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Numerical Weather PredictionNumerical Weather Prediction
Moist ThermodynamicsMoist Thermodynamics
Peter Bechtold and Adrian Tompkins
Can we represent clouds in a GCM Can we represent clouds in a GCM ieie. Moisture transport and phase changes ?. Moisture transport and phase changes ?
e.g. T799 36h forecast from 20080525
Meteosat 9 versus forecasted satellite image
What is the annual global mean cloud cover ?
The same for GOES12 versus IFSThe same for GOES12 versus IFS! !
Tropical Cyclone (Gustaf), Tropical continental convection Stratocumulus , Cirrus
Overview of Overview of Clouds/ConvectionClouds/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)
Moist ThermodynamicsMoist Thermodynamics
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”
Textbooks
The First and Second LawThe 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 radiationAnd 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 workAnd there'll be perfect peace
Really?Yeah, that's entropy, maan!
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 rulerAnd 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!
The First and Second LawThe First and Second Law
Authors: M. Flanders (1922-1975) & D. Swann (1923-1994)From "At the Drop of Another Hat“
Moist ThermodynamicsMoist ThermodynamicsIdeal Gas LawIdeal Gas Law
• Assume that “moist air” can be treated as mixture of two ideal gases: “dry air” + vapour
TRp ddd Dry air
equation of state:
Gas Constant fordry air = 287 J Kg-1 K-1Pressure
Density
TTRe dRvvv
Water Vapour
equation of state:
Vapourpressure
vapourdensity
Gas constant forVapour = 461 J Kg-1 K-1
0.622
Temperature
What is definition of ideal gas?
Pressure, partial pressures and Pressure, partial pressures and gas law for moist airgas law for moist air
; ;
( )
d d d v
d
d
d d dd d v v
v v
p p e p pN e p N
dpdp de
p p e
p V m R TpV T m R m R
eV m R T
Partial pressures add if both gases occupy
same volume V. Nx are the mol masses
First law of thermodynamicsFirst law of thermodynamics
Heat supplied by diabatic process
Change in internal Energy
Work doneby Gas
1; ;
Vde dQ dw dQ pd dQ Tds
m
Energy conservation and Heat
All quantities are per unit mass (specific)dQ is not a perfect differential, but ds (change in entropie is !)
Can write as
vde c dT dQ pd 5
2v d
e Qc R
T T
Specific Heat atconstant volume
First law of thermodynamics: First law of thermodynamics: Enthalpy Enthalpy and Legendre and Legendre transformationtransformation
Changing variables
( ) pdh d e p c dT dQ dp
p v d
p
h Qc c R
T TSpecial processes: “Adiabatic Process” dQ=0 or better ds=0
ln ln
p
p d
c dT dp
c d T R d P
Special significance since many atmospheric motions can be approximated as adiabatic
( )pd d p dp
Enthalpy and flow processEnthalpy and flow process
1
dUp p
dt
dUh T s h
dt
In isentropic (adiabatic) flow
velocity
Summary :Potentials and Maxwell Summary :Potentials and Maxwell relationsrelations
s ps
T pT
de Tds pd dh Tds dp
T p T
s p s
df sdT pd dg sdT dp
s p s
T p T
Internal Energy Enthalpy
Helmholtz free Energy Gibbs free Energy
Conserved VariablesConserved Variables
Using enthalpy equation and
integrating, obtain Poisson’s equation
0 0
0 0
d p
p v
R C
C C
T por
T p
p
p
Setting reference pressure to 1000hPa gives the definition of potential
temperature for dry airpd CR
p
pT
0
Conserved in dry adiabatic motions, e.g. boundary layer turbulence
What is the speed of sound?
4. Mixing ratio
Humidity variablesHumidity variablesThere are a number of common ways to describe There are a number of common ways to describe vapour content etcvapour content etc
1 kgkg
vv
m
V
t lq q q
1. Vapour Pressure
2. Absolute humidity 3 mkg
3. Specific humidityMass of water vapour per unit moist air
(1 )v v
d v
m e eq
m m p e p
1 kgkg
5. Relative humidity (or )s s
e qRH
e q
v v
d d
m e er
m p e p
1 kgkg
Pa
Mass of water vapour per unit dry air
6. Specific liquid water contentl
lq 1 kgkg
7. Total water content
e / 0.622d vR R
Humidity variablesHumidity variablesHow to define “moist quantities” and how to switch How to define “moist quantities” and how to switch from mixing ratio to specific humidity.from mixing ratio to specific humidity.
(1 )
1
1
v dd v v v d d v d
d v d v
v d
d v d
m mm m m m
m m m m
q q
or dividing by m rr
1 1
r qq r
r q
For any intensive quantity we have
Virtual temperatureVirtual temperature TTvv
d v
P P
R T RT
Another way to describe the vapour content is the virtual temperature , an artificial temperature.
1 (1 )1 1 (1 0.608 )
(1 )v
rT T q T T q
r
By extension, we define the virtual potential temperature, which is a conserved variable in unsaturated ascent, and related to density
0
d pR c
v v
pT
p
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
Definition:1(1 ) (1 )v d dR q R q R R q
The Clausius-Clapeyron The Clausius-Clapeyron equationequation
)(
1
wv
vs L
TdT
de
• For the phase change between water and water vapour the equilibrium pressure (often called saturation water vapour pressure) is a function of temperature only
water
air+water vapourConsider this closed system in equilibrium:T equal for water & air, no net evaporation
or condensationAir is said to be saturated
2TR
eL
dT
de
v
svs •with v >> w, and the ideal gas law v=RvT/es
The Clausius-Clapeyron The Clausius-Clapeyron equation - Integration 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.
TTR
L
e
e
v
v
s
s 11ln
00
Nonlinearity has consequences for
mixing in convection
es0 = 6.11 hPa at T0=273 K
Meteorological energy Meteorological energy diagramsdiagrams
Total heat added in cyclic process:
(ln )d
p
R dpdTp pT C pdQ c T c Td
Thus diagram with ordinates T versus ln will have the properties
of “equal areas”=“equal energy”
Called a TEPHIGRAM
rota
te to h
ave p
ressure (alm
os
t) ho
rizon
tal
Dry adiabatic motion
Pressure
T
T
Tephigram (II)Tephigram (II)
pressure
r
Saturation specific
humidity
ss
s
er
p e
(1 )s
ss
eq
p e
Saturation mixing ratio
Function of temperature and pressure only – tephigrams
have isopleths of rs
Using a Using a TephigramTephigramAt a pressure of
950 hPa
Measure T=20 oC r=10 gkg-1
plot a atmospheric sounding
Enthalpy and phase changeEnthalpy and phase change
p sdh c dT Tds dp Ldq
Have we been a bit negligeant ? Yes, more precisely
( ) ( ) ( )
( ) ( )
d pd v pv d v d d v v v
pd pv d v s
dpm c m c dT T dS dS m R m R T Ldm
p
dpc rc dT R rR T Ldr
p
Divide by md (or md+mv) and assume adiabatic process
Ways of reaching saturationWays of reaching saturation
Several ways to reach saturation:
• Diabatic Cooling (e.g. Radiation)
• Evaporation (e.g. of precipitation)
• Expansion (e.g. ascent/descent)
Cooling: Dew point temperature TdTemperature to which air must be cooled to reach saturation, with p
and r held constant
All of these are important cloud
processes!!!
Evaporation: Wet-bulb temperature TwTemperature to which air
may be cooled by evaporation of water into
it until saturation is reached, at constant p
( ( ) )p v s v sc dT L dq L q T q iteration
Will show how to determine graphically from tephigram
Ways of reaching saturation:Ways of reaching saturation:Expansion: (Pseudo) Adiabatic Expansion: (Pseudo) Adiabatic ProcessesProcesses
As (unsaturated) moist air expands (e.g. through vertical motion), cools adiabatically conserving .
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.
Ways of reaching saturation:Ways of reaching saturation:Expansion: (Pseudo) Adiabatic Expansion: (Pseudo) Adiabatic ProcessesProcesses
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
T
Tephigram Tephigram (III)(III)
pressure
r
Pseudoadiabat
(or moist adiabat)
Remember: Involves an arbitrary “cloud
model”
Cooling: (Isobaric process)
gives dew point
temperature
parcel mixing ratio=5g/kg
Expansion, (adiabatic
process) gives condensation temperature
Wet BulbTemperature
Moist Adiabat
Parcel at 850 hPa, T=12.5oCr=6 g/kg
Te
Raise parcel pseudoadiabatically
until all humidity condenses and then
descend dry adiabatically to
reference pressure
e (=315K)
Equivalent Potentialtemperature
Tc
rL
ep
vv
econserved in adiabatic motions
Summary: Conserved Summary: Conserved VariablesVariables
Dry adiabatic processes Moist adiabatic processes
p
dc
R
p
pT
0
potential temperature
Tc
rL
ep
vv
e
equivalent potential temperature
p v vh c T gz L q moist static energy
t v lq q q total water specific humidity
; :
0
p
p
p
s c T gz proof
dh c dT dp g dz
ds c dT gdz
In HYDROSTATIC ATMOSPHERE: dry static energy
vq
Specific humidity
Last not least: how to compute Last not least: how to compute numerically saturation numerically saturation (adjustment)(adjustment)
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 !!)
* *( ) ( ( ) ( ) (2) )sp v s
T
dqc T T L q q T T T O
dT
T,qv
T*,qvs(T*)
qv
A few Unofficial Social Tips…A few Unofficial Social Tips…see also www.reading-guide.co.uk For Music listings: pick up “Bla see also www.reading-guide.co.uk For Music listings: pick up “Bla Bla”Bla”
Station
“Sw
eeni
e an
d T
odds
” P
ie P
ub
“RISK”: Salsa 19.30 lessons £5/9, free dancing after 21:30 Tuesday (upstairs) International drinks
“Thai corner”
Igua
na: C
offe
e+co
ckta
ils (
upst
airs
)
“Bei
jing
nood
le h
ouse
”
Purple Turtle, open until 3am
Jazz club – live music on Thurs £7 (incl 2 drinks)
Zero D
egrees:Best P
izza and brewery
Gulshan Indian Restaurant
Wagamamas (Oracle by canal), Asian noodle Chain
Abbey Ruins, Reading’s (only) historical part!
AR
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cin
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at R
ead
ing
Uni
vers
ity,
Tue
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e re
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mth
eatr
e.co
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For
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the
atre
, ch
eck
out
OF
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IAL
ha
lf-pr
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ticke
t b
ooth
in L
eice
ster
Squ
are
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
Gospoda-Polish Pub, Oxford Street Karaoke on Thursdays
Mango
Revolution