GLOBAL SCALE ASSESSMENT OF CLIMATE

SENSITIVITY IN RELATION TO CARBON

DIOXIDE CONCENTRATIONS

Climate model Group IOscar Bjornham, Paul Constantine, Robert Kakarigi, Dag Lindbo,

Emanuel Rubensson, Michael Stockli, and Sara Zahedi

Summer School on Multiscale Modeling and Simulation in ScienceBoson, Stockholm, Sweden, June, 2007

Project supervisors: PhD Heiner Kornich and Prof. Erland Kallen, MISU(Meteorologiska Institutionen, Stockholm Universitet)

The climate system consists of the atmosphere, the oceans, the cryosphere(land ice, snow, sea ice), the lithosphere, and the biomass. The behavior ofthe individual components of the system is governed by processes occurringover a broad range of time and space scales. The components are coupled byphysical, biological, and chemical processes, and the coupled system seemscapable of undergoing fluctuations on all time scales. In addition to theseinternal climatic processes, external processes (such as variability in the so-lar irradiance or human activities) must also be considered.Kutzbach J.E., Quat. Res. 6, 471-480 (1976).

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1 Introduction

Global concentrations of carbon dioxide (CO2), methane (NH4) and dini-trogen monoxide (nitrous oxide, N2O) have increased noticeably as a resultof anthropogenic activities since pre–industrial times (1750). The globalincreases in CO2 concentration are due mostly to fossil fuel utilization andland–use change. Agricultural practices are predominantly responsible forthe increase of NH4 and N2O (IPCC WG1, 2007). Certain greenhouse gases(GHGs) have greater radiative forcing per unit than others (e.g. NH4 is 21times more effective than CO2 at heating the atmosphere). In this reportwe are going to use CO2 equivalent values representing GHGs. In this wayemissions are calculated in terms of how much CO2 would be needed toproduce a similar warming effect. For example, 10 tones of NH4 would havea CO2 eq. of 10 · 21 = 210 tones.

CO2 is the key anthropogenic GHG. According to IPCC WG1 (2007)the global concentration of CO2 in the atmosphere has increased from thepre–industrial times value of approximately 280 ppm to 379 ppm in 2005.The ice core analyses revealed that the atmospheric concentration of CO2

in 2005 exceeds by far the natural range over the last 650,000 years (180 to300 ppm). The yearly CO2 concentration growth–rate was larger during thelast decade (1995 – 2005 average: 1.9 ppm per year), than it has been sincethe beginning of continuous direct atmospheric measurements (1960 – 2005average: 1.4 ppm per year) even though there is year–to–year variability ingrowth rates (IPCC WG1, 2007).

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2 Socioeconomic scenarios for CO2 emission

Before we present our simulation results we will consider possible socioe-conomic scenarios. These scenarios involve different expectation values forCO2 emissions which in turn results in different probability distributions forthe global CO2 stabilization levels. These stabilization levels will make upour starting point for simulations.

Figure 1: CO2 emissions, CO2 concentrations, CO2 equivalent radiativeforcing and global mean temperature predictions for six possible future sce-narios described in the text. Shaded areas around the expectation valuesplotted with solid lines indicate standard deviations for the model tunings.

In consistence with Intergovernmental Panel on Climate Change (IPCC)reports, we are going to use the Special Report on Emissions Scenarios(SRES). The SRES scenarios (Figure 1) assume that no climate policiesare implemented and they represent outcomes of distinct storylines of eco-nomic development and demographic and technological change (Nakienoviand Swart, 2000). Approximate CO2 equivalent concentrations correspond-ing to the computed radiative forcing due to anthropogenic greenhouse gasesand aerosols in 2100 for the SRES B1, A1T, B2, A1B, A2 and A1FI illus-trative marker scenarios are about 600, 700, 800, 850, 1250 and 1550 ppmrespectively (IPCC WG1, 2007). There are some uncertainties associatedwith the 2100 values due to model parametrizations. For example, the un-

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Figure 2: Several steps from emissions to climate response contribute tothe overall uncertainty of a climate model projection. In a comprehensiveclimate model, physical and chemical representations of processes permit aconsistent quantification of uncertainty. Planet Simulator Model is based onthe marker scenarios for CO2 derived from the modeled radiatively activespecies step. Middle row graphs are for illustration only. Modified fromIPCC WG1, 2007.

certainties are −12% to +10% for the ISAM model and −14% to +31% forthe Bern–CC model. These uncertainties reflect incomplete understandingof climate sensitivity and the carbon cycle at the end of the 20th centuryIPCC TAR WG1 (2001). According to IPCC WG1 (2007) uncertainty inclimate change modeling arise in various consecutive steps. Figure 2 showsseveral steps taken in the modeling that contribute to the overall uncer-tainty. At the same time, the figure illustrates the step of modeled concen-trations of radiatively active species (e.g. CO2) where our marker scenariosfor CO2 originate and indicates the relative position of our simulations withthe Planet Simulator. In Section 6 we will use beta distributions of CO2 sta-bilization levels that roughly correspond to the scenarios above to computefirst and second moments of equilibrium temperature and globally averagedtemperature.

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The Emission Scenarios of the IPCC SRES for future worlds

A1. World with very rapid economic growth and the rapid introductionof new and more efficient technologies. Global population peaks inmid-century and declines thereafter. Major underlying themes areconvergence among regions, capacity building and increased culturaland social interactions, with a substantial reduction in regional differ-ences in per capita income. The A1 scenario family develops into threegroups that describe alternative directions of technological change inthe energy system: fossil intensive (A1FI), non-fossil energy sources(A1T), or a balance across all sources (A1B) (where balanced is definedas not relying too heavily on one particular energy source).

A2. A very heterogeneous world. The underlying theme is self relianceand preservation of local identities. Fertility patterns across regionsconverge very slowly, which results in continuously increasing popula-tion. Economic development is primarily regionally oriented and percapita economic growth and technological change more fragmentedand slower than other storylines.

B1. A convergent world with the population that peaks in mid-century anddeclines thereafter, as in the A1 storyline, but with rapid change ineconomic structures toward a service and information economy, withreductions in material intensity and the introduction of clean and re-source efficient technologies. The emphasis is on global solutions toeconomic, social and environmental sustainability including improvedequity.

B2. A world in which the emphasis is on local solutions to economic, so-cial and environmental sustainability. It is a world with continuouslyincreasing global population, at a rate lower than A2, intermediatelevels of economic development, and less rapid and more diverse tech-nological change than in the B1 and A1 storylines. While the scenariois also oriented towards environmental protection and social equity, itfocuses on local and regional levels.

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3 Mathematical and Numerical Overview

3.1 The Planet Simulator

When meteorologists and climatologists speak of climate models they talkabout what physical processes the model simulates, such as clouds, precip-itation, radiation and winds. From a more general mathematical and nu-merical point of view this says relatively little of the simulation as a whole.The purpose here is to show how our climate modle is constructed, whatit’s mathematical structure is, how it is computed and how the inherentmulti-scale features of the climate enter.

The simulation tool used in this work is called the Planet Simulator. Atthe macroscopic, global, level the model contains four systems

• Atmosphere

• Ocean

• Ice

• Vegetation.

These are separate modules computationally, but they are coupled at themacro level by transfer of energy in different forms. In the code that we haverun the atmospheric simulation is fairly advanced, while the ocean, ice andvegetation simulators are more rudimentary. Each of the modules contain

their own separate model for large scale dynamics, described by systems ofPDEs. Here we shall take a look at the atosphere as an example.

3.2 Example: Atmosphere

The Planet Simulator includes humidity and boundary layer in addition toPUMA-2 model as one of the major components. PUMA stands for thePortable University Model of the Atmosphere. Basically this is a circulationmodel written in Fortran 90 and developed at the Meteorological Instituteof the University of Hamburg. PUMA originated in a numerical predictionmodel that was altered to include only the most relevant processes in theatmosphere.

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The basic macro model is the Navier-Stokes compressible equations ona sphere,

∂ζ

∂t= −~ν · ∇(ζ + f)− ω

∂ζ

∂p− (ζ + f)∇ · ~ν + ~k · (∂~ν

∂p×∇ω) + Pζ

∂D

∂t= ~k · ∇ × (ζ + f)~ν −∇ · (ω∂~ν

∂p)−∇2(φ +

~ν2

2) + PD

∇ · ~ν +∂ω

∂p= 0.

Together with the hydrostatic approximation and laws from thermody-namics the system can be closed and simplified. Of particular interest arethe source terms, Pζ and PD. Energy transfer from e.g. atmosphere to oceanenters the mathematical model by

• Source terms (e.g. water vapor formation)

• Boundary conditions (e.g. wind stress).

3.2.1 Adding complexity to the atmosphere: Macro-Micro ex-change

The atmospheric model in our simulation, PUMA-2, takes many physicalprocesses into account. These include

• Long and short wave radiation

• Clouds

• Precipitation

• Horizontal and vertical diffusion

Most of these have their own mathematical models in terms of ODEs orsimple energy equations. The question is then how these enter into themacro scale calculations.

The answer is again in the source terms of the PDEs. At a particular timestep the cloud formation, radiation etc. are computed from the macroscopicquantities. The results from these micro scale calculations are then fedinto the source terms of the PDE at the next time step. The number andcomplexity of these physical processes determine the overall complexity ofthe climate model.

3.3 Numerics

3.3.1 Discretization of macro PDEs

In the case of the atmospheric model, the PDEs are computed using

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• Spectral transform method.

• Finite differences in the vertical direction.

• Semi-implicit time stepping with mode splitting. The fast propagationof the gravity waves reduce the time step of explicit schemes. Thedifferential equations are separated into fast (linear) gravity modescalculated implicitly in the spectral space and slower non-linear termswhich are solved explicitly in the physical space.

3.3.2 Time scales

The relevant time scales in the different modules are

• Wind speeds ∼ 100 m/s ⇒ 15 min. timesteps in atmosphere.

• Ocean and Ice: Timesteps ∼ 2 hours.

• Vegetation: Timescales are even longer, ∼ 1 week.

This means that from the point of view of the atmosphere, the ocean isconstant for 32 time steps before the ocean is updated.

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0 10 20 30 40 500

5

10

15

20

Year

Yea

rly a

vera

ge te

mpe

ratu

re [

o C]

CO2 [ppm] =1620CO2 [ppm] =1440CO2 [ppm] =1260CO2 [ppm] =1080CO2 [ppm] =900CO2 [ppm] =720CO2 [ppm] =540CO2 [ppm] =360CO2 [ppm] =180

Figure 3: Yearly averaged temperatures for a simulation running for fiftyyears for different levels of CO2. All simulations were started at an initialstate at rest.

4 Climate equilibrium simulations

Predictions resulting from simulations with global climate models can bedivided into three classes.

1. Short–range weather predictions. Initial value problem.

2. Medium–to–long range weather or climate predictions. Mixed initialand boundary value problem.

3. Climate equilibrium simulations. Boundary value problem.

In this work we focus on equilibrium simulations for different rates of CO2concentrations. We have therefore fixed the CO2 concentration in a rangebetween 180–1620 ppm and have run simulations until equilibrium and con-tinue for some years and save the data at the end of each month to getstatistical data. In Figure 3 it can be seen that the temperature stabilizesafter simulations for ≈ 10−20 years except for the lowest CO2 concentration.Hence, the final 30 years can be used for averaging.

Temperature In Figure 4 we compare our equilibrium temperature changesrelative to 1990 for different CO2 stabilization levels with the data reportedby IPCC. The dashed line was obtained from a cubic spline fit of our data.The results agree well. Figure 5 shows our modeled equilibrium temperature

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(a) IPCC results

500 600 700 800 900 10000

1

2

3

4

5

6

7

8

9

10

CO2 stabilization level [ppm]

Tem

pera

ture

cha

nge

rela

tive

to 1

990

[ o C]

(b) This work.

Figure 4: Predicted temperature change relative to 1990. Comparison be-tween IPCC predictions, panel (a), and our predictions modeled with thePlanet Simulator, panel (b)

for two different CO2 concentrations. The left panel approximately repre-sents what the Earths temperature distribution looks like today. With in-creasing CO2 levels, the Earths equilibrium temperature distribution shouldrise. This is what is indicated in the plot on the right hand side where theimage represents the equilibrium temperature distribution for the CO2 levelof 1000 ppm. As we can see, the African plains are the most affected regionby this increase in CO2 levels. At the same time, the temperature increaseseverywhere on Earth giving rise to overall higher mean temperature values.Of special concern is the warming of the poles which can lead to melting ofthe sea ice cover.Note that there is a slice of data missing in the plots of the global images.This is due to a quirk in the MATLAB M MAP library which was used forvisualizing the data 1.

Sea ice Ice plays an important role in our climate. When ice melts, wa-ter becomes warmer and warm water expands causing a rise in sea–levels.Figure 6 show the sea ice cover as percentage of the amount of sea that iscovered by ice over a grid cell. Here we see a general receding of the sea icecover distribution over the globe. This sea ice undergoes a phase change asthe CO2 levels increase and thus causing the sea to warm and expand.

1http://www.eos.ubc.ca/∼rich/map.html

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Figure 5: Images of temperature distribution across the globe. Left panel:CO2 ∼390 ppm corresponding roughly to the value of today. Right panel:CO2 ∼1000 ppm

Figure 6: Images of sea ice cover distribution across the globe. Left panel:CO2 ∼390 ppm corresponding roughly to the value of today. Right panel:CO2 ∼1000 ppm

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(a) Short–wave radiation is reflected bythe atmosphere and the surface of theearth.

(b) Long–wave radiation from the atmo-sphere and the earth.

Figure 7: Short–wave and long–wave Radiation

5 Reflectivity & Emissivity

A continuous spectrum of radiation wave lengths contributes to the energybalance of the earth. It is feasible to divide this spectrum into two categories-short-wave and long-wave radiation. In general short-wave is constituted bysolar radiation while long-wave radiation originates from thermal radiation.

5.1 Short–wave radiation

The incoming solar radiation is short-wave with about half its intensity inthe visible range (400-700 nm). A fraction of the incoming radiation isreflected due to the presence of atmospheric and surface albedo and theozone layer. These effects can be summarized by one planetary albedo value,αP ,

RTot = αP SIn (1)

where RTot is the total reflected radiation and SIn is the total incomingradiation.

The reflectivity for the atmosphere can be extracted by introducing anatmospheric reflective constant, αA, and take multiple bouncing of the ra-diation into the account. In this model a albedo for the surface, αS , isintroduced and absorption only take place at earth surface. See figure 7(a).

RTot =∞∑i=0

Ri = R0 + R1

∞∑i=1

(αAαS)i−1

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R1 = SInαS(1− αA)2

5.2 Long–wave radiation

Perfect black bodies radiate according to the Stefan-Boltzmann law J = σT 4

where J is the radiated flux per square meter and σ is the Stefan-Boltzmannconstant. Energy balance is required in steady state which can be formulatedby two equations for this system.

Earth SIn(1− αP ) = σT 4S − bσT 4

A

Atmosphere bσT 4S = 2bσT 4

A

Figure 7(b) illustrates schematically how the long-wave radiation system isconstructed. The earth is assumed to be a perfect black body of temperatureTS while the atmosphere is a grey body described by the emissivity constantb and temperature TA. This system can be reduced to an effective emissivityconstant for the surface temperature for the planet.

εEff = 1− b

2

Seen from outside the earth is then a grey body with emissivity εEff and aradiating flux J = εEffσT 4

S per square meter.

5.3 Albedo sensitivity to CO2 concentration

An increase of CO2 will cause an increase of the atmospherical albedo. In-terestingly the surface albedo responds in the opposite way. This is a selfaccelerating process since an increase of the temperature will cause a meltingprocess of the polar ice. Since ice and snow reflects light to a high degree thismelting process increases the surface albedo even more. An observer situ-ated outside the earth atmosphere will only recognize one planetary albedothat describes the ratio of incoming radiation that is reflected back intospace. The three albedos are coupled together as Figure 8(a) show. Despitethe increase of the atmopheric albedo the planetary albedo decreases whenCO2 increases. All three albedos are plotted in Figure 8(a).

In the long–wave region the system can be described by only one parameter–the emissivity of the atmosphere, b. Incoming solar radiation that is ab-sorbed by the earth must be balanced by a net thermal radiation outwards.The emissivity of the atmosphere affects at which temperature the earthsurface and atmosphere finds a new equilibrium. In the same manner as forthe short-wave radiation it is possible to take a step out and look at theearth and its atmosphere as one object with one effective emissivity. In thisperspective the earth is a grey body although the earth surface was seen asa perfect black body. The change of the effective emissivity follows qualita-tively the results reported by Kallen et al. as is presented in Figure 8(b).

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200 400 600 800 10000.15

0.2

0.25

0.3

0.35

0.4

CO2 concentration [ppm]

Alb

edo

PlanetarySurfaceAtmospheric

(a)

0 500 1000 1500 20000.55

0.6

0.65

0.7

CO2 concentration [ppm]

Em

issi

vity

Our resultsKallen

(b)

Figure 8: Global average albedo and emissivity

Figure 9 shows the modeled surface albedo distribution over the globefor two different CO2 concentrations. The values increase monotonicallyfrom the equatorial regions where it takes the lowest values to the polarregions where the largest values occur. The maximum observed in the polarregions are mainly associated with the snow cover and weaker solar radiationcompared to the equatorial regions. It can also be seen that the albedo inthe equatorial regions is smaller in the oceans compared to the continents.We note also that the albedo is larger in the Sahara desert.

Figure 9: Images of the surface albedo distribution across the globe. Leftpanel: CO2 ∼390 ppm corresponding roughly to the value of today. Rightpanel: CO2 ∼1000 ppm

In Figure 10 the planetary albedo is further investigated. We see that ifthe ice cover and ice thickness decreases, the planetary albedo will decrease.Decrease of the planetary albedo leads to less solar radiation being reflectedand temperature increase leading to further decrease of the ice cover. Wehave also studied the impact of clouds on the planetary albedo. For simplic-

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5 10 150.29

0.3

0.31

0.32

0.33

0.34

Yearly average temperature [ oC]

Yea

rly a

vera

ge a

lbed

o

(a)

0.51 0.52 0.53 0.54 0.550.29

0.3

0.31

0.32

0.33

0.34

Yearly average total cloud cover [frac.]

Yea

rly a

vera

ge a

lbed

o

(b)

0.05 0.1 0.15 0.20.29

0.3

0.31

0.32

0.33

0.34

Yearly average sea ice cover [frac.]

Yea

rly a

vera

ge a

lbed

o

(c)

0 0.5 1 1.50.29

0.3

0.31

0.32

0.33

0.34

Yearly average sea ice thickness [m]

Yea

rly a

vera

ge a

lbed

o

(d)

CO2[ppm] =1620CO2[ppm] =1440CO2[ppm] =1260CO2[ppm] =1080CO2[ppm] =900CO2[ppm] =720CO2[ppm] =540CO2[ppm] =360CO2[ppm] =180

Figure 10: Global average planetary albedo plotted for different CO2 stabi-lization levels against temperature (a), total cloud cover (b), sea ice cover(c), and sea ice thickness (d). Each dot represents the global average fromone year of the simulation. The CO2 stabilization levels are indicated withdifferent colors, see panel (d).

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ity we have only studied the total cloud cover and cannot see any correlationbetween total cloud cover and the planetary albedo. However, a completeinvestigation of the correlation between clouds and planetary albedo shouldcontain studies of different types of clouds, cloud heights, cloud liquid andice water content, and the size of cloud particles. The planet simulator usedin our work cannot handle phase changes of convective or large scale precip-itation within the atmosphere or the condensation growth of cloud droplets.However, distinction between rain and snow fall is made at the surface.

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6 Stochastic Analysis

I. Global temperature sensitivity to CO2 concentrationi. Model the CO2 concentration as a Beta distributed random variable

ii. Want to quantify the uncertainty in temperature given the uncertaintyin the CO2 concentration.

II. Compute approximate moments of temperature and globally averagedtemperature

i. Use stochastic collocation, i.e. quadrature rules in the random space,to compute first and second moments.

III. Approximate density function of temperaturei. Use a truncated polynomial chaos representation of temperature. ii.

Sample from the Beta distribution to approximate the density density func-tion of temperature.

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7 Concluding Remarks

Looking at the data, we can see that we are dealing with a feedback system.As we increase the concentration of CO2 in the atmosphere, the warmer ourplanet becomes which leads to sea ice cover melting which in turn leads toa lower albedo which means that less energy will be reflected away from theearth which means that the temperature increases etc...

So every time the CO2 levels rise, our climate system attempts to finda new equilibrium point. This equilibrium point seems to move further andfurther away from a comfortable future as the CO2 levels increase meaninga warmer climate for everyone.

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