level the temperature will become equal to the dewpoint.
p
T
4. The air parcel is saturated.
The pressure where this occurs is called the lifting condensation level. Further ascent
means that water must condense, and the air parcel becomes cloudy. It now rises along a
line of constant w (a saturated adiabat).
D
T
w
T
Normand's construction.
Now suppose that the air parcel were returned to the ground, but that during the descent,
it remained in contact with some source of water.
1. As the parcel moves to warmer levels in the atmosphere, water re-evaporates into it.
2. The air parcel is cooled since the latent heat of evaporation must come from its
internal energy.
3. The parcel will descend along the saturated adiabat, ending up at a temperature at the
ground which is intermediate between T and Td . This is called the “thermodynamic wet
bulb temperature”, Tw, and is very close to the wet bulb temperature measured in a
meteorological screen.
This diagram in fact illustrates a very useful construction on the tephigram called
“Normand's construction”. This states that:
The line of constant rs passing through the dewpoint temperature, the line of constant w
passing through the wet bulb temperature and the line of constant  passing through the
dry bulb temperature all meet at the lifting condensation level.
Thus
knowing
any
two
of
TD , Tw , T and plc , the remaining
quantities can be read off from the
tephigram.
And the lifting condensation level
is where you would find the cloud
base if you could raise an air
parcel to that level by some
means.
MT11B Weather Systems Analysis
T4 5 - Stability:
A radiosonde records the temperature, dewpoint and pressure as the balloon ascends.
These data are plotted on a tephigram and the resulting curve is called the “environment
curve”. The environment curve tells us about the stability of the atmosphere, which in
turn tells us about what would happen to an air parcel if we lifted it in that atmosphere
(remembering that in each case the air parcel would rise along a dry or a saturated
adiabat). We recognize four different cases.

Case S - “Absolutely stable”: The environment curve lies to the right of both the
dray and saturated adiabats. If an air parcel is lifted, it follows either the dry adiabat if
it is unsaturated or the saturated adiabat. In either case, it will find itself colder and
therefore denser than the environment at the new level. It will tend to sink back to its
original level. The environment profile is absolutely stable.

Case U - “Absolutely unstable”: The environment curve lies to the left of the dry
adiabat. In this case, the lifted air parcel will be warmer and less dense than the
environment. It will therefore be more buoyant, and it will tend to continue moving
away from its original level. The profile is said to be absolutely unstable. This is a
rather unusual situation, generally confined to a shallow layer immediately above a
heated surface. Otherwise, the instability leads to very rapid mixing which renders the
profile neutral.

Case N - “Neutrally stable”: The environment curve lies along the dry adiabat. The
displaced parcel will have exactly the same temperature as the environment, and
experiences zero buoyancy force. A neutral layer is commonly observed in the
atmospheric boundary layer, below the lifting condensation level.

Case C - “Conditionally unstable”: In the most complicated case, the environment
curve lies between the dry and saturated adiabats. The displaced parcel will be
unstable if it is saturated but stable otherwise, a situation called conditional instability.
Generally, the instability will be realized if the parcel is moistened or if it can be
displaced a finite distance upwards so that it becomes saturated. Most of the tropical
troposphere is conditionally unstable.
Eleanor Highwood, December 2003
MT11B Weather Systems Analysis
Eleanor Highwood, December 2003
Q. If we displace an unsaturated air parcel from its home level in stable or
conditionally unstable conditions what will happen to it?
We can put a number on this using the Brunt-Vaisala frequency, N/2 where:
N2 
g 
 z
and we can calculate this using the tephigram (example in EC8.)
Generally N is large when the atmosphere is stable (especially in the stratosphere) and
smaller when it is unstable.