climate sensitivity to changes in solar insolation in a simple coupled climate model

J. Bendtsen

Climate sensitivity to changes in solar insolationin a simple coupled climate model

Received: 31 August 2000 /Accepted: 17 August 2001 / Published online: 18 December 2001� Springer-Verlag 2001

Abstract A simple coupled ocean, atmosphere and sea-ice model is presented. The idealised model consists of azonally averaged land and ocean strip of constant an-gular width extending from pole to pole. The meridionalenergy transport in the ocean is modelled by contribu-tions from the large scale thermohaline overturning cellsand from horizontal diffusive fluxes. The atmosphericmeridional energy transports are parametrised as diffu-sive fluxes in addition to advective transports in theHadley domain. This parametrisation resolves theequatorward moisture transport as well as the polewardtransport of potential energy in the upper branch of theHadley circulation. The model reproduces the annualaveraged meridional energy transports in the climatesystem with a small number of free model parameters.The basic feedbacks between the three climatic compo-nents are studied by investigating the model’s sensitivitytowards reductions in the solar insolation. It is foundthat the meridional energy transport in the ocean doesnot amplify the ice albedo feedback. This has importantimplications for modelling the climate sensitivity in at-mosphere-only models, as these would exaggerate thesensitivity to changes in the solar insolation if theirparametrisations of the meridional energy transport areconstrained by surface temperatures. The role of thedependence of the atmospheric transports on the me-ridional temperature gradient is shown to have a sig-nificant influence on the sensitivity on the coupledmodel, and the inclusion of seasonal cycles greatly in-crease the models sensitivity. The Hadley circulationdoes significantly alter the strength of the ice-albedofeedback in the coupled model. The idealised configu-ration of the model makes it a useful tool for studying

the feedbacks in the ocean-atmosphere-sea ice system inthe context of the ‘‘Snowball Earth’’ hypothesis.

1 Introduction

To understand the basic workings of the climate onEarth, a hierarchy of models have been used rangingfrom simple zero-dimensional models describing theglobal mean temperature to the complex three dimen-sional coupled ocean-atmosphere-sea ice models whichresolve many details of the physical processes in theclimate system. An influential group of such simplemodels originate from the energy balance models (EBM)of Budyko (1969) and Sellers (1969). The EBMs considerthe conservation of the vertical integrated energy in acolumn of the atmosphere-land-ocean system which isdetermined from the energy fluxes at the top-of-the-at-mosphere (TOA) and the divergence of the meridionalenergy fluxes in the ocean and in the atmosphere. TheBudyko-Sellers energy balance models only use the lat-itudinal distribution of the surface temperature Ts as astate variable and all energy fluxes and the extension ofice caps or sea ice are expressed in terms of Ts. Althoughthe details of the models of Budyko (1969) and Sellers(1969) differ, they essentially contain the same para-metrisations of the energy fluxes, in particular theparametrisations of the meridional energy transportwhich is calculated as a diffusive flux dependent uponthe local latitudinal gradient of Ts (�Ts) (North 1975).The models reproduce the latitudinal distribution of theannual mean surface temperature, and more importantfor this study, they also have a very high sensitivity tochanges in the solar insolation. Both models predictedthat only a few percent reduction of the solar insolationwould lead to a completely ice-covered Earth. This so-lution also appears as a stable solution in both modelsdue to the high albedo from sea ice and ice caps.As the solar luminosity has increased from a 25%

lower value of its present luminosity during Earth

J. BendtsenDanish Center for Earth System Science,Niels Bohr Institute for Astronomy,Physics and Geophysics,University of Copenhagen, Juliane Maries vej 30,2100 Copenhagen O, DenmarkE-mail: [email protected]

Climate Dynamics (2002) 18: 595–609DOI 10.1007/s00382-001-0198-4

history, the high sensitivity to reductions in the solarinsolation predicted by these simple models would havebeen a constant threat to life on Earth. This aspect haslead to numerous studies of the nature of the instabilitymechanism, as well as studies of how further improve-ments of the EBMs would alter the sensitivity tochanges in the solar insolation (Held and Suarez 1974;North 1975; Hartmann and Short 1979; Lindzen andFarrell 1977; North et al. 1981. The simplicity of theEBMs make them a suitable tool for studying thepaleoclimate on Earth, and in particular their highsensitivity to solar insolation have been used to supportrecent interpretations of the geologic record from750 - 550 million years ago during the Neoproterozoicperiod which argues for periods with a global glaciatedEarth, i.e., the Snowball Earth hypothesis (Hofmannet al. 1998; Hyde et al. 2000). A global glaciation couldalso have a profound influence on the evolution of life,as the harsh climatic conditions would act as an envi-ronmental filter promoting the rapid evolution ofmulticellular organisms during the Cambrian explosionwhich appear in the fossil record after the last globalglaciation (Hofmann et al. 1988). The Snowball Earthhypothesis raises a number of fundamental questions tothe workings of the climate on Earth, in particular towhether external forcing or internal feedbacks in theclimate system can initiate a global glaciation. Answersto these questions could also shed light on why theEarth, according to the Snowball Earth hypothesis, onlyexperienced a global glaciation a few times during thetime interval from 1 billion - 500 million years ago andnone afterward.The high sensitivity seen in EBMs is dependent upon

the efficiency in which energy is transported polewards(Held and Suarez 1974): a large meridional energytransport leads to an ice margin located close to thepole, but also to an increase in the sensitivity to changesin the solar insolation due to the relatively weak tem-perature gradient near the ice margin. As the sea icemargin is determined from the surface temperature inthese models, a small reduction in the insolation wouldcause a lower global mean temperature (Tm) and therebya relatively large equatorward displacement of the icemargin. This would result in an increase in the planetaryalbedo and a further decrease in Tm. In this way thepositive ice-albedo feedback is amplified by the merid-ional energy transport. This feedback amplification is aprominent feature of the Budyko-Sellers type EBM andits strength is primary determined by the parametrisa-tion of the meridional energy transport. Parameter val-ues have been estimated by fitting the model to theobserved global surface temperature. This temperaturedepends on energy transports in the atmosphere andocean but the simple EBM approach lumps togetherthese transports. Therefore the individual role of theocean and the atmosphere in the climate system cannotbe determined or compared in these EBMs. The im-portance of analysing the energy transport from theocean and the atmosphere separately was pointed out in

the early study of Held and Suarez (1974). Since the sea-ice formation is tightly coupled to the oceanic state, thestrength of this coupling could also alter the stabilityproperties of the coupled system.Here I develop and apply a simple coupled model to

quantify the relative interactions between the atmo-spheric and oceanic energy transports and the influenceof sea ice. This model builds upon the results from theformer simple energy balance models, but takes accountexplicitly of the atmosphere, ocean and sea-ice compo-nents of the climate system.Here the attempt is to identify basic mechanisms and

feedbacks in the climate system and to point towardscritical subprocesses which are crucial for the instabilityto take place, rather than determine the absolutethreshold values of the bifurcation points. Therefore themodel has a simple idealised configuration, andattempts to describe essential characteristics of theclimate system, e.g. the meridional energy transports,with a minimal number of global free parameters. Alsothe configuration and sensitivity studies are carried outin the context of the Snowball Earth hypothesis ratherthan estimating the sensitivity of the present-dayclimate. In Sect. 2 the ocean, sea ice and atmosphericcomponents are described and coupled. In Sect. 3 thefree parameters are scaled by using idealised observa-tional data from present-day climate as constraints, andin Sect. 4 the role of the free parameters and the mod-el’s, sensitivity to those are studied. Section 5 is a studyof the models sensitivity towards reductions in the solarinsolation when different model components are chan-ged, and Sect. 6 summarises the conclusions of thiswork.

2 Model description

The idealised model consists of zonally averaged atmosphere,ocean and sea ice components. The model is divided into 29% landand 71% ocean coverage corresponding to the present land-oceanfraction. The land-ocean distribution has been relatively stableduring the last billion years, over which the continental crust hasbeen increasing from about 80% of its present distribution (Taylorand McLennan 1995). The latitudinal distribution of land andocean can have a considerable influence on the radiation balanceon Earth (Crowley and Baum 1993), in particular if the land areasare covered by snow or ice, but for the purpose of this simplemodel, the ocean and land area is here simply confined to a singlestrip of constant angular width from pole to pole.

2.1 The ocean model

The ocean model is based on the zonally averaged, two-dimen-sional model of Marotzke et al. (1988). The model reproducesqualitatively the zonally averaged thermohaline circulation and hassimilar stability behaviour to changes in the boundary conditionsas other zonally averaged models used for paleoclimatic studies(e.g. Wright and Stocker 1991). The model does not explicitly re-solve the wind-driven circulation, so the influence from the wind-driven gyre motion and the intense western boundary currents isroughly parametrised by a constant horizontal diffusivity which isadded to the original model of Marotzke et al. (1988). In this model

596 Bendtsen: Climate sensitivity to changes in solar insolation in a simple

the zonally averaged thermohaline circulation is parametrised interms of the meridional pressure gradient. This parametrisationrelies on the assumption that the meridional velocity in the large-scale thermohaline circulation is related to the east-west pressuredifference across an ocean basin, and that this is related to themeridional density gradient. This relation is furthermore assumedto be local which leads to a diagnostic equation (Marotzke et al.1988):

vzz ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p

A�q0apx ð1Þ

where v is the meridional velocity, A* is a friction parameter relatedto the large-scale flow, q0 is the ocean reference density, p is thepressure, a is the radius of the Earth and x= sin(/) where / is thelatitude. Index z and x denotes the partial derivatives with respectto z and x. From the zonally averaged continuity equation themeridional and vertical velocities (v, w) can be expressed in terms ofa stream function w and Eq. (1) can be rewritten in terms of w andthe meridional density gradient. The density is determined from athird order polynomial approximation to the equation of state ofseawater to one atmosphere pressure (Winton and Sarachik 1993),which includes the temperature dependence of the thermal expan-sion coefficient.

The conservation equation for potential temperature (h) andsalinity (S) is given by:

Ut þ1

aðv

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

pUÞx þ ðwUÞz

¼ ðkvUzÞz þ1

a2ðkhð1� x2ÞUxÞx þ qU ð2Þ

where F= (h, S). Index t denotes the partial derivative with respectto time, the horizontal and vertical eddy diffusion coefficients aredenoted by kh and kv respectively, and qF is a source term due toconvective adjustment, which causes a complete vertical mixingwhen the stratification becomes unstable.

The boundary conditions for the model are assumed to be : w=0 at all boundaries assuring no fluxes through the boundaries, afree slip condition at the surface wzz= 0, and a no slip condition atthe bottom wz = 0. The boundary conditions at the surface aredetermined by the vertical diffusive flux of energy (Fo) and thevirtual salt flux (Fs), which accounts for the net freshwater fluxthrough the surface. There is no diffusive flux through the oceanboundaries so hz = Sz = 0 at z = )D and hx = Sx = 0 atx=±1. The ocean model is solved on an Arakawa C-grid using aforward in time and upstream scheme on a grid with 20 equallyspaced grid-points in the vertical and 50 equally spaced grid-pointsin the horizontal in terms of x. This means that each grid-pointcover the same area, whereas the meridional resolution varies from2.3� at the equator, 3.7� at 50�N,S to 11.5� at the pole. The modeldomain is from 90�S to 90�N, and the ocean is 4000 m deep. Thehorisontal resolution is the same for the atmosphere and the sea-ice model, and the model is integrated with a time step of threedays.

2.2 The atmosphere model

The fluxes of energy into a vertically integrated air column abovethe ocean/land surface is shown in Fig. 1. It is assumed that theenergy balance of the column can be related to the surface tem-perature Ts:

0 ¼ SW � LW � ðHAÞx þ xF ð3Þwhere SW is the incoming absorbed shortwave radiation, LW is theoutgoing longwave radiation at the top-of-the-atmosphere (TOA),(HA)x is the meridional divergence of the atmospheric meridionalenergy transport HA, and F is the energy flux at the bottom of theatmospheric air column. In case of ice free conditions F is equal tothe energy flux from the ocean Fo, otherwise F equals the energy

flux from the sea ice Fi described in the next section. It is assumedthat energy fluxes across the land-atmosphere interface can beneglected. The energy balance is calculated for a zonal belt aroundthe Earth, so x is the ratio of the ocean area to the total area in thatlatitude belt, which is assumed to be a constant equal to 0.71. Theland temperature is assumed to equal the ocean temperature, whichis a good zero order assumption as the heat capacity of theatmosphere is low compared to the ocean surface layers and thezonal transport in the atmosphere smooth out the temperaturedifference between land and ocean very efficiently. Due to thisrelatively low heat capacity of the atmosphere compared to theocean mixed layer, the atmospheric heat capacity is neglected in themodel (Nakamura et al. 1994; Marotzke and Stone 1995; Bendtsen1998).

The model includes the seasonal cycle of the incoming short-wave radiation SW. The solar insolation depends on the latitudex and the declination of the Earth (see e.g. Peixoto and Oort,1992):

SW ðx; tÞ ¼ S0sðx; tÞð1� aðx; xdÞÞ ð4Þ

where S0 is the solar constant, s(x,t) is the dependence of thesolar insolation on latitude and the time of the year (declination),and a is the planetary albedo dependent on the latitude and thesea-ice extent xd. The eccentricity of the Earth’s orbit around theSun is neglected so the Sun-Earth distance is set equal to itsmean distance, resulting in a symmetrical radiative forcing of thetwo hemispheres. The eccentricity leads to variations in the solarinsolation of ±3.4% throughout the year, which is small com-pared to the seasonal changes in the insolation. The planetaryalbedo a is high at high latitudes due to both ice and snowcovered areas, as well as increased cloudiness and insolationangle. This is taken into account by using an albedo for ice andsnow free conditions, which depends on the latitude. From themodel calculations of Hartmann and Short (1979) of the surfacealbedo without snow cover, a third order polynomial fit isdetermined:

aðxÞ ¼ 0:249þ 0:183jxj � 0:662jxj2 þ 0:787jxj3 ð5Þ

For ice covered areas the albedo is 0.6. The land area is assumed tobe covered with snow in the same latitudinal band as the ocean iscovered by sea ice.

Fig. 1. The energy fluxes into an atmospheric air column above theocean. The energy fluxes at the top-of-the-atmosphere are theabsorbed short-wave radiation SW and the outgoing longwaveradiation LW calculated as a function of the surface temperatureTs. The divergence of the atmospheric energy transport HA and theenergy flux from the ocean Fo closes the energy budget for theatmospheric air column. The upper ocean temperature h inthe ocean model is extrapolated to the surface to obtain Ts

Bendtsen: Climate sensitivity to changes in solar insolation in a simple 597

The outgoing longwave radiation at the TOA is related to thesurface temperature as in Budyko (1969):

LW ¼ Aþ BTs ð6Þwhere Ts is the surface temperature and the values for A= 212 Wm)2 and B = 1.7 W m)2�C)1 are taken from Nakamura et al.(1994). This simple relationship has been used frequently in EBMs,and it is based on a correlation between the surface temperatureand either the calculated LW (Budyko 1969) or the observed LW.From satellite observations of LW Cess (1976) found that B did notdepend on latitude and that the temporal correlation at 35�N gaveresults consistent with the results obtained from the spatial corre-lation including data from all latitudes. However, as the longwaveenergy loss from the surface actually decreases with increasing SSTfor high temperatures (Hartmann and Michelsen 1993), the for-mula shown should be used with care, in particular when tropicalregions are considered. A correlation of LW and Ts in the timedomain for all latitudes would take this effect into account, butnon-local dependencies of LW on Ts would still not be contained inA and B. Such a non-local correlation could be the dynamicalconnection between the radiative cooling in the descending branchof the Hadley circulation and the tropical SSTs. The sensitivitystudies presented here reduce the global mean temperature due todecreased insolation and thereby reduce the moisture content of theatmosphere. This reduction in the moisture content are in favour ofthe parametrisation in Eq. (6). With these precautions and to fa-cilitate comparison with former energy balance models the simpleparametrisation of LW is used here.

The atmospheric moist static energy E per unit mass is definedwith E= cp T+ Lvl+gz, where T is temperature, Lv is the latentheat of vaporisation, l is the mass-mixing ratio for water vapourand z is the height. The meridional energy transport is dominated bythe first two terms in the extra-tropics, whereas all three terms areimportant in the tropical and the subtropical areas (see e.g. Hart-mann 1994, Fig. 6.9). Also the meridional energy transport can beseparated into different dynamical regimes. In the extra-tropics thetransport by large-scale, synoptic eddies contribute the most to themeridional energy transport, whereas in the tropics the meridionaltransport is dominated by the Hadley circulation. These generalcharacteristics provide the motivation for considering the transportin the Hadley domain separately. This domain is also importantwhen the overall stability of the climate system is studied, as thisdomain represents the final obstacle for a progressive sea-ice coverbefore the Earth become completely ice covered.

Disregarding the eddy transport of potential energy, the generalexpression for the meridional energy transport can be written as:

HAðxÞ ¼ affiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p Z2p0

Z10

qaðLvðv0l0 þ vlÞ

þ cpðv0T 0 þ vT Þ þ gvzÞdz dk ð7Þ

where cp and qa are the specific heat capacity and density for dry airrespectively, v is the meridional velocity, k is longitude and v0T 0 andv0l0 are the Reynolds terms corresponding to the eddy fluxes ofsensible and latent energy, where the over-bar denotes the timemean average, and the primes denote deviation from the time meanvalue. Outside the Hadley domain the energy and freshwatertransport Fw are approximated by :

HAðxÞ ¼ affiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p Z2p0

Z10

qaðLvm0l0 þ cpm0h0Þdz dk ð8Þ

qwFwðxÞ ¼ affiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p Z2 p

0

Z10

qav0l0dz dk ð9Þ

Assuming that the meridional eddy-fluxes of energy and moisturearise from baroclinic instability of atmospheric waves, theoretical

and empirical studies have related the fluxes to the meridionalsurface temperature gradient (Stone and Miller 1980; Nakamuraet al. 1994:

qwFwðxÞ ¼ �ð1� x2ÞCleLv

RvðTsþ273ÞTxjTxjn ð10Þ

HAðxÞ ¼ LvqwFwðxÞ � ð1� x2ÞCT TxjTxjn ð11Þ

where Rv is the gas constant for water vapour. The meridionalsurface temperature gradient is denoted by Tx and it is raised tothe power n+1 which is set equal to 2 here (n=1) (Stone 1972;Held 1999). The role of this power dependence will be studied inSect. 5. The difference between CT and Cl is related to the relativesmall vertical scale on which latent energy is transported com-pared to the transport of sensible energy, so even though it is thesame physical process which transports the two forms of energy,they are transported with different efficiencies. The exponentialterm arises from the Clausius-Clapeyron equation and takes intoaccount the dependence of the atmospheric moisture content onsurface temperature, by assuming a vertical profile of l(z) and afixed relative humidity (Stone and Yao 1990; Wang et al. 1999).

The meridional transport in the Hadley domain is confined tothe interval )xH £ x £ xH where xH is the poleward extension ofthe Hadley cell. The Hadley circulation is assumed to be symmetricaround equator, and to be characterised by a vertical velocity w(x)across an interface h(x) separating the upper and lower branch ofthe Hadley cell as shown in Fig. 2. The vertical velocity is assumedlinear as w(x) = a|x|+b and it is zero at x ¼ � 1

2 xH , it has amaximum (upward motion) at x = 0 and a minimum (downwardmotion) at x = ±xH. By considering the vertical averagedmeridional flow in the lower branch of the Hadley cell andimplying continuity, the meridional transport V(x) defined by:

V ðxÞ ¼ZhðxÞ0

qaðzÞvðzÞdz ¼Zx0

qaðx; hðxÞÞwðx; hðxÞÞdx ð12Þ

can be determined by:

V ðxÞ ¼

4V0xxH

� �2� xxH

!: x � 0

4V0 � xxH

� �2þ xxH

!: x < 0

8>>>>>><>>>>>>:

ð13Þ

Fig. 2. The parametrisation of the meridional energy and moisturetransport in the Hadley domain defined by the interval [0, xH]assumes that the vertical velocity w(x) at the top of the lowertroposphere h(x), decreases linearly from x = 0 to xH. V(x) is theresulting meridional transport in the Hadley domain


where V0 is the maximal equatorward transport at x ¼ � xH inthe lower troposphere. The Hadley circulation is mainly drivenby the release of latent energy in rising convective air parcels inthe tropics. The vertical integral of this energy release due tocondensation is proportional to the precipitation assuming thatthe water storage in the air column is negligible. The precipita-tion then equals the sum of local evaporation and the conver-gence of the meridional moisture transport. As the Earthapproaches a global glaciation the atmospheric water vapourcontent would decrease significantly and this would alter thestrength of the hydrological cycle and also the dynamics of theHadley circulation. This effect is not accounted for in the modelas there is no quantitative estimate of this effect to constrain aparametrisation. The structure of the meridional transport in theHadley domain in Eq. (13) impose a two cell structure on theHadley circulation corresponding to the annual mean transportsin the Hadley domain. The large seasonal change in the Hadleycirculation, which is charaterised by a dominating single cellcirculation with a maximum equatorward transport during win-ter time is not resolved, and the role of this simplification isdiscussed in Sect. 6.

The atmospheric meridional energy transport in the Hadleydomain is given by Eq. (7) and the corresponding freshwatertransport is given by:

qwFwðxÞ ¼ affiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

p Z2p0

Z10

qaðv0l0 þ vlÞdzdk

¼ ð�ð1� x2ÞClTxjTxjn þffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

pVðxÞÞe�

LvRvðTsþ273Þ ð14Þ

The parametrisation for the meridional velocity is here defined asV(x) = V0 V(x)/V0 where V0 is a free parameter. Thecontributions to the meridional transports can be separated intoan ‘eddy’-term and an ‘advective’-term, corresponding to the twoterms on the right hand side of Eq. (14) respectively, and simi-larly the atmospheric energy transport can be written as:

HAðxÞ ¼ LvqwFwðeddyÞ þ LvqwFwðadvectiveÞ

� ð1� x2ÞCT TxjTxjn þffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

pVðxÞDhc

þffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

pVðxÞP

ð15Þ

Here the advective sensible energy transport is proportional tothe temperature difference between the upper and the lowerbranch of the Hadley cell (Dh), and the transport of potentialenergy is determined by the free parameter P. As the transportof latent and sensible energy only differ by their different verticaldistribution characterised by the constant c, it is assumedthat the same vertical scale difference which were argued toexist outside the Hadley domain, is similar in the Hadleydomain:

HAðsensible; eddyÞHAðlatent; eddyÞ

HAðsensible; advectiveÞHAðlatent; advectiveÞ

ð16Þ

From this relationship the value of c in Eq. (15) can be de-termined:

c ¼ CTCl Dh ð17Þ

The resulting net evaporation (E)P) can be calculated from Fw(x)

in Eq. (14) and converted to a virtual salinity flux, using a ref-erence salinity of 34.7 psu, which are used to force the oceanmodel. The ocean receives the atmospheric freshwater indepen-dent of whether sea ice is present or not.

These parametrisations have introduced four free parametersfor the atmosphere model : CT, Cl, V0 and P. These global con-stants are constrained by fitting the model solution to a set ofidealised observations.

2.3 The sea-ice model

The sea ice has the potential to alter the radiation balance of theplanet significantly on a very short time scale due to its high albedo(a) of about 0.6 compared to typical ocean and land albedos of 0.06and 0.1–0.3 respectively. Sea ice also influences the ocean circula-tion by insolating the ocean surface layer efficiently from theatmosphere due to the low heat conductivity of sea ice. Also theformation of sea ice enrich the surface waters with salt due to brinerejection. This leads to an increase in the density of the surfacewater which tends to destabilise the stratification of the watercolumn. On the surface of the sea ice, the temperature is no longer‘locked’ to the freezing point of seawater, and therefore the tem-perature can be very low, whereas it now is constrained by an upperlimit given by the freezing temperature of sea ice, which is 0 �C asthe salinity of the ice is assumed to be zero. This decoupling of thesurface temperature from the ocean changes the outgoing longwaveradiation significantly. During the melting or formation phases ofsea-ice energy is used or released respectively. These features areincluded in the sea-ice model described late.

The sea-ice model is shown in Fig.3. The energy balance in theatmospheric air column is now calculated with the energy fluxthrough the sea ice Fi. The temperature in the sea ice is assumed tobe in a steady state and advective and horizontal diffusive trans-ports are neglected, so the temperature can be described by the onedimensional steady state heat equation:

0 ¼ Tzz ð18Þ

The energy flux Fi through the sea ice is then given by

Fi ¼ �kiTz ¼ �kiTs � Tb

dð19Þ

where ki = 2.034 W m)1 K)1 is the thermal conductivity of sea iceand Ts and Tb are the temperature at the top and at the bottom ofthe sea ice of thickness d.

The prognostic equation for the sea-ice thickness d is deter-mined by energy conservation within the ice:

qiLi _dd ¼ Fi � Fo ð20Þ

where qi is the density of sea ice, Li is the latent heat of fusion forseawater and _dd is the time derivative of d. The formation of sea ice,in which the salinity is assumed to be zero, creates a salt flux intothe ocean (brine rejection) which is added to the salinity flux Fs at

Fig. 3. As in Fig. 1, but here the surface is covered with sea icewith a thickness d. The surface energy flux is now Fi and Fo is theocean energy flux to the sea ice, hm is the mixed layer temperatureand the fixed temperature at the bottom of the sea ice is denoted Tb


the surface of the ocean. The energy flux from the ocean surfacelayer to the sea ice Fo is parametrised as:

Fo ¼ �kiwðTb � hmÞ ð21Þ

where hm is the mixed layer temperature below the sea ice. Thislayer is assumed to be 50 m deep and hm is extrapolated from theocean temperatures below. The role of the exchange parameter kiwis discussed in Sect. 5.

The salt flux due to the formation of sea ice (brine rejection) isdetermined from

F iceS ¼ �Sref _dd ð22Þ

2.4 Coupling the model components

Following the assumption of no heat capacity in the atmosphericair column or in the sea ice, the energy fluxes into the air columnshould equal zero at any moment. At a given time tn the surfacetemperature T n

s and the sea ice thickness dn are known, and thebalance equation for the energy fluxes above the ice free areas isgiven by Eq. (3):

0 ¼ SW nðdnÞ � LW nðT ns Þ � ðHn

AðT ns ÞÞy þ F n

o ð23Þ

From Eq. (23) F no is determined and the ocean component can be

integrated to the next time-step tn+1. In case the surface is coveredwith sea ice, the ocean energy flux F n

o is calculated from Eq. (21).Now the surface flux in Eq. (23) is replaced with the flux throughthe sea ice F n

i , which is given by Eq. (19). Substituting this intoEq. (23) one obtains an expression depending on the now unknownsurface temperature T n

s :

FðT ns Þ¼ SW nðdnÞ�LW nðT n

s Þ�ðHnAðT n

s ÞÞy

�kiT ns �TBdn

¼ 0 ð24Þ

The expression F(T ns ) = 0 is solved for Ts by integrating

FðT ns Þ ¼ c _TT s during each time step for the ocean component,

using T n�1s as an initial guess and a small heat capacity corre-

sponding to 1 m of water for the constant c. All ice covered gridpoints are integrated simultaneously. In the end of the integrationin each time step the Ts-field converge towards the solution ofEq. (24), with a maximum residual of less than 0.05 �C day)1 inthe reference setup shown. When the new field of T n

s is deter-mined, the sea ice thickness dn+1 can be calculated from Eq. (20).From the definition of the energy flux through the sea ice inEq. (19) it is seen that this expression will go to infinity if d is zeroor very small. To avoid this behaviour the sea ice is allowed tobuild up to a thickness of 1 cm following Eq. (20), assuming thatthe surface temperature Ts is equal to the freezing point of sea-water and diagnosing the sea ice energy flux F n

i directly fromEq. (24). With this procedure the energy fluxes into the air col-umn balance according to Eqs. (23) or (24), and the ocean modelis driven by the resulting ocean surface energy flux, the net pre-cipitation on the ocean surface from Eq. (14) and the formationof sea ice calculated from Eqs. (20) and (22).

3 Scaling the model

The model outlined depends on eight free parameters :A*, kh, kv, kiw, CT, Cl, V0 and P0. These are all chosento be scalars, so that any latitudinal or vertical depen-dent parameters are avoided. This results in a verysimple model formulation, where the results are easilyinterpretated, but on the other hand this simplicity is

obtained at the expense of the model’s ability to fit theconstraints perfectly. This simple approach ensures thatthe number of free parameters are at least an order ofmagnitude less than the number of observations towhich the model is fitted. The free parameters are con-strained by the latitudinal fields shown in Fig. 4. Thesefields are obtained from present-day zonally averagedobservations which are symmetrised around the equator.These idealised symmetrised observations (SOD) arechosen to scale this idealised model instead of the zon-ally averaged fields, as these are strongly influenced bythe asymmetry between the Northern and SouthernHemisphere land distributions. The best fit values of thefree parameters are given in Table 1. With these pa-rameters the model has at least three stable steady states.One stable steady state does not have any sea ice, onehas sea-ice coverage down to x = ±0.86 and the laststable steady state is the complete ice-covered solution.Further steady states might exist but this issue is notconsidered any further here. The partly ice-covered stateis chosen as a reference state in the following investi-gation.The model solution of the annual averaged fields are

shown in Fig. 4. The equator to pole gradient in themodel (solid) is like the SOD (dash-dotted), except in thepolar areas where the cold temperatures on Antarcticacauses the SOD to be below )10 �C. The surface tem-perature is slightly asymmetrical around the equator,due to a weak asymmetrical circulation in the oceandiscussed later. The meridional energy and moisturetransports are also in overall agreement with the SOD.The moisture transport resolves the southward transportby the Hadley circulation and also the maximum size ofthe poleward transport agrees well with the SOD,though the maximum transports in the model are dis-placed equatorwards compared to the SOD. The abruptincrease of the atmospheric transports at the sea-icemargin is discussed later. Despite the simplicity of themodel, it resolves the main features of the atmosphericconstraints in Fig. 4a–c.The globally averaged vertical ocean temperature

distribution is shown in Fig. 4d. The deep ocean tem-perature is close to the observations and also the depthof the main thermocline is in accordance with the ob-servations, but it is a little too warm. The salinity dis-tribution does not resolve the relatively freshintermediate waters at about 800 m depth. As thesewaters are formed by combined wind and buoyancyforcing, these features cannot be resolved by the zonallyaveraged model which disregards the influence from thewind and the three dimensional structure of the winddriven gyre circulation.The oceanic meridional energy transports are shown

in Fig. 4f. The SOD for these transports have a maxi-mum of 2.5 PW at / = ±20�, whereas the model’smaximum of 2.0 PW is located at / = ± 40�. Theoceanic energy transport is due to the combined ther-mohaline and wind driven circulation, and in particularat low latitudes the wind driven circulation contributes


significantly to the transport by the western boundarycurrents and the poleward Ekman transport in the sur-face layers in the subtropical gyres (Bryden et al. 1991).This feature can not be resolved by the simple model, asthe influence from the wind driven circulation is notincluded. At mid-latitudes the ocean transport is com-parable to the SOD, but the transport in the zonallyaveraged model becomes very small poleward of the sea-ice margin, as this acts as a barrier for the thermohalinecirculation.In Fig. 5 the models oceanic (dashed line) and at-

mospheric (dotted line) energy transport are shown, andit is seen that the sudden decrease of the ocean trans-ports at the sea-ice margin is almost compensated by anincrease in the atmospheric transport, so the total energytransport changes smoothly across the sea-ice edge (solidline), and it is comparable to the total energy transportof the SOD (dashed-dotted line). The small change in theslope of the total transport across the sea-ice marginshows that the atmospheric transport can not compen-sate completely for the ocean energy transport. Thislimited atmospheric compensation across the sea-ice

margin will be shown to have important implications forthe coupled models sensitivity to reduced solar insola-tion.The transport streamfunction, potential temperature

and salinity are shown in Fig. 6. There are two slightlyasymmetric overturning cells with sinking at the poleswith a maximum transport of 18 Sv (1 Sv = 106 m3 s)1).This can be compared with estimates of the north At-lantic overturning of 27 Sv at 45�N and 12 Sv at 10�N(Macdonald and Wunsch 1996). The model cannotsustain a one-cell circulation where the only sinking is atone pole, in contrast to the behaviour of the uncoupledversion of the ocean model (Marotzke et al. 1988).Marotzke et al. (1988) showed that an equatorial sym-metrical circulation in a zonal averaged model wasunstable with finite amplitude perturbations underrestoring boundary conditions on surface temperatureand a fixed flux boundary condition on the salinity,whereas an equatorial symmetric circulation was shownto be stable for small perturbations in a three-dimen-sional primitive equation model with a similar set ofboundary conditions on temperature and salinity (Bryan

Fig. 4. The symmetrised obser-vational data (SOD) (dashed-dotted line) and the best fitmodel solution (solid line). a Thesurface temperature Ts. Theobserved surface mean temper-ature obtained from averaging40 years of the NCEP reanalysisdata. b The atmospheric fresh-water transport. The observedmeridional atmospheric fresh-water transport is obtained byintegrating the (E-P) valuesfrom Table 7.1 in Peixoto andOort (1992). c The meridionalatmospheric energy transportand the SOD obtained fromTrenberth and Solomon (1984).d, e The oceanic global meanpotential temperature and sa-linity and the globally averagedprofiles from Levitus et al.(1982) f The ocean meridionalenergy transport. The SOD arefrom Trenberth and Solomon(1994)


1986). Bryan (1986) also showed that the mixedboundary conditions on temperature and salinity causedmultiple equilibria where an equatorial asymmetricalcirculation with sinking at only one hemisphere alsoresulted in a stable solution. The imposed boundaryconditions for the salinity was obtained from theobserved salinity distribution in the ocean averagedaround the equator. Thus, it implicitly contained thestrong freshwater forcing in the North Atlantic, becauseof the large catchment area of North America andSiberia into the relatively small area of the northernNorth Atlantic. This means that the positive advectivesalinity feedback, where a reduced circulation leads toan increase in the residence time of the surface watersand thereby a decrease of the surface density due to thefreshwater forcing which then weakens the circulationfurthermore (Stommel 1961; Rooth 1982; Warren 1983;Marotzke 1990), is then strong enough to counteract thestrong thermal forcing at high latitudes in the noncon-vective hemisphere. This amplification of the freshwaterforcing at high latitudes due to a large land/ocean areafraction is not contained in the idealised configurationof the model. In addition the use of mixed boundaryconditions has been found to exaggerate the influencefrom the freshwater forcing as the surface energy fluxbecomes unrealistic when the ocean circulation changessignificantly (Rahmstorf and Willebrand 1995). Eventhough the positive salinity feedback is not strong en-ough to sustain a pole to pole circulation cell, it causesthe circulation to be slightly asymmetrical around theequator. No symmetric at solutions could be found,unless the equation of state of seawater was replacedwith a linear approximation neglecting the reduced in-fluence from temperature on the density at cold tem-peratures.The temperature distribution in Fig. 6b shows a

gradual decrease of the temperature in the main ther-mocline down to the relatively homogeneous deepocean. The salinity distribution in Fig. 6c shows thesaline subtropical surface areas, which appear as a resultof the large divergence of the atmospheric freshwatertransport, and consequently the relative fresh water inequatorial regions arises due to the convergence of themoisture transport here.

The sea-ice thickness decreases from 1.4 m at thepoles to temporary ice coverage at x = ±0.86 duringwinter time as shown in Fig. 7. The ice coverage of x=0.9 has a yearly mean value of 0.38 m and a seasonalcycle between 0.56 m and 0.08 m.

4 The role of the atmospheric free parameters

As the atmospheric module includes several new para-metrisations, the role of the four free parameters CT, Cl,V0 and P are analysed in detail here. If only the diffu-sive terms are included in the atmospheric energytransport, e.g. V0 = 0, the solution in Fig. 8 (dashedline) shows that the resultant poleward energy transportdecreases a little and because of this the surface tem-perature at the equator increases slightly. The majorchange is seen in the moisture transport where thetransport is polewards at all latitudes. Setting V0 = 0corresponds to the parametrisations used in formersimple energy balance models, e.g. North et al. (1981). Ifonly the energy transport is considered these models doresolve the main features of the observed transport butthe simple parametrisations used in EBMs fail to resolvethe equatorward moisture transport from the subtropicsto the tropics. As the freshwater is crucial fordetermining the strength of the thermohaline circulation,a proper description of the freshwater transport isalso important also at low latitudes in this type ofsensitivity experiment, where the sea-ice marginapproaches the equatorial regions before the glaciationis complete.When the advective terms from the Hadley circula-

tion are included and the transport of potential energyis disregarded, e.g. P=0, the poleward energy trans-port is reduced compared to the case where V0 = 0, asenergy now is advected towards equator in the Hadley

Table 1. Best fit values of the free parameters used in the model

Freeparameter

Descriptionof the parameter

Best fit value

A* Ocean friction 1.0Æ103 m2 s)1

kv Ocean vertical diffusion 2.0 Æ10)4 m2 s)1

kh Ocean horizontal diffusion 3.0 Æ103 m2 s)1

kiw Ocean-sea ice energyexchange

1.5 Æ102 W m)2 K)1

CT Sensible energy transport 1.5 Æ1012 W m K)2

Cl Latent energy transport 8.0 Æ1013 m kg K)2 s)1

V0 Transport in the Hadleycirculation

4.5 Æ1016 kg s)1

P Potential energyin the Hadley circulation

)9.0 Æ10)2 J kg)1

Fig. 5. The annual averaged energy transports in the reference caseof the ocean (dashed line), atmosphere (dotted line) and the sum ofthe oceanic and atmospheric energy transports (solid line). TheSOD are from Trenberth and Solomon (1994) (dashed-dotted line)


domain (dotted line). This results in a significantincrease in the tropical surface temperatures. Despitethe equatorward advection of moisture, the moisturetransport is still poleward. This is due to the diffusivefluxes which now are relatively large in the tropicalregions compared to the case where V0 = 0, becausethe convergent energy transport results in a steeptemperature gradient from the tropics towards thesubtropics, thereby increasing the poleward diffusivetransport. In the final case where the energy is trans-ported away from equator, e.g. both V0 and P equalstheir best fit values (solid line), the tropical temperatureprofile becomes flat and the diffusive fluxes diminish inthe equatorial regions. Thereby the advection ofmoisture becomes the dominant term for the transportof moisture and the subtropical divergence in the

freshwater transport is simulated well by the model.The four free parameters in the atmospheric formula-tion include several important aspects of the meridionalatmospheric circulation, and may constitute a minimalnumber of free parameters necessary to resolve themain features of the meridional moisture and energytransport in a zonally averaged model of the climatesystem.

5 Sensitivity to reduced insolation

The model is started from the reference state and thenthe solar constant is gradually decreased at a rate of10)6yr)1 from its present value, until the model statebecomes completely covered with sea ice.

Fig. 6. The ocean annual aver-aged fields in the reference state.a The transport stream function(Y) in the model domain.b Potential temperature (h).c Salinity (S)


Figure 9 shows the model’s sensitivity to reductions(�) in the solar constant when the model’s best fit valueof (Ch, Cl) is changed by a factor k ranging from 0.5 to

1.5. With the reference values of the atmospherictransport parameters CT and Cl, e.g. k = 1, the modelbecomes totally ice covered when the insolation isreduced with 4.2% (solid line), whereas changing thevalues by 50%, e.g. k = 0.5 and 1.5, the bifurcationtakes place at 4.8% and 4.3% respectively. The globalmean temperature shown in Fig. 9b is directly related tothe sea-ice extent due to the linear dependence of LW onTs in Eq. (6). The coupled model’s sensitivity to �decreases for small values of k. This result is consistentwith the findings in simple EBMs, whereas the sensitivityof the coupled model is seen to be lower when k = 1.5,contrary to the sensitivity found in EBM’s (Held andSuarez 1974). A large meridional transport causes thesea ice extent to remain at high latitudes as seen fromFig. 9a, and the warmer temperatures at high latitudesincrease the transport of latent energy, which also tendsto reduce the sea-ice extent.The critical reduction in the solar insolation �c,

defined by � at the bifurcation point between the partlyice-covered and complete ice-covered steady states, isshown in Fig. 10 for different values of k. The referencecase (solid line) shows little variation of �c between4.2% and 4.8%. This reference case is compared to acase where the atmospheric transports are only pro-portional to the meridional temperature gradients to thefirst power, e.g. n = 0 in Eqs. (10) and (11). To makethe two cases comparable the values of (Ch, Cl) is scaledin the case with n = 0, such that the yearly mean valueof HA at 35�N is similar to the reference case. Thesensitivity shown for the case n = 0, is seen to be muchless in Fig. 10 (dashed line), and also the purely diffu-sive case has a much stronger dependence on k than inthe reference case, ranging from �c = 10.4% for k =0.5, to �c = 5.4 % at k = 1.5. The stronger dependenceof HA on the meridional temperature gradient in thecase of n = 1 is a strong negative feedback on themeridional temperature gradient as argued in Stone(1973), so the critical value �c changes only little forvariations in k compared to the case with n = 0. Thissensitivity is in contrast to the findings in simple EBM’swhere the dependence on n was shown not to be sig-nificant (Held and Suarez 1974). The sensitivity is lessdependent on the value of k in the case of n = 1, as thetemperature gradient near the sea-ice margin is deter-mined primarily by the energy transports’ power de-pendence on Tx.To compare the behaviour of the coupled model to

models which do not explicitly resolve the oceandynamics and energy transport, the ocean depth wasdecreased to 100 m and the advective, convectiveand horizontal diffusive transports in the ocean wereneglected, whereas the vertical diffusivity coefficient wasincreased to 10)2 m2 s)1. This corresponds to a diffusivetime scale for the mixed layer of the ocean of about 10days, and an essential homogeneous water columnresults from this large vertical mixing. All other modelcomponents were left as in the reference case. Theresultant surface temperature, energy and moisture

Fig. 7. The seasonal changes of the sea-ice thickness at the twomost equatorward ice-covered grid points located at x = 0.90(upper curve) and x = 0.86 (lower curve)

Fig. 8a–c. The solution with the reference values of CT, Cl, V0

and P (solid line) and the symmetrised averaged observations(dashed-dotted line). The case with V0=0 is shown with a dashedline and the case with P=0 is shown with a dotted line. a Surfacetemperature Ts. bMeridional freshwater transport Fw. cMeridionalenergy transport HA


transports are shown in Fig. 11 for this slab ocean ver-sion of the model (dashed line), the coupled model (solidline) and the SOD (dash-dotted line). The surface tem-peratures in Fig. 11a are a few degrees warmer, due tothe lack of upwelling of cold deep water which cools thetropics in the coupled model. The atmospheric energytransport shown in Fig. 11c almost compensates for thelack of ocean energy transport. The slightly warmerequatorial region leads to an enhanced moisture trans-port in the Hadley domain and the steeper equator topole gradient also causes the transport of sensible andlatent energy to increase at mid latitudes as shown inFig. 11b, c. The sensitivity of the slab ocean version ofthe model to changes in k is shown in Fig. 10 (dash-dotted line), and it ranges between 3.4% for k = 0.5 to2.3% for k = 1.0.When the model is forced with the annual mean in-

solation, the critical insolation increases to �c = 5.3%,

which demonstrates the importance of the seasonal cyclein destabilising the model towards reduction in the solarinsolation (Fig. 12, dashed line). A similar decrease ofthe sensitivity is found for the slab-ocean model, wherethe sensitivity decreases from 2.3% to 4.3%. The valueof the exchange parameter kiw in Eq. (21) was found tohave a minor influence on �c, as the small ocean energytransport across the sea-ice margin in the zonally aver-aged model allows the mixed layer temperature toapproach the freezing temperature under the sea ice andthereby the influence from the energy flux through thesea ice becomes very small. There is no significantchanges of the model’s sensitivity when the influence ofsea-ice formation on the surface salinity is suppressedcompared to the reference case.With an expanding sea-ice coverage of the Earth in

an early phase of a Snowball Earth scenario, the sea-icemargin at some point would reach the poleward rim ofthe Hadley cell and also approach the wind-driven,subtropical gyres in the ocean. This might be the lastobstacle for the ice before the complete glaciation ofthe Earth. Lindzen and Farrell (1977) considered thisfeature in the context of Budyko’s (1969) EBM. Theyreplaced the latitudinal dependent insolation by aconstant in the Hadley domain, thereby implicitlyimposing an energy transport by the Hadley circulationand this caused a weak meridional temperature gradi-ent in the Hadley domain. As a result, their EBM wasless sensitive to reductions in the solar insolation. Tounderstand this result in terms of the ice-albedo feed-back, one has to focus on the discontinuity which arisesat the poleward rim of the Hadley domain in theirmodel. The Hadley circulation by itself tends toweaken the meridional temperature gradient within theHadley domain and this would strengthen the sea-icealbedo feedback, but at the discontinuity point, located

Fig. 9. The sensitivity of the model to reductions in the solarinsolation for three different values of k scaling the exchangeparameters (Ch, Cl). The reference case k = 1 (solid line), k = 0.5(dashed line) and k = 1.5 (dashed-dotted line). The reduction isexpressed in terms of 1)�, where the reduction of the solarinsolation � is defined by : �= 1 )S0 / S0 (present-day). a The sea-ice extent xd is determined by the grid point covered with sea iceclosest to the equator. b The global mean surface temperature. Tm

and xd are annual averaged values

Fig. 10. The critical insolation �c (see text) as a function of thestrength of the meridional circulation exchange parameters (Ch, Cl)which are scaled with k. The reference case (n=1) is shown by asolid line, a linear diffusive atmospheric energy transport (n=0) isshown by a dashed line and a slab-ocean version of the model isshown with a dashed-dotted line


at the poleward extension of the Hadley domain, theHadley circulation causes a strong local meridionaltemperature gradient and this gives the decreasedsensitivity.To investigate this feature in the model, the Hadley

transportV0 is scaled with a constant v, and the criticalinsolation �c for v ranging between 0 and 1.5 is shown inFig. 13 (solid line). The presence of the Hadley circula-tion does have a significant influence on the modelssensitivity, as �c increases from 3.3% for v = 0 to 4.8%for v= 1.5, in qualitative agreement with the findings ofLindzen and Farrell (1977) for a simple EBM. Forcomparison the sensitivity to v is determined in the caseswhere the model is forced with the annual mean inso-lation (dashed-dotted line) and the slab ocean version ofthe model (dashed line). Both cases show the samedependence on v but the sensitivity levels are shifted asdiscussed already.In Fig. 14 the general behaviour of the model is

shown in four extreme cases. The slab ocean case (dotted

line), similar to the case discussed in Figs. 10 and 11,corresponds to the extreme case of no oceanic transport.The other extreme is the case with only ocean transports,e.g. Ch = Cl = V0 = 0, and it is seen that this case isvery stable towards reduction in the solar insolation, asthe sea-ice edge gradually approaches the equatorwithout any critical latitude. The model now becomesice covered with �c = 32.0%. In the absence of atmo-spheric energy transports, the convergence of the oce-anic energy transports at the sea-ice margin, which actsas a barrier for the oceanic thermohaline circulation, willcreate a very steep local temperature gradient which willstabilise the model.In the last two cases shown in Fig. 14, the ice-cov-

ered area has the same albedo as the ice-free area, sothe ice albedo feedback is absent by definition. In onecase the Hadley circulation is suppressed and the iceedge gradually approaches the equator which isreached when � is increased to 27%. When the Hadleycirculation is present the model becomes glaciatedwhen � is increased to 17.5%. These cases show muchlarger amplitudes of internal variability as seen fromthe irregular curve, due to the missing albedo differencebetween ice and ocean which otherwise tends to dampthe internal oscillations. The presence of the Hadleydomain creates an instability in the absence of the ice-albedo feedback. As the sea ice approaches the Hadleydomain, the surface temperature at the poleward rim ofthe Hadley domain will approach the freezing temper-ature, and as the Hadley circulation efficiently smoothsout the temperature difference within the domain, thetemperature will be close to the freezing point here.Therefore any further reduction in the insolation causesthe whole domain to freeze over. This points towards acritical role of the Hadley circulation in a SnowballEarth scenario when the sea-ice edge approaches theequator.

Fig. 11. The solutions of the slab-ocean version of the model(dashed line), the coupled model (solid line) and the SOD (dashed-dotted line). a Surface temperature, b meridional freshwatertransport and c meridional energy transport, and the sum of theSOD for the atmosphere and the ocean (dashed-dotted line)

Fig. 12. The sensitivity of the model to reductions in the solarinsolation � in the reference case (solid line), the case with noseasonal cycles (dashed line), the slab-ocean model (dotted line) andthe slab-ocean model with no seasonal cycles (dashed-dotted line)


6 Discussion

A simple coupled model consisting of a dynamic ocean,sea ice and atmosphere component has been developed.The model can quantitatively reproduce the main char-acteristics of the meridional energy and freshwatertransports in the atmosphere and in the ocean, the oceantemperature field, the surface salinity field and the sea-ice distribution with a small number of free modelparameters. The model is used to study basic feedbacksin the climate system, in particular the role of the ice-albedo feedback.The ice-albedo feedback’s dependence on the strength

of the meridional energy transport found in Budyko-Sellers type EBMs (Budyko 1969; Sellers 1969; Held andSuarez 1974) is also found in the coupled model.However the ocean and atmosphere have very differentroles in setting the strength of this feedback. The me-ridional energy transport in the ocean was shown not todestabilise directly the model, so there is no amplifica-tion of the ice-albedo feedback from the ocean trans-ports alone. The thermohaline circulation transportsenergy to the sea-ice margin, which acts as a barrier forthe ocean circulation. Consequently, the energy releasedby convection right at the sea-ice margin tend to weakenthe ice albedo feedback due to the steep temperaturegradient created just in front of the sea-ice margin.Therefore the efficiency of the local atmospheric trans-port across the sea-ice margin becomes an importantcomponent for determining the stability of the com-bined system. In the coupled model the atmosphericenergy transport was not able to compensate completelyfor the lack of ocean energy transport across the sea-icemargin.These different roles of the ocean and atmospheric

energy transports motivated a reconsideration of the

results from simple EBMs which lump the oceanic andatmospheric transports together. Such models wouldconsequently exaggerate the sensitivity of the modelsresponse to changes in the solar insolation due to astrong ice-albedo feedback. This is illustrated in Fig. 10where the critical reduction of the solar insolation of4.3% in the coupled model is reduced to 1.4% in a slab-ocean version of the model.The strength of the Hadley circulation has a sig-

nificant influence on the stability of the model, as astronger Hadley circulation causes the meridionaltemperature gradient at mid-latitudes to becomesteeper, and thereby the Hadley circulation worksagainst the positive ice-albedo feedback (Lindzen andFarrell 1977). This role of the Hadley circulation canbe related to the possible role of the wind drivenocean circulation which is not resolved in the model,as the subtropical gyres also transport energy from thetropics to the subtropics. In the current ocean-atmo-sphere system the oceanic low-latitude meridional en-ergy transport exceeds the atmospheric energytransport (see Fig. 4c, f), so a more dynamical for-mulation of the wind driven ocean circulation and theHadley circulation is a prerequisite for determining thestability of the climate system when the sea ice ap-proaches equator. The seasonal change in the Hadleycirculation could also influence the stability, becausethe meridional transports in the Hadley domain has itsmaximum during winter time when the sea ice formsat the sea-ice margin. When the ice-albedo plays aminor role for the radiative energy balance on Earth,e.g. in a scenario with an excessive low-cloud coveragewhere the higher albedo from the clouds would reducethe influence from the high sea-ice albedo, the Hadley

Fig. 13. The critical insolation �c as a function of the strength ofthe Hadley circulation V0 scaled with v. The critical reduction isshown for the coupled model (solid line), the slab-ocean version ofthe model (dashed line) and the case where the model is forced withthe annual mean insolation (dashed-dotted line)

Fig. 14. The model behaviour in the reference case (solid line), theslab ocean version (dotted line) and the case with no atmosphericmeridional energy and freshwater transport (dashed line). Twocases with large internal variability are shown where there is noalbedo difference between ice-covered and ice-free regions. The casewithout a Hadley circulation approaches the equator gradually,whereas the case with a Hadley circulation becomes ice coveredwhen the insolation is reduced by 17.5%


circulation and presumably also the wind driven oceancirculation would work in the opposite direction andincrease the sensitivity of the climate system, favouringa global glaciated state due to the homogenisation ofthe surface temperatures in the tropics, as shown inFig. 14. The role of the clouds is not resolved in thecurrent simple model, but it might be of first orderimportance for the sensitivity of the climate system. Ina study by Hyde et al. (2000) an atmosphere/ice-cap/slab ocean model shows the existence of near-equa-torial, ice-free areas in an otherwise glaciated world.This model solution was obtained with an atmosphericCO2 concentration 2.5 times higher than the present-day level and a strong reduction in the planetaryalbedo due to reduced cloudiness in an almost glaci-ated world. The inclusion of a dynamical ocean andsea-ice model would have the instability behaviourshown in Fig. 14 within the Hadley domain, and couldwork against near equatorial ice-free areas.The influence from the brine rejection during sea-ice

formation on the thermohaline circulation was shownnot to be significant. More important was the inclusionof seasonal cycles in the solar forcing, as the sensitivityof the model was significantly larger when the seasonalcycles were included compared to the case with onlyannual mean solar forcing. The larger sea-ice extentduring winter time and the steeper equator to pole gra-dient enhance the positive ice-albedo feedback when themodel is forced with a seasonal varying insolation,compared to the case with a constant solar insolationthroughout the year.The model contain a simple dynamical description

of the interaction between the ocean, atmosphere andsea ice and therefore it is a suitable tool for studyingthe feedbacks in the climate system in the context ofthe Snowball Earth hypothesis. This hypothesisinvolves huge changes in the climate on Earth, andtherefore this hypothesis should be seen as a greatchallenge for climate modelling. This issue willbe pursued with the simple coupled model presentedhere.

Acknowledgements I wish to thank Gary Shaffer for comments onthe manuscript, Rodrigo Cabellero for stimulating discussions andJochem Marotzke for providing the original version of the oceanmodel. This work was funded by the Danish Natural ScienceFoundation.

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climate sensitivity to changes in solar insolation in a simple coupled climate model

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