aa r X i v : . [ phy s i c s . pop - ph ] A ug Climate Modeling and Bifurcation
Mayer Humi ∗ Department of Mathematical SciencesWorcester Polytechnic Institute100 Institute RoadWorcester, MA 01609September 4, 2019
Abstract
Many papers and monographs were written about the modeling the Earth climateand its variability. However there is still an obvious need for a module that presentsthe fundamentals of climate modeling to students at the undergraduate level. Thepresent educational paper attempts to fill in this gap. To this end we collect in thispaper the relevant climate data and present a simple zero and one dimensional modelsfor the mean temperature of the Earth. These models can exhibit bifurcations fromthe present Earth climate to an ice age or a ”Venus type of climate”. The modelsare accompanied by Matlab programs which enable the user to change the modelsparameters and explore the impact that these changes might have on their predictionson Earth climate. ∗ e-mail: [email protected]. Introduction
The Earth climate and its variations has always been of great interest to Humans as it hasmajor impact on Human activities. In this paper we present prototype models for the meantemperature of the Earth which will enable us to explore possible bifurcations in the Earthclimate which might due in part to anthropogenic emissions and natural forcing processes.These model are based in part on the approach presented in [1, 4]. More elaborate andsophisticated models are available in the literature [3, 5, 4].Much was written lately about the gradual change in the mean temp of the earth due toHuman intervention which is expected to range by 1-2 degrees by the end of this century.However the real danger of these changes is that they may lead to a rapid major change inthe Earth climate (viz. climate bifurcation [6]) in the same way that a rubber band snapssuddenly when it is over stretched or a sudden snow avalanche occurs on the slopes of amountain.The Earth climate system is composed of the following components: land, biosphere,atmosphere, ocean and the cryosphere (ice and frozen water), These components display abroad range of variability on temporal and spatial scales such as the Dansgaard-Oeschgercycles which occur quasi-periodically on a millennial time scale or the El Nio-SouthernOscillation in the equatorial Pacific. Thus a refined model of such a complex system requiresvast amounts of data and elaborate sophistication. In the following we stick to the basics.The plan of the is as follows: In Sec 2 we introduce the basic concepts needed for climatemodeling. Sec 3 presents the relevant climate data. The zero dimensional model and itpredictions are discussed in Sec 4. The one dimensional climate model is presented in Sec 5.Sec 6 discusses some attempts to mitigate the effects of the greenhouse effect. We end up inSec 7 with some conclusions. 2
Some Basics
In this section we introduce some basic concepts and data needed for the modeling of theEarth climate.
When radiation impinges on a (perfect) mirror all of this radiation is reflected back and thetemperature of the mirror remains unchanged. On the other hand is such radiation impingeson a (perfect) black body all of the radiation is absorbed by that body. In general howeversome of this radiation is absorbed by the body and some is reflected back. We say that in thiscase we are dealing with a ”grey body”. In general the albedo of a body is the percentage ofradiation that is reflected back by the body into space. Thus for a perfect mirror the albedois 1 and for a black body the albedo is zero. For a grey body the albedo is a number betweenzero and one.The Earth is a grey body. However it albedo changes with time due to snow and icecover, vegetation and Human activities (e.g. the paving of asphalt roads)
Radiation from the Sun reaches Earth (mostly) in the visible part of the spectrum (viz. wavelengths in the range of 380 × − - 750 × − meters). Radiation with shorter wavelengthsis absorbed (mostly) by the Van-Allen belts. The Earth absorbs part of this radiationand emit it back in the infra red part of the spectrum (with wave lengths n the range of7 . · − − · − meters. If the Earth had no atmosphere this radiation will be reflected intospace. However in the Earth atmosphere some trace gases such as Carbon dioxide ( CO ),Methane ( CH ) and water vapor can absorb this radiation and reflect it back to Earth [4].This leads to a warming of the Earth surface. (Fig. 1 presents the absorption spectra ofseveral trace gases- source of this figure is unknown).NOAA Earth System Research Laboratory (Global Monitoring Division) has been mon-3toring the concentration of the trace gases in the atmosphere from an observation stationon Mona Loa mountain in Hawaii and globally[7, 8, 9] by averaging these concentrationsas measured by observation stations throughout the globe. This data is shown in Figs.2 , , CO and CH over many years which can be attributedHuman activities.The effect of clouds in this scenario is dual. First they block the Sun radiation and reflectit into space. At the same time they also reflect the Earth infra red radiation back to Earth.There is still on going debate in the scientific community as to which of these two process ismore prominent for the determination of the Earth energy balance. StefanBoltzmann states that the power radiated from a black body as a function of itstemperature is given by P = σAT where A is the body surface area, T is it temperature and σ = 5 . × − W atts/ ( m K )is the StefanBoltzmann constant.For a grey body P has to multiplied by the “grayness” coefficient of the body. We enumerate here the data that is needed to model the mean temperature of the Earthand then assemble these into model equations.1.
The Sun Forcing
The rate at which energy from the Sun reaches the top of Earths atmosphere is calledtotal solar irradiance (or TSI)[10, 16, 17]. TSI fluctuates slightly from day to day andweek to week. In addition to these rapid, short-term fluctuations, there is an 11-year4ycle in TSI measurements related to sunspots (a part of the Suns surface that istemporarily cooler and darker than its neighboring regions).The evidence shows that although fluctuations in the amount of solar energy reachingour atmosphere do influence our climate, the global warming trend of the past sixdecades cannot be attributed to changes in the Sun output. Fig. 5 which was compiledby Professor G. Kopp[17] (and reproduced here with his permission) is a graph of TSIfor the last 400 years. It shows clearly the ”small ice age” during the 17th century.This TSI average is referred to as the ’solar constant’, in power (watts) per squaremeter.From a practical point of view we can consider the Sun rays arriving in parallel toEarth. This energy flux was measured to be F s ≈ W att/m [10, 16, 17]. Howeverthe cross section of the Earth to this flux is πR while it surface area is 4 πR . Thereforewe divide this flux be 4 when we compute the average temperature of the Earth.The average albedo of the Earth from the upper atmosphere, its planetary albedo, is3035% because of cloud cover, but widely varies locally across the surface because ofdifferent geological and environmental features.2. the Albedo The albedo of the earth is constantly changing. We shall model it as having two values: α M = 0 . , α m = 0 . α M is the albedo for Earth covered by snow,ice (frozen Earth), and α m = 0 .
25 isthe albedo for the part of the Earth that is not covered by snow, ice etc (i.e not frozen).The Earth will be considered as frozen if it mean temperature is below T = 240 Kelvinand unfrozen if it mean temperature is over T = 275 Kelvin. In between T and T we use linear interpolation to compute the albedo α ( T ) = α m − α M T − T ( T − T ) + α M (3.1)5Observe that the effect of changes in the vegetation of the Earth is not taken (directly)into consideration in this model)More elaborate models for the earth albedo are available[2, 4] and satellites are beingused currently to give accurate real time data on its value. It is estimated that theaverage albedo of the Earth (viz. planetary albedo) is 30% to 35% because of cloudcover and the effect of trace gases, but it varies widely across the surface because ofdifferent geological and environmental features.3. Clouds and the Greenhouse factor
Let T be the ambient temperature for green house effect, and κ ( T ) the cloud coverdue to the green house effect. (Without the greenhouse effect the mean temperatureof the Earth will be ≈
240 Kelvin and effectively there will no cloud cover)The incoming radiation from the Sun that is absorbed by Earth is R in = F s A − α ( T ))where A is the surface area of the Earth. The outgoing infra red radiation from Earthis R out = σAg ( T ) T where g ( t ) is a lump sum parameter that represents the impact of the greenhouse gasesand clouds. This parameter was modeled (empirically) by Sellers[1] as g ( T ) = 1 − κ tanh[( T /T ) ] , κ = 0 . T ≈
275 Kelvin. This expression implies that as T increases g ( T ) decreases andthe greenhouse effect becomes stronger ( R out decreases)6 Zero-dimensional Climate Model
If we denote by C the heat capacity of the Earth then by the law of energy conservation itfollows that C dTdt = R in − R out (4.2)at equilibrium (viz. Steady state) dTdt = 0 and we must have therefore that R in − R out = 0. If the Earth atmosphere was totally transparent with no greenhouse gases (or if the Earthhad no atmosphere) then g ( T ) = 1. The percentage of Sun energy that is reflected bythe Earth into space (viz. albedo) is estimated to be α ( T E ) = 0 .
3. The equilibrium meantemperature of the Earth T E will have to satisfy then F s A − α ( T E )) − σAg ( T E ) T E = 0 (4.3)The solution of this equation yields T E = 255 Kelvin which is well below the freezing pointof water. Thus the Earth will be covered by snow and ice. It follows then that the differencebetween this value of T E and the (estimated) current mean temperature of T = 284 . g ( T ) that leads to this equilibriumtemperature we substitute T = 284 . g (with α ( T E ) = 0 . g is 0 . mean .m that implements it. At first we used this program, with the modelparameters that are quoted above[1, 4]. Using this program with an initial (mean) tempera-ture of 285 Kelvin we obtained Fig. 6 (20 years simulation). This initial mean temperatureis based on data which was collected by NASA which shows that the Earth mean surfacetemperature in 2017 was 14 . C . Fig. 6 shows a steady increase in the average temperatureof the Earth for the first few years which then stabilizes around 305 Kelvin. (This programis available for download from the author web page [14]).7any of the original parameters[1, 4] used in this program are estimates (at best) and arechanging with time. Therefore one can use this program to probe also for the dependence(or sensitivity) of the results to the values of the various parameters of this model. Asan example we varied the value of T in (3.1) from 275 (original value) to 296 .
11 with nosubstantial changes in the results. However if we increase T to over 296 .
12 we obtain Fig 7where the Earth mean temperature plunges to an ice age. Thus the model exhibits a climatebifurcation if T is in between [296 . , . In this model the Earth temperature is considered to be a function of time and latitude θ .Using this one dimensional model it is possible to study the oscillation of the Earth climatebetween ice ages in the past.To present this model we need to develop first, two mathematical formulas. We want to find a formula for the area of a strip on a circle of radius R that is enclosedbetween the equator and south of latitude line θ (see Fig 8). This area equals the sum ofthe area of the rectangle 2 R cos θ sin θ and the area enclosed by equator and the two arcs ofthe circle. Hence the total area of the strip is A ( θ ) = 2 R cos θ sin θ + 2 Z RRcosθ √ R − x dx = R (cos θ sin θ + θ ) = R (cid:18) sin(2 θ )2 + θ (cid:19) . (5.4)It follows that the area of a strip on the circle between θ and θ + dθ is dA = dA ( θ ) dθ dθ = R (cos(2 θ ) + 1) dθ. (5.5)8 .2 Area of a ”polar cap” on a Sphere The formula for the area north of a line of latitude θ can be computed directly as in theprevious subsection. It is given by, S ( θ ) = 2 πR (1 − sin( θ )) (5.6)The area enclosed by a strip on the sphere between θ and θ + dθ is∆ S = 2 πR (1 − sin( θ )) − πR (1 − sin( θ + ∆ θ ) (5.7)= 2 πR (sin( θ + ∆ θ ) − sin( θ )) = 2 πR sin( θ + ∆ θ ) − sin( θ )∆ θ ∆ θ which in the limit ∆ θ → dS = 2 πR cos( θ ) dθ (5.8) To derive a one dimensional steady state model for the Earth temperature as a function of θ we consider a strip on the Earth between θ and θ + dθ . The cross section of this strip tothe sun radiation is given by (5.5) . The incoming radiation from the Sun that is absorbedby this strip on Earth is R in = F s (1 − α ( T ( θ )) dA = F s (1 − α ( T ( θ )) R (cos(2 θ ) + 1) dθ (5.9)The outgoing infra red radiation from Earth from this strip is R out = σg ( T ( θ ) T ( θ ) dS = σg ( T ( θ )) T ( θ ) (2 πR cos( θ ) dθ ) (5.10)Hence when the system is out of equilibrium and T = T ( t, θ ) we have by the law of energyconservation C ∂T∂t = k ∇ T + R in ( θ ) − R out ( θ ) (5.11)where R in and R out are given by (5.9) and (5.10) respectively. The term k ∇ T represents theheat transport (or diffusion) due to temperature difference (viz. gradient) in the different9atitudes (due to atmospheric and ocean currents). The coefficient k is called the diffusioncoefficient. Without this meridional heat transport the equator will become increasinglywarmer while the poles increasingly colder. An approximation of this term (that is usedby some authors) is to replace it by a term proportional to the temperature difference¯ k ( T ( t, θ ) − T ave ) where T ave is an estimate for the global temperature average.We observe also that this model equation does not take into account the change in theEarth tilt with respect to the Sun at different seasons.Neglecting the diffusion term in (5.11) we must then have at the steady state R in = R out and the following expression for T ( θ ) F s (1 − α ( T ( θ )) cos( θ ) = πσg ( T ( θ )) T ( θ ) (5.12)This equation have to be solved either graphically or iteratively since the values of α ( T )and g ( T ) depend on the final equilibrium temperature.The results of this model are rather unusual. If one starts the simulation from above thefreezing temperature (e.g. 280 Kelvin) one obtains Fig 8 where at least part of Earth is wellabove the freezing point. On the other hand if one starts the simulation from 240 Kelvinone finds that the Earth is a frozen ball (see Fig 9). This demonstrates that this modelexhibits the existence of two stable climates, one is ”warm” and the other is ”frozen”. Thefirst corresponds to the present climate while the second corresponds to an ice age. Matlabprograms mod03.m and mod04.m which implement these results can be downloaded from[15]If the parameters of the system change (e.g if F S changes) ”slightly” and the Earth isat the warm stable climate it will stay at warm temperatures. On the other hand if it is inthe cold stable climate it will remain at an ice age. The transition from one stable climateto another occurs due to variations in the solar output and the CO (and other greenhousegases) concentration cycle in the atmosphere.In this model the average Earth temperature as a function of the system variables followa hysteresis diagram and the transition between these two stable climates can happen withina very short period of time. This is illustrated in Fig. 10. In this (“illustrative”) plot the red10ine denote a ”warm climate” while the blue line ”ice age”. In the interval [ − . , . − . . In one dimension (and spherical coordinates) the diffusion term for a strip between θ and θ + dθ in (5.11) has the form k ∇ T = kR dTdθ dθ . Using (5.9) and (5.10), the steady state equation(5.11) becomes K dTdθ = F s (1 − α ( T ( θ ))(1 + cos(2 θ )) − σg ( T ( θ )) T ( θ ) (2 π cos( θ ) (5.13)where K = kR . We simulated this equation with K = 1 , . , . and initial condition T = 280Kelvin at the equator ( θ = 0). The solution(s) are depicted in Fig. 11. Matlab programdiffusion.m that was used to obtain these results can be downloaded from [15] Several countries signed the Paris agreement to curb the concentration of greenhouse gases inthe atmosphere. However this agreement is ”not totally binding” and there is little politicalwill to abide by this agreement by some countries.Several ideas were considered in the last few years to reverse the possible effects of climatechange. One of these is to release a compound into the stratosphere that would reflect someof the Suns energy back into space.To experiment and gauge the impacts of such a program Harvard researchers [11, 12]sent a balloon into the stratosphere, where it released about 100 grams of calcium carbonate11his compound was chosen since it could stay in the upper atmosphere for a long period oftime and reflect sunlight back into space.It is expected that this small scale experiment will provide data about the risks andrewards of a large scale geoengineering programs.Carbon sequestration is another scheme that was floated as a way to reduce greenhousegas emissions into the atmosphere. Carbon capture and storage (CCS) (or carbon captureand sequestration) is the process of capturing waste carbon dioxide, transporting it to astorage site, and depositing it where it will not enter the atmosphere, normally into un-derground geological formation [14]. In some cases the captured carbon dioxide is pumpedunderground to force more oil out of oil wells. This set-up allows companies to actually makemoney from capturing carbon rather than it just being a financial burden. Carbon capturehas seen the most success in the United States, where so far projects have stored nearly 160million metric tons of carbon dioxide in underground geological formations.
The models presented in this paper are prototype models for the steady state temperatureof the Earth. They ignore many features that control the Earth’s climate. In spite of thisthey provide a ”window” to the most important factors that influence the Earth climate viz.Albedo and greenhouse effects. They provide a warning signal about the possible impact ofhuman activities on global warming and climate.12 eferences [1] W.D. Sellers -A climate model based on the Energy balance of the Earth-Atmospheresystem, J. Appl. Met, 8, pp.392-400, 1969[2] W.D. Sellers - A New Global Climate Model, J. Appl. Met., 12, pp.241-254 ,1973[3] K. McGuffie and A.H. Sellers -A Climate Modeling Primer, Wiley, 3rd Ed. NY, 2005.[4] M. Ghil and S. Childress- Topic in Geophysical Fluid Dynamics, Chapter 10, Springer,NY. 1987.[5] D.J. Wilson, J. Gea-Banacloche - Simple model to estimate the contribution of atmo-spheric CO ch CO Sequestration in Deep Sedimentary Formations,Elements, 4, pp. 325-331, 2008. 1315] https://users.wpi.edu/ mhumi/model/.[16] O. Coddington, J.L. Lean, P. Pilewskie, M. Snow, D. Lindholm - A solar irradianceclimate data record, Bull. American Meteorological Soc., 2015, doi: 10.1175/BAMS-D-14-00265.12.[17] G. Kopp and J. L. Lean - A new, lower value of total solar irradiance:Evidenceand climate significance, Geophysical Research Letters VOL. 38, L01706, 2011,doi:10.1029/2010GL045777. 14igure 1: Absorption Spectra of some Trace Gases15igure 2:16igure 3:17igure 4:18igure 5: Solar Irradiance19 time(years) equ ili b r i u m t e m pe r a t u r e - K e l v i n Equilibrium Temperature vs. Time, T2=275 K
Figure 6:20 time(years) equ ili b r i u m t e m pe r a t u r e - K e l v i n Equilibrium Temperature vs. Time, T2=296.12 K
Figure 7:21 (radians) T e m pe r a t u r e K e l v i n s Figure 8:22 (radians) T e m pe r a t u r e K e l v i n s Figure 9:23 x Hysteresis -0.3849 0.3849
Figure 10:24
10 20 30 40 50 60 70 80 90 (degrees) t e m pe r a t u r e
1D Climate Model with Diffusion k=1 redk=1.5 greenk=2 bluek=1 redk=1.5 greenk=2 blue