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 ?