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CONSIDER A SPHERICAL EARTH
Stephen E. Schwartz
Upton, New York
Ithaca, New York19 November 2007
http://www.ecd.bnl.gov/steve
GLOBAL ENERGY BALANCEGlobal and annual average energy fluxes in watts per square meter
Schwartz, 1996, modified from Ramanathan, 1987
RADIATIVE FORCING
A change in a radiative flux term in Earth’s radiationbudget, ∆F, W m-2.
Working hypothesis:On a global basis radiative forcings are additive andfungible.
• This hypothesis is fundamental to the radiativeforcing concept.
• This hypothesis underlies much of the assessment ofclimate change over the industrial period.
CLIMATE RESPONSEThe change in global and annual mean temperature,∆T, K, resulting from a given radiative forcing.
Working hypothesis:The change in global mean temperature isproportional to the forcing, but independent of itsnature and spatial distribution.
∆T = λ-1∆F
CLIMATE SENSITIVITYThe change in global and annual mean temperature perunit forcing, λ, K/(W m-2),
λ-1 = ∆T/∆F.
Climate sensitivity is not known and is the objective ofmuch current research on climate change.
Climate sensitivity is often expressed as thetemperature for doubled CO2 concentration ∆T2×.
∆T2× = λ-1∆F2×
∆F2× ≈ 3.7 W m-2
CLIMATE SENSITIVITY ESTIMATESTHROUGH THE AGES
Estimates of central value and uncertainty range from majornational and international assessments
6
5
4
3
2
1
0
∆T2×
, Sen
sitiv
ity to
2 ×
CO
2, °C
190018901880
Arrhenius
Stefan-Boltzmann
2010200019901980
CharneyNRC – – – – IPCC – – – –
1 sigma> 66%
"Likely"
1.5
1.0
0.5
0.0
Sensitivity, K
/(W m
-2)
Despite extensive research, climate sensitivity remains highly uncertain.
COMMITTED WARMING IN CLIMATE MODEL RUNSAtmospheric composition held constant at 2000 value (IPCC, 2007)
models models
} Committedwarming
Temperature continues to increase for composition held constant.
Projected incremental 21st century is 50% beyond warming alreadyrealized.
“COMMITTED WARMING,” “THERMAL INERTIA,”“WARMING IN THE PIPELINE”
“ Additional global warming of ... 0.6˚C is “in the pipeline” and willoccur in the future even if atmospheric composition and other climateforcings remain fixed at today’s values.
Hansen et al, Science, 2005
“ Even if the concentrations of greenhouse gases in the atmosphere hadbeen stabilized in the year 2000, we are already committed to furtherglobal warming of about another half degree.
Meehl, Washington, et al., Science, 2005
“ Even if atmospheric composition were fixed today, global-meantemperature ... rise would continue due to oceanic thermal inertia. Thewarming commitment could exceed 1˚C.
Wigley, Science, 2005
“COMMITTED WARMING,” “THERMAL INERTIA,”“WARMING IN THE PIPELINE” (cont’d)
“ Because of the long time scale required for removal of CO2 from theatmosphere as well as the time delays characteristic of physicalresponses of the climate system, global mean temperatures are expectedto increase by several tenths of a degree for at least the next 20 yearseven if CO2 emissions were immediately cut to zero; that is, there is acommitment to additional CO2-induced warming even in the absenceof emissions.
Friedlingstein and Solomon, PNAS, 2005
WHAT CAN WE LEARN FROMENERGY BALANCE MODELS?
EMPIRICAL DETERMINATIONOF EARTH’S CLIMATE
SENSITIVITY
JUST PUBLISHED IN JGR
STOVE-TOP MODEL OFEARTH’S CLIMATE SYSTEM
STOVE-TOP MODEL OF EARTH’S CLIMATE SYSTEM
dH
dtC
dT
dtQ k T T= = − −( )amb
H = heat content T = temperature
C = system heat capacity
Q = heating rate from stove
Tamb = ambient temperature
Steady State T: T TQ
k∞ = +amb
let Q Q F→ + : ∆TF
k∞ =
Sensitivity: λ− ∞≡ =1 1∆T
F k
Time constant: τ λ= −C 1
Tem
pera
ture
Time
T = T0 + (Tf - T0)(1-e-t/τ )
0 τ
T0
Tf
τ is the time constant of the system response to a perturbation.
DEPENDENCE OF RESPONSE ON SYSTEM HEAT CAPACITY
Tem
pera
ture
Time 0 τ
T0
Tf
T
empe
ratu
reTime
0 τ
T0
Tf
For constant k, ∆T∞ and λ−1 are independent of system heat capacity C.
Time constant τ varies linearly with heat capacity: τ λ= −C 1
Sensitivity can be inferred from τ and C as λ τ− =1 / C.
BILLIARD BALL MODEL OFEARTH’S CLIMATE SYSTEM
BILLIARD BALL TEMPERATURESENSITIVITY AND TIME
CONSTANTEvaluated according to the
Stefan-Boltzmann radiation law
Global energy balance: dH
dtQ E Q T= − = − σ 4
Initially Q T0 04= σ
Temperature sensitivity: ∆ ∆T Qss = −λ 1 ; ∆ ∆T t Q e t( ) = −− −λ τ1 1( )/
For Stefan-Boltzmann planet sensitivity is λS-B-1 =
T
Q4
Relaxation time constant is τ λS-B S-B= = −T C
QC0
00
1
4 , or λ τS-B
S-B,0
1− =C
BILLIARD BALLTEMPERATURE SENSITIVITY
Evaluated according to theStefan-Boltzmann radiation law
For Q S0 0 4= γ / where S0 is the solar constant = 1370 W m-2
and γ is global mean co-albedo = 0.69
Climate sensitivity is λS-B-1 = 0.27 K/(W m-2)
For 2 × CO2 forcing F2× = 3.71 W m-2, ∆T2× = 1.0 K
ENERGY BALANCE MODEL OFEARTH’S CLIMATE SYSTEM
ENERGY BALANCE MODEL OFEARTH’S CLIMATE SYSTEM
Global energy balance: CdT
dt
dH
dtQ E J Ts
s4= = − = −γ εσ
C is heat capacity coupled to climate system on relevant time scale
Ts is global mean surface temperature H is global heat content
Q is absorbed solar energy E is emitted longwave flux
J is 14
solar constant γ is planetary co-albedo
σ is Stefan-Boltzmann constant ε is effective emissivity
ENERGY BALANCE MODEL OFEARTH’S CLIMATE SYSTEM
Apply step-function forcing:
At “equilibrium”
F Q E= −∆( )
∆T Fs( )∞ = −λ 1
Tem
pera
ture
Time 0 τ
T0
T(∞)
λ−1 is equilibrium climate sensitivity λγ
−1 04
= 0 S
fT
J K / (W m )-2
f is feedback factor fd
d T
d
d T= − +
−1
14
140 0
1lnln
lnln
γ ε
Time-dependence: ∆T t F e ts( ) ( )/= −− −λ τ1 1
τ is climate system time constant τ λ= −C 1 or λ τ− =1 / C
TEMPERATURE RESPONSE TOLINEARLY INCREASING
FORCINGβ = dforcing/dtime
Energy balance:
Time-dependence:
CdT
dtt J Ts
S s4= + −β γ εσ
∆T t t e ts( ) [( ) ]/= − +− −βλ τ τ τ1
Tem
pera
ture
Time2τ 3τ 4τ τ 0 . .
∆ T = βλ-1τ Teq
T
λ−1 and τ are the same as before: λ τ− =1 / C
For t /τ >∼ 3, ∆T t ts( ) ( )= −−βλ τ1
Temperature lags equilibrium response by: ∆Tlag = −βλ τ1
APPROACH
Empirically determine heat capacity C andtime constant τ of Earth’s climate systemfrom observations over the instrumentalperiod.
Evaluate sensitivity as λ-1 = τ/C.
DETERMINING EARTH’SHEAT CAPACITY
BY OCEAN CALORIMETRY
HEAT CAPACITY OF EARTH’SCLIMATE SYSTEM FROM GLOBAL
MEAN HEAT CONTENT ANDSURFACE TEMPERATURE TRENDS
CdH dt
dT dt
dH
dT= =/
/s s
C: Global heat capacity
H: Global ocean heat content
Ts: Global mean surface temperature
ZONAL AVERAGE HEAT CONTENT TREND (1955-2003)1018 J (100 m)-1 (1˚ latitude)-1 yr-1
• Heating is greatest in upper ocean, with downwelling plumes.
HEAT CONTENT OF WORLD OCEANS, 1022 J
Levitus et al., 2005
HEAT ABSORPTION BY COMPONENTSOF EARTH’S CLIMATE SYSTEM
The world ocean is responsible for ~84% of the increase in globalheat content. Levitus et al., 2005
GLOBAL TEMPERATURE TREND OVER THE INDUSTRIAL PERIOD
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
Tem
pera
ture
Ano
mal
y, K
2000
2000
1990
1990
1980
1980
1970
1970
1960
1960
1950
1950
1940
1940
1930
1930
1920
1920
1910
1910
1900
1900
1890
1890
1880
1880
1870
1870
1860
1860
GISS, Goddard Institute for Space Studies, NASA, USA CRU, Climatic Research Unit, East Anglia, UK
EMPIRICAL DETERMINATION OF OCEAN HEAT CAPACITY
CdH dtdT dt
= //s
Surfacetemperature Ts:
GISS, CRU
Ocean heatcontent H:
Levitus et al., 2005 -10
-8
-6
-4
-2
0
2
4
6
Oce
an H
eat C
onte
nt A
nom
aly
H, W
yr
m-2
20001990198019701960
1.5
1.0
0.5
0.0
-0.5
Tem
perature anomaly ∆
T, K
Heat Content← L300 L700
L3000
Temperature→
GISS CRU
0.200.150.100.050.00
Fraction equilibrated
3500
3000
2500
2000
1500
1000
500
0
Dep
th, m
150100500
Equlibrated depth, m
3500
3000
2500
2000
1500
1000
500
0
Dep
th, m
20151050
Heat capacity C, W yr m-2 K-1
3500
3000
2500
2000
1500
1000
500
0
Dep
th, m
• ~50% of heat capacity is between surface and 300 m.
• Other heat sinks raise global heat capacity to 17 ± 7 W yr m-2 K-1.
CHARACTERISTIC TIME OFEARTH’S CLIMATE SYSTEM
FROM TIME SERIES ANALYSIS
DETERMINATION OF TIME CONSTANT OF EARTH’S CLIMATESYSTEM FROM AUTOCORRELATION OF TIME SERIES
Recipe (GISS annual global mean surface temperature anomaly Ts)
0.4
0.0
-0.4T
emp.
Ano
mal
y, K
2000198019601940192019001880
1. Remove long term trend; plot the residuals:210
-1-2
Nor
m. R
esid
ual
2000198019601940192019001880
2. Calculate autocorrelogram (& standard deviations; Bartlett, 1948):
-1.0-0.50.00.51.0
Aut
ocor
rela
tion
r
20151050Lag time ∆t, yr
Recipe for determining climate system time constant, continued
3. Examine the lag-1 autocorrelation:
2
1
0
-1
-2
Lag1
Nor
m. R
esid
ual
210-1-2
Normalized Residual
4. Remove the trend; plot the residuals:210
-1-2
Nor
m. R
esid
ual
120100806040200Time, yr
5. Examine for any remaining autocorrelation:
-1.0-0.50.00.51.0
Aut
ocor
rela
tion
2520151050Lag time ∆t, yr
Recipe for determining climate system time constant, continued
6. If no residual autocorrelation (Markov process) calculate timeconstant τ for relaxation of system to perturbation:
r t e t( ) /∆ ∆= − τ or τ( ) / ln ( )∆ ∆ ∆T T r T= − (Leith, 1973)
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
• Time constant τ increases with increasing lag time.
• Implies coupling of Ts to a system of longer time constant.
• On decadal scale time constant asymptotes to 5 ± 1 yr.
• This is the e-folding time constant for relaxation of global meansurface temperature to perturbations on the decadal scale.
THIS RESULT IS ROBUST1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
GISS 1880-2004
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
CRU 1880-2004
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
GISS 1880-1942
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
CRU 1880-1942
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
GISS 1943-2004
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
CRU 1943-2004
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
GISS NH 1880-2005
1086420
Tim
e C
onst
. τ, y
r20151050
Lag time ∆t, yr
CRU NH 1880-2005
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
GISS SH 1880-2005
1086420
Tim
e C
onst
. τ, y
r
20151050Lag time ∆t, yr
CRU SH 1880-2005
SUMMARY RESULTSQuantity Unit Value 1 σ
Effective global heat capacity C W yr m-2 K-1 17 7
Effective climate system time constant τ yr 5 1
Equilibrium climate sensitivity λ τ− =1 / C K/(W m-2) 0.30 0.14
Equilibrium temperature increase for 2 × CO2,∆T2×
K 1.1 0.5
Total forcing over the 20th century,F T20 20
1= −∆ / λW m-2 1.9 0.9
Forcing in 20th century other than GHGs(mainly aerosols), F F F20 20 20
other ghg= −W m-2 -0.3 1.0
Lag in temperature change, ∆Tlag K 0.03
COMPARISON WITH PREVIOUS RESULTS
6
5
4
3
2
1
0
∆T2×
, Sen
sitiv
ity to
2 ×
CO
2, °C
190018901880
Arrhenius
Stefan-Boltzmann
2010200019901980
CharneyNRC – – – – IPCC – – – –
1 sigma
> 66%
"Likely"
This study
1.5
1.0
0.5
0.0
Sensitivity, K
/(W m
-2)
Sensitivity obtained in this study is much lower than that fromclimate models and paleo studies.
WHAT MIGHT BE WRONG WITH THISANALYSIS?
• Ocean heat capacity too great, resulting in low sensitivity.
Erroneous or nonrepresentative data.
Obtaining heat capacity from measurements.
• Time constant too short, resulting in low sensitivity.
Time series too short to give true time constant.
Detrending emphasizes the rapid fluctuations.
• Earth’s climate system is much more complex than can berepresented by a single-compartment model.
Multiple time constants, multiple heat capacities.
CLIMATE SENSITIVITY AND INFERRED20th CENTURY TOTAL AND AEROSOL FORCING
Inverse calculation of forcing as function of climate system time constant τ
λ τ− =1 / C F T C T20 201
20= =−∆ ∆/ /λ τ F F Faer GHG= −20
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Clim
ate
sens
itivi
ty λ
-1, K
/(W
m-2)
16141210864
Climate system time constant τ, yr
5
4
3
2
1
0 ∆
T2×
, K
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
Aerosol forcing over 20
th century, W m
-2
161284
Climate system time constant τ, yr
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Tot
al fo
rcin
g ov
er 2
0th c
entu
ry, W
m-2
Time constant from autocorrelation is τ = 5 ± 1 yr.
Submitted comment suggests τ too small because of length of data record.
Climate sensitivity and inferred forcing depend strongly on time constant.
SUMMARY• Despite intense research Earth’s climate sensitivity
remains uncertain to at least a factor of 2.
• Energy balance considerations and empirical observationsmay usefully refine sensitivity estimates.
• Climate sensitivity can be determined as time constantupon heat capacity.
• The effective heat capacity of Earth’s climate system is17 ± 7 W yr m-2 K-1 ≈ 150 m of the world ocean.
The time constant of Earth’s climate system is 5 ± 1 years.
• Climate system response to greenhouse forcing is innear steady state, with little further warming (due topresent GH gases) “in the pipeline.”
• The equilibrium sensitivity of Earth’s climate system is0.30 ± 0.14 K / (W m-2); ∆T2× = 1.1 ± 0.5 K.
CONCLUDING OBSERVATION
• The time constant, heat capacity and sensitivity of Earth’sclimate system are important integral properties that mightinstructively be examined in model calculations as well asin observations.
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