Title: A dynamical-system perspective on carbon and water vulnerabilities: views at global and local scales
1A 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
2Outline
- 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
3Global 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
4Emissions, 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)
5Present radiative forcing
IPCC AR4, WG1 SPM, second draft (24-mar-2006)
6The 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
7Terrestrial C vulnerabilities
- Drivers
- A atmospheric composition
- B climate
- C land use
8Vulnerable 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.
9Vulnerabilities in the carbon cycle a simple
model
- Dynamic equations for 8 state variables
10Forcing CO2 emission flux
11Temperature, CO2 data and predictions
?TA (degK)
- Global temperature record
- Amospheric CO2 record
- Climate sensitivity to CO2
- ?CO2 0.008 K/ppm
CA (ppm)
12Vulnerability 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
13Vulnerability 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
14Outline
- 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
15Potential 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)
16Response 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
17Global 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
18Spatial 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)
19Spatial 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
20Observed precipitation trends (1900 to 2000)
IPCC (2001) Third Assessment
21Partition 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)
22Steady 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
23Steady water balance similarity approach
- E/Q as a function of P/Q
- Sensitivity of runoff to P to Q
24Outline
- 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
25Annual mean temperature Australia and global land
26Australian 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)
27Correlation between temperature and rainfall
Maximum temperature and rainfall Cloudless days
are rainfree and hot
Minimum temperature and rainfall Cloudless nights
are rainfree and cool
28Water 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
29Test area Murrumbidgee basin
Murrumbidgee basin
3081
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
3181
82
83
84
Murrumbidgee Total Evaporation (mm d-1)
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
00
01
02
03
04
05
32Predicted 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
33Australia vegetation greenness trends 1990-2005
NDVI, FC
NDVI (various)
fraction cover FC from GlobCarbon LAI
34Summary 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
35Outline
- 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
36Modelling 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
37Basic 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
38Dynamics 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
39Simplified 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)
40Simplified terrestrial BGC model trajectories
41Simplified terrestrial BGC model equilibrium
points
- At equilibrium, x2 and x1 satisfy
- Either 1 or 3 equilibrium points (A, B, C)
42Simplified 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
43Simplified 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
44Final 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 46Wetland 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)
47The 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
48Vulnerabilities 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
49Vulnerabilities in the carbon cycle a simple
model
- Phenomenological equations
50Is 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
51Carbon 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
52Biosphere-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
53Basic 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
54Basic 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)
55Extended 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
56Extended BH model dimensionless form and
equilibrium points
- Dimensionless forms of b, h, t
- Dimensionless form of extended BH model
- Equilibrium points
57Extended BH modelbehaviour of coexistence
equilibrium point (B)
- Dependence on W (resource condition index) and a1
(biomass limitation of NPP)
58Extended 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
59Extended BH model trajectories on (b,h) plane
increasing growth rate
declining resource condition
increasing resource limitation on harvest
increasing resource limitation on production
60Extended 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