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Climate transitions on long timescalesPeter D. Ditlevsen a

a Centre for Ice and Climate, The Niels Bohr Institute, University of Copenhagen, Copenhagen,Denmark

To cite this Article Ditlevsen, Peter D.(2009) 'Climate transitions on long timescales', Contemporary Physics, 50: 4, 511 —532To link to this Article: DOI: 10.1080/00107510902840313URL: http://dx.doi.org/10.1080/00107510902840313

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Climate transitions on long timescales

Peter D. Ditlevsen*

Centre for Ice and Climate, The Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100Copenhagen, Denmark

(Received 30 January 2009; final version received 20 February 2009)

The climate has changed through the history of the Earth as evidenced in the geological records. Today we might beexperiencing a climate change of the same magnitude as the transition into an ice age caused by very rapid burningand emission to the atmosphere of a substantial part of the fossilised carbon. Whether this leads to a gradualwarming or if we will experience a transition into a different climatic state is presently unknown. The present daystate-of-the-art numerical climate models are capable of producing fair representations of the current climate and areas such trusted to also predict the climate changes due to increasing atmospheric concentrations of greenhouse gases.However, the models are not presently capable of reproducing the rapid transitions from one climatic state, such as aglacial climate, into another, such as the present climate. The reason for this is unknown. The transitions areinherently ‘non-linear’ and thus not accessible through linear response theory. The term ‘non-linear’ is in this contextdefined as the phenomenon that the response of the system to a change in the forcing of the system is not linearlyproportional to the forcing. This would happen if a threshold is reached such that the state of the system becomesunstable and the system bifurcates into a different state. There are strong indications in the geological records of thiskind of behaviour for the climate. These dynamics can be understood in the context of fairly simple models of theclimate.

Keywords: climate change; climate models

1. Introduction

The climate is governed by a hierarchy of physicalprocesses determining how the incoming solar radia-tion and the outgoing long wave radiation arebalanced. A very useful way of viewing the dynamicsof the climate system is through analysing the energyflows. The incoming solar radiation is partly scatteredback into space, partly absorbed into the atmosphereand partly absorbed into the surface of the planet. Thesurface is thus heated and subsequently releasing heatinto the atmosphere both directly (sensible heat) andthrough evaporation of water, which release the heatwhile re-condensing in clouds (latent heat). Theincoming energy flux to the surface depends on latitudethrough the variation in inclination of the Sun, thus‘inclination’ and ‘climate’ have the same ontologicalroots. The differential heating leads to the generationof mechanical energy through the buoyancy forcescaused by the expansion of air due to heating. This inturn redistributes the heat vertically through convec-tion and horizontally through the winds. The mechan-ical energy is dissipated mainly through surface drags,some deposited as ocean surface waves, while at theend the energy is lost to space through long waveblack-body radiation. The connection between the

long wave radiation and the surface temperaturedepends on the vertical structure of the atmosphere,cloud processes and the amount of greenhouse gases inthe atmosphere.

We shall in a loose sense consider the climate stateto be the statistically steady state represented by somewell defined variable, such as the global mean surfacetemperature. Already in doing this we must becautious, since in principle, the system could reside indistinct climatic states, i.e. characterised by differentmeridional (equator to pole) temperature gradientswith the same mean temperature. With this pragmaticdefinition of the climate the very first question onewould ask is: ‘What determines the mean surfacetemperature?’. Secondly, if this is indeed determinedby, say, the incoming solar radiation: ‘What makes itstable and what makes it change?’. These questionsturns out to be much harder to answer than what onewould perhaps expect. A comparison of the present-day surface temperatures on Venus, Earth and Marsshows that these differ much more than what can beexplained by the distances from the planets to the Sun.Figure 1 shows the solar flux as a function of thedistance to the Sun. The curve shows what the surfacetemperature would be if conditions were as they are on

*Email: [email protected]

Contemporary Physics

Vol. 50, No. 4, July–August 2009, 511–532

ISSN 0010-7514 print/ISSN 1366-5812 online

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the Earth for planets at a given distance. As shown, thesurface temperature on Venus is about 400 K warmerand Mars about 20 K colder than should be expected.This is an illustration that the temperature stronglydepends on the planetary atmosphere and is not setsolely by the incoming solar flux. The incoming solarflux has, due to the increase with time of nuclear fissionin the interior of the star, risen by about a third duringthe lifetime of the solar system. This has remarkably notcaused a drastic change in the surface temperature onEarth. On the contrary, there is good evidence for theexistence of liquid water on the surface of the Earth formore than 4 billion years. This indicates that there aresome mechanisms in the system to stabilise the existingclimate state over long geologic times. Today we havepretty good ideas of what these mechanisms are.

Moving from the billion years timescale to the tensof millions of years, there is disturbing evidence ofglobal glaciations, with freezing all the way to theequator, happening a few times and lasting for tens ofmillions of years just prior to the development ofmulticellular life. These are the so-called SnowballEarth events, which we shall see to be a consequence ofmultiple steady states in the radiative balance of theEarth. Since the last Snowball Earth event the climatehas been warm and stable enough for the developmentof complex life forms.

For the last 20 million years the Earth has been socold, that permanent ice sheets are found at the poles.Antarctica especially has been permanently icecovered. This is due to an unprecedented low atmo-spheric concentration of the greenhouse gas CO2. This

climatic period is sometimes dubbed the ‘IcehouseWorld’ (in contrast to the ‘Greenhouse World’). TheIcehouse World is characterised by periodic waxingand waning of the large glaciers between ice ages andwarm periods. It has long been known that the comingand going of ice ages is related to periodic changes dueto the perturbations from the other planets in Earth’sKeplerian orbit around the Sun. However, the climaticresponse to the changing solar insolation is notcompletely understood. As we shall see later this isalso a strongly non-linear response phenomenon.

Finally, ice core records show that even within thelast glacial period there are variations at the thousandsof year timescale with abrupt jumps between twodistinct quasi-stable climate states. These jumpsapparently occur at random which indicates thatstochastic dynamics might be at play in the climatesystem. This may lead in the end to a speculation aboutthe predictability of climate.

2. The radiative energy balance

The climate system and the biosphere, for that sake, isdriven by the balance of the incoming low entropicshort wave radiation from the Sun and the outgoinghigh entropic long wave black-body radiation to space.The possibility of driving the climate and biological‘engines’ comes from the big temperature differencesbetween the surface of the Sun (*6000 K), the surfaceof the Earth (*300 K) and outer space (*3 K). Thesurface temperature of the Earth depends on thisbalance. Firstly, the amount of incoming radiationdepends on the planetary albedo. The albedo of anobject is the fraction of sunlight hitting the objectwhich is reflected, thus it is the complement to theabsorption. The planetary albedo is not a constantfactor, it depends through the amount of clouds andice, on the state of the climate itself. The feedback ofclouds on temperature is very complicated. It dependson the height in the atmosphere where the clouds areformed and the state of the atmosphere surroundingthe clouds. The clouds cool by reflecting the incomingradiation and they heat by trapping the outgoingradiation. These are both strong factors, of equalmagnitude, thus partially cancelling each other out,resulting in a net cooling effect. Ice and snow on thesurface unambiguously cool by reflecting the incomingshort wave radiation. The amount of ice and snowinfluences the planetary albedo. This effect we candescribe in a model of the climate represented by justone parameter, the mean surface temperature T. Wethus need to determine the functional relationshipbetween this temperature and the long wave radiationand the albedo, respectively. The amount of ice andsnow is larger when the temperature is lower, so the

Figure 1. The surface temperature on a planet as a functionof distance to the Sun. The curve shows how the temperaturewould be if conditions were identical to those of the Earth.Venus is much warmer due to an extremely thick greenhouseatmosphere. Mars is colder even though its atmospherecontains mainly CO2. That is because the atmosphere is verythin, only about 1% of that on Earth.

512 P.D. Ditlevsen


lower the temperature the higher the albedo, and wecan write the albedo a(T) as a function of thetemperature. If the temperature is below some lowtemperature T1 the planet will be completely icecovered and a further decrease in temperature cannotincrease the albedo. If the temperature is above someother high temperature T2 the ice is completely meltedand a further increase in temperature will not lead to adecrease in albedo. The simplest functional form is alinear dependence of the albedo on temperature inbetween these two temperatures. This is the mostreasonable choice when no other information isavailable a priori. We then have the relation:

aðTÞ ¼a1; T � T1;ðT2 � TÞa1 þ ðT� T1Þa2

T2�T1; T1 < T � T2;

a2; T > T2:

8<:

ð1Þ

The change of the temperature T is determined by thedifference in incoming and outgoing radiation [1,2],

cdT

dt¼ Ri � Ro ¼ ð1� aðTÞÞS� � sT4

eff; ð2Þ

where c is the heat capacity, S* ¼ S/4 is a quarter ofthe solar constant. (The quarter comes from the ratioof the surface of the sphere to the cross-sectional areablocking the sunlight.)

Teff is the temperature at the level in the atmo-sphere from where the black-body radiation is emittedto space. Since the atmosphere is opaque in the infra-red wave-band due to greenhouse gas absorbers thislevel is estimated as one optical dept into the atmo-sphere in this band. The greenhouse effect, the changein cloudiness and other factors must all be expressedthrough the connection between the surface tempera-ture T and the temperature Teff. The more greenhousegas in the atmosphere the higher in altitude is theeffective level of emission. The temperature decreaseswith height by convective adjustment, such that theatmosphere is stable with respect to vertical adiabaticmotion. The greenhouse warming can then be definedas DT ¼ T 7 Teff, such that the last term in Equation(2) becomes Ro ¼ s(T 7 DT)4. The value of theplanetary albedo can be obtained elegantly by compar-ing the bright and the dark side of the Moon at halfMoon, the bright side is lit by the Sun, while the darkside is lit by sunlight reflected off the Earth. From this(and from satellite measurements) the planetary albedois measured to be approximately 0.31; using this andsolving (2) with respect to temperature we get thepresent-day greenhouse warming to be DT � 32 K.

The greenhouse warming will in general be afunction of T mainly through the temperature

dependent water vapour (a greenhouse gas) concentra-tion and cloud formation.

The behaviour of (2) is easily understood from agraphic representation. Figure 2 shows the incomingand outgoing radiation as a function of temperature.There are three temperatures Ta, Tb, Tc for which thecurves cross such that the incoming and outgoingradiations are in balance. These points are the fixedpoints of (2). Consider the climate to be at point Ta. Ifsome small perturbation makes the temperature lowerthan Ta we will have Ri 4 Ro ) cdT/dt 4 0 and thetemperature will rise to Ta. If on the other hand, theperturbation is positive and the temperature is a smallamount larger than Ta we have Ri 5 Ro ) cdT/dt 4 0 and the temperature will decrease to Ta again.Thus, Ta is a stable fixed point. The same analysisshows that for Tb a small perturbation will grow intime and the temperature will move away from Tb.Thus, Tb is an unstable fixed point. A similar analysisshows that Tc is a stable fixed point. The tendencies areindicated by arrows in Figure 2. The model has twostable climate states Ta and Tc. So if the temperature atsome initial time is lower than Tb it will eventuallyreach the temperature Ta and if it is higher than Tb itwill reach the temperature Tc. The present climate isthe climate state Tc where the ice albedo does not playa significant role in cooling the Earth.

The radiation might not be constant in time butdepends on some external factors. The solar flux variesboth because of changes in solar output and because ofchanges in the Earth’s inclination and orbit around theSun. The solar constant can then be expressed as mS*,where m 4 0 is a time dependent parameter so we haveRi ¼ (17a)mS* in (2). Consider the climate repre-sented by Tc in the situation m 4 1. Then for m not toosmall, corresponding to the dashed curve in Figure 3,

Figure 2. The global mean surface temperature isdetermined by the balance between incoming and outgoingradiation. The net incoming flux depends on temperaturethrough the decreasing planetary albedo with temperature.The outgoing flux is determined by the black-body radiationand the greenhouse effect as a function of the surfacetemperature.

Contemporary Physics 513


the equilibrium temperature is lowered a little, as wewould expect when the solar input decreases. However,if m becomes smaller than some value m0 the two curvesdo not cross anymore and there is no stable fixed pointnear Tc. The climate will then run into the only stablefixed point Ta which is still present. A bifurcation hasoccurred and there is a large change in climate. Thiskind of behaviour is for obvious reasons sometimesalso called catastrophic. If m grows again, the climatestate Tc will not recover until m exceeds some othervalue m00 4 1, where Ta disappears. This is the dot-dashed curve. For each value of m we have either oneor three fixed points and we can plot the fixed points asa function of m. This is a bifurcation diagram (seeFigure 4). The two full curves represent the stable fixedpoints and the middle dashed curve represents theunstable fixed point. As can also be seen from Figure 4the unstable and one of the stable points coincide at thebifurcation points m0 and m00. Assuming that the solarflux changes periodically in time as m(t) ¼ (m0 þ m00)/2 þ e sin ot where e 4 jm0–m00j the climate will jumpperiodically between the states Ta and Tc. The evolutionis represented by arrows in Figure 4, this is a hysteresisloop. If the climate T(t) is considered as the response tothe forcing function m(t) the response is strongly non-linear, which means that at some times there is a largeresponse to a small change in forcing, which wasoriginally motivating the model as an explanation forglacial cycles as a response to the weak forcing from theperiod changes in Earth orbit around the Sun. The timeevolution is schematically shown in Figure 5.

3. The Snowball Earth and the ‘Faint young Sun

paradox’

The stable climate state Ta corresponds to a totally icecovered planet and as such is not a realistic theory ofthe glacial cycles. The totally ice covered planet is the

Snowball Earth [3]. There is geological evidence ofsuch an extreme ‘deep freeze’ climate several times inthe Neoproterozoic period earlier than 700 millionyears ago. This is based on synchronous findings ofglacial deposits like moraine in many places which atthose times were near the equator. The speculated wayout of the deep freeze is the following: the balance ingeological timescales between weathering bindingatmospheric CO2 into rocks and volcanic out-gassingof CO2 was changed during the deep freeze. Due to thecold conditions the atmosphere dried out and weath-ering was reduced. Unchanged volcanic out-gassingresulted in an ever increasing amount of CO2 in theatmosphere. At some point, after about 80 millionyears, this would result in a greenhouse warmingstrong enough to melt the ice. This scenario corre-sponds to gradually moving the Ro curve upwards inFigure 2 until the stable fixed point Ta disappears(together with Tb) through a saddle-node bifurcation,and the only remaining fixed point is Tc. The warmingwould then be almost explosive with globalmean temperatures going from some 7408 C to

Figure 3. The incoming radiation change as a function ofchanges in the solar constant, represented by a controlparameter m. For some larger value m > m10 only the upperstable fixed point exists. For some smaller value m 5 m onlythe lower stable fixed point exists.

Figure 4. The fixed points plotted as a function of thecontrol parameter m. At m% m10 the lower state disappearsthrough a saddle-node bifurcation, while at m&m0 the upperstate disappears. A cyclic change in m leads to a hysteresisloop.

Figure 5. A small periodic variation in the forcing can leadto a strongly non-linear large response through a hysteresisloop.

514 P.D. Ditlevsen


some þ508 C within a few years. Few organisms wouldbe able to survive such a dramatic climatic jump, andwe could speculate that this is a reason why it tooksuch a long period for multicellular life to evolve.Multicellular life came into existence some 600 millionyears ago in the Cambrian explosion shortly after thelast deep freeze. To substantiate such a speculation ofcourse, unsuccessful development of multicellular lifeterminated by a Snowball Earth event must be found.

The fact that the Earth was not frozen in its veryearly history despite the fact that the Sun was about30% fainter is known as the ‘Faint young Sunparadox’. The most probable solution to the puzzle(which is not really a paradox) is that the geochemicalcarbon cycle was balancing the out-gassing of CO2

from volcanoes with CO2 removal through weathering,where carbonic acid rain slowly dissolves silicate rocks[4]. The balance between these two processes deter-mines the atmospheric CO2 concentration and thus thetemperature through greenhouse warming. Obviously,this can only work as a regulator of temperature if oneof these processes depends on the temperature itself [5].This is the case for weathering, since it dependsstrongly on precipitation. The regulation acts likethis negative feedback loop: more atmospheric CO2!higher temperature ! more rain ! stronger weath-ering ! less atmospheric CO2. The reason whySnowball Earth events have not happened since beforethe Cambrian explosion is thought to be a pole-wardcontinental drift changing the hydrological cycle andthe weathering of atmospheric CO2.

4. One-component dynamical systems

Models like the Bodyko–Sellers climate model intro-duced above are very useful in the study of climatedynamics. It is therefore relevant to exploit thebehaviour in mathematical terms in a more generalsetting. Consider the system as described by thevariable x by the equation

_xðtÞ ¼ fðxðtÞ; mÞ; ð3Þ

xð0Þ ¼ xi; ð4Þ

where as usual the dot means differentiation withrespect to time and m is some parameter called acontrol parameter. This is the simplest case where thesystem is described by only one dependent variable, sowe can call the system a ‘one-component’ dynamicalsystem. Equation (3) is an autonomous (f does notdepend explicitly on time, disregarding for now anypossible time dependence of m), first order (onedifferentiation with respect to time), ordinary (nopartial derivatives) differential equation. It has aunique solution with the specified initial condition (4)

where the subscript ‘i’ indicates ‘initial’. For themoment we will suppress the dependence on theexternal parameter m. Equation (3) can be integrated,

dx

dt¼ fðxÞ )

Z xðtÞ

xi

dx

fðxÞ ¼Z t

0

dt ¼ t ð5Þ

to obtain x as an implicit function of t, the integrand isonly defined for f(x) 6¼ 0. If f(x0) ¼ 0 then x0 will be afixed point where the solution will reside and theintegral above is not defined.

The fixed points of Equation (3) are determined asthe set of solutions to the equation

fðxÞ ¼ 0: ð6Þ

Having found a fixed point x0 the next thing to do is todetermine the stability of x0. When x is close to x0 wecan expand f(x) in a Taylor series around x0: f(x) ¼f(x0) þ f 0(x0)(x7x0) þ O((x7x0)

2) ¼ f 0(x0)(x7x0) þO((x7x0)

2). Inserting this into (3), defining the smallperturbation y ¼ x–x0 we get to first order,

_y ¼ f 0ðx0Þy) yðtÞ ¼ yð0Þexp ðf 0ðx0ÞtÞ: ð7Þ

If f 0(x0) 4 0 the perturbation y(0) will grow exponen-tially with time and the fixed point x0 is unstable. Iff 0(x0) 5 0 the perturbation will decrease exponentiallyin time and the fixed point is stable, so the stability isdetermined by the first derivative f 0(x0) at the fixed point.The interval of initial conditions xi which eventually willend up in (or arbitrarily close to) the stable fixed point x0is called the domain of attraction of x0. Consider x1 5 x2to be two stable fixed points and assume that there are noother stable fixed points between x1 and x2. If x1 5x 5 x2 is a point close to x1 then x will approach x1 withtime and thus f (x) 5 0. On the other hand, if x is close tox2 it must approach x2 with time and f(x) 4 0. Thus,there must exist a point x3 separating the domains ofattraction of x1 and x2 such that x1 5 x3 5 x2 andf(x3) ¼ 0. This fixed point must be unstable since weassumed that there were no stable fixed points between x1and x2. This shows that any two stable fixed points mustbe separated by an unstable fixed point, and by acompletely similar argument two unstable fixed pointsmust be separated by a stable fixed point. Therefore, inthe stability analysis of the climate model above it sufficesto see that Ta is a stable fixed point. From this it followsthat Tb must be unstable and Tc must be stable. Thisanalysis is easily understood if the equation of motion (3)is expressed in terms of a potential,

_x ¼ � dUðxÞdx

; ð8Þ



where U(x) ¼Rxf(y)dy is the potential corresponding

to the force f(x). The fixed points correspond tolocal extrema for the potential. The stability of the fixedpoints are determined by the curvatures f0(x0) ¼7U00(x0). So if U00(x0) 4 0 then U(x0) is a minimumand x0 is a stable fixed point. If U00(x0) 5 0 thenU(x0) is a maximum and x0 is an unstable fixedpoint. Figure 6 shows the potential for (2) which is adouble-well potential. It is now trivial, why twostable fixed points must be separated by an unstablefixed point.

5. Stochastic climate dynamics

The climate system obviously contains very manyvariables interacting in complex ways, so when theextreme reduction in representing only one variablegoverned by the energy balance we should considerEquation (2) as an effective equation and seek a way ofincorporating the effect of the unresolved variables.The dynamics is characterised by the interactionbetween components with very different typical time-scales. The atmospheric variations, weather patterns,are typically of days to weeks duration while variationsin ocean currents are on much longer timescales, yearsto centuries, buildup of ice sheets takes severalmillennia and even changes in tectonics and geochem-istry on millions of years timescales are influential inthe climate.

The timescales where the slow climate variableschange appreciably are beyond the correlation time forthe fast variables. At these timescales the fast variablesare effectively decorrelated and could then be describedas being stochastic [6].

Assume that we can describe the system by a set ofgoverning equations for the system variables that canbe split in separate variables represented by the vectors

x ¼ (x1, . . . , xi, . . .) and y ¼ (y1, . . . , yj, . . .). Thenthe system differential equations can be written interms of two sets of functions fi and gj as

_xi ¼ fiðx; yÞ; ð9Þ

_yj ¼ gjðx; yÞ; ð10Þ

where we can associate a typical timescale txi and tyj,respectively, to each variable such that txi � tyj forall (i, j). Equation (10) describes the dynamics of thelarge scale observable. For brevity we drop vectornotations, and consider x and y as scalar variables.The extension to more dimensions is mostlystraightforward. In the effective dynamics for y wecan write the small scale variable as x ¼ hxjyi þ x0,where the brackets denote the average of x condi-tioned on y. Inserting this into the second equation,using that x varies much faster than y, we canapproximate

_y ¼ gðhxjyi þ x0; yÞ � gðhxjyi; yÞ þ @xgðhxjyi; yÞx0

¼ geffðyÞ þ sðyÞZ: ð11Þ

For the short time correlated variations x0(t) we havesubstituted a stochastic white noise s(y) Z(t) withhZ(t)Z(t0)i ¼ d (t–t0).

Equation (11) is a Langevin equation. In thesimplest form we will assume the noise intensity tobe independent of the climate state y. Writing F(y) forgeff (y) we have

_y ¼ FðyÞ þ sZ ¼ �dU=dyþ sZ; ð12Þ

where the drift ‘force’ F(y) is defined as (minus) thegradient of an effective potential U(y). If the noiseintensity is low the y component will for a long timesettle around a stable fixed point y0 providedF(y0) ¼ 0 and F0(y0) ¼ 7a 5 0. The notion of lownoise intensity must be defined in the sense of anArrhenius escape time, T * exp(h/s2), where h is thebarrier height in the potential U separating the stablefixed point y0 from other stable fixed points. Withoutloss of generality we can take y0 ¼ 0 and expandF(y) in (12) to first order

_y ¼ �ayþ sZ: ð13Þ

A process satisfying this linear stochastic equation iscalled an Ornstein–Uhlenbeck process or a red noiseprocess [7]. Equation (13) can easily be solved for theauto-correlation function, c(t) ¼ hy(t)y(t þ t)i:

cðtÞ ¼ hy2iexp ð�ajtjÞ;

Figure 6. The double-well potential corresponding to theforcing in Equation (2). In this case m0 5 m 5m1 such thatboth states with temperatures Ta and Tc are stable. Tb is thetemperature of the separating unstable fixed point staterepresented by the dashed line in Figure 4.

516 P.D. Ditlevsen


where the correlation time is T ¼ 1/a. The power-spectrum is the Fourier transform of the autocorrela-tion function, which becomes,

PðoÞ ¼ s2=ðo2 þ a2Þ; ð14Þ

and the connection between the intensity of the processand the intensity of the noise is hy2i ¼ s2/(2a). This isthe simplest form of the fluctuation-dissipationtheorem.

The red noise process (13) is quite a good modelof the variations of climate series in our presentclimate. Figure 7 shows the power spectrum of thelast 4000 years of a Greenland ice-core proxy record[8] of temperature. The smooth curve is given byEquation (14) with a correlation time of 5 years. Thepeak, indicated by the arrow is the annual cycle intemperature, which is not included in the simpledescription (13). The correlation time of 5 yearsshould be expected from the typical timescales ofvariations in surface water temperatures in the NorthAtlantic ocean influencing the air temperatures inthat region.

The Langevin Equations (11) or (12) are stochasticand will as such have a solution which depends notonly on the initial condition y0(t0), it will also dependon the specific realisation of the noise. A deterministicequation for the conditional probability p(x,tjx0,t0),which is the probability density for x at time t, given

the process is at x0 at time t0, can be derived fromEquation (12) [7]:

@tpðx; tjx0; t0Þ ¼ �@x½FðxÞpðx; tjx0; t0Þ�

þ @2xs2ðxÞ2

pðx; tjx0; t0Þ� �

: ð15Þ

This is the Fokker–Planck equation for the conditionalprobability density.

6. Ice ages

The climate has varied periodically between ice agesand warm periods for the last few tens of millions ofyears. A continuous record of these variations is foundin sedimentation in the deep sea. The sediments areformed from sinking planktonic organisms (forameni-fera). The deep sea is biologically a desert, thus thesedimentation rate is extremely low, of the order of ametre in a million years. The oxygen isotopes in thecalcium-carbonate shells of these organisms is a proxyfor the isotopic composition of the ocean water, andsince light isotopes evaporate easier a proxy for theamount of water deposited in ice sheets. A compositeof deep sea records is shown in Figure 8 [9]. Sincegoing deeper into the sediments corresponds to goingback in time, the geological convention is to invert thetime axis. So keep in mind that time grows from rightto left in the figure. The convention for the units ‘yearsBP’ is ‘year before present’, with ‘present’ ¼ 1950. Thegraph should be read as a proxy-temperature record,showing cycles of warm (high values) and cold periods(low values).

The glacial cycles are attributed to the climaticresponse of the orbital changes in the irradiance to theEarth [10,11]. These changes in the forcing are toosmall to explain the observed climate variations assimple linear responses. Non-linear amplifications arenecessary to account for the glacial cycles. Thedominant orbital periods in solar insolation is the41 kyr obliquity cycle (tilt of rotational axis, determin-ing the meridional gradient in insolation) and theprecessional cycles (determining the season when Earthis closest to the Sun) which decompose into 19 kyr and23 kyr periods. However, through the last 800 kyr–1 Myr the dominant period for the glacial cycles isapproximately 100 kyr similar to the one order ofmagnitude weaker eccentricity cycle (determining thesemi-annual difference in distance to the Sun). Theweakness of this climatic forcing is referred to as the100 kyr problem of the Milankovitch theory [12]. It isnow generally accepted that the 100 kyr glacial time-scale cannot be attributed to the eccentricity cycle [9].In the Plio- and early Pleistocene, 3–1 Myr BP, the

Figure 7. The power-spectrum of the last 4000 years of aGreenland ice-core proxy record of temperature (Dansgaardet al. [8]). The smooth curve is given by Equation (14) with acorrelation time of 5 years. The peak, indicated by the arrow,is the annual cycle in temperature. The annual cycle is notincluded in the red noise process (13).



dominant period of variation was indeed the 41 kyrobliquity variation [13,14].

Different mechanisms have been proposed toexplain the occurrence of the 100 kyr glacial cycle.These range from self-sustained non-linear oscillators[15–17], forced non-linear oscillators [18] to theglaciations being a resonance phenomenon. Stochasticresonance, which we shall return to later, wasoriginally constructed as an explanation of ice ages[19].

Combined evidence from records of glaciations onland and deep sea records suggest that the climate hasshifted between different quasi-stable states charac-terised by the mode of the global ocean circulation andthe degree of glaciation [20]. These states may originatesimilar to the stable states in the simple ocean boxmodel which we shall introduce later, or the energybalance model introduced above. By comparisonbetween the paleoclimatic record and a non-linearstochastic model, it is demonstrated that the record canbe generated by the forcing from insolation changesmainly due to the obliquity cycle through the fullrecord including the last 1 Myr. It has been longknown that the ‘100 kyr world’ is not linearlyresponding to the orbital forcing [21], but even in the‘41 kyr world’ the climate response to the orbitalforcing is non-linear [22]. The assumption here is thatthe orbital forcing resulted in periodic jumps betweentwo stable climate states. What happened approxi-mately 800 kyr–1 Myr ago was that a third deepglacial state became accessible resulting in a change inlength of the glacial cycles. The reason for this mid-Pleistocene transition (MPT) is unknown, andattributed to a gradual cooling due to a decreasingatmospheric CO2 level [23] or a change in the bedrockerosion (the regolith hypothesis) [24].

Since in the energy balance model (2) of the Earth,the climate is characterised by only one variable, theglobal mean temperature, there is ambiguity inascribing the orbital forcing from the time and spacevarying insolation field across the globe.

Huybers et al. [13,25] argue that the ice melt-offdepends on the integrated summer insolation indicat-ing that this could be the relevant measure of orbitalforcing [13,25]. This is closely related to the concept ofdegree days, which is the annual number of days withtemperatures above freezing. This measure is domi-nated by the obliquity cycle since the increasedinsolation when Earth is close to the Sun in its orbitis compensated by shorter time spent there due toKepler’s second law. Thus the total insolation duringthe degree days becomes independent of the preces-sional cycle.

In contrast to this, Paillard [26] shows using asimple rule-based model of jumping between threedifferent quasi-stationary climate states, that theclimate record can be a response to thesummer solstice insolation at 65N. The two proposedforcings (degree day insolation and June 26 insola-tion) are different, since the latter has a strongcomponent of the precessional cycle. Using the June26 insolation as the better proxy for the forcingcan be rationalised from the point of view of athreshold crossing dynamics, since the extremalvalues (mid-summer insolation) would then be thegoverning parameter. However, since we cannotdecide between the two within the framework of asimple model, we can take the alternative approachof assuming the linear combination of the two,considered as a first order expansion, which gives thebest fit for the observed record as a response to theforcing.

Figure 8. The (normalised) paleoclimatic isotope record from a composite of ocean cores. The record is a proxy for the globalsea level or minus the global ice volume. The curve shows changes between ice ages and warm periods through the last twomillion years. Note that the geological convention of time increasing from right to left is followed, thus present time is at theorigin.

518 P.D. Ditlevsen


7. The non-linear stochastic ice age model

Due to the high dimensionality and the stochasticnature of the climate fluctuations it is highly unlikelythat regular periodicities can result from internaloscillatory modes alone. It is much more plausiblethat non-linear responses to weak external periodicforcing would lead to periodic behaviour. There isevidence from observations as well as models thatmultiple states exist in the climate system [20]. Thissuggests a possible scenario of periodically induceddestabilisations of three quasi-stable climate states [27].

The three states can be identified with the oneslabelled G (deep glacial), g (pre-glacial), and i(interglacial), respectively, by Paillard [26]. Paillardobserved in the paleoclimatic record that there seemsto be ‘forbidden’ transitions between the three states.In the period 2–1 Myr BP the record shows regularoscillations between only the two states i and g, whilein the period 1–0 Myr BP there is only a specificsequence of occurrences: i ! g ! G ! i permitted.The underlying bifurcation structure provides adynamical explanation of this observation.

From the observed climate record we derive anempirical model. This assumes that the climatedynamics are reflected in a single variable x(t), whichis taken to be the global mean surface temperatureanomaly represented by the deep sea oxygen isotopeproxy record shown in Figure 8. The dynamics aredescribed by an effective non-linear stochastic differ-ential equation,

_x ¼ faðx; mÞ þ sZ; ð16Þ

where the white noise term Z with intensity s describesthe influence of the non-resolved variables and theinternally generated chaotic climate fluctuations. It iswithin this framework that the roles of the orbitalforcing and internal stochastic forcing are investigated.The deterministic part, fa(x, m), of the dynamicsdepend on the external orbital forcing, labelled by asingle control parameter m and internal parameters,represented by a. Note that Equation (16) is a non-autonomous generalisation of the Equation (12), sincem and a are time-dependent.

The full climate dynamics can obviously not becompletely reconstructed by such a single valuedfunction. However, since stability and bifurcationsare topological quantities it could be robust withrespect to the detailed dynamics modelled. It is thus thebifurcation structure of fa(x, m), with respect to thecontrol parameter m, which determines the climatedevelopment.

Guided by the observed record and the transitionrules we can empirically construct a bifurcation

diagram: Figure 9(a), shows the bifurcation diagramfor the drift function fa(x, m) as a function of m at thetime interval 2–1 Myr BP. The bifurcation diagramshows the curves {x0(m)jf(x0, m) ¼ 0}. The fat curvesare the stable fixed point curves for which @xf 5 0,while the thin curves are the unstable fixed point curvesfor which @xf 4 0. Thus, in the case of no additionalnoise (s ¼ 0 in Equation (16)) the state of the systemx(t) is uniquely determined from the initial state x(0)and the development of the forcing m(t).

In the real climate system the internal noise issubstantial and the system will not reside exactly in the

Figure 9. The bifurcation diagram for the ice age model.Along the x-axis is the forcing represented by the controlparameter m, along the y-axis are the fixed points{x0(m)jf(x0,m) ¼ 0} of the drift function fa(x, m). The driftfunction is simply approximated by a fifth order polynomial,with the roots determined by the fixed points. The horizontaldashed line segments indicate (real part of) sets of complexconjugate roots. The fat curves show the stable fixed points.The bifurcation point a is the point where the deep glacialstate G disappears. The arrows indicate the hysteresis loop asthe forcing parameter is changed. Upper panel: the glacialstate G is not accessible. Lower panel: now the location of thebifurcation point a has changed in such a way that the deepglacial state G is accessible.



steady states determined by the bifurcation diagram.Thus, the full drift function needs to be parameterised.The simplest way to parameterise the drift function inaccordance with the bifurcation diagram is as a fifthorder polynomial:

faðx; mÞ ¼Y5j¼1ðx� xjaðmÞÞ; ð17Þ

where xjaðmÞ is the j th steady state (zero-points) in thebifurcation diagram. As labelled in the figure theparameter a determines the position of the lowerbifurcation point.

It should be noted that this is of course not the onlypossible drift function corresponding to this bifurca-tion diagram. In order to reconstruct the drift functionfrom the observed realisation, one could in principleobtain the stationary probability density pm0(x) bysorting x(t) according to m(t) ¼ m0. Assuming that m(t)is changing slowly in comparison to the timescale forx(t) to drift to a stationary state x0 (fa(x0, m) ¼ 0), onecould then obtain fa(x, m0) by solving the Fokker–Planck Equation (15) associated with Equation (16) forfixed m ¼ m0. This would require a very long data seriesand complete absence of additional non-climatic noisein the proxy data. This is not the case for the existingpaleoclimatic record.

The climate forcing is, as mentioned before, takento be a linear combination of the summer solstice 65Ninsolation (fss) and the integrated summer insolation at65N (�fI), where the summer period is defined as theperiod where the daily mean insolation exceedsI ¼ 200 W m72. The model results are robust withrespect to the threshold I chosen in a rather broadinterval. The forcing, f ¼ l�fI þ (17l)fss, is shown inFigure 11, second panel, in the next section [28]. Valuesof l around 0.5 gives the best result, l ¼ 0.5 is used.This assignment might, within the framework of thenon-linear model, be interpreted as an empiricaldetermination of the dominating components of theorbital forcing.

8. The hysteresis behaviour

The diagram in Figure 9(a), shows the fixed points offa(x, m) as a function of the deterministic forcing m. Thethree branches of stable fixed points xj(m) for thefunction, such that fa(x

j, m) ¼ 0 and @xfa (xj, m) 5 0,are indicated by fat curves. The specification of thexj(m)’s and Equations (16) and (17) completely definesthe model, see [27] for more details. Since x is a proxyfor the global mean surface temperature anomaly, orminus the global ice volume, the lower branchcorresponds to the deep glacial state G. The middle

branch corresponds to the climate state g and theupper branch to the interglacial state i. The thin curvescorrespond to the separating unstable fixed points. Thedashed line-segments correspond to pairs of complexconjugate roots in the fifth order polynomial. Noteagain that assuming a polynomial drift function, this isuniquely determined from the roots, except from atrivial multiplicative constant.

Suppose that the climate is in either of the states gor i and the climatic noise is too weak to induce acrossing of a barrier separating the stable states. Thenthe only way a forcing induced shift between theclimate states can occur is through bifurcations and ahysteresis loop i ! g ! i as sketched by the arrows.Clearly the climate state G is unreachable.

Assume now that the lower bifurcation point,indicated by a0 in Figure 9(a), moves toward largervalues of m indicating that a stronger forcing isneeded in order to destabilise the deep glacial state. Inthis case, a0 ! a1 shown in Figure 9(b), the glacialstate G is now reachable and a hysteresis loopi ! g ! G ! i will appear. The central postulate ofthe model is the change in this bifurcation structurerepresented by the shift of the point a (from a0 to a1on the m-axis) at the mid-Pleistocene transition. Thisconstitutes a dynamical explanation for the ‘rules offorbidden transitions’ which are apparently obeyed bythe observed record.

The change in the position of the lower bifurcationpoint is modelled such that a ¼ a1 when the climate isin state i. When the state G is reached through twobifurcations, a is gradually changing. The gradualchange in the bifurcation diagram is modelled as arelaxation, da/dt ¼ 7(a7a0)/t, where a0 is the earlyPleistocene equilibrium value and t is a relaxationtime. When the climate bifurcates through the rapidtransition G ! i, the parameter a again changes to a1.

In order for the climate to skip the 41 kyr obliquitypacing of deglaciations the timescale t governing thebifurcation structure must be considerably longer than41 kyr. The model results are quite insensitive to thespecific value of t in the interval 70–130 kyr, and is setto be 100 kyr. It is a major challenge to interpret thebehaviour of the bifurcation point a, governed by sucha long timescale in terms of real climate dynamics. Onecould speculate that it is linked to the carbon cycleor with erosion of continents on these long timescales.

The presence of the stochastic forcing implies thatthe climate evolution is not fully deterministic. Figure10, first panel, shows a particular realisation of themodel. The second panel shows the forcing, the redcurve shows the value of a, which is defined as theposition of the lower bifurcation point in Figure 9along the axis of the forcing (the x-axis). Note that atransition G ! i without noise assistance is only

520 P.D. Ditlevsen


possible when the forcing exceeds the value of a (that iswhen the blue curve is above the red curve in thesecond panel).

The composite Atlantic ocean sedimentation re-cord in Figure 8 is repeated in the bottom panel forcomparison. The curve is plotted with normalisedvariance and the mean subtracted. The dating of the

record is based on a depth–age model independentfrom astronomical tuning.

The differences between the single records compris-ing the stacked record gives an estimate of the additionalnoise from bioturbation and other factors that make therecord different from a true record of ice volume. So inorder to compare the model with the observed climate

Figure 10. The top panel shows a realisation of the model. Second panel shows the orbital forcing driving the model. The redcurve shows a(t), where the jumps to a ¼ a1 are triggered by the transition G ! i. The next transition is in the low noise limit,only possible when the blue curve is above the red curve. The third panel shows a ‘pseudo paleorecord’, where a red noisecomponent representing the non-climatic noise, is added to the model realisation in the top panel. Lower panel is the same recordas in Figure 8 for comparison.

Figure 11. Five realisations of the model with the same orbital forcing and different stochastic forcing. The first panel shows arealisation without stochastic forcing. This is the purely deterministic climate response to the orbital forcing. The bottom fourrealisations have a noise intensity s ¼ 0.8 K kyr71/2. It is seen that only in the last part of the 100 kyr world the timing of theterminations are independent from the noise. In all five realisations an additional non-climatic ‘proxy noise’ is added a posteriori.



record an additional red noise, of the same magnitude asthe difference in deep sea records is added to the model.This is shown in the third panel, which should becompared with the observed record in the fourth panel.

In order to investigate the role of the stochasticnoise, a set of realisations of the model is presented inFigure 11. For the comparison between the model andthe proxy climate records we focus on the rapidtransitions G ! i, called terminations [29,30]. The toppanel shows a realisation with no stochastic noise. Thisis the deterministic climate response to the orbitalforcing. It is seen that the last five terminations arereproduced as observed, but there are fewer inter-glacials in the earlier part of the late-Pleisocene period(1000–500 kyr BP) than in the observed record. FromFigure 10, second panel, it is seen that the amplitude ofthe orbital forcing is low in this period. The bottomfour panels show different realisations with a moderatestochastic forcing (s ¼ 0.8 K kyr71/2). In these rea-lisations the added noise induces additional termina-tions at different times, suggesting a fundamentalunpredictability in glacial terminations.

9. Noise induced transitions within the last glacial

period

The last glacial period showed millennium scaleclimatic shifts between two different stable climatestates. The state of thermohaline ocean circulationprobably governs the climate, and the triggeringmechanism for climate changes is random fluctuationsof the atmospheric forcing on the ocean circulation.

The high temporal resolution paleoclimatic datafrom ice cores are consistent with this picture and a bi-stable climate pseudo-potential can be derived. It isfound that the fast timescale noise forcing the climatecontains a component with an a-stable distribution,also called a Levy flight [31]. As a radical consequencethe abrupt climatic changes observed could in principlebe triggered by single extreme events. These events arerelated to ocean-atmosphere dynamics on annual orshorter timescales and could indicate a fundamentallimitation in predictability of climate changes.

Paleoclimatic records from ice cores [8] show thatthe climate of the last glacial period experienced rapidtransitions between two climatic states, the cold glacialperiods and the warmer interstadials (Dansgaard–Oeschger (DO) events). Deep sea sediment cores [32]and coral records [33] indicate that the ocean circula-tion is a key player in these climatic oscillations [34].Ocean circulation models, from the most simpleStommel type [35] to the complex circulation models[36] show that different flow states can exist as stableclimatic states. To show this we shall shortly divertinto the simplest possible ocean model.

10. The thermohaline circulation

The large scale ocean circulation is as important asthe atmospheric circulation for redistribution of heaton the planet. This is part of the cause for theclimate belts not simply following the latitudes. Theclimate belts are traditionally defined by the occur-rence of different species of tropical, subtropical,temperate or arctic plants. The ocean circulations aredriven by shear from the atmospheric winds, fromtidal forces and by gravity through the buoyancy ofthe water itself [37]. These are in turn ‘the winddriven circulations’, ‘tides’ and ‘the thermohalinecirculation’. The thermohaline circulation is drivenby the sinking of heavy water and the rising of lightwater. The density of ocean water depends on twoparameters; temperature T and salinity S: warmerwater is lighter and saltier water is heavier. Thedensity can then, to first order in the differencefrom the mean state, represented by (T0, S0), bewritten as

rðT;SÞ ¼ rðT0;S0Þ þ @TrðT0;S0ÞðT� T0Þþ @SrðT0;S0ÞðS� S0Þ¼ r0ð1� aDTþ bDSÞ; ð18Þ

where we have introduced the coefficients of expan-sion 7ar0, br0 with respect to temperature andsalinity [38]. The Stommel model is the simplestpossible non-trivial model describing the dynamics ofthe thermohaline circulation [39]. We assume that theocean can be split into two boxes A and B, eachascribed a temperature and a salinity TA, SA and TB,SB, respectively. These could be seen as meanquantities like TA ¼

RAdxT(x). The boxes are con-

nected by an upper and a lower hydraulic link, eachfrom continuity carrying the same current q inopposite directions, see Figure 12. The current willbe driven by buoyancy and will depend on the densitydifference between the boxes. Heavy water will betransported in the lower link from the box containingthe heavier water to the box containing the lighterwater and vice versa for the upper link. For simplicitywe can assume a linear relationship,

q ¼ CðrA � rBÞ ¼ Cr0½bðSA � SBÞ � aðTA � TBÞ�;ð19Þ

where C is an empirical constant. Here a variable, inthis case q, depending on the whole fluid flow,temperature – and salinity fields not resolved in themodel – is substituted by a functional relationshipbetween variables in the model. This kind of modelling

522 P.D. Ditlevsen


is termed a parameterisation. The governing equationsfor the model are

VA_TA ¼ jqjðTB � TAÞ þ kTð ~TA � TAÞ;

VB_TB ¼ jqjðTA � TBÞ þ kTð ~TB � TBÞ;

VA_SA ¼ jqjðSB � SAÞ þ kSð ~SA � SAÞ;

VB_SB ¼ jqjðSA � SBÞ þ kSð ~SB � SBÞ: ð20Þ

Here VA and VB are the volumes of the two basins,which we for simplicity will regard as having same sizeVA ¼ VB ¼ 1 and neglect all together. The first termson the right-hand sides are simply the exchanges ofwater between the boxes, while the second termsparameterise all other forces restoring to some meantemperatures ~TA, ~TB and salinities SA, SB, respectively.This latter parameterisation is termed a Newtoniancooling. The parameters kT and kS can be regarded aslinear response coefficients. They signify the (inverse)timescales associated with restoring to the mean statesfrom perturbations in temperatures or salinities. Thefour equations in (20) are linearly dependent: byintroducing T ¼ TA7TB and S ¼ SA7SB, subtractingthe second equation from the first and the fourth fromthe third we get

_T ¼ �jqjTþ kTð ~T� TÞ; ð21Þ

_S ¼ �jqjSþ kSð ~S� SÞ; ð22Þ

with q ¼ (bS7aT). The constant C is triviallyeliminated by rescaling of time, kT and kS. For afurther reduction of notation we introduce x ¼ aT andy ¼ bS so (21) and (22) become,

_x ¼ �jy� xjxþ kTð~x� xÞ; ð23Þ

_y ¼ �jy� xjyþ kSð~y� yÞ: ð24Þ

The forcing terms could be chosen in other ways, sayby ascribing a differential heating H to the temperatureequation and a fresh water forcing F to the salinityequation. The fresh water forcing represents the waterevaporated from the warmer box transported by theatmosphere and precipitated into the colder box,leading to a net transport of salt in the oppositedirection. The different forcings on the two equationsare termed mixed boundary conditions. Had theforcings been the same: kT ¼ kS ¼ k the equations(23) and (24) could be further reduced. Introducing thevariable z ¼ y7x:

_z ¼ �jzjzþ kð~z� zÞ; ð25Þit is easy to see that this model has a single necessarilystable fixed point. So in order to observe any non-trivial behaviour it is crucial that the model has mixedboundary conditions. The mixed boundary conditionsin the model are a natural consequence of the fact thatthe timescale for restoration of the salinity is muchlonger than the timescale for restoration of thetemperature, kS � kT. The model has three fixedpoints which can be seen in the phase space portraitof the tendencies in Figure 13.

The phase space for a set of two coupled first ordernon-linear Equations such as (23) and (24) is verysimple. Since trajectories cannot cross, most of theglobal information is obtained by identifying the fixedpoints. A short exhaustive list of possible fixed pointsis obtained from the linear stability of the fixed point.There will be two eigenvalues for the linearised system,assuming non-degeneracy of the eigen-space, thesecorrespond to two perpendicular eigenvectors. If theeigenvalues are real, there are three possibilities: twonegative eigenvalues corresponding to a stable fixedpoint, a negative and a positive eigenvalue correspond-ing to a saddle point and finally two positiveeigenvalues corresponding to an unstable fixed point.

Figure 12. The Stommel model of the Atlantic ocean. Theocean is represented by two boxes: a warm tropical basin (A)and a cold extra-tropical and polar basin (B). These areconnected by hydraulic links with a flux q, one in the surfaceocean and one in the deep ocean. The state of the system isrepresented by the two mean temperatures in the boxes, TA

and TB and the two mean salinities SA and SB.

Figure 13. The phase space for the two-box Stommelmodel. The dots are the fixed points. As is seen from thephase space flow, temperature adjusts much faster thansalinity.



If the eigenvalues are complex this corresponds tostable, neutral or unstable foci, depending on the signof the real part of the eigenvalues. The orbits along theeigenvectors, beginning and ending at fixed points, arethe homoclinic (beginning and ending at the same(saddle) fixed point) and heteroclinic orbits (connect-ing stable and unstable fixed points or stable/unstablefixed points with saddle points). These orbits separatethe phase space in basins of attraction for the stablefixed points. If an orbit is closed without passingthrough a fixed point it is a limit cycle.

In this case we can understand the behaviour of themodel by assuming that temperature will equilibrateinstantaneously, such that we can solve (23) in the case_x ¼ 0 and obtain the equilibrium temperature as afunction of salinity. From this an effective potentialdepending on the salinity alone can be constructed byintegrating the force along the curve (S, T(S)), asshown in Figure 14.

The changes in the thermohaline circulation of theAtlantic has probably been such that North Atlanticdeep water (NADW) is produced in the warm periodsand North Atlantic intermediate water in the coldperiods [40,41]. The key question is then whether theswitching between two stable climatic states of theoceanic flow can be internally triggered by the randomforcing within the ocean–atmosphere system [42].

11. Noise induced climate changes

Taking the ice core record to be a climatic proxyresulting from the climate dynamics described through aLangevin equation of the type (12), we have to confirmthat this is a consistent description and from the analysisto observe the structure of the noise driving the system.This provides strong constraints on the types of possiblemore realistic models of the underlying triggeringmechanisms for the observed climatic shifts.

The calcium signal from the GRIP ice core is thehighest temporal resolution glacial climate record

which exists [43]. The logarithm of the calcium signalis (negatively) correlated with the d18O temperatureproxy with a correlation coefficient of 0.8 [44], thus weuse the logarithm of calcium as a climate proxy. Thisproxy has a very high temporal resolution, since it isrelated to dust in the ice and it is therefore not diffusingin the firn (the snow-pack before compactified to ice)as the d18O isotope signal does. The temporalresolution of log(Ca) is about annual from 11 kyr to91 kyr BP (80,000 data points). This is an order ofmagnitude higher than that of d18O. The calcium signalfrom the GRIP ice core is shown in Figure 15(a). Thetypical waiting time for jumping from one state to theother is between 1000 and 2000 years. The probabilitydistribution for the waiting times between the begin-ning of the glacial – and the beginning of the followinginterstadial states – is shown in Figure 16. The straightline is an exponential distribution with mean waitingtime of 1400 years. This is expressed as P(T 4 t) ¼exp(7t/t), where t ¼ 1400 yr is the mean waitingtime. The probability density function (PDF) of thesignal, Figure 17, shows a bimodal distribution withpeaks corresponding to the warm interstadials and theglacial state.

From the premise of the stochastic dynamics andthe data we can now uniquely determine the climatepseudo-potential, U(y), and the structure of the noiseterm. The noise term (diffusion term) is to first order,neglecting the drift term, defined as the derivative ofthe signal estimated as (ytþDt7yt)/Dt, shown inFigure 15(b). This signal is stationary except for aslow trend through the record which is partly due tosmoothing with depth in the ice core. The intensity ofthe noise is thus approximately independent of theclimate state itself.

The noise is approximately a white noise (notcorrelated in time) and has a strongly non-Gaussiandistribution. Figure 18(a) shows the cumulated Prob-ability on a scale on which a gaussian distribution is astraight line (probability paper scale). Figure 18(b)shows the two distribution tails on a log–log plotmagnifying the behaviour of the tails. This has in anintermediate range a power function scaling with apower of about 2.75 and an additional extreme tail.The signal can only be described consistently by aLangevin equation if it is driven by two noise terms:

dy ¼ �ðdU=dyÞdtþ s1dxþ s2dL: ð26Þ

The first noise component, s1dx, is generated by an addi-tional Langevin equation, dx ¼ 7xdt þ (17x2)1/2dB,where x is an (unobserved) independent variable anddB is a unit variance Brownian noise. Here we haveintroduced the mathematically more correct notation ofincrements dy, dB etc. The notation is more correct

Figure 14. The effective potential U(S) along the curve (S,T(S)) shown as the fat curve in Figure 13.

524 P.D. Ditlevsen


since the process (12) strictly speaking is not differenti-able in time.

The stationary distribution for x is a t-distributionwhich fits to the observed tail distribution for the noise

on y. This term describes the forcing from theatmosphere. That the noise is white means that it isuncorrelated between consecutive data points. Thatdoes not exclude that the noise is red on shorter(unresolved) timescales. Actually the term 7xdt givesa correlation time of one year. It should be noted thatthe same signal could be scaled with a factor r:dx ¼ 7r2 xdt þ r(1 þ x2)1/2dB consistently with thedata as long as r 4 1. r71 signifies the correlationtime (which is shorter than one year). The reason that

Figure 15. (a) The logarithm of the calcium concentration as a function of time (BP) in the GRIP ice core. The dating of thisupper part of the record is rather precise. The temporal resolution is about 1 year, much better than the d18O record since thedust does not diffuse in the ice. The signal is a proxy for the climatic state. (b) The derivative of the signal in (a). Thisapproximately stationary signal is strongly intermittent.

Figure 16. The jumping between the glacial and interstadialstates is well described as a Poisson process. The waitingtimes are defined as the times between consecutive first upcrossings through the level log(Ca) ¼ 2 (glacial state) andfirst down crossings through the level log(Ca) ¼ –0.6(interstadial state). The waiting times have an exponentialdistribution (diamonds). The full line is an exponentialdistribution with a mean waiting time of 1400 years. The datarecord is not long enough to determine if the waiting timesfor the interstadials and the glacial states are significantlydifferent. The exponential distribution is consistent with thestochastic dynamics. The triangles, which are verticallyshifted for clarity, are results from simulation (see text).

Figure 17. The probability density function (PDF) of thelog(Ca) signal shows the bimodal distribution. The leftmaximum corresponds to the interstadial state and the rightmaximum corresponds to the glacial state. The thin curve isthe PDF of the simulated signal (Figure 6).



this noise term is not just a Gaussian white noise termis that the intra-annual variability is strongly depen-dent on season, with much stronger intensity in winterthan in summer, and that the inter-annual correlationleads to the red noise signal, where the ‘non-Gaussianity’ survives the annual averaging.

The second noise term is an a-stable noise withstability index a ¼ 1.75 [45]. The a-stable distributions,which are also called Levy flights, have cumulativeprobability tails which scale as x7a, for a 5 2,implying that only moments of order less than a exists(hjxjbi ¼ ? for b � a. The a-stable distributions fulfilla generalised version of the central limit theorem,namely that the distributions of sums of identicallydistributed random variables with cumulative

distribution tails scaling as x7a1!a1 converges to ana-stable distribution with a ¼ a1. These distributionshave very fat tails, meaning that the probability ofextreme events is high, such that single extreme eventswithin a period over which the variable is averaged willshow up also in the distribution of the averages. The a -stable distributions were first observed in hydrologicalrecords of river flow [46], and have later been observedin various different physical systems [47] such asturbulent diffusion [48] and vortex dynamics [49]. Tothe present there is still no full theoretical under-standing of why these distributions are observed, andto what extent it has importance in climate dynamics.

A generalisation of the Fokker–Planck equationfor the two coupled Langevin equations with a-stablenoise excitations connects the stationary densitysolution to the pseudo-potential U(y). However, onlythe marginal distributions are known. For y this is thePDF for log(Ca) shown in Figure 17. The pseudo-potential, shown in Figure 18, is thus determinediteratively by simulation starting from a solution to thestationary one-dimensional Fokker–Planck equationusing the marginal distribution.

In order to validate that the log(Ca) signal can bedescribed by (1) a consistency check must beperformed. This is done by simulation. Using thederived pseudo-potential, Figure 19, fitting s1 and s2from the noise structure of the signal, Figure 20, showsa realisation of (1). This should be compared with thelog(Ca) signal, Figure 15(a).

The thin lines in Figures 17 and 18 are derived fromthe simulated signal. The stationary Fokker–Planckequation does not contain information about thetimescales for jumping, Figure 16, and temporalharmonic decomposition of the sample as represented

Figure 18. (a) The probability density of the noise (Figure2(b)). (b) The cumulated distribution of the noise. The scale isa ‘probability paper scale’ where a gaussian distributionshows up as a straight line. This signal is strongly non-gaussian. (c) The two tails of (b) on a log–log plot. For theupper tail the probability of values larger than the abscissa isshown. The thin curves are from the simulation, showing thatthe signal is well described as containing a t-distributed noisecomponent and an a-stable noise component.

Figure 19. The climate pseudo-potential is a double-wellpotential with the left well representing the interstadial stateand the right well representing the full glacial state. Thepotential is obtained from a generalised stationary Fokker–Planck equation.

526 P.D. Ditlevsen


by the power spectra which are compared in Figure 21.These constitute independent verifications. As seen inall the figures the agreement is astonishing. Judgedfrom different simulated realisations the two signalsonly deviate within the statistical uncertainty.

This is not merely an advanced curve fittingroutine. If the calcium data is assumed to be generatedby the dynamics described through a Langevinequation, the driving noise must be of the formdescribed here. In order to understand the underlyingclimate dynamics it is important to establish theconnection between this climatic proxy and theclimate. It is especially important to interpret the twonoise terms and connect them to the atmosphere–ocean dynamics. The noise term ‘s1dx’ is probablyrelated to the ‘normal’ atmospheric fluctuations. Sincethe sampling is coarse on the timescales of thesefluctuations, there is is an indeterminacy in the noisestructure on timescales shorter than about one year. Inthe model this reflects itself in the invariance of thisnoise term with respect to r. The noise term s2dLrepresents extreme events and calls for attention. Thea-stable noise seems to occur in dynamical systemswith many different timescales where the dynamicsbecomes strongly intermittent.

12. Stochastic resonance in climate transitions

In the previous analysis the paleoclimatic record wasdescribed as resulting from noise induced jumpingbetween two stable climate states, where the jumpingprobability only depends on the present state of thesystem. Thus, it is a Markov process, for which thedistribution of waiting times becomes exponential, asindicated by the straight line in Figure 16. Theobserved record is short in the sense that only of theorder 25 DO events occurred in the last glacial period.The limited length of the record leaves the possibilitythat other models might fit the data just as well. It wasrecognised from spectral analysis that there seems to bea significant power at a frequency corresponding to a

period of 1470 yr for transitions into the interstadialclimate state [50]. By fitting the phase and overlaying aperiodic set of vertical lines on the climate record, seeFigure 22, a striking regularity emerges [51]. This hasbeen proposed to be due to a stochastic resonanceresponse to an (unknown) periodic forcing [52] or aghost resonance response to the beating frequencycombination of two much shorter periods in the solarintensity [53].

The strikingly regular timing needs to be testedstatistically. This is not completely straightforward:the general problem is that when observing apattern in a data set, the significance of the patterncan be very difficult to assess a posteriori unless thespace of possible outcomes for ‘striking patterns’ isknown.

Figure 20. An artificial log(Ca) obtained from simulating a sample solution to the Langevin equation using the climate pseudo-potential, an a ¼ 1.75 white noise and s1/s2 ¼ 3. This should be compared to Figure 16(a). The two signals are statisticallysimilar, showing that the log(Ca) signal can be generated by the stochastic dynamics.

Figure 21. The temporal harmonic resolution as expressedthrough the power spectrum is an independent measure ofthe signal. The top curve is the power spectrum of log(Ca).This is a red noise spectrum without significant peaks. Thebottom curve is the power spectrum of the simulated signalvertically shifted for clarity.



We shall denote the identified time sequence forjumps as ti, i ¼ 1, . . . , N. A preferred periodicity inthe time sequence can be detected by Rayleigh’s Rmeasure defined as RðtÞ ¼ ð1=NÞj

Pj exp 2ptj=tj,

where obviously R(t) 2 (0, 1) [13]. This measure iseasy to understand if we define the angles yi ¼ 2pti/tand plot the angles on the unit circle. If the timesequence is multiples of the time t modulo an(unknown) phase, all angles will be located near thesame point on the unit circle and R(t) � 1. Converselyif the data points do not cluster on the unit circle wehave R(t) � 0.

In Figure 23(b), the value of R(t) as a function of tis shown. The period of 1470 yr shows the largest valueR ¼ 0.65. The angles with respect to the 1470 yr periodof the time sequence of DO jumps are plotted on theunit circle in Figure 23(a). The mean phase is indicatedby the radial line segments, the length is equal to R(1470 yr), indicated by the arrow in (b), the mean phasedefines the vertical lines plotted in Figure 22.

The next, and necessary, step in the analysis is totest the significance of the periodicity found in thedata [54]. This can only be done by assuming a test-model generating the data. Given such a model, wemay choose any measure derived from the data, xdto compare with the same measure derivedfrom similar realisations of the test model, xm. Thenull hypothesis is then that the data series is aspecific realisation of the model. It is important tonote that a null hypothesis can only be rejected andnot confirmed. That is, the value of the chosenmeasure for the data may well be within the highlikelihood region for the model, but this does notprove that the data cannot be generated fromanother (competing) model with same highlikelihood for the chosen measure. On the contrary,only if the measure for the data falls within a lowlikelihood region, say with probability measurep � 1, the model can be rejected with probability17p.

Figure 22. A part of the NGRIP oxygen-18 record, where the dating is done by annual layer counting from the top. This is aproxy for temperature, where the transition into the present warm period is seen to the left. The last 10 Dansgaard–Oeschger(DO) climate events are shown. The vertical red lines shows an apparent periodicity in the initiations of DO events. Thesignificance of this regularity must be checked statistically.

Figure 23. The Rayleigh R for different periods. The maximum is obtained for the period t ¼ 1470 yr, indicated by the arrow.(a) Shows the timing of the onsets tn plotted on the unit circle using the transformation yn ¼ 2ptn/t. The numbering is thestandard numbering of DO events going back in time. The segments of radians points at the mean phase, corresponding to thevertical bars in Figure 22.

528 P.D. Ditlevsen


The simplest possible model which can be chosenfor the statistical test is that the DO events occurrandomly, without a memory, on the millennialtimescale. This is consistent with the analysis presentedabove. The waiting times follow an exponentialdistribution corresponding to a Poisson process. Themean waiting time can be assumed to be 2800 years.This is obtained as an estimate from the mean waitingtimes for the 14 DO events in the period 10–50 kyr.This is also the estimate obtained from the best fit to anexponential distribution of all DO events in the fullglacial period shown as the straight lines in Figure 16.

To test the data against this model we use theRayleigh R obtained from the data. For each of themeasures a probability density for a sample, similar tothe observed record, is obtained from a Monte Carlogenerated ensemble of 10000 realisations. The result isshown together with the measure from the data record inFigure 24(a). From the figure it is obvious that theobservations fall within the high likelihood region of theexponential distribution. The 90% (dashed) and 99%(full) confidence levels are shown in the figure as verticalblue bars. Thus there is no basis for rejecting thehypothesis of no-periodicity for the data.

The climate system is dominated by internalnoise masking possible periodic components. Thus,a reasonable assumption is that a periodicity iscaused by an internal non-linear amplification of aweak external periodic forcing. This could bedescribed by a stochastic resonance as proposed byAlley et al. [52].

The stochastic resonance model [19] is defined bythe governing equation:

dx ¼ @xUaðx; t; tÞdtþ sdB

¼ f�2ðx3 � xÞ þ a cos ð2pt=tÞg dtþ sdB;

where a particularly simple form of the drift potentialUa(x, t, t) is chosen here. The potential is a double-wellpotential, which changes periodically with period tfrom having a shallow well (s) to the right and a deepwell (d) to the left to the opposite situation. The ratioof the barrier heights Hs!d/Hd!s is determined by theamplitude parameter a. The timescales for jumpingfrom the shallow well to the deep well is given by anArrhenius formula; Ts!d * exp(Hs!d/2s), and simi-larly for Td!s. The criterion for resonance, where thesignal x is most periodic, is Ts!d � t�Td!s. Thisdetermines the noise intensity s. The differencebetween a hysteresis loop through a set of bifurcationsand a stochastic resonance is seen in comparing theschematics in Figures 25 and 26.

The proposition of rejecting a stochastic resonance(SR) model for the ice-core data is more tricky, sincethere exists a continuum of SR models with waitingtime distributions from the exponential to the delta-distribution for the perfect periodicity [55]. However,the only spectral weight notably above the continuumis at t71 ¼ 1470 yr71 and not at the mean waitingtime 2800 yr71. Near the stochastic resonance oneshould expect the same order of magnitude ‘earlyjumps’ (corresponding to a noise induced jump fromthe deep well to the shallow well) as ‘late jumps’(corresponding to missing a jump from the shallow tothe deep well). The mean waiting time being about

Figure 24. (a) By Monte Carlo an ensemble of 1000realisations of waiting times in a 40 kyr period has beengenerated from an exponential distribution with meanwaiting time of 2800 years, corresponding to 14 DO-eventsin 40 kyr. This gives probability densities for the maximalRayleigh’s R(t) in the range 500 yr 5 t 55000 yr. The redbar gives the values for the observed ice-core record. Theblue bars are 90% (dashed) and 99% (full) confidence levels.(b) Same as in the top panel, with distribution functionsobtained from stochastic resonance models with period of1470 years. From light to dark green the model parametersare: a ¼ 0.1, 0.2, 0.4 and s ¼ 0.38, 0.35, 0.27 (see text), whichgenerates on average 11 DO-events in 31 kyr. The importantdifference from the case shown in the panel above is that theRayleigh’s R in this case are calculated for the fixed period of1470 yr. The red bar is the ice-core observation as above. Theblack curve is the distributions for the exponential modelrepeated from (a). This shows that the SR model witha ¼ 0.1 cannot be identified in a sample of this length, sincespurious coincidental periodicities will give a better match tothe data than the 1470 yr cycle.



twice the observed spectral period indicates that apossible SR is ‘off the resonance’ with a too low noiselevel. In terms of SR parameters, this means that thecriterion; Ts!d � t � Td!s (Ts!d being the meanwaiting time for a transition from the shallow to thedeep well), is not fulfilled. We rather see t 5 Ts!d.

Here we test against three SR models with the periodt ¼ 1470 yr and a ¼ 0.1, 0.2, 0.4. The mean numberof DO events being 11 events/31 kyr corresponding tothe climate record. This determines the noise intensityto be s ¼ 0.38, 0.35, 0.27 for the three models.

An ensemble of 1000 simulations with same lengthas the data records were generated and the same threesignificance tests were performed. The results for thethree models; a ¼ 0.1 (light green), a ¼ 0.2 (mediumgreen), a ¼ 0.4 (dark green) are shown in Figure 24(b).The distribution in the top panel, for the exponentialdistribution, is over-plotted in black. It is seen that thefirst model, a ¼ 0.1, apparently has less periodicity,represented by Rayleigh’s R, than the purely exponen-tial model. This is because in the case of the SR model

the distribution is of R(1470 yr), while in the case ofthe exponential waiting time distribution (correspond-ing to a ¼ 0), the distribution is for the largest value ofR found in the sample. This means that for the SRmodel with a ¼ 0.1, the period will not be identified incomparison to other spurious coincidental periodici-ties. We have thus identified the ‘weakest’ SR modelwhich may be identified for a sample of the size of therecord.

So the statistical tests show that the waiting times forDO events are within the high likelihood region of theexponential distribution. This distribution implies thatthere is no long term memory in the climate system orunknown 1470 years periodic forcing triggering theclimate shifts. By the nature of the statistical test we canonly reject the hypothesis of a periodic component whenthe period is sufficiently above the noise level. For SRmodels with too low a strength of the periodiccomponent, the period would with high probabilitynot be detected in comparison to detecting a spuriouscoincidental periodicity in the sample.

Figure 26. For a weak periodic variation of the potential internal noise is required to induce a transition. For the rightcombination of noise intensity and barrier height the response to the periodic change is still periodic. This is the stochasticresonance.

Figure 25. A strong periodic variation of the potential leads to hysteresis behaviour through successive saddle-notebifurcations.

530 P.D. Ditlevsen


13. Summary

The paleoclimatic records show both remarkableclimate stability on geological timescales and dramaticchanges between different climate states. The transi-tions most likely results from combinations of desta-bilisation through bifurcations and transitions inducedby internal variation, which can be well described asstochastic noise. The current state-of-the-art generalcirculation climate models do a fair job in integratingthe flow equations for the atmosphere and oceans,admittedly at a coarse resolution. They also incorpo-rate many physical and chemical interactions involvingthe cryosphere (ice masses), the lithosphere (the landmasses) and the biosphere (vegetation) and give arealistic representation of the present climate. How-ever, the models are far from being able to simulate theobserved past climate transitions. It is even not knownif the models possess a non-trivial bifurcation struc-ture. The identification of the dynamical bifurcationdiagram from observations should thus be a guidelinefor verification of realistic glacial climate models.

The presence of a fat tailed noise component couldimply that the triggeringmechanisms for climatic changesare rare extreme events. Such events, being on thetimescale of seasons, are fundamentally unpredictableand never captured in the numerical circulation models.The lack of dynamical range might be due to under-estimation of internal variability in too coarse resolution,thus the climate noise is too weak to induce transitionsfrom one stable climate state to another. This could bepart of the explanation why these models have yet neversucceeded in simulating shifts between climatic states.

Notes on contributor

Peter Ditlevsen is Professor ofTheoretical Physics at the NielsBohr Institute, University of Co-penhagen. His main interests arein statistical physics, turbulenceand climate dynamics. His invol-vement in climate research beganwhen he was researcher for a yearat the Danish Meteorological In-stitute. Since then he has becomedeeply involved in interpretationof ice-core records obtained from

drilling in Greenland. He obtained his Ph.D. from Den-mark’s Technical University in solid state physics and did hisDoctorate at the University of Copenhagen in turbulenceand climate dynamics.

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