Meteorology ENV 2A23
Stability and Moisture
Lectures
• If ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally
stratified
• the lapse rate is equal to the dry adiabatic lapse rate (DALR)
• Γd = g/cp ≈10 K km-1 i.e. the temperature decreases by 10 K
every km.
.
• If ∂θ/∂z > 0, i.e. θ increases with height, the atmosphere is said
to be stable, statically stable, or stably stratified.
• Γ < Γd
• If an air parcel is moved upwards adiabatically, it will follow
the DALR, so be colder (therefore denser) than its
environment and so sink back down.
• Conversely …
• In other word, the atmosphere is stable to small perturbations.
• If ∂θ/∂z < 0, i.e. θ decreases with height, the atmosphere is said
to be unstable, statically unstable, or unstably stratified.
• Γ > Γd
• If an air parcel is moved upwards adiabatically, it will follow the
DALR, be warmer (less dense) than its environment and so keep
on rising.
• In other word, the atmosphere is unstable to small perturbations.
If ∂θ/∂z = 0, the atmosphere is said to be
neutral,or neutrally stratified, and the
lapse rate is equal to the dry adiabatic
lapse rate (DALR) Γd ~= 10 K km-1 i.e. the
temperature decreases by 10 K every km.
If ∂θ/∂z > 0, i.e. θ increases with height, the
atmosphere is said to be stable, statically
stable, or stably stratified.
If ∂θ/∂z < 0, i.e. θ decreases with height, the
atmosphere is said to be unstable, statically
unstable, or unstably stratified.
• The observed atmospheric lapse rate is closer
to the saturated adiabatic lapse rate than the
dry adiabatic lapse rate
• Typically Γ ≈ 7 K km-1
• The difference is due to moisture…
Brunt-Väisälä frequency
• Another measure of
stability
• Also known as static
stability frequency
• N is the frequency with
which a vertically
displaced air parcel
would oscillate
• Typically N = 10-2 s-1
g 
N 
 z
2
Moisture in the Atmosphere
• See Ahrens Chapter 4
Water can exist in 3 phases in the atmosphere:
• Gas
– water vapour
• Liquid
• Solid
Water can exist in 3 phases in the atmosphere:
• Gas
• Liquid
– Cloud droplets
– Rain drops
• Solid
Water can exist in 3 phases in the atmosphere:
• Gas
• Liquid
• Solid
– Ice crystals
– Snow
– Hail
Phase-transition equilibria
Fig – Tsonis, Ch 6.
Latent heat
• When water changes
phase a large amount of
energy, latent heat, is
either released or must be
supplied.
• In liquid or solid state the
H2O molecules tend to
align themselves by
charge (to minimise their
potential energy)
• In gas phase water
behaves as an ideal gas.
http://www.its.caltech.edu/~atomic/snowcrystals/ice/ice.htm
Latent heat of fusion Lf = 3.34x105 J kg-1
melting
FUSION
Solid (ice)
Latent heat of
sublimation
Ls = 2.83x106 J kg-1
freezing
Gas
(water
vapour)
Liquid (water)
Latent heat of
vapourisation
Lv = 2.50x106 J kg-1
Note all latent heats are functions of temperature, values here are for T=0oC
• Lv must be supplied from the environment when
water goes from liquid to gas
– e.g. during evaporation, heat is taken out of the ocean
and the ocean cools
• Lv is released to the environment when water
goes from gas to liquid,
– e.g. condensation in cumulus clouds.
– In the tropics this is a major driver of atmospheric
motion
– Also in tropical cyclones
Hydrological cycle
Measures of Water Vapour I
• There are many ways of measuring and expressing water
content of the atmosphere.
• One of the simplest measures is as a mixing ratio:
mw
r
md
Mass of water vapour/ mass of dry air
Units: kg/kg, or more usually g/kg
Measures of Water Vapour I
• There are many ways of measuring and expressing water
content of the atmosphere.
• One of the simplest measures is as a mixing ratio:
mw
r
md
Mass of water vapour/ mass of dry air
Units: kg/kg, or more usually g/kg
mw mw / V  w
r


md md / V  d
Measures of Water Vapour I
• Almost equal to r is q the specific humidity:
mw
w
w
q


md  mw d   w 
mass of water vapour/
total mass of air
Units: kg/kg, or more
usually g/kg
Measures of Water Vapour I
• Almost equal to r is q the specific humidity:
mw
w
w
q


md  mw d   w 
w
r
q

r
d   w 1  r
mass of water vapour/
total mass of air
Units: kg/kg, or more
usually g/kg
,as r is small
Measures of Water Vapour I
• Almost equal to r is q the specific humidity:
mw
w
w
q


md  mw d   w 
w
r
q

r
d   w 1  r
mass of water vapour/
total mass of air
Units: kg/kg, or more
usually g/kg
,as r is small
• q and r are constant for an “air parcel” if further water is
not added or subtracted.
Climatological distributions of q and r:
Fig - Hartmann
Climatological distributions of q and r:
Climatological distributions of q and r:
Equation of state for moist air
• For thermodynamic purposes we can treat water vapour as a
separate gas (c.f. Dalton’s Law)
• The partial pressure of water vapour will obey ideal gas law:
e = (pw) =ρwRwT,
where ρw is density w.v.
Rw is specific gas constant for w.v.
Rw = R*/Mw = 461 J kg-1 K-1
• The equation of state for dry air is
p - e = ρdRdT
Equation of state for moist air
Equation of state for moist air
• Moisture can be measured as the ratio of the
partial pressure of water vapour (e) to that of
dry air (p-e):
e
 w RwT
Md r

r

p  e  d Rd T
Mw 
, where ε = Mw/Md
= 18.02/28.99 = 0.6222
Equation of state for moist air
• We can combine ideal gas laws for dry air &
water vapour:
p  (  d Rd   w Rw )T
r
p   d Rd (1  )T
p


r
(1  r )
Rd (1  )T

1 r  
p  Rd 
T
 1 r 
1 r  
p  Rd T , whereT  
T
 1 r 
*
*
T* is the virtual
temperature
Equation of state for moist air
1 r  
p  Rd T , whereT  
T
 1 r 
*
*
• Note: T* is just a convenience, it is not a true thermodynamic
variable, i.e. it is not a measure of the internal energy of a
moist air parcel.
Saturated vapour pressure
vapour
• Consider a closed system of fixed volume,
which contains some liquid water
• The average internal energy of the water
molecules in the liquid is proportional to the
temperature.
• Imagine the no. molecules in vapour state
increases, i.e. vapour pressure (e)
increases, then no. of molecules reentering liquid will increase.
• At a certain critical vapour pressure, system
will come into equilibrium,
– i.e. condensation = evaporation
• e = esat, the saturated vapour pressure
liquid
Saturated vapour pressure
• As the internal
energy of the liquid
water molecules
depends on
temperature, it is
intuitive that esat will
increase with
temperature.
• Determined by the
Claussius Clapeyron
equation
es is a strong function of temperature and be using the first and second laws of
thermodynamics it can be shown that:
des
Lv

dT T ( w   l )
where Lv is the latent heat of vapourisation,
αw is the specific volume of water vapour, and
αl is the specific volume of liquid.
This is the Clausius-Clapeyron equation.
es is a strong function of temperature and be using the first and second laws of
thermodynamics it can be shown that:
des
Lv

dT T ( w   l )
where Lv is the latent heat of vapourisation,
αw is the specific volume of water vapour, and
αl is the specific volume of liquid.
This is the Clausius-Clapeyron equation.
It can be simplified, given αw >> αl to be
des
Lv es

dT RwT 2
es is a strong function of temperature and be using the first and second laws of
thermodynamics it can be shown that:
des
Lv

dT T ( w   l )
where Lv is the latent heat of vapourisation,
αw is the specific volume of water vapour, and
αl is the specific volume of liquid.
This is the Clausius-Clapeyron equation.
It can be simplified, given αw >> αl to be
Can be solved to give:
 L
es  es 0 exp
 Rw
des
Lv es

dT RwT 2
 1 1 
   

 T0 T  
where e = es0 at T = T0. If T0 is
taken as 0 oC (the triple point)
then es0 = 6.11 mb.
es is a strong function of temperature and be using the first and second laws of
thermodynamics it can be shown that:
des
Lv

dT T ( w   l )
where Lv is the latent heat of vapourisation,
αw is the specific volume of water vapour, and
αl is the specific volume of liquid.
This is the Clausius-Clapeyron equation.
It can be simplified, given αw >> αl to be
Can be solved to give:
Or approximated as
 L
es  es 0 exp
 Rw
des
Lv es

dT RwT 2
 1 1 
   

 T0 T  
 L


es  es 0 exp 
T ' 
2
 RwT0 
where e = es0 at T = T0. If T0 is
taken as 0 oC (the triple point)
then es0 = 6.11 mb.
where T’ = T-T0.
Vapour pressure & boiling
• As water boils, bubbles of
vapour rise to the surface and
escape
 esat = atmospheric pressure
and esat proportional to T
 where lower atmos. pressure
 lower esat
 lower boiling point
 food takes longer to cook
Ahrens – Ch 5
Measures of Water Vapour II
• Relative humidity
• RH = vapour pressure/ saturated
vapour pressure
e
RH 
 100
esat
Measures of Water Vapour II
• Dew Point Temperature (TD)
• TD is the temperature at which air becomes saturated if
cooled at constant pressure.
• Hence e(T) = esat(TD)
Measures of Water Vapour II
• Wet Bulb Temperature (TW)
• TW is the lowest that can be reached by evaporating water
into the air (at constant pressure)
• TD ≤ TW ≤ T, with equality at saturation
• TW is the temperature of an evaporating cloud droplet of
rain drop
• It is a good measure of how comfortable for humans, as for
TW << 37 oC allows lots of evaporation to take place.
Measures of Water Vapour II
• Wet Bulb Temperature (TW)
• TW is the lowest that can be reached by evaporating water
into the air (at constant pressure)
• TD ≤ TW ≤ T, with equality at saturation
• TW is the temperature of an evaporating cloud droplet of
rain drop
• It is a good measure of how comfortable for humans, as for
TW << 37 oC allows lots of evaporation to take place.
rs (T )  r
TW  T 
c p Lrs (T )

L
RwT 2
Moist Adiabatic Processes
• The relatively small
amounts of water vapour in
the atmosphere means that,
in practice, the moist
adiabatic lapse rate for
unsaturated air is close to
the dry adiabatic lapse rate.
• It can be shown that:
i.e. Γm ≈ Γd , as r is small.
g
m 
 d (1  0.9r )
c pm
Saturated Adiabatic Processes
• For saturated air, the effects of
latent heat release when water
changes phase are important.
• Consider an air parcel, rising
adiabatically:
• Condensation takes place and
latent heat is released:
 the (saturated) adiabatic lapse rate
is lower than the dry adiabatic
lapse rate,
 i.e. Γs < Γd.
Saturated Adiabatic Processes
• From the first law of thermodynamics
(and using hydrostatic balance) it is
possible to show that the saturated
 Les 
adiabatic lapse rate is
d 1 

RTp 

• Typically Γs is 6-7 K/km in the
s 


troposphere

L
de
s
1 



• Note Γs(T,p)
pC
dT
p


• Hence saturation adiabats, i.e. lines that
follow the saturated adiabatic lapse rate,
are marked as curved lines on a
tephigram.
• A saturated air parcel will follow a
saturated adiabat.
plus saturated adiabatics (lines of constant equivalent potential temperature).
Plus the international standard atmosphere.
Normand’s Rule
• Uses a tephigram to find the condensation
level (or lifting condensation level LCL) for
an “air parcel”
• Lets follow an air parcel rising through the
atmosphere:
• Air parcel at p0 has temp.
T & dew point temp. TD
• If unsaturated the air
parcel rises along DALR
• At pN it becomes saturated
(thereafter it follows
saturated adiabat)
• Normand’s point, at
pressure pN, is the
intersection of r line from
TD and the DALR from T.
• This is the LCL, the Lifting
Condensation Level.
• From the LCL, if the air
parcel rises, it will follow
saturated adiabat.
Normand’s Rule
• From LCL following
Normand’s Rule
saturated adiabat down
tells us temperature air
would have if water is
continually evaporated into
it so it remains saturated;
i.e. the wet bulb
temperature TW
• Hence saturated adiabats
are lines of constant wet
bulb potential temperature
θW
• Higher up, θW and θ
curves become parallel.
• θW asymptotes to the
equivalent potential
temperature θe
• θe is the potential temp an
air parcel would have if all
its water vapour were
condensed out.
• θW and θe are conserved
Normand’s Rule
by an air parcel, if there
are no external heat
sources or sinks
• Hence θW and θe are often
used to indentify “air
masses”.
Atmospheric Stability II
• We can now reconsider atmospheric
stability for a moist atmosphere
• Latent heat release is important for
modifying the stability of the atmosphere
• Recall for a dry atmosphere:
If ∂θ/∂z < 0, i.e. θ decreases with height, the
atmosphere is said to be unstable, statically
unstable, or unstably stratified.
If ∂θ/∂z = 0, the atmosphere is said to be
neutral,or neutrally stratified, and the
lapse rate is equal to the dry adiabatic
lapse rate (DALR) Γd ~= 10 K km-1 i.e. the
temperature decreases by 10 K every km.
If ∂θ/∂z > 0, i.e. θ increases with height, the
atmosphere is said to be stable, statically
stable, or stably stratified.
(1) If Γ > Γd or ∂θ/∂z < 0,
the atmosphere is said to be unstable, statically
unstable, or unstably stratified.
(2) If Γ = Γd or ∂θ/∂z = 0, the
atmosphere is said to be neutral,or
neutrally stratified
(3) If Γs < Γ < Γd or ∂θ/∂z > 0 and ∂θW/∂z < 0 (or ∂θe/∂z < 0)
the atmosphere is said to be conditionally unstable
(4) If Γs = Γ or ∂θW/∂z = 0 (or ∂θe/∂z = 0) the atmosphere is
said to be saturated neutral or neutral to moist convection.
(5) If Γ < Γs or ∂θw/∂z > 0 (or ∂θe/∂z > 0) the atmosphere is
said to be absolutely stable
Cloud Development
• Clouds form as air rises, expands and
cools
– Once T cools to TD then condensation can
take place
– Water vapour condenses onto particles
(called cloud condensation nuclei) in the
atmosphere to form cloud droplets
• What causes the air to rise?
Cloud Development
• Convectively forced
cloud development
• CCL (Cloud
Condensation Level) is
point when T = TD
• Upward motion in
thermal
• Downward motion to
sides
– Continuity
– Colder, denser air at
cloud edge due to
evaporation
• The erosion due to
evaporation and
entrainment leads to the
clumpy form of
convective cumulus
• Downward motion
(subsidence) is slower,
hence occurs over a
larger area
• Subsidence inhibits
convection
• Shade surface, so cut off
further clouds, until blown
away
• Cumulus advected by
wind
Trade-wind cumulus clouds over the South Pacific Ocean
Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal
Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal
Small cumulus clouds over EastIsle
Africa.
of Wight
Development of cumulus cloud
• Environmental stability is critical for depth of Cumulus
development
• Cumulonimbus can reach as high as the tropopause,
where the high stability of the stratosphere blocks further
ascent and causes spreading into an ‘anvil’
Cumulus humilis
Cumulus congestus
Cumulonimbus
Spectacular cumulonimbus anvil
Development of Topographic Cloud
Orographic stratus. View south to Quinag, north-west Scotland, from B869 road,
16:15 GMT.
Orographic cumulus and the Matterhorn
Foehn conditions, Leukerbad, Switzerland. View northwards from the southern
outskirts of Leukerbad towards the Gemmi Pass, which is covered in cloud. The
mountain almost completely covered in cloud in the centre of the picture is
2,600m high. The white mountain (Rinderhorn) is about 3,450m high.
Development of lenticular clouds
Altocumulus lenticularis.
Cumulus, cirrus, cirrocumulus and altocumulus lenticularis.
Picture from the Royal Meteorological Society's historic Clarke and Cave Collection.
Further Reading for reading week:
C. Donald Ahrens (2007) Meteorology Today. Eighth Edition, Brooks/Cole.
ISBN 0-534-37201-5
Chapter
1. Earth and its atmosphere – composition, vertical structure, instruments,
etc
2. Energy: Warming the Earth and Atmosphere – temperature, heat
transfer, radiation, radiative processes, the sun.
3. Seasonal and daily temperatures – orbital patterns etc.
4. Atmospheric Moisture – water in the atmosphere, measures of water, etc
5. Condensation: dew, fog and clouds
6. Stability and cloud development – stable air, unstable air, tephigrams,
cloud development.