Michael Raupach and Pep Canadell CSIRO Marine and Atmospheric Research, Canberra, Australia

A dynamical-system perspective on carbon and water vulnerabilities: views at global and local scales

Michael Raupach and Pep CanadellCSIRO Marine and Atmospheric Research, Canberra, Australia

Global Carbon Project (IGBP-IHDP-WCRP-Diversitas)

Canberra, 5-9 June 2006

Outline

Vulnerabilities in the global carbon cycle

Vulnerabilities in the global water cycle

Regional scale vulnerabilities (mainly Australia)

• Water cycle

• Vegetation responses

A dynamical systems framework

• Example: biosphere-human system

Global atmospheric carbon budget

http://lgmacweb.env.uea.ac.uk/e415/co2/carbon_budget.htmlCorinne LeQuere

Data Sources: Land Use:

Houghton (1999) Tellus

Fossil Fuel: Marland et al (2005) CDIAC

Ocean: Buitenhuis et al (2005) GBC

Atmosphere: Keeling and Whorf (2005) CDIAC

Terrestrial: difference

Emissions, CO2, temperature

150-year records of:

Anthropogenic CO2 emissions from fossil fuel burning

Changing atmospheric CO2 concentrations

Changing global mean temperatures (from instrumental record with effects of urbanisation removed)

0

1

2

3

4

5

6

7

1850 1900 1950 2000Year AD

Fo

ssil

Fu

el E

mis

sio

n (

GtC

/y)

Emissions

280

300

320

340

360

380

1850 1900 1950 2000Year AD

Atm

oap

her

ic [

CO

2] (

pp

mv) [CO2]

2 ppm/year

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1850 1900 1950 2000Year AD

Tem

per

atu

re (

deg

C)

Temperature 0.02 C/year= 0.2 C/decade

Present radiative forcing

IPCC AR4, WG1 SPM, second draft (24-mar-2006)

The changing carbon cycle 1850-2100

C4MIP = Coupled Climate Carbon Cycle Model Intercomparison Experiment

Intercomparison of 8 coupled climate-carbon cycle models

Uncertainty (range among predictions) is comparable with uncertainty from physical climate models and emission scenarios

Friedlingstein et al. 2006, in press

Atmospheric CO2

Land C uptake

Ocean C uptake

NOW

temperature implication: 2 to 3 degC

present land sink (2 to 3 GtC/y) becomes a source

Terrestrial C vulnerabilities

Drivers:

A: atmospheric composition

B: climate

C: land use

Vulnerable land and ocean carbon pools (2000-2100)

Gruber et al. (2004)In: Field CB, Raupach MR (eds.) (2004) The Global Carbon Cycle: Integrating Humans, Climate and the

Natural World. Island Press, Washington D.C. 526 pp.

Vulnerabilities in the carbon cycle: a simple model

Dynamic equations for 8 state variables

22

non CO Global temperature: radiative forcing

Atmospheric C:

Terrestrial biospheric C:

Ocean (marine) C:

Marine pCO

A ACO

A B M P FEmis LUC

BNPP R LUC

MMA

dT dC

dt dtdC dC dC dC dC

F Fdt dt dt dt dtdC

F F FdtdC

Fdt

2 22

pCO pCO:

Peatland C:

Frozen (tundra) C:

Terrestrial mineral N:

M

M

PPA

FFA

BNDep NLoss NUptake NMob

d dC

dt C dt

dCF

dtdC

FdtdN

F F F Fdt

Forcing: CO2 emission flux

1800 1850 1900 1950 2000 2050 2100

5

10

15

20

25

30

Emission: GtCy data scenarios: ROGBA1,A2,B1,B2

data

A1

A2

B1

B2FEmis (

Pg

C/y

)

ta degK: data, model

1850 1900 1950 2000 2050 2100

0

1

2

3

4

5

1850 1875 1900 1925 1950 1975 2000

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Temperature, CO2 : data and predictions

Global temperature record

Amospheric CO2 record

Climate sensitivity to CO2 :

CO2 = 0.008 K/ppm

TA (degK)

CO2 ppm: data, model

1750 1800 1850 1900 1950 2000 2050 2100

300

400

500

600

700

800

900

1750 1800 1850 1900 1950 2000

260

280

300

320

340

360

380

400CA (ppm)

Vulnerability of peatland and frozen C:effect on CO2

CP0 = 400 PgC, CF0 = 500 PgC, kPT = kFT = 0.001 [y1 K1]CO2 ppm: data, A1:ref, A2, B1, B2, A1peat

1750 1800 1850 1900 1950 2000 2050 2100

400

600

800

1000

1200

1750 1800 1850 1900 1950 2000 2050 2100

260

280

300

320

340

360

380

400

A1A2

B2

B1

A1 + vulnerable peatland C, frozen C: extra 100 ppm of atmospheric CO2

CA (ppm)

Vulnerability of peatland and frozen C:effect on temperature

CP0 = 400 PgC, CF0 = 500 PgC, kPT = kFT = 0.001 [y1 K1]ta degC: data, A1:ref, A2, B1, B2, A1peat

1850 1900 1950 2000 2050 2100

0

1

2

3

4

5

6

7

1850 1900 1950 2000 2050 2100

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

A1A2

B2

B1

A1 + vulnerable peatland C, frozen C: extra 0.8 degK warming

TA (degK)

Outline




• Water cycle




Potential vulnerabilities in the water cycle

1. Changes in global mean precipitation

2. Changes in large-scale spatial distribution of precipitation

3. Changes in temporal distribution of precipitation Interannual variability, seasonal cycling, frontal and convective rainfall

4. Changes in partition of precipitationCompetition for soil water (transpiration, soil evaporation, runoff, drainage)

Response of global precipitation to global temperature change(IPCC Third Assessment Report, WG1)

Figure 9.18: Equilibrium climate and hydrological sensitivities from AGCMs coupled to mixed-layer ocean components; blue diamonds from SAR, red triangles from models in current use (LeTreut and McAvaney, 2000 and Table 9.1)

Source: IPCC (2001) Climate Change 2001: The Scientific Basis, p. 560

1.2% per deg C

Global equilibrium evaporation?

Physical result: For any semi-closed system supplied with energy, the evaporation rate settles to equilibrium evaporation in the long-term limit

• High generality: any mixing, any spatial distribution of evaporating surfaces

Hypothesis: the main evaporating parts of the atmosphere are approximately thermodynamically closed, and therefore evaporate at the equilibrium rate.

Global water balance:

• A = available energy flux, = dimensionless slope of saturation humidity

A simple sum:

Choosing T: Global average Bowen ratio = 7/24 = 0.29; 1/ = 0.29 at 28 oC

?Precipitation Evaporation

1A

Equilibrium evaporation:Raupach (2001) QJRMSRaupach (2000) BLM

aerosol andthermodynamic response cloud feedback

1 1 1 1

1

dP dE d dA

P dT E dT dT A dT

1.2% per C at 28 C ???

Spatial distribution of precipitation:Present global and continental water budgets

Global precipitation = evaporation (PrecGlobe = EvapGlobe)

(AreaGlobePrecGlobe = AreaOceanPrecOcean + AreaLandPrecLand) (likewise for Evap)

(PrecLand = EvapLand + RunoffLand) (likewise for ocean)

-200

0

200

400

600

800

1000

1200

1400

1600

1800

An

nu

al

Wate

r F

lux (

mm

)

PrecipitationRunoffEvaporation

-200

0

200

400

600

800

1000

1200

1400

1600

1800

An

nu

al

Wate

r F

lux (

mm

)

PrecipitationRunoffEvaporation

JJA DJF

Can

adia

n:

CG

CM

1H

adle

y: H

adC

M2

Spatial distribution of precipitationPrecipitation change through 21st century = (Y2100 - Y2000)/Y2000 (%)

US National Assessment of the Potential Consequences of Climate Variability and Change (2003)http://www.usgcrp.gov/usgcrp/nacc/background/scenarios/found/fig20.html

IPCC (2001)Third Assessment

Observed precipitation trends (1900 to 2000)

Partition of precipitation Quasi-steady-state water balance: a similarity approach

Time averaged water balance in the steady state:

Dependent variables: E (mean total evaporation) R (mean total runoff)

Independent variables: P (mean precipitation) = water supplyQ (mean potential evaporation) = water demand

Similarity assumptions (Fu 1981, Zhang et al 2004):

Solution (Fu 1981, Zhang et al 2004) finds E and R (with parameter a):

1

1

,

,

aa a

aa a

E P Q P Q P Q

R P Q P Q Q

1 1

2 2

, with 0 0 (wet limit)

, with 0 0 (dry limit)

E P f Q E P f

E Q f P E Q f

0dW dt P E R

Normalise with potential evap Q:plot E/Q against P/Q

Normalise with precipitation P:plot E/P against Q/P

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

P/Q

E/Q

NECoastSECoastTasAgricNtropicsArid2345

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

Q/P

E/P

NECoastSECoastTasAgricNtropicsArid2345

Steady water balance: similarity approach

dry wet

wet dry

a=2,3,4,5

a=2,3,4,5

Steady water balance: similarity approach

E/Q as a function of P/Q

Sensitivity of runoffto Pto Q

1 2 3 4 5PQ1

2

3

4

5sensR to P

1 2 3 4 5PQ

-5

-4

-3

-2

-1

sensR to Q sens to =

P RR P

R P

sens to =Q R

R QR Q

1 2 3 4 5PQ0.2

0.4

0.6

0.8

1

EQa=2,3,4,5

dry wet

Outline




• Water cycle




Annual mean temperature: Australia and global land

Temperature anomalies 1900-2004: Australian continent and global averageAustralia: http://w w w .bom.gov.au/cgi-bin/silo/reg/cli_chg/timeseries.cgi

Globe: http://cdiac.esd.ornl.gov/trends/temp/jonescru/data.html

-1

0

1

2

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

tem

pe

ratu

re a

no

ma

ly:

de

g C Australia

Globe

Australian climate variability over 100 yearsRainfall

Sources:

Lavery, B., Joung, G. and Nicholls, N. (1997). An extended high-quality historical rainfall dataset for Australia. Aust. Meteorol. Mag 46, 27-38

BoM climate data set (http://www.bom.gov.au/cgi-bin/silo/reg/cli_chg/timeseries.cgi)

SILO gridded data set (Queensland Department of Natural Resources, Mines and Energy)

BoM gridded data set (Jones, Plummer et al 2005, part of Australian Water Availability Project)

Annual rainfall: Australia

0

100

200

300

400

500

600

700

800

900

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

rain

fall

(mm

/y)

Lavery et al 1997BoM climateSILO griddedBoM gridded

http://www.bom.gov.au/cgi-bin/silo/reg/cli_chg/timeseries.cgi

Correlation between temperature and rainfall

Australia

13.5

14

14.5

15

15.5

16

16.5

300 400 500 600 700 800

rainfall (mm/y)

min

imu

m t

emp

(d

egC

)

1910-1960

1961-1995

1996-2005

Australia

27

27.5

28

28.5

29

29.5

30

300 400 500 600 700 800

rainfall (mm/y)

max

imu

m t

emp

(d

egC

)

1910-1960

1961-1995

1996-2005

2001

2002

2003

2004

2005

2000

19981996

1997 1999

2001

2002

2003

2004

2005

2000

1998

1996 19971999

Maximum temperature and rainfall Cloudless days are rainfree and hot

Minimum temperature and rainfall Cloudless nights are rainfree and cool

Water and carbon balances: dynamic model

Dynamic model is of general form dx/dt = f(x, u, p)

All fluxes (fi) are functions fi(state vector, met forcing, params)

Governing equations for state vector x = (W, Ci):

Soil water W:

Carbon pools Ci:

Simple (and conventional) phenomenological equations specify all f(x, u, p)

Carbon allocation (ai) specified by an analytic solution to optimisation of NPP

precipitation transpiration runoff to drainage tosoilChange of soil

rivers groundwaterevaporationwater store

P T S R DdW dt Q Q Q Q Q

heterotrophicChange of C allocated

respirationin pool NPPpartitioned NBP

i i NPP i i

i

dC dt a F k C

11.051.11.151.21.251.31.351.41.451.51.551.61.651.71.751.81.851.91.9522.052.12.152.22.252.32.352.42.452.52.552.62.652.72.752.82.852.92.9533.053.13.153.23.253.33.353.43.453.53.553.63.653.73.753.83.853.93.9544.054.14.154.24.254.34.354.44.454.54.554.64.654.74.754.84.854.94.9555.055.15.155.25.255.35.355.45.455.55.555.65.655.75.755.85.855.95.9566.056.16.156.26.256.36.356.46.456.56.556.66.656.76.756.86.856.96.9577.057.17.157.27.257.37.357.47.457.57.557.67.657.77.757.87.857.97.958

U rban

H orticu lture

C ropping

FertilisedG razing

W oodland &R angeland

Forest

W ater

D ra inageBasins

R oads

R ivers

Melbourne

Adelaide CanberraNarrandera

Hay

Shepparton

Ballarat

Albury

Renmark

100 100 200 km0

W agga

Test area: Murrumbidgee basin

Murrumbidgee basin

J F M A M J J A S O N D

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

Murrumbidgee Relative Soil Moisture (0 to 1)

J F M A M J J A S O N D

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

MurrumbidgeeTotal Evaporation(mm d-1)

0

1

2

3

4

5

6

Predicted and observed discharge 11 unimpaired catchments in Murrumbidgee basin

25-year mean: Jan 1981 to December 2005Prior model parameters set roughly for Adelong, no spatial variation

0

50

100

150

200

250

300

350

400

0 100 200 300 400

ZDisCM = Predicted Discharge [mm/y]

AD

isC

M =

Ob

serv

ed D

isch

arg

e [m

m/y

]

Adelong:410061

Goobarragandra:410057

Both ZDisCM and ADisCM are conditioned on ADisCM>=0 (discharge data avai;able)[m/mth] [m/mth] [mm/y] [mm/y]ZDisCM ADisCM ZDisCM ADisCM

410044:ZDisCM 0.001445 0.00398 17.34367 47.76458 410044 MuttamaCreek@Coolac410038:ZDisCM 0.012694 0.015991 152.3273 191.8946 410038 AdjungbillyCreek@Darbalara410047:ZDisCM 0.003045 0.008373 36.53617 100.4813 410047 TarcuttaCreek@OldBorambola410048:ZDisCM 0.002893 0.00471 34.71892 56.52 410048 KyeambaCreek@Ladysmith410057:ZDisCM 0.015951 0.032463 191.4125 389.5508 410057 GoobarragandraRiver@Lacmalac410061:ZDisCM 0.016078 0.018988 192.932 227.8604 410061 AdelongCreek@BatlowRoad410059:ZDisCM 0.03018 0.03469 362.1584 416.2745 410059 GilmoreCreek@Gilmore410097:ZDisCM 0.001358 0.004217 16.29116 50.60565 410097 BillabongCreek@Aberfeldy410033:ZDisCM 0.009444 0.005276 113.3247 63.31652 410033 MurrumbidgeeRiver@MittagangCrossing410141:ZDisCM 0.007159 0.003185 85.90418 38.21652 410041 AdjungbillyCreek@Darbalara222007:ZDisCM 0.000616 0.001175 7.395969 14.10417 222007 WullwyeRiver@Woolway

Australia: vegetation greenness trends 1990-2005

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1990 1992 1994 1996 1998 2000 2002 2004 2006

ND

VI

Australian continent averageNDVI (MODIS)NDVI (AVHRR BPAL)NDVI (AVHRR CATS2a; BRDF)NDVI (SeaWiFS)NDVI (SpotVGT)FC (from GlobCarbon LAI)

fraction cover FC from GlobCarbon LAI

NDVI (various)

ND

VI,

FC

Summary so far


• BGC vulnerabilities comparable with physical climate and human dimensions

• Quantitative analysis using perturbation of simple carbon-climate model

• Example: vulnerability to peatland, frozen C is ~ 100 ppm or 0.8 degK


• Four kinds of change in water availability through precipitation:Global meanSpatial distributionTemporal distributionPartition


• Current trends are not the same as trends over past 100 years

• Consequences of hot droughts for water availablity and vegetation state

Outline




• Water cycle




Modelling water, carbon and nutrient cycles:Dynamical systems framework

Variables: x = {xr} = set of stores (r) including all water, C, N, P, …

stores f = {frs} = set of fluxes (affecting store r by process s)

m = set of forcing climate and surface variablesp = set of process parameters

Stores obey mass balances (conservation equations) of form (for store r)

Equilibrium solutions:

Fluxes are described by scale-dependent phenomenological equations of form

1 2r r r rss

x t f f f

, ,rs rsf f x m p

, , 0f x m p

, ,t

x f x m p

f x

Basic dynamical systems theory:equilibrium points and local stability

Dynamical system:

Equilibrium points satisfy:

Determine local stability near equilibrium points by solving the linearised system around an equilibrium point xQ:

Solutions:

Stability criteria:

• all λm have negative real parts => xQ is a stable equilibrium point

• Imaginary parts of λm determine oscillatory behaviour of solution near xQ

d dt x f x

0Q f x

is the perturbation of about, where

and is the Jacobian matrix

Q Q

ik i k

d dtJ f x

x x x x xx Jx

J

eigenvalues of

exp , whereeigenvectors of

mm m

m m

t t

Jx v

v J

Dynamics at small and large scales

Most of the systems we study have small-scale and large-scale dynamics

Often we need to infer large-scale dynamics from small-scale dynamics

Small-scale dynamics Large-scale dynamics

Relationship between phenomenological laws [f(x)] at small and large scales:

d dt x f x , withd dt X F X X x

2

prob ...i ji j i j

d x xx x

x X

fF X f x x x f X

x

x

Ff

prob(x)x

Ff

prob(x)

x

Simplified terrestrial biogeochemical model

Pools: (x1, x2) = (plant C, soil C)

Parameters: q1 = 1, q2 = 1 = scales for limitation of production by x1 and x2

k1 = 0.2, k2 = 0.1 = rate constants for fast, slow poolss1 = 0.01 = seed production (constant)

This is the test model used in the Optimisation Intercomparison (OptIC): comparative evaluation of parameter estimation and data assimilation methods for determining parameters in BGC models (see GlobalCarbonProject.org)

1 21 2

1 2 2

1 1 21 1 1

1 1 2 2 Decay from Small "seed to production"Primary Production, modified by both and

21 1 2 2

Flux from Decay out to of

x xx x

x x x

dx x xF t k x s

dt x q x q

dxk x k x

dt

Simplified terrestrial BGC model: trajectories

1 2 3 4 5x1

2

4

6

8

10x2 vary q1 from 0. to 1.

1 2 3 4 5x1

2

4

6

8

10x2 vary q2 from 0. to 1.

1 2 3 4 5x1

2

4

6

8

10x2 vary s1 from 0. to 0.1

1 2 3 4 5x1

2

4

6

8

10x2 vary k2 from 0.05 to 0.2

Simplified terrestrial BGC model: equilibrium points

At equilibrium, x2 and x1 satisfy

Either 1 or 3 equilibrium points (A, B, C)

2 1 1 2

2 31 0 1 1 2 1 3 1

21 2 2 1 10

21 1 2 2 1 1 2 1 2 11

2 0 1 1 2 2 1 1

3

0

with

1

x k x k

j x c c x c x c x

q q k s kc

q k q k s q q k k kc

c F q k q k s kc

f(x) = dx/dt

x

x1Q (stable)

0

x2Q (unstable) x3

Q (stable)

f(x) = dx/dt

x

x1Q (stable)

0

x2Q (unstable) x3

Q (stable)A: stable B: unstable C: stable

Simplified terrestrial BGC model:cubic defining the equilibrium points

Three equilibrium points: A (stable)B (unstable)C (stable)

If seed production s1 = 0: point A is at the origin (stable "extinction")

If seed production s1 > 0: point A has x1Q (A) > 0 (stable "quiescence")

1 1 2 3 4x1

8

6

4

2

2

4

6

8j

0.1 0.1 0.2 0.3 0.4x1

0.04

0.02

0.02

0.04

0.06

0.08

0.1j

C

A, B BA

Simplified BGC model:effect of random forcing

"Log-Markovian" random forcing F(t)(Mean = F0, SDev/mean = 0.5)

k1 = 0.2, s1 = 0.01

k1 = 0.4, s1 = 0.01

k1 = 0.5, s1 = 0.01

k1 = 0.5, s1 = 0 Forcing F(t)

System flips randomly between active and quiescent stable states

• "Blip and Flip" chaos

• NOT Lorenzian chaos 200 400 600 800 1000

12345678

Forcing

200 400 600 800 1000

2.55

7.510

12.515

17.520

k1:0.4, s1:0

200 400 600 800 1000

2.55

7.510

12.515

17.520

k1:0.5, s1:0.01

200 400 600 800 1000

2.55

7.510

12.515

17.520

k1:0.4, s1:0.01

200 400 600 800 1000

2.55

7.510

12.515

17.520

k1:0.2, s1:0.01

Final summary


• BGC vulnerabilities comparable with physical climate and human dimensions

• Quantitative analysis using perturbation of simple carbon-climate model

• Example: vulnerability to peatland, frozen C is ~ 100 ppm or 0.8 degK


• Four kinds of change in water availability through precipitation:Global mean, spatial distribution, temporal distribution, partition


• Consequences of hot droughts for water availablity and vegetation state

Dynamical systems

• Equilibria, stability, cycles, trajectories, thresholds, phase transitions

• Example: simplified BGC model (used in OptIC project)

• "Flip and blip" chaos is some circumstances

Hilary Talbot

Wetland and frozen terrestrial C pools

• 200-800 PgC in wetlands and peatlands• Tropical, temperate, boreal

• CO2, CH4 exchanges both important

• Vulnerable: ~ 100 PgCeq

Gruber et al. (2004, SCOPE-GCP)

• 200-800 PgC in frozen soils• Warming => melting

• CO2, CH4 exchanges both important

• Vulnerable: ~ 100 PgCeq

The nitrogen gap

Modelled terrestrial sink through 21st century (CO2 + climate):

• 260 to 530 PgC

• 16 to 34% of anthropogenic emissions

N required: 2.3 to 16.9 PgN

N available: 1.2 to 6.1 PgN

Vulnerability (as foregone terrestrial C uptake):~ 200 to 500 PgC

Production of New N to 2100Hungate et al. (2003) Science


Aim of analysis: study process perturbations in carbon cycle modelling

Given a trajectory XR(t) from integration of the reference model, can we find properties of a similar perturbed model, if the reference and perturbed phenomenological laws FR(XR) and FP(XP) are similar in some sense?

Reference model:

• Simple C model which approximately replicates mean of C4MIP simulations

Perturbed models:

• Same simple model, including C release from peatland C, frozen C

How results are interpreted:

• Difference XP(t) XR(t) is a measure of the vulnerability associated with extra processes included in FP(XP) beyond FR(XR)

• BUT XR(t) from simple model is not an independent carbon-climate prediction

Reference model:

Perturbed model:

R R R

P P P

d dt

d dt

X F X

X F X


Phenomenological equations

0 2

2

2

1 0 2 2 0

0

2

CO emission from fossil fuels: prescribed

CO emission from LUC: prescribed

Terrestrial NPP:

Terrestrial respiration: 2

Ocean-air CO flux:

A A T

Emis

LUC

NPP B NPP A NPP NPP

T TR B B

MA Ocean

F

F

F f N F f C F F

F k C

F A v

2 2

2 0

2 0

pCO pCO

Peatland-air CO flux:

Tundra-air CO flux:

N deposition flux: prescribed

Plant uptake N flux:

Net N mobilisation flux: 1

Piston HenryM A

PA P PT A A

FA F FT A A

NDep

NUptake NC NPP

NMob Gas

k

F C k T T

F C k T T

F

F r F

F f

Net N loss flux:

Loss NC R

NLoss N B

r F

F k N

Is terrestrial C currently vulnerable?Observed vegetation greenness trends (2)

Gains from earlier onset of growing season are almost cancelled out by hotter and drier summers which depress assimilation

Suggests a decreasing net terrestrial C sink

Angert et al. 2005; Dai et al. 2005; Buermann et al. 2005; Courtesy Inez Fung 2005

1980s: d(NDVI)/dt Summer 1982-1991

1990s: d(NDVI)/dt Summer 1994-2002

Carbon consequences of vegetation greenness changes

Model

Let biospheric C obey rate equation dC/dt = FC kC, with mean turnover rate k. If NPP changes suddenly by FC, then while t << 1/k, the change in C is

Assume NPP ~ green leaf cover fraction:

Then biospheric C change associated with a perturbation in green leaf cover is

Numbers

• Take t = 1 year; FC = 1 GtC/y; fGL/fGL = 0.2 (a low value)

• => C = 0.2 GtC = 0.2 PgC = 200 MtC = 730 Mt CO2

• Compare: Australian GHG emissions (2002 NGGI) were 550 Mt CO2eq

C C GL GLF F f f

C GL GLC t F f f

CC t F

State variables: b(t) = biomassh(t) = human population

Equations:

Model for extraction of biomass by humans:

• more humans extract more biospheric resource

• each human extracts more as b increases (b is surrogate for quality of life)

Example of a resource utilisation system: familiar from dynamical ecology

Biosphere-human interaction: basic BH model

;db dh

p kb cbh r cbh mhdt dt

Primary production of

biomass

Extraction of biomass by humans

Respiration of biomass

Surplus in biomass

extraction

Population growth rate

extraction cbh

Basic BH model: equilibrium points

Equilibrium points:

• Point A = biosphere-only equilibrium: unstable to perturbation in h

• Point B = coexistence equilibrium: stable to all perturbationsrequires km/(cp) < 1

Resource condition index W = (biomass B) / (biomass A):

Three dimensions: biomass [B], humans [H], time [T]

Five parameters:• p [B T1]: biomass production• k [T1]: biosphere decay rate• c [H1 T1]: rate of biomass extraction per human• m [B H1 T1]: human maintenance requirement• r [H B1]: growth rate of human population per unit biomass surplus

Two (= 5 3) dimensionless groups:

For the basic BH model, resource condition index is W = U

,U km cp V rm k

Point A: , 0

Point B: ,

QA QA

QB QB

b p k h

b m c h p m k c

QB QAW b b

0.2 0.4 0.6 0.8 1 1.2b

0.1

0.2

0.3

0.4

0.5

0.6

h vary c

0.2 0.4 0.6 0.8 1 1.2b

0.1

0.2

0.3

0.4

0.5

0.6

h vary r

0.2 0.4 0.6 0.8 1 1.2b

0.2

0.4

0.6

0.8

1

h vary p

0.2 0.4 0.6 0.8 1 1.2b

0.2

0.4

0.6

0.8

1

h vary mIncrease p (primary

production)Decrease m (human

maintenance requirement)

Increase c (extraction of biomass by humans)

Increase r (growth rate of humans in response to surplus)

Basic BH model: trajectories on (b,h) plane

Extended BH model

Extend the BH model by including limitation and saturation of both the production and harvest fluxes with respect to biomass:

Three dimensions (B, H, T); seven parameters (p, k, c, m, r, bP, bH)

Four independent dimensionless groups:

Resource condition index:

H

P H

H

H

bdb bp kb cbh

dt b b b b

bdhrcbh rmh

dt b b

1 2, , ,QA

PQA QA

HP

bm m rm bU V a a

cb k b bc p k b

21

QB

QA

b UW

b a U

Extended BH model: dimensionless form and equilibrium points

Dimensionless forms of b, h, t:

Dimensionless form of extended BH model:

Equilibrium points:

1 11 1 21 1 2 1

1 1 2 1

2 1 22 1 2 2

2 1

1,

1

,1

a xdx Vx xf x x x

ds x a U a x

dx Vx xf x x Vx

ds U a x

1 2

1 2

1 21

Point Z: , 0, 0

Point A: , 1, 0

1Point B: , ,

QZ QZ

QA QA

QB QB

x x

x x

W Wx x W

V a W

1 2, ,QA QA

P P

b b h bx x s kt

b p k b rb r p k b

Extended BH model:behaviour of coexistence equilibrium point (B)

Dependence on W (resource condition index) and a1 (biomass limitation of NPP)

0.2 0.4 0.6 0.8 1x1

0.2

0.4

0.6

0.8

1x2

curves: a1 = 0 to 1

Resource condition index W varies parametrically from 0 to 1 along each curve

Extended BH model: flow fields

Flow fields on (x1, x2) plane

• W = 0.2 (system -> point B)

• W = 0.5 (system -> point B)

• W = 1.0 (system -> point A)

Details:

• Parameters: V = 1, a1 = a2 = 0.5

• x1 (horizontal) axis: 0 to 1.2x2 (vertical) axis: 0 to 0.5

Extended BH model: trajectories on (b,h) plane

0.2 0.4 0.6 0.8 1x1

0.1

0.2

0.3

0.4

0.5

x2 vary a1 from 0. to 2.

0.2 0.4 0.6 0.8 1x1

0.1

0.2

0.3

0.4

0.5

x2 vary a2 from 0. to 2.

0.2 0.4 0.6 0.8 1x1

0.1

0.2

0.3

0.4

0.5

x2 vary w from 0.1 to 1.

0.2 0.4 0.6 0.8 1x1

0.1

0.2

0.3

0.4

0.5

x2 vary v from 0.5 to 2.

declining resource condition

increasing resource limitation on

production

increasing growth rate

increasing resource limitation on harvest

Extended BH model: limit cycles

More oscillatory tendency in the extended BH model than in the basic BH model

Limit cycles occur at:

• small W (poor resource condition)

• large a2 (strong limitation of harvest by biomass)

0.2 0.4 0.6 0.8 1x1

0.2

0.4

0.6

0.8

1

x2 vary w from 0.3 to 0.1

0.2 0.4 0.6 0.8 1x1

0.2

0.4

0.6

0.8

1

x2 vary a2 from 1 to 4

declining resource condition

increasing resource limitation on harvest

Michael Raupach and Pep Canadell CSIRO Marine and Atmospheric Research, Canberra, Australia

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Transcript of Michael Raupach and Pep Canadell CSIRO Marine and Atmospheric Research, Canberra, Australia