G Thermodynamic Machinery of Planet Earth//gaia.mpg.de MPI-BGC Jena Gaia vs. Maximum Entropy Production B I O S P H E R I C T H E O R Y A N D M O D E L L I N G Lovelock: chemical disequilibrium

MPI-BGC

Jenahttp://gaia.mpg.de

BIOSP

HER

IC THEORY AND MO

DELLING

Axel [email protected]

Max-Planck-Institute for BiogeochemistryJena, Germany

Stochastic Methods in Climate Modelling26 August 2010

Life, Hierarchy, and the Thermodynamic Machinery of

Planet Earth

http://gaia.mpg.de

MPI-BGC

Jena

Gaia vs. Maximum Entropy ProductionBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

http://gaia.mpg.de

MPI-BGC

Jena

Gaia vs. Maximum Entropy ProductionBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (µ

mol

)sh

allo

w o

cean

sm

O2 (µ

mol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (

µmol

)sh

allo

w o

cean

sm

O2 (

µmol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

http://gaia.mpg.de

MPI-BGC

Jena

Gaia vs. Maximum Entropy ProductionBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (µ

mol

)sh

allo

w o

cean

sm

O2 (µ

mol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (

µmol

)sh

allo

w o

cean

sm

O2 (

µmol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

Earth: evolution away

from equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Gaia vs. Maximum Entropy ProductionBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (µ

mol

)sh

allo

w o

cean

sm

O2 (µ

mol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (

µmol

)sh

allo

w o

cean

sm

O2 (

µmol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

MEP: fastest approach

to equilibrium

Earth: evolution away

from equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Gaia vs. Maximum Entropy ProductionBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (µ

mol

)sh

allo

w o

cean

sm

O2 (µ

mol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (

µmol

)sh

allo

w o

cean

sm

O2 (

µmol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

MEP: fastest approach

to equilibrium

???

Earth: evolution away

from equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Importance of Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

thermodynamicequilibrium

far fromequilibrium

thermodynamics and variability

http://gaia.mpg.de

MPI-BGC

Jena

Importance of Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

thermodynamicequilibrium

far fromequilibrium

uniform, constantspatial and temporal

variability

thermodynamics and variability

http://gaia.mpg.de

MPI-BGC

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Importance of Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

thermodynamics and variability

power required to maintain disequilibrium

thermodynamicequilibrium

far fromequilibrium

http://gaia.mpg.de

MPI-BGC

Jena BIOSP

HER

IC THEORY AND MO

DELLING

How does the Earth system generate and maintain thermodynamic disequilibrium?

… and why should we care?

http://gaia.mpg.de

MPI-BGC

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OutlineBIO

SPH

ERIC THEORY AND M

ODELLING

1. Background:

• How to maintain disequilibrium?

• How much power can be generated?

2. Earth system: Power generation, transfer, hierarchy and maximization

3. Implications: So what?

http://gaia.mpg.de

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first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining Disequilibrium BIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

dQ = 0 J = k · (Th − Tc)

first law: second law:dSh

dt= − J

Th

dSc

dt=

J

Tc

dStot

dt= J ·

�1

Tc− 1

th

�= σ ≥= 0

c · dTh

dt= −J

c · dTc

dt= J

c ·�dTh

dt+

dTc

dt

�= 0

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining Disequilibrium BIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

dQ = 0

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining Disequilibrium BIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

initial temperaturegradient is dissipated

to a state of thermodynamic

equilibrium

dQ = 0

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining Disequilibrium BIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

JheatdQ �= 0

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining Disequilibrium BIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

JheatdQ �= 0

disequilibrium can bemaintained by entropy

exchange across systemʼs boundary

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

dStot =dU + dW

T! dSheat + dSdiseq

combined:

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

dStot =dU + dW

T! dSheat + dSdiseq

combined:

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

dStot =dU + dW

T! dSheat + dSdiseq

combined:

entropy change due to changes in

heat content

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

dStot =dU + dW

T! dSheat + dSdiseq

combined:

entropy change due to changes in

heat content

entropy change due to work done on/by the system

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

or, using Helmholtz free energy A:

2

dA = dU − TdStot = −dW (1)

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

characterization of disequilibrium:

dSdiseq = !dA

T

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

http://gaia.mpg.de

MPI-BGC

Jena

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Free

Ene

rgy (

109 J

)

a. equilibrium setting b. disequilibrium setting

Time (years)

Entropy (106 J K

-1)

Time (years)

Free

Ene

rgy (

109 J

) Entropy (106 J K

-1)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat

isolated system non-isolated systemFr

ee E

nerg

y (10

9 J)

a. equilibrium setting b. disequilibrium setting

Time (years)

Entropy (106 J K

-1)

Time (years)

Free

Ene

rgy (

109 J

) Entropy (106 J K

-1)

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

external fluxes result in heating gradients

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE extraction of

power from aheat gradient

…

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE extraction of

power from aheat gradient

…

… generates free energyand disequilibrium

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

thermodynamic gradientsare dissipated…

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

thermodynamic gradientsare dissipated…

… resulting inentropy

production…

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

thermodynamic gradientsare dissipated…

… resulting inentropy

production…

… and dissipative

heating

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

entropy production …

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

entropy production …

… is constrained

by net entropy exchange …

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

entropy production …

… is constrained

by net entropy exchange …

… but free energy alters heat fluxes and NEE

http://gaia.mpg.de

MPI-BGC

Jena

2

dA = dU − TdStot = −dW (1)

dA

dt= P − T · dSdiseq

dt(2)

first law:

second law:

Maintaining DisequilibriumBIO

SPH

ERIC THEORY AND M

ODELLING

Helmholtz free energy A:

dU

dt= Jnet ! (P ! D)

dS

dt= ! ! NEE

disequilibrium

entropyproduction

power

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

dU = dQ ! dW

dS = dQ/T ! 0

Jin Jout

Pex = dW/dt

Tin Tout

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

dS = dQ/T ! 0

Jin Jout

Pex = dW/dt

Tin Tout

Jin = Jout + Pex

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

Jin Jout

Pex = dW/dt

Tin Tout

Jin = Jout + Pex

Jin − Pex

Tout− Jin

Tin≥ 0

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

Jin Jout

Tin Tout

Jin = Jout + Pex

Jin − Pex

Tout− Jin

Tin≥ 0

Pex ≤ Jin · Tin − Tout

Tin“Carnot limit”

http://gaia.mpg.de

MPI-BGC

Jena

but in the Earth system…

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLING

Jin Jout

Tin Tout

Pex ≤ Jin · Tin − Tout

Tin“Carnot limit”

1. temperature gradient responds to flux;2. power extraction competes with other irreversible processes (radiative, diffusion)3. free energy is dissipated in steady state

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat“2-boxclimatemodel”

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat

Figure 3: Simple two-box model (as in Fig. 1) to illustrate the maintenance of disequilibrium in a non-isolated systemresulting from entropy exchange at the system boundary. The entropies Sh, Sc and Stot are expressed as a fraction of thevalue representing thermodynamic equilibrium. After [1].

Jin,h and Jin,c in the two boxes at temperatures Th and Tc respectively:399

σmix,h = Jin,h ·�

1Th

− 1Tin

�σmix,c = Jin,c ·

�1Tc

− 1Tin

�(9)

and by the mixing associated with heat flux Jheat from the warm box to the cold box, as before:400

σheat = Jheat

�1Tc

− 1Th

�(10)

The entropy budget of the whole system is given by:401

dStot

dt=

dSh

dt+

dSc

dt= σmix,h +σmix,c +σheat −NEE (11)

where NEE = (Jout,h/Th + Jout,c/Tc)− (Jin,h + Jin,c)/Tin is the net entropy exchange across the402

system boundary.403

What we notice in Fig. 3 is that the initial gradient in temperature is depleted as in the404

previous setup, but it does not vanish in the steady state. Instead, a non-zero heat flux transports405

heat from warm to cold that acts to deplete the temperature gradient that is continuously built up406

by the differential heating Jin,h − Jin,c. This heat flux produces entropy by the depletion of the407

temperature gradient, but instead of increasing the entropy of the system, the produced entropy408

in steady state is exported by the enhanced export of entropy associated with the outgoing heat409

flux Jout,h + Jout,c. The enhanced entropy export results from the overall lower temperature at410

which the total amount of received heat is exported to the surroundings.411

14

σheat = Jheat ·�

1

Tc− 1

Th

�

σheat = Jheat ·Th − Tc

ThTc

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat“2-boxclimatemodel”

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat

Figure 3: Simple two-box model (as in Fig. 1) to illustrate the maintenance of disequilibrium in a non-isolated systemresulting from entropy exchange at the system boundary. The entropies Sh, Sc and Stot are expressed as a fraction of thevalue representing thermodynamic equilibrium. After [1].

Jin,h and Jin,c in the two boxes at temperatures Th and Tc respectively:399

σmix,h = Jin,h ·�

1Th

− 1Tin

�σmix,c = Jin,c ·

�1Tc

− 1Tin

�(9)

and by the mixing associated with heat flux Jheat from the warm box to the cold box, as before:400

σheat = Jheat

�1Tc

− 1Th

�(10)

The entropy budget of the whole system is given by:401

dStot

dt=

dSh

dt+

dSc

dt= σmix,h +σmix,c +σheat −NEE (11)

where NEE = (Jout,h/Th + Jout,c/Tc)− (Jin,h + Jin,c)/Tin is the net entropy exchange across the402

system boundary.403

What we notice in Fig. 3 is that the initial gradient in temperature is depleted as in the404

previous setup, but it does not vanish in the steady state. Instead, a non-zero heat flux transports405

heat from warm to cold that acts to deplete the temperature gradient that is continuously built up406

by the differential heating Jin,h − Jin,c. This heat flux produces entropy by the depletion of the407

temperature gradient, but instead of increasing the entropy of the system, the produced entropy408

in steady state is exported by the enhanced export of entropy associated with the outgoing heat409

flux Jout,h + Jout,c. The enhanced entropy export results from the overall lower temperature at410

which the total amount of received heat is exported to the surroundings.411

14

σheat = Jheat ·�

1

Tc− 1

Th

�

Th − Tc = (∆Jin − 2Jheat)/kb

σheat = Jheat ·Th − Tc

ThTc

http://gaia.mpg.de

MPI-BGC

Jena

first law:

second law:

Limits to Power Generation and TransferBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat“2-boxclimatemodel”

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures

Th

JTc

S

a. 2-box model

c. entropies

Time (years) Time (years)

Figure 2: Simple two-box model to illustrate the evolutionary trend towards a state of maximum entropy of an isolated

system. The entropies Sh, Sc and Stot are expressed as a fraction of the value representing thermodynamic equilibrium.

After [1].

decrease in entropy production σheat in time. Initially, the warm box stores more heat and has382

hence a higher entropy than the cold box. The heat flux Jheat depletes the entropy of the hot383

reservoir and by adding heat to the cold reservoir, entropy is increased in that box. Due to the384

mixing of the heat contents, the increase of entropy in the cold reservoir is greater than the385

reduction of entropy in the hot reservoir. Hence, the overall entropy of the system Stot increases386

with time.387

3.3. Maintaining thermodynamic disequilibrium388

When entropy exchanges are allowed for at the system’s boundary, the system can be main-389

tained away from a state of thermodynamic disequilibrium. Fig. 3 a shows a setup that resembles390

a case with entropy exchanges of both boxes with the surroundings. While this setup is not the391

simplest possible, this setup is chosen because it is used later to describe the differential heating392

of the Earth between the tropics (warm box) and the poles (cold box).393

To account for energy and entropy exchange across the system boundary, we need to alter the394

energy balances of the simple system considered before (eqns. 4 and 5) to:395

c · dTh

dt= Jin,h − Jout,h − Jheat c · dTc

dt= Jin,c − Jout,c + Jheat (7)

and396

dSh

dt=

Jin,h

Tin−

Jout,h

Th− Jheat

Th+σmix,h

dSc

dt=

Jin,c

Tin− Jout,c

Tc+

Jheat

Th+σmix,c +σheat (8)

where the entropy associated with the incoming energy is associated with a characteristic tem-397

perature Tin. In this setup entropy is being produced by the mixing of the incoming energy fluxes398

13

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat

Figure 3: Simple two-box model (as in Fig. 1) to illustrate the maintenance of disequilibrium in a non-isolated systemresulting from entropy exchange at the system boundary. The entropies Sh, Sc and Stot are expressed as a fraction of thevalue representing thermodynamic equilibrium. After [1].

Jin,h and Jin,c in the two boxes at temperatures Th and Tc respectively:399

σmix,h = Jin,h ·�

1Th

− 1Tin

�σmix,c = Jin,c ·

�1Tc

− 1Tin

�(9)

and by the mixing associated with heat flux Jheat from the warm box to the cold box, as before:400

σheat = Jheat

�1Tc

− 1Th

�(10)

The entropy budget of the whole system is given by:401

dStot

dt=

dSh

dt+

dSc

dt= σmix,h +σmix,c +σheat −NEE (11)

where NEE = (Jout,h/Th + Jout,c/Tc)− (Jin,h + Jin,c)/Tin is the net entropy exchange across the402

system boundary.403

What we notice in Fig. 3 is that the initial gradient in temperature is depleted as in the404

previous setup, but it does not vanish in the steady state. Instead, a non-zero heat flux transports405

heat from warm to cold that acts to deplete the temperature gradient that is continuously built up406

by the differential heating Jin,h − Jin,c. This heat flux produces entropy by the depletion of the407

temperature gradient, but instead of increasing the entropy of the system, the produced entropy408

in steady state is exported by the enhanced export of entropy associated with the outgoing heat409

flux Jout,h + Jout,c. The enhanced entropy export results from the overall lower temperature at410

which the total amount of received heat is exported to the surroundings.411

14

σheat = Jheat ·�

1

Tc− 1

Th

�

Th − Tc = (∆Jin − 2Jheat)/kb

σheat = Jheat ·Th − Tc

ThTc

Pex = Jheat ·∆Jin − 2Jheat

kb · Th

Pex ≤ ∆Jin8

· Th,0 − Tc,0

Th,0

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

sensitivity to Jheat

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

sensitivity to Jheat

ηmax =1

8· 313K − 268K

313K= 1.8%

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)

b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

entropy production and power

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)

b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

entropy production and power

maximum power generation rate≈ 2 W m-2

or ≈ 900 TW

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)

b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

entropy production and power

maximum power generation rate≈ 2 W m-2

or ≈ 900 TW

maximum efficiency

2 %

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MPI-BGC

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

Sh

Sc

Stot

0.98

0.99

1.00

1.01

1.02

0 12 24 36 48

Entro

py (f

rac.

max.)Th

Tc

-10

0

10

20

30

40

Heat Flux Jheat (W m-2)0 12 24 36 48

Tem

pera

ture

(°C)

b. entropiesa. temperatures

ex

Pex

0

2

4

6

8

10

0

0.5

1.0

1.5

2.0

0 12 24 36 48

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1)

Power (W m

-2)

c. entropy production and power

Free

Ene

rgy

(MJ m

-2) Entropy

(MJ m-2 K

-1)

d. free energy and disequilibrium

0 12 24 36 48

Heat Flux Jheat (W m-2)

Heat Flux Jheat (W m-2) Heat Flux Jheat (W m-2)

Tem

pera

ture

(°C)

Flux

(W m

-2)

Entro

py P

rodu

ctio

n(m

W m

-2 K

-1) Entropy

(frac. max)

b. temperatures c. entropies

Time (years) Time (years)

Jheat !heat

Th Tc

a. 2-box modelJin,h Jout,cJin,cJout,h

Jheat

extracted powergenerate motion

motion transports

heat

radiative forcing creates heating

gradient

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ODELLINGKleidon (in prep.)

Time

powe

r, di

ssip

atio

n(W

m-2)

c. power, dissipation

a. 2-box model

d. free energy, disequilibrium

!h vh

JmomSmomPin Dmom

!l vl

Time

Entropy (103 J m

-2 K-1)

Free

Ene

rgy

(106 J

m-2)

vh

velo

city (

m s-1

)

100 * vl

b. velocities

Time

momentum balance model

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a. box-model b. power

vout

ρmass

Jmass

L

H

vin

; ρ i

n

vout

ρout

PmassPlift

Pout

0

20

40

60

80

100

0

0.1

0.2

0.3

export velocity vout (m/s)0 0.2 0.4 0.6 0.8 1.0

Po

we

r (W

)

Pm

ass (W

)

motion can be used to power transport of suspended solids

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ODELLING

M

TcTh

heat engine:large scale circulation driven by differential radiative heating between tropics and poles

Kleidon (in prep.)

radiative gradient

motion

maximumpower

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ODELLING

M

TcTh

heat engine:large scale circulation driven by differential radiative heating between tropics and poles

Kleidon (in prep.)

µw

µv

M

dehumidifier:circulation acts to dehumidify the atmosphere and runs the water cycle

radiative gradient

motion

water cycling

power transfer

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ODELLING

M

TcTh

heat engine:large scale circulation driven by differential radiative heating between tropics and poles

Kleidon (in prep.)

radiative gradient

motion

µw

µv

M

dehumidifier:circulation acts to dehumidify the atmosphere and runs the water cycle

water cycling

transporter:water cycle runs thecycling of rocks(dissolution, suspension)

geochemical cycling

power transfer

power transfer

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

casc

ade

of p

ower

tra

nsfe

rca

uses

dis

equi

libriu

m

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Power Generation within the Earth SystemBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

casc

ade

of p

ower

tra

nsfe

rca

uses

dis

equi

libriu

mcascad

e of effects cause interactions and

feedb

acks

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SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

KE: Kinetic EnergyPE: Potential EnergyCE: Chemical Energy

1 W/m2

≈ 510 TWclimate system:

atm. circulation 900 TW (KE) water cycling 558 TW (PE)

desalination 39 TW (CE) cont. runoff 13 TW (KE)

dissolution <1 TW (CE)

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ERIC THEORY AND M

ODELLINGDyke et al., Earth System Dynamics Discussion, in press.

ESDD1, 1–55, 2010

Surface life andinterior dynamics of

planet Earth

J. G. Dyke et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

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Dyke, Gans, Kleidon: Surface life and interior dynamics of planet Earth: 7

That is, σv is a quadratic function of ∆T , and since ∆T issome function of Jv , there is an optimum value of Jv thatmaximizes σv . The MEP principle applied to this examplestates that convection adopts this optimum flux Jv,opt thatmaximizes σv . The associated maximum rate of work doneand dissipation by the convective flux is then given in steadystate, as above, by Pmax = Dmax = Tc · σv,max.

3.2 Overview of the models

Our three models are set up to correspond to three thermody-namic subsystems that exchange heat at their boundaries. Weneglect exchanges of mass for simplicity. The boundaries areillustrated in the conceptual diagram of the rock cycle shownin Fig. 3.

In the mathematical formulation of the models, we use thenaming convention for parameters and variables as shownTable 1. The indices used to identify variables in the dif-ferent subsystems is given in Table 3.2. An overview of allvariables used in the following is given in Table 3.

Table 1. Naming convention used in the model formulations.

symbol property unitρ density kg m−3

k conductivity W m−1K−1

g gravitational acceleration m s−2

η viscosity kg m−1s−1

fc fractional coverage of continents -fo fractional coverage of oceans -F force kg m−1s−2

P power WJ heat flux W m2

J(m) mass flux kg m2s−1

D dissipation WT temperature Kσ entropy production W K−1

S entropy J K−1

NEE net entropy exchange W K−1

3.3 Model 1: mantle convection

Figure 3.3 represents the components of mantle convection.In this model we are concerned with capturing the dynamicsof the flux of heat from the base of the mantle to the bottomof the lithosphere. For the purposes of this model we assumea uniform rate of heat production via the decay of radioac-tive elements within the mantle and latent heat produced bythe freezing of the liquid outer core. Our results and analy-sis still apply if the mantle is instead subject to greater heat

Fig. 3. The rock cycle’s major components of: mantle convection,oceanic crust recycling and continental crust recycling are shown.The subsystem boundaries are delineated with dashed black lines.

Index Componenta atmospherec continental crusts sedimentso oceanic crustm mantle

Table 2. Convention for the use of indices to identify subsystemsas shown in Fig. 3.

input from the core/mantle boundary and continental crust ismodelled with higher concentrations of radiogenic elements.Also, while the conductivity of mantle material will vary astemperature varies, such changes in conductivity are suffi-ciently small to be ignored so that conductivity can be fixedfor the range of temperatures under consideration. The pro-duction of entropy via mantle convection is conceptually thesame as the simple system shown in Fig. 2. Reservoir 1 is theouter core, reservoir 2 is the lithospere. Heat is transportedvia conduction and convection within the mantle. While lab-oratory experiments can provide estimates for the rate of con-duction through mantle rock, determining the rate of con-vection can be problematic. This is because the mantle overgeological timescales behaves like a liquid with temperaturedependent viscosity; the hotter it is, the more vigourous it

Fig. 3. The rock cycle’s major components of: mantle convection, oceanic crust recycling andcontinental crust recycling are shown. The subsystem boundaries are delineated with dashedblack lines.

45

ESDD1, 1–55, 2010

Surface life andinterior dynamics of

planet Earth

J. G. Dyke et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

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tion. Equations (9) and (10) and energy conservation Φh =∇ ·Jh together in sphericalcoordinates yield the following heat conduction differential equation:

Φh =−2kmN

r∂Tm

∂r−kmN

∂2Tm

∂r2(11)

The analytical solution of the diffusion equation in steady state (∂Tm/∂t = 0) is givenby:5

Tm(r)= Tcore−Φh

6kmNr2 (12)

with the convective heat flux Jm,v given by:

Jm,v =−km(N−1)∇T =Φhr(N−1)

3N(13)

We now have an expression for temperature within the mantle as a function of theNusselt number which in turn is a function of mantle convection. By altering the rate of10

mantle convection, we are able to produce different temperature structures within theEarth. In the following sections we will calculate rates of entropy production via mantleconvection and then find that value of mantle convection that produces maximum ratesof entropy production.

3.3.2 Entropy balance15

We consider two mechanisms for entropy production within the mantle: conductiveand convective heat flux. Calculating entropy produced via conductive heat flux shouldstraightforward as rates of conduction will be an immediate result of the particular prop-erties of the mantle (if we make the first order assumption that conduction does not varywith varying temperature). Convective heat flux and its associated entropy production20

is more challenging because rates of convection will vary with varying temperature and

18

ESDD1, 1–55, 2010

Surface life andinterior dynamics of

planet Earth

J. G. Dyke et al.

Title Page

Abstract Introduction

Conclusions References

Tables Figures

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tion. Equations (9) and (10) and energy conservation Φh =∇ ·Jh together in sphericalcoordinates yield the following heat conduction differential equation:

Φh =−2kmN

r∂Tm

∂r−kmN

∂2Tm

∂r2(11)

The analytical solution of the diffusion equation in steady state (∂Tm/∂t = 0) is givenby:5

Tm(r)= Tcore−Φh

6kmNr2 (12)

with the convective heat flux Jm,v given by:

Jm,v =−km(N−1)∇T =Φhr(N−1)

3N(13)

We now have an expression for temperature within the mantle as a function of theNusselt number which in turn is a function of mantle convection. By altering the rate of10

mantle convection, we are able to produce different temperature structures within theEarth. In the following sections we will calculate rates of entropy production via mantleconvection and then find that value of mantle convection that produces maximum ratesof entropy production.

3.3.2 Entropy balance15

We consider two mechanisms for entropy production within the mantle: conductiveand convective heat flux. Calculating entropy produced via conductive heat flux shouldstraightforward as rates of conduction will be an immediate result of the particular prop-erties of the mantle (if we make the first order assumption that conduction does not varywith varying temperature). Convective heat flux and its associated entropy production20

is more challenging because rates of convection will vary with varying temperature and

18

energy balance:

analytical solution:

ESDD1, 1–55, 2010

Surface life andinterior dynamics of

planet Earth

J. G. Dyke et al.

Title Page

Abstract Introduction

Conclusions References

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as neither the temperature nor rate of convection is known, the problem is poorly de-fined. Application of the MEP allow us to make predictions for rates of convection byassuming it is that rate which produces maximum entropy. Entropy production for themantle system is:dSm

dt=NEEm+σm (14)5

where Sm is the entropy of the mantle, σm is the total entropy production within themantle, and NEEm is the net entropy exchange of the mantle to its surroundings. Atsteady state, σm =−NEEm. Entropy is exchanged with the surroundings by the heatingrate h (entropy import) and by the export of entropy by the heat fluxes across themantle-crust boundary. The entropy export is the heat flux out of the surface divided10

by the surface temperature. JsAs/Ts. The calculation of the entropy import is nottrivial because the temperature at which heat is added to the system is not constant.Consequently it is necessary to integrate over the whole interior, and the entropy fluxinto the system is:

�V h/TdV . This leads to the formulation for entropy production in

steady state as:15

σm =JsAs

Ts−�

V

Φh

TdV (15)

By definition of the Nuesselt number, the contribution of entropy production just bymantle convection is given by (N−1)/N ·σm.

3.3.3 Maximum entropy production due to mantle convection

It is possible to formulate entropy production within the mantle as a function of mantle20

convection with Eq. (15). Figure 5 shows entropy production as a function of Nusseltnumber. When the Nusselt number ≈ 7.5 the greatest rates of entropy are produced.This equates to mantle conduction of ≈ 3 WK−1. and convection of ≈ 21 WK−1. Whenthese values are used in Eq. (12) a temperature structure of the internal Earth can beconstructed as shown in Fig. 5.25

19

entropy production:

=> max. entropy production with respect to Nusselt number N

after Lorenz (2002)

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That is, σv is a quadratic function of ∆T , and since ∆T issome function of Jv , there is an optimum value of Jv thatmaximizes σv . The MEP principle applied to this examplestates that convection adopts this optimum flux Jv,opt thatmaximizes σv . The associated maximum rate of work doneand dissipation by the convective flux is then given in steadystate, as above, by Pmax = Dmax = Tc · σv,max.

3.2 Overview of the models

Our three models are set up to correspond to three thermody-namic subsystems that exchange heat at their boundaries. Weneglect exchanges of mass for simplicity. The boundaries areillustrated in the conceptual diagram of the rock cycle shownin Fig. 3.

In the mathematical formulation of the models, we use thenaming convention for parameters and variables as shownTable 1. The indices used to identify variables in the dif-ferent subsystems is given in Table 3.2. An overview of allvariables used in the following is given in Table 3.

Table 1. Naming convention used in the model formulations.

symbol property unitρ density kg m−3

k conductivity W m−1K−1

g gravitational acceleration m s−2

η viscosity kg m−1s−1

fc fractional coverage of continents -fo fractional coverage of oceans -F force kg m−1s−2

P power WJ heat flux W m2

J(m) mass flux kg m2s−1

D dissipation WT temperature Kσ entropy production W K−1

S entropy J K−1

NEE net entropy exchange W K−1

3.3 Model 1: mantle convection

Figure 3.3 represents the components of mantle convection.In this model we are concerned with capturing the dynamicsof the flux of heat from the base of the mantle to the bottomof the lithosphere. For the purposes of this model we assumea uniform rate of heat production via the decay of radioac-tive elements within the mantle and latent heat produced bythe freezing of the liquid outer core. Our results and analy-sis still apply if the mantle is instead subject to greater heat

Fig. 3. The rock cycle’s major components of: mantle convection,oceanic crust recycling and continental crust recycling are shown.The subsystem boundaries are delineated with dashed black lines.

Index Componenta atmospherec continental crusts sedimentso oceanic crustm mantle

Table 2. Convention for the use of indices to identify subsystemsas shown in Fig. 3.

input from the core/mantle boundary and continental crust ismodelled with higher concentrations of radiogenic elements.Also, while the conductivity of mantle material will vary astemperature varies, such changes in conductivity are suffi-ciently small to be ignored so that conductivity can be fixedfor the range of temperatures under consideration. The pro-duction of entropy via mantle convection is conceptually thesame as the simple system shown in Fig. 2. Reservoir 1 is theouter core, reservoir 2 is the lithospere. Heat is transportedvia conduction and convection within the mantle. While lab-oratory experiments can provide estimates for the rate of con-duction through mantle rock, determining the rate of con-vection can be problematic. This is because the mantle overgeological timescales behaves like a liquid with temperaturedependent viscosity; the hotter it is, the more vigourous it

Fig. 3. The rock cycle’s major components of: mantle convection, oceanic crust recycling andcontinental crust recycling are shown. The subsystem boundaries are delineated with dashedblack lines.

45

entropy production:

temperature profile:

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0

1e+10

2e+10

3e+10

4e+10

5e+10

6e+10

1 10 100 1000

EP [W/K]

EP [W/K]

Nuesselt number

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

1 10 100 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Core temperature

Fraction of convective heat flux

Nuesselt number

Fig. 5. Top plot: Core temperature and fraction of convective heatflux with varying Nusselt number. The Nusselt number is a dimen-sionless value of the proportion of convection to conduction. Coretemperature is plotted with a solid line (units on left vertical axis).Fraction of convective heat flux is plotted with a dashed line (unitson right vertical axis). Bottom plot: Entropy production via mantleconvection with varying Nusselt number.

Fig. 6. MEP mantle convection temperature structure with threedifferent Nusselt number values. Depth beneath the surface of theEarth is shown on the horizontal axis. Temperature in degreesKelvin is shown on the vertical axis. With no mantle convection(N=1 solid line) the core temperature is > 40,000 degrees Kelvin.With high rates of mantle convection (N=100, dotted line) the coretemperature is < 1000 degrees Kelvin. When N is set to the MEPvalue of 7.6 (dashed centre line) the core temperature is ≈ 6000degrees Kelvin.

Therefore, 12TW is the maximum amount of work that canbe performed by the mantle convection system.

3.4 Model 2: Oceanic crust cycling

The processes of mantle convection and conduction deliv-ers an amount of heat to the base of the lithosphere whichfinds its way to the surface and then radiates out into space.In model 2 we consider how the recycling of oceanic crusttransports a proportion of this heat from mantle to surface.Continental crust is rigid and its thermal properties reason-ably well known, so it is relatively straightforward to calcu-late rates of heat flux through the surface of continental crustas a function of upper mantle temperature. Oceanic crusttransfers heat both via conduction and also via the bulk trans-port of heat as hot mantle material from the asthenosphererises to the surface at mid oceanic ridges. The production ofmid oceanic basalt (MORB) and its eventual subduction backinto the mantle releases a significant proportion of heat fromthe interior. This process is conceptually similar to mantleconvection in that an eddy convection process will transporta certain amount of heat given a certain temperature gradient.We will show in following sections that the rate of oceaniccrust recycling has a significant affect on the temperature ofthe asthenosphere and so mantle convection.

Figure 6 is a schematic representation of oceanic crust cy-cling. To parametrize the heat flux through the oceanic crustwe use the so called “half space cooling model” (Stuwe,2002). Hot MORB cools in contact with the cold ocean wa-ter. As new material is produced from mid oceanic ridgespreviously extruded material is pushed away from the ridge.Consequently, the distance from the ridge, the temperatureand the time on the surface for oceanic crust are correlated.

3.4.1 Energy balance

We start with the heat balance between oceanic and continen-tal crust. We assume that the heat flux through the surface ofthe Earth equals the heat flux from continental and oceaniccrust.

Jh(re) = Jcc + Joc (17)

The total heat flux through continental crust is a linear func-tion of the temperature difference, volume and thermal prop-erties of continental crust

Jcc = fckcTmc − Tca

∆zc(18)

Heat transport through oceanic crust is modelled as heat dif-fusion

∂T

∂t= κ

∂2T

∂z2(19)

where κ is the heat diffusivity of oceanic crust which is theratio of its density and heat capacity. We ignore any horizon-tal diffusion of heat though the crust, so the time-dependenttemperature profile, T (z, t), is entirely determined by thevertical heat diffusion. We assume that the temperature of

Fig. 5. Top plot: core temperature and fraction of convective heat flux with varying Nusseltnumber. The Nusselt number is a dimensionless value of the proportion of convection to con-duction. Core temperature is plotted with a solid line (units on left vertical axis). Fraction ofconvective heat flux is plotted with a dashed line (units on right vertical axis). Bottom plot:entropy production via mantle convection with varying Nusselt number.

47

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Fig. 5. Top plot: Core temperature and fraction of convective heatflux with varying Nusselt number. The Nusselt number is a dimen-sionless value of the proportion of convection to conduction. Coretemperature is plotted with a solid line (units on left vertical axis).Fraction of convective heat flux is plotted with a dashed line (unitson right vertical axis). Bottom plot: Entropy production via mantleconvection with varying Nusselt number.

0

1000

2000

3000

4000

5000

6000

7000

0 1000 2000 3000 4000 5000 6000

Temperature (K)

Depth (km)

N=1N=7.6N=100

Fig. 6. MEP mantle convection temperature structure with threedifferent Nusselt number values. Depth beneath the surface of theEarth is shown on the horizontal axis. Temperature in degreesKelvin is shown on the vertical axis. With no mantle convection(N=1 solid line) the core temperature is > 40,000 degrees Kelvin.With high rates of mantle convection (N=100, dotted line) the coretemperature is < 1000 degrees Kelvin. When N is set to the MEPvalue of 7.6 (dashed centre line) the core temperature is ≈ 6000degrees Kelvin.

Therefore, 12TW is the maximum amount of work that canbe performed by the mantle convection system.

3.4 Model 2: Oceanic crust cycling

The processes of mantle convection and conduction deliv-ers an amount of heat to the base of the lithosphere whichfinds its way to the surface and then radiates out into space.In model 2 we consider how the recycling of oceanic crusttransports a proportion of this heat from mantle to surface.Continental crust is rigid and its thermal properties reason-ably well known, so it is relatively straightforward to calcu-late rates of heat flux through the surface of continental crustas a function of upper mantle temperature. Oceanic crusttransfers heat both via conduction and also via the bulk trans-port of heat as hot mantle material from the asthenosphererises to the surface at mid oceanic ridges. The production ofmid oceanic basalt (MORB) and its eventual subduction backinto the mantle releases a significant proportion of heat fromthe interior. This process is conceptually similar to mantleconvection in that an eddy convection process will transporta certain amount of heat given a certain temperature gradient.We will show in following sections that the rate of oceaniccrust recycling has a significant affect on the temperature ofthe asthenosphere and so mantle convection.

Figure 6 is a schematic representation of oceanic crust cy-cling. To parametrize the heat flux through the oceanic crustwe use the so called “half space cooling model” (Stuwe,2002). Hot MORB cools in contact with the cold ocean wa-ter. As new material is produced from mid oceanic ridgespreviously extruded material is pushed away from the ridge.Consequently, the distance from the ridge, the temperatureand the time on the surface for oceanic crust are correlated.

3.4.1 Energy balance

We start with the heat balance between oceanic and continen-tal crust. We assume that the heat flux through the surface ofthe Earth equals the heat flux from continental and oceaniccrust.

Jh(re) = Jcc + Joc (17)

The total heat flux through continental crust is a linear func-tion of the temperature difference, volume and thermal prop-erties of continental crust

Jcc = fckcTmc − Tca

∆zc(18)

Heat transport through oceanic crust is modelled as heat dif-fusion

∂T

∂t= κ

∂2T

∂z2(19)

where κ is the heat diffusivity of oceanic crust which is theratio of its density and heat capacity. We ignore any horizon-tal diffusion of heat though the crust, so the time-dependenttemperature profile, T (z, t), is entirely determined by thevertical heat diffusion. We assume that the temperature of

Fig. 6. MEP mantle convection temperature structure with three different Nusselt number val-ues. Depth beneath the surface of the Earth is shown on the horizontal axis. Temperature indegrees Kelvin is shown on the vertical axis. With no mantle convection (N=1 solid line) thecore temperature is >40 000 degrees Kelvin. With high rates of mantle convection (N = 100,dotted line) the core temperature is <1000 degrees Kelvin. When N is set to the MEP value of7.6 (dashed centre line) the core temperature is ≈6000 degrees Kelvin.

48

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That is, σv is a quadratic function of ∆T , and since ∆T issome function of Jv , there is an optimum value of Jv thatmaximizes σv . The MEP principle applied to this examplestates that convection adopts this optimum flux Jv,opt thatmaximizes σv . The associated maximum rate of work doneand dissipation by the convective flux is then given in steadystate, as above, by Pmax = Dmax = Tc · σv,max.

3.2 Overview of the models

Our three models are set up to correspond to three thermody-namic subsystems that exchange heat at their boundaries. Weneglect exchanges of mass for simplicity. The boundaries areillustrated in the conceptual diagram of the rock cycle shownin Fig. 3.

In the mathematical formulation of the models, we use thenaming convention for parameters and variables as shownTable 1. The indices used to identify variables in the dif-ferent subsystems is given in Table 3.2. An overview of allvariables used in the following is given in Table 3.

Table 1. Naming convention used in the model formulations.

symbol property unitρ density kg m−3

k conductivity W m−1K−1

g gravitational acceleration m s−2

η viscosity kg m−1s−1

fc fractional coverage of continents -fo fractional coverage of oceans -F force kg m−1s−2

P power WJ heat flux W m2

J(m) mass flux kg m2s−1

D dissipation WT temperature Kσ entropy production W K−1

S entropy J K−1

NEE net entropy exchange W K−1

3.3 Model 1: mantle convection

Figure 3.3 represents the components of mantle convection.In this model we are concerned with capturing the dynamicsof the flux of heat from the base of the mantle to the bottomof the lithosphere. For the purposes of this model we assumea uniform rate of heat production via the decay of radioac-tive elements within the mantle and latent heat produced bythe freezing of the liquid outer core. Our results and analy-sis still apply if the mantle is instead subject to greater heat

Fig. 3. The rock cycle’s major components of: mantle convection,oceanic crust recycling and continental crust recycling are shown.The subsystem boundaries are delineated with dashed black lines.

Index Componenta atmospherec continental crusts sedimentso oceanic crustm mantle

Table 2. Convention for the use of indices to identify subsystemsas shown in Fig. 3.

input from the core/mantle boundary and continental crust ismodelled with higher concentrations of radiogenic elements.Also, while the conductivity of mantle material will vary astemperature varies, such changes in conductivity are suffi-ciently small to be ignored so that conductivity can be fixedfor the range of temperatures under consideration. The pro-duction of entropy via mantle convection is conceptually thesame as the simple system shown in Fig. 2. Reservoir 1 is theouter core, reservoir 2 is the lithospere. Heat is transportedvia conduction and convection within the mantle. While lab-oratory experiments can provide estimates for the rate of con-duction through mantle rock, determining the rate of con-vection can be problematic. This is because the mantle overgeological timescales behaves like a liquid with temperaturedependent viscosity; the hotter it is, the more vigourous it

Fig. 3. The rock cycle’s major components of: mantle convection, oceanic crust recycling andcontinental crust recycling are shown. The subsystem boundaries are delineated with dashedblack lines.

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Variable Description ValueToa − Tmo Temp gradient mantle-oceanic crust 730K

Joc Heat flux through oceanic crust 34TWJcc Heat flux through continental crust 16TWvo Oceanic crust recycling velocity 20cm−1y−1

Table 4. Values for oceanic crust recycling when ‘diffusion’ param-eter γ is selected to produce MEP.

8.6e+10

8.8e+10

9e+10

9.2e+10

9.4e+10

9.6e+10

9.8e+10

1e+11

1e-12 1e-11 1e-10

EP [W/K]

EP [W/K]

Diffusion parameter

3e+13

3.2e+13

3.4e+13

3.6e+13

3.8e+13

4e+13

4.2e+13

4.4e+13

4.6e+13

1e-12 1e-11 1e-10 300

400

500

600

700

800

900

1000

1100

1200

1300

Oceanic Crust Heat Flux (W)

Temperature gradient (K)

Diffusion parameter

Fig. 8. Top plot: Oceanic crust heat flux plotted with solid line(units on right horizontal axis) with varying values for the ‘Diffu-sion parameter’, γ. Increasing γ increases the heat flux throughoceanic crust. Temperature gradient, the difference in temperaturebetween surface of oceanic crust and upper mantle, plotted withdashed line (units on left vertical axis) with varying values for γ .Increasing γ decreases the temperature gradient. Bottom plot: En-tropy production in oceanic crust recycling as a function of γ.

To calculate the maximum amount of work that can bedone by oceanic crust recycling we multiply the entropy pro-duction by the surface temperature:

P = σTs = 0.097TWK−1 · 293K ≈ 28TW (30)

Therefore, 28TW is the maximum amount of work that canbe performed by the oceanic crust recycling system.

3.5 Model 3: uplift and erosion

As continental material is eroded away into the sea, the massof continental crust decreases and this reduction in massleads to mantle pressure pushing the continental crust up.Erosion and uplift are related in that higher rates of erosionwill lead to higher rates of uplift, with maximum rates ofuplift being determined by the material properties of the as-thenosphere.

Fig. 9. Model 3: A simple model of the mass balance of conti-nental crust driven by uplift and erosion. Weathering and erosionprocesses transfer continental crust material to the ocean where itis deposited as sediment. Continental crust material moves back tothe continent though the process of subduction.

Fig. 9 is a schematic representation of continental crustuplift and erosion. Model 3 characterises the process ofcontinental crust uplift and erosion in terms of a compet-ing processes that move material away and towards thermo-dynamic equilibrium. Mountains are manifestation of non-equilibrium geological processes in that they are not at iso-static equilibrium and so over time will sink back down intothe asthenosphere. They also represent an energy gradientthat erosion dissipates; material is moved from high abovethe surface of the Earth to the lower sea floor. Isostatic im-balance and erosion will both lead to a decrease in the heightof any mountain. In the following sections, we will quantifytheses processes in thermodynamic terms that will includethe production of entropy via uplift and erosion.

3.5.1 Potential energy balance

Density differences are responsible for uplift as the densityof continental crust is less than oceanic crust (which includessediments and sedimentary rock). We ignore fluxes of heatas uplift and erosion are effectively irrelevant in determiningthe temperature of continental crust. The potential energy ofcontinental crust material at the surface of the continents isexpressed using the notion of a geopotential µca:

µca = g · zca (31)

where of g is gravity and zca height above the mantle. Thegeopotential of continental crust material at the crust/mantle

Fig. 8. Top plot: oceanic crust heat flux plotted with solid line (units on right horizontal axis) withvarying values for the “Diffusion parameter”, γ. Increasing γ increases the heat flux throughoceanic crust. Temperature gradient, the difference in temperature between surface of oceaniccrust and upper mantle, plotted with dashed line (units on left vertical axis) with varying valuesfor γ . Increasing γ decreases the temperature gradient. Bottom plot: entropy production inoceanic crust recycling as a function of γ.

50

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0

5e+07

1e+08

1.5e+08

2e+08

2.5e+08

3e+08

3.5e+08

0 5e-16 1e-15 1.5e-15 2e-15 2.5e-15 3e-15

EP [W/K]

Erosion constant kcs

FrictionErosion

Total

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

0 5e-16 1e-15 1.5e-15 2e-15 2.5e-15 3e-15 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Continental Crust height (m)

Uplift velocity (mm/yr)

Erosion constant kcs

Fig. 10. Top plot: Continental crust height plotted with solid line(units on left horizontal axis) and uplift velocity plotted with dashedline (units left horizontal axis) with varying values for erosion con-stant kcs. Increasing kcs decreases continental crust height and in-creases uplift velocity. Bottom plot: Entropy production by erosionas a function of erosion rate, kcs. Entropy produced by frictionplotted with a solid line, erosion plotted by a dashed line and totalentropy production with a dashed line (top line).

4 Discussion

In this section we first include the maximum power estimatesfrom the three models into a work budget of the Earth’s in-terior and the global rock cycle and compare them to bioticactivity. We then show the sensitivity of the oceanic crustcycling and mantle convection models to continental crustthickness in order to substantiate our main hypothesis thatbiologically-mediated surface processes affect interior pro-cesses.

4.1 The work budget of interior processes and the rockcycle

We now summarize our results in the form of a work budgetof the global rock cycle and interior processes. This workbudget is summarized in Fig. 11. The maximum power asso-ciated with mantle convection (12 TW), oceanic crust cycling(28 TW), and continental uplift (< 1 TW) is taken from theprevious three sections.

Also shown in the work budget are processes driven pri-marily by the climate system. For comparison we show the900 TW of power involved in driving the global atmosphericcirculation (Peixoto and Oort, 1992). This power drives thedehumidification of atmospheric vapor and therefore the hy-

drologic cycle. The strength of the hydrologic cycle is rele-vant here in that it (a) distills seawater, (b) lifts vapor into theatmosphere, and (c) transports water to land. The precipita-tion on land then contains chemical and potential free energy.The chemical free energy inherent in precipitation is used tochemically dissolve rocks and bring the dissolved ions to theoceans. The potential energy in precipitation at some heightof the land surface generates stream power which can be usedto mechanically transport sediments.

To estimate the available power to chemically weatherrock by abiotic means, we consider the work necessary todesalinate the water when evaporated from the ocean. Givena salinity of 3.5 %, the work required to desalinate a literof seawater is about 3.8 kJ. For a net moisture transport of37×1012m3yr−1, this corresponds to a power of about 4TW. This power is potentially available to dissolve rock andbring the precipitated water to saturation with the continen-tal rocks. However, since most of the salinity of the ocean issodium chloride, which is only a relatively minor product ofchemical weathering, the actual power for chemical weather-ing should be much less.

To estimate the power inherent in the potential energy inrunoff and for a maximum estimate for physical weatheringof continental rocks, we use estimates from a spatially ex-plicit land surface model with realistic, present-day climaticforcing. With this model we estimate the power in conti-nental runoff to be 13 TW, which sets the upper limit onthe power available for sediment transport. The potential tophysically weather bedrock by seasonal heating and coolingand freeze-thaw dynamics is less than 50 TW (Gans et al.,in prep.). This latter number is an upper estimate in that itassumes bedrock to be present at the surface.

The power associated with biotic activity is derived fromestimates of gross primary productivity (GPP) of 120 peta-grams yr−1 on land and 50 petagrams yr−1 in the oceans(Solomon et al., 2007). If we assume all carbon is producedvia photosynthesis and that all photosynthate is glucose, then1.67× 1015 and 6.91× 1015 moles of glucose are producedeach year on land and in the oceans respectively. One moleof glucose contains 2874 kJ. This gives an energy produc-tion for land and in the oceans of 4.79 × 1021 J yr−1 and2.11 × 1021 J yr−1 respectively or 152 TW on land and 63TW in the oceans giving a global biological power of 215TW which is an order of magnitude greater than the powergenerated by mantle convection. From this energy budget,approximately 50% of land GPP and 20% of oceanic GPPwill be used to drive autotrophic metabolisms, mainly viathe process of respiration. The remaining energy is availableto grow and to concentrate, move and transform geochemicalmaterial.

Fig. 10. Top plot: continental crust height plotted with solid line (units on left horizontal axis)and uplift velocity plotted with dashed line (units left horizontal axis) with varying values forerosion constant kcs. Increasing kcs decreases continental crust height and increases upliftvelocity. Bottom plot: entropy production by erosion as a function of erosion rate, kcs. Entropyproduced by friction plotted with a solid line, erosion plotted by a dashed line and total entropyproduction with a dashed line (top line).

52

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temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

KE: Kinetic EnergyPE: Potential EnergyCE: Chemical Energy

1 W/m2

≈ 510 TW

Earth’s interior: mantle convect. 12 TW (KE)

oceanic crust 28 TW (KE) continental crust <1 TW (PE)

climate system: atm. circulation 900 TW (KE)

water cycling 558 TW (PE) desalination 39 TW (CE) cont. runoff 13 TW (KE)

dissolution <1 TW (CE)

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Power Generation within the Earth SystemBIO

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ERIC THEORY AND M

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temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

alterationof rates

alterationof rates

bioticactivity

nutrients

nutrients

photo-synthesis

KE: Kinetic EnergyPE: Potential EnergyCE: Chemical Energy

1 W/m2

≈ 510 TW

mantle conv. 21 TW (KE) oceanic crust 2 TW (KE) oceanic uplift 2 TW (PE)

continental crust 1 TW (KE) continental uplift 1 TW (PE)

biosphere:biotic activity 215 TW (CE)

Earth’s interior: mantle convect. 12 TW (KE)

oceanic crust 28 TW (KE) continental crust <1 TW (PE)

climate system: atm. circulation 900 TW (KE)

water cycling 558 TW (PE) desalination 39 TW (CE) cont. runoff 13 TW (KE)

dissolution <1 TW (CE)

http://gaia.mpg.de

MPI-BGC

Jena

Power Generation within the Earth SystemBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

alterationof rates

alterationof rates

bioticactivity

nutrients

nutrients

photo-synthesis

KE: Kinetic EnergyPE: Potential EnergyCE: Chemical Energy

1 W/m2

≈ 510 TW

human system:from biotic activity 25 TW

from fossil fuels 17 TW

biosphere:biotic activity 215 TW (CE)

Earth’s interior: mantle convect. 12 TW (KE)

oceanic crust 28 TW (KE) continental crust <1 TW (PE)

climate system: atm. circulation 900 TW (KE)

water cycling 558 TW (PE) desalination 39 TW (CE) cont. runoff 13 TW (KE)

dissolution <1 TW (CE)

http://gaia.mpg.de

MPI-BGC

Jena

Implications of a “Powerful” PerspectiveBIO

SPH

ERIC THEORY AND M

ODELLING

• a. Earth system modeling: Do models transfer power adequately? Most probably not…

• b. Gaia: How does this hierarchy of power generation and transfer relate to the Gaia hypothesis?

• c. Human imprint: Humans as a planetary force

http://gaia.mpg.de

MPI-BGC

Jena

Thermodynamics and GaiaBIO

SPH

ERIC THEORY AND M

ODELLING

MEP: maximum power

transfer enables higher rates of entropy production in a

hierarchy

http://gaia.mpg.de

MPI-BGC

Jena

Thermodynamics and GaiaBIO

SPH

ERIC THEORY AND M

ODELLING

MEP: maximum power

transfer enables higher rates of entropy production in a

hierarchy

Earth: power generation and

transfer evolves variables away from

equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Thermodynamics and GaiaBIO

SPH

ERIC THEORY AND M

ODELLING

MEP: maximum power

transfer enables higher rates of entropy production in a

hierarchy

Life: adds substantial amount

of free energy to geochemical cycling(215 TW >> 1 TW)

Earth: power generation and

transfer evolves variables away from

equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Thermodynamics and GaiaBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to widespread life

MEP: maximum power

transfer enables higher rates of entropy production in a

hierarchy

Life: adds substantial amount

of free energy to geochemical cycling(215 TW >> 1 TW)

Earth: power generation and

transfer evolves variables away from

equilibrium

http://gaia.mpg.de

MPI-BGC

Jena

Thermodynamics and GaiaBIO

SPH

ERIC THEORY AND M

ODELLING

Lovelock: chemical disequilibrium in the Earth’s atmosphere

due to life

MEP: maximum power

transfer enables higher rates of entropy production in a

hierarchy

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (µ

mol

)sh

allo

w o

cean

sm

O2 (µ

mol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

phosphorites and marine manganese deposits surelyreflects the complexity of their development and theincomplete overlap between the factors that controltheir formation (Roy 1992, 1997).

7. SUMMARY AND CONCLUSIONSFigure 10 summarizes the proposed evolution of theatmospheric pressure of O2 and the concentration ofO2 in the shallow and deep oceans during the last3.8 Gyr of Earth history. During stage 1, from 3.85 to2.45 Ga, atmospheric O2 was almost certainly less thana few parts per million, except possibly during theperiod between 3.0 and 2.8 Ga. The oceans werealmost certainly anoxic except perhaps in oxygen oaseswithin the photic zone. Such oases may have beenpresent during the last 200–300 Ma of the Archaean.They probably existed earlier if cyanobacteria evolvedbefore 2.8 Ga. The deep oceans were almost certainlyanoxic during all of stage 1.

Atmospheric O2 rose during the GOE between ca2.4 and 2.0 Ga. Its value at 2.0 Ga is still poorlydefined. It was probably higher than 10% PAL

(0.02 atm) but significantly lower than 1 PAL(0.2 atm). The concentration of O2 in much of theshallow oceans was probably close to equilibrium withatmospheric O2 levels, but the deep oceans were almostcertainly anoxic during much, if not all of stage 2(2.45–1.85 Ga).

Stage 3 (1.85–0.85 Ga) seems to have been ratherstatic. There is no evidence for large changes in the O2

content of the atmosphere or in the shallow oceans. Insome areas, the O2 minimum zone must, however, havebeen anoxic or euxinic as indicated, for instance, by thehighly reduced ca 1.5 Gyr sediments of the McArthurBasin, Australia (Shen et al. 2002). Such spatialheterogeneities in the O2 content of the shallow oceanare not represented in figure 10. The deep oceansappear to have been mildly oxygenated during much, ifnot all of stage 3.

Stage 4 (0.85–0.54 Ga) saw a further rise in the levelof atmospheric O2 and in the concentration of O2 in theshallow oceans. However, anoxic and euxinic con-ditions prevailed in the shallow oceans at least locally,as on the Yangtze Platform at the end of the

stages

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

0.5

0.4

0.3

0.2

0.1

0

atm

osph

ere

PO

2 (atm

)de

ep o

cean

sm

O2 (

µmol

)sh

allo

w o

cean

sm

O2 (

µmol

)

?

?

500

400

300

200

100

0

500

400

300

200

100

03.8 3.0 2.0 1.0 0

Figure 10. Estimated evolution of atmospheric PO2and the concentration of O2 in the shallow and deep oceans.

912 H. D. Holland Oxygenation of the atmosphere and oceans

Phil. Trans. R. Soc. B (2006)

evolution towards greater disequilibrium in Earth’s history => more power generation?

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Global ChangeBIO

SPH

ERIC THEORY AND M

ODELLINGKleidon (in prep.)

temperature gradients

gain o

f ener

gy

(sunli

ght)

phot

osyn

thes

is

abso

rptio

n emissionrespiration

loss of energy

(infrared radiation)

radiative gradients

temperature gradients

motion

hydrologic cycling

geochemical cycling

heating

buoyancy

dehumidification,desalination

dissolution,transport

heattransport

transformation of atmosphere

mantle convection

oceanic crust cycling

continental crust cycling

geochemical cycling

buoyancy

transformationof crust

heat generation

(radiogenic, crystallization)

buoyancy

subduction

heattransport

rockformation

alterationof rates

alterationof rates

bioticactivity

nutrients

nutrients

photo-synthesis

KE: Kinetic EnergyPE: Potential EnergyCE: Chemical Energy

1 W/m2

≈ 510 TW

human system:from biotic activity 25 TW

from fossil fuels 17 TW

biosphere:biotic activity 215 TW (CE)

Earth’s interior: mantle convect. 12 TW (KE)

oceanic crust 28 TW (KE) continental crust <1 TW (PE)

climate system: atm. circulation 900 TW (KE)

water cycling 558 TW (PE) desalination 39 TW (CE) cont. runoff 13 TW (KE)

dissolution <1 TW (CE)

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Global ChangeBIO

SPH

ERIC THEORY AND M

ODELLING

globalwarming tropical

deforestation

watercrisis

biodiversityloss

populationexplosionfood

supply

renewableenergy

geo-engineering

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Global ChangeBIO

SPH

ERIC THEORY AND M

ODELLINGimage: wikimedia.org; porsche.de

how to characterize a car:temperature of the engine

orthe power of the engine?

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Global ChangeBIO

SPH

ERIC THEORY AND M

ODELLINGimage: NASA

temperature

power

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Global ChangeBIO

SPH

ERIC THEORY AND M

ODELLING

globalwarming tropical

deforestation

watercrisis

biodiversityloss

populationexplosionfood

supply

renewableenergy

geo-engineering

power?

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Renewable EnergiesBIO

SPH

ERIC THEORY AND M

ODELLING

M

TcTh

µw

µv

M

power transfer

wind power(wave power, ocean power)

hydropower,osmotic power

solar power

power transfer

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Renewable EnergiesBIO

SPH

ERIC THEORY AND M

ODELLING

+–M

TcTh

µw

µv

M

=> impacts areunavoidable!

power transfer

power transfer

power transfer

impacts

impacts

http://gaia.mpg.de

MPI-BGC

Jena

Oceans of OpportunityHarnessing Europe’s largest domestic energy resource

A report by the European Wind Energy Association

Human Imprint and Renewable EnergiesBIO

SPH

ERIC THEORY AND M

ODELLINGMiller, Gans, Kleidon (submitted)

MPI-BGC

Jena

extracted wind power

intensity of wind removal

naturaldissipation

(= generation)

http://gaia.mpg.de

MPI-BGC

Jena

Human Imprint and Renewable EnergiesBIO

SPH

ERIC THEORY AND M

ODELLINGMiller, Gans, Kleidon (submitted)

wind power extractionreduces power availability and generation within the Earth system

solar power enhances absorption of sunlight and thereby can increase power generation within the Earth system

http://gaia.mpg.de

MPI-BGC

Jena

OutlineBIO

SPH

ERIC THEORY AND M

ODELLING

1. Background: • link between disequilibrium, spatiotemporal variability

and power generation => limits to stochastic forcing?• thermodynamic limits to power generation << Carnot• max. power generation = max. dissipation ≈ MEP

http://gaia.mpg.de

MPI-BGC

Jena

OutlineBIO

SPH

ERIC THEORY AND M

ODELLING

1. Background: • link between disequilibrium, spatiotemporal variability

and power generation => limits to stochastic forcing?• thermodynamic limits to power generation << Carnot• max. power generation = max. dissipation ≈ MEP

2. Earth system: • hierarchy of power generation and transfer• global work budget: the missing budget

http://gaia.mpg.de

MPI-BGC

Jena

OutlineBIO

SPH

ERIC THEORY AND M

ODELLING

1. Background: • link between disequilibrium, spatiotemporal variability

and power generation => limits to stochastic forcing?• thermodynamic limits to power generation << Carnot• max. power generation = max. dissipation ≈ MEP

2. Earth system: • hierarchy of power generation and transfer• global work budget: the missing budget

3. Implications:• do models adequately capture power transfer?• life as a substantial power generator• humans as planetary dissipator• limits and impacts of renewable energy

http://gaia.mpg.de

MPI-BGC

JenaText

BIOSP

HER

IC THEORY AND MO

DELLING