1. Stability and Moisture Lectures
Meteorology ENV 2A23
Stability and Moisture
Lectures

2. 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.

3. In other word, the atmosphere is stable to small perturbations.
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.

4. 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.

5. 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.

6. 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…

7. 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

8. Moisture in the Atmosphere
See Ahrens Chapter 4

9. Water can exist in 3 phases in the atmosphere:
Gas
water vapour
Liquid
Solid

10. Water can exist in 3 phases in the atmosphere:
Gas
Liquid
Cloud droplets
Rain drops
Solid

11. Water can exist in 3 phases in the atmosphere:
Gas
Liquid
Solid
Ice crystals
Snow
Hail

12. Phase-transition equilibria
Fig – Tsonis, Ch 6.

13. 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.

14. Latent heat of fusion Lf = 3.34x105 J kg-1
melting
FUSION
Solid (ice)
Liquid (water)
freezing
condensation
VAPOURISATION
SUBLIMATION
Latent heat of sublimation
Ls = 2.83x106 J kg-1
sublimation
evaporation
Latent heat of vapourisation
Lv = 2.50x106 J kg-1
Gas
(water
vapour)
Note all latent heats are functions of temperature, values here are for T=0oC

15. 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

16. 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

17. Hydrological cycle

18. 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:
Mass of water vapour/ mass of dry air
Units: kg/kg, or more usually g/kg

19. 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:
Mass of water vapour/ mass of dry air
Units: kg/kg, or more usually g/kg

20. Measures of Water Vapour I
Almost equal to r is q the specific humidity:
mass of water vapour/
total mass of air
Units: kg/kg, or more usually g/kg

21. Measures of Water Vapour I
Almost equal to r is q the specific humidity:
mass of water vapour/
total mass of air
Units: kg/kg, or more usually g/kg
,as r is small

22. Measures of Water Vapour I
Almost equal to r is q the specific humidity:
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.

23. Climatological distributions of q and r:
Fig - Hartmann

24. Climatological distributions of q and r:

25. Climatological distributions of q and r:

26. 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

27. Equation of state for moist air

28. 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):
, where ε = Mw/Md
= 18.02/28.99 =

29. Equation of state for moist air
We can combine ideal gas laws for dry air & water vapour:
T* is the virtual temperature

30. Equation of state for moist air
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.

31. Saturated vapour pressure
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 re-entering liquid will increase.
At a certain critical vapour pressure, system will come into equilibrium,
i.e. condensation = evaporation
e = esat, the saturated vapour pressure
vapour
liquid

32. 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

33. es is a strong function of temperature and be using the first and second laws of thermodynamics it can be shown that:
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.

34. where Lv is the latent heat of vapourisation,
es is a strong function of temperature and be using the first and second laws of thermodynamics it can be shown that:
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

35. where Lv is the latent heat of vapourisation,
es is a strong function of temperature and be using the first and second laws of thermodynamics it can be shown that:
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
where e = es0 at T = T0. If T0 is taken as 0 oC (the triple point) then es0 = 6.11 mb.
Can be solved to give:

36. where Lv is the latent heat of vapourisation,
es is a strong function of temperature and be using the first and second laws of thermodynamics it can be shown that:
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
where e = es0 at T = T0. If T0 is taken as 0 oC (the triple point) then es0 = 6.11 mb.
Can be solved to give:
Or approximated as
where T’ = T-T0.

37. 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

38. Measures of Water Vapour II
Relative humidity
RH = vapour pressure/ saturated vapour pressure

39. 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)

40. 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.

41. 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.

42. 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.

43. 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.

44. Saturated Adiabatic Processes
From the first law of thermodynamics (and using hydrostatic balance) it is possible to show that the saturated adiabatic lapse rate is
Typically Γs is 6-7 K/km in the troposphere
Note Γs(T,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.

45. plus saturated adiabatics (lines of constant equivalent potential temperature) a quantity which is conserved for a saturated air parcel
plus saturated adiabatics (lines of constant equivalent potential temperature).

46. Plus the international standard atmosphere.

47. 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:

48. Normand’s Rule 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.

49. From LCL following 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.
Normand’s Rule

50. θW and θe are conserved by an air parcel, if there are no external heat sources or sinks
Hence θW and θe are often used to indentify “air masses”.
Normand’s Rule

52. 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:

53. 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.

54. (1) If Γ > Γd or ∂θ/∂z < 0,
the atmosphere is said to be unstable, statically unstable, or unstably stratified.

55. (2) If Γ = Γd or ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally stratified

56. (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

57. (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.

58. (5) If Γ < Γs or ∂θw/∂z > 0 (or ∂θe/∂z > 0) the atmosphere is said to be absolutely stable

60. 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?

61. Cloud Development

62. 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

63. 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

64. Trade-wind cumulus clouds over the South Pacific Ocean

65. Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal

66. Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal
Small cumulus clouds over East Africa.
Isle of Wight

67. Development of cumulus cloud

69. 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’

70. Cumulonimbus
Cumulus congestus
Cumulus humilis

71. Spectacular cumulonimbus anvil

72. Development of Topographic Cloud

73. Orographic stratus. View south to Quinag, north-west Scotland, from B869 road, 16:15 GMT.

74. Orographic cumulus and the Matterhorn

75. Foehn conditions, Leukerbad, Switzerland
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.

76. Development of lenticular clouds

77. Altocumulus lenticularis.

78. Cumulus, cirrus, cirrocumulus and altocumulus lenticularis.
Picture from the Royal Meteorological Society's historic Clarke and Cave Collection.

79. Further Reading for reading week:
C. Donald Ahrens (2007) Meteorology Today. Eighth Edition, Brooks/Cole. ISBN
Chapter
Earth and its atmosphere – composition, vertical structure, instruments, etc
Energy: Warming the Earth and Atmosphere – temperature, heat transfer, radiation, radiative processes, the sun.
Seasonal and daily temperatures – orbital patterns etc.
Atmospheric Moisture – water in the atmosphere, measures of water, etc
Condensation: dew, fog and clouds
Stability and cloud development – stable air, unstable air, tephigrams, cloud development.