J. Meteor. Res.  2014, Vol. 28 Issue (5): 747-761   PDF    
http://dx.doi.org/10.1007/s13351-014-4027-1
The Chinese Meteorological Society
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Article Information

LI Yaokun, CHAO Jiping. 2014.
Two-Dimensional Energy Balance Model and Its Application to Some Climatic Issues
J. Meteor. Res., 28(5): 747-761
http://dx.doi.org/10.1007/s13351-014-4027-1

Article History

Received 2014-4-23;
in final form 2014-7-8
Two-Dimensional Energy Balance Model and Its Application to Some Climatic Issues
LI Yaokun1 , CHAO Jiping2    
1. College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875;
2. National Marine Environmental Forecasting Center, Beijing 100081
ABSTRACT:Based on a two-dimensional energy balance model, the studies on some climatic issues such as the relationship between ice cap latitude and solar constant, desertification, and the warming effect of carbon dioxide, have been reviewed and discussed. The phenomenon that a fixed solar constant might correspond to different equilibrium ice cap latitudes is determined by the continuity of albedo distribution. The discontinuity in albedo distribution increases the number of equilibrium ice cap latitudes. Desert would expand both northward and southward when desert surface albedo is increasing. This would deteriorate the ecological environment in border regions, and then threaten the existence of local inhabitants. Melting of the polar ice would not be accelerated, with increasing carbon dioxide concentration. The ice cap latitude would move northward slowly, with some "hiatus" periods, under the slowly increasing global average surface temperature. According to the current research, future development of the two-dimensional energy balance model and possible progress are also forecasted.
Keywordstwo-dimensional energy balance model     multiple equilibria     ice cap latitude     desertification     warming effect of carbon dioxide    
1. Introduction

Global change is one of the most active and fastestgrowing fields in the scientific research, and climatechange is one of the major research priorities in globalchange(Ye et al., 2002). Changes in the radiationbalance of the earth-atmospheric system caused bygreenhouse gases and aerosol concentrations will alterthe energy balance of the climate system and causeglobal climate change(Qin et al., 2007). The nega-tive impacts of climate change on natural ecosystems, economy, and national security system gain far moreattention than its positive impacts(Qin, 2004). A se-ries of serious environmental issues accompanied withclimate change has threatened the development of hu-man society. It has become an urgent need for thehuman survival and development to avoid possible en-vironmental disasters and to achieve the sustainableeconomic and social development through scientific and rational way(orderly human activities)(Ye et al., 2001).

Mankind has not yet come to really underst and the driving forces of climate change on differenttimescales and the complex response-feedback pro-cesses in the climate system, even though the contri-bution of increasing greenhouse gases to global warm-ing has already been recognized(Liu, 2006). For oneexample, ice core data analysis showed that there isno direct causal relationship between greenhouse gasesconcentration and atmospheric temperature and tem-perature change could even lead the concentrationchange of greenhouse gases(Cui et al., 2012). Foranother example, the linear trend of global averagetemperature during 1999-2008 is close to zero, signif-icantly lower than previous projection(Knight et al., 2009); however, concentration of greenhouse gases has steadily increased. Therefore, it not only has impor-tant theoretical significance but also provides somepractical guidance to the orderly human activities todeeply analyze and underst and the various driving fac-tors and feedback processes which contribute to theglobal climate change.

Climate and earth system models are powerfulresearch tools which could be used to quantitativelyanalyze the impacts of certain driving factors or feed-back processes on global or even regional climate. Thiscould no doubt deepen the underst and ing of climatechange, project for future change, and provide refer-ence for decision makers. In addition, some simpletheoretical models with solid physical basis could alsobe used to study climate change. These models couldhighlight the driving factors for climate change, ana-lyze in-depth the physical processes, supplement othermodels' results theoretically and physically, and im-prove and enrich the climate change research. Thesimplest approach is to consider the earth system asa particle satisfying the energy conservation, whichcould be used to solve the average temperature of thesystem. This forms the zero-dimensional energy bal-ance model(EBM). It is easy to introduce the green-house effects and ice albedo feedback mechanism intothe zero-dimensional EBM through revising some rel-ative parameters(Budyko, 1969; North et al., 1981).

One-dimensional EBM could be derived easilyby introducing the latitudinal temperature distribu-tion into the zero-dimensional EBM. The classicalstudies by Budyko(1969), Sellers(1969), and North(1975a)are most representative and have had a pro-longed impact on ensuing research. Two-dimensionalEBMs(Sellers, 1976; Chao and Chen, 1979)couldalso be constructed by further introducing longitu-dinal or vertical physical processes. The role ofoceans could also be considered(Shi et al., 1996).Extending the zero-dimensional EBM vertically getsone-dimensional radiative-convective model(Manabe and Strickler, 1964; Manabe and Wetherald, 1967)which could not be solved through analytical meth-ods. To obtain the analytical solution, it is necessaryto simplify the long-wave radiation absorption spec-trum. For example, Chao and Chen(1979)developed a two-dimensional EBM by using the simplified long-wave radiation absorption scheme introduced by Kuo(1973). Based on this two-dimensional EBM, theystudied the relationship between ice cap latitude and solar constant(Chao and Chen, 1979), desertification(Chao and Li, 2010a), and warming effects of carbondioxide(Chao and Li, 2010b). Their studies would beintroduced, improved, and developed in the followingsections.

2. Two-dimensional EBM

The average atmospheric temperature field is con-trolled by many factors, such as the latitudinal distri-bution of solar radiation, radiation propagation and conversion processes, turbulent heat exchange, latentheat caused by condensation and evaporation pro-cesses, and the heat exchange caused by cold and warmadvection. The temperature equation could be writtenas(Ye and Zhu, 1958)

where ρa is air density, Cp is specific heat of air, T isair temperature, and p is the pressure; ε1, ε2, and ε3are the heat flux caused by radiation, turbulence, and condensation or evaporation, respectively.

Set kj as the longwave radiation absorption coeffcient at the wavelength λj; k' as the mean radiation absorption coeffcient for solar radiation; Aj(Bj)as the downward(upward)longwave radiation duringthe wavelength interval Δλi;Ej as the black body radiation during the same wavelength interval Δλi;Qas the downward solar radiation. Then, the heat fluxcaused by radiation could be written as

According to Bouguer-Lambert law

The heat flux caused by turbulence could be expressed as

where Kh is horizontal turbulent thermal diffusivity(κh = ρaCpKh is called horizontal turbulent thermalconductivity); K is vertical turbulent thermal diffusivity(κz = ρaCpKh is called vertical turbulent thermalconductivity); is the horizontal Laplacian operator.Note that Kh and K are set to constants. Tempera-ture Eq.(1)could be simplified by the assumptionthat the radiation transfer processes are balanced byturbulent heat flux. The zonal mean form of Eq.(1)under spherical coordinates could be written as

where x = sin φ, φ is latitude, and a is the radius of theearth. To solve the above equation, integral along theentire radiation absorption spectrum is needed, whichis complex and inconvenient for theoretical analysisthat copes with the radiation transfer processes an-alytically or semianalytically. Kibel(1943)had established strict temperature distribution theory in theearly 1940s, and Blinova(1947)further developed and refined the theory. They considered the entire radiation absorption spectrum as a whole to avoid complicated integral calculation and eventually derived afourth-order partial differential temperature equationby assuming radiation absorption coeffcients as con-stants. Detailed introduction could be found in Ye and Zhu(1958). Kuo(1973)further divided the en-tire longwave radiation absorption spectrum into twoparts(strong and weak absorptive zones, presented bysubscripts s and w)by using the criterion kjd/dz.The mean radiation absorption coeffcients are defined

Meanwhile, we define Es = rE, Ew =(1-r)E, and E = σT4.

according to the simplified scheme, it could be derivedfrom Eq.(7)that

when

and is the mean solar radiation arriving at the upper bound of atmosphere. Compared with κz, κr canbe called the equivalent radiation exchange coeffcient, which represents the radiative effect in the strong absorptive zone. N2 is called Newtonian radiative cooling coeffcient, which represents the radiative effect inthe weak absorptive zone. Brunt(1934)pointed outthat the role of the strong absorptive zone is much likethermal conduction process and the role of weak absorptive zone could take the form of Newtonian cooling(quoted from Chen and Chao, 1988). Kuo(1973)unified these two effects into one simple equation. However, he left the specific form of the integral constant Cunsolved. Chao and Chen(1979)derived the integralconstant C by integral over the globe, which could bewritten as

The upper and lower boundary conditions are taken as

where 0 = 1 - Γp and 1 = 1 - Γp -(1 - e-ξ0)arenet solar radiation proportion arriving at the upperbound of atmosphere and at the ground, respectively.0 = 0S(x)dx and 1=1S(x)dx are their global integral values. Chao and Chen(1979)took 0 = 1, which means that the planetary albedoat the upper boundary is neglected.

Use Legendre series to solve Eq.(10). Set

extend relevant variables into Legendre series and solve the coeffcients E(n)(ξ), and then the an-alytical solution could be obtained. This two-dimensional EBM is then called the Kuo-Chao-Chenmodel(KCCM). Chen(1982a, b)further analyzed thestability of the solution and the sensibility of the pa-rameters. The above KCCM has a wide spread appli-cation in the climate change field, such as ice cap sen-sitivity against solar constant(Chao and Chen, 1979), planetary atmospheric temperature distribution(LÄu and Chao, 1981), desertification(Chao and Li, 2010a), and warming effects of carbon dioxide(Chao and Li, 2010b).

3. Relationship between ice cap latitude and solar constant

Many scientists studied the reasons of the coldperiod in the 1960s. The studies of Budyko(1969), Sellers(1969), and North(1975a)are most represen-tative. Ice cap latitude is very sensible against solarconstant in one-dimensional EBM(Fig. 1). Only a2% decrease in solar constant would push the ice capto move southward till around 50ffN(glacial climate), even around the equator(a snowball earth). On theother h and , a certain solar constant(for example, thecurrent value)might correspond to multiple ice caplatitudes, which means multiple climate states. Thesetwo phenomena could be interpreted into two ques-tions. One is the sensitivity of the ice cap againstsolar constant and the other is the stability and phys-ical meanings of different equilibrium climate states.

Fig. 1. The ice cap latitude xs(sine of latitude)as a function of the solar constant in unit of the present solar constant(Q0). [From North, 1975a]

Lindzen and Farrell(1977)pointed out that thedependence of the ice cap on solar constant could bereduced just by adding some more realistic heat trans-port processes on the one-dimensional EBM. Lian and Cess(1977)found that the dependence could be re-duced by correcting the relationship between temperature and albedo. Making a more reasonable revi-sion on the parameterization between longwave radi-ation and temperature through satellite data, Oerle-mans and Van Den Dool(1978)suggested that glacialclimate occurs only when solar constant reduces itsvalue by about 9%. Generally speaking, due to lack ofvertical energy rearrangement, one-dimensional EMBsare more sensible to solar constant variation. Chao and Chen(1979)re-examined the issue with KCCM and noted that about a 15% decrease in solar con-stant could push the current climate into a glacial one, which means that two-dimensional EBM could greatlyreduce the sensibility.

Budyko(1972)firstly discovered the multipleequilibria characteristic in the simple one-dimensionalEBMs. He pointed out that a stable equilibrium pointsatisfies dQ/dxs > 0 and an unstable one correspondsto dQ/dxs < 0. subse-quently analyzed the multiple equilibria phenomenonin ice cap solution. North(1975a, b)found two equi-librium ice cap latitudes, which were south near thepolar region and close to each other. He thought thatthe unstable equilibrium state closer to the polar pointa little unreasonable and removed it by adjusting thezonal distribution of solar radiation. Held and Suarez(1974)analyzed the albedo feedback mechanism in onesimple EBM and his conclusion supported the resultsof Budyko(1972). They contributed the occurrenceof the two equilibria near the polar region to the in-troduction of the temperature-related diffusive term.When diffusion was small enough, there was no signif-icant physical discrepancy between the two equilibria.Ghil(1976)found there were three equilibria corre-sponding to inter-glacial, glacial, and snowball earthclimate, respectively. Drazin and Griffel(1977)fur-ther analyzed the characteristic of the multiple equi-librium solutions and indicated that one equilibriumsolution would be unstable if its corresponding eigen-values were smaller than zero. Lin(1978)removed theunstable equilibrium closer to the polar region by pre-scribing the nonlinear turbulent exchange coeffcients.Cahalan and North(1979)suggested that the unstableequilibrium solution was caused by the step functionof the albedo distribution and any form of smoothing of the albedo near the ice cap latitude could removethe unstable equilibrium solution, which is also men-tioned by Coakley(1979).

The unstable equilibrium means that ice capmoves southward when solar constant increases itsvalue. It seems unreasonable. Therefore, in the re-view paper about EBMs, North et al.(1981)disqual-ified the realistic existence possibility of the unstableequilibrium solution. They contributed it to the arti-ficial step albedo function introduced for the mathe-matical facilitation. However, the results of Held and Suarez(1974) and Ghil(1976)seemed to have no suchmeanings. In fact, actual albedo distribution could ex-perience large discontinuity, such as the great albedodiscrepancy between Antarctic ice sheet and the sur-rounding sea water, in which situation the step func-tion form of albedo seems more reasonable. There-fore, the unstable equilibrium might characterize oneclimate evolution under certain circumstance. It is in-appropriate to consider the unstable equilibrium as asolution with no physical meaning and then directlyremove it.

Chao and Chen(1979)contributed the ice cap lat-itude sensitivity to solar constant to ignoring the ver-tical heat transport in one-dimensional EBMs. Theybuilt the KCCM which could cope with the vertical en-ergy transport to study the question further. Accord-ing to their analysis, the temperature is insensitive tothe solar constant. A glacial climate needs 15%"20%decrease of the solar constant. Meanwhile, there wouldbe no significant difference in solutions when other pa-rameters vary 20% of their values(Chen, 1982b). Likethe one-dimensional model, albedo distribution in thetwo-dimensional model is taken as

where xs = sin φs, φs is the ice cap latitude and is de-termined by T = -10‰℃; α1 and α2 are the albedo dis-tributions of the ice and non-ice surface, respectively, i.e., they could take continuous(North and Coakley, 1979), semi-continuous(Coakley, 1979), or discontin-uous(Budyko, 1969)forms such as α1(x)= α2(x)=α02P2(x); α1(x)= 0:62, α2(x)= α02P2(x); and α1(x)= 0:62, α2(x)= 0:132, respectively. The distribution of the ice cap latitude in each case is shown in Fig. 2.

Fig. 2. The ice cap latitude as a function of the solar constant in unit of the present solar constant. The right and left y-axis coordinates are latitude and its sine, respectively. The solid, dashed, and dash-dotted lines meanthat the albedo distribution is taken as α1(x)= α2(x)= α0 + α2P2(x); α1(x)= 0:62, α2(x)= α0 + α2P2(x); and α1(x)= 0:62, α2(x)= 0:132.

In the continuous albedo distribution case(solidline), the relationship between the ice cap latitude and the solar constant is monotonous. Polar ice sheetwould shrink northward until emergence of an ice-free earth, with increasing solar constant. An ice-freeearth needs a 10% increase of the solar constant and a glacial climate(50°N)needs a 20% decrease of thesolar constant. In the semi-continuous case(dashedline), the monotonous relationship is destroyed. Acritical ice cap latitude(near 30°N)divides the relationship into two parts. The upper part remains thesame with the continuous case and the lower part sug-gests that bigger solar constant corresponds to largerpolar ice range. The solar constant has to increase10% to heat an ice-free earth and it has to decrease16% to cool the earth into glaciation. In this case, onesolar constant might correspond to two ice cap lati-tudes.

In the discontinuous case(dash-dotted line), thesituation is similar to the semi-continuous case butwith a more complex variation. There are two criticalice cap latitudes(near 30° and 60°N), which divide therelationship into three parts. The relationship for the30°-60°N part means that bigger solar constant wouldmelt more polar ice while for the other two parts, big-ger solar constant accompanies with larger polar icecoverage. A given solar constant(such as the currentvalue)could correspond to three equilibrium ice caplatitudes(near 72°, 50°, and 16°N), which seem tohave occurred in the earth's evolutionary history. TheArctic ice sheet is around 72°N in the current climate;it marched to the midlatitude in the Carboniferous-Permian glaciation and Quaternary glaciation; and itmight have arrived in low latitude, even the equator(a snowball earth), in the new Proterozoic glaciations(Sumner et al., 1987; Schmidt et al., 1991; Schmidt and Williams, 1995; Sohl et al., 1999).

To conclude, the relationship between ice cap lat-itude and solar constant is determined by the distri-bution of surface albedo. The relationship would bemonotonous if given a continuous albedo distribution.That is, bigger(smaller)solar constant correspondsto smaller(larger)polar ice range. This monotonouscharacteristic would be destroyed by a discontinuousalbedo distribution. One given solar constant might belinked with two even three ice cap latitudes. There-fore, the circumstance that bigger(smaller)solar con-stant could correspond to larger(smaller)polar icerange could occur. Although such a circumstancemight be not easily understood and unstable, it is stillretained for discussion. More detailed global surfacealbedo products derived from satellite data would con-tribute to a better underst and ing of this issue.

The theoretical calculation reproduces the surfacetemperature distribution well except for the highertemperature in the tropical region(Fig. 3a). It might be caused by not reasonably refiecting the tropicalheat transport by Hadley cell. If more realistic heattransport is introduced(Lindzen and Farrell, 1977), the temperature simulation in the tropics could beimproved. Significant discrepancy between the theo-retical and observed vertical temperature distributionscould be resulted if the definition of optical depth usingEq.(9)is directly applied. Revised by the observeddata, the main features of the temperature distribu-tion in the troposphere could be well simulated al-though the simulation is not perfect in the lower trop-ical troposphere and in the upper polar troposphere.Since the EBMs do not have the capacity to directlycope with the influence of the atmospheric dynamicalprocesses on temperature distribution, they have donetheir best.

Fig. 3. Distributions of(a)surface temperature and (b)vertical temperature. Solid lines represent the theoreticalresults and the asterisk in(a) and the dotted lines in(b)are the temperature distribution derived from observation. The x-axis in(a)is temperature and (b)the sine of latitude. The y-axis in(a)is the sine of latitude and (b)vertical levels.
4. Desertification

The United Nations Convention to Combat De-sertiflcation(UNCCD)deflnes the term desertiflcationas "l and degradation in arid, semi-arid, and sub-humidareas resulting from various factors including climaticvariations and human activities." Seventy percent ofthe world's dry-l and s(excluding hyper-arid deserts), or some 3600 million hectares, are degraded(UNCCD, 2000). North and Northwest China are vulnerableto climate change due to severe water shortage(Ye, 1986). By the end of 2009, total 2623700-km2 l and , accounting for 27.33% of the total l and area of China, mainly the northwestern and northern areas, experi-enced desertiflcation(State Forestry Administration, 2011). The drought in northern China has exacerbatedin the last several decades and the arid and semiaridzones have exp and ed southward and eastward. Per-sistent drought has led to a series of local environ-mental problems(such as desertiflcation), which havebecome a serious obstacle to economic development(Ma and Fu, 2005, 2006; Fu and Ma, 2008). How doesthe drought in northern China form and evolve? An-swers to its eco-social impacts as well as coping strate-gies have become national dem and s. These problemshave been investigated by Chinese scientists and muchprogress has been made, such as analysis of the three driving factors of the drought formation and develop-ment in northern China, i.e., 1)the natural law of thewet and dry variation of the monsoon system, 2)theabnormal response of the monsoon system to globalchange(mainly global warming), and 3)the impact ofthe human activity on local environment(Fu and An, 2002; Fu and Wen, 2002).

Charney(1975)analyzed the impact of vegeta-tion at the desert margin on climate. He pointed outthat a reduction of vegetation, with a consequent in-crease in albedo, at the southern margin of the Saharawould cause sinking motion, and additional drying, and would therefore perpetuate the arid conditions.Dickinson(1984)developed parameterization for thecalculation of evapotranspiration, which distinguishesbetween evaporation from the ground and evapotran-spiration from plant foliage. His pioneer work pro-moted the development and maturity of l and surfacemodels. Now, desertiflcation is investigated mostlywith the aid of l and surface models or climate mod-els, and theoretical analysis is relatively scarce. Chao and Li(2010a)hypothesized that the separating linebetween vegetation and desert could be determinedby a given temperature value. Then, they discussedthe evolution of desertiflcation by taking albedo dis-tribution as piecewise function(albedo values in ice, desert, and vegetation covered areas are different and discontinuous). As an application of KCCM in climatechange, their work will be introduced and updated inthis paper.

Although the surface albedo in different regionsvaries largely(Fig. 4a), its zonal averaged value showscertain rules. For example, in the Northern Hemisphere, the albedo value decreases, increases, and thendecreases from north to south, corresponding to polarice, high-latitude vegetation, midlatitude desert, and low-latitude vegetation(Fig. 4b). This characteristic is most signiflcant in the West Asian and NorthAmerican continents. In the Southern Hemisphere, the zonal mean albedo value changes similarly fromsouth to north. The varying albedo value correspondsto Antarctic ice, sea water(the albedo value could betaken as 0.07), desert, and tropical vegetation, respectively. According to the observation data, the surface albedo distribution used in the two-dimensional modelcould be concluded as

Fig. 4.(a)Spatial distribution of the surface albedo and (b)its zonal mean values. The surface albedo data are downloaded from the ESA GlobAlbedo Project website http://www.GlobAlbedo.org.

This means that the albedo distribution is divided intofour parts according to different l and surface types;i = 1, 2, 3, 4 means polar ice, high-latitude vegeta-tion, midlatitude desert, and low-latitude vegetation, respectively; and xs, xd, xv are the sine of the bound-ary latitude between polar ice, desert, and vegetation, respectively. Set α1 = 0.75, α2 = α4 = 0.1, α3 =0.25, xs ≈ 0.95, xd ≈ 0.766, xv ≈ 0.5, and then themean surface albedo value is around 0.15, close to theactual situation. According to the former discussion, the ice cap latitude is determined by temperature Ts= -10‰℃. Similarly, compared with the observed zonalmean temperature distribution, xd(north edge of thedesert zone) and xv(south edge of the desert zone)aredetermined by Td = 5‰ and Tv = 19‰℃, separately.Then, the evolution and development of desertificationcould be featured by the variation of the desert sur-face albedo value α3 in the desert zone(see Fig. 5). In Fig. 5a, when desert surface albedo α3 increases, bothxs(solid line) and xd(dashed line)retreat northward, implying that polar ice sheet would shrink northward and the north edge of the desert zone would exp and northward. However, the south edge of the desert zonexv(dash-dotted line)would move northward and thensouthward slightly. The size variation of polar icesheet, vegetation, and desert could change the plan-etary albedo of the earth(Fig. 5b). Planetary albedoΓp increases and then decreases with increasing desertalbedo α3. When α3 is smaller, ¡p would decreasewith increasing α3. This is because the increment ofplanetary albedo caused by α3 itself could not bal-ance the reduction caused by polar ice sheet shrink.The decreasing trend could last until α3 is about 0.26.When α3 is larger than 0.26, the reduction of albedocaused by shrinking polar ice sheet is not significantdue to its smaller range. Therefore, bigger α3 as wellas its larger area would result in increasing planetaryalbedo. Planetary albedo change would affect the vari-ation of temperature eventually. As shown in Fig. 5c, global integral mean surface temperature exhibits op-posite changes against the planetary albedo.

Fig. 5. Variations of(a)xs, xd, xv(solid, dashed, and dash-dotted lines, respectively), (b)global integrated planetaryalbedo Γp, and (c)global integrated surface temperature T, with desert albedo α3. The y-axis coordinates in(a)arelatitude and its sine.

Figure 6 shows the surface temperature derivation from the current climate(α3 = 0.25). When the desert surface albedo α3 is smaller(larger)thanpresent, the temperature in high latitude is lower(higher)due to larger(smaller)range of polar ice. Thetemperature in low latitude declines with a moderaterange when α3 is increasing. Distribution of the temperature anomalies seems to indicate that the modulation of temperature by the ice albedo feedback is more significant in high latitude.

Fig. 6. Surface temperature discrepancy with the current climate states(α3 = 0.55). The y-axis coordinates are latitude and its sine.

The northward movement of the north edge ofthe desert zone(xd)implies that desert would invadeinto the high-latitude vegetation zone, which woulddeteriorate the local living environment, and harmthe life and production of the local residents. Mean-while the northward movement of the ice cap lati-tude(xs)means that high-latitude vegetation wouldexp and closer to the polar region. Therefore, desertifi-cation would mainly shift the high-latitude vegetationzone northward rather than sharply reduce its areasize. The south edge of the desert zone(xv)slightlymoves northward, and then southward with increasingdesert surface albedo(α3). When the desertificationis less severe, the northward movement of xv meanslow-latitude vegetation would exp and , which is bene-ficial for the local environment; when the desertifica-tion is well developed, the southward movement of xvjeopardizes the local environment in turn. It could beseen that increase in the desert albedo would make thedesert zone exp and both southward and northward.Therefore, the ecological environment in the border area between desert and vegetation zone is inherentlyvulnerable. On the other h and , decreasing the desertalbedo would reduce the area of the desert zone, henceimproving the local ecological environment. The sur-face albedo is associated with the physical and ecolog-ical status of the underlying surface. It is necessaryto study the climate sensitivity to local albedo vari-ations. It would tell us to what extent the orderlyhuman activities(such as irrigation and afforestation)could affect the regional and even global climate and how we could use these activities to benefit the sus-tainable development.

5. Warming effect of carbon dioxide

Greenhouse gases keep the mean surface temper-ature of our earth in a suitable range by absorbingthe upward longwave radiation. Excessive greenhousegases would warm the earth. The carbon dioxide emit-ted by human production and industrial activities isthought to be an important element driving the globalwarming. The concentration of carbon dioxide in theatmosphere has been steadily increasing since the In-dustrial Revolution. According to a report by theWorld Meteorology Organization(WMO), the dailyaverage concentration of carbon dioxide has broken400 ppm at several stations, climbing up the historicalmaximum record in the past 3 million years(WMO, 2013). At the current increasing rate, the annualmean concentration of carbon dioxide would exceed400 ppm in 2014 or 2015. It is widely believed thatwith the increasing concentration of carbon dioxide, the global average surface temperature will continue torise. However, it seems inconclusive about the rate ofthe surface temperature increase, especially during thelast decade when steady increasing concentration ofcarbon dioxide did not correspond to a steady warm-ing globe; instead, the global mean temperature showsa so-called "hiatus"(Kerr, 2009; Knight et al., 2009).Therefore, the relationship between concentration ofcarbon dioxide and global temperature variations isstill obscure and various research methods and anglesare still necessary.

Liu et al.(2002)discussed the saturation of green-house effect due to atmospheric carbon dioxide. They pointed out that absorption has saturated in the centerof the carbon dioxide 15-μm b and indeed. However, absorption does not saturate for the wings of the 15-¹m b and and the other b and s in the near future. Thisindicates that potential warming effect of carbon diox-ide still cannot be ignored. Climate models are usefultools to investigate the relationship between concen-tration of carbon dioxide and temperature variation.Meanwhile, some simple models with solid physicalprocesses are useful supplements that could be used todeepen our underst and ing about the issue. For example, Liu(2006)analyzed the physical processes con-trolling the glacial and interglacial cycles during thepast several hundred thous and years by using the sim-ple, one-dimensional Budyko model(Budyko, 1969).

Chao and Li(2010b)discussed the response ofatmospheric temperature to concentration of carbondioxide with the KCCM. Here, more reasonable physical consideration is added to update their work. Theatmospheric absorption coeffcient kj for longwave radiation could be written as

where η = 5/3 is the scatter factor, α is the mass absorption coeffcient, ρ is the concentration of absorbingmedium, subscripts h2O and CO2 represent vapor and carbon dioxide. The absorption coeffcient in long-wave b and (4-100 μm)could be obtained from HI-TRAN ON THE WEB website(http://hitran.iao.ru/)which calculates the absorption coeffcient accordingto HITRAN 2004 high-resolution spectral data. Ab-sorption coeffcients in different b and s can vary sev-eral orders of magnitude, which is very complex fortheoretical analysis. For convenience, the absorptionsimplification scheme in longwave b and s used by Kuo(1973)is still adopted, in which the entire long-waveb and is divided into strong and weak absorption zones.

The variations of absorption coeffcients in strong and weak absorption zones with concentration of car-bon dioxide show different features(Fig. 7). The ab-sorption coeffcient in the strong absorption zone willdecline gradually with increasing concentration of car-bon dioxide until 400 ppmv, higher than which it de-creases slightly and then even increases slightly whenconcentration is higher than 600 ppmv(dash-dotted line). The absorption coeffcient in the weak absorp-tion zone varies dramatically when concentration ofcarbon dioxide is lower than 200 ppmv, higher thanwhich it decreases overall but with many small fluctu-ations(solid line). Although both strong and weak ab-sorption coeffcients decrease overall, the energy pro-portion absorbed by the strong absorption zone in-creases steadily with increasing concentration of car-bon dioxide(dashed line). Chao and Li(2010b)hy-pothesized that the strong and weak absorption zoneeach absorbs half of the entire energy. Therefore, the proportion equals 0.5. This assumption is a littlecoarse albeit simple and intuitive. As an improvementto the previous work, more accurate HITRAN 2004high-resolution spectral data are used here to calculatethe strong and weak absorption coeffcients accordingto Eq.(8). The coeffcients vary with air tempera-ture and pressure but with no obvious discrepancy inthe calculation of air temperature distribution, whichcould be close to the observed data through adjustingthe turbulent exchange coeffcients.

Fig. 7. Absorption coeffcients in strong and weak ab-sorptive zones and the proportion of the strong absorptivezone in the total absorption as a function of the concen-tration of carbon dioxide. The reference temperature and pressure are set to 290 K and the st and ard atmosphericpressure.

Referring to the former discussion, the surfacealbedo is divided into two parts, namely, α1 = 0.62 and α2 = 0.132. The variation of ice cap latitude with con-centration of carbon dioxide is then calculated(Fig. 8). In the equilibrium representing the current cli-mate(Fig. 8a), ice cap retreats northward slowly withmany small °uctuations when concentration of carbondioxide is increasing. Ice cap latitude varies little insome concentration ranges such as 400-500 ppmv, and even moves southward slightly in some ranges suchas 900-950 ppmv. Polar ice would melt completely(ice-free earth)only when the concentration increasesto about 1000 ppmv. Global mean surface temper-ature increases gradually but with a slowing rate inthe current climate(Fig. 9a). When concentration ishigher than 600 ppmv, there is almost no temperaturerise, that is, the "saturation" state. In the equilib-rium representing the glacial climate, ice cap latitudewould march southward gradually with increasing con-centration of carbon dioxide(Fig. 8b). Temperaturedecreases in the glacial climate(Fig. 9b). The meansurface temperature could be even lower than -10‰when carbon dioxide exceeds 600 ppmv. Carbon dioxide does not act as a driving factor for global warming.Oppositely, it accompanies with global cooling. In theequilibrium representing "snowball earth", the ice caplatitude is situated in the tropics. Polar ice wouldshrink its range when higher concentration of carbondioxide is added into the atmosphere(Fig. 8c). Eventhough temperature is as low as about -55‰ in thesnowball earth, it would still decline with increasingcontent of carbon dioxide(Fig. 9c). The discrepa-ncy in the three equilibrium climatic states might be caused by the relative importance of albedo feedback and greenhouse effect. The climate would tend tobe cooling if albedo feedback overweighs the green-house effect, and be warming if greenhouse effect dom-inates. It should be noted that the missing values ofice cap latitude in the current and glacial climates sug-gest that no stable ice cap latitude correspond to theCO2 concentration ranges of 0-120 ppmv and 180-270ppmv, in which the two equilibrium states converge to-gether and there is no real roots(ice cap latitudes)forthe system.

Fig. 8. Variations of the three equilibrium ice cap latitudes with the change of carbon dioxide concentration.

Fig. 9. Variations of the global mean surface temperature in the three equilibrium states with the change inconcentration of carbon dioxide.
6. Concluding remarks

The development of two-dimensional EBM and its application to some climatic issues have been sys-tematically reviewed in this paper. Though simple, two-dimensional EBM has a solid physical basis. Itcould deepen the underst and ing of climate change and supplement the results from numerical climate models.The climatic issues discussed in this paper are of greatconcern currently, but not yet completely understood.The answers to these issues are likely to facilitate cer-tain progress and breakthroughs in the future, whichto some extent represents the direction of theoreticalresearch in the climate change field. Besides, paleo-climatic data and climate models are needed to further verify the theoretical results. We hereby proposethat future topics in these regards include but are notlimited to the following.

(1)The relationship between albedo distribution and ice cap latitude equilibria as well as the stabilityof the equilibria. The numbers of equilibrium are de-termined by the continuity of the surface albedo distri-bution. Smoother albedo helps reduce the equilibriumnumbers, and vice versa. The actual albedo distribu-tion is continuous in some regions but discontinuous inother regions. It is still unclear which situation is morelikely to occur. In addition, questions such as how theequilibrium stability is, whether unstable equilibriumhas any physical meaning, and how the unstable equi-librium evolves, all need further study. The answers tothese questions would help to underst and which statethe climate system is currently evolving in and its de-velopment direction.

(2)Desertification and its impact on the globaltemperature distribution and atmospheric motion.Current researches mainly focus on the impact of sur-face albedo distribution on desertification and air mo-tion in the planetary boundary layer. It is neces-sary to analyze the impact of desertification on at-mospheric circulation by coupling the two-dimensionalEBM with the atmospheric motion equations in thefree atmosphere. Meanwhile, it is still simple and cer-tainly artificial to fix the albedo distribution and thesouth/north edge of the desert zone; it is still unclearhow sensitive the desertification is to these parame-ters; it is still unknown how to appropriately definethe l and surface characteristics that can both re°ectthe physical nature and facilitate the theoretical cal-culation. All these questions need to be addressed.

(3)The impact of concentration of carbon dioxideon global mean surface temperature. According to thetheoretical calculation, the relationship between con-centration of carbon dioxide and global mean surfacetemperature is not monotonic. Increasing carbon diox-ide might correspond to upward or downward temper-ature trend. Temperature increment caused by CO2concentration increase tends to become zero(or calledwarming saturation)in the current climate state. It isstill uncertain about whether such a conclusion is reasonable or not, and whether such a phenomenon couldoccur or not in more realistic l and cover distribution.All these equations deserve further study.

Besides the three questions discussed above, two-dimensional EBM could also be used on other im-portant climatic issues, such as the climatic e®ect ofaerosols and so on.

Acknowledgment:

The l and surface albedodata are downloaded from the ESA GlobAlbedoProject website http://www.GlobAlbedo.org. Theauthors gratefully thank three reviewers for their con-structive comments and suggestions.

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