1. Numerical Weather Prediction Moist Thermodynamics
ECMWF Training Course
May 2001
Numerical Weather Prediction Moist Thermodynamics
Peter Bechtold and Adrian Tompkins
Clouds 3

2. Can we represent clouds in a GCM ie
Can we represent clouds in a GCM ie. Moisture transport and phase changes ?
e.g. T799 36h forecast from
Meteosat 9 versus forecasted satellite image
What is the annual global mean cloud cover ?

3. The same for GOES12 versus IFS!
Tropical Cyclone (Gustaf), Tropical continental convection Stratocumulus , Cirrus

4. Overview of Clouds/Convection
ECMWF Training Course
May 2001
Overview of Clouds/Convection
Introduction
Moist thermodynamics
Parametrization of moist convection (4 lectures, Peter),
Theory of moist convection
Common approaches to parametrization including the ECMWF scheme
Cloud Resolving Models (1 lecture - Richard)
Their development and use as parametrization tools
Parametrization of clouds (4 lectures - Richard)
Basic microphysics of clouds
The ECMWF cloud scheme and problem of representing cloud cover
Issues concerning validation
Planetary boundary-layer (4 lectures – Martin+Anton)
-- Surface fluxes, turbulence, mixing and clouds
Exercise Classes (1 afternoon, Peter and Richard)
Clouds 3

5. Moist Thermodynamics Textbooks The great Belgian tradition:
“Thermodynamique de l’atmosphere”
Dufour and v. Mieghem (1975)
Most recent:
Maarten Ambaum (2010) “Thermal physics of the atmosphere”
For simplified Overview:
Rogers and Yau (1989) “A short course in cloud physics”
Thermodynamics and Kinematics:
K. A. Emanuel (1994) “Atmospheric Convection”

6. The First and Second Law
-----
The First Law of Thermodynamics:
Heat is work and work is heat.
Heat is work and work is heat
Very good! The Second Law of Thermodynamics:
Heat cannot of itself pass from one body to a hotter body,
Heat cannot of itself pass from one body to a hotter body
Heat won't pass from a cooler to a hotter,
Heat won't pass from a cooler to a hotter
You can try it if you like but you'd far better notter,
You can try it if you like but you'd far better notter
'Cos the cold in the cooler will get hotter as a ruler,
'Cos the cold in the cooler will get hotter as a ruler
'Cos the hotter body's heat will pass to the cooler,
'Cos the hotter body's heat will pass to the cooler
Good, First Law:
Heat is work and work is heat and work is heat and heat is work
Heat will pass by conduction,
Heat will pass by conduction
And heat will pass by convection,
Heat will pass by convection
And heat will pass by radiation,
Heat will pass by radiation
And that's a physical law.
Heat is work and work's a curse,
And all the heat in the universe,
Is gonna cooooool down
'Cos it can't increase,
Then there'll be no more work
And there'll be perfect peace
Really?
Yeah, that's entropy, maan!

7. The First and Second Law
And its all because of the Second Law of Thermodynamics, which lays down:
That you can't pass heat from a cooler to a hotter,
Try it if you like but you far better notter,
'Cos the cold in the cooler will get hotter as a ruler,
'Cos the hotter body's heat will pass to the cooler.
Oh you can't pass heat, cooler to a hotter,
Try it if you like but you'll only look a fooler
'Cos the cold in the cooler will get hotter as a ruler
And that's a physical Law!
Oh, I'm hot!
Hot? That's because you've been working!
Oh, Beatles - nothing!
That's the First and Second Law of Thermodynamics!
Authors: M. Flanders ( ) & D. Swann ( )
From "At the Drop of Another Hat“

8. Moist Thermodynamics Ideal Gas Law
What is definition of ideal gas?
Assume that “moist air” can be treated as mixture of two ideal gases: “dry air” + vapour
Density
Dry air equation of state:
Temperature
Gas Constant for
dry air = 287 J Kg-1 K-1
Pressure
Water Vapour equation of state:
Vapour
pressure
vapour
density
Gas constant for
Vapour = 461 J Kg-1 K-1
0.622

9. Pressure, partial pressures and gas law for moist air
Partial pressures add if both gases occupy same volume V. Nx are the mol masses

10. First law of thermodynamics
ECMWF Training Course
May 2001
First law of thermodynamics
Energy conservation and Heat
All quantities are per unit mass (specific)
dQ is not a perfect differential, but ds (change in entropie is !)
Change in internal Energy
Heat supplied by diabatic process
Work done
by Gas
Specific Heat at
constant volume
Can write as
Clouds 3

11. First law of thermodynamics: Enthalpy and Legendre transformation
ECMWF Training Course
May 2001
First law of thermodynamics: Enthalpy and Legendre transformation
Changing variables
Special processes: “Adiabatic Process” dQ=0 or better ds=0
Special significance since many atmospheric motions can be approximated as adiabatic
Clouds 3

12. Enthalpy and flow process
ECMWF Training Course
May 2001
Enthalpy and flow process
velocity
In isentropic (adiabatic) flow
Clouds 3

13. Summary :Potentials and Maxwell relations
ECMWF Training Course
May 2001
Summary :Potentials and Maxwell relations
Internal Energy Enthalpy
Helmholtz free Energy Gibbs free Energy
Clouds 3

14. ECMWF Training Course
May 2001
Conserved Variables
Using enthalpy equation and integrating, obtain Poisson’s equation
What is the speed of sound?
Setting reference pressure to 1000hPa gives the definition of potential temperature for dry air
Conserved in dry adiabatic motions, e.g. boundary layer turbulence
Clouds 3

15. ECMWF Training Course
May 2001
Humidity variables There are a number of common ways to describe vapour content etc e
1. Vapour Pressure
2. Absolute humidity
3. Specific humidity
Mass of water vapour per unit moist air
4. Mixing ratio
Mass of water vapour per unit dry air
5. Relative humidity
6. Specific liquid water content
7. Total water content
Clouds 3

16. ECMWF Training Course
May 2001
Humidity variables How to define “moist quantities” and how to switch from mixing ratio to specific humidity.
For any intensive quantity we have
Clouds 3

17. Virtual temperature Tv
It describes the temperature required for dry air, in order to have at the same pressure the same density as a sample of moist air
Another way to describe the vapour content is the virtual temperature , an artificial temperature.
Definition:
By extension, we define the virtual potential temperature, which is a conserved variable in unsaturated ascent, and related to density

18. The Clausius-Clapeyron equation
ECMWF Training Course
May 2001
The Clausius-Clapeyron equation
water
air+water vapour
Consider this closed system in equilibrium: T equal for water & air, no net evaporation or condensation
Air is said to be saturated
For the phase change between water and water vapour the equilibrium pressure (often called saturation water vapour pressure) is a function of temperature only
with av >> aw, and the ideal gas law av=RvT/es
Clouds 3

19. The Clausius-Clapeyron equation - Integration
ECMWF Training Course
May 2001
The Clausius-Clapeyron equation - Integration
The problem of integrating the Clausius-Clapeyron equation lies in the temperature dependence of Lv.
Fortunately this dependence is only weak, so that approximate formulae can be derived.
Nonlinearity has consequences for mixing in convection
es0 = 6.11 hPa at T0=273 K
Clouds 3

20. Meteorological energy diagrams
Total heat added in cyclic process:
rotate to have pressure (almost) horizontal
Thus diagram with ordinates T versus ln q will have the properties of
“equal areas”=“equal energy”
Called a TEPHIGRAM
Pressure T
Dry adiabatic motion q

21. Tephigram (II) T q r Saturation specific humidity
Saturation mixing ratio
pressure
Function of temperature and pressure only – tephigrams have isopleths of rs

22. plot a atmospheric sounding
Using a Tephigram
At a pressure of
950 hPa
Measure
T=20 oC
r=10 gkg-1
plot a atmospheric sounding

23. Enthalpy and phase change
ECMWF Training Course
May 2001
Enthalpy and phase change
Have we been a bit negligeant ? Yes, more precisely
Divide by md (or md+mv) and assume adiabatic process
Clouds 3

24. Ways of reaching saturation
Several ways to reach saturation:
Diabatic Cooling (e.g. Radiation)
Evaporation (e.g. of precipitation)
Expansion (e.g. ascent/descent)
All of these are important cloud processes!!!
Cooling: Dew point temperature Td
Temperature to which air must be cooled to reach saturation, with p and r held constant
Evaporation: Wet-bulb temperature Tw
Temperature to which air may be cooled by evaporation of water into it until saturation is reached, at constant p
Will show how to determine graphically from tephigram

25. Ways of reaching saturation: Expansion: (Pseudo) Adiabatic Processes
As (unsaturated) moist air expands (e.g. through vertical motion), cools adiabatically conserving q.
Eventually saturation pressure is reached, T,p are known as the “isentropic condensation temperature and pressure”, respectively. The level is also known as the “Lifting Condensation Level”.
If expansion continues, condensation will occur (assuming that liquid water condenses efficiently and no super saturation can persist), thus the temperature will decrease at a slower rate.

26. Ways of reaching saturation: Expansion: (Pseudo) Adiabatic Processes
Have to make a decision concerning the condensed water.
Does it falls out instantly or does it remain in the parcel? If it remains, the heat capacity should be accounted for, and it will have an effect on parcel buoyancy
Once the freezing point is reached, are ice processes taken into account? (complex)
These are issues concerning microphysics, and dynamics. The air parcel history will depend on the situation. We take the simplest case: all condensate instantly lost as precipitation, known as “Pseudo adiabatic process”
Pseudo
adiabatic
process

27. Remember: Involves an arbitrary “cloud model”
Tephigram (III) T q r
Pseudoadiabat
(or moist adiabat)
Remember: Involves an arbitrary “cloud model”
pressure

28. parcel mixing ratio=5g/kg
Expansion, (adiabatic process) gives condensation temperature
Cooling: (Isobaric process) gives dew point temperature

29. Moist Adiabat
Wet Bulb
Temperature

30. qe Equivalent Potential temperature Te (=315K)
Raise parcel pseudoadiabatically until all humidity condenses and then descend dry adiabatically to reference pressure
Equivalent Potential
temperature
conserved in adiabatic motions
Parcel at 850 hPa, T=12.5oC r=6 g/kg Te qe
(=315K)

31. Summary: Conserved Variables
ECMWF Training Course
May 2001
Summary: Conserved Variables
Dry adiabatic processes
Moist adiabatic processes
potential temperature
equivalent potential temperature
In HYDROSTATIC ATMOSPHERE: dry static energy
moist static energy
Specific humidity
total water specific humidity
Clouds 3

32. Last not least: how to compute numerically saturation (adjustment)
ECMWF Training Course
May 2001
Last not least: how to compute numerically saturation (adjustment)
T,qv
T*,qvs(T*) qv
given T, q
check if q > qs(T) then
solve for adjusted T*,q* so that
q* = qs(T*)
ql = q-qs(T*)
using cp dT = -Lv dqs
either numerically through iteration or with the aid of a linearisation of qs(T*) (see Excercises !!)
Clouds 3

33. ECMWF Training Course
May 2001
A few Unofficial Social Tips… see also For Music listings: pick up “Bla Bla”
Gulshan Indian Restaurant
Jazz club – live music on Thurs
£7 (incl 2 drinks)
“Thai corner”
Purple Turtle, open until 3am
Station
If the Weather is nice, don’t forget to take a walk by Thames, off the top of this map, or take a train to Pangbourne or Goring nearby and see the Thames there
For London theatre, check out OFFICIAL half-price ticket booth in Leicester Square
ARTS cinema at Reading University, Tues/Thurs, see readingfilmtheatre.co.uk
“Beijing noodle house”
Abbey Ruins, Reading’s (only) historical part!
Revolution
Iguana: Coffee+cocktails (upstairs)
Zero Degrees:Best Pizza and brewery
“Sweenie and Todds” Pie Pub
Mango
Wagamamas (Oracle by canal), Asian noodle Chain
Gospoda-Polish Pub, Oxford Street Karaoke on Thursdays
“RISK”: Salsa lessons £5/9, free dancing after 21:30 Tuesday (upstairs) International drinks
Clouds 3