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

1
A dynamical-system perspective on carbon and
water vulnerabilities views at global and local
scales
Michael Raupach and Pep Canadell CSIRO Marine and
Atmospheric Research, Canberra, Australia Global
Carbon Project (IGBP-IHDP-WCRP-Diversitas)
Canberra, 5-9 June 2006
2
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

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Global atmospheric carbon budget

• http//lgmacweb.env.uea.ac.uk/e415/co2/carbon_bud
  get.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

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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)

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Present radiative forcing
IPCC AR4, WG1 SPM, second draft (24-mar-2006)
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The changing carbon cycle 1850-2100
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
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Terrestrial C vulnerabilities

• Drivers
• A atmospheric composition
• B climate
• C land use

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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.
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Vulnerabilities in the carbon cycle a simple
model

• Dynamic equations for 8 state variables

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Forcing CO2 emission flux
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Temperature, CO2 data and predictions
?TA (degK)

• Global temperature record
• Amospheric CO2 record
• Climate sensitivity to CO2
• ?CO2 0.008 K/ppm

CA (ppm)
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Vulnerability of peatland and frozen Ceffect on
CO2

• CP0 400 PgC, CF0 500 PgC, kPT kFT
  0.001 y?1 K?1

CA (ppm)
A1 vulnerable peatland C, frozen C extra 100
ppm of atmospheric CO2
A2
A1
B2
B1
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Vulnerability of peatland and frozen Ceffect on
temperature

• CP0 400 PgC, CF0 500 PgC, kPT kFT
  0.001 y?1 K?1

?TA (degK)
A1 vulnerable peatland C, frozen C extra 0.8
degK warming
A2
A1
B2
B1
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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

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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 precipitationCompetiti
  on for soil water (transpiration, soil
  evaporation, runoff, drainage)

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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
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Global equilibrium evaporation?
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

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Spatial distribution of precipitationPresent
global and continental water budgets

• Global precipitation evaporation (PrecGlobe
  EvapGlobe)
• (AreaGlobePrecGlobe AreaOceanPrecOcean
  AreaLandPrecLand) (likewise for Evap)
• (PrecLand EvapLand RunoffLand)
  (likewise for
  ocean)

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Spatial distribution of precipitationPrecipitatio
n 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/backgroun
d/scenarios/found/fig20.html
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Observed precipitation trends (1900 to 2000)
IPCC (2001) Third Assessment
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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)

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Steady water balance similarity approach

• Normalise with potential evap Qplot E/Q against
  P/Q
• Normalise with precipitation Pplot E/P against
  Q/P

a2,3,4,5
a2,3,4,5
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Steady water balance similarity approach

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

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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

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Annual mean temperature Australia and global land
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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-bi
  n/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)

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Correlation between temperature and rainfall
Maximum temperature and rainfall Cloudless days
are rainfree and hot
Minimum temperature and rainfall Cloudless nights
are rainfree and cool
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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

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Test area Murrumbidgee basin
Murrumbidgee basin
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Murrumbidgee Relative Soil Moisture (0 to 1)
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90
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92
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97
98
99
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01
02
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81
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Murrumbidgee Total Evaporation (mm d-1)
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96
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01
02
03
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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

Goobarragandra410057
Adelong410061
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Australia vegetation greenness trends 1990-2005
NDVI, FC
NDVI (various)
fraction cover FC from GlobCarbon LAI
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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

35
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

36
Modelling water, carbon and nutrient
cyclesDynamical 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 variables p 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

37
Basic dynamical systems theoryequilibrium
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 ?m have negative real parts gt xQ is a stable
  equilibrium point
• Imaginary parts of ?m determine oscillatory
  behaviour of solution near xQ

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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

x
x
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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)

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Simplified terrestrial BGC model trajectories
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Simplified terrestrial BGC model equilibrium
points

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

42
Simplified terrestrial BGC modelcubic 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 gt 0 point A has x1Q (A) gt
  0 (stable "quiescence")

C
A, B
B
A
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Simplified BGC modeleffect 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

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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 precipitationGlobal 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

45

• Hilary Talbot

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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 gt melting
• CO2, CH4 exchanges both important
• Vulnerable 100 PgCeq

Gruber et al. (2004, SCOPE-GCP)
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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

Hungate et al. (2003) Science
48
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

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Vulnerabilities in the carbon cycle a simple
model

• Phenomenological equations

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Is terrestrial C currently vulnerable?Observed
vegetation greenness trends (2)
1980s d(NDVI)/dt Summer 1982-1991
1990s d(NDVI)/dt Summer 1994-2002

• 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
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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 ltlt 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)
• gt DC 0.2 GtC 0.2 PgC 200 MtC 730 Mt
  CO2
• Compare Australian GHG emissions (2002 NGGI)
  were 550 Mt CO2eq

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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
Extraction of biomass by humans
Respiration of biomass
Surplus in biomass extraction
Population growth rate
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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) lt 1
• Resource condition index W (biomass B) /
  (biomass A)
• Three dimensions biomass B, humans H, time
  T
• Five parameters
• p B T?1 biomass production
• k T?1 biosphere decay rate
• c H?1 T?1 rate of biomass extraction per
  human
• m B H?1 T?1 human maintenance requirement
• r H B?1 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

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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)
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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

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Extended BH model dimensionless form and
equilibrium points

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

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Extended BH modelbehaviour of coexistence
equilibrium point (B)

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

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Extended BH model flow fields

• Flow fields on (x1, x2) plane
• W  0.2 (system -gt point B)
• W 0.5 (system -gt point B)
• W 1.0 (system -gt point A)
• Details
• Parameters V  1, a1  a2  0.5
• x1 (horizontal) axis 0 to 1.2x2 (vertical)
  axis 0 to 0.5

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Extended BH model trajectories on (b,h) plane
increasing growth rate
declining resource condition
increasing resource limitation on harvest
increasing resource limitation on production
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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