New Newtonian Alchemy: Turning Noise into Signal

Peter CoxMathematics Research Institute

University of Exeter

With thanks especially to

Chris Huntingford, Ben Booth, Roddy Dewar, Tim Lenton

Outline

Maximum Entropy Production – applications to surface-to-atmosphere turbulent fluxes.

Interannual variability of CO2 as a constraint on the sensitivity of tropical land carbon to climate change

Amazon Forest Dieback as a Climate Tipping Point – Time-series Precursors ?

Maximum Entropy Production

– Applications to Climate

(Dewar, Niven, Jupp, Gregory)

Maximum Entropy Production : Application to the Climate System

1960s : Ed Lorenz suggests that the climate system maximises “work”. (E. Lorenz, 1960)

1970s : Garth Paltridge develops successful climate model based on the assumption that heat transports maximise the rate of entropy production. (Paltridge 1975; 1978)

2003 : Ralph Lorenz et al. show that MEP is consistent with the observed equator-to-pole temperature contrasts on Titan and Mars (as well as Earth). (R. Lorenz et al., 2003).

Equator-Pole heat flux F = 2D (T0 – T1)

D chosen to maximise the rate of Entropy Production

Entropy Production Rate = F { 1 / T1 – 1 / T0 }

Lorenz et al., 2003

Entropy Production by Zonal Heat Transport

Observed Equatorial Temperature

Observed Polar Temperature

Modelled Polar and Equatorial Temperatures

Entropy Production Rate due to equator-pole heat transport

Maximum Entropy Production (MEP) fits observations

Application of MEP to Equator-Pole Temperature difference on Saturn’s Moon Titan

Lorenz et al., 2003

2-box MEP model including dynamics

(Jupp + Cox, Proc Roy Soc B, 2010)

Solve for flow U, with surface drag CD as free parameter

Maximum Entropy Production : Application to the Climate System

1960s : Ed Lorenz suggests that the climate system maximises “work”. (E. Lorenz, 1960)

1970s : Garth Paltridge develops successful climate model based on the assumption that heat transports maximise the rate of entropy production. (Paltridge 1975; 1978)

2003 : Ralph Lorenz et al. show that MEP is consistent with the observed equator-to-pole temperature contrasts on Titan and Mars (as well as Earth). (R. Lorenz et al., 2003).

2003: Roddy Dewar derives the MEP principle from Information Theory, in a manner similar to the information approach to the second law of thermodynamics. (Dewar, 2003, 2005).

“Dangerously Seductive”...but sometimes it’s nice to be

seduced...

Atmospheric Energy Balance on Earth

TOAFluxes

TurbulentFlux

Leaky Greenhouse Model

Surface, Ts

Absorbed SW Radiation

Rs

Atmosphere absorbs

a fraction of the radiation from the

surface

Ts

4

Surface radiates as a blackbody

Atmosphere, Ta

Ts4 Ta

4

Atmosphere emits

with emissivity

Ta4

Turbulent Heat Flux

F

+ Turbulent Flux by MEP

Top-of-the-atmosphere energy balance:

Rs = (1-) Ts4 + Ta

4

Equations of the “Model”

Surface energy balance:

Rs + Ta4 = Ts

4 + F

Assume F maximises the rate of entropy production:

dS/dt = F {1/Ta – 1/Ts}

Entropy Production vs Turbulent Flux and Emissivity

MEP Condition

Turbulent Transfer Coefficient at MEP versus Emissivity

Tu

rbu

len

t F

lux

/ A

bso

rbed

SW

IR Optical Depth

2-box MEP Solution

Lorenz&

McKay2003

Osawa et al. 1997

Simple Models for Turbulent Flux

Tu

rbu

len

t F

lux

/ A

bso

rbed

SW

IR Optical Depth

2-box MEP SolutionEarth

Mars

Lorenz&

McKay2003

Osawa et al. 1997

Simple Models for Turbulent Flux

Ongoing Work

Extension to include cloud cover feedbacks

Comparison of MEP predictions to climate observations and models, e.g. Clement et al., 2009:

Modelled

Climate-Carbon Cycle

Feedbacks

Standard Climate Change Predictions

Fossil Fuel + Net Land-use

CO2 Emissions

Online

OfflineCLIMATE

OCEAN LAND

CO2

Greenhouse Effect

CO2 Uptake by Land / CO2-fertilization of

plant growth

CO2 Uptake by Ocean / CO2 buffering effect

Fossil Fuel + Net Land-use

CO2 Emissions

Online

OfflineCLIMATE

OCEAN LAND

CO2

Greenhouse Effect

Climate Change effects on

Solubility of CO2

Vertical MixingCirculation

Climate Change effects on plant productivity, soil

respiration

Climate Change Predictions including Carbon Cycle Feedbacks

Predictions of extra CO2 due to climate effects on the carbon cycle

Friedlingstein et al., 2006

Prediction of Amazonian Forest Dieback due to Climate Change in Hadley Model

1850 2000 2100

World Bank Amazon Dieback Project - PIKRammig et al. (2010)

Tropical forest

Deciduous forest

Open forest

Woodland

Shrub

Savannah

Projection of Amazon Vegetation Uncertain due to Uncertainties

in Rainfall Change

LPJ Model for SRES A1B Scenarios

HadCM3

CCSM3

ECHAM5

1981-2000

HadCM3

CCSM3

ECHAM5

2081-2100

Observed Changes in the

Carbon Cycle

(Global Carbon Project)

Atmospheric CO2 Concentration

Data Source: Pieter Tans and Thomas Conway, 2010, NOAA/ESRL

1970 – 1979: 1.3 ppm y-1

1980 – 1989: 1.6 ppm y1

1990 – 1999: 1.5 ppm y-1

2000 - 2009: 1.9 ppm y-1

2009 1.622008 1.802007 2.142006 1.842005 2.392004 1.602003 2.192002 2.402001 1.892000 1.22

December 2009: 387.2 ppmSeptember 2010 (preliminary): 389.2 ppm39% above pre-industrial

Annual Mea Growth Rate (ppm y-1)

GLOBAL MONTHLY MEAN CO2

2006 2007 2008 2009 2010 2011

Nove

mbe

r 201

0Parts

Per

Milli

on (p

pm)

390

388

386

384

382

380

378

Updated from Le Quéré et al. 2009, Nature Geoscience; Data: NOAA 2010, CDIAC 2010

Key Diagnostic of the Carbon CycleEvolution of the fraction of total emissions that remain in the atmosphere

Total CO2 emissions

Atmosphere

CO2 P

artit

ioni

ng (P

gC y

-1)

1960 20101970 1990 20001980

10

8

6

4

2

Time (y)

Modelled Natural CO2 Sinks

Updated from Le Quéré et al. 2009, Nature Geoscience

Land

sin

k (P

gCy-1

)

5 m

odel

s

1960 20101970 1990 20001980

0

2

-2

-4

-6

Oce

an s

ink

(PgC

y-1)

4

mod

els

Time (y)1960 20101970 1990 20001980

0

2

-2

-4

-6

What does Variability in CO2 Growth-Rate

tell us about climate-carbon cycle feedbacks ?

Fossil Fuels

Total

Land-use Change

2003 Anomaly

Years after Volcanic EruptionsEl Chichon

PinatuboMt Agung

Jones and Cox, 2005

Interannual variability of CO2 as

a constraint on the sensitivity

of tropical land carbon to

climate change

(Booth, Huntingford)

Decompose the Land Carbon Sink into

CO2

T

Degenerate unless interannual variability is included !

(MPI Model)

Relationship betweenInterannual Variability in

Tropical Land Carbon Sink and Temperature (1960-2010)

(MPI Model)

Best linear fit to detrended T

Residual Trend is assumed to be due to CO2 fertilization

(MPI Model)

Good fit to logarithmic dependence on CO2

dCL/dt + CL = CO2(0) log{CO2/CO2(0)} + T

Simple Model for Evolution of Tropical Land Carbon Sink

CL is the change in tropical land carbon

is derived from interannual variability

is from fit to non-T related trend

is the mean turnover time of tropical land carbon

For dCL/dt << CL and CO2 /CO2(0) ~ 1 this model

reduces to CL = {CO2 - CO2(0)} + T

As Friedlingstein et al., 2006

Simple Model fit to GCM (MPI)- Tropical Land Carbon Sink

Period of Fit

Simple Model fit to GCM (MPI)- Change in Tropical Land Carbon

HADLEY

IPSLCCSM1

LLNL

Fails to PredictStrongNon-linearity

Ongoing Work

Extension of simple model to capture non-linearities.

Estimation of climate impacts on tropical land carbon from the Mauna Loa CO2 record.

Constraints of climate-carbon cycle model projections.

Amazon Forest Dieback as a

Climate Tipping Point –

Time-series Precursors ?

(Huntingford, Lenton, Vitolo)

Map of potential policy-relevant tipping elements in the climate system, updated from ref. 5 and overlain on global population density

Lenton T. M. et.al. PNAS 2008;105:1786-1793

©2008 by National Academy of Sciences

Tipping Points (Lenton et al., 2008)

Slide from Tim Lenton

Evolution and Variability of the Tropical Land Carbon

Sink(Hadley Model)

Tipping Point Precursors ?(Hadley Model)

Ongoing Work

Spatial correlation as a possible precursor of Amazon dieback (Huntingford, Lenton).

Test of “slowing down” as forewarning of Amazon dieback against an ensemble of climate-carbon cycle projections (Booth, Cox).

Development of simple models to demonstrate different types of tipping points and to investigate their precursors (Vitolo, Ashwin).

Any Questions ?