the compost-bomb instability and the petmumn mathematics of climate seminar november 12, 2013 1 the...

The Compost-Bomb Instability and the PETM

Jonathan Hahn

UMN Mathematics of Climate Seminar

November 12, 2013

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The Paleocene-Eocene Thermal Maximum

2

The Paleocene-Eocene Thermal Maximum

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Carbon Mass Balance

Mfinal ⇥ �13Cfinal = Minitial ⇥ �13Cinitial +Madded ⇥ �13Cadded

Mfinal = Minitial +Madded

CIE = �13Cfinal - �13Cinitial

Madded = -CIE ⇥ Minitial�13

Cfinal-�13

Cadded

�13Csample = (Rsample/Rstandard - 1)

Rsample = C13/C12

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�13C of Carbon Sources

Source

Methane clathrates:

Wildfires:

Drying of inland seas:

Thermogenic methane:

Permafrost and peat:

�13C

-6%

-2.2%

-2.2%

-3%

-3%

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Estimating the CIE

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Mass of Carbon Added to the Atmosphere

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Source of carbon unclear

Zeebe et. al.:

Panchuck et. al. and Pagani et. al.:

Wright & Schaller:

Most geologists:

must have been methane

definitely not methane

probably a comet

unlikely to be a comet

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CIE time frame

This data suggests the onset happens in about 20,000 years

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CIE time frame

Wright and Schaller say:

13 years

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Peat and Permafrost

DeConto et. al. use GCM with orbital forcing

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Compost-Bomb Instability

Luke & Cox:

The compost bomb: thermal instability in peatland soils

dC

dt= ⇧- Cr(T )

✏dT

dt= Cr(T )-

�

A(T - Ta)

CTTa

r(T ) = r0

e↵T

Vertically integrated soil carbon content (kg/m2)Soil temperatureAtmospheric TemperatureSoil respiration rate

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Compost-Bomb Instability

dC

dt= ⇧- Cr(T )

✏dT

dt= Cr(T )-

�

A(T - Ta)

r(T ) = r0

e↵T

r0

= 0.01(yr-1); ↵ = ln(2.5)/10Pi = 1.055(kgm-2yr-1)� = 5.049 ⇤ 106 (Jyr-1m-2

degC-1)

A = 3.9 ⇤ 107(Jkg-1)µ = 2.5 ⇤ 106(Jm-2 degC-1)✏ = µ/A = .064

microbial respiration parameterslitter fall from plantssoil-to-air heat transfer coe�cientspecific heat from microbial respirationsoil heat capacity

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Compost-Bomb Instability

Equilibrium analysis

0 = dCdt = ⇧- C eqr(T eq)

0 = dTdt = C eqr(T eq)- �

A(Teq - Ta)

C eq = ⇧/r(T eq)T eq = AC eqr(T eq)/�+ Ta

C eq = ⇧/r(T eq)T eq = A⇧/�+ Ta

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Compost-Bomb Instability

Equilibrium analysis

0 = dCdt = ⇧- C eqr(T eq)

0 = dTdt = C eqr(T eq)- �

A(Teq - Ta)

C eq = ⇧/r(T eq)T eq = AC eqr(T eq)/�+ Ta

C eq = ⇧/r(T eq)T eq = A⇧/�+ Ta

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Compost-Bomb Instability

Equilibrium analysis

0 = dCdt = ⇧- C eqr(T eq)

0 = dTdt = C eqr(T eq)- �

A(Teq - Ta)

C eq = ⇧/r(T eq)T eq = AC eqr(T eq)/�+ Ta

C eq = ⇧/r(T eq)T eq = A⇧/�+ Ta

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Compost-Bomb Instability

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Compost-Bomb Instability

J =

0

@- �

A✏ + C↵r(T)✏

r(T)✏

-C↵r(T ) -r(T )

1

A

det(J - I�) = �2 + �

✓�

A✏-

C↵r(T )

✏+ r(T )

◆+

�

A✏r(T )

Instability condition:�

A✏-

C eq↵r(T eq)

✏+ r(T eq) < 0

�

A✏-

⇧↵

✏+ r(T eq) < 0

Stable!

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Compost-Bomb Instability

J =

0

@- �

A✏ + C↵r(T)✏

r(T)✏

-C↵r(T ) -r(T )

1

A

det(J - I�) = �2 + �

✓�

A✏-

C↵r(T )

✏+ r(T )

◆+

�

A✏r(T )

Instability condition:�

A✏-

C eq↵r(T eq)

✏+ r(T eq) < 0

�

A✏-

⇧↵

✏+ r(T eq) < 0

Stable!

16

Compost-Bomb Instability

J =

0

@- �

A✏ + C↵r(T)✏

r(T)✏

-C↵r(T ) -r(T )

1

A

det(J - I�) = �2 + �

✓�

A✏-

C↵r(T )

✏+ r(T )

◆+

�

A✏r(T )

Instability condition:�

A✏-

C eq↵r(T eq)

✏+ r(T eq) < 0

�

A✏-

⇧↵

✏+ r(T eq) < 0

Stable!

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Compost-Bomb Instability

However, Luke & Cox claim there is instability if the decrease insoil carbon cannot keep up with the increase in soil temperature.

Assume C = C0

is constant (dCdt << dTdt ).

Then according to the previous condition, the system is unstable if:

�

A✏-

C0

↵r(T eq)

✏+ r(T eq) < 0

T eq > (1/↵) ln

✓-�/(r

0

A✏)

1- C0

↵/✏

◆

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Compost-Bomb Instability

Equilibrium for T are given by: T eq - AC0

r(T eq)/� = Ta

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Canard Trajectories

Wieczorek et. al. examine the same system but including:

dTa

dt= v

In this system, a “canard explosion” occurs for v > vcritical .

Canards: trajectories which transition from small amplitude limitcycles to large ones with a small perturbation in a parameter.

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Canard Trajectories

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Rate-induced tipping

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Credits

McInerny & Wing: “The Paleocene-Eocene Thermal

Maximum: A Perturbation of Carbon Cycle, Climate, and

Biosphere with Implications for the Future”

Luke, C. M, & Cox, P.M.: “Soil carbon and climate change:

from the Jenkinson e↵ect to the compost-bomb instability”

Wieczorek et. al.: “Excitability in ramped systems: the

compost-bomb instability”

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