and p - Numericable

On the use of different forms of 

Available Energies 

or Exergy 

in Meteorology 

University of Reading, 

13th of February, 2012 

Pascal MARQUET 

Météo-France. Laboratory of Forecasting

Motivity 

W. Thomson / Lord 

Kelvin (1853) 

F( X ) X ln( 1 

X ) X 

2 

/ 2 

Available Energies 

and Exergy? 

Edward N. Lorenz (1955) 

APE cycle 

J. W. Gibbs (1873) 

E 

tot 

 

E 

Stot tot 

tot 

1 

 

T 

Stot 

0 

Available 

Energy 

S 

 

W 

max 

Ther. diagrams 

Total Entropy 

J. C. Maxwell (1871) 

Available 

Energy

Contents 

1) Introduction: Energetics = quadratic functions!? 

2) Margules / Lorenz : Available (kinetic) Potential Energies… 

APE (Lorenz) Cycle… Limitations? 

3) Exergy functions: (Thermodynamics) Carnot (1823), Thomson-Kelvin 

(1853-79), Maxwell (1871), Gibbs (1873), EXERGY Rant (1956) / Bejan 

(1987) … (Atmosphere) Normand (1946), Keenan (1956), Dutton (73,76), 

Pichler (77), Pearce (78), Blackburn (83), Karlsson (90), Marquet (90,91,93,…), 

Sheperd (95), Kucharski (97), Fortack (98), Bannon (2005), Tailleux (09), … 

ex: A h versus APE Cycles? 

4) Summary / Outlook : Moist Entropy & Enthalpy & 3D-Diagrams 

/ CAPE / N.H. / … 

3

Contents 









ex: A h versus APE Cycles? 


/ CAPE / N.H. / … 

4

Energetics: quadratic functions! 

Special 

2 1 v 

E E 

1 

Relativity : 0 m c 

 

m 

2 2 

 

1 

v / c 2 

Pendulum : 

A + ekin constant (if no dissipation) 

A E 

p 

m g R 

E p 0 

z z 

A m g 

 

m 

g 

R 

0 

5 

2 

1 cos 

 

2 

 

/ 2

Energetics: quadratic functions? 

Why is it important? 

It allows the study of the cascade of energy: mean eddies 

ekin emean 

eeddies 

 

m 

with 

( v) 

2 

h cp 

T cp 

( T T 

') 

h cp 

v 

2 

 

m 

( v') 

 

Compressible fluids / Atmosphere / Ocean : enthalpy ? 

no eddy term, because 

T 

( T') 

 

( v') 

2 

0 

0 

2 

only linear in T ! 

6

Contents 









ex: Ah versus APE Cycles? 


/ CAPE / N.H. / … 

7

Max Margules 

(1856–1920) 

MARGULES 

Available 

Kinetic 

Energy 

Equivalent 

Work 

LORENZ 

Available 

Potential 

Energy 

Edward Norton Lorenz 

(1917–2008) 

8

Contents 











/ CAPE / N.H. / … 

9

Max MARGULES (1901-1903) 

Equivalent Work? 

10

X 

p 

p 

0 p0 

H( 

X ) 

p0 


Differences in pressure 

Useful Work? 

1 X ln1 

X 

H( X ) 

X 

11 

X 

2 

2


Compute the maximum 

Available Kinetic Energy 

among all possible 

reordering of a column? 

Differences in temperature and pressure 

Kinetic Energy (Storms) 

12


Available Kinetic Energy? 

Air initially at rest, but in disequilibrium… 

Air start to move, it tends to its equilibrium position 

Equilibrium position: horizontal layers are isobaric, 

with entropy (potential temperature) increasing with 

height, and with a Maximum of kinetic energy 

Trajectory toward the equilibrium position: adiabatic 

(isentropic) motions, conservation of the mass 

E p Ei 

E 

init p Ei 

end 

K 

 

max 

13

E 

i 

psfc 

p0 Max MARGULES (1901-1903) 

Available Kinetic Energy? 

E p Ei 

E 

init p Ei 

end 

K 

 

max 

( c T) 

dp / g E ( g z) 

dp / g 

Dry air + 

Integration 

by parts + 

Hydrosstatic 

E 

p 

v 

E 

p 

 

c R 

v 

psfc 

Ei 

p 0 

p 

psfc 

p0 p p 

p0 

p0 

sfc sfc 

z p ( RT 

) dp / g 

( c 

c 

p 

p 

T) 

dp 

g 

[ z p ] 0? 

 

H 

H: Total heat? 

Enthalpy? 

H K 

max 

init 

H 

end 

14

Sir Charles Normand (1946) 

1) Account of Margules’s work 

2) Promote: Total heat / Enthalpy 

15

Contents 











/ CAPE / N.H. / … 

16

Meteorology: APE = « Available Potential Energy » 

Margules-Lorenz isentropic redistribution of mass 

1 

2 

1 

2 1 

1 

The actual state The reference state 

APE H H 0 with 

ref 

1 

E APE 

k 

Edward N. Lorenz 

(Tellus, 1955) 

C 

ste 

Is the APE 

quadratic? 

17

( 

p) 

 

Meteorology: APE = « Available Potential Energy » 

g p 

R 

G C D 

APE 

APE 

R 

 

 

cp 

V 

( T T 

) 

2 

T ( p) 

T 

( p) 

 

 

p p 

 

K 

Edward N. Lorenz 

(W.M.O., 1967) 

Yes APE is quadratic! 

2 

 

dxdy 

dp 

 

g T 

R 

APE / K :10 5 J m -2 

G / C / D : W m -2 

OK, APE quadratic: but mean/eddies? 

 

0 

ln( 

 

) 

ln( 

p) 

 

 

g T 

R c 

p 

s 

ln( p) 

18

APE 

A = A Z + A E 

A Z 

A E 

V 

Meteorology: APE - E.N. LORENZ 

APE quadratic: but mean/eddies? 

( T T 

) 

2 

2 

dxdy 

dp 

K = K Z + K E 

K Z 

K E 

 

0 

T T T 

' 

u u 

v v 

u' 

v' 

19

A Z 

A E 

K Z 

K E 

Meteorology: APE - E.N. LORENZ 

G Z 

C A 

G E 

A Z 

A E 

APE / K :10 5 J m -2 

G / C / D : W m -2 

C E 

Baroclinic path of energy! 

C Z 

K Z 

K E 

D Z 

C K 

D E 

20

Meteorology = problems with the APE? 

Pb1: Enthalpy (c p T) different from "c v T + g z" except 

(and only) for Global applications / What about local ones? 

Pb2: the conversion term is 

/ What about 0 for local applications? 

R T / p R T / p R 'T' 

/ 

Pb3: APE division by or 

/ p 

/ What about ? (PBL) 

Pb4: APE = a dry-air approach / What about a moist version of it? 

Pb5: APE = several approximations (TOM) / Impacts? 

Question: is there other more exact 

and local definitions for 

Available energy? 

0 

Answer : YES EXERGY 

(1988-1990 research … PhD 1994) 

21 

p

Contents 




3) Exergy functions: (Thermodynamics) Carnot (1823), Thomson- 

Kelvin (1853-79), Maxwell (1871), Gibbs (1873), EXERGY Rant (1956) / 

Bejan (1987) … (Atmosphere) Normand (1946), Keenan (1956), Dutton 

(73,76), Pichler (77), Pearce (78), Blackburn (83), Karlsson (90), Marquet 

(90,91,93,…), Sheperd (95), Kucharski (97), Fortack (98), Bannon (2005), 

Tailleux (09), … ex: A h versus APE Cycles? 


/ CAPE / N.H. / … 

22

(1823) 

Sadi CARNOT: « puissance motrice 

du feu » = Available Energy / 1823 

The “Available Energy” is universal. 

It is independent of the actual 

machine or apparatus which deliver it. 

It only depends on the 

difference in temperature (and p, ….) 

23

W. Thomson / Lord Kelvin 

W. Thomson (Trans. Roy. Soc. Edinb. - Vol 16 - 1849) 

Account of Sadi Carnot’s work 

+ corresponding computations 

24

W. Thomson / Lord Kelvin : « the work available » 

W. Thomson (Phil. Mag., Feb. 1853) 

25



T 

T 

t 

T 

26


Modern notations : 


t T 0 T T 0 J 1 

W cp 

W p 

c T0 

 

( T T 

) T 

logT 

T dV 

0 0 

0 

 

FX 

dV 

( T T T X 

2 

X X ln( 1 

X ) X / 2 

F 

F(X) quadratic! 

0 ) 

0 

T 0 

T 0 

T 

T 0 

27

W. Thomson / Lord Kelvin (1879) : « Motivity » 

P. G. Tait W. Thomson (Phil. Mag., 1879) 

W p 

X dV 

c T0 

F 

0 0 ) ( T T T X 

 

28

Maxwell / Gouy / Stodola 

England: James Clerk MAXWELL 

"Available Energy" (The Heat,1871) 

Germany: Aurel Boleslav 

STODOLA "Maximal Available 

Work" (1898, 1903) 

(Enthalpy and Entropy) 

France: Louis Georges GOUY 

"Available Energy" (C.R.A.S. + 

J. Ph.,1889) 

Wmax 

( U U0 

) T0 

( S S0) 

W H T 

S 

max 

0 

29

J. W. Gibbs (1873) 

USA: Josiah Willard GIBBS "Available Energy" (1873) 

30

Etot 

( T0 

) 

Stot 

J. W. Gibbs (1873) 

 

E 

Stot tot 

tot 

1 

 

T 

0 

S 

 

W 

max 

A W T S 

max 0 

tot 

31

A cp 

T 

0 

( T T T X 

0 ) 

F( 

X ) 

2 

X X ln( 1 

X ) X / 2 

F 

T 0 

T 0 

T 

T 0 

0 

Q 

Kelvin / Maxwell / Gibbs ? 

Wmax 

( U U0 

) T0 

( S S0) 

T 0 

They seem different? 

Thermostat 

(reservoir) 

A W T 

max 0 

In fact: equivalent! 

32 

Stot 

For instance, at constant pressure: 

Q cp 

dT 

U c T X 

 

U 0 p 0 

S S cp 

T ln( 1 

X ) 

0 

Stot cp 

 

0 

F(X 

)

Contents 




3) Exergy functions: (Thermodynamics) Carnot (1823), Thomson- 

Kelvin (1853-79), Maxwell (1871), Gibbs (1873), EXERGY Rant (1956) / 

Bejan (1987) … (Atmosphere) Normand (1946), Keenan (1956), Dutton 

(73,76), Pichler (77), Pearce (78), Blackburn (83), Karlsson (90), Marquet 




/ CAPE / N.H. / … 

33

Zoran RANT (1956) 

34

Adrian BEJAN (1987) 

But this pressure term? 

2 

X X ln( 1 

X ) X / 2 

F 

( T T T X 

0 ) 

0 

35

Contents 







Pichler (77), Pearce (78), Blackburn (83), Karlsson (90), Marquet 




/ CAPE / N.H. / … 

36

John DUTTON (1973, 1976) 

Conservations of 

Mass and Energy 

Gibbs 

it is just rearranged and 

divided by T 0 

37

Robert P. PEARCE (1978) 

d 

dt 

H H ) T 

( S S ) 

( 0 0 0 

Available Enthalpy 

 

(Enthalpy) 

dH 

dt 

T 

0 

dS 

dt 

38

Let us define locally: 

ah ( h hr 

) Tr 

( s sr 

) 

ah T p 

( T, 

p) 

a ( T) 

a ( p) 

aT ( T) 

cp 

Tr 

P. Marquet (CRAS 1990, QJ 1991) 

FX 

 

 

( p) 

R Tr 

p X 

p 

ap r 

H ( ) 


T 

X ( T T 

) Thomson (1853) 

r Tr 

2 

X X ln( 1 

X ) X / 2 

F 

p 

X ( p p ) 

r 

r 

( T / T 1) 

T T 

a ( T) 

c T log 

a ( p) 

RT 

log 

p 

r 

p 

1 X ln1 

X 

With a “dead state”: 

H( X ) 

X 

r 

C T 

r 

r 

p p 

r 

ste 

C p 

Margules (1901) 

r 

X 

2 

39 

ste 

2 

r


ah ( h hr 

) Tr 

( s sr 

) 

da r 

h R 

T T 

1 

q 

dt p T 

dek R 

T 

B() 

d 

dt p 

d 

 

dt 

 

B() 

t 



dt 

d 

dt 

h e 

q 

d 

k 

With a “dead state”: 

C T 

A local Bernoulli equation: 

d r 

h k 

 

t 

 

t 

T 

 

T 

a e 

1 q 

d 

Carnot’s factor Differential heating 

r 

ste 

C p 

r 

 

 

 

40 

ste


ah ( h hr 

) Tr 

( s sr 

) 



Average values: / 1/ 

T 

leading to: 

T r 

The “dead state” ? 

1 and log p r log( p) 

T r 

 

251 

K 

or possibly: 273 K 

Pearce (1978): 

T r 

/ 

/ 

T 

p r 

p r 

r 

 

C 

ste 

367 

1000 

C p 

r 

hPa 

hPa 

41 

ste


ah ( h hr 

) Tr 

( s sr 

) 

F X X ln( 1 

X 

X T / T 1 

S 

X B 

) 

r 

T 

/ T 1 

T '/ 

T 

Pearce (1978) 



T T'( 

x, 

y, 

p) 

T 

( p) 

 

Tr 

X FX 

S X B X S X B 

X FX 

S FX 

B X S X B 

X FX 

FX 

X 

F 

F 

The “dead state” ? 

T 

constant values… 

A lack of generality? 

Tr 

F S B ; B 0 

self-similarity + another one for H X … 

r 

/ 

C 

ste 

C p 

r 

 

42 

ste

an Available Enthalpy 

“a h” Cycle? 

P. Marquet (PhD 1994, QJ 2003ab) 

: the “a h” Cycle? 

43


the “a h” Cycle: huge amount of computations… but exact! 

General properties… 

44



Start of the computations for “a Z” … 

45



computations 

for “a Z” continue… 

46



+ other computations 

for “a S” “a E” 

and “k S” “k Z”“k E” … 

end of the computations for “a Z” only … 

47

P. Marquet (QJ 2003a) 

the “a h” cycles for a pressure level 

of a limited area domain (with !) 

the global “a h” cycles: 

48

Global 

P. Marquet (QJ 2003b) 

Global 

Mid-tropo 

Thorncroft and Hoskins (1990) 

with the Arpege model 

Strato 

PBL 

49

P. Marquet (QJ 1995) 

Non-canonical Hamiltonian systems 

Casimir invariants 

T 

s ? 

ah ( h hr 

) Tr 

( s sr 

) 

K r 

“Pseudo-activity” Shepherd (1993) 

a Link with 

Lorenz’ APE… 

50

P. Marquet (QJ 1995) 

Non-canonical Hamiltonian systems 

Casimir invariants 

ah ( h hr 

) Tr 

( s sr 

) 

K 

Tr 

Available 

Enthalpy 

“Pseudo-activity” Shepherd (1993) 

s? 

X min 

0 X 

Etot 

H0 

H 

( T0 

) 

Gibbs (1873) 

Stot 

51 

X 

 

 

K 

K0

Contents 








Sheperd (95), Kucharski (97), Fortack (98), Bannon (2005), Tailleux (09), … 


4) Summary / Outlook: Moist Entropy & Enthalpy & 3D- 

Diagrams / CAPE / N.H. / … 

52

Summary / Outlook 

http://perso.numericable.fr/~pmarquet/ 

pascal.marquet@meteo.fr 

1) APE in Meteorology (Margules-Lorenz) … but also Dutton, 

Pearce, … (flowing / non-flow) Exergy in Thermodynamics: 

no need to seek for the “best” or the “right” one / different aspects 

2) Let us use Enthalpy for itself, with “g z” having is own impact 

on (local) atmospheric energetic (do not mix with internal energy). 

3) No need to use “adiabatic redistribution of the mass” 

4) A possible exact and local Available Enthalpy “a h” Cycle. 

5) F(X) and H(X) + separating properties F(X) = F(X 1 ) + F(X 2 ) 

6) Other Exergy approaches: Fred Kucharski (U.R. with A. Thorpe) 

/ Sten Karlsson (Sweeden) Kullback information i p i ln ( p i / p i0 ) 

7) Next related studies: Moist Exergy “a m” (1993) / Absolute moist 

Entropy (2011) & Enthalpy + CAPE + BVF (2012) + PV ... 

53

Motivity 

W. Thomson / Lord 

Kelvin (1853) 

F( X ) X ln( 1 

X ) X 

2 

/ 2 

Thanks a lot / Questions? 

http://perso.numericable.fr/~pmarquet/ 

pascal.marquet@meteo.fr 

APE cycle 

Edward N. Lorenz (1955) 

J. W. Gibbs (1873) 

E 

tot 

 

E 

Stot tot 

tot 

1 

 

T 

Stot 

0 

S 

 

W 

Available 

Energy 

max 

Ther. diagrams 

Total Entropy 

J. C. Maxwell 

(1871) 

Available 

Energy

and p - Numericable

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