Climate Modelst 002 Warming A Selective Review and Summary AMERICANPETROLEUM INSTITUTE MARCH 1982 - . API PUB. NUMBER 4347 American Petroleum Institute 2101 Street, Northwest. Washington, DC. 20037 0-0 Lamont-Doherty Geological Observatory (Columbia University) Palisades, New York 10964 MODELS AND 002 WARMING A Selective Review and Summary Prepared for the American Petroleum Institute by Alan Oppenheim and William L. Donn LamOnt-Doherty Geological Observatory L(Columbia University) Contribution No. 3310 - - March 16, I982 CONTENTS II II. The Nature of Climate and Climate A. The Lowest Order: Radiation Balance Radiative D. E. General Circulation V. Assessment of the A. Common B. Energy Balance C. Radiative-Convective D. The Thermodynamic (Adem) E. General Circulation F. Future Figures - Figure l. (bmparison of Solar and Terrestrial Radiation Figure 2. Spectral Distribution of Infrared Absorption and Emission in the Atmosphere - Figure 3. Comparison of Radiation Balance?Derived .Atmospheric Lapse Rates with Adiabatic Lapse Rates. SUMMARY This report is a selective summary and discussion of the types of models that have been applied to the prediction of the anthropogenic warming of the atmosphere. All of the quantitative predictions involve only the CO effect although it is recognized that other trace gases may contribute from 53 to 100 percent additional warming. The types of models discussed are, from simple to complex: radiation balance, energy balance, radiative-convective, thermody- namic and general circulation models. The results given in the summary table below have been generated by the most physically complete versions of the models in each category. Additional results, discussed in the report, are omitted from this selection. It seems clear from the discussion herein that all models are still sufficiently unrealistic that.a definitive evaluation of the problem requires continued effort. Table 1. 1 Model Predictions of 2 C02 warming of Atmosphere Model Type and Source Mean Surface Effect High-Latitude Effect Radiation Balance Jason Group Globe: Energy Balance Jason Group Globe: 2.4? Budyko [12] Globe: 3.l? 9.0 Radiative-Convective Augustsson Ramanathan [18] Globe: 2? Hansenletual3 [17] Globe: 2.8? Hummel and Rack [39] Globe: 2.1? Thermodynamic Adem (Preliminary) Hem: 0.6? 2? General Circulation Model_ Manabe Stouffer [26] Globe I 7? Hansen Etna}: Globe 3.5? 7? I. INTRODUCTION In this report we summarize and discuss with some qualitative evaluation the types of models that have been applied to the prediction of,the atmospher- ic warming consequent upon the increasing 002 content of the atmosphere. The examples selected in each of the climate model categories are those that have 'been developed to the highest state of the art in each category at this time. The report is organized so as to first explain 'the problem, including -an explanation of what is meant by climate, followed by a qualitative summary of the types of models in order of increasing complexity. This is followed by a discussion of the key scientific problems involved, namely the quantitative- evaluation of the increased "greenhouse effect". After this discussion a more detailed presentation of the appropriate climate models is given followed by an appraisal of each of the models and categories diacussed. carbon dioxide is the most studied of the combustion gases since it plays an important role in the interaction between the sun's radiation and the atmo- sphere. The carbon.dioxide concentration has increased steadily since the beginning of the industrial revolution from about 290 parts per million (ppm) to about 340 today (1981). it is expected to double some time in the next century. Just when depends on-the particular estimate of the level of in- creasing energy use per year and the mixture of carbon based fuels None of these estimates includes the role of the oceans and the biosphere as sources or sinks of 002 since the mechanisms of their exchange with the atmo- sphere are too uncertain at this time. Climate modelers begin. with the assumption that atmospheric 002 will double (with corresponding increases in other combustion gases) and try to predict what climate changes will occur. a They all predict some kind of increase in temperature within a global mean range of The conSensus is that high latitudes will be heated more than the equator and the land areas more than the oceans. Such a warming can have 'serious consequences for man?s comfort and survival since patterns of aridity and rainfall can change, the height of the sea level can increase considerably and the world food supply can be affected. The detailed consequences of a 002 warming are not yet knOWn. The conclusion is that optimum forecasting of cli- mate changes is a necessity for any realistic long term planning by government and industry., II. THE NATURE OF CLIMATE AND CLIMATE MODELS An operational definition of 'climate' is that it is a time?mean state of the atmosphere Time means are averaged over a given period of time such as 30 days, a season, a year, etc. 'Climate changes' refer to changes of state of the atmosPhere over intervals greater than the-averaging-time.? 'Climate models'_ are mathematical descriptions of the earth-a?tmoSphere system as it is driven by the radiation'from the sun. The models range in complexity-from a crude picture of the earth as a uniform rotating ball in radiation balance with the solar radiation to the complex general circulation models (GCM) that treat atmospheric dynamics by numerical solutions of fluid dynamical equations. IRegardless of (ummlexity, all climatefmodel studies5 indicate that a doubling of C02 will produce a significant increase in the glabal and annual mean temperature of the earth. Climate model predictions range from to over 49C, depending more on the physical aesumptions than upon the complexity of the model. Several empirical studies 5] give a lower estimate of about .There is sufficient uncertainty in the range of predictions to leave the consequences of the 002 doubling in considerable doubt. The difference between the low end of and the high end of g4.?C has obvious consequences regarding the amount and speed of polar ice melting and the degree of sea level rise. Other uncertainties are the effect of the warming on: snow and sea?ice cover, the-distributions of temperature, rain? fall and aridity over the globe, the ocean'circulation and oxygen budget, and the related world food supply A summary of the models to be given further elaboration in Section Ill is given here for a quick overview of approaches to the C02 problem. These-do not exhaust the different kinds of models used for climate studies but they are the principal tools, either singly or in combination, applied for pre- dicting.the outcome of a C02 doubling. (Schneider and Dickinson.[3] give a 'thorough and readable survey of the many approaches to climate modeling that is still current.) A. The Lowest Order: Radiation Balance Model In this model the earth is treated as a uniformly rotating sphere with a homogeneous atmosphere in radiative equilibrium with the sun's energy flux. What this means is that the sun bathes the earth with radiation (that is_pri- marily in the visible), that the earth absorbs some of that radiation and in the equilibrium state reradiates it back to Space with a spectrum that is pri~ marily in the infrared. All that is needed to compute the "effective radia- tion temperature" of the earth Te is to give the known solar energy flux at the top-0f the atmosphere and that fraction of the solar flux absorbed by the earth-atmosphere system and radiated back as infrared radiation. It turns out that the mean surface temperature of the earth TS is about larger than Te. This is accounted for by the "greenhouse gases? that absorb and trap thermal (infrared) energy in the atmOSphere. Some atmospheric structure must be added to the originally simple picture to compute the ab- sorption of the constituent gases and to-maintain the machanical stability of the lower troposphere. .The model as amended is used to estimate a global warming due to the doubling of C02 when the remaining physics is unchanged. B. Energy Balance Models (EBM) These models add a latitude dependence by dealing with quantities such as surface temperature, heat capacity and albedo that are averaged over a com- plete latitude strip. One more term is needed to balance the energy equation in each strip. The new term represents horizontal heat transported out of the. strip by fluid motions. In equilibrium, the energy balance for each strip wouLd now read: (net transport out) (infrared out) (solar in). -The transport term is usually parameterized as'a diffusion operator (familiar from ordinary heat transfer theory). - Energy balance models have the added attraction that they can be used to estimate the latitude distribution of :1 002 warming with its effects on the_ polar ice caps. As with the original radiation balance model, separate calcu- lations with vertical structure in the atmOSphere are required to give the infrared terms. The EBM diffusion equation itself, however, admits only a 'latitude dependence in the equilibrium case with the addition of a time depen- -dence when seasonal and other time varyingvconditions are treated. C. Radiative - Convective Medels Radiative-convective models are more complex versions of the radiation balance model described above. IPhysical parameters are- treated as global averages with Spatial variation only in the vertical direction. Mathematical computations are performed for convenience in. a plane parallel configura- tion. The atmosphere is divided up into many uniform layers for numerical computation. For suCh models the principal equations to be solved are those I of radiative transport with the convective part appearing as adjustments to the temperature structure 'to prevent the lower tropOSphere from becoming mechanically unstable. These models are applied principally to numerical experimentation in which atmospheric parameters can be varied one at a time to estimate the sensitivity of the atmOSphere to change. (These include experi- ments with radiative models of clouds, changes in distribution of'partiCulate matter with height, the effects of?CD2 and other combustion gases, solar con-? stant, volcanic dust, aerosols and atmospheric chemistry. Occasionally exper- iments are performed in which the physical parameters of a particular latitude strip are taken as the global mean values in order to get a sense'of the changes in vertical structure to be expected at different latitudes and in particular to estimate how the structure varies under perturbation. Much of the mathematical details of the radiative transport calculations of these nmdels can, with some modification, be used in tandem wi?n those models that treat horizontal transport since the radiative part is calculated separately to give heating rates. D. The Thermodynamic (Adem) Model The thermodynamic model of Adem is an operating climate model developed on the basis of the assumption that for periods of a month or longer climate can best- be described by determining the mean thermal state of the atmo- sphere. The model uses two basic equations, one for the conservation of ther? mal energy for the atmospheric layer and one for the conservation of thermal energy for the ocean layer. The two layers are coupled in the full model. In each_equation there is a two_dimensional horizontal eddy diffusion term (the two dimensional version of the one dimensional transport term in the energy balance equation of section II.B). In addition to the eddy diffusion terms there is'a term that parameterizes the transport by the mean winds in the atmospheric equation and a3 temm that parameterizes transport by horizontal ?currents in the ocean equation. Continents and oceans have appropriate-values of the albedo - the fraction of solar energy reflected to space - as functions of position on the globe. The chief effect of the continents is a varying albedo due to changing snow and ice cover. Although an initial input-vari- able, albedo-feedback permits computation of changes in snow.cover extent. . At present the model computes thermal and solar radiative absorption for _each grid point as-a function of time with a computed cloud cover.: As with all global models the amount of atmOSpheric structure required for transport by diffusion and advection need not be the same as for the radiative terms, Isince the t?me scale for radiative equilibrium is so much smaller. 10 E. General Circulation Models (GCM) General circulation models differ from the previously described models in that they treat horizontal momentum directly. That is, they use the analogue of Newton's second law relating the change in momentum to the applied for? ces. The atmosphere is broken up into several layers and in each.layer there Hare equations for the energy and the horizontal components of the wind velocir ty. _As with the previous models the radiative transport equations must be solved in order to get the radiative heating or cooling at each layer and at. each grid point. In order to follow the horizontal velocity components the' GCM requires much shorter time steps than is necessary for the.other models. The GCM also differs from the prior models in an initial condi- tion and then integrating forward in time from basic principles. It-uses more explicit dynamics and less parameterizations then other types of models. Compared with GCMs, the thermodynamic model differs significantly in? method, purpose and simplicity. The GCM aattempts to predict- climate from first principles while Adem?s model generates climate anomalies based on the use of the stored data fields. These data fields represent analog solutions that contain implicitly whatever scales of motion contribute to climate. The' model is limited by the quality and extent of the data fields and related parameterizations. In predicting perturbations from normal climate by the subtraCtion of computed normal from actual fieldst the model avoids common - biases or errors that may be introduced by computational schemes and common parameterizations. 11 THE A. Explanation of the Effect The sun supplies the energy that drives the motions of the atmosphere and oceans. Solar radiation enters the top of the atmosphere with a characteris- tic that is mostly in the visible. A fraction of this radiation, called the planetary albedo, is reflected to space. The remainder is absorbed and transformed as it interacts with the earth and reaches equilibrium. The transformed radiation has a characteristic spectrum that is mostly in the in- frared. It is referred to variously as long-wave or thermal radiation. The spectra of both incoming solar and outgoing thermal radiation are defined by the equilibrium black body emission function (the Planck formula). This is a strictly thermodynamic relation because radiation and matter come to equilib?_ rium on time scales much shorter than those-of the macroscOpic motions. The Planck formula depends on the absolute temperature of the emitting surface and wavelength. The solar radiation, shown in Fig. 1, has the spec? trum of an emitting surface.with a mean temperature of For contrast the spectrum corresponding to the present mean temperature of the earth of_ about appears on the right. Both curves have been normalized relative to their respective maxima so they appear to have the same height.; The actual intensity of the solar curve is about four million times that?of the thermal curve. Thermal radiation, for temperatures characteristic of the earth and its-atmosphere, is predominantly in the range 4 to 40 um. As will be seen in Section IV.A where we elaborate on the lowest order radiation balance model, if we assume that the planetary albedo of the earth is 0.30, then the remaining fraction, 0.70, of the solar radiation will be INTENSITY 10*3 RADIATION TERRESTRIAL SOLma?a-inh'thiOEldS I00 (Micrometers) WAVELENGTH Figure 1. Comparison of solar spectral intensity (left?side scale) with terrestrial spectral intensity (right?side scale). 12 absorbed and transformed into thermal radiation with a mean temperature of (or Te is called the-effective radiation temperature of the earth. Just as the sun appears to be radiating at the relatively cool effec- tive temperature of its outer layers there appears to be an effective radiating temperature for the earth from its relatively cooler upper tropo- sphere The surface temperature of the earth is The reason that the surface temperature is warmer is due in part to the presence of the so called greenhouse gases that absorb strongly within the 4 to 40 pm thermal spectrum. Abacrbing gases very quickly reradiate the energy they get from the surface of the earth,-but in the atmOSphere a gas radiates both upward and downward, while the surface merely radiates upward. The spectrum of the equi? librium radiation.in the atmOsphere will be determined by the actual tempera- ture of the air. It is the mediating effect of the atmospheric_absorbersIand the fact that-they radiate downward as well as upward that provides an addi? tional bath of heat for the surface. This heating is called the "greenhouse effect" simply because the role of the absorbing gases resembles what was once thought to be the role of the glass of the greenhouse in trapping infrared_ radiation. I A key scientific problem to be solved is the evaluation of the increased greenhouse effect resulting from an increase in combustion gases in the atmOSphere. B. The Greenhouse Gases The expected role of the greenhouse gases can be illustrated with use of Fig. 2. The figure shows part of the envelope of the thermal emission curve for from 4 to 20 um. Estimates of the absorption curves C02 30- EMISSION CURVE . L9 .- NJ 2: L?sow- Lu . 2 - \k 10WAVE LENGTH (MICROMETERS) Figure 2. The spectral distribution of emitted and absorbed radiation in the atmosphere. The region of strong absorption by water vapor and CO is horizontally lined. The main CO absorption band is shown by thezbracket centered on 15 micrometers. Weak C02 absorption is shown by the stippled region. . .5. -13 and 03, which are sketched from computations based on data in Kondratyev correspond well with a similar figure in Ref. The horizontally striped region indicates roughly: the main absorption regions of water vapor between 4~um and 8 um and_above 17 um; and of 002 between 12.5 um and 17 pm. The principal infrared window to space is between 8 pm and 12 pm. Within the win? dow are an ozone (03) absorption band and a relatively unimportant (for pres?- ent climate) set (us 002 bands (stippled).' If 002 were to increase in the atmosphere the principal effect would be a filling in Of the presently weak band in the window to space as well as a filling in to the left of_ the strong absorption band that begins at 12.5 pm. If the 002 is more absorb- .ing, it will reradiate more both upward and downward. If all of the-other physics remains the same, Ithe effect of Ithis lincreased downward radiation would be to further increase the temperature of the surface. If the water vapor content were to increase there would be a; similar filling in of the window region_ with Inore absOrbing gases. The other' greenhouse gases with_ absorption bands within the window include NOX, methane (CH4), ammonia (NH3), thefhalomethanes (freons), carbon monoxide (CO) and hydrocarbons. Moreover, increased emission of and CD can increase CH4 and 03 in'the tropOSphere 9] via chemical reactions that compete with reactions that would otherwise remove them. Most attention has been directed to the roles of_H20 and 002 principally because the data base on them is far_more extensive and because the steady increase of C02 has been well established while the future increase seems a' rational-projection. According to the Jason group the freons have absorbing bands in the. middle of the atmospheric window so that if they were'to increase by a factor 'of about one hundred they could contribute strongly to the greenhouse effect. 14 N?o, CH4 and NH3 appear to.be marginally important at present. These gases would fill in the window from the left as C02 does from the right if they were 'to increase. Most recently Lacis_gt calculated that the warming from the increase in trace gases, and chlorofluorocarbons during 1970-1980 amounts to 50% to 100% of that due to C02 in the same period. I Apparently? then, all increases in greenhouse gases plus aerosols should be considered in the total anthropogenic effect. IV. DESCRIPTION OF THE MODELS -A. Radiation Balance Model - The description and results given here for this most primitive of models follOws the standard derivations and results of others, eg Chamberlain Picture the earth as a uniformly rotating ball with a homogeneous atmo- sphere in-radiative balance with the mean solar heat flux at the top of the atmOSphere of'SO 1367 watts/meterz. Moreover, suppose that in equilibrium. the earth radiates as a black body at the effective temperature Te. The sun's radiative flux is plane parallel with a certain fraction, the albedo a, being 2 refleCted int0'space. The amount (1 - a) 50 is absorbed; FR is the ef- fective area for plane parallel rays striking a spherical earth with radius R. The-absorbed energy is reradiated according to the radiation balance-equation; dh a ?E,de di (1) (4nR2) aTe?-= (sz) (l where the Stefan-Boltzmann constant 0.56687 10-7 watts/m2/?K4. For the value-a 0.3, the effective radiation temperature of the earth Te 2 absorbing material in the infrared so that 15 A simple relation between surface temperature, TS-and Te can be derived the following conditions: the thermal radiation is in local thermody- namic equilibrimn with the atmospheric gases; the absorptidn coefficient is? independent of frequency (the grey gas approximation); the atmOSphere can be treated in a1 plane parallel geometry; and the thermal radiation flux goes either vertically up or vertically down. From this it is simple to derive [10] the relation l"l? TS. Te (1 0.75 1g), (2) where $8 is the effective optical depth of the atmosphere - a non dimensional measure of the opacity or absorbing capacity of the atmosphere. So far the theory does not give a value for s" From Eqs. (1) and_(aTe? - - (3) where AS 18. If, as is the present practice, one assumes that 3 remains constant if C02 is doubled, then '1?e remains unchanged. However, 1' is a measure of 8 must increase with the C02 in- 3 crease and with it, AS. The ratio must remain constant by assumption so that TS4 will increase just enough to compensate for the increase in AS. A more asphisticated calculation is required to get I . The Jason group 8 in a calculation that summed up the_contributions of the absorption coef- 16 ficients of the major atmospheric constituents found a mean 1 of 0.748 for 8 the present C02 and 0.828 for double C02. Substitution of these values gives TS (present C02) and TS (double'COZ) The actual values of TS are not important but any change in T5 i?_important; The warming In a more complete calculation involving nine absorption bands, the Jason group obtained a change. The result for a model like this can be only suggestive.' The model producing these results gives the wrong temperature lapse rate- for the lower troposphere. The lapse rate is the rate at which temperature falls with altitude. the case of purely radiative equilibrium, the lapse rate in the lower troposphere will be steep enough for the upper air to be colder and denser than the lower air. Because this is mechanically unstable (a convective instability) the upper air will tend to sink and mix with lower air.until the lapse rate reaches a stable value, that is; equal to or less than the adiabatic curves shown in Fig. 3. BeCause a solution with a stable lapse rate is required for a meaningful result it is necessary to modify the procedure. The simple precedure followed is the replacement of the lower portion Of the initiallyederived lapSe rate curve one of the adiabatic curves. i.e. the dashed curve of Number 3. The use of such an atmospheric structure Would supply enough thermal radiation from the lower trepOSphere to maintain the radiative profile above the point of intersection This TEMPERATURE Figure 3. The vertical temperature lapse rate curve resulting from radiation balance solutions (solid curve). Dry adiabatic (constant) lapse rate-curves are Shown by broken lines 1,2 and 3. Below A the lapse rate curve is unstable. Adiabatic curve 3 i - - is substituted for the lower portion of the lapse rate curve below the point of intersection at to achieve_convective stab- ility and the maintenance of the radiation profile above C. l7 converts the radiation balance model into the crudest of the radiative- convective models (to be given more attention in Section C). B. Energy Balance Models Energy balance models are the next in complexity. They add a latitude dependence and therefore the capacity to treat snow?ice feedback. Morepver,? they are capable of analytical solutions limited to highly idealized situa- tions. For these models [11] the globe is divided into latitude strips or zones over which a balance equation states that the solar heat energy flux entering each latitude belt is exactly balanced by the loss rate. In the steady state, the equation for latitude belt i is (net transport out)i (infrared out)i (solar energy in)i. (4) 'Each term would be in units of energy/second/area after the common area of the latitude belt has been divided out. Recall that Eq. (3), the corresponding, equation for the radiation balance model has the form (infrared out) (solar energy in). The eXtra term in (4) represents horizontal transport of heat carried by fluid motions. The properties for each latitude belt.are average quantities. The only spatial variable is the quantity sine where is the latitude. In (3) the quantity O.l5rg) represents the total infrared radi- ation badk to space. One could use such a formula for each strip.' However, it is more convenient to convert TS to Celsius degrees by 273 . 18 T(x) where T(x) is in and replace the infrared radiation term on the left side of Eq. (3) with the linear_version on the left hand side of (5): - . (5) Where A and B, in the linearization, absorb the constants of but are given experimentally derived values below. So(x) is the average-value of the Isolar flux and a(x) the average albedo at altitude x. The variable is con- venient for this work because the differential-dx is proportional-to the area of the latitude circle. If (5) is 'multiplied by dx: and integrated over latitude the result would be equivalent to (3). Present cloud cover, varia-- tion of water vapor content over the globe and the presencekof greenhouse gases are accounted for in (5) by writing A(x) 0.7518) (6) B(x) 0.75Tg) where A(x) and B(x) are deduced from 'measurements. The Jason. group computes mg by using the U.S. Standard Atmosphere for 1976 and allowing the assumed structure to radiate to space. They break the thermal-region into nine frequency bands, compute the flux for each band separately and then sum than for the total. Thus 1 accounts for the presence of.the greenhouse 8 gases. Then-a(x) and b(x) account for variations in cloud cover and water vapor via the measured quantities A(x) and B(x) in (6). If one assumes that a(x) and b(x) do hot change up to a doubling of 002 then a computation of for the caserof'COZ doubled (all other quantities 8 remaining the same) will give a new set of A(x) and B(x) through Eqs. (6). 19 .This approach is not based on- sOlving the equations of 'radiative transfer hence, again, an estimate for the greenhouse effect rather than a precise? number results. I 'The energy balance equations deal only with the energy content in the .total column of atmosphere above the complete latitude strip. These are rep- resented by an average heat capacity multiplied by the surface temperature Since there are no horizontal velocities in this picture we have to_ parameterize "net transport out" in terms of T(x) and its derivatives. -The I parameterization uses the form of molecular diffusion but here the diffusion te?m has to account for turbulent and eddy transport of heat. ;For energy balance models the-parameterization takes the form of Eq. (7) (or alternate 9 but similar forms) a [net transport_out] (7) where D, the diffusion coefficient, is at our disposal. It can be a,global; constant or vary with or be prescribed in any way that tunes solutions to 'present climate. At'present, whatever method is taken for fixing D, it is _kept.the same_foriestimates_of doubled C02. ?In steady'state, the full equation has the form _dx dT(x) .dx _Acx.) spell??24311 I (8) ..-. - 20 In (8) the surface albedo can. be preacribed or .it can. be written in order to account for variations in the ice line if one perturbs the climate. The Jason group uses this parameterization in different notation - -. {3:2 it; as where 2(a) is a correction for variation of zenith angle with x. I Eq. (8) and its time varying counterpart are useful for numerical experi- mentation. In their most realistic experiment the Jason group finds that by doubling C02 an increase of resulted when water vapor increased (con? stant relative humidity) from the effect of temperature feedback and sea.ice was allowed to shrink to zero. High latitudes increased by The cor- responding results from the original budyko version of this type of model [12] was for the global-change and at high latitudes. C. Radiative-Convective Models Manabe and his collaborators [13, 14, 15] developed these models as-a prelude to the incorporation of radiative transfer into general circulation models. All of the physical parameters in these models are taken as global Iaverages and all of the computations are in one spatial dimension with Ivariation only in the vertical (zuaxis) measured upward from the surface. The equations of radiative transfer can be considered as bookkeeping relatidns that keep track oft the change in radiation intensity I(f) .(in watts/meterz/unit solid angle/unit frequency interval about the frequency'f) in some distance dz. They have the form: the change in intensity in distance 21 dz - the amount absorbed - the amount scattered out of the beam the amount 'scattered into the beam the amount emitted.by matter into the beam. In principle the equations are first selved for each narrow frequency interval df with boundary conditions Specified at the top and bottom of the atmOSphere: lat the top the net upward flux of thermal radiation must equal the net down? ward flux of solar radiation; at the surface the net ?upward thermal flux equals the net downward solar flux. To reduce the computational burden [14, 16] the solar and long wave fluxes are given different emphasis. That is, the solar radiation may be absorbed, scattered or reflected but emission is ne- glected, while the long wavelength thermal radiation may be absorbed or may be emitted but scattering is neglected. ?Molecular absorption is a process in which the molecules absorb radiation and go into excited energy states; they then reradiate_energy in all directions. Quantum mechanics locates the energy levels and thermodynamics shows how the absorption_coefficient is broadened by' pressure and temperature variations. Further simplifications involve breaking the frequency spectrum into representative bands and computing mean absorption Coefficients over each band. This reduces considerably the number of inter- vals for which the radiative_transfer equations need solution. In radiative-convective models the atmosphere is. divided into uniform layers for numerical solution of the radiative transfer equations._ carbon dioxide is treated as uniformly miXed but water vapor and ozone are given :vertical distributions appropriate to present mean climate. Clouds strongly influence both solar and thermal fluxes. The distribution of clouds as high, . I medium and low is generally prescribed according to present climate atatiSf tics. HanSen at for example, take climatological cloud cover to be' 50% with distributions in the_fraction: 0.1 for high, 0.1 for medium and 0.3 for low clouds.' (Low clouds cool the surface while high clouds warm it.) 22 -. Wavelength dependences of cloud and aerosol prOperties are-included in some of the later R-C models The essential point is that the radiative transport equations are treated with considerable detail in models. lAt any level they give the net heat- ing or cooling after summing over all frequencies. In Section IV A we saw that the atmosphere is unstable under purely radi- ative balance for the very simple model described. To avoid this problem in models another-set of conditions is required for stability. There must be some mechanism to transport heat upward from the_surface (a convective adjust- ment) so that a stable temperature lapse rate will exist. One of the.several alternative methods for performing these adjustments is via the time-stepping procedure of Refs. [14, 17]. (In this and the more complex models that fol- low, artificial time steps can be used to go from an initial state to the desired equilibrium state without changing the external conditions or solar forcing.) One begins with a standard atmosphere having a given composition and temperature structure subject to the given incoming solar flux and suit- able boundary conditions at the surface. Then the equations of radiative transfer are solved to compute the radiative heating terms at each atmospheric level. Given these, the density of the air, the width of the level, and a suitable time step At one can compute the temperature adjustment within the ith level. If the lapse rate exceeds -4 a standard normal for I mid?latitudes used in most R-C models -- the atmosphere is unstable and enough 'heat (or equivalently a convective adjustment to AT) must be added to ensure that the lapse rate will be or less. With the new temperature struc- ture the radiative transport equations are solved again subject to the same boundary conditions to get new heating rates. The procedure is iterated until the atmosphere is in radiative and convective balance. 23 Hansen gt [17] performed different experiments for doubled C02 -- the' differences are with cloud parameterization, relative humidity, snow and ice and vegetation albedo feedback. With relative humidity and cloud temperatures fixed at present values and with the limiting lapse rate they find that the mean surface warming is ATS with an uncertainty of a_factor of 2. In contrast, Augustsson and.Ramanathan [18] give ATS for a dif- ferent cloud parameterization. The_quoted numbers are those preferred by the authors of [17] and [18] out of sets of runs with a variety of parameteriza- tions. However, similar-assumptiOns lead to similar results (see-table of- Ref. [17].and table 2 of Ref. [1d] -- a clear indication of the sensitivity of the results to assumptions about the cloud physics. Hummel and Beck [39] improved on previous radiative-convective models by adding water vapor tranSport to their?version of the early Manabe?WEtherald model thus permitting calculation of cloud location and thickness.- Prior models used a constant relative humidity profile_and cloud distribution. a doubled (:02 content and a standard cloud cover input their modification Igives an increase of surface temperature.of 2.05? compared with 1.71? for the Manabe?Wetherald model. This difference is due to a larger, more realistic _water content. References [14, 16, 17] give a balanced and clear picture_of the tech? niques of radiative-convedtive modeling.- D. The Thermodynamic (Adam) Model The basic assumption of this model is that for periods of a umnth or longer the mean state of the atmosphere depends primarily on the thermodynam- ics of the atmosphere and seems to be but weakly dependent on the dynamical 0] - Mir-m motions -- those governed by the fluid dynamics version of Newton's laws. The model follows the time evolution of-the thermodynamic state of an atmosphericl layer about 10 km high that includes a?cloud layer, an ocean layer of 50 to 100 meters in depth and a continental layer of negligible depth and heat stOrage. It also includes a layer of snow and ice over the continents and oceans. The basic prognostic equations used_in this system are those of Con? servation of thermal energy applied to variables that are time averaged over a prescribed interval. It is assumed that the equations of hydrostatic equilib- _rium, the perfect gas law, and the continuity equation are valid for the time averaged variables. The thermal energy equations for atmosphere and oceans-are integrated over their about 10 km in the atmosphere and 50 to 100 in the oceans. The resulting equations follow the mean energy of the atmosphere chosen proportional to the absolute temperature Tm at an altitude equal to one half of the mean height of the atmosphere, and the mean energy of 'the oceans chosen prOportional to their surface temperature TS. Three basic equations'are use to describe the atmosphere, oceans and continents respectively._ For the atmosphere and oceans, the equations (which have-the same form.and are coupled in a full solution) have_terms on the left side for local rate of change of thermal energy, heat transport by eddy diffu-' sion, heat tranSpOrt by mean winds (currents) and vertical heat tranSport (below the ocean mixed layer). These are balanced on the right side by the 'heat snurces and sinks: radiation, latent heat and sensible heat. For the -continents, which have insignificant storage depth and mobility, the left side terms are zero and only the balance among the heat sources and sinks is con- sidered. 25 -Albedo feedbaok for snow and ice is obtained by'adjusting the snow-ice margin to a selected isotherm (currently by iterative solutions until convergence is obtained. All of the terms of the atmosphere-earth radiation balance are also computed internally in the model solutions. Changes in cloud amount are computed as a function of latent heat changes. Solutions of the linearized differential equations are carried out at 512 grid points over the hemisphere. Output is in the form of hemispheric maps for surface (land and water) and mid-tropospheric temperatures. In addition,? all' of the diagnostic terms involved are also printed out in separate charts. These include evaporation at the surface, latent heat of condensation in the atmosPhere, computed meridional and zonal winds and associated heat tranSports, absorbed surface radiation, long wave and net radiation and cloud. cover. The model has been applied with success to the calculatiOn of absolute and anomalous values of all of the above terms. It has also been applied with good results to the computation of known long?range climate changes during geologic time. Although this model was developed primarily for prediction of current climate, it can be modified to be applied to predict the anthropogenic changes in climate for increased combustion gases. The current model has been used to get a provisional prediction of a doubled C02 effect by using published values of the changes that would occur in black?body emission. A mean increase_of 0.69C is predicted in the experi- ment with a high-latitude change of 2? and a low to mid-latitude change_of about These values may be revised when the model is optimized for the experiment. The mathematical version of the model is given below. reads: 26 In the' present _form of the model the equations for Tm (the mid- tropOSpheric temperature) and (theT surface temperature) both have_ the general form_ gE-+ - KVZT heat sources and sinks (11) The first term is proportional to the local time rate of change of the energy. The second term represents the transport of heat by mean motions: for the atmOSphere is determined by thermodynamic relations -- the geo? strophic wind relations and for the oceans is determined from the surface wind Speeds [19].i The third term represents horizontal diffusion of heat by. eddy and turbulent motions. The right hand sides, the heat sources and sinks (HSS) are: HSS for the atmosphere Et G5 G2, (12) and H83 for the oceans ES GB - G2. (13) In addition, with neglect of heat storage in the continents the third equation . (14) for the continental surfaces._ - .. 27 In (12) Et is the heat energy added to the atmosphere by radiation, G5 is the heat added by condensation of water vapor in the clouds and G2 is the heat added by vertical turbulent transport from the surface parameterization of the convective transport.that is dealt with by-the convective adjustments in radiative?convective models. In (13) and (14) E?.is the rate at which energy is added to the surface by radiation, and G3 is the rate at which heat is lust by_evaporation. The parameterization of Et and ES is based on assumptions that the cloud layers and the earth radiate as black bodies and that the clear sky atmOSphere' has a window for between 8 pm and-13 um. -They are given in terms of T5, Tm, the insolation I at the given latitude, the cloud cover, the-total radiation received at the surface (fOr ES) under clear Sky conditions and the surface albedo a. Over the oceans the quantities Gz and G3 are parameterized' in terms of measured normals, departures of (TS - Tm) from their normal values -and normal surface wind Speeds. Over the land G2 has the same parameteriza- tion while G3 simply'depends on empirical normals, and a known function of map. coordinates. ?Similarly, the heat gained by condensation of water vapor in the clouds GS is given in terms of its normal seasonal values GSN and (Tm and its first order derivatives with reapect to map coordinates. (The sub- script indicates normal values.) 'The cloud cover_E is a variable given by: 8 2 SN GSN) where a is the normal cloud cover and I) is a constant. .Details of-the 2 parameterizations are given in Refs. [20, 21, 22, 23]. E. General Circulation Mbdels (GCM) The design of general circulation models begins with the basic equations- governing large scale atmospheric nmtions and follows by transforming them into some kind of finite differences scheme suitable for solution on digital computers. In this process, the original equations and boundary conditions (or the finite differences scheme itself) would be altered to remove the phys? ical mechanisms responsible for wave motions that would otherwise be generated by errors in initial data. These waves would be spuriously amplified by com-- _putational rather than actual physical instabilities and ultimately swamp the motions under study The first of the basic equations is the equation of conservation "of mass. It states that if you follow an individual parcel of gas in time the _total mass of the parcel remains constant.. This becomes: The fractional change in density of the parcel with time the negative of the fractional rate of change of its volume with (16) (Thus a fractional increase in volume of 1% wOuld be accompanied by a frac- tional decrease in density of The next equation describes the evolution of the internal energy per unit mass of the parcel in time. In terms of the absolute temperature and CV the specific heat at constant volume: a (17) 28 29 The energy equation (the First Law of Thermodynamics) reads: The time rate of change of internal energy the rate of working by the fluid system the rate at which heat is added to the system. (18) Equation (8) of the energy balance models is derived as an approximation' to Eq. (18). For Adem's model, Eqs. (16) and (18) together with a parameteri- zation of the mean motion reduce to equations (11) and (12) fer the atmosPhere Hand (11) and (13) for the oceans. The new equations for GCM are the fluid dynamical versions of Newton's . . Second Law ma. In cartesian coordinates, for the horizontal west to east coordinate and velocity and the south to north coordinate and velocity the equations read(19) and (20) [dt u] where the symbol gE-stands for the time rate of change as we follow the given parcel of fluid, is the pressure, girand gg-are components of the pressure gradient (the force termS), is the density of the parcel, the (briolis parameter is equal to 29 sin? where is the angular rotation of the earth (2n? Iradians/day) and is the latitude. Eqs. (19) and (20) When multiplied by the volume of the parcel are in the form of Newton's Law ma . (21) in a rotating frame. Vertical accelerations (which would be perturbations on the fundamental hydrostatic .pressure balance equation of- the atmopshere) are important on smaller space-scales than the motions followed in meteorology and muSt be parameterized in order to include anadequate treatment of convective heat -transport from the surface to the atmosphere. Ademls model uses a thermodynamic parameterization of (19) and (20) by setting the d/dt terms equal to zero and computing and from ?1 -1 - (pf) ap/ay, (pf) ap/ax. (22) These are the geostrophic winds and are used wherever advection terms are used. For large scale motions of the kind used in climate studies this is a reasonable approximation since the geostrOphic ?wind approximates the true horizontal velocity to within about 15% in midlatitudes. For GCM the full-set of equations must be transformed into some version of a finite differences scheme suitable for solution by a digital computer. Typical horizontal grid spacings might range from a-4? 5? net to an 8? 10? net.' For the vertical structure, 2, 7, 9 or more levels are used. With each choice of net there are of motions-smaller than the grid spacings that cannot be resolved. Because these subgrid motions are important trans- port mechanisms for energy and momentum their effects must be parameterized in terms of the grid scale variables and their derivatives. At present the GCM use oceans without surface currents as sources and sinks of heat. In addition to the convection of moisture and sensible heat 31' (described above for the thermodynamic model) the COM must also provide for convection of momentun. I I In a survey of this size it is difficult to describe in. detail the cemputational complexities of the major 'models or the various schemes for parameterization of physical processes that can not be treated directly or simply. It is relevant to note that all of these are active areas of present research and that year by year -mode1s undergo modification. to 'accomodate changes'in.hnowledge. Reference contains a summary of predictions of the outcome of ?02 warmings from two of the principal GCM modeling groups: the group led-by Hansen at the Goddard Institute for Space Studies in New York and the group led by Manabe at the GeOphysical Fluid Dynamics Laboratory at Princeton, N.J. The global mean warming for the most complete of the two sets of models is for Manabe_et El. and for_Hansen_et_al., quoted in As might be expected different parameterizations and different feedback mecha? nisms produce different results within the above range. In the latest published GCM experiment, Gates [27] used a two layer atmospheric model which included an ocean constrained with prescribed climatological temperature and obtained a global surface air temperature warm? ing of only and a surface warming of only 0.l?C. This low result is primarily a function of the use of a prescribed sea surface temperature. The magnitudes_of the warmings by GCM, as with all models are higher at high latitudes. The maximum value, according to Ref. is-between and in polar and adjacent-regions for the annual mean surface AT. All models also indicate increased warming in summer and over land, but the magnitudes" differ. - 32 V. ASSESSMENT OF THE MODELS A. Common Assumptions In climate modeling certain physical processes must be parameterized in order to make computations tractable on current generation computers. This includes eddy diffusion of heat and momentmn, convective transport of heat, moisture and_momentum, and the radiative properties of the atmosphere. In Imany of these parameterizations a particular constant (the parameter) is given a value that tunes the final result to present climate. This is a perfectly .respectable procedure and enables one to perform experiments involving small changes in solar constant or some physical input with a reasonable expectation that the constants will remain valid for the altered climate state. Common practice involves the assumption that present parameterizations will be valid for a doubling of 002. Some of these parameterizations have a very strong influence on the outcome of the doubling. One such assumption-is that the mean planetary albedo a_and consequently Te, the effective radiation temperature of the earth, remain unchanged. An example of this in the simplest case is seen with equation (3): 0330/4: . where As 1 0.75 Tg. It is clear that depending on whether a goes up or down Te can be colder or warmer. Another assumption is that the present mean distribution of relative humidity remains fixed for a 002 doubling. ~Since a C02 warming will increase the water vapor content of the atmOSphere, if relative humidity remains con- 33 stant, the greenhouse effect of water vapor in the atmospheric window will result in a strong positive feedback. Thus, the Jason group model pre? dicts an additional warming of about 50% of the bare 2 002 warming. How- ever, in the complex feedback mechanism, increased 032 .leads to increased i *temperature with a consequent increase in evaporation and increased moisture content of the atmosphere. Although this effect leads to a further warming, the probable increase in cloud cover would increase valbedo and offset the warming effect The true effects of all of these sensitive relationships are not yet known. I Wherever the models.agree on these two assumptions it is likely that the predicted global warnings will be close simply because the final results are very sensitive to the planetary albedo and the relative humidity. The assump? tions-serve as constraints and as modeling efferts evolve theSe constraints will-be relaxed. B. Energy Balance models Energy balance.models are extremely tractable for both analytical and nu?. merical treatment [11, With them one can follow the lowest order effects of climate. change on the ice line and the latitudinal distribution. of a warming. The ice line separates the region of snow and ice where the albedo is high from the region of bare earth where it is low. In these models, the ice line can be made a function of surface temperature and will shift latitudinally with the surface temperature. '34 The principal difficulties with this class of models are: 1. they define all physical variables over a complete latitude strip so that there is no proper-Separation of oceans and continents; ii. they have absolutely no advection parameterization -- an important tranSport mechanism for mid?latitudes; they have no hydrological cycle and iv. any interaction between oceans and air would be too crude to offer any reliable time estimates for a warming. In summary, the models are extremely useful for preliminary experiments since they are fast and simple but they can give no definitive answer about a climate warming. C. Radiative-Convective Medals Radiative-convective models are one dimensional representations of the earth?s atmosphere with Variation possible only in the vertical and in time. They were designed originally as precursors for the incorporation of radiative transfer into but have served for a considerable amount of interesting experimentation. I The principal weakness of these models are: . 1. they treat a mean earth; 2. they have no horizontal heat transport and 3. the convective adjustments are very crude mechanisms introduced to maintain mechanical stability of the lower troposphere. ?35 Treatments with horizontal transport and realistic oceans and continents could modify any conclusions drawn from radiative?convective models. Nonetheless, these models can be powerful tools for exploring radiative properties of the atmosphere especially parameterizations of cloud cover and dynamics and the effect of industrial pollutants- The fact of a predicted change wi11_be im? portant rather than the magnitude. For accurate magnitudes the radiative come putation package must be appended to models with two dimensional horizontal variation and a realistic geography. Rack and collaborators have studied the effects of a wide range of indus- trial pollutants on climate with a version of the Manabe-Wetherald radiative- convective model. These include the effect of aerosols on climate [28, 29,. 30] and the effects of the freons on atmospheric surface temperature These and related numerical experiments and others on 002 warmings [16, 32] are suggestive rather than definitive at this time. -D. The-Thermodynamic (Adam) Model The thermodynamic model of Adam is the only operating climate model that gives reasonable forecasts of current temperature anomalies. It has also been used on a quasi-Operational basis to predict climate with very good performance and has been successful in simulating past climates related to ice ages and different continental locations [33, Since early in 1980 it has been generating forecasts with good skill for the northern Hemisphere [35, 36]. Other strong points of the model are: 36 i i. it is fast a one month forecast takes about 1 minute on i an ii." it is the only eaisting model with a realistic mean ocean having wind driven currents parameterized in a useful way and I I 3 it generates cloud-cover, the radiation balance, snow-ice I I feedback and sea surface temperatures internally. Sagqp The weak point of the model is that many of its parameterizations, while adequate 'for present climate. predictions, require adjustment for Optimum application of the model to the 002 warming problmn. As with all models, parameterization of physical processes require better and more fundamental understanding of the role of.cloud physics, the distribution of moisture in the atmosphere, and the way subgrid scale motions contribute to time mean 'lmotions followed by the equations of motion. Further work is necessary on this model to optimize it for application to the C02 problem. E. General Circulation Models Despite the fact that General Circulation Models include simultaneously details of those processes that control climategthey may not be,at least at this time, the appropriate vehicles for predicting long term climate change for the following reasons (paraphrased from Refs. and i. the computing time for current GCM could take from a half Iof a year to a full year to calculate a century of climate for a single combination of initial conditions or prescribed external parameters; Mil . . - . - - . . . . order to be useful for climate studies it may require i Both Ref. and suggest that simpler models would be more tractable - - . I . iH. calculation. of statistics from an. ensemble rs? numerical and USEEUI for Probing long term variations 1? Climate-- I: .. l integrations and iE it is difficult to track down cause and effect - - F- Future Directions I relationships with the many degrees of freedom involved in II I GCMI I -Most models prescribe the data for the radiation balance the atmosphere. Involved with this is the tuning to present conditions of No GCM (as of this writing) can predict present climate or give even a two- covers ralative humidity and Planetary albedo- These terms'have a Pro? 5 week forecast. There is a phenomenon of "intrinsic stochasticity" referred to 5~ II found influence on the warming due to a Change in combUStion 835 It 13 in Ref. which refers to a kind of internal chaotic motion (not driven by essential-that these terms ?01% in some Parameterized way, with changing i external noise) that occurs even in simple dynamical systems that are deter- I I Climate- In this connection WE note that 0hrng and [33] deduced from Eh I ministic' certain values of the constants in the equation these Systems I observations that the'net albedo cooling effect of clouds is greaterI i become extremely sensitive to initial conditions so that closely neighboring . than the greenhouse warming effect. - .I I initial physical states can evolve into quite different final states_ SinceI In addition to the problems connected with the meteorological parameters, i GUM have so many constants and adjustable parameters the authors oeref. I I a 5331C PrOblem appears to in the 0f heating rates-I This i - IIHII expect a good amount of such chaotic behavior. I computation reQuires the solution of.the equations of radiative transfer, a Ij ? In addition to these difficulties, it is muCh more difficult to assess if. I process which currently uses in part, analytical expressions devised prior to the effects of any input assumptions, including errors, on results of the coma IQ the advent of present high?speed computers. The magnitude of potential errors I plex GCM than it is for simpler models. Example of effects already detected I in the above Procedure be evaluated and computational strategies de? II I in which large changes occur are evident in the change from a swamp ocean (no I- I vised to determine transmission in sufficiently small frequency bends over the I heat storage) to a 'mixed?layer' (heat storage) ocean inI the models -of I SPECtrum 0f int?r35t? Manabe and WEtherald [37] and Manabe and Stouffer The warming of 30 in In addition to the above fundamental areas, necessary improvement must I i: - the former cgse falls to 2? in the latter, And in'the case OfJGateS 25 El: .. igII still be carried out for many of the parameterizations of both atmOSpheric and 'ii [27] who used effectively an ocean of infinite heat storage a warming of only 'gpbh II ocean terms. I I 0.1.. resulted. These are gross effects. Changes due to the many more subtle I I Despite all of the uncertainties in the classes of models described, aspects-are much more difficult to trace. I I their very errors serve toIgive outside limits of global warming from a doubling of 002.. Effects of trace gases, referred to earlier, w; I I mightIleadIto a near doubling of these numbers. nll'rm 39 References Ad Hoc Study Grbup on Carbon Dioxide and Climate (Jule G. 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