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

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

Outline

Michael Raupach and Pep Canadell

CSIRO 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.html

Corinne 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)

Present radiative forcing

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

The changing carbon cycle 1850-2100

Atmospheric CO2

NOW

Land C uptake

Ocean C uptake

temperature implication: 2 to 3 degC

- 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

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

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

Temperature, CO2 : data and predictions

TA (degK)

- Global temperature record
- Amospheric CO2 record
- Climate sensitivity to CO2 :

CO2 = 0.008 K/ppm

CA (ppm)

Vulnerability of peatland and frozen C:effect on CO2

- CP0 = 400 PgC, CF0 = 500 PgC, kPT = kFT = 0.001 [y1 K1]

CA (ppm)

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

A2

A1

B2

B1

Vulnerability of peatland and frozen C:effect on temperature

- CP0 = 400 PgC, CF0 = 500 PgC, kPT = kFT = 0.001 [y1 K1]

TA (degK)

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

A2

A1

B2

B1

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

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 precipitationInterannual 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)

1.2% per deg C

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

Global equilibrium evaporation?

???

1.2% per C at 28 C

Equilibrium evaporation:

Raupach (2001) QJRMS

Raupach (2000) BLM

- 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

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)

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

DJF

JJA

Canadian: CGCM1

Hadley: HadCM2

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

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 supply Q (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):

Steady water balance: similarity approach

dry wet

wet dry

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

a=2,3,4,5

a=2,3,4,5

Steady water balance: similarity approach

a=2,3,4,5

dry wet

- E/Q as a function of P/Q
- Sensitivity of runoff to P to Q

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

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)

Correlation between temperature and rainfall

Maximum temperature and rainfallCloudless days are rainfree and hot

Minimum temperature and rainfallCloudless 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

Test area: Murrumbidgee basin

Murrumbidgee basin

J

F

M

A

M

J

J

A

S

O

N

D

81

82

83

84

Murrumbidgee Relative Soil Moisture (0 to 1)

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

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

Goobarragandra:410057

Adelong:410061

Australia: vegetation greenness trends 1990-2005

NDVI, FC

NDVI (various)

fraction cover FCfrom GlobCarbon LAI

Summary so far

- Vulnerabilities in the global carbon cycle
- 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
- Vulnerabilities in the global water cycle
- Four kinds of change in water availability through precipitation: Global mean Spatial distribution Temporal distribution Partition
- Regional scale vulnerabilities (mainly Australia)
- Current trends are not the same as trends over past 100 years
- Consequences of hot droughts for water availablity and vegetation state

- 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

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

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

Dynamics at small and large scales

f

F

f

F

x

prob(x)

prob(x)

x

- 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:

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 pools s1 = 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)

Simplified terrestrial BGC model: equilibrium points

A: stable

B: unstable

C: stable

- At equilibrium, x2 and x1 satisfy
- Either 1 or 3 equilibrium points (A, B, C)

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")

C

A, B

B

A

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

Final summary

- Vulnerabilities in the global carbon cycle
- 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
- Vulnerabilities in the global water cycle
- Four kinds of change in water availability through precipitation:Global mean, spatial distribution, temporal distribution, partition
- Regional scale vulnerabilities (mainly Australia)
- 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

Wetland and frozen terrestrial C pools

- 200-800 PgC in wetlands and peatlands
- Tropical, temperate, boreal
- CO2, CH4 exchanges both important
- Vulnerable: ~ 100 PgCeq

- 200-800 PgC in frozen soils
- Warming => melting
- CO2, CH4 exchanges both important
- Vulnerable: ~ 100 PgCeq

Gruber et al. (2004, SCOPE-GCP)

The nitrogen gap

Production of New N to 2100

- 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

Hungate et al. (2003) Science

Vulnerabilities in the carbon cycle: a simple model

- 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

Vulnerabilities in the carbon cycle: a simple model

- Phenomenological equations

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

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

- 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

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

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

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 dFC, then while Dt << 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 Dt = 1 year; FC = 1 GtC/y; dfGL/fGL = 0.2 (a low value)
- => DC = 0.2 GtC = 0.2 PgC = 200 MtC = 730 Mt CO2
- Compare: Australian GHG emissions (2002 NGGI) were 550 Mt CO2eq

Biosphere-human interaction: basic BH model

- State variables: b(t) = biomass h(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

Primary production of biomass

Respiration of biomass

Extraction of biomass by humans

Population growth rate

Surplus in biomass extraction

Basic BH model: equilibrium points

- Equilibrium points:
- Point A = biosphere-only equilibrium: unstable to perturbation in h
- Point B = coexistence equilibrium: stable to all perturbations requires 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

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

Decrease m (human maintenance requirement)

Increase p (primary production)

Increase c (extraction of biomass by humans)

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

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:

Extended BH model: dimensionless form and equilibrium points

- Dimensionless forms of b, h, t:
- Dimensionless form of extended BH model:
- Equilibrium points:

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

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

curves: a1 = 0 to 1

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

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

increasing growth rate

declining resource condition

increasing resource limitation on harvest

increasing resource limitation on production

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)

increasing resource limitation on harvest

declining resource condition

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