MONTHLYWEATHERREVIEWVOLUME 93, NUMBER 14DECEMBER 1965L RESULTS FROM A NINE-LEVEL GENERAL CIRCULATION MODELOF THE ATMOSPHERE` JOSEPH SMAGORINSKY, SYUKURO MANABE, ond J. LEITH HOLLOWAY, Jr.Geophysical Fluid Dynamics Laboratory, Environmental Science Services Administration, Washington, D.C. The "primitive equations of motion'' are adopted for this study. The nine levels of the model are distributedso as to resolve surface boundary layer fluxes as well as radiative transfer by ozone, carbon dioxide, and water v8por.The lower boundary is a kinematically uniform land surface without any heat capacity. The stabilizing effect ofmoist convection is implicitly incorporated into the model by requiring an adjustment of the lapse rate whenever itexceeds the moist adiabatic value. The numerical integrations are performed for the mean annual conditions overa hemisphere starting with an isothermal atmosphere at rest. The spatial distribution of gaseous absorbem is assumedto have the annual mean value of the actual atmosphere and to be constant with time. A quasi+quilibrium is attained about which a cyclic energy variation occurs with an irregular period of about2 weeks. The dominant wave number of the meridional component of the wind is 5 to 6 in the troposphere but isreduced to about 3 in the stratosphere. The gross structure and behavior of the tropopause and stratogphere below30 km. agree reasOn8blg well with observation. The meridional circulation obtained from the computation haa a3-cell structure in the troposphere and tends toward a 2-cell structure with increasing altitude in the stratosphere.Although the level of the jet stream as well as that of the maximum northward transport of momentum coincideswith observation, the intensity of the jet stream turns out to be much stronger than the observed annual mean.In the stratosphere the temperature increases with increasing latitude because of the effect of 18rge-scde motion.The magnitude of the increase, however, is smaller than that observed. A detailed study of the vertical distribution of the budget of kinetic energy, of 8VSik3ble potentii energy, ofheat, and of angular momentum is made. The mechanism for maintaining the kinetic energy of the jet streamand of the stratosphere is discussed. It is concluded that in the model the kinetic energy in the stratosphere ismaintained against ita conversion into potential energy and dissipation through interaction with the troposphere,which is in qualitative agreement with the resulta derived from an analysis of the stud atmosphere. In the tropo-sphere, the convemion of potential energy reaches a maximum at about the 500-mb. level. This energy is thentransferred to the level of the jet stream and to the surface boundary layer by the so-called pressure interaction term,thus providing the source of kinetic energy for these two levels at which dissipation is predominant. As with theresults of Phillips (271 and Smagorinsky [37], the ratio of eddy kinetic energy to zonal kinetic energy and that ofeddy to zonal avai~ab~e potential energy are computed to be much smaller than those of the actual atmosphere.728 MONTHLY WECONTENTSPage72772872972972973 173 173173273373473473573773974 174374474474474574674874 874975 17517537567577597617627637637647647647647657657657667671. INTRODUCTIONThe work to be reported upon in this paper represent,sthe results of the initial phase of a comprehensive long-term research program in the dynamics of the generalcirculation. This program, conceived in its present formin 1958, is a natural outgrowth of the two principalantecedents : (1) Phillips's [27] two-level quasi-geostrophicmodel in a zonally periodic domain on a @-plane, whichinitially demonstrated the feasibility of numerically simu-lating the general circulation, and (2) Smagorinsky's [37]primitive equation two-level model with motion within aATHER REVIEW Vol. 93, No. 12spherical zonal strip, which considerably generalized thehydrodynamic framework. These models have alreadyprovided extremely useful information regarding ourability to simulate some of the important gross propertiesof the general circulation and the processes by which theyare maintained.Both of these models had in common the gross over-simplification of the vertical structure of the atmosphereand all that it implies; namely, the necessity for strongparameterization of the planetary boundary layer, theradiative transfer, convective transfer, and the verticalstatic stability. The role of fronts was highly over-simplified, and the effects of the tropopause and thestratosphere were ignored. The interaction of the hydro-logic cycle was for t,he most part also ignored.The ultimate objective of this work is to understand thebasic mechanisms responsible for maintaining the generalcirculation and the climatology of the atmosphere and tosimulate the essential (macroscopic) features of the generalcirculation with a minimum of parametric constraints.It was, however, apparent that such a complete step at theoutset would inevitably end in failure and would only beunraveled by going back to less complex versions. Thestrategy adopted anticipated such difficulties and took along-range view. We first constructed with considerablecare, and in fact programed, the most general of a hierrtrchyof models in order to uncover in some detail the body ofphysics needed, to determine where the obvious weaknesseswere, and to give us some idea of the computational limi-tations we could expect. The perspective thus gained wasinvaluable. We then laid out a program of simplifiedmodels which can be' constructed as a sub-set of the mostgeneral one. The mn requirements were (1) that eachmodel represent a physically realizable state, (2) that theycould be constructed computationally bp program by-passes, and (3) that they collectivelp would provide astep-by-step study of the behavior of new processes andtheir influence on the interactive system. Hence, manyof the intermediate models in t.hemselves may lack detailedsimilitude to the atmosphere but provide the insightnecessary for careful and systematic scientific inquirv.In this connection me have carried from OUT previousexperience the practice of performing comprehensivedynamic diagnoses of the balances of energy, heat, andanwlar momentum. Although this requires what mayappear to be an inordinate and inconpous effort, we feelthat the apparent simulation of the synoptic manifesta-tions does not in it,self constitute an understanding. It isnot inconceivable that the results superficially may becorrect but for the wrong reason. Diagnostic inteP1techniques provide a very sensitive measure of the mecha-nistic similitude of the model to the atmosphere. In ourview they provide the type of insight from numericalstudies that one normally expects to derive from analyticalDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 729computation of radiative transfer. It, is clear that asimple parameterization of the calculation of radiativetransfer requires that. we implicitly presume a knowledgeof the vertical thermal structure of the real atmosphere.To avoid the parameterization, we decided to compute theradiative flux as a function of an arbitrary vert,ical dis-tribution of the atmospheric absorbers, such as watervapor, carbon dioxide, and ozone. For the sake ofsimplicity of computation we adopted a scheme whichrequires the results of low resolution measurements ofabsorptivity. Preceding the present study, extensivecomputations of thermal equilibrium were carried out bythe use of this scheme. Refer to Manabe and Strickler[17] for the detailed description of these calculations.One of the most important factors which influences, ifnot cont.rols, the general circulation of the atmosphere ist,he process of condensation. The explicit incorporationof this process into the general circulation model, however,initially overcomplicates the scheme a great deal andmakes the analysis of results more difficult. Therefore,we have introduced a simple process of moist convectiveadjustment as a substitut,e for the moist convection, theadjustment being carried out whenever the lapse rateexceeds the moist adiabatic value. It must be kept inmind, however, that^ although this process roughly cor-responds to the stabilizing effect of moist convection, itdoes not simulate the northward transport of latent energywhich could be of major importance. Therefore, thisnumerical study could be regarded as an intermediate stepbefore we attempt to study the behavior of a more generalmodel involving the process of condensat~ion. We shalldiscuss the results obtained from the time integration ofthe general circulation with the hydrologic cycle in apaper which follows [IS].In dealing with a lower boundary of uniform heatcapacit-y, there are two extreme choices in simulating thethermal properties. One can choose an ocean-coreredearth of infinite heat. capacity for which one must specifythe quasi-equilibrium surface temperature. In t,his case,the t,emperature specified at the ocean boundary would allbut control the latit~udinal temperature gradient of theatmosphere by convect,ion.Another choice for the uniform lower boundary con-dition is a land surface with no heat. capacity. In thiscase, the temperature of the earth's surface is determinedby the heat balance among solar radiation, long-waveradiation, and the turbulent flux of enerm from theearth's surface to the atmosphere. By adopting thislower boundary condition, we implicitly neglect theeffect of the heat transport by ocean currents. Thenumerical experiment with this boundary conditionconstitutes a logical step before the study of the modelwith the effect of energy transport by ocean currents.Therefore, we adopted this 1att.er boundary condition fort,he present study.It was our intention to permit sufficient vertical resolu-tion to:(1) make unnecessary the Ekman approximation be-t,ween the lowest level and the earth's surface-namely,the boundary layer is thin enough so that the rotationalforces may be assumed negligible with respect to theviscous forces,(2) have the upper level high enough to represent themain ultraviolet absorption of ozone,(3) describe the gross thermal structure of the tropo-pause and stratosphere, and .(4) provide tropospheric resolution great enough toaccount for the major role of frontal dynamics and ofcondensation processes to be incorporated later.2. SYSlEh4 OF PROGNOSTIC EQUATIONSA. DYNAMIC AND THERMODYNAMIC EQUATIONSAdopting pressure normalized by surface pressure as thevertical coordinate, me write the equations of motion on thestereographic map projection as follows (refer to the papersby Phillips [28], [30]):-[zn sin (VX-UY) PJT11The continuity equation is:The hydrostatic equation combined with the equation ofstate is:" a4 ETdQ"QThe thermodynamical energy equation is:The not.ations used in these equations are as follows:X abscissa of stereographic rectangular CoordinateI' ordinate of stereographic rectangular coordinatet timeU earth velocity component in X-diition730 MONTHLY WEATHER REViEW Vol. 93, No. 1 PV earth velocity component in Y-directionP atmospheric pressureP* atmospheric pressure at the lower boundaryT temperature4 geopotential height of -surfaceQ PIP, (P: pressure) (see section 5 for definition of-level spacing)0 dQ/dt0 dP/!i rate of non-adiabatic heatinge latitudem map scale factor for stereographic projection=FX ,F, rate of momentum changes due to ReynoldsP* P, stress in the X- and Ydirections- FT rate of temperature change due to hteralP* subgrid-scale diffusion of heatR gas constant of airCP specific heat of air at constant pressure8 angular velocity of earth-? -a radius of earthvFx vF, rate of momentum changes in the X- and Y-P* P, directions due to vertical diffusion-7 -B. SUBGRID-SCALE MIXINGThe momentum change due to the Reynolds stress andthe temperature change due to subgrid-scale mixingmay be separated into two parts, i.e.,F= gF+ vFwhere subscripts H and V denote the contribution ofhorizontal diffusion and vertical mixing, respectively.According to Smagorinsky [37], the non-linear lateral diffusionmay be formulated on the basis of Heisenberg's similarityhypothesis. If we ignore the density variation on theconstant Q-coordinate surface, we may take the viscousforce due to lateral stress and the temperature changedue to lateral diffusion to beIn this equation, 101 is the total deformation defined byA is the grid length; ko is a nondimensional parameterof order unity analogous to the Karman constant and istaken to be 0.4 in this paper. The frictional force dueto vertical mixing is(2Bll)where p and g are the density of air and the acceleration ofgravity respectively. Based upon the mixing lengthhypothesis applied to the boundary layer, the upwardflux of momentum is computed by usingavv7= PKV (2B12)andKv=12i$l (2B13)where 1 is the mixing length. As Rossby and Montgomery[32] suggested, we assumed the following simple law forthe vertical variation of 1:l=ko(Z+Zo) ZSh(2B14)(2B15)1 =ko(h+Zo) 6 - H"z hZHH-hl=Q HSZ (2B16)where the roughness parameter =l cm. According toRossby and Montgomery [33], the thickness of the loga-rithmic boundary layer, h, is 50 to 100 m. In this experi-ment h=75 m. and H=2.5 km.The upward flux of momentum v~ and heat ,H at theHFT=m*P', [ (P, -I)+=$ (p,KH by is given bywhere CD is the drag coefficient at the ground surface and(2335) (2B19)whereb(mU) b(mV) and T+ is the temperature of the earth's surface. Since be 75 m., these fluxes are a function of the velocity and theD --.--"___T- bX bY (2B6) the height of the lowest level of our model is designed tob(mV) b(mU) tempeure of this level. In order to incorporate thebX +bY (2B7) effect of free convection, a very large value of CD is adoptedDa=-December 1965 J. Snagorinsky, S. Manabe, and J. L. Holloway, Jr. 731for the computation of vertical heat flux, (vZT)Q+ whenthe stratification is unstable. The large coefficientprevents the large temperature gradient at the earth'ssurface which otherwise appears in lower latitudes.C. BOUNDARY CONDITIONS(1) Laleral Boundary Condith.-At the lateral bound-ary, an insulated free-slip wall is assumed to exist. Inother words, the exchange of momentum and heat withthe equatorial boundary is zero andv,=o, (2C1)where V,, is the component. of the wind normal to the wall.(2) Vertkat Boundary Conditions.-At the top and bot-tom of the atmosphere, the vertical -velocity is zero, i.e., Q=O at Q=O, 1 (2C2)Also, at t,he top of the atmosphere, the vertical flux ofmomentum due t.0 subgid-scale mixing is zero, i.e.,(V7)qPO=0 (2C3)This condition is automaticdly satisfied by applyingequation (2 B 16).The specification of F~ and ,H at the eart!h's surfRce(=1) has already been done in section B.(3) Bounday Con.ditwn .for Raciiati7?lc Traqfer.-.4t.the top of the atmosphere, the downward long-wareradiation is assumed to be zero, and a solar constantof 2 ly./min. is adopted.At the bottom of the atmosphere, the temperature ofthe earth's surface, T+ , is determined such that it satisfiesthe requirement of the heat balance tit tjhe earth's sur-face. If we Assume that, the heatb capacity of the ettrt.his zero, the balance equation of heat iss*+(DLR)*=gE+(vH)Q=l (2cz)where S* and (DLR)* me the net downward solar in-solation and the downward long-wave radiation atthe earth's surface, respectively, and u is the Stefan-Boltzman constant. The solution of this equation yieldsa T* which satisfies the heat balance condition at theearth's surface. Since we have eliminated the dailyvariation of temperature from our model by adoptingthe effective mean zenith angle of the sun, the assumptionof no downward conduct,ion int.0 the soil may not produceany serious difficulties.D. MOIST CONVECTIVE ADJUSTMENTAs explained in the introduction, we avoided the ex-plicit incorporation of the condensation process for thesake of simplicity of the model. A complete lack ofthe condensation process, however, would make theatmospheric static stability too unrealistically unstableto compare the results of the experiment with geoph.ysicaldata. In order to simulate the stabilizing effect of moistconvection as simply as possible, a convective adjustmentof the lapse rate to a critical value was made wheneverthe lapse rate exceeded that value, with the requirementthat the total potential energy be unaltered. Implicitly,it is assumed that the kinetic energy of moist convectiveeddies produced as a result of the overturning of airis dissipated into heat as soon as it is produced. Themoist-adiabatic lapse rate, which is dependent uponthe ambient temperature of the air, is adopted as thecritical lapse rate for the adjustment. It must be em-phasized that, in the present model, the effect of lateraltransport of latent energy is completely neglected. Referto reference [18] in which this effect is incorporated bymaking tho mixing ratio of water vapor an independentvariable of the model.E. RADIATIVE TRANSFERThe temperature change due to radiative tra.nsfer iscomputed as a function of the vertical distribution ofatmospheric absorbers as well as of temperature. Theatmospheric absorbers which are taken into considerationare water vapor, carbon dioxide, ozone, and clouds. Forsimplicity of computation, low resolution measurementsof band absorptivity are used instead of the line intensitydatta. The details of the computation scheme hare beendescribed previously by Manabe and Strickler [17].Hence only a brief outline of the computation methodis given here.In the thermodynamical equation, the heat sourceterm ma,? be separated into three parts, i.e.:where qSR and qLR are the heating due to the absorptionof solar radiation and of long-wave radiation, respectively;and is t,he heating due to condensation. (In thisstudy we do not incorporate t,he effect of qc explicitly.Instead, we adopted the simple process of moist-convec-tive adjust,ment which WAS described in section 2D.)qLR is computed by use of the following equation:where F4 is the net upward flux of long-wave radiationat level Q and is obtainable as a function of the verticaldistribution of absorbers and temperature. qSR is rep-resented b\7:where 8, is the downward insolation at level Q, andRS, is the reflected upward solar radiation at the samelevel. In order to avoid the complication in the heatbalance of the earth's surface due to the dnily variationof solar insolation, it was assumed that the zenith angleof the sun is constant with time. The effective meancosine of the zenith angle 5 is comput,ed by732 MONTHLY WEATHER REVEW Vol. 93, No. 11$4".-".-+-.--.-..... Bndy......... C-+............................................................................................E nv.""v,T, x-2 dP ,074- d 3 .189dt -336"""""""""-p/p*~4""--""5 .5006 .664""""8 .9 26I_ ""Z (std 131.60 km.a18.0012.008.305.503.301.700.64....................................................991 0.071.Ooo 0.00................... - P/P* '0, P* d dtPRANDTL BOUNDARY LAYER THEORY......................................~j\11,......................................? I \ FIGURE 2F2.-Diagram of Q levels and their approximate heights.;I Notations are made as to where the model variables are predicted.Pole, Ij,I"were performed showing one quarter of the stereographic projec-tion plane of the Northern Hemisphere (Ar=20). The free-slip,insulated boundary (Bndy.) is located one-half-grid-pointdistance outside the envelope (Env.) of the southernmost gridpoints in the Northern Hemisphere, not including the four pointson the Equator (Eq.) at the intersections of the coordinate axes.The absorptivities of gaseous absorbers under differentpressures, and the reflectivities as well as absorptivitiesof high, middle, and low clouds adopted for this studyare described in the paper by Manabe and Stricklcr (171.Refer to the same paper for the climatological verticaldistributions of water vapor, carbon dioxide, ozone, andclouds. All the above and the surface albedo are speci-fied as a function of latitude only.F. COMPUTATIONAL SPACE MESHRefer to figure 2Fl for a diagram of one quadrant of thegrid including this boundary.The so-called "a-system" of Phillips [28] is used for thevertical coordinate; that is, dah levels are defined asbeing at specified fractions of the surface pressure P*rather than at fixed pressures. Furthermore, c0ntra.qto normal procedure with this u-system, the nine datalevels used in this model are not evenly spaced withrespect to pressure but are arranged so as to give maxi-mum resolution in pressure at the ext,remes of the atmos-phere, namely near the ground and in the stratosphere.The fractions used, denoted as Q values, are defined asn=d(3-220-,) (2F1)where 2k-1a= -18 9 (k=l, 2,., 9) (2mreported on here we have taken-N=20 which correspondsto an earth distance between grid points of about 320 km.at the equator, 540 km. at 45', and 640 km. at the pole.In order to restrict this study to the Northern Hemi-sphere an insulated, free-slip boundary is constructed justone-half-grid-point distance outside the envelope of thesouthernmost grid points in the hemisphere. An excep-tion is made at the coordinate axes where the boundaryis one-half grid point inside the hemisphere. ThisTABLE 2F.-Q levels used in modelLevell 0 I Q I apDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 733hadway between the u values corresponding to fulllevels.A diagram of these Q levels and their approximateheights is shown in figure 2F2. The horizontal com-ponent of the wind, temperahre, and w are predicted onfull levels whereas 0 is computed on half levels. Surfacepressure and temperature are forecast on level 93: whereQ=l.O.3. INITIAL CONDITIONS AND TIME INTEGRATIONThe initial condition adopted for the time integrationis a resting isot-hemal atmosphere at 289OK. In orderto save computation time, the integration from 0 days to78 days was performed by use of a coarse grid system(..V=lO); further integrations were then continued byuse of the her grid system (N=20). Refinement of thegrid was performed by linearly interpolating wind velocity,temperature, and pressure. During the initial period ofinvestigation, a Hadley regime emerges. A circumpolarwesterly vortex develops in the upper part of the atmos-phere and an easterly vortex appem near the earth'ssurface, while a single cell meridional circulation pre-dominates. At 78 days of the N=10 calculation themeridional temperature gradient of 45OC. does not yieldbarocliiic instability because the horizontal resolutionis inadequate. However, about eight days after therefinement to N=20 (a total lapse time of 86 days), wavesstart to develop in the surface pressure pattern, indicatingthe beginning of baroclinic instability. In the followingtwo weeks, westerly waves are formed and the polarHigh penetrates into low latitudes of the subtropics atvarious longitudes and forms several subtropical Highs.Meanwhile, the westerly flow extends to the earth'ssurface in middle latitudes and the single Hadley cellevolves into a threwell meridional circulation.The erolution from an isothermal atmosphere into thestratosphere-troposphere system is very similar to theresults of the time integration preformed by ,l-lanabe andStrickIer I171 without the effect of largwcale motion.Refer to figure I of their paper for further details of thisevolution. Usually, it takes 200 days before a state closeto thermal equilibrium is reached. However, the clouddistribution of the model was changed at 176 days of ourtime integration in order to bring it to closer correspond-ence to the real atmosphere. It was tEherefore necessaryto continue the computation to 300 days by which timestatistical equilibrium was well established. Figure 3.1shows the time series of the hemispheric mean temperaturefor the period from 175 to 300 days. The 70-day periodof 231-300 days wm adopted for our detailed analysis.Based upon the time series of temperature shown infigure 3.1, it may be concluded that we reached a stateclose enough to the state of thermal equilibrium beforethe beginning of this period.In order to evaluate the degree of convergence towardthe state of quasi-equilibrium, the hemispheric mass-weighted integrals of various quantities other thanIIIIIIIIIIIIIIIIIII11111180 IW 200 210 210 230 140 150 260 270 280 290 300DAYFIGURE 3.L-Time variation of hemispheric mean temperature forthe period of 175-300 days.13 H2 "' izl,I.IcI.I.I.I.I240 m ia 2l8 2tn m 011 3oEFIQIJRE 3.2.-1n the upper part of the figure, the solid line showsthe time variation of the hemispheric .mean of kinetic energy forthe period of 231-300 days, and the long-dashed and short-dashedlines show the time variation of the hemispheric mean of con-version and of dissipation, respectively. In the middle part ofthe figure, the time variation of total available potential energyand pss static stability are shown by solid and dashed linesrespectively. The rate of generation (source) and conversion ofavailable potential energy are also shown by solid and dashedlines respectively. In the lower part of the figure, the solid lineand long-dashed line show the time series of the hemisphericmean of absolute angular momentum and of relative angularmomentum for the period of 231-300 days, respectively. Theshort-dashed line shows the time variation of the hemisphericmean of surface torque.temperature were also obtained for each time step in theintegration. It turned out that these integrals are veryuseful as indicators of the consistency of the model.The upper part of figure 3.2 shows the time variation ofkinetic energy, that of conversion from potential tokinetic energy, and that of dissipation of kinetic energyfor the 70-day period chosen for our analysis. Sincethe magnitude of the conversion term varies violentlywith time in the model, the variation with period shorterthan 1 day is removed by a rlrnning mean. Vote thevariation of kinetic energy is consistent with the difference734MONTHLY WEATHER REVIEWVol. 93, No. 12TABLE 3.1.-Hemispheric budget of kinetic energy jot 931-3000-day periodM- conversion of potential energy.. . . - __. - - - - - - - - - - - - - - .... - ._ .. 2901 erg cm.-r sec."Mean dissipation of kinetic energy _.._________.____...-..........-. 2939 erg an.-* sx-1Mean rate of change of kinetic energy. ___________..._.........--..- 4 erg cm.+ sx-1between the dissipation and conversion. Also, we com-puted the budget of kinetic energy for this 70-day periodby performing the time integration of conversion anddissipation. Table 3.1 shows the result. These resultssuggest that the hemispherical integral of the contribu-tion of the inertial terms almost vanishes as it shoulddespite the truncation error of our time integration.In the middle part of figure 3.2, the time variation ofavailable potential energy as defined by equation (AIIS),of the net generation and of conversion as defined byequations (AII11) and (AII10) are shown. According tothis figure, the rate of net generation of available potentialenergy varies little with time partly because cloudiness isindependent of tune in the present model. Again, thedifference between conversion and net generation approxi-mately corresponds to the variation of available potentialenergy which is out of phase with that of kinetic energy(and gross static stability). Table 3.2 shows the budgetof available potential energy during the 70day period.These results suggest that, in our finite-difference system,the budget of avadable potential energy is consistentlymaintained.In the lower part of figure 3.2 the time variation of thehemispheric integral of absolute angular momentum andthat of surface torque are shown. Again, we notice thatthere is a rather good correspondence between the changeof absolute angular momentum and surface torque, thoughthe correspondence is not as good as in the case of thebudget of kinetic energy. The budget of absoluteangular momentum for this 70day period is tabulated intable 3.3.This result suggests that there is a fictitious source ofangular momentum in our model. If one compares itsmagnitude with the latitudinal distribution of surfacetorque shown in figure 6A2, one finds that the relativemagnitude of this fictitious source is practically negligible.The mean value for the area of negative torque is about3.5X108 dyne/cm., whereas the magnitude of fictitioustorque is 5 X loe dynelcm. Therefore, we shall disregardthis discrepancy in further discussions.4. STATE OF QUASI-EQUILIBRIUMIn this section, detailed descriptions of the state ofquasi-equilibrium obtained from the time integration aremade. As we mentioned in the previous section, the 70-day period of 231-300 days was chosen for this analysis.Figures and numerical results which are discussed in thisand following sections were obtained for this period exceptTABLE 3.3.-Hemiaphetie budget of absolute angular momentum for 851-303Oo-day periodfor the case specified otherwise. Da were stored oncea day and averaged.Since on the stereographic projection adopted for ourtime integration few grid points lie on common latitudecircles, it was necessary to perform a linear interpolationto obtain the zonal mean of any quantity at a given lati-tude. In order to avoid the error produced by interpola-tion, the zonal mean of product terms such as the transportof angular momentum, heat, and kinetic energy, which arediscussed in the following sections, were computed by theprocesses described below.(1) Compute the product terms using the same finite-difference method as was adopted for the time integrationof the model.(2) Obtain the average value of the products thus com-puted for each of a number of narrow latitude belts.A. ZONAL MEAN TEMPERATUREIn figure 4A1, zonal mean temperatures obtained atthree latitudes from the numerical integration of ourmodel are shown on an adiabatic diagram together withthe observed annual mean temperatures. According tothis comparison, the general features of the stratosphere-troposphere system are simulated very well by our generalcirculation model. The pole-to-equatorial difference ofthe height of the tropopause in the model is about 7 km.which is somewhat smaller than, but not far from, theobserved difference of about 10 km. In the numericalintegration, a stable layer appears at high latitudes becauseof the stabilizing effect of baroclinic waves, northwardadvection of heat by the large-scale eddies, and the largealbedo of the polar cap. Recently, Manabe and Strickler[17] computed the vertical distribution of the temperatureof the atmosphere in thermal equilibrium for variousalbedos of the earth's surface. (Refer to fig. 11 of thatpaper.) Although they performed the computation forthe case of large albedo, the temperature inversion on theearth's surface did not appear. These results suggestthat not only the large surface albedo but also the effectof large-scale motion, particularly that of the horizontaladvection of heat by the large-scale eddies, is indispensablefor the maintenance of the polar inversion. Though thestable layer of the polar region, which is obtained in thepresent computation, is in qualitative agreement with theobserved features, its static stability is less than that ofDecember 1965 J. Sagorinsky, S. Manabe, and J. L. Holloway, Jr. 735FIGURE 4Al.-The vertical distributions of mnal mean temperatureobtained from our computation for various latitudes and that ofthe actual atmosphere are shown on the left- and rightihandsides of the figure, respectively. Light solid line marked SAindicates the ICAO standard atmosphere..ow \ '.""_ """"> "" 23 "-3022Lthe actual atmosphere. Further study is desirable beforewe can decide on the exact cause of this discrepancy.Figure 4A2 shows the latitude-height distribution ofzonal mean temperature obtained from our model com-pared with that of the actual atmosphere. In general,the similarity between the two distributions is remarkable,particularly near the earth's surface and in the upperstratosphere. In our model, the pole-to-equatorial tem-perature difference is about 42O C. at the earth's surfaceand coincides well with the observed difference. How-ever, in the upper troposphere of middle latitudes, thelatitudinal gradient of the temperature of our model ismuch larger than observed, and this is one of the majordiscrepancies between our model and the actual atmos-phere. Tn the lower stratosphere of the model, t8hetemperature increases by about 14' C. from the equatorto middle latitudes, but decreases again with a furtherincrease of latitude with the result that the net increase oftemperature from equator to pole is only 5' C. It isencouraging, however, that we get significant latitudinalincrease of temperature in the lower latitudes. At thetop model level in the upper stratosphere (f/P*=0.009),the decrease of temperature from equator to pole is about28' C. which is close to the observed decrease of annualmean temperature (24' C.). In general, the thermalstructure of our model atmosphere looks very similar tothat of the observed winter atmosphere in spite of thefact that the annual mean solar insolation was given.In their study of thermal equilibrium, Manabe andStrickler [17] obtained the state of the radiative equilib-rium of the stratosphere overlying the troposphere witha realistic distribution of temperature. (Refer to fig. 13of their paper.) According to this figure, the equilibnumtemperature of the stratosphere increases slightly witahincreasing latitnde for a July case, but the equilibrium792-51'7 O"tX5-2nW \.YY IFIGURE 4A2.-The Istitude-height distribution of the zonal meantemperature obtained from our computation and that of theactual atmosphere are shown in the upper and lower parts of thefigure, respectively. The observed distribution is bsed on theresults obtained by Peixoto [26] for the troposphere and by 001%1241 and Kochanski [IO] for the stratosphere. The tropopause isindicated by a heavy line broken in middle latitudes.temperature obtained for January decreases monotonicallywith increasing latitude. Therefore, the decrease oftropopause height with increasing latitude is si@cantlyless t.han that obtained from the present study. Theimprovement accomplished by the present study is dueto the incorporation of the effect of large-scale motion.Refer tosection 5 (Heat Balance) for the discussion of thiseffect .B. MEAN FLOW FIELDIn figure 4B1, the zonal mean of zonal current obtainedfrom the present computation is compared mth that ofthe actual atmosphere. The data obtained by Buch [4],and Oort 1241 are used for determinmg the observedannual mean. The distributions for January and Julyare taken from the results obtained by Kochanski [lo] for80' W. longitude. According to this comparison, thelatitude of maximum west wind in the upper troposphereof the model coincides well with the observation. Theintensity of the jet, however, is much stronger than theobserved annual mean. By the thermal wind relation-ship, this result is consistent with t,he excessive latitudinal736 MONTHLY WEATHER REVIEW Vol. 93, No. 1 PFIGURE 4Bl.-In the upper left, the zonal mean of the zonal current of the model atmosphere obtained from our computation is shownIn the lower left, the annual mean of the observed wind obtained by Buch [41 and 001% (241 is shown. The intensity of the zonalcurrent at 80' W. obtained by Kochanski [lo] for both January and July is shown in the upper right and lower right, respectively. (Units: m./sec.)gradient of temperature in the upper troposphere of themodel atmosphere which was pointed out in the previoussection. A similar tendency appears in the results ob-tained by Smagorinsky [37] and Phillips [27]. One of thereasons may be the absence of a hydrologic cycle in themodel. We shall discuss this possibility in a companionpaper [ 181.Another possibility is that the observed annual mean isderived from an atmosphere subjected to the annual vari-ation of solar radiation including a polar night. On theother hand, our calculations represent the response to theannual mean solar radiation. There may be a fundamentalfallacy in comparing these two. It is noteworthy that themaximum intensity of the annual mean zonal current ismuch less than the average of the maximum intensity at80' W. in July and in January. In general, the computeddistribution is close to the observed distribution in Jan-uary though the tilt of the axis of the maximum westerliesin the stratosphere is much less than observed. In thetropical upper strat.osphere of the model atmosphere, aweak easterly current appears. Its intensity, however, ismuch weaker than the observed intensity which prevailsin tropical latitudes. Refer to figure 4C2 for the longi-tudinal distribution of these easterlies.Next, we shall examine the structure of the mean merid-ional circulation, since it plays a major role in the generalcirculation of the atmosphere. In figure 4B2, are shownthe zonal mean of the meridonal wind component and thatof the vertical wind component obtained from our model.According to this figure, a three-cell circulation appears inthe troposphere. In the stratosphere, the circulationtends toward two cells with increasing altitude; that is,t.he direct polar cell is squeezed out by a poleward expan-sion of the equatorial Hadley cell. Accordingly, the axisof maximum meridional velocity tilts northward in thestratosphere. Mintz and Lang [19] computed the dis-tribution of the meridional component of the wind for thetroposphere based upon an angular momentum balanceDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 737.009 ,- 30In I0991FIQURE 4B2.-The zonal mean of the vertical component of thewind (cm./sec.) and of the meridional component of the wind(m./sec.) of the model atmosphere are shown in the upper andlower prrrta of the figure, respectively. Positive values are upward and northward.requirement. More recently, Teweles [41] applied a verysimilar technique to the stratosphere. Figure 4B3 showsthe combined results. The axis of the maximum merid-ional component of the wind also tilts northward in thestratosphere, however the degree of tilt is somewhat largerin the actual at,mosphere than computed in our model.(Refer also to the distribution of vertical motion obtainedby Miyakoda 1211 for winter.)Another characteristic feature of our results is the nar-row belt of strong meridional wind component near theearth's surface. A somewhat similar belt is noticeablein the results of Mintz and Lang 1191. As the result ofthis phenomenon, the meridional circulation cell ob-tained by the model is highly eccentric. It is clear fromour study of the momentum balance in section 6, that theeccentricity of the meridional circulation depends uponthe assumed distribution of t,he vertical mixing coefficient.This is because the change in angnlar momentum in theEkman boundary layer resulting from the surface torquemust mainly be compensated by the change of relative90' 8il" 70" 60" 50" 4b" 30" 20' lb. 0" "LAmlFIGURE 4B3.-The meridional component of the wind (m./sec.)obtained from the requirement of momentum balance of theactual atmosphere. The distribution in the stratosphere (30-mb., 50-mb., and 100-mb. level) was obtained by Teweles [41]and that in the troposphere was obtained by Mintz and Lang(191.angular momentum due to the meridional circulation.Further study of observat,ional data is necessary for amore definitive evaluation of the simulated meridionalcirculation.C. SYNOPTIC MANIFESTATIONSFigure 4C1 displays an example of t.he synoptic distri-bution of temperature, pressure, and wind in our modelatmosphere. In this figure the synoptic distributions ofthe geopotential lines and isotherms are shown by solidand dashed lies respectively, and the areas of southerlyflow are shaded. These shaded areas or the geopotentialgradient at these levels remind us of the actual situa-tion which prevails during the winter. According tofigure 4C2, which shows one example of the calculatedsynoptic distribution of the zonal wind component atPjP* =0.009, a. very strong zonal wind develops in middlelat~itudes. On the other hand, patches of easterly windappear at low latitudes. According to figure 4B1, thecalculated zonal mean easterly wind is much weaker thanobserved. It is encouraging, however, that some easterlywind even appears in the simulated tropical stratosphere.In figure 4C1, the msps of the tropospheric levels showthe same features as those observed in the actual atmos-phere: for example, the tilt of the troughs with respectto the meridians and the lag of the thermal trough behindthe trough of geopotential height. This is obviouslynecessary for a correct transfer of angular momentumand heat. Accordhg to figure 4C3, the occlusion ofisot,herms into low centers develops at various places.Also at a low level a high pressure belt develops in thesubtropics as is found in the actual atmosphere. Figure4cQ shows the zonal mean of surface pressure in ourMONTHLY WEATHER REVIEW738Vol. 93, NO.FIGURE 4CI.--In each map, the solid and dashed lines show the contour heights of isobaric surfaces (every 100 m.) and isotherms (every 5O C.) on the 259th day, respectively. The areas of eoutherly flow are shaded.December 1%5 J. Snagorinsky, S. Manabe, and J. L. Holloway, Jr.739FIGURE 4C2.-Distribution of the intensity of zonal wind at thefirst model level (P/P*=O.OoQ) on the 259th day. The contourinterval is 5 m./sec. The areas of easterlies are shaded.FIQURE 4C3.-Surface isobars (every 5 mb.) and surface isotherms(every 5* K.) on the 259th day are shown by solid and dashedtines, respectively.model? The latitude of the subtropicd High and thatof the lowest zonal mean pressure coincide well withobservation. The magnitude of latitudinal variation ofpressure turned out to be somewhere between the observedvalue of the Northern Hemisphere and that of themrved because the model has the same mm of air as the sctual atmoJphere but it hasThe mesn srufece pressnm of the model stmoaphen? la abaut 28 mb. Iomr than obno monntafns.IIunm'./'990FIGURE 4C4.-The latitudinal dist.ribution of the zonal mean Ofsurface pressure. Computed distribution is shown by a thicksolid line (scale on right), and observed distributions for theYorthern and Southern Hemispheres are shown by thin solidand thin dashed linea, respect'ivelg (scale on the left).Southern Hemisphere. Further improvement in thesimilitude of the surface pressure field may be accom-plished by considering global motions and admittingland-sea contrast. Recently, Mintz [20] successfullysimulated the distribution of the zonal mean of surfacepressure in both hemispheres by taking into considerationthe effect of the land-sea distribution.The synoptic distributions of vertical motion are shownin figure 4C5. Above the level of the tropopause, theintensity of the vertical motion decreases sharply withincreasing height. (Xote that the contour intervals arenot the same for all maps.) Some features of the large-scale vertical motion in the troposphere, however, areidentifiable in the stratosphere despite the strong damping.D. HARMONIC ANALYSISIn figure 4D1 the energy spectra of the calculated andobserved meridional and zonal components are shown.As indicated bp the synoptic charts, the meridionalcomponent of the wind in our model atmosphere has amaximum energy at, wave number 6 in t,he troposphere,and this wave number of maximum energy decreaseswith inrreasing altitude in the stratosphere. As theright-hand side of the figure indicates, the same wavenumbers prevail in the actual troposphere. In the tropo-sphere the energy spectrum of t,he zonal component of thewind of our model atmosphere has relatively largevalues at wave numbers from 1 to 4, whereas a sharpmaximum exists at wave number 1 in the actual atmos-phere. This discrepancy is partly a result of ignoring740MONTHLY WEATHER REVIEWVol. 93, No. 12P-= .034 SCALE: 2MB/DAY--H = .259 SCALE: 100 MB/DAYpx- H - .126 SCALE:.20 MB/DAYE-%---H = .583 SCALE: 100 MB/DAYp*FIGURE 4C5.-The distribution of Q on the 259th day. The areas of positive (downward motion) are shaded. Note that the scale (contour interval) is different for different levels..009 .9912 k 5 810121 TEWELESWSMVfDwart. IMR Awart. I~LR +FIGURE 4Dl.-The vertical distribution of the energy spectra ofthe zonal and meridional wind components are shown by solidand dashed lines, respectively. The computed values are hemi-spheric means and the observed data are the average between15" N. and 80' N. The numbers beside the computed distribu-tions are PIP* ratios.the effects of the land-sea distribution and the effectsof mountains. If the flow circling around the pole hassome eccentricity due to the lack of uniformity of theearth's surface, the eccentricity of the zonal current isregarded as an eddy of wave number 1. It is interesting,however, that the energy spectrum of t:he meridionalcomponent of wind is quite different from that of thezonal components of wind for both the actual atmosphereand the model. In the model, the energy in low wavenumbers is probably transferred from that in higher wavenumbers by the non-hear interaction between waves.In order to examine the vertical variation of wavenumbers further, the hemispheric mean effective wavenumber 'i at various altitudes is computed for the modeland for the actual atmosphere. ii is defined as follows:?;=(f.dEc4)/(fdE(d) (4D)where E(n) is the energy spectrum. The spectra obtainedby Teweles [41] for the period from July 1957 to June 1958are used for obtaining ii for the actual atmosphere.Figure 4D2 shows the results. The ?i of the meridionalwind component of the model is close to observation, theDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Hoiloway, Jr. 741400c V COYIONENT 'II1 Iit 144ej U CDMPOllfWT i..I -6i 7WAVE NUMBER +FIGURE 4D2.-The vertical variation of energy-weighted wavenumber Ti. Dotted, dashed, and solid lines show the verticaldistributions of the effective wave number Fi of the meridionalwind component (hemispheric mean), of the zonal wind com-ponent, and of the total wind component, respectively. In thelower part of the figure is shown the vertical variation of Ti com-puted from the results of the harmonic analysis of the observedwind field, which was performed by Teweles [41] for the periodJuly 1957 to June 1958. The domain of averaging ranKes from15O N. to 80' N.difference being only Xi of a wave number. On the otherhand the 5i of the zonal wind component of the model islarger than that of t,he actual atmosphere by about. onewave number. The general height dependence of E,however, is simulated very well. As we shall show later,the kinetic energy of the stratosphere is maint.ained byenergy from t3he troposphere supplied mainly by the pres-sure int,eraction term. According to Charney andPedlosky [5], the damping of the pressure interactiont.em with altitude is proportional to the st,ability of thelayer, to wave number, and to the degree of baroclinicinstability. Thus t.he theory is consistent with the factF~GURE 4D3.-The latitudinal variation of the effective meanwavelengths of the zonal wind component and of the meridionalwind component are shown by solid and dashed lines, respectivelg.that long waves predominate in the stratosphere. Thissubject will be discussed further in the section on thebalance of kinetic energy.For reference, the latitudinal variation of the energyspectra is shown in figure 4D3. For convenience in com-parison, the spectrum is plotted versus wavelength in-stead of wave number. In high latitudes the eddy kineticenergy of the actual atmosphere is much larger than thatof our model atmosphere. Again, this discrepancy maypartly be due to the eccentricity of t.he circumpolar vortexin the observed at,mosphere. It is noteworthy that thecharacteristic wavelength increases slightly with decreas-ing latitude for both the act,ual atmosphere and our model.E. DlSTRlBUTlON OF KINETIC ENERGYIn this sec,t,ion we shall examine the latitude-heigh tdistribution of eddy kinet,ic energy. Figure 4E1 showsthe vertical distribution of the hemispheric mean eddykinet~ic energy of om model atmosphere. In the samefigure we also show values of the hemispheric mean eddykinetic energy for the ac,tual atmosphere obtained byvnrious authors. According t~ this comparison, the ob-served values are mnch larger than the c.alculat.ed ones.Although t,hep may be somewhat overestimated becauseof the geostrophic assumption, there must, be ot,her reasonsfor this discrepancy. The level of the marrimurn eddykinetic energy lies a,t, approximately 200 mb. in the modelatmosphere and cointides with that. in the actualatmosphere.The latitude-hei9ht~ distribution of eddy kinetic energyis shown in figure 4E2. The lstit,ude of the maximum ofeddy kinet,ic energy falls approximately at 40 X. andcoincides with that of the maximum zonal velocity. InfiFwe 4E3 the latitudinal dist.ribution of eddy kineticenergy at the 500-mb. level obtained by Salt,zman [35], iscompared w6t.h our results. In high lat,itudes t.he eddykinetic energy of the act.urtl atmosphere is much Ia.rFer742 MONTHLY WEATHER REVlEWVol. 93, No. 12WEa0m:P? ,SALTLIAN *WIM - NILLSCN 0lo00 1 1 I00.5 1.0 1.5 2.0 2.5 3.0x 10" jwlt/ca2, nb$'IGURE 4El.-The vertical distribution of the hemispheric mean ofeddy kinetic energy is shown as a function of altitude. Thearea mean of eddy kinetic energy obtained by Wiin-Nielsen [431,Teweles [41], and Saltzman [35] for the actual atmosphere are alsoplotted for the sake of comparison. The latitudinal ranges of thearea means for these studies [43,41, and 351 are 16.75'-88.75" N.,15-800 N., and 15O-80" N., respectively.UTlTllDLFIQURE 4EfL.-Latitud~~height distribution of eddy kinetic energy obtained by the model.-I-,/' \c. 0'`. 1I e"""% AA""" ""-I I I Im I M y1 4a 30 m 100UTlTuMFIGURE 4E3.-The latitudinal distribution of eddy kinetic energyat the 5Wmb. level. The annual mean value obtained by Salt-[35] for 1951 is also plotted for comparison.DAYSFIGURE 4E4.-1n the top of the figure the time variations of totalkinetic energy, of zonal kinetic energy, and of eddy kinetic energyfor the period of 231-300 days are shown by short dashed,dashed, and solid lines, respectively. (Units: joule/cm.*) Inthe center, the time variation of latitudinal distributions of bothzonal and eddy kinetic energy are shown (units: joule/cm.z).At the bottom, the time variation of the vertical distribution ofeddy kinetic energy is shown. The units are joule cm.-*mb." (Hemispheric mean).than that of our model probably because of the irregu-larity and the eccentricity of the circumpolar vortexcaused by the kinematic and thermal asymmetries of thelower boundary, i.e., land and sea.The upper part of figure 4E4 shows the time variationof the total kinetic energy, the zonal kinetic energy, andthe eddy kinetic energy. As expected, during the 70-dayperiod chosen for our extensive analysis, the variation ofzonal kinetic energy is mainly out of phase with that ofDecember 1965 J. Snagorinsky, S. Manabe, and J. L. Holloway, Jr. 743TOTAL AVAILABLE POTENTIAL ENERGYTOTAL.! .b l!O 112 l!4 1!6 L'SJWLESICM~,MBZONAL AVAILABLE POTENTIAL ENERGY.~Ct.m-L@?O .: .! .: .d l!O 1!2 114 1!6 118yJOULE SIC^, MBEDDY AVAILABLE POTENTIAL ENERGY. loo.#x).m--.6uJ.m-.m.m- I-IEDDYm P2 D4 96 ps .lo -12 .I4JWLESICM~, MBFIGURE 4Fl.-The vertical distribution of total, zonal, and eddy available potential energy are shown in the left, center, and right side of this figure, respectively. The corresponding distributions obtained by Wiin-Nielsen [43] for April 1963 and the whole pear of 1962 are also shown.eddy kinetic energy. Since the variation of eddy kineticenergy is much larger than that of the zonal kinetic energy,the time variation of the eddy kinetic energy and that ofthe total kinetic energy are very similar. Although themagnitude of the kinetic enerm varies with a period ofapproximately two weeks ns Smagorinsky [37] pointed outin his earlier work, it is not possible to find a distinctperiod. In the center of the figure, the time variation ofthe lat,itudinal distribution of zonal and eddy kineticenergy are shown. The latitude of maximum energychanges wit,h time. The range of fluctuation, which isseveral degrees of latitude, is somewhat, smaller than therange observed in the actual atmosphere.In the lower part of this figure, the time variation ofthe vertical distribution of eddy kinetic energy is shown.This result shows that, t'he level of maximum eddy kineticenergy hardly changes with time during this portion ofthe numerical simulation.F. DISTRIBUTION OF AVAILABLE POTENTIAL ENERGYAccording to Lorenz [15], simplified versions of thetotal, the zonal, and the eddy available potential energymay be defined as the mass integral of AT, A,, and A,defined as follows:where ( )" denotes the deviation of the temperature onan isobaric surface from its hemispherical mean, and792617 0-65- 3where 6 is the potential temperature of ths air and Poodenotes the base pressure, 1000 mb., and ( ) andn de-note the hemispheric mean and zonal mean respectively.In figure 4F1, the vertical distributions of total, zonal,and eddy available potential energy obtained from ourcomputation are shown together with those computedfrom actual data by Wiii-Nielsen [43]. In general, thezonal available potential energy obtained from our modelis much larger than that of the actual atmosphere (annualmean value). This large available pot,ential energy isconsistent with the very large latitudinal temperaturegradient of t,he upper troposphere which appeared in ourcalculation (see section 4A). On the other hand, theeddy available potential energy of our model atmosphereis significantly smaller than that of the actual atmosphere(annual mean value). This discrepancy may result partlyfrom the lack of mountains and of land and sea contrast.In otjher words, the asymmetries of the lower boundarycreate the eccentricity in the latitudinal gradient of tem-perature and alter the partitioning of the available po-tential energy in favor of the eddy potential energy.Furthermore, the magnitude of the energy components issensitive to the horizontal resolution of the model andthe magnitude of the subgrid-scale mixing coefficient.One must also expect, that the eddy amplitudes dl de-pend on whether or not heat may be transferred in latentas well as sensible form.The level of maximum zonal available potential energylies at about the 350-mb. level, which issomewhat higherthan in the real atmosphere (400 mb.). Again, this is-Ii744 MONTHLY WEATHER RWEW Vol. 93, No. 14consistent with the model error in the latitudinal temper-ature gradient. The maximum eddy available potentialenergy lies at approximately 400 mb., which is reasonablyclose to the pressure of the observed maximum.5. HEAT BALANCEThe study of heat balance of the atmosphere has beenperformed by many authors (e.g. Houghton [7], London[14], Ohring [23], Manabe and Moller [16], and Davis [SI).In this section, detailed investigation of the heat balanceof the model atmosphere is made and compared with thatof the actual atmosphere. Based upon this comparison,the causes of the coincidences or discrepancies of thevarious features of the thermal structure of the modelatmosphere with the observed features are discussed.A. HEMISPHERIC MEAN OF HEAT BALANCE COMPONENTSThe hemispheric means of various heat balance compo-nents at the top and bottom of the model atmosphere arecompared in table 5A with those obtained by London [14]for the actual atmosphere. The coincidence betweenthem is encouraging and is the natural consequence ofthe successful simulation of the temperature field as awhole. Note that all the net radiation received by theearth's surface is transferred to the atmosphere in theform of sensible heat because of the lack of evaporation.TABLE 5A.-Hemispheric mean of heat balance components (unita are ly./min.)-Model Actualatmosphere atmosphereNet solar radintion ._________ -. 237 -. 232Net solar radiation __________ .324 .329Net long-wave radiation-.". "0.324 "0.328Earth's surface.. __________ Net long-wave radiation _____ . OB1 .ow) Turbulent energy flux _______ .147 ,154Top of the atmosphere------B. LATITUDINAL DISTRIBUTION OF HEAT BALANCE COMPONENTSThe latitudinal distribution of net upward radiativefluxes at the top of the atmosphere and those of net upwardradiative fluxes and turbulent energy fluxes at the earth'ssurface are shown in figure 5B1. For the sake of compari-son, the corresponding quantities obtained by London [14]for the actual atmosphere are plotted in the same figure.The general agreement between the net fluxes of our modeland those of the actual atmosphere is very good. In figure5B2, the northward flux of energy expected from theradiative imbalance of the earth-atmosphere system isshown together with the northward fluxes obtained byHoughton [7] and London. According to this figure, thetot,al northward transport of energy of our model turnedout to be somewhat smaller than the annual mean fluxobtained by either Houghton [7] or London [14] mainlybecause of the slight difference in the latitudinal gradientof net outgoing radiation. As we described in section 4A,.5[IISR)y.2 .'iFIGURE 5Bl.--The upper half of the figure shows curves of thelatitudinal distribution of net upward long-mve radiation(NLR)T and that of net downward solar radiation (NSR)T atthe top of the atmosphere. The lower half of t,he figure showsthe latitudinal distribution of the net upward long-wave radia-tion (NLR)ER, of net upward solar radiation (NSR)ER, and ofthe upward turbulent flow of heat (SHFX)ER at the earth'ssurface. The corresponding quantities obtained by London [14]are plotted in the same figure as stars, dots, and triangles forcomparison.the latitudinal gradient of temperature obtained from ourmodel is significantly larger than the annual mean gradientof the actual atmosphere and this is responsible for thepresent discrepancy.In the lower part of figure 5B2, the northward eddyflux of heat obtained from our model is shown togetherwith those obtained by Starr and White [39] for theannual mean and Peixoto [26] for winter. According tothis comparison, the computed eddy flux is significantlylarger than observed. This result does not necessarilycontradict the results shown in the upper part of thisfigure. Since the effect of the northward transport oflatent energy is not incorporated in this model, it may bereasonable to have a large eddy flux of sensible heatdespite the relatively small energy flux required fromradiative imbalance. This will be discussed further ina companion paper 1181 on general circulation simulationswith a simple hydrologic cycle.The latitudinal distributions of various heat balancecomponents in the atmosphere are plotted in figure 5B3.In general, the heating effect of convection from theearth's surface is mostly compensated for by the coolingDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 745 effect of radiation. The remaining imbalance is takenA LlBaA flI)EmlA A up by the effect of large-scale motions. In the Tropics,1.0 the cooling resulting from the meridional circulation plays4L* ..asX"A AA an important role; in the middle latitudes the effect of the compensate each other; and in higher latitudes the heating produced by the convergence of eddy flux of.I- A ,.` ``.6 -A F"" meridional circulation and that of the eddies more or less.4 - / "\ \A of the large-scale motion is cooling at low latitudes and/ % A`\ AA 0 sensible heat predominates. Accordingly, the net effect/\A0\ heating at high latitudes. Note that the temperature \\ eddies involves the effect of adiabatic heating as well90- jr jb. 1$. io- The subgrid-scale transport of sensible heat plays a minor.2 - A/' \\\ \A change caused by the meridional circulation and by the/\0 as that of the convergence of sensible heat transport.UllWE role in the heat budget.IC. VERTICAL DISTRIBUTION OF HEAT BALANCE COMPONENTSIn ne 5C1 are shown the calculated vertical distri-butions of the hemispheric mean values of various heatbalance components. According to this figure, thestratosphere as a whole is in almost complete radiativeequilibrium while in the troposphere the hemisphericmean temperature is maintained as the balance amongconvective heat,ing, radiative cooling, and the s0mewha.tsmaller effect of larze-scale motions. In figure 5C2, theFIQURE 5B2.-1n the upper part of the figure, the northward flux ofthe total energy expect.ed from radiative imbalance is shown by adashed line. Also, the flux obtained by Houghton [7] and thatcomputed from the results of London [14] for the actual at-mosphere are plotted. In the lower part of the figure, thenorthward flux of heat due to large-scale eddies and that due tosubgrid-scale diffusion are shown by solid and dotted lines,respectively. The northward flux of heat due to large-scaleeddies in the actual atmosphere for wint,er (Peixoto [26]) andthe annual mean (Stan and White [39]) are also plott.ed.e5 .4c3 -..5 -.bFI~WRE 5B3.-The latitudinal distribution of temperature changedue to convection, radiation, meridional circulation (M.C.),Iarge-ecale eddies (EDDY), and horizontal subgrid-scale mixing(N.D.) are shown.Ivertical distributions of heat flux resulting from. themeridional circulation and large-scale eddies are showntogether wit.h the temperature change caused by adiabaticexpansion and divergence of heat flux. As one mightexpect from the theory of baroclinic instability, t,helarge-scale eddies transport heat upward except, in thestratosphere a,nd tend to stabilize the static stsabilityof the troposphere and counteract the net effect of radi-ation and convection. The hemispheric mean tempera-ture change caused by the meridional circulation isrelatively small.RATE Of TEMRRATUkf CHANGE e WYI -FIGURE 5CL"The simulated vertical distribution of the hemi-spheric mean of the temperature change due to radiation,convection, and large-scale motion (advection and adiabaticheating) are shown.746MONTHLY WEATHER REVIEW Vol. 93, No. 19.a09I I I I I I I II I I",\' 'RATE OF TEMPERATURE CHANGE P C/DAY)VERTICAL FLUX OF HEAT (JOULES/DAY) -+FIGUIW 5C2.-0n the left side of the figure are shown the verticaldistributions of the rate of net temperature change (on the con-stant -surfaces) due to adiabatic expansion and divergence ofheat flux by the meridional circulation (M.C.) and by large-scaleeddies (EDDY). On the right side, the vertical distributionsof heat flux on constant -surfaces due to meridional circulation(M.C.), large-scale eddies (EDDY), and to both (NET FLUX)are shown. Units: joules cm.-e day," right side. Upward fluxis positive.D. LATITUDE-HEIGHT DISTRIBUTION OF HEAT BALANCECOMPONENTSFigure 5Dl shows the latitude-height distribution ofnet temperature change caused by radiation. Com-paring this result with the corresponding distributionobtained by London [14] for the actual atmosphere, onecan find many common features, i.e., the rapid decreasewith increasing altitude of radiative cooling of the uppertroposphere, the relatively large cooling in the tropicaltroposphere, and the strong cooling in the low cloud layer.This strong cooling due to the existence of low cloud issomewhat exaggerated in our calculation since we assumedthe low clouds to be too thin. In the stratosphere weakheating occurs at low latitudes and cooling at higherlatitudes. Similar features appeared in the heat balancecomputations performed by Ohring [23], Manabe andM6ller [l6], Davis [6], and Kennedy [9] for the actualatmosphere. The rate of cooling in high latitudes forthe computed stratosphere is much smaller than thatobtained in any of these studies. The failure to get asufficiently warm stratosphere in higher latitudes (seesection 4A) is the major reason for this discrepancy. (Thecooling rate obtained by Ohring [23] and Davis [S] in thehigher latitude stratosphere is somewhat larger than thatobtained by Manabe and Moller [16] or by Kennedy 191mainly because of their assumption of a humid strato-sphere in higher latitudes.)In the middle and lower parts of figure 5D1 are shownthe latitude-height distribution of the rate of temperaturechange due to solar radiation and that due to long-waveradiation. In the top stratospheric level, heating resultingfrom the absorption of solar ultraviolet radiation by ozoneCB 4500.926*'" 90" 80" 70" 60" 50" 40" 30" 20" 10" 0"0UNTWE"1.0--1.5*e-.2Y10 =0UllfllOE' FIGURE BDl.-Upper, middle, and lower parts of the fiem givethe latitudeheight distributions of calculated temperature change(OC./day) due to the net rate of radiation, the Rolar radiationonly, and long-wave radiation only.is compensated for by the cooling caused by the long-waveradiation of carbon dioxide and water vapor. Aroundthe level of the tropopause, both the effect of solar radra-tion and that of long-wave radiation are small, and theyare in rather delicate balance. In the troposphere, theDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 747-074 1 IFIQURE 5D2.-The latitude-height distribution of calculated nettemperature change (OC./day) due to conduction from the earth'ssurface and to the moist adiabatic adjustment.,009 I I 1"7 IAI io ournFIQURE 5D3.-The upper part of t,he figure shows t.he latitude-height distribution of tshe calculat.ed poleward eddy transport ofheat on isobaric surfaces. In the lower part of the figure t,henorthward heat transport obtained by Starr and White 1391 forthe gear 1950 is shown. Unit's are 10x7 joule mb." day."cooling due to the long-wave radiation of water vtrporoutweighs the heating due to t,he solar radiat,ion absorptionby water vapor. The net radiative cooling in t,he t.ropo-sphere is compensated for by convective heat,ing originat-ing from the earth's surface. Refer to the paper bvManabe and Strickler [17] for the details of the verticaldistribution of the contributions of various gases.The latitude-height distribution of the rate of temper-ature change due to conduction at the earth's surface andmoist convective adjustment is shown in figure 5D2. Asone would expect, the thickness of the moist convectivelayer is deeper in the Tropics than in high latitudes. Thedepth of this layer is about 12 km. in the Tropics andmonotonically decreases with increasing latitude. In ourmodel, the moist convective adjustment is very active insubtropics where the observed rainfall is at a minimum.This unreasonable result is the direct consequence of theadjustment process which disregards the effect of relativehumidity on the moist convective process. We will beable to discuss this matter in more detail in the followingcompanion paper [l8] on the general circulation modelwith a hydrologic cycle.So far we have examined the latitude-height distributionof temperature change due to radiation and convectiveadjustment. Another important factor is the effect oflarge-scale eddies. Figure 5D3 shows the latitude-heightdistribution of the northward transport of heat both in ourmodel and in the actual atmosphere. In the real atmos-phere, the area of the maximum transport of heat appearsin the lower and upper part of the mid-latitude tropo-sphere, whereas our model has a maximum in the lowertroposphere but fails to have another maximum in theupper t,roposphere. In the tropical troposphere of ourmodel atmosphere, extremely weak southward transportappears. Recently Starr and Wallace 1401 pointed out theesistence of a. counter-gradient flux of heat in the t.ropica1troposphere. Since the magnitude of the negative fluxobtained from our computation is very small, t,his coinci-dence with observation mag not be significant.The latitude-height distribution of the calculat,ed verti-cal heat flux is shown in figure 5D4 toget,her with that ofthe actual atmosphere obtained by Jensen [SI for April.According to this figure, the upward flux of heat predomi-nates in the troposphere except in the Tropics where 8 veryweak downward flu.. of heat appears. In the stratosphereof our model atmosphere, the area of downward eddy heattransport occupies most of the area. Siar features arenoticeable in the results of Jensen [8].The net effect of these eddy fluxes of heat (including theeffect of adiabatic heating or cooling caused by eddies)upon t,he rate of temperature change is shown in figure5D5. In middle latitudes large-scale eddies make a signifi-cant contribution t,oward stabilizing t,he atmosphere. Ac-cording to our results, the height of maximum stgabfiationdecreases with increasing 1at.itude. In higher 1at.itudes itis very close to the earth's surface and contribut,es signifi-cantly to t,he maint.enance of the inversion in the lowert)roposphere of high latit,udes. Despite this st,abiiizingeffect, the calculated area of stable layer in high latitudesis much smaller t,han the observed area. The existenc,e ofa relatively warm sea surface around the polar cap may bepartly responsible for t,he maintenance of the strong inver-748 MONTHLY WEATHER REVIEW Vol. 93, No. $9a* -074-b+ .183-Ai364.926.991 90"80" 70" 60" 50" 40" 30" 20" 10" 0"0LATITUDE-uuY I I- 30+ .189 .33680' 70" 60" 50" 40" 30' 20"LATITUDE110"AFIGURE 5D4.-The upper part of the figure gives the latitude-height distribution of the calculated vertical eddy flux of heaton isobaric surfaces. The lower part gives that obtained byJensen [SI for the actual atmospheric transient eddy componentfor April. Units are joule cm.-* day.-*sion in the actual winter atmosphere. Further study onthis subject is desirable.Next we shall examine the heat balance of the strato-sphere. Figure 5D6 shows the latitudinal distribution ofthe rates of temperature change due to various heatbalance components at model level 2 (P/P*=0.074) inthe stratosphere. According to this figure, the directmeridional circulation in lower latitudes contributes tothe latitudinal increase of temperature there. The large-scale eddies transport heat northward (counter-gradientin lower latitudes) and help maintain the high tempera-tures in higher latitudes. The net effect of the large-scaleeddies is cooling in the subtropics and heating in highlatitudes. The magnitude of this effect, however, is farfrom sufficient to produce the observed temperaturereversal in the stratosphere because of the counteractingeffect of the polar cell, The temperature of this levelincreases about 14" C. from the equator to middle lati-tudes, but decreases with increasing latitude in the higher30"20"10".a0930"O $0" 50" 40" 30" 20" 10" 0'LATITUDEFIGURE 5D5.-The latitude-height diributions of the rate of tem-perature change ("C./day) due to large-scale eddies and subgrid-scale mixing are shown in the upper and lower parts of the figure,respectively.temperature from equator to pole is only 5O C., which ismuch smaller than the observed annual mean increaseof 20" C. For comparison, the annual mean rate ofradiative temperature change obtained for the observeddistribution of temperature is shown by the trianglesymbols in figure 5D6. The cooling rate in the higherlatitudes is much larger for the actual atmosphere thanfor the model atmosphere because of the higher observedtemperatures in high latitudes. In summary, there issome tendency toward a latitudinal increase of temper-ature in the model's stratosphere, but further refinementof the model is needed to simulate the full magnitude ofthis property.6. ANGULAR MOMENTUM BALANCEA. LATITUDINAL DISTRIBUTION OF ANGULAR MOMENTUM BUDGETThe distributions of northward transport of angularDecember 1965 J. Smagorinsky, S. Manabe, and 1. L. HollowaY, Jr.7490.5 I 1VI0.2 ---0.1 -PD 80c 0.1 -e-0.1 --0.2 I I IPD 80 m M 40 20 10 0LAltTUMFIQURE 5D6.-1n the upper part of the figure, the latitudinaldistributions of the rate of temperature change due to variousheat balance components at the second model level (k=2,P/P*=0.074) are shown. The triangles show the annual meanrate of radiative temperature change at this level computed forthe observed temperature distribution. In the lower part of thefigure, the net temperature change due to large-scale eddies andt,he meridional circulation ie shown.circulation, and subgrid-scale mixing are shown in figure6A1 together with the eddy transport of the actualatmosphere obtained by Starr and White [39] and Buch 141.According to this figure, the calculated latitude of maxi-mum eddy transport is about 30 N. and coincides withthat of the actual atmosphere. The magnitude of themaximum eddy flux, however, is somewhat larger thanthat of the actual atmosphere. The contribution ofsubgrid-scale mixing turned out to be about % of that ofthe large-scale eddies. Figure 6A2 is presented to showhow the angular momentum balance is maintained ateach latitude by the above mechanisms. According tothis figure, the positive angular momentum, which issupplied from the earth's surface in tropical latitudes andin very high latitudes, is transported into middle latitudesmainly by the large-scale eddies, and the accumulatedmomentum is returned to the earth's surface there. Inmiddle latitudes, the westerly momentum is supplied bythe convergence of absolute angular momentum by thelarge-scale eddies, and in the subtropics it is supplied bythe meridional circulation.B. LATITUDE-HEIGHT DSTRIBUTION OF ANGULARMOMENTUM BUDGnIn figure 6B1 the latitude-height distribution of thenorthward eddy transport of angular momentum obtainedfrom our model is compared with the observed transportobtained by Buch [4] Oort [24]. Acc.ording to this figure,the height and latitude of the maximum northward eddytransport coincide very well with those of the actual at-mosphere. The negative eddy transport in high latitudes,UrmDEFIGURE 6Al.-The latitudinal dist.ribution of northward trans-port of angular momentum due to various components. Thetransports due to meridional circulation, large-scale eddies,horizontal subgrid-scale mixing are shown by dotted, solid, anddashed lines, respectively. The annual mean values of momen-tum flux obtained by Starr and White [39] and those obtained byBuch [4] are plotted by stars and squares, respectively.unmFIGURE 6A2.--The rates of the change of angular momentum dueto meridional circulation, large-scale eddies, subgrid-scale dif-fusion, and surface torque are shown as functions of latitude.In the lower part of this figure the distribution of computedsurface torque (solid line) is cont,rasted with the estimate ofsurface torque by Priestley 1311 for the actual atmosphere (codedcircles). The Priestley data are means for two hemispheresaveraged by seasons.7 50.009-2 0074-k+ 0189-0336-.5OO -MONTHLY WEATHER REVIEW Vol. 93, No. 12 UTINOE.009 I I I IIIII,OBSERVED- 30[BUCH L OM1 IE.LY-10 =0336-0664.926 7'.99100UNITS ~10~~~,~~,s~c/m~.~ayFIGURE GB1.-The upper part of the figure shows the calculatedlatitude-height distribution of northward transport of angularmomentum due to large-scale eddies. The lower part shows thecorresponding distribution obtained by Buch [4] and Oort [24]for the actual atmosphere..m ,80 70 6050403020 10 0UmUDE30420-e*Ic=Y10 =*991 90" 80" 70" 60" 50" 40" 30" 20" 10" 0"0*00g*30d ,074FIGURE 6B3.-The latitude-height distribution of the rate ofchange of relative angular momentum due to the large-scaleeddies, the meridional circulation, horizontal and vertical subgid-scale mixing are shown in the upper, middle, and lower parts of thefigure, respectively. Units are 101* gm. cm.2 aec.--l. mb." day- 1.FIGURE GB2.-htitude-height diatribution of vertical transportof angular momentum by the largegcale eddies. Unit is 1018gm. fIec-1 day.+December 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 751BUDGET OF ABSOLUTE ANGULAR MOMENTUMI I2POLE +TORQUE - TOSQUE + TOYQUE EQUATOR'00MBFIGURE GB4.-Schematic representation of the flow of absolute angular momentum in the model atmosphere.which is obtained in our model, does not appear clearly inthese data for actual atmosphere. However, the resultsobtained by Obasi [22] for the Southern Hemisphere showsignificant negative transport at about 62O S.The latitude-height distribution of vertical flux ofangular momentum by large-scale eddies is shown infigure 6B2. It is interesting to note t,hat the angularmomentum is transported upward in middle latitudeswhere the jet stream predominates. In other words,it is transported against the gradient of momentum belowthe jet stream. Since this flux decreases with increasingaltitude at the level of the jet stream, it supplies themomentum to the zonal current. The relative magnitudeof this contribution is about, 30 percent of that of thehorizontal eddy transport.In figure 6B3 the latitude-height distributions of therate of the change of relative angular momentum dueto the large-scale eddies, the meridional circulation, thehorizontal subgrid-scale mixing, and the vert-ical mixingare shown. According to this figure, the absolute angularmomentum created by surface torque in the tropical regionis transported upward by the direct tropical cell andsupplies the relative angular momentum in the uppertroposphere near the equator. This relative angularmomentum then is transported northward by the large-scale eddies. (Note that the eddy transport of angularmomentum is at a maximum in the upper troposphere.)This eddy northward flus of relative angular momentumconverges in middle latitudes, counterbalances the sinkof relative angular momentum resulting from the indirectmeridional circulation in middle latitudes, and therebyhelps maintain the jet, stream. Figure 6B4 gives aschematic representation of the flow of absolute angularmomentum described here.Finally, we should comment on the dependence of ourresults upon the vertical mixing coefficient adopted for ourcomputation. As figure 6B3 shows, the change of angularmomentum attributed to vertical mixing is very large inthe planetary boundary layer and is compensated formainly by the change of relative angular momentum due tothe meridional circulation. This is why the very narrowT82J17 "Abelt of the meridional component of the wind sticks to theearth's surface as figure 4B3 shows. In other words, thedegree of eccentricity of the meridional circulation of theatmosphere is mainly dominated by the thickness of theboundary layer or by the specified distribution of thevertical mixing coefficient near the earth's surface.7. BUDGET OF KINETIC ENERGY AND AVAILABLE POTENTIAL ENERGYThe release of available potential energy in the tropo-sphere has been the subject of many studies [42], [43],[ll], [35]. According to Jensen [8] the maximum re-lease takes place at about the 500-mb. level. It is veryimportant to know how the energy, which is convertedfrom potential energy, is distributed in the atmosphere.In connection with this problem, the mechanism for main-taining the kinetic enerm in the stratosphere has receiveda great deal of attention recently. (See, for example,Barnes [2], Oort [24], Reed et al. [32], Miyakoda 2211.)According to these authors, the release of eddy potentialenergy is negative and the kinetic energy of the strato-spheric motion is maintained against, dissipation by actionfrom the troposphere. In this section we shall analyze thebudget of svailable potential energy and that of kineticenergy in the model atmosphere in detail and comparethem with those in the actual atmosphere whenever it ispossible. It is hoped that the present analysis is usefulfor forming a coherent picture of the energy balance ofthe atmosphere.A. ENERGY BALANCE FOR THE WHOLE ATMOSPHEREBefore becoming immersed in a detailed study of theenergetics of the atmosphere, we constructed an energydiagram similar to the four-box diagram constructed byPhillips [27] and Smagorinsky [37]. Figure 7A1 shornthe comparison among the energy diagram for our result:that obt>ained by Smagorinsky and Phillips, and thatcompiled by Oort for the actual atmosphere. Accordingto this comparison there is general qualhative agreementbeheen the energy flow of the model atmosphere andthat of the actual atmosphere. There are, however, thefollowing quantitative discrepancies.(1) The generation of available potential energy, andtherefore the conversion of potential energy into kinet,icenergy, in the model atmosphere is significantly largerthan that obtained by Oort [25] for the ac8tual atmosphere.This discrepancy results from the absence of polewardtransport of latent heat, of condensRtion, and of t,he oceanheat transport. All these act to decrease the latitudinaltemperature gradient. Although the model of Phillipsdoes not incorporate these effects, his heating function isdeliberately modified to partially t.ake these effects intolinearized and is deAned by equation (4F1). Thts linearization is probably responsiblea The available potential energy adopted far the wustructlm of this energy di- isfor the relatively poor balance between the net generation and the conversion ofthe wail-able potential energy hecause the non-linearized version, which is defined by equntion(AIW, mainttdns a wry rood balance 85 table 3.2 shows.752MONTHLY WEATHER REVIEWVol. 93, No. 1 PColrWTEOPRESENT RESULTSOBSERVE0OORTEltIFIQURE 7Al.-In the upper and middle partsof the figure the energy diagram of the modeland that compiled by 001% 1251 for the actualatmosphere are shown. In the lower partthe energy diagram obtained by Smagorin-sky [37] is compared with that obtained byPhillips [27] (in parentheses in black boxes).@ and @ E are hemispheric means of zonaland eddy available potential energy, gzand E are hemispheric means of zonal andeddy kinetic energy, Q: denotes the changeof available potential energy by radiationand convection, Hdenotea the destructionof available potential energy by horizontalsubgrid-scale mixing, and F, IIF, and VFindicate the total dissipation, dissipationby horizontal mixing, and that by verticalmixing respectively. The unit of energytransformation is 104 joule cm." mb."day-' and that of energy itself is joule/cm.*December 1 %5 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 753account. Accordingly, the magnitude of the generationof available potential energy of his model is somewhatsmaller than in our model atmosphere.(2) The ratios XE/Xz and @,/@, are computed to bemuch smaller than the observed values, although theyare somewhat larger in our results than in the earlier two-level calculations (Smagorinsky [37]). One of the reasonsfor this discrepancy may be the absence of surface asym-metries as discussed earlier.(3) The transfer of eddy kinetic energy into zonalkinetic energy in our model atmosphere is larger thanthat estimated by Oort 1251 for the actual atmosphere,though it is somewhat smaller than the energy transferobtained by either Phillips or Smagorinsky. This resultis consistent with the very strong jet stream which weobtained in our computation. (See section 4B.)B. TOTAL KINETIC ENERGY BUDGETFor the sake of simplicity of interpretation of results,a pressure coordinate system is used instead of a a-coor-dinate system. The equation for the rate of change ofkinetic energy in the pressure coordinate syst.em iswhere V is earth velocity, F is frictional force, and.m.loo.m.m.m$+1'I= .m.m.7oo.m.900Using the continuity equation, we obtain-V.v+=-~a-(v.(Ym)+~ b (dl) (7B3)The equation for the rate of change of hemispheric meanof K is317" --a --a a -a -bt bP~ ="(WK) "war " (*) +v.P (7B4)aPwhere nE denotes the hemispheric mean; Q is the specificvolume of air, CP is geopotential height on an isobaricsurface.In equation (7B1) we shall refer to -V.V+ and V.Fas the source term and the sink term of kinetic energyrespectively. In equation (7B4) the term -Tp(*) a -Hand the term -- (uK) may be called the pressure inter-action term and the transport term of total kinetic energy,respectively, and the term -w(TH will be referred to as theconversion term of potential energy. Figure 7B1 displaysthe vertical distribution of the contribution of these termsto the rate of change of total kinetic energy. Accordingtro this figure the level of the maximum release of potentialenergy occurs at about 500 mb. As figure 7B2 shows theenergy thus converted is transferred upward to the levelof the jet stream and downward to the level of the surfaceb --aap-IIt-- [BUDGET OF TOTAL KINETIC ENERGY]\8I--""I """"9'-"rc I I I I 1" -,- "3-15 .10 .05 .oo .05 .10 .15lJOULE/CMZ. ME, DAY.)FIQURE 7B1.-Verticsl distributions of the rate of change of kinetic energy on each isobaric surface due to various mechanisms.7 54MONTHLY WEATHER REVIEW' Vol. 93, No. 12SUPPLY OF KINETIC ENERGY*loot.2w-.3w -,400-,I .5w -.600 -.7m.-(JOULE/CM2, DAY) -11. 2.3. 4.5.6.7. 8. 9.10.lllllll~~l= v .v+-/"z3al/.10"""""Wd .IS. 00.05-.05-oay-b*V+ (JOULEICM , MB, DAY)- H 2"FIGURE 7B2.-Vertical distribution of -V.V+ , -d , and --(w on an isobaric surface.-H H -Hboundary layer by the mechanism corresponding to thepressure interaction term -wQ. As a result of theseenergy transfers, the source term of kinetic energy"v. v4 is a maximum at these two levels. Accordingto figure 7Bl these sources of kinetic energy are almostexactly compensated for by the sink of kinetic energyV. f, i.e., by dissipation. The effect of vertical trans-port of kinetic energy is negligibly small. Recently,Rung I121 computed the vertical distribution of the sourceterm of kinetic energy over the North American Continentfrom observed wind data. It is encouraging that his re-sults have a clear double maximum in the source term atthe level of the jet stream and in the boundary layer as wepredicted. According to Brunt [3] the dissipation ofkinetic energy in the free atmosphere is approximatelyequal to that in the boundary layer. In the model"a-atmosphere, the ratio of the dissipation above the 811-mb. level to that below the 811-mb. level is 1.60 to 1.00.The equation for the change of the vertically integratedkinetic energy iswhere v is the meridional component of the earth windvelocity and ( ) and ( ) denote the mean operation withrespect to pressure and longitude, respectively. In the-P -ADecember 1965 J. Smagorinsky, S. ManabeERGICM~. sec:, and J. L. Holloway, Jr. 755FIGURE 7B3.-1n the upper part of the figure the distribution ofthe source t.erm and sink term (dissipation) of total kinet.icenergy are shown as functions of latitude. In the lower part ofthe figure, the latitudinal distributions of northward dux of totalkinetic energy due to the meridional circulation (M.C.), due tothe large-scale eddies (EDDY), and due to both effects (TOTAL)are shown.upper and lower parts of figure 7B3 the latitudinal dis-tribution of the source and the sink of kinetic energy areshown respec,tirelp. According t,o this figure, the latit,udeof maximum dissipation is about 45' N., and the latitudesof the primary and secondary maxima of the source termare about 32' X. and 62' h'. As one may expect, the lati-tudinal distribution of the northward transport of kineticenergy, shown in figure 7B3, is very similar to that, of theangular momentum shown in figure 6A1. Accordingly,the kinetic energy is transferred from the source regionin the subtropics (or from high lat,itudes) int,o the sinkregion in middle Mtudes. This transport compensat,esfor the imbalance between the source and sinks of kineticenergy. Saltzman, Gottuso, and Fleisher 1341 computedthe latitudinal distribution of t-he nort,hward transport ofkinetic energy by the large-scale eddies at the 500-mb.* .074- i i ,3n--20 -EI-.LcCDw-10 =FIGURE 7B4.-The latitudeheight distribut,ion of the source termof kinetic energy is shown in the upper part of the figure. Thatof the sink term (dissipation) is shown in t.he lower part of thefigure. Unit: 104 joule mb." day".level of the actual atmosphere. Their latitude of maximumpoleward transport is approximately 40' X. and coincideswit,h that of the model atmosphere. However, the weaksouthward t:ransport, which appears in the model atmos-phere around the latitudes of 50' to 70, is missing intheir distribution. The magnitade of maximum pole-ward transport is about 6X1Ol5 joules day" mb."(annual mean) and coincides reasona-bly well with that ofthe model atmosphere at 500 mb. in which it is approxi-mately 8X 1015 joules day" mb.-l (See fig. 7B5.)The latitude-height distribution of the sink of kineticenergy (V F) which is shown in t,he lower half of figure7B4 is rev similar to that of the eddy kinetic energyshown by figure 4E2 except in the area near the earth'ssurface, where t,he dissipat,ion in the boundary layerpredominat,es. It is int.eresting t,hat the sourm termhas a negative value in t.he upper troposphere of themiddle lat,iitudes. The kinetic energy is transported int,o756MONTHLY WE.336e500,664-4 e81 1.926.!??I1 90"i '\I80"-70" 60'.fm -E.IIkCDw10 =50" 40" 30" 20" 10" 0"0" ..UTffflBfFIQURE 7B5.-The latitude-height distribution of the northwardtransport of total kinetic energy due to large-scale eddies is shownin the upper part of this figure (Unit: 1017 joule mb." day-1) andthat of the vertical transport of total kinetic energy due to large-scale eddies is shown in the lower part (Unit: lo+ joule cm."day").this sink region from the source regions located on bothsides. For comparison, the distribution of the polewardand vertical transports of total kinetic energy by thelarge-scale eddies is shown in figure 7B5.C. EDDY AND ZONAL KINETIC ENERGY BUDGETThe equations for the rate of change of the hemisphericmean of eddy kinetic energy and of zonal kinetic energymay be written as follows:where u is the zonal component of the earth wind velocity.We shall refer to -da' and -w a as the eddy con-version term and the zonal conversion term, "v' Frg-E"x-'-Eand --=" as the eddy dissipation cind the zonal-dissipation, and -w'+' and -a 6 as the eddy pressureinteraction and zonal pressure interaction term, re-spectively. (. KB)2 represents the energy transferfrom zonal kinetic energy into the eddy kinetic energyattributed to the vertical interaction term (Reynoldsstress by the Iarge-smle eddies) and (Kz. rep-resents the remaining part of the transfer from Kz to K,.In the upper part of figure 7C1, the vertical distributionsof these terms contributing to the change of eddy kineticenergy are shown. As the upper part of figure 7C2 shows,the net effects of the eddy conversion of potential energy4 -"and the eddy pressure interaction term create doublemaxima in the distribution of the source of eddy kineticenergy at the level of the jet stream (200-mb. level) andin the boundary layer. According to figure 7c1, theupper tropospheric maximnm in the source of eddykinetic energy is mainly compensated for by dissipationthrough horizontal mixing and the maximum in the bound-ary layer is mainly compensated for by dissipation throughvertical mixing. Although the magnitude of the eddypressure interaction term decreases remarkably withincreasing altitude at the level of the jet stream, itsupplies a significant amount of energy to the stratosphereand constitutes a major source of kinetic energy there.In the lower half of figure 7Cl are the vertical distri-butions of the terms contributing to the change of zonalkinetic energy. As we stated already, the maximumtransfer from eddy kinetic energy into zonal Gneticenergy takes place at about the ZOO-mb. level and wnsti-tubs a major source of zonal kinetic energy. Part ofthis zonal kinetic energy is dissipated at the same leveland part is transferred by the zonal pressure interactionterm into the surface boundary layer where dissipationpredominates. (See the lower half of fig. 7C2.) Sincethe zonal pressure interaction term is estimated-as a smalldifference between two larger quantities, the accuracy ofthe computation of its contribution is rather low. It ishoped, however, that the general qualitative featuresof its contribution are correctly evaluated. Figure 7C3shows the summary of the budget which we have describedso far.December 1 %5 J. Snagorinsky, S. Manabe, and J. L. Holloway, Jr. 757JOULES/CM2, MB, DAY JOULEIMB, CM2, DAYFIGURE 7Cl.-The vertical distributions of the rate of change of eddy kinetic energy due to various terms on an isobaric surface are shown in the upper part of the figure, and those of the rate of change of zonal kinetic energy are shown in the lower part.Finally, we shall examine the latitude-height distribu-tion of some of these terms. In the upper and lower partsof figure 7C4, the latitude-height distribution of the eddyconversion term and that of tshe eddy presure interactionterm are shown, respectively. According to this figure,the eddy conversion of potential energy to kinetic energyhas a maximum at about the 500-mb. level in middlelatitudes. In the stratosphere, it hkq a small negativevalue except around 50 latitude where some very weakpositive conversion appears. Similar distributions of theeddy conversion term were obtained by Jensen [8] andMiyakoda [21]. In the tropical troposphere, there are re-gions of extremely weak conversion of potentid energy.Further study is desirable to determine how significantthese negative regions are. It is reasonable that the eddyintermtion term -d+' has a maximum value in theupper troposphere of middle latitudes or just below the-Hregion of maximum eddy kinetic energy where dissipationmay predominate.D. TOTAL AVAILABLE POTENTIAL ENERGY BUDGETThe equation for the rate of change of the hemisphericmean values of AT (see equation 4F1) in the pressure co-ordinate system may be written as follows:* Thisequattmandequations VEl) and VEZ) ap~lyto$obsric~unfn~p~by the earth's surf%. The part of the chaw@ of Ar due to the change of I is neglected.758MONTHLY WEATHER REVIEWVol. 93, No. 12SUPPLY OF EDDY KINETIC ENERGY-mH (JOULE/CM~, DAY) -.OOo, -; -f -; -,2 -,1-4 1 2 3 4 5 6 7 8 9f.looti\.2000-11 o_*I.5OO.600-tH -H 2a, -viV# (JOULE/CM , MB, DAY)-SUPPLY OF KZH-x-x (JOULE/CM ,DAY) -2-0 qJ-5-4-3-2-1 0 12 3 4 5 6 7.ooo.loo -1 I I I I I I I I.200-H-300-.-05 .-04 .-03 .-02 .-01 .OO .01 .02 .03 .04 SO5 -06 a07 -06FIGURE 7C2.-The vertical distributions of -mH, -w'a' , and -w'4' are shown in the upper part of the figure and thoee of "tiA.Vg , -W a , and -0 4 are shown in the lower part of the figure. These terms are on isobaric surfaces.-.. -..--AR--ADecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 759OTBUDGET OF KINETIC ENERGY( )" denotes the deviation from the hemispheric mean.The first term on the right-hand side of equation (7D1)corresponds to the effect of the redistribution of totalavailable potential energy by vertical motion. The thirdterm denotes the net generation of available potentialenergy caused by the effect of radiation, convection, andhorizontal diffusion. (CR) is the extra term which arisesfrom the simplified definition of available potential energy.In general, this term is very small and, therefore, it will beexcluded from further discussion.In the upper part of figure 7D2 are shown the verticaldistributions of the net generation of total availablepotential energy by convection, radiation, and subgrid-scale horizonta.1 mixing. According to this figure, con-vection generates the available potential energy andradiation and subgrid-scale diffusion destroy it. Figure7D1 shows that the net generation by t.hese three processesis mostly compensated by the conversion of total availablepotential energy, Le., by war The conversion term hasa maximum value at about the 500-mb. level which issomewhat lower than the level of maximum net generation.Vertical transport of AT adjusts the imbalance betweenthem.-CW. KETEDDY PRESSUREINTERACTIONMAX. COW. ZONAL PRESSUREOF 'E INTERACTIONEDDY WlESSUREINTERACTION+TEDDY PRESSUREINTERACTIONMAX. COW.OF 'EEDDY WlESSUREINTERACTIONZONAL PRESSUREINTERACTIONIMAX. DISS. Of KE MAX. DISS. OF KzFIGURE 7C3.-Schematic diagram showing the vertical distiihtions of the budget components of kinetic energy.30E. ZONAL AND EDDY AVAILABLE POTENTIAL ENERGY BUDGETThe equations for the rate of change of the hemisphericmeans of Az and Ae (see equations 4F2 and 4F3) in apressure co0rdinat.e system may be written as follows:c=CDY.lo =_"dAz"- ="b-at bP (wA~)~+S~+~ P (??)'I. (6 )-A 'IR-i 10 0and-0FIGURE 7C4.-The latitude-height distributions Of -xR (in The krms on the ripht,-hand sides of equations (7~1)units of joule cm.-* mb." day-') and that of -wT" (in units of10-4 joule cm.-* day") on the isobaric surface are shown in the and (7E2) are the zonal and eddy generation of avail-upper and lower parts of the fiqre, respectively. nble potential enev, and A~.A" is the conversion760 MONTHLY WEATHER REVIEW Vol. 93, No. 12.m,-,2006-1-- .ml- ,800RATE OF CHANGE IJWLEIMB,DAYII \I[GENERATION1... la," !.5m.Tt, .W[GENERATIONRATE OF CHANGE IJWLESIOAY, MBI:p-818 -.016 -.014 -.Of -.OlO-.MIiFIGURE 7Dl.-The vertical distribution of the budget componentsRAE OF CHANGE IJOULElDAY. KBIof total available potential energy, of zonal available potentialenergy, and of eddy available potential energy on an FIGURE 7D2.-The vertical distribution of the generation or de-surface are shown in the upper, middle, and lower psrts of the struction of available potential energy due to convection, radiation,figure, respectively. and horizontal subgrid-scale mixing. The distributions for the total available potential energy, zonal available potential energy,and eddy available potential energy are shown in t,he upper, middle,and lower parts of the figure, respectively. Note the change ofhorizontal scale on the eddy graph.-~December 1965 J. Snagorinsky, S. Manabe, and J. L. Holloway, Jr. 761from zonal to eddy available potential energy. (CRZ)and (CRW are terms arising from the assumption adoptedfor the linearization of the definition of available potentialenergy. Again, we shall disregard these terms in thefollowing discussion.According to figure 7D1, the net generation of zonalavailable potential energy has a maximum in the uppertroposphere. This generation is mostly compensatedfor by the transformation of zonal available potentialenergy into eddy available potential energy, which isalso at a maximum in the upper troposphere. Figure7D2 shows that the major mechanism which generatest3he zonal available potential energy is the heat,ing causedby vertical Convective mixing (conduction from theearth's surface and moist convective adjustment).The budget components of eddy available potentialenergy are shorn in figure 7D1. According to thisfigure the energy supply from the zonal available po-tential energy constit>utes a major source of eddy arail-able potential energy. On t,he other hand, major sinksare the transformat,ion into eddy kinet.ic energy and thenet destruction by radiation, convection, and horizontaldiffusion. Figure 7D2 shows that the convective adjust-ment dest'roys some of the eddy available pot.entia1energy new the earth's surface. The vertical transport,of A, (see equation 4F3) compensates for the height,difference bet,ween the level of maximum supply andt,hat of maximum loss.Because the latitudinal temperature gradient is toolarge in the upper troposphere of our model, the sharpmaximum of energy transfer between zonal availablepotential energy and eddy available potentiai energy inthe upper troposphere may be exaggerated in our cal-culation. The general qualit.ative features of the energybudget described above, however, may not be too farfrom reality. It would be desirable to carry out a similarstudy of the vertical dist,ribut,ion of energy budget forthe real atmosphere.F. ENERGY BALANCE OF ME STRATOSPHEREIn order to investigate the energetics of the strato-sphere obtained from ourmode1,we constructed energy dia-grams for a stratospheric layer (P/P*=0.034 to 0.126, i.e.,t.he second layer from the top). Figure 7F1 shows the two-box energy diagram. The terms which are written outsidethe domain enclosed by dashed lines represent the in-teraction with other layers. Since the interaction betweenthe second layer and the lower layer is much larger thanthat between the second layer and the top layer, we mayregard t,hese terms outside the domain as the interactionbetween this strat,ospheric layer and the lower layers ofthe atmosphere. According to this figure, the energytransfer from below is mainly accomplished by t.hepressure inberaction term -c,P+~. This energy supplycounterbalances the loss of kinetic enem by the con-version from kinetic energy to potential energy and bydissipation. The available potential energy thus con--r-----F"""1IIII1 I1I 1 1 t1InI AFIGURE 7Fl.--?tvo-box energy diagrams for the model stratosphere(approximatdg the 34-126-mb. layer). Qg, generation of avail-able potent.ial energy due to radiat,ion and convection from theearth's surface. oA, the effect of the vertical transport of avail-able potential energy. wK, the effect of vertical transport ofkinetic enerpv. 4, effect of the pressure interaction term. Theunit of energy transformation is 10-3 joule mb." cm.+ day-l andthe unit of energy itself is joule cm.-* mb.". For further expla-nation refer to figure 7A1. NOTE: Value of @Z should be ,064instead of 372.verted from kinetic energy is destroyed by the subgrid-scale thermal diffusion and by radiation. These resultsare in qualitative agreement witjh the conclusion obtainedby Barnes [2], Miyakoda [21], and 001% [24] for the actualatmosphere.In order to compare our results witah the energp diagranobtained by Oort [24] for t,he act,ual atmosphere, a fourbox energy diagram is shown in fqure 7F2 together wit,]the diagram of Oort [24]. According to this diagram, t,heddy kinetic energy of the stratosphere is supplied by t.hlower at-mosphere through the eddy pressure int.eractioiterm. The eddy kinet.ic energy t.hus produced is t'rans.762 MONTHLY WEATHER REVIEWVol. 93, No. 12COMPUTEDmP1"""" "_"""" - """"U w!W vi"""" "_"""" - """"1I II II---; I I""L """" --"""-1OBSERVEDOORTII 1I1IIL""JI I"" """""""_FIGURE 7F2.-Four-box energy diagram for the stratosphere of the model atmosphere (approximately 3126-mb. layer) and of the actus1atmosphere (30-100-mb. layer) as compiled by 001% [24] are shown in the left and right-hand sides of the figure, respectively. Theenergy exchange with the other layers of the atmosphere is shown by extending the arrow outside the domain enclosed by the dashedline. R.S.=the mass integral of the Reynolds stress term Kz.KB>* (see equation (7C6).). The unit of energy transformation isjoule mb." cm1.-2 day-', and the unit of energy itself is joule cm.-* mb.-l For further explanation, refer to fiaures 7A1 and 7F1.Nom: value of @ should be .056 instead of .064.ferred into zonal kinetic energy and then into the zonalavailable potential energy. The transfer of energy de-scribed so far is in qualitative agreement with the resultsof Oort [24] obtained for the actual atmosphere. Quanti-tatively, the magnitude of the eddy pressure interactionfrom the lower atmosphere and that of energy exchangebetween zonal and eddy kinetic energy are much largerfor our model atmosphere than for the actual atmosphere.There are ot.her disagreements. According to this dia-gram, small amounts of zonal available pot.entia1 energyare transferred into eddy available potential energy, inqualitative disagreement with the results Oort [24] ob-tained for the actual atmosphere. Also, the degree ofdestruction of zonal available potential energy by radiationis much smaller than that for the actual atmospbere.These comparisons suggest that the energy cycle in themodel stratosphere is significantly different from, at least,Oort's results for the stratosphere despite some qualitativesimilarity. In order to simulate the stratosphere morerealistically, the following improvements of the model arebeing planned :(1) lmprovement of vertical resolution of the model inthe neighborhood of the tropopause. This improvementmay enable us to represent the vertical variation of thepressure interaction term more accurately.(2) Incorporation of the effect of condensation. Thisimprovement may significantly alter the meridional tem-perature gradient of the atmosphere and therefore changethe magnitude of the zonal kinetic energy in the strato-sphere as well as in the troposphere.(3) Incorporation of the seasonal variation of radiation.Since the condition of the stratosphere varies drasticallyfrom one season to another, it is essential to include thiseffect for the successful simulation of the climatology ofthe stratosphere.8. CONCLUDING REMARKSIn this study, we achieved a rektonable degree of successin simulating t,he vertical structure of t,he atmosphere byadopting a model with a relatively high vertical resolu-tion. The general thermal structure of stratosphere-troposphere syst,em is simulated. The pole-to-equatordifference in the height of the tropopause turned out to beapproximately 7 km., which is reasonably close to theobserved difference of 10 km. The latitude of the tropo-spheric jet coincides with the observation. The generalfeatures of the distributions of radiative flux and convec-tive flux are very similar to what London obtained for theactual atmosphere. Also, the characteristic features ofwave distnrbance are successfully simulated. The waveDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 763number of these disturbances decreases very sharply wit'hincreasing altitude around the level of the tropopause,and very long waves predominate in the stratosphere.The level of maximum eddy kinetic energy is around the200-mb. level and coincides with the level of maximumin the actual atmosphere. The vertical dist.ribution ofthe budget of kinet.ic energy is in qualitative agreementwith the recent results of analyses of the actual atmos-phere. For example, the kinetic energy of our modelstratosphere is maintained by the supply of energy fromthe troposphere through the pressure interaction termin agreement with the results obtained by analysis of theactual atmosphere.On the other hand, there are many discrepanciesbet~ween our model atmosphere and the actual at,mosphere.Many of the discrepancies pointed out by Phillips [27]and Smagorinsky [37] in their numerical experiments arestill present. For example, the ratio of eddy kineticenergy to tshe zonal kinetic energy and that of eddy avail-able potential energy t,o its zonal value are much smallerfor our model atmosphere than for the actual atmosphere.Also, the latitudinal gradient, of temperature of our modelin middle latitudes is much larger than the observedgradient. Accordingly, the intensity of t.he jet streamwhich we obtained from our model is much t,oo strong.In the stratosphere, the latitudinal increase of tempera-ture with increasing lat,itude is insufficient. Thus, thest,rat*ospheric westerlies of our model atmosphere turnedout. to be much stronger than the observed annual mean.The amount. of energy supplied from the troposphere intothe stratosphere also seems to be much larger for the modelthan for the acttlal atmosphere. The eddy kinetic, energycreated by this energy supply is used to maintain tvhe verylarge zonal kinetic, energy of our model stratosphere.These features of t,he stratosphere obt,ained from our com-putation are much closer t,o t,hose of the winter strato-sphere than to those of the annual mean stratospheredespite the fact. we adopted the annual mean insolationfor onr computation.One of the major factors which may be responsiblefor these discrepencies is tha.t~ the condensation processin t,he atmosphere is missing in the model. For example,the modification of tzhe latit,udinrtl distribution of heatingby condensatdim could aker significantly t.he t,emperaturegradient. in middle latitudes and accordingly the intensityof the zonal current. Recent,ly, we have completed a.preliminary integration of the general circulation modelwhich includes a simple hydrologic cycle. The compara-t,ive result is discussed in a companion paper [18].Also, the effects of the asymmetriesof the lower bound-ary such as mountains and land-sea distribution couldbe important. Recently, Mint,z [20] successfully simi-lated the longitudinal as well as latitudinal distributionsof temperature and pressure at the earth's surface by incor-porating these effects. As we discussed already, t,hey maysignificantly increase the ratio of eddy t,o zonal kineenergy.Another factor of importance is the horizontal andvertical resolution of the model. According to the pre-liminary results of an integration of our model withhigher horizontal resolution (N=40), the magnitude ofthe northward transport of total energy (or momentum)due to the meridional circulation and that due to thelarge-scale eddies are significantly different from theresults obtained from the present model (N=20). Re-cently, Smagorinsky and staff members [38] performeda series of forecast experiments with real initial data andconcluded that the increase of resolution greatly improvestheir forecasts. Therefore, it seems mandatory to performexperiments with greater resolution; otherwise, the quan-titative comparison of our results with the feat~ures ofthe actual atmosphere will be of limited value.In this study, a great deal of effort is devoted to theanalysis of the budget of kinetic energy of our modelatmosphere. We attempted to show how the kineticenergy of the jet, stream, the large eddy kinetic energyin the upper troposphere, and the kinetic energy of thest,ratosphere are maintained against dissipation. Al-t.hough some of the results we obtained are in qualitativeagreement with analyses of observed data, further studyof the budget of kinetic energy of the actual atmosphereis needed for the satisfactory verification of simulation.Finally, it must be pointed out, t,hat. part of the successin simulating the atmosphere is due to the fact that weadopted the climatological distribution of atmosphericabsorbers for the computat,ion of radiative transfer. Itis a further goal of our studv t,o perform the numericalint,egration without assuming the climatological distri-but,ion of water vapor, ozone, and clouds.APPENDIX I.-FINITE DIFFERENCE EQUATIONS A. SPACE DIFFERENCEThe finitse difference represent.at,ion of the non-linearterm, which satisfies some of the integral requirements ofinertia terms and avoids some of the possibilit#ies of non-linear instability (pointed out. by Phillips [29]) mas recentlyproposed by Arakawa [l]. Basing it upon the same prin-ciple, Lilly' proposed R general energy and momentumconserTing represent,ation of t.he non-linear t>em which weadopt.ed for the present, st.udp.In order to facilitate the display, we shall first, definet$he following sum and difference operators adopted byShuman and T'anderman [36] and Lillp [13].where cp is any function of t3he variable X. For example,the notdon cp signifies that. the value of t$ at, t,he adjacentlevels is averaged.47 Pemnal mmmunic9tion.164 MONTHLY WEATHER REVIEW Vol. 93, No. 12The finite difference forms of equations of motion inthis notation arewhere is the Q-thickness of a layer. The kite dif-ference form of the continuity equation is-x -YP*=-m2[Sx(%) bt +6y ('e? ] A""P* 6QQ - 42 (AI5)The thermodynamica.1 equation is- (P,T)=-m2bat (qx Tx)+Sy (Pzy .Ty)] A-IAs we mentioned before, the free-slip wall is assumed atthe side boundary. On a wall parallel t,o the Y-axis,(AIS)andOn a wall parallel to the X-axis,6y (P* g)=oandvyP,- =om(AI10)(AI11)For the configuration of the side wall, refer to section 2F.B. TIME DIFFERENCEIn general, we adopt a centered difference for the timeintegration. The viscous force term and the thermaldiffusion term, however, are computed by forward timedifferences to guarantee computational stability. Thetime interval adopted for the integration is 10 min. Inorder to avoid the growth of a computational mode, t$hewind, temperature, and pressure fields at three consecutivetime steps are averaged by the weights of 0.25, 0.5, and0.25 every 53 time steps.APPENDIX II."3-DIMENSIONAL INTEGRALPROPERTIESHere, we shall obtain the expressions for the massintegrals of the changes of kinetic energy, angular momen-tum, available potential energy, and gross stat.ic stabilitywhich are used in section 3.A. KINETIC ENERGYThe mass integral of the equation describing the changeof kinetic energy giveswhere(AII1)where 4, is the geopotential of t,he earth's surface, K isthe kinetic energy per unit mass, and f ( )dA denotesthe area integral. C and D denote the conversion frompotential to kinetic energy and the dissipation by thesubgrid-scale mixing, respectively. In the present com-putation, the second term on the right-hand side of theequation (AII3) is zero because we adopted a flat lowerboundary.B. ABSOLUTE ANGULAR MOMENTUMThe mass integral of the equation describing the changeof absolute angular momentum gives- [ s( s' 5 AAQ) dA]= (MTQ) +(ST) (AI 15)at 09aDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr.765(MTQ) and (STQ) are torque due to mountains and sur-face drag, respectively. 0, +* and (T~~)~,~ denote latitude,the geopotential of the earth's surface, and the surfacestress respectively. In the present computation (MTQ)is zero.C. AVAILABLE POTENTIAL ENERGYAccording to Lorenz [ 151, the available potential energycan be defined as follows?I= @+I) -@+a (Am)where E and I denote potential and int.ernal energiesrespectively. The operator - denotes the quantitieswhich would be obtained after redistributing the atmos-phere until the isent,ropic surfaces are horizontal. Theequation describing the change of available pot,ent.ialenergy is(AIIS)whereY- " "* dA (AII11). 9 dtG is the net generation of avdable potent,ial energy byradiative heating GRAD, by convection qcv, and by GKD,which is the equivalent heat source corresponding to theeffect of horizont,al diffusion. Second terms on the right-hand side of the equations (AIIIO) and (AII11) are zeroin the present comput,a.tion because of the lack ofmountains.D. GROSS STATIC STABILITYhlodifying slightly the definition given by Lorenz, wecan define static st,abilit,y of the atmosphere as follows+-(E+I)+(E+E (AI1 12)where = denoh the quant,ities of the idealized atmos-phere whose pot.ent.ia1 t.emperature is constant everywhereand whose mean potential temperature is equal to that ofthe act,ual at,mosphere.The equation describing the change of gross st,aticst,ability of the atmosphere is(AII13)where GG is the net rate of destabilizat.ion of t,he grossstdc stability of the atmosphere by qmD1 qcv, and @HD,a.nd is represent.ed by the following equation JAgain, the second term on the right-hand side of theequation is zero in the present calculation.APPENDIX Ill.-SUBGRID-SCALE MIXING COEFFICIENTThe coefficient of diffusion KK in the subgrid-scalemixing used in the model is computed from the relation:where the notation is that, of section 2@). The onlyfactor in this equation not exactly specified by t,he theoryis the Karman constant ko. We used a value of 0.4 in thelong-time integrat,ion report,ed on here. However, weperformed extensive test integrations using a simple two-level stereographic model with four values of ko around0.4 to make sure we adopted a reasonable value for thisparameter. The simplified model was essentially t,hesame as used in the general circulation experimentexcept for modifications to the surface drag and convectiveadjustment made necessary because of the low resolutionin the vertical. A simple parametric heating functionwas used instead of the explicit radiation computationof the regular nine-level model.The model was st,arted from rest and an isothermalstate and run with k0=0.4 until a more-than-adequatepole-to-equator temperatsure gradient htbd developed andbaroclinic waves were beginning t,o form. Then an arbi-t>rary time step u-as chosen as an init,ial condition, and themodel was run 11 days beyond this point four times wit,hfour different values of ko (0.2, 0.283, 0.4, and 0.566)but with no other changes.Aft.er 11 days an inspection of the maps of winds,pressure, and temperature indicat>ed that those computedwith ko=0.4 exhibited the best, compromise between toomuch "noodling" in small-scale features and too littledevelopment of baroclinic waves. Space does not permit'the printing of all the maps that contributed to thisdecision, but figure A-I11 shows cross-sections of tempera-ture and meridional wind at, the top level (250 mb.) donga peat circle passing through the Borth Pole. Cert,ainlyit is clear from these curves that ko=0.566 gives too muchsmoothing and ko=O.2 not enough. However, whetherko=0.4 is the optimum value is still in doubt, but thisvalue WHS chosen for the longtime integration because ofits historical precedent.766MONTHLY WEATHER REVIEW Vol. 93, No. 123' \MERIDIONAL WIND VELOCITY AT250 MBIll IIIII I1 I1 II II I1 II I124'N 50'N NORTH POLE GRID POINTS50N 24'NFIGURE A-111.-Cross-sections of meridional wind and temperature at 250 mb. along a great circle passing through the North Pole computedin a simple twc-level baroclinic mod61 solved on a stereographic projection. Four runs of 11 days each were made in which the onlydifference in the model was the value of the Karman constant (ko) used in the horizontal diffusion of momentum and heat. These fourvalues were 0.2 (short-dashed curves with stars), 0.283 (dash-dotted curves with triangles), 0.4 (solid curves with solid circles), and0.566 (long-dashed curves with open circles).ACKNOWLEDGMENTSThe research reported on here was such a large undertaking thatit could only have been accomplished by extraordinary teamworkon the part of a great number of talented and conscientious workers.It would be utterly impossible in a few words to acknowledgeproperly the superlative assistance we have received during thepreparat.ion of this paper.We are especially indebted to Dr. Patrick H. Sterbenz, Dr.Donald A. Quarles, Jr., Dr. Kurt Spielberg, and other membersof the Mathematics and Applications Department of the I.B.M.Corporation for planning and writing the original machine-languageSTRETCH computer program for the time integration.The following present or former members of the Geophysical FluidDynamics Laboratory staff contributed substantially to the successof this project: Mr. Robert F. Strickler who provided very knowl-edgeable assistance with the radiation codes and related phases ofthe program; Mrs. W. M. Carlton and Mrs. E. A. Staley who gavemeticulous attention to details in the systems proe;rming; Mrs.C. J. Hiland and Mrs. E. C. Arnold who wrote the extensive diag-nostic programs for analysis of our results; Dr. Kirk Bryan whosemany useful questions and suggestions influenced the course of ourresearch a great deal; Dr. Douglas Lilly who developed the modifiedversion of the energy-conserving system of equations suitable for ourmodel; Drs. Yoshio Kurihara and Kikuro Miyakoda who mademany useful suggestions regarding numerical techniques andmethods of analysis; Mr. J. M. Kennedy who prop;ramed mallmodels for test computations described in the Appendix; Nr. JohnYoung who assisted with the formulation of the energy integds;Mr. H. H. Engelbrecht and his computer operators who workedaround the clock expertly running our machine pme;rSms; Mrs.Marylin Varnadore who made very accurate and fast hand calcula-tions and data analysis; Mr. R. D. Graham who handled the in-numerable administrative details which a project of this sizeDecember 1965 J. Smagorinsky, S. Manabe, and J. L. Holloway, Jr. 767encounters; and Mrs. J. A. Snyder and Mr. E. W. Rayfield whoassisted with the preparation of the manuscript and figures,respectively.Finally, we are indebted to Dr. Sidney Teweles of ESSA forkindly supplying the results of harmonic analyses of the wind.REFERENCES1. A. 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Preliminary results of this investigation were first presented at the International Symposium on Dynamics of Large-Scale Processes, Boulder, Colo., September 3–7, 1963.