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The Effect of Vertical Finite Difference Approximations on Simulations with the NCAR Community Climate Model

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Abstract

Two commonly used vertical finite difference approximations produce markedly different simulations when adapted to the nine-level Community Climate Model assembled at the National Center for Atmospheric Research. The differences are conveniently illustrated by considering the zonal average temperature and zonal wind, but these different zonal averaged are also associated with differences in the stationary and transient waves in the model. The hydrostatic equation and vertical temperature advection are the main contributors to the differences in the simulations. Other terms produce only minor differences. Except above the equatorial tropopause, the two schemes converge to the same solution with significantly higher vertical resolution. In many respects, this convergent simulation is closer to that produced by one of the approximations on the original nine levels than to that produced by the other. However, the resemblance is not adequate to justify use of that scheme on the coarse grid when other aspects of the simulation are also considered. Higher resolution should be used so that the simulation becomes insensitive to the vertical finite difference approximations.

Abstract

Two commonly used vertical finite difference approximations produce markedly different simulations when adapted to the nine-level Community Climate Model assembled at the National Center for Atmospheric Research. The differences are conveniently illustrated by considering the zonal average temperature and zonal wind, but these different zonal averaged are also associated with differences in the stationary and transient waves in the model. The hydrostatic equation and vertical temperature advection are the main contributors to the differences in the simulations. Other terms produce only minor differences. Except above the equatorial tropopause, the two schemes converge to the same solution with significantly higher vertical resolution. In many respects, this convergent simulation is closer to that produced by one of the approximations on the original nine levels than to that produced by the other. However, the resemblance is not adequate to justify use of that scheme on the coarse grid when other aspects of the simulation are also considered. Higher resolution should be used so that the simulation becomes insensitive to the vertical finite difference approximations.

40 JOURNAL OF CLIMATE VOLUMEThe Effect of Vertical Finite Difference Approximations on Simulations with the NCAR Community Climate Model DAVID L. WILLIAMSONNational Center for Atmospheric Research,* Boulder, Colorado(Manuscript received lg November 1986, in final form 7 August 1987)ABSTRACT Two commonly used vertical finite difference approximations produce markedly different simulations whenadapted to the nine-level Community Climate Model assembled at the National Center for Atmospheric Research.The differences are conveniently illustrated by considering the zonal average temperature and zonal wind, butthese different zonal averages are also associated with differences in the stationary and transient waves in themodel. The hydrostatic exluation and vertical temperature advection are the main contributors to the differencesin the simulations. Other terms produce only minor differences. Except above the equatorial tropopause, thetwo schemes converge to the same solution with significantly higher vertical resolution. In many respects, thisconvergent simulation is closer to that produced by one of the approximations on the original nine levels thanto that produced by the other. However, the resemblance is not adequate to justify use of that scheme on thecoarse grid when other aspects of the simulation are also considered. Higher resolution should be used so thatthe simulation becomes insensitive to the vertical finite difference approximations.1. Introduction The spectral transform method has been widelyadopted for the horizontal aspects of numerical weatherprediction and general circulation models. Examplesof the former are the operational models developed atthe National Meteorological Center (Sela, 1980), theEuropean Centre for Medium Range Weather Forecasts (Baede et al., 1979) and the Division de Rechercheen Prtvision Numtrique (Daley et al., 1976). The latterinclude the original spectral transform model of theAustralian Numerical Meteorological Research Center(Bourke, 1974; Bourke et al., 1977; McAvaney et al.,1978), the model developed at the University of Reading (Hoskins and Simmons, 1975), and that of theGeophysical Fluid Dynamics Laboratory (Gordon andStem, 1982). For all practical purposes these modelsuse the same horizontal spectral transform algorithmand therefore for the same horizontal resolution incurthe same errors from this aspect. All of the models listed above also use finite difference.approximations in the vertical direction, but herethe Situation is much less standard. While the horizontal spectral transform technique has become moreor less standard following Bourke's (1972) implemen * The National Center for Atmospheric Research is sponsored bythe National Science Foundation. Corresponding author address.' Dr. David L. Williamson, NationalCenter for Atmospheric Research, Box 3000, Boulder CO, 803073000.ration, the vertical approximations remain very different, with each modeling group choosing unique approximations for various, sometimes arbitrary, reasons.The choices involve placement of vertical discrete levelswithin the particular sigma coordinate used as well asthe actual approximations on that grid. Both of theseaspects can significantly affect the simulations producedby a model when, for economic reasons, the grid cannotbe made fine enough so that the discrete solutions converge to the continuous solution. In the process of assembling the Community ClimateModel (CCM) at the National Center for AtmosphericResearch (NCAR), two different vertical approximations adopted at other centers were found to give verydifferent simulations when all other components of themodel, including the grid level placement, were unchanged. The first of these approximations was developed at the Australian Numerical Meteorological Research Center (ANMRC; Bourke, 1974; Bourke et al.,1977) and was later adopted in the CCM (version 0)at NCAR (Pitcher et al., 1983). This Australian schemecontains standard, straightforward approximations tothe derivatives and integrals involved in the equationswith no secondary considerations such as conservationof energy. We will refer to these approximations asscheme X. The second set of approximations was ouradaptation of the-general form of approximations inthe first European Centre for Medium Range WeatherForecasts (ECMWF) operational grid point model(Burridge and Haseler, 1977) and in a matching research spectral transform model (Baede et al., 1979).These approximations have the same formal accuracyc 1988 American Meteorological SocietyJANUARY 1988 DAVID L. WILLIAMSON 41as scheme X, but were also designed to maintain energyconservation. This scheme was applied to our ninelevel vertical grid which was the same as that used inthe Australian model, but was very different from thatused by the ECMWF, both in the number of levels andin the vertical distribution of the levels. We refer tothese approximations as scheme Y. Within the NCAR CCM0 which incorporates thestandard nine vertical levels originally adopted at theGeophysical Fluid Dynamics Laboratory (GFDL;Smagorinsky et al., 1965) and intermediate levelsadopted by ANMRC, these two sets of approximationsproduce notable differences in the simulation of thezonal wind profile and of the tropical tropopause temperature. With scheme X the model simulates a strongseparation between the wintertime westerly jets in thetroposphere and stratosphere with zonal wind speedsin the stratosphere in agreement with observations.However, energy is not conserved in this simulationand the effective imbalance is around 10 W m-2, onethird of which can be attributed to the moisture component of the system. With scheme Y the model simulates minimal separation of these jets with an unrealistically strong polar stratospheric jet and associatedoverly cold polar stratospheric temperatures. This errorstructure corresponds to the error noted by Bengtsson(1985) in ensembles of medium range forecasts produced by the ECMWF model and is typical of manyglobal circulation models. The energy, on the otherhand, is well conserved in the simulation with schemeY. In section 2, we will show that this polar stratospheric error structure can be initiated by the verticaldifferences and then possibly exacerbated by the dynamical processes. In addressing the question of which components of the vertical approximations are responsible for the dif ferences in the simulation, we will concentrate on ex amining the zonal average wind and temperature fields. The major differences in these fields occur in the stratosphere and may at first be thought to be of little importance to the troposphere. Boville (1984) has shown that this is not the case. The polar night jet structure in an atmospheric model influences not only the stationary planetary waves in the troposphere but also the transient eddies of all scales. Therefore, al though we concentrate on the zonal average fields in the stratosphere to demonstrate the primary effect of the approximations, it should be kept in mind that these structures have a pronounced effect on the tro pospheric waves and therefore they are very relevant to the tropospheric prediction and simulation prob lems.2. Basic model The starting point for the examination of the variouscomponents in the vertical finite differences is thestandard NCAR CCM0 which uses scheme X. It isTABLE 1. ~ levels used in simulations.9 levels 19 levels 37 levels.009 .009 .009 .017125 .02525 .02525 .033375 .0415 .0415 .048625 .05775 .05775 .065875.074 .074 .074 .10275 .1315 .1315 .16025.189 .189 .189 .22575 .2625 .2625 .29925.336 .336 .336 .377 .418 .418 .459.500 .500 .500 .541 .582 .582 .623.664 .664 .664 .70075 .7375 .7375 .77425.811 .811 .811 .83975 .8685 .8685 .89725.926 .926 .926 .94225 .9585 .9585 .97475.991 .991 .991described in detail in Pitcher et al. ( 198 3) and Williamson (1983). The model is formulated in the ~ coordinateof Phillips (1957), ~ = PIPs where p is pressure and Psis surface pressure. In its standard form and for thefirst experiments discussed here it has nine unequallyspaced levels in the vertical (given in Table 1). Intermediate levels used by the finite difference approximations are taken as the arithmetic average of thestandard levels. The model uses the spectral transformtechnique in the horizontal with wavenumber 15rhomboidal truncation. A horizontal X72 diffusion termis applied over the top half of the rhomboidal spectraldomain in the troposphere and over the entire spectraldomain in the stratosphere (top two levels). The physical parameterizations include the radiationand interactive cloud routines described in Ramanathan et al. (1983), and moist and dry convective adjustment, stable condensation, vertical diffusion, surface fluxes and surface energy balance prescription developed at GFDL (Smagorinsky, 1963; Manabe et al.,1965; Smagorinsky et al., 1965; Holloway and Manabe,1971). These physical parameterizations, horizontal42 JOURNAL OF CLIMATE VOLUME Iapproximations and horizontal diffusion remain unchanged in the following nine-level simulations.3. Simulations with nine levels We first show the simulations resulting from the twocomplete sets of finite differences, that of scheme Xand that of scheme Y. These are applied to the ninelevels in the left column of Table 1. A set of intermediate or half levels between those of Table I are alsoused by the approximations. These are taken to be thearithmetic average of the levels shown, i.e., halfwaybetween in a. This intermediate set is the form adoptedin the ANMRC model and in the NCAR CCM0, butdoes not correspond to the original GFDL definition.The ECMWF did not adopt this particular nine-levelgrid for most of its work, nor did they choose the intermediate set to be the arithmetic average as ANMRCdid. As noted above and as will be seen below, thecharacteristics of the simulation are dependent on theactual grid used. Thus they do not necessarily reflectthe ECMWF model as it was originally implemented. Of the following simulations, all were run for 150model days with perpetual January external parameters, starting from day 200 of a previous perpetual January CCM0 model simulation. Therefore, the initialconditions represent a state from the CCM0 (schemeX) model climate. The first 60 days of each simulationwere ignored and the figures show averages over thelast 90 days of each run. The 60-day spinup periodbefore averages are taken is adequate to capture themajor numerical-dynamical adjustments to the newmodel climate. It may not be long enough to capturecompletely secondary radiative responses to the newstate. Nevertheless, these averages are adequate to indicate the major model response to the various numerical approximations. Figure 1 shows the 90-day zonal average temperatureand zonal wind fields produced by the two vertical finitedifference schemes and the difference between them.Although we do not perform a proper statistical significance test here, several regions show differenceswhich are very large compared to the standard devia-tion in an ensemble of 90-day averages from a verylong run of the basic CCM0 (Williamson and Williamson, 1984, Fig. II.A.2 and 4). The tropical topopausetemperature is warmer by 10 K with scheme Y thanwith scheme X. This can be compared to a standarddeviation of the 90-day averages of 0.5 K in this region.The winter polar stratospheric temperature differenceis 26 K. Although this is a region of larger variation inthe ensemble of 90-day averages, the standard deviationof the 90-day average is still only 4 K there. Thesetemperature differences create stronger latitudinal gradients poleward of 50-N in the upper three levels ofthe model. The 90-day average zonal average of thezonal wind shows a corresponding difference. Thestratospheric polar night jet is stronger by 40 m s-~with scheme Y and has only a very weak separationfrom the tropospheric jet. The standard deviation ofthe 90-day averages of the zonal wind is 5 m s-~ in thatregion. The stratospheric equatorial easterlies are alsoweaker.by about 5 m s-~ with scheme Y. The standarddeviation of the 90-day average in this region is 1.5 ms-l, so the significance of this difference is questionable.In the following sections, we will concentrate on theseaspects of the zonal average fields. Notice that the differences appear primarily in thetop two levels of the model. As we noted above, differences in the simulation at these levels produce differences in other aspects of the simulation at lowerlevels. Boville (1984) shows how differences only in thehorizontal diffusion in the top two levels producechanges in the polar night jet similar to those shownin Fig. 1. These stratospheric differences produce a difference in the stationary planetary waves in the troposphere and also in the transient eddies. For example,although we do not show it here, the difference in the500 mb height field produced by the two simulationsin Fig. I looks exactly like that in Fig. 3 of Boville(1984). The troposphere.in our case is responding tochanges in the stratosphere exactly as it did in Boville'sexperiments in which the stratospheric differences werecaused by a completely different mechanism. Several aspects of the vertical numerical approximations contribute to these differences. In subsections3a-c we consider various components of the approximations independently. Each component is changedto the scheme Y form, one at a time, and simulationswith only that component changed are compared tothe complete scheme X simulation described above. Itis important to remember that in the following ninelevel simulations only the component being discussedis different from the scheme X simulation. The changesare not made in a cumulative manner and the basis ofcomparison is always the complete scheme X simulation.a. Hydrostatic equation The hydrostatic equation for the geopotential gak atthe level ak is ga~ = gas - R Tdlna, (1)=1where T is temperature, gas is the surface geopotential,and R is the gas constant for dry air. A general difference approximation to (1) is 1 gak = gas + R ~ B~j Tj (2) j=Kwhere the vertical grid index runs from I at the toptemperature level to K at the bottom (Fig. 2). The matrix Bnj is often triangular (Bej = 0, j < k); however,there may be exceptions when extrapolation is involved. The hydrostatic, integrals from scheme X andJANUARY 1988 DAVID L. WILLIAMSON 43xm b~; x 74500991T ff l'i/ ,~ 2' I II./111I1.~111 I ,//1~ 71/x X _~__.~.~~ ~ I IIIl\\\\\~:/llllll III I l ' ~ ~', ',",,,',..'..",'",',',,'","/m I ll\\\\\\X.'"Jlllllll\il',H?, / I;,'i I \ .ll\\\\~.z,,/lllllllliii!,,??,,,,,,;/ \ I\\\\\\x."-'/I/\/llll! ?.o_., //-,,/ \ I \\\\\x,,TM I/////lil ',. '-.,'.,'/I ,,~ \i - " 9~189991 90 70. 50 30 I0 -I0 -;SO -50 -70-900 70 50 30 I0 -I0 -$0 -50 -70-90 LATITUDE LATITUDE FIG. 1. Time averaged, zonal averaged temperature (left) and zonal wind component (right) from nine-level simulations with(top) scheme X and (middle) scheme Y approximations, and (bottom) the difference scheme X-scheme Y. Contour intervals are5 K and 5 m s-t for total fields and 2 K and 4 m s-~ for differences. Negative contours are dashed.44 JOURNAL OF CLIMATE VOLUME Ischeme Y are illustrated graphically in Fig. 2, the geopotential at ak being given by the enclosed area plus~bs. The ordinate is lna and the abscissa is T. Thescheme Y hydrostatic equation assumes constant temperatures within layers (e.g., from ak-~/2 to ~+~/2) andtakes the geopotential at the full levels (dashed lines)to be the average of that at adjacent half levels (solidlines). Note that scheme Y never uses informationabout the location of the main full vertical grid levels,only the intermediate half levels. In fact, the ECMWFdid not define a full level set for the purpose of thenumerical approximations; only for the physical parameterizations was the full level set identified. The'scheme X form assumes linear (in lnv) variation between grid points with extrapolation to the surface(yielding the nontriangular B since ~0x depends on Txand Tx-~). Except at endpoints, the forms of the discreteB operators applied to our particular grid, in which theintermediate level a values are taken to be the arithmetic average of the main levels, arescheme Y:scheme X:B&j '~' lntrj+l/2 -- lnaj_~/2, 1 (In aj+l - lnaj_l).B~,j ~ ~(3)(4)If either the grid were equally spaced in lna, or theintermediate levels were taken to be the geometric average of the full levels rather than the arithmetic average, these two forms would be identical (except forthe extrapolation to the surface). However, the gridadopted here is not equally spaced and the two approximations integrate to different heights. Figure 3 shows the 90-day average zonal averagetemperature T and zonal wind u from a case in whichthe hydrostatic equation was of the scheme Y form,but all other terms were of the scheme X form. Thedifference of the complete scheme X case minus thiscase is also shown. The tropical tropopause temperaturefrom this.combination is 192 K, closer to that of thecomplete scheme Y case in Fig. 1, while the winterstratospheric temperature is 195 K, midway betweenthat of the two schemes. The stratospheric polar nightjet from the combination is overly strong as in thescheme Y case. When integrating to the upper levels of the model,the difference in the geopotential height given by thetwo hydrostatic approximations isqb&(X) - qbk(Y) '-, (rx- rx-i) x lnax-~- lnax (-lnax) + ~ (Tt+~- Tt) l=kk- I/2k In o-k T, ~k+ I/2k+l In O'k+I T, q~ 1,4, InO-K_I T, ~K-I/2K IncrK T, ~K+I/2 qSs ----- = SCHEME Y -- SCHEME XFIG. 2. Graphical representation of vertical grid and hydrostatic integrals.JANUARY 1988 DAVID L. WILLIAMSON 45X [lno. t+w2_ ~1 (lna/+ rk[~(lna~_w2+ lnat,+l/2)--lna~] k>l-,(X) - &l(Y) ,'-- (TK- TK-O [~ lnaK . . .] K-,X lnag~ --- lnaK [--lnaK)] +,=,~ (T,+, - rt) I at+O]. (5)X [lnal+W2 - ~ (lna/+All terms in the square brackets multiplying the temperatures are positive for our choice of a levels. Therefore, the difference in the north-south geopotentialgradient in the upper levels of the model has the form0_~_~ IT'_o~, (x) - ~ ,=~ \' ~ o~,! x (positive coefficient). (6)In either of the two simulations shown in Fig. 1, themagnitude of the temperature ~adient in the noah~1~ re,on incre~s M~ heist in the top three levels,i.e., ~ OT~+~ OT~ < < 0. (7)Thus, the geopotenfial ~adients resulting from the hydrostatic equations Mll mtisfy a- a- (x) 0, (8) ~ (Y) <a~ <and the co~esponding geostrophic zonal Mnd MI1 ~tisfy U~o(Y) > U**o(X) > 0, (9)T U ,, ' !' /x ,'' ~O 70 50 30 I0 -I0 -30 -50 -TO -~O ~0 70 50 30 I0 -IO -30 -50 -~0 -gO LATITUDE LATITUDE ~G, 3. As in Fi8. 1 cx~pt fo~ (top) ~hemc ~ hydros~fic mat~x B and (boSom) ~fferen~ ~ ~heme X, i.e. ~heme X-~heme Y (B).46 JOURNAL OF CLIMATE VOLUMEwhich is consistent with the figures. This relationshipholds with the temperature gradient the same in bothcases and thus partially explains the increase in thestratospheric zonal jet with the scheme Y approximations. However, the temperature gradient also increases even more in the scheme Y case, exaggeratingthe effect on the zonal wind. The geostrophic relationused to explain the increase in zonal wind does notexplain the temperature gradient increase. Figure 3shows that with the scheme Y hydrostatic matrix thestratospheric temperature decreases poleward of 60-and the tropopause temperature increases equatorwardof 60-. It is not possible to explain this temperatureincrease as a first order response to the finite differences.Given the same nonzero constant lapse rate, the integrals of the two hydrostatic equations will producedifferent heights and thus the temperatures at thosedifferent heights will also be different. However, witha 6.5 K km-~ lapse rate, this effect only directly explainsabout a 1 K increase for the scheme Y hydrostaticequation whereas the maximum increase is 8 K between 20- and 30-N. Boville (1984) has argued that when the stratosphericjet increases, the refractive properties of the mean flowchange such that upward propagating energy between65- and 75-N is refracted equatorward near the tropopause more with the stronger jet. Such a processmight result in the relative warming at the tropopausefrom 50-N to the equator and relative cooling in thepolar stratosphere when the hydrostatic equation ischanged. The scheme Y hydrostatic equation primarilyproduces a stronger jet which then changes the refractive properties of the mean flow which feeds back togive a stronger temperature gradient and an evenstronger geostrophic jet. The temperature difference inFig. 3 resulting from the change in the hydrostaticequation is very similar to the temperature differencein Fig. 2 of Boville (1984), both in the winter polarstratospheric cooling and in the warming of the midlatitude and tropical tropopause temperature.b. Energy conversion term The energy conversion term K(Tw/p) is intimatelyrelated energetically to the hydrostatic equation. Theconversion term isK -- = t~T V. V lnps - - (b + V. X7 lnps)da p a (10)where V is the horizontal velocity, 5 the divergence, wthe pressure ve~ic~ velocity, and ~ the ratio of the ~consent for d~ ~r to the specific heat capacity of d~~r at consent pressure (R/c.). In discrete fo~ ~~/~ ~T~ r~. v lne, - Z C~(*~ + %. V lne~). j=l (ll)For conse~afion of energy in the discrete system theconve~ion matrix C is related to the hydrostatic matrix8 by Aaj aj+l/2 -- aj_l/2 'Ckj = Bjk ~,~ = Bjk (12) ffk+l/2 -- ffk-I/2This relationship is satisfied in the fore chosen byECMWF (Bu~dge and Haseler, 1977) and adopted inscheme Y. The scheme X fore, however, is chosen ~a str~tfo~d approfimation to the inte~ ~thoutany energetic considerations (Bourke et ~., 1977). Thetwo fores with the NCAR inteme~ate a level choiceare as follows:Scheme YScheme XCkj= [~k](O'j+l/2-- O'j-I/2)el I ~' [~1] (~1 -- 0)(13)In both,, the terms in square brackets are approximations to 1/a and those in parentheses are approximations to Aa. Note again that the particular form ofthese approximations given here follows from the definition of the vertical grid and is not necessarily thesame for other choices of the relationship between mainand intermediate levels such as that chosen by theECMWF. The largest differences in. the coefficients C of theintegral in the conversion term occur at the top of themodel where C~ = 1 in the scheme X form and C~l= 1.5 in the scheme Y form with the nine vertical levelgrid used here. Elsewhere, the two forms of the generalnon-end term C~,j differ by only 1% for this vertical gridexcept at the second level from the top where theJANUARY 1988 DAVID L. WILLIAMSON 47scheme X exceeds the scheme Y form by 5%. For thegeneral endpoint of the integral, Ckk, the scheme Yform exceeds the scheme X form by 35% in the lowestlayers. This difference decreases as the endpoint becomes the next higher level and changes sign arounda = 0.5, then increases again. These differences areclearly highly dependent on the particular vertical gridchosen, making a general statement comparing the twoforms applied to other discrete grids difficult. Figure 4 shows the time averaged, zonal average ofthe conversion term K Tw/p from the scheme X simulation. It also shows the difference between this conversion term and that which is computed by the schemeY approximations from the same state variables (i.e.,the scheme Y conversion term shown in this figure wasnot actually produced by a simulation; rather, it wasdiagnosed from data produced by the scheme X simulation and does not feed back to any simulation). The KTo) ~ ~;i I ' ' ' ~ ' I' ., 500m ~ 74L~I 0 50O 991 :' I~ 90 70 50 30 10 -I0 -30 -50 -70 -90 LATITUDE ~G. 4. Time aver~, zon~ avemg~ ~nve~ion te~s ~T~/pfrom (top) ~heme X ~m~a6on and (~Rom) ~ffe~n~ ~ ~hemeY approxima~ons appli~ to s~te v~ab]es From ~heme X simulations. Contour inte~s ~e ]2 X 10-6 K s-~ for field and 2 X 10-~K s-~ for different. Ne~ve contou~ a~ dmhcd.largest differences are at the top of the model and aresuch that the scheme Y form has increased heatingfrom 30- to 65-N and increased cooling from 65-Nto the pole. This structure would increase the polartemperature gradient with the scheme Y approximations. Figure 5 shows the time averaged, zonal averagedtemperature and wind fields from a simulation usingthe scheme X approximations except for the matrix Cwhich is of the scheme Y form and the difference fromthe complete scheme X form. (Note that this form ofC is not a priori conservative when coupled with thescheme X hydrostatic equation; it is conservative onlywhen coupled with the scheme Y hydrostatic equation.)The difference shows that with the scheme Y form thetop layer is warmer around 50-N with minimal changeselsewhere. (This difference is not simply that ofp. For example, the polar stratospheric cooling indi.cated in Fig. 4 does not appear in the actual simulationdifference.) The tropical tropopause temperature (10-Sto 10-N, a = 0.074) is 3- colder with the scheme Yconversion term. The conversion term in Fig. 4 showslittle difference at this level. However, at the next lowerlevel (a = 0.189) the scheme Y form produces increasedcooling, decreasing moist convective adjustment andproducing less heating at the a = 0.074 level. As illustrated here, the interpretation of the simulation differences in terms of the discrete approximations is madedifficult by physical patameterizations behaving differently in response to differences produced by numerical approximations. The hydrostatic and conversion approximations donot by themselves completely explain the differencesbetween the simulations with the two complete formsof approximations. The scheme Y hydrostatic equationgives a colder polar stratospheric temperature while itsconversion term tends to warm so that both terms together would be expected to produce a simulation intermediate to those with the complete scheme Y orscheme X forms. This is, in fact,.the case. A simulationwith the scheme Y hydrostatic equation B and conversion term C and differences with the scheme X simulation are shown in Fig. 6. Again, this simulation fallsbetween those from the complete scheme Y andscheme X shown in Fig. 1.c. Vertical advection Another aspect of the numerical approximations thatcontributes to the difference in the simulations is thevertical advection. In general, in scheme Y and schemeX the vertical advection of some quantity ~k has theform\O0'/k+l/2 + (1 - aO a~-,a 0'-~ ~-,/248 JOURNAL OF CLIMATE VOLUME 1 T 9 ~ ~'~1~"+.-~-1.~.~-4~' I I I I I~~oo~o_ 901 9 % r-_ld~'l/9'/lilll II ~1 ~ ' 11 ~ I ' /// X x~/ / / ' N~ // ///-, --.3--;?y/_ / / ,,,.'~'~ " "// / / ,'~ ~ 74~ 0 x~, 8 ;~ ~m 189 500 :> . ~ 991 - ~ ~~1 ~ I ~ I ~ I ~ I ~ q 9070 50 30 I0 -I0 -30 -50 -70 -9090 70 50 30 I0 -I0 -$0 -50 -71 -90 LATITUDE LATITUDE FIG. 5. As in Fig. I except for (top) scheme Y energy conversion matrix C and (bottom) difference with scheme X, i.e. scheme X-scheme Y (C).in which(~) - '-"~+~' - -* (15) ~ffffk+l/2 O'k+l -- O'k -The endpoint values are not needed since b~/2 = bK+l/2= 0. Both scheme Y and scheme X applied t% ourparticular grid use this form for the vertical advectionof u, v, q, and T except the weights a are exactly opposite:ak(X) = 1 - ak(Y) =O'k -- ffk-12A~1 - a~,(X) = an(Y) = ak+~ - an 2Aak 1Ao'k = ffk+l/2 -- tYk-I/2 = '~ (ffk+l -- O'k-l), (16)since, as mentioned earlier, the half-level sigma valuesare taken as the arithmetic average of the full indexvalues. The scheme X form may be thought of as aninterpolative approximation since the weights are inversely proportional to the distance of the half levelfrom the point k whereas the scheme Y form may bethought of as an integral approximation wherein theweights are proportional to the mass or thickness ofthe layers. Both these schemes formally are of the sameorder accuracy. The scheme Y form applied to temperature coupled with ,the corresponding continuity orsurface pressure tendency equation conserves energywith a general grid, whereas the scheme X form doesnot necessarily do so. When the grid spacing is uniformin a, the two forms are identical. All differences insimulations attributable to this term are due to choosinga nonuniform grid and thus the differences dependgreatly on the actual grid spacing chosen. Both schemesformally have the same order accuracy, but in praticalJANUARY 1988 DAVID L. WILLIAMSON 49application the grid is not fine enough to be near theconvergent limit. There are additional differences between the twoschemes besides the weights. Although these other differences are relatively unimportant, they are summarized here for completeness. The approximations tothe integrals in ~ are slightly different: b = a (~ + V. V lnps)da- (b + V. V lnps)d,.The integrals are illustrated graphically in Fig. 7. Aswith the hydrostatic integral, the scheme Y form assumes constant values within layers while scheme Xassumes linear variation between data values with constant values at the two ends. Because ffk+l/2+ *k+0, the integral from 0 to I is the same for anychoice of ak and the integral from 0 to a differs onlyby the small triangular piece from a~ to ~rk+~/2. All the derivatives (O~/Off)k+l/2 are the same exceptscheme X assumes a logarithmic variation for q and Tin the bottom layer. The bottom layer is thin enoughthat this difference has little effect. For the temperatureadvection only, scheme X in addition uses b~ in placeof ~k+~/2 and bk-m. The weights, however, remain asdescribed before; ~ is calculated by stopping the integral illustrated in Fig. 7 at ak rather than ak+U2. ASmentioned above, simulations (not discussed in detailhere) have shown that these last differences describeddo not significantly affect the simulation. The only important difference is introduced by the choice of theweights themselves. In addition, only the vertical temperature advection is important, not that of momentum or moisture. Figure 8 shows the temperature and wind'fields whenthe vertical advection is changed from the scheme Xform to the conserving scheme Y form and the differences of that simulation from the complete scheme Xsimulation. Comparison with Fig. I shows that a largeT U 'tr ~>~'~"~ '"' i~l~glll~,','"~',x'x''-'''',';'';'';'.'' ~ I tlll[x..//tt//t /' xXXt~/I iXXXXX,,,x !,~ \\\\\~-~ I II/\\\~i ~ ~tjl ~ - ' '"" \\\\",~ , , t , Ifl~ J ~ / IIII I / 111 I ~ .. ,,,,...,;,)Y~>- ~o \9x~- -~o / z/~ 8 \~ '"':-/"'~' "'- ~~-~~~'(~-~1 I ~\1i\ k,~5~._.,.' ~00 ~ ! Q ~.~' ) .~, k,--~ k~-X,-.5.~ ,-_~,'NN_/.-r",, *'"7[[llllli ":/'i '~ 1 \x,~/I'1 'i' ",' ' ' I ""~" '"";'"'"";'"'"'[~\',_,':~,,l///'i", ,/, \"' ' ' J' ~ ' ' ' '[ 711111[", / !/~//" .... I ,,,.,...,,,.,,,,,~ ~J//~,i'-; i~--~/ ':'-~-'"'"'" ,, ...... ,',',',--'; I',111\/X ',',',_.'\J l(~)I I I I H I \ ~x~.~,~ '/IX , ~ I / I,, ,,~ ~ '~ ~I - ~ i ~''x I t .-_ I \ 'x I I=~ ~'~- /IT', , '. )'.. '~.'- .... :-I -",,"-' JI I , I -' '- '--.. / ~x ~e, ~,._.' y ---"x" _, ,~O0 ~,, :N .Xx~, ,,.,/h ,,..,, ~, ,,, , ~O 70 50 ~0 I0 -I0 -~O -O -?O -9090 ?O O $O IO -I0 -30 -50 LATITUDE LATITUDE FIG. 6. As in Fig. I except for (top) scheme Y hydrostatic matrix B and energy conversion matrix C and (bottom) difference with scheme X, i.e. scheme X-scheme Y (B + C). m-70 -9050 JOURNAL OF CLIMATE VOLUME 1I/2I11/22A0-1,0-1A0-2 , 0'2k-I/2kk+l/2A0'k~ 0-kEquol AmosK-I/2KK+I/2A0'K, 0'K0'=1 ------ = SCHEME Y = SCHEME XFIG. 7. Graphical representation of vertical grid and b integrals.part of the total difference is attributable to the verticaladvection. As stated above, this difference is, in fact,attributable to the vertical advection of.temperature.Another case (not shown), in which vertical advectionof momentum and moisture uses the scheme Y formbut that of T retains the scheme X form, does not showthis difference. In Fig. 8 the tropical tropopause temperature is 8 K warmer with the conserving scheme Yform for vertical advection and the north polar temperature at the top of the model is colder by 16 K. Thepolar night jet increases to 55 m s-~ and extends downto a = 0.074 eliminating the separation with the tropospheric jet. Note also that the stratospheric easterliesare weaker. Figure 9 shows, the average heating by b(OT/Oa) fromthe complete scheme X simulations and that value minus the average heating from the scheme Y approximations applied to the scheme X simulated state (i.e.,these scheme Y values were diagnosed from the statesproduced by the scheme X simulations and are notthe result of--nor were they fed back into--an actualsimulation). Negative values of b(OT/Oa) representheating. The scheme Y forms gives more heating at a= 0.189 in the equatorial region (larger negative valuesand therefore positive difference) and less cooling at a= 0.074 at 30-N (smaller positive values and thereforepositive difference), yielding in both cases a warmeratmosphere in those regions. In addition, at the toplevel from 50- to 90-N, the scheme Y form acts toincrease the north-south temperature gradient relativeto the scheme X form. Notice, however, that the temperature difference in Fig. 8 does not only depend onthe difference in the vertical advection term. Othercomponents of the model are interacting with it. Forexample, in the equatorial regions (10-S to 10-N) thescheme Y advection by itself (Fig. 9) gives less heatingat a = 0.336 and more heating at a = 0.926, yet theactual simulation (Fig. 8) is warmer everywhere in thatJANUARY 1988 DAVID L. WILLIAMSON 51,500 "r~711111 ' ' '~,L '//' ' 7' ' 'k"' / /HH /-... ~ ,,' ,,, .... ////I/ ...... ~111.- ..... , ' '~~ ~ /,' ,,~: /1~ i ~ ~ ~ i ~ ~1~ ~ / x z4- , , ,-e- ~ , , , ,,, ~ I~ ,' ,' ~ ~ '',%~~ -, , ~ 9 ,, ,," "-' x. I I~g --: , , I ( '~ '-, "~ X . ~,' , ._.' _ ', N~ ~- ,,' ,,-'-'-.~ ',~-.N ~ ~ .... ~ ~ ~ ( ~ ~ ~ . '~ _.-'' '~ ~ ..- i I "- -- ~' ~x - / ~ -- 500 - . ..... . ' .-' , ~-~ ~ x ~% -- ~/ x ~ ?~x . 991 -'n'~F%::~' ~ I ~ ~ I ~'-[- ~ I ~ I "~ 90 _70 50 30 IO -IO -30 -50 -?O -90 90 70 50 30 IO -I0 -30 -50 -70 -90 LATITUDE LATITUDEFK~. 8. As in Fig. I except for (top) scheme Y vertical advection b(OT/O~) and (bottom) difference with scheme X, i.e. scheme X-scheme Y [b(0T/0a)].column when the scheme Y approximations are used.In this case, because of the additional heating at ~= 0.926 the convective adjustment produces greaterheating in the simulation at a = 0.336 which compensates the cooling from the advection. In summary, the differences in the simulations produced by the scheme X and scheme Y vertical approximations are primarily due to the vertical advection of temperature (Fig. 8) and to the hydrostaticequation (Fig. 3). Either of these terms by itself resultsin warmer tropical tropospheric temperatures and acolder polar stratosphere with an overly strong stratospheric polar night jet extending down and mergingwith the tropospheric jet. The energy conversion termon the other hand has only a small effect on these properties. The difference between the hydrostatic equations orbetween the vertical advection can be eliminated byaltering the vertical grid. As shown above, the differencebetween hydrostatic equations is eliminated if the gridis made equally spaced in lna or if intermediate ~ valuesare the geometric average of full levels. The differencebetween the vertical advection is eliminated by takingthe grid to be equally spaced in a, retaining the arithmetic average for intermediate levels. These two modifications are mutually exclusive so some differencesmust remain in one of these two terms as long as thegrid remains relatively coarse. We have performed one additional experimentcombining the hydrostatic matrix B, the energy conversion matrix G, and the vertical advection of schemeY with a vertical grid in which the intermediate levelsare the geometric average of the full levels. Recall, inthis case, that the scheme Y and the scheme X hydrostatic equations become identical, but of course theyare no longer the same as either on our standard grid.The results of the simulation are somewhere betweenthose shown in Fig. 1 with the original grid, but tendingto be more like those of scheme X. The polar night jetis stronger than that of scheme X, but not as strong as52 JOURNAL OF CLIMATE VOLUME 1that of scheme Y; however, the separation from thetropospheric jet remains clear.4. Simulations with 19 levels We have seen in section 3 that with just nine verticallevels the simulation is strongly dependent on thechoice of the discrete vertical approximations. Although formally the schemes considered have the sameorder accuracy, the actual errors with the coarse, nonuniform grid can be quite different. Therefore, it isuseful to know how the schemes converge as the gridresolution is increased. With this in mind we have performed simulations with 19 vertical levels, essentiallyreducing the grid interval by a factor of 2. This 19level grid is defined by adding a level halfway (in a)between the original nine sigma levels, leaving the lowx ~ .I III500 ,, ./ -..~~'~991 '~-*"~9 ' / I ~/' [ ' I[' ']',2'' ] ' t [ '/I L/ /.~ ,~Jo~ 74 I oxoot,s - 189=o ~oo:, ~91 ffi 70 50 30 I0 -I0 -30 -50 -70 -90 LATITUDE ~O. 9. Time averaged, Zonal averaged ve~ic~ tem~mture adve~ion b(OT/Oa) from (top) ~heme X simulation and (bottom) diffe~n~ ~ ~heme Y approfimafions app~ to state v~abl~ ~om~heme X ~mulafions. Note ~at n~five v~u~ of ~OT/Oa) repentheating. Contour inte~s are 10 X 10-~ ~s for the to~ field and1.8 X I0-~ K/s for the ~fference. Ne~tive contoum ~e d~h~.est level as, = 0.991 and the highest as a = 0.009.Because of the sensitivity at the top of the model, threeequally spaced (in a) levels were added between thetop two levels of the original grid. The actual a valuesare given in Table 1. The intermediate half-levels areset to the average (in a) of the adjacent full levels. Inthese 19-level simulations, the horizontal diffusion wasapplied over the entire rhomboidal spectral domain inthe top five levels so that it is applied over the samephysical domain as in the nine-level cases. Otherwise,all other aspects of the model are formally unchanged,only applied over the finer grid. Figure 10 shows the zonally averaged temperatureand wind fields from simulations with 19 levels usingthe complete scheme X and scheme Y approximationsand the difference between them. The only significantdifference in the temperature field is at the tropicaltropopause where the difference is 12 K, similar to thedifference in the nine-level cases. This difference ismainly the result of the minimum associated with thetropical tropopause forming at a = 0.132 with schemeY rather than at a = 0.074 as with scheme X. Detailsof the difference are thus highly dependent on the exactdistribution of levels in thatarea. The major differencein the north polar stratospheric temperatures seen inthe nine-level simulations has been eliminated by increasing the vertical resolution to 19 layers. The schemeX approximations give a polar temperature there whichis 4 K colder than that of scheme Y, with scheme Ygiving colder temperatures by 4 K at 70-N. Note thatthis north-south polar stratospheric temperature gradient in the difference is opposite that from the ninelevel simulations in Fig. 1. However, these differencesare much less than the natural variability of the originalnine-level scheme X simulations. The polar night jetis slightly faster in the scheme Y case even though thetemperature gradient is less. This difference is likelydue to the different hydrostatic approximations described earlier. In any case, with just this pair of experiments it is impossible to distinguish this differencein the jets from the natural variability in that region,assuming that the variaiblity in the 19-level simulationsis the same as that of the nine-level. The previous experiments were carded out by simplyincreasing the vertical resolution to 19 levels, with allparameterizations applied to this new grid. This impliesthat the forcing due to physical parameterizations suchas the radiation and clouds was done on a finer scalethan the original nine-level experiments. The resultsare affected by this difference in the physical parameterization as well as by the numerical approximationsfor the dynamics. In an attempt to keep the forcing onthe original nine-level scale and to consider the convergence of the numerical dynamics with finer resolution, we repeated the previous experiments using 19layers for the dynamics and the original nine-layer'subset for the radiation. This is an attempt to provideinformation about the convergence of the vertical finiteJ^NU^RY1988 DAVID L. WILLIAMSON 53Xu,J b'~ xg~741895OOT U 9 ;~\~,~/~- ~, ~/x~, ~, ~, ~ , ' ~1 l\',J//--', / \ ..2 ,; l\ I>' . (, , \LL-J ~~ ' ', ~ i~ '~'_::--'-\ /-/~: ' -.,; / ....... -'=--. '-~.&-t _ ~ / / ~--- ~_~ ~ ~ ~~ b --::- < ',, 'Z- :-::-~q'x x, _-~ .=_:~ / / ~ ~ "--'.=::::-'~-;, / ~. --I x 74 ~ '',"~"xx~ -"-:~ ~ )~ ) /' ,,--'-'~--~--I I g .'",,.,/ / ~ ....... :_'-'.- \ Ix~ ~ ~ .... ---,-.~.--- (~ ~ 500 ~ 991 ~ I 90?O 50 30 I0 -I0 -30 -50 -70-90 70 50 30 IO -I0 -30 -50 -70 -90 LATITUDE LATITUDE FIG. i0. As in Fig. I except for 19-1ev-1 simulations with (top) scheme X and (middle) scheme Y approximations, and (bottom) the difference scheme X-scheme Y.difference approximations while keeping the forcing atthe original nine-level resolution. Some subtle aspectsare associated with such an attempt. The state variablesrequired by the radiation are taken to be the valuesfrom the common nine levels. The heating rates at the19 levels are calculated from the nine-level subset usinglinear in'lne interpolation. The convective adjustmentis applied to the complete 19-level set. The parameters54 JOURNAL OF CLIMATE VOLUME 1from these processes needed for the cloudiness calculation must be reduced to the equivalent nine-levelsubset. Therefore, for this purpose, we assume that ifadjustment took place in a subset of one of the originalnine layers, for the purpose of nine-level cloudiness weassume it took place throughout the layer. We will refer.to these experiments as 19/9 level simulations. Figure 11 shows the zonal average T and u fieldsfrom 19/9 level simulations with the complete schemeX and scheme Y and the differences between, them.As with the regular 19-level simulation, the large difference in the polar stratospheric structures that waspresent with nine levels is not present with the additional resolution. The temperatures are quite close,While scheme Y produces a slightly stronger jet thanscheme X, but with just one simulation for each, thisdifference cannot be distinguished from the naturalvariability. The tropical tropopause temperatures remain different by about 10 K. Again, this difference isassociated with the tropopause forming at a lower levelwith scheme Y than with scheme X. In both the 19level and 19/9 level simulations the vertical advectiongives more cooling in the equatorial troposphere in thescheme X simulation than in the scheme Y simulation.The vertical gradient of the cooling between a = 0.811and a = 0.236 is larger in the scheme X case as well.This relative cooling structure would le~d to more vigorous moist convective adjustment in the scheme Xcase in the equatorial region. In addition, scheme Ytends to give relatively more cooling at a = 0.132 thanat ~ = 0.074 compared to scheme X. This cooling differential may provide enough stabilization to preventthe moist convective adjustment from penetrating upto ~ = 0.074 in the scheme Y case, leading to the lowertropopause and thus temperature differences in thatregion. Comparison of Fig. 11 with Fig. 10 shows that thetropical tropopause temperature differs by 6 to 7 Kwith the different radiation scheme, i.e. the 9- or 19level radiation coupled to the 19-level dynamics. Evidently, the actual temperature is also highly dependenton the physical parameterizations and, therefore, it isdifficult to isolate the convergence properties of thenumerical approximations themselves.5. Simulations with 37 levels We have also performed a pair of 37-level simulations by placing a level halfway (in a) between each ofthe 19 levels, leaving the first and last levels as in the19- and 9-level cases. These 37 levels are also shownin Table 1. The horizontal diffusion was applied overthe entire rhomboidal spectral domain in the top eightlevels so that again it is applied over the same physicaldomain as the 9- and 19-level.cases. In the 37-levelcase, the nine-level radiation and clouds were used inthe same way they were used in the 19/9 level experiments, i.e., the radiation and clouds were calculatedonly on the nine-level subset. Again, this strategy waschosen to attempt to keep the forcing on the originalnine-level scale and thus to provide information aboutthe convergence of the vertical finite difference approximations. Figure 12 shows the zonally averagedtemperature and wind fields from these 37-level simulations using the complete scheme X and scheme Yapproximations and the difference between them. Thesimulations have not converged beyond that seen inthe 19-level simulations. In fact, the difference in thetemperature field has increased to 24 K above the tropical tropopause, although the tropopause temperatureitself has converged to within I K at a = 0.103. Notethat this level is the one added between a = 0.074 and0.132 of the 19-level cases, and that this tropopausetemperature in the 37-level simulation is between thoseof the 19/9-level simulations in Fig. 11. The equatorialstratospheric eastefiies in the 37-level simulations alsodiffer with scheme Y being about 10 m/s less at a= 0.074 and 25 m/s less at a = 0.009. These differencesare possibly significant. Scheme Y at the other resolutions has consistently tended to give weaker equatorial stratospheric eastefiies than scheme X. We havenot emphasized this, since to say they are significantwould require a large number of additional simulations. The differences in the north polar stratospheric temperature and wind are also larger between the 37-levelsimulations than between the 19-level simulations.However, these differences between the 37-level casesare comparable to various extremes found within verylong simulations with the standard nine-level versions,namely, 14 K temperature differences and 35 m/s winddifferences. In fact, the two cases look like the oppositeextremes observed in 90-day averages from a very longnine-level CCM0 simulation. Scheme X gives thewarmest temperature (205 K) and weakest winds (20m/s) and scheme Y the opposite (190 K and 55 m/s).With only one sample of each case we are unable toargue that these differences are significant and thereforewe assume the simulations have converged in this region. Since the 19-level simulations seem to indicate convergence, the differences that appear with the 37-levelcase are somewhat troublesome. They may be indicative of a shortcoming in our experimental strategywhich attempts to allow finer-scale numerical-dynamical motion while retaining the original coarse scaleforcing, which nevertheless remains interactive. Thephysical forcing calculated on the coarse grid cannotadjust to the fine scales of the dynamics. The 19-levelexperiments were an attempt to validate this approachand seemed to do so, but the difference in scales between 37-level dynamics and 9-level physics may betoo great for it to remain valid.6. Best nine-level approximation to convergent solution Assuming that the 37-level simulations represent aconvergent solution of the discrete problem exceptJANUARY 1988 DAVID L. WILLIAMSON 55T U Ixbx 74~/JjjJlitllllllll ",,]1 ,~ Ill ?1 II ~l' I' III/111!~ - II~,l I / llll i~ I I j '//fillll.,,,,, ..., ,,, ~I/Hill/I/; I - II, ~/11111!~ ~ "-, / I~; / /fill',','.--. ', ,.' ,,.,,'ILk,',,','-,,',,-',',',',' .--~%'x'%--", '- -', ,'~--. ~ J / ~ xX\\\ ', '.'.'.,F/.-~ ~,189500991 901 ~///j~/~ I"'- I ' I ~ I ~ I ~ I I I i.... ::::-.,,,%,,//1.,'. ~:...:. _ . : __. ', ,,--.J/!t;':-.~-::-'---=-----=~'.'. '. '.~ I1~1 I '.-~-'-----'-- ''~'' ~ ~ x \ " . I1!I' ~ \ *"\ ~ ~.% ~ \ "'----. I ~',~, ,._ ~ ~;);.~ - ---.,~?\~ .,-:zz---_-:;..'.:.,~',.-~ ~ ~ ', '- :-_E-_C-:=:.; -/.' / I ,. l! I~ ~\ I I 70 50 ~ I0 -I0 -$0 -50 -70 -90LATITUDEFIG. 11. As in Fig. 1 except for 19/9-level simulations (i.e., radiation and clouds evaluated on a nine-level subset)with (top) ~chcme X and (middle) scheme Y approximations, and (bottom) the diffcrenbc scheme X-scheme Y.above the equatorial tropopausc, it is of interest to knowwhich scheme applied at the original nine levels is thebest approximation to this convergent solution. As faras the stratospheric polar night jet is concerned, schemeX seems clearly to be a closer approximation to theconvergent solution although a little on the weak side.56JOURNAL OF CLIMATEVOLUME 1The nine-level scheme Y simulation consistently givesthe extremely strong stratospheric polar night jet. Infact, the two schemes converge in this region when theresolution is doubled to 19 levels. On the other hand,in the nine-level simulations, scheme Y conserves energy in producing its simulation whereas scheme X has.~.o O741895OO991T U:/ I~ I" ~[; I I T"I' \ I [ 'PI[ I1, l! P% //~li~\\ I Ill! I 1IIiIIt%'~\\ I I1'1 ~ I I I,~oo991 91~/ff~ __~-~--~->-~-'~'~--~.' ,I ' I i~titq','~/l~il' i' '' 'l;"[V'% ' 'i ~' ~ / l ~3:~z:~LCz--~:~- - .... :===-~. -, ,, ,,~, ~,,,,, , ,, , ~ ,:; , , J,,,,,,,.,.,,.,,,,, ,,,,, ,,,,, ~/J/4~ ~i h~"k ,'ht't ~ ,'., ; i ',__/ i / , ~ I~/Z~$)v';:-::'~':-'--~'"x.----.----- -.x xx"~' ~ ~-, -~ I~h~t",zilll ..../'., -~%. , / t ~ /~,,~,', '.;:::'::====:-~eqx ~ , ~ ~-' ~ F~ ~X'-~gll I ,'/,"-/' .'.",' ." .~. ~...~M!/~i~, ~,, .... ::::::~..,.. ,',, - 1~ ;'~ - .... ~LUzl/~tlt ~ q F~',~x~o~ttt ~ t ~ x_ /~- .z / ~ P /liJhli kXxx~_Lt ...... ~/~//~;~1/5~1// i I q PI~x Iiii I I x x_/ //.- / i ~ (l[l[[l' x---ee----e.g~fi///. / t d N~~' I,I ~ ~ '---// / / ~x~',,: >~ .... ~=-:d---' P ~',X ~ i, , '-, ./- .... / ~k:Q~ r x-.-; , ~ ' -'_, k ' _--. /~4 ~.~~_, K ~.X i , ~ ~- ,/,, ~.. ,. / kXX', ll~k I ,'r % / ( i ) ' %J I ~ Ime ~, ~ ~ ~, ,I ( ~ l ~L ,~ m' In f ,.: V/if' i kL , ) , m , m , m , ~ ~L~( J m ~m ~ ,k~ , m , m , gO ?O ~O )O IO -me -)o -~o -lo -go9o 7o ~o )o IO -IO -)O -~o -?O -~o LAT mTUDE L~TmTUoE~o. 12. As in ~ig. 1 except for 37/9-1oval simulations (i.e., m~ation and clouds ev~uated on a nine-level sub,t)~th (top) ~heme X and (mid~e) ~heme Y approximaUons, and (bottom) the difference scheme X-~hemo Y.JANU^RY 1988 DAVID L. WILLIAMSON 57an unaccountable energy loss of about 10 w/m2 withone-third of that attributable to the moisture component of the system. A cursory look also indicates that scheme X betterprovides the tropical tropopause temperature. However, the actual value of the temperature is establishedby a very delicate balance which is critically dependenton the exact placement of the a levels and on the physical parameterization. Therefore, this conclusion is lessdefensible.7. Conclusions The time averaged, zonal averaged state producedby the nine-level NCAR Community Climate Modelis highly dependent on the choice of vertical finite difference approximations. We have compared the simulation produced by the standard CCM0 approximations, taken to be those developed at 'ANMRC anddenoted scheme X here, with a set obtained by applyingthe general form developed at the ECMWF to our particular vertical grid; The latter approach is denotedscheme Y. We note that our results are highly dependent on the vertical grid structure and that our resultsare therefore not necessarily representative .of whatwould be produced by the ECMWF model itselfi With the standard CCM0 nine vertical levels, schemeY produces minimal separation between the winterstratospheric and tropospheric jets with an unrealistically strong polar stratospheric jet and associated overlycold winter polar stratospheric temperatures. SchemeX produces a strong separation between these jets withzonal wind speeds and zonal average stratospherictemperatures in agreement with observations. In addition, the tropical tropopause temperatures differ byabout 10 K in these two simulations. On the otherhand, scheme Y conserves energy in the simulationwhile scheme X has a 10 w/m2 unaccountable energyloss. In considering the effect of the discrete vertical approximations, we concentrate on the examination ofthe zonal average wind and temperature fields sincethese fields provide a summary of the primary differences in the simulation. The differences in these fieldsappear mostly in the stratosphere, but it must be remembered that these structures have a strong influenceon the troposphere. As discussed in the Introduction,the polar night, jet structure in an atmospheric modelinfluences both stationary and transient waves in thetroposphere (Boville, 1984). The stratospheric differences considered here are very relevant for the prediction and simulation of the troposphere. The various components of the vertical differenceswere considered independently to establish which wereprimarily responsible for the differences in the simulations. The individual components were changed to thescheme Y form, one at a time. Simulations with anindividual component changed were then comparedto the complete scheme X simulations. It is importantto remember that only the component being discussedin the following paragraphs is different. The changeswere not made in a cumulative manner. The basis ofcomparison is always the complete scheme X simulation. The discrete scheme Y hydrostatic equation wasshown to produce stronger polar stratospheric geostrophic jet speeds from typical stratospheric temperatures than the scheme X equation. The geostrophicargument by itself, however, does not completely explain the differences in the simulation. It was then argued, following Boville (1984), that such a stronger jetwould change the refractive properties of the mean flowresulting in even stronger temperature gradients andjet speeds in the stratosphere. The discrete form of the energy conversion termK Too/p has only a minimal influence on the simulations.The largest differences in the integrals involved are dueto the approximations at the ends of the integrals. Thescheme Y form produces a slightly warmer top levelaround 50-N, and a slightly colder tropical tropopausetemperature. The tropopause temperature differencearises from the interaction of the moist convective adjustment with the differences caused by the numericalapproximations rather than directly from the numericalapproximations. This illustrates the difficulty in isolating the direct effect of the numerical approximationsin a complicated nonlinear model such as the CCM. A large part of the difference in the simulations produced by the two schemes is also attributable to thevertical advection of temperature. The scheme Y vertical temperature advection results in warmer tropicaltropopause temperatures and colder winter polarstratospheric temperatures with a large increase in thestratospheric jet speeds, Scheme Y tends to increasedirectly the north-south stratospheric temperaturegradient. Experiments were performed at higher vertical resolution to assess the convergence of the two schemes.Doubling the resolution to 19 levels resulted in convergence of the winter polar stratospheric simulation.'Scheme X on the nine-level grid approximated thisconvergent state much more than scheme Y. The tropical tropopause temperature did not yet converge withthe 19-level resolution. The tropopause formed at different levels in the two schemes, exaggerating the difference. Doubling the resolution once more to 37 levelsresulted in convergence of the tropical tropopausetemperature; however, the tropopause formed at a levelunique to the 37-level case. The temperature above thetropical tropopause did not converge. In fact, the difference became larger than in the 19-level cases. Whenconsidered only on the nine-level subset, the convergentsolution again matches the nine-level scheme X more -than scheme Y. This is perhaps a less meaningful comparison, however, since the physical parameterizationsplay such an important role in the delicate .balanceresulting in the tropopause formation.58 JOURNAL OF CiLIMATE VOLUME 1 Although scheme X at nine levels most closely resembles the higher resolution simulations, retainingscheme X on the nine-level grid does not seem to bea satisfactory basis for climate modeling. It does notconserve energy in the long simulations and the energyimbalance of 10 w/m2 exceeds the signal of some phenomena to which investigators would like to apply suchmodels. It is very disturbing to have the simulationdepend so highly on the discrete approximations. Amore desirable basis is to increase the vertical resolutionand decrease the dependency on the discrete approximations. The complete 19 levels used here may not.be required. Most of the difference arises at the topand is associated with the nonuniformity in the ninelevel grid there. It may prove satisfactory to increasethe resolution and equalize the grid only in the upperpart of the domain. Acknowledgments. The model code used for the experiments reported here originated with an adiabatic,inviscid version of the spectral model developed at theECMWF by A. P. M. Baede, M. Jarraud, and V. Cubasch. I would like to thank the ECMWF for makingthat code a,~ailable to us. I would also like to thank B.Boville for helpful discussions; R. Dickinson, R. Errico,P. Rasch, D. Burddge, C. Girard, and R. Rosen forcomments on the original manuscript; G. Williamsonfor running and processing all the experiments; andM. Niemczewski and R. Bailey for typing the manuscript.REFERENCESBaede, A. P. M., M. Jarraud and V. Cubasch, 1979: Adiabatic for mulation and organization of ECMWF's spectral model. ECMWF Tech. Rep. No. 15, 40 pp. [Available from ECMWF,Shinfield Park, Reading, Berkshire, RG2 9AX, England.]Bengtsson, L., 1985: Medium-range forecasting--the experience of ECMWF. Bull. Amer. Meteor. Soc., 66~ 1133-1146.Bourke, W., 1972: An efficient, one-level, primitive-equation spectral model. Mon. Wea. Rev., 100, 683-689.--, 1974: A multilevel spectral model. Part I: Formulation andhemispheric integrations. Mon. Wea. Rev., 102, 687-701.--, B. McAvaney, K. Puri and R. Thurling, 1977: Global modeling of atmospheric flow by spectral methods. Methods in Compu tational Physics, Vol. 17 of General Circulation Models of the Atmosphere, Academ~c Press, 267-324.Boville, B. A., 1984: The influence of the polar night jet on the tro- pospheric circulation in a GCM. J. Atmos. Sci., 41, 1132-1142.Burridge, D. M., and J. 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