442 MONTHLY WEATHER REVIEW VOLUb4E 114Surface Temperature Calculation in Atmospheric Circulation Models with Coarse Resolution of the Boundary Layer YVES DELAGERecherche en pr~vision numdrique, Service de !'Environnement atmosphdrique, Dorval, Quebec. Canada(Manuscript received 24 January 1985, in final form 16 September 1985) ABSTRACt A para.meterization of the surface nocturnal inversion of temperature is p~oposed to enable atmosphericcirculation ,models to handle the surface energy budget without having to resolve the boundary layer. Thescheme allows a wide range of vertical resolutions for the host model. The principle is to replace the l~aditionalinstantaneous flux-profile relationship in the surface layer by a time integrated heat conservation equationlinking the surface hen! flux to the amplitude and thickness of the temperature inversion. The model is able toreproduce successfully the mean diurnal cycle of the Wangara data, using the observed vertical profiles tosimulate atmospheric models with various resolutions. Unbiased surface temperature results are obtained fromruns in which the information from the host model is taken at a heisht ranging from 40 to 625 m above thesurface.1. Introduction The surface temperature over land is closely connetted with most of the components of the surfaceenergy budget, namely the outgoing long-wave radiation, the sensible heat flux, the latent heat flux, theground flux and the melting heat, if any. It also affectsthe surface stress by defining the thermal stratificationnear the surface. The determination of a correct surfacetemperature is therefore very important in atmosphericcirculation models (ACMs), for which the main drivingforces originate at the surface. This importance hasbeen recognized by most modelers in their effort toresolve the boundary layer with more levels than inthe rest of the troposphere. An increased resolutionnear the surface allows a more correct determinationof surface fluxes and surface temperature, but at a highcost since a sizeable portion of the model is then devoted to the boundary layer and the time step may berestricted. We present here a formulation that links the surfacetemperature and heat flux to the temperature at thelowest level in the ACM. This level is not restricted tolie within the surface layer, as is usually the case whensurface layer similarity theory is used, but may be ashigh as several hundred meters above the surface. Otherformulations (e.g, Deardorff, 1972; Suarez et al., 1983)use bulk properties of the boundary layer to derive thesurface fluxes, but the present scheme differs in thatonly the lowest ACM layer is considered. Also, the widerange of possible thicknesses for that layer allows maximum freedom for the vertical structure of the hostACM. An earlier study (Delage, 1985) provided diagnostic relationships between surface fluxes and interiorquantities; in this paper, special attention is given tothe nocturnal inversion, and a time integral ofthe surface heat flux is used to produce a more realistic diurnalvariation of surface temperature. This feature is of particular interest for forecast models but the scheme maybe used also in general circulation models where highresolution is often prohibitively expensive. (The termACM stands here for both types of models.) Since ACM variables from only the lowest layer ofarbilrary thickness are required by the scheme, theWangata data (Clarke et al., 197l) with appropriatevertical averaging will be used to test the formulationat various resolutions. The present formulation is nota complete boundary-layer parameterization; it essentially links surface quantities with variables at the firstinterior level of an ACM. Suitable eddy diffusion orother parameterizations must be incorporated into theACM to simulate the vertical exchanges between theinterior levels.2. A model of the nocturnal inversion The traditional relationship between the surface kinematic heat flux Qo and the potential temperaturedifference 0 - 0o between a level z and the surface isthe following: Oo = -u, Cr(O - 0o), ( l )where u, is the friction velocity and Cr is a transfercoefficient dependent on the thermal stratification ofthe surface layer and on the height z above the surface.In unstable stratification, Ca- approaches an appropriateconstant as z increases, a formulation consistent withthe well-mixed structure of the daytime boundary layerc 1986 American Meteorological SocietyFEBRUARY 1986 YVES DELAGE 443in which 0 is a constant above the surface layer (see,e.g., Delage, 1985). As long as z lies within the unstableboundary layer, (1) can therefore be used to link thesurface heat flux to the temperature. The situation is different in stable stratification. According to surface layer theory, Cr is given by anexpression such as (e.g., Delage, 1985)Cr = k(l - a Ri)/In(z/zo),(2)where k is the von Kfirmfin constant (=0.4), z0 theroughness length, Ri the bulk Richardson number(=gz(O - Oo)/OU2), U the wind speed at level z, a thecoefficient of the linear term in the log-linear profile(m5) and g is the acceleration due to gravity. It is clearfrom (2) that Cr becomes zero as Ri approaches 1/aand that (2) is meaningless for Ri > l/a. Delage (1985)has shown that the frequency of events where Ri exceeds l/a in the Wangara data increased with heightand reaches 40% at about 100 m. If an ACM is runwith its lowest level around or above 100 m, theexpression (1) with Cr given by (2) is dearly inadequate. To alleviate this difficulty, two approaches will betested here: one is to replace (2) by an expression forCr which remains positive for all values of Ri (e.g.,Deardorff, 1972; Louis, 1979; Delage, 1985); anotheris to replace the instantaneous relationship (1) by aformulation in which Cr does not appear. We presentin this section the second approach: an alternativeto(l). - The proposed relationship (similar to the one described by Stull, 1983) links the cooling that has takenplace in the surface inversion from the beginning ofthe night (ti) to In, the surface heat flux integrated overthat period. It is expressed as (0 - O,)dz = I, - Qodt, (3)where 0 - Or is the temperature change from a referencepotential temperature profile Or unaffected by the turbulent transfer to the surface. To ensure the validityof(3) the turbulent heat flux must vanish at the top ofthe nocturnal boundary layer (NBL), h, at all times;this constrains h to be a monotonic function of time.The reference temperature profile 0r is modified by factors other than turbulent transfer to the surface, whichdo not appear explicitly in (3), e.g., advection, radiation,horizontal diffusion and transfers to the free atmosphere. We now make the assumption that Or is constantin the vertical up to h and we can rename it 0n sinceby definition 0 = 0r at z = h:Or = Oh. (4)The assumption (4) is somewhat crude in that it negiects the vertical structure of the radiative cooling,often more pronounced near the top of the turbulentNBL (Andr6 and Mahrt, 1982) or very near the surface(Garratt and Brost, 1982). To some extent these deficiencies will be compensated for by our choice of thetemperature profile to be imposed on the inversion.Also, (4) affects the host ACM only within its lowestlayer (as will become clearer shortly); if more than oneACM level is present within the NBL, a different cooling (radiative or other) is allowed in each layer. A universal shape for O(z) is now specified so thatthe left-hand side of (3) can be evaluated to yield anequation involving only the surface temperature andthe thickness of the NBL. A few of the functions whichhave been proposed for O(z) are plotted in Fig. 1. Oneis Stull's (1983) exponential profile, another is Yamada's (1979) cubic profile. Wetzel (1982) and Delageand d'Amours (1979) used a linear profile. The linearcurve is closer to the response to turbulent cooling alone(e.g., Delage, 1974), while the other two simulate theeffect of radiative cooling at the top of the NBL (Garmttand Brost, 1982; Andr6 and Mahrt, 1982) by a smoothmatching with the temperature profile above the NBL.A linear profile was first chosen (Delage and D'Amours,1979) for consistency with observations and modelingresults in the turbulent NBL. In our experiments withthe Wangara data (sections 3 and 4), the profile in(z/h)t/2 given byNOCTURNAL INVERSION PROFILES ........... stun Oaea)1.4 Yamada (1979)1.2 Wetzel (1982)1.0 present paper0.O0.4 ~,7."0.3o,o , ....0.0 0.51.0NORMALIZED POTENTIAL TEMPERATURE FIG. 1. Various functions used to define the nocturnal potentialtemperature profile normalized by the amplitude of the inversion.The ordinate y may be taken to be z/h although the relative scalingof the functions was arbitrarily done by the author. The functionsare 0 --- t - e-ay (Stull, 1983), 0 = 1~- (t - 2y/3)3 (Yamada, 1979),0 = 4y/3 (Wetzel, 1982) and 0 = -y (present paper).444 MONTHLY WEATHER REVIEW VOLUME 114 l Oo + (On - Oo)(z/h)'/*, z < h 0 = ton, z >~ h, (5)and shown in Fig. I will be seen to yield better results.(See Fig. 4.) The larger vertical variation near the surface in (5) compensates for the neglect in (4) of a stronger radiative: cooling in this layer (Garratt and Brost,1982). Also, the logarithmic profile of the surface layeris better simulated by (5) than by a linear profile. Both(5) and the linear profiles have a distinct NBL top,although the higher height h in (5) may exceed thetruly turbulent NBL. Note that this profile is not imposed on the ACM; it is used to derive a relationshipbetween the surface temperature, the mean temperature of the lowest ACM layer and the time integratedsurface flux. With this definition of 0, the integral on the lefthand side of (3) is performed yielding h ~ (0h - 0o) = -1.. (6) When compared to (1), the new integral relationship(6) has replaced the dubious transfer coefficient Cr byh, the thickness of the NBL which is yet to be determined. But another unknown appears, 0h, the potentialtemperature at the top of the NBL (or reference temperature), which must be related to the temperature inthe ACM. This question will be addressed first. To make this inversion model applicable to ACMswith various vertical discretizations, let us define 0z asthe mean potential temperature of a layer of thickness2z adjacent to the surface; this thickness normally representing the layer of influence of the lowest level inthe ACM: Oz = ~z 0az. (7)Since the ACM is affected by all' of the factors changingthe temperature, including the turbulent cooling to thesurface, 0z is the actual mean potential temperaturewhile 0z - Oh represents the vertical average over 2z ofthe effect of surface turbulent cooling during the night. Using (5) in the definitio, n (7) of 0z, the followingrelationships can be obtained: 0~ - 00 ~ = F, (8) On - Oowhere f(8z/9h)m, z ~< h/2 F 1-h/6z, z>~h/2.The fraction F is schematically illustrated in Fig. 2 .where 0z separates the hatched area into two equal parts.Note that h and z are completely independent of eachother. Ifz is small compared to h, the ACM can resolvepart of the inversion and the present parameterizationis used only to extrapolate below level z. If 2z > h (asheight~.zho Oo x*, x%,-0 FiG. 2. Diagram showing the relationship (8) between the meanpotential temperature Oz averaged over a layer of thickness 2z andthe potential temperature On at the top of the NBL h. The 0~ approaches0k when h is small compared to z while 0z approaches 0o when z issmall compared to h.in Fig. 2) the surface inversion is completely eml~xldedin the lowest layer of the ACM. Using (8) to replace On - Oo by 0~ - 0o in (6), weobtain the final expression for the nocturnal inw:rsionmodel: -3Fin 0z -0o - h (9) Equation (9) states that the difference between themean potential temperature of the first layer m theACM and the surface potential temperature increasesas the negative of I,, i.e., as the energy extracted byturbulence from the atmosphere during the night buildsup. The factor Flimits this difference to the appropriatefraction of the inversion occupied by the first ACMlayer, e.g., F would be small for a high vertica/l resolution ACM while it approaches unity at low resolution.Hence, the errors introduced by the crudeness of themodel decrease as the resolution of the ACM increases.But the usefulness of the scheme also decreases at highresolution since more and more of the NBL structureis then governed by the exchange between the levelsin the ACM. Finally, the NBL thickness h sets the., depthof the layer in which the inversion will develop as aresponse to the thermal forcing I~; the larger h is, thesmaller the temperature difference will be since thegiven amount of energy will be spread to a greaterdepth. To complete the formulation, we must define a suitable expression for the NBL top. Since h is neededfrom the beginning of the night when the NBL growsrapidly (Delage, 1974), we choose a prognostic equationof the form (Nieuwstadt and Tennekes, 1981):FEBRUARY 1986 YVES DELAGE 445 Oh he- h ~, (10) 0t rwhere he is the equilibrium value and r a suitable timescale. We tested the following two formulations for headapted from the expression for h in Arya (1981): he = h, + c,u,/lf, (11) [ Lu. \m he = h2 + c2[,i-~] , (12)wherefis the Cofiolis parameter and L the Obukhovlength. The constants h~, c~, h2 and c2 together with rand the initial value of hi at t = ti will be determinedfrom our experiment described in section 3. Strictlyspeaking, since only turbulent transfer is involved in(9), the thicknesses of both the inversion layer and theturbulent layer should be given by h. In actual fact,because of the choice of the profile for 0, Eq. (5), andour assumption (4) which tend to underestimate 00,the NBL thickness h used in this model must be greaterthan the extent of the fully turbulent layer. The valuesof h~ or h2 are therefore expected to be larger thanthose found by Arya (1981). (See section 4.) Althoughthey are probably also site dependent, it was not possible to determine their geographical variation in thisstudy. The nighttime phase of the diurnal cycle has beendefined by (9), (10) and (11) or (12). To be complete,our model of the nocturnal inversion must also describethe relationship between the surface heat flux and thetemperature during the morning phase when the inversion is being dissipated. In Fig. 3 the area under thecurve to the left of 0b is proportional to the heat supplied by the surface since t = t~ when Q0 became positive. This is given by f (0~) - Oj)dz = Im ~ Oodt, (13)where 0j is the potential temperature profile at t = tjand ht is the top &the well-mixed layer with a constantpotential temperature equal to 0b. Here, Im is differentfrom In only in the origin of the integration: the nighttime cumulative energy In (a negative quantity) originates at t = ti while the morning energy Im (positive)originates at tj. The use of Oh instead of 0o indicates thereference to a fixed temperature 0z~ at t~. The true surface temperature 0o is taken with reference to the current 0z in the following way: Oo = ob + o~ - 0~ - U~/2z).(14)The last term removes from 0~ the contribution fromthe surface flux, which is already included in 0~. Therelation between ht and hi, the top of the inversion atthe beginning of~e morning phase (ti), is geometricallyfound to be 0b - 0oj 2 h, = h,(oh). (15)height FIG. 3. The potential temperature structure during the morningphase of the diurnal cycle when a mixed layer of thickness ht is growing. The hatched area is proportional to the heat which has beentransferred to the atmosphere from the surface since the beginningof the morning phase when the surface temperature was #0j. Thetemperature difference #~ - #0j as well as the height h~ at sunrise areheld fixed during the morning phase. To solve for 0~, 0h -- 00j is replaced by the knownquantity (0~ - Oo)/F using (8), 0j is given its analyticprofile from (5) and the left-hand side of (13) is evaluated to yield Oo) V Ira) . 16) Ob = 0o~ + \ ~,Th~ ( It might be worth noting that during the morningphase the surface heat flux is upward in spite of thefact that the bulk potential temperature gradient seenby the ACM (0~ - 00) is positive. This is not possiblewith the conventional formulation (1), where thedownward heat flux causes an unrealistic warming atthe surface. This problem is discussed in the followingsections where the present inversion model will betested against the Wangara data and compared with aformulation using (l) but with an expression for Crdifferent from (2).3. The experiment In this section we describe the numerical experimentthat was designed to test the inversion model. TheWangara data are presented, additional equations forthe surface energy balance, evaporation, surface stressand transfer coefficient for the unstable case are discussed and three model configurations are defined inalgorithmic form. The purpose of modeling the nocturnal inversion isto enable ACMs to realistically manage the surface en446 MONTHLY WEATHER REVIEW VOLUME 114ergy budget without the need of extra levels near theground. It is therefore important to test the ability ofthe model to correctly respond to a wide range of vertical resolutions. The Wangara data (Clarke et al., 197 l)offer that possibility since wind, temperature andmoisture are available up to 2 km, together with information on the net radiative flux at the surface andthe ground heat flux. The vertically averaged 0z, simulating the mean potential temperature for the lowestlayer of an ACM, is thus available up to 1 km. Thereader is referred to Delage (1985) for details on theaveraging. The complete diurnal cycle is simulated inthe experiment and verification is done hourly againstthe given surface temperature. Two periods were chosen in which the few missing pieces of informationcould easily be interpolated: 0900 LST on day 6 to1200 LST on day 21 and from 1500 LST on day 25to 0200 LST on day 41, about 31 days in all. Windinformation is given hourly, and the boundary layertemperature and specific humidity were interpolatedat every hour from the three-hourly ascents using cubicsplines. The surface net radiative flux Rn and the fluxinto the ground G are given hourly. These quantitiesare interpolated to the time steps of the models(20 min). The surface energy budget is given by an - G - pcvQo - pL,,Eo = O, (17)where E0 is ihe kinematic water vapor flux, Lv thevaporization heat, p the density and cv the specific heatof air at constant pressure. The moisture flux is givenby Eo = -u,Crw(qz - q0), (18)where qz is the mean specific humidity for the layerextending from the surface to level 2z and q0 is thesaturation specific humidity at the surface. The soilmoisture parameter w is the fraction of the potentialevaporation estimated at 0.05 for the Wangara site.When qz > qo, i.e., when condensation occurs, w isequal to unity. The friction velocity is given by u. = C~tlUzl, (19)where Uz is the mean wind averaged in the same wayas 0z and qz. The transfer coefficients CM and Cr are derived from surface layer theory for the unstable case (Delage, 1985) and calculated as in Mailhot and Benoit (1982) but using qbM and qbr from Dyerand Bradley (1982). They are expressed as , F(~o + 1)(~o + 1) k + mL'~ + 17] + 2(tan-~- tan-~0) , [~'0 -t- 1\2-[-1with A = ln(z/zo), ~ = (1 - 28z/L)~/4, ~0 = (1 - 2:gz0/L)TM, ~' = (1 -- 14z/L)~/2 and ~'o = (1 - 14go/L)~/2. Forthe stable case C~i and Crare taken from Delage (1985):k 1 za Ri 2Az ~1/2 -5, (21)with za a constant = 2285 m. The roughness length isset to 0.0012 m (Clarke et al., 1971) for C~and to 1.5m for Cr since in this experiment the surface temperature is verified against the screen-level temperature. In one version of the model (M l), the ground heatflux is assumed known (actually provided by measurements), while in another version (M2) G is calculatedusing the force-restore method (Bhumralkar, 1975): G=fi ~-+0o-Tv , , (22)in which w is the diurnal angular frequency (=2.~/24hours), Tv the deep soil temperature, ~ a thermal parameter = (wXp~G) with X the soil conductivity, c~ thesoil specific heat capacity and p~ the soil density. TheM2 version is more similar to an ACM, in which Ghas to be calculated; the appropriate values for Tv andfi were found to be 10-C and 6.5 W m-2 K-~, respectively, when using the observed temperature at 1.5 mfor 00. For comparison, we examine a simpler model,M3, which consists of using (1) and (2 I) in M2 insteadof the inversion model represented by (3) to (16). The algorithm for the most complex model (M2) isnow summarized. The superscripts t - 1 and t indicatetwo successive time steps. We first define a net availablesurface heat flux, Fn, which will be used to control thebranching into one of the three phases: Fn -- Rn - pLvEo - fi(Oo - Tp). (23)a. Night phase (Fnt-I < O, O0t-I '< Ozt-l)The value ht is calculated from (10):ht: ht_~+ At[her_l I ~. ? 5 (ht + ht-~)Int and Oot are obtained by solving the linear systemconsisting of(9) at time t and the following relationshipInt + pCpWO0t= Int-I + ~l (AtFnt-~ + ~ Oot-~) , (24) pcp wwhich is obtained by differentiating the right-.mostexpression in (3), discretizing it and (22) as follows:Int - Int-t = AtQot-~,(25)Gt-I fi(--t- --t-I ) = +0ot-~-Tp , (26) wAtFImRU^RY 1986 YVES DELAGE 447and finally eliminating Qot-~ from (25) and (17) andusing (23), the definition of Fnt-~; the terms u, and Eoare calculated using (18), (19) and (21).b. Morning phase (Fnt-I ~ 0, O0t-I ~ O~t-I) - Oot and Imt are obtained by analytically solving thecubic system composed of (24) (with Ir~ replacing I,),(14) and (16). (The solution is given in the Appendix.) - u, and Eo are calculated as above.c. Afternoon phase (00t-I ~ Ozt-l) - Eqs. (26), (23), (17) and (1) are combined to give Fnt-1 q- (~Oot-1)/(wAt) + pCpCTt-lu~-IOzt Oot = (27) ~/o~At + ocpCrt-~u~-~ - To arrive at (27), the heat flux in (17) has beenexpressed as QO ~ --CTt-lu~-l(Ozt -- O0t)to make the model valid even at small (or null as inM 1 ) values of fi; - CM and CT are obtained as a function of git by aniterative method which uses (20); - u, and Eo are calculated using (18) and (19). To obtain the solution for M 1, R, is replaced by R,- G and/~ is set = 0. For M3, there remains only the"afternoon phase" in which C~ and Cr are given by(20) or (21) depending on the sign of Ri.4. The results In the first runs, we used M I with z = 213 m todetermine the sensitivity of the model to the parametersr, hi, c~, or h2 and c2. The best simulation of the surfacetemperature time evolution was obtained with a valueoft around 3.5/f or ~ 12 hours, in agreement with theresults of Nieuwstadt and Tennekes (1981). The initialvalue of h at the beginning of each night was set to 70m. Both (11) and (12) gave good results but (12) hada slight edge as measured by the surface temperaturerms error, which was equal to 1.40-C compared to1.45-C with (11). Consequently, (12) was used in allsubsequent runs with M1 and M2. This minimumvalue for the rms error was obtained with h2 = 435 mand c2 = 0.7. Since the mean value of the equilibriumheight, he, is around 550 m, the constant term accountsfor almost 80% of the contribution. This behavior isnot particular to the chosen form of the temperatureprofile. When (5) was replaced by a linear profile inz/h (let us call this model M 1L), the minimum rmserror was found for h2 = 275 m and c2 = 0.5 while he~ 360 m. The ratio h2/he is about the same as before.It appears that for the Wangara data our effective NBLheight h is not very sensitive to the surface stress. The above results have been obtained at a singlelevel (z --- 213 m), chosen for convenience in the middleof the range of values ofz for which the present schemeis proposed. At that. level the parameter h2 is chosento remove the mean error. To test the formulation weverify how well the surface temperature is simulatedwhen the ACM variables are taken at different heights.In Fig. 4, the mean and standard error of surface temperature for M 1, M 1L and M2 are plotted as a functionof the height z of the first interior level in the ACM,as simulated by the Wangara data. Let us compare M 1and M 1L first. The standard error increases with z similarly in both models with less than 0.1 -C difference,M 1 being superior. However, there is a difference inthe mean error. While M 1 has remarkedly little errorin the mean below 600 m, M 1L shows a positive biasat small values of'z. Though the bias is small, it confirms the hypothesis that a potential temperature profilein (z/h)u2 fits the Wangara data better than a linearprofile. For the rest of the study, we will limit our discussionsto results of M 1, M2 and M3. Comparing M 1 and M2in Fig. 4, one finds the same unbiased results for z lessthan about 600 m, but positive mean errors increasingabove that level, the error being almost twice as largefor M 1. This warm bias is explained by the fact thatthe assumption of a potential temperature constantwith height above the surface layer (daytime) or above 1100 , .............. ,,,,, ....~tooo - / / //.] "~ 800 ?00 ] . .o 400 F,.~ 300 , 200 I 100 i 0-.5''' 0.0 0.5 1.0 ' 1.5 2.0 SURFACE TEMPERATURE ERROR - FIG. 4. Surface temperature error for 31 days as a function of theheight of the first interior level as simulated by the Wangara data forM 1 (dashed lines), M 1L (broken lines) and M2 (solid lines). For eachmodel the curve on the left is the mean while the other is the standarddeviation of the error.448 MONTHLY WEATHER REVIEW VOLUME 114the inversion (nighttime) is less and less valid as z increases, and a positive bias is introduced by the warmerair above the boundary layer. (This interpretation willbe confirmed by Fig. 6e, where 0z at 975 m is largerthan the observed 00 even during daytime.) The smallererror for M2 results from the term/~(0o - T~,) in thecalculation of the ground heat flux (22), which resiststhe warming as 0o rises above Tp. This restoring forceis absent in M 1, in which G is given independently of00. The same mechanism largely explains the smallerstandard error of M2: it acts at night in preventing theminimum temperature on individual days to becometoo low. In Fig. 5, the standard deviation of the surfacetemperature error at z = 388 m is plotted as a functionof the time of day. The maximum errors occur nearthe time of minimum temperature with M 1 half a degree higher. Daytime errors are lower and similar inboth models, reflecting the predictable, nature of thewell-mixed layer. The term with O0o/Ot in (22) also decreases the error in M2 in producing a temperaturecurve which is smoother in time. On the same graphthe standard deviation of the observed surface temperature about the mean value is also drawn. This curverepresents the limit of the skill for the models since itcorresponds to forecasting the climatological temperature for each hour (in our experiment, the mean of31 days). The error associated with M2 is about halfof the natural variability at z = 388 and increases slowlywith height as seen in Fig. 4. Since the proposed parameterization is intended, for ACMs in which G willbe calculated, the results from M2 are more relevantthan those of M 1. The mean diurnal variation of the surface temperature is presented for various values of z in Figs. 6ae. The curves for M 1, M2 and M3 are compared with3.53.02.52.01.51.00.50.0 0STANDARD ERROR .............. ..... ..... ..... th ..... TIME OF DAY (HOURS) FIG. 5. Standard deviation of observed surface temperature (brokenline) and of surface temperature errors from M1 (dashed line) andM2 (solid line) as a function of time of day for the 31 days of theWangara simuhttion with z = 388 m. MEAN SURFACE TEMPERATURE 16 ................. , ..... 15 z= 38 ml -~?~'' 14 13 - ~e ~'r.,~llO 8 'i 7 ., oo ! 6 ~ ~. - ,' ~ 40 ..... ~ ..... 1~~ .... 1~ .... ~4 TIME OF DAY (SOURS) FiG. 6a. Thimy-one day mean surface tem~rature hour by hourfor Ml (~hed line), M2 (solid line) and M3 (broken line) forz = 38 m. The obeyed mean suda~ tem~rature is indicated byla~e dots. The o~n circles are 8~.the observed value represented by large dots. The diurnal cycle is clearly well simulated by M 1 and M2 upto 625 m, especially by M2, while the already discussedpositive bias is apparent at z = 975 m. Notice the similarity of the surface temperature curves for widely different values ofz. For example, 0z at 6 h increases from16151413121110 9 8 7 6 5 4 0MEAN SURFACE TEMPERATURE Iz=213 mlO o- o o O O O '-.~/..... ~ ..... 1~ ..... 1~' TIME OF DAY (HOURS) F~G. 6b. As in Fig. 6a except z = 213 m.FEBRUARY 1986 YVES DELAGE 44916151413MEAN SURFACE TEMPERATURE76540 ..... ~ ..... 1~2 ..... 1~' ' ' TIME OF DAY (HOURS) FiG. 6c. As in Fig. 6a except z = 388 m.2416151413 74MEAN SURFACE TEMPERATURE o ~975 ml O O Oo O Oo o O0 00I Io,,:,X/ i1'12 18 24 TIME OF DAY (HOURS)FiG. 6e. As in Fig. 6a except z = 9'/5 m.7-C at z = 38 m (Fig. 6a) to more than 12-C at 625m (Fig. 6d), yet 00 from M 1 and M2 are unchanged.This supports the validity of the formulation, at leastfor the Wangara data. A different response is given by M3, with too rapidcooling at the beginning of the night and too rapidwarming in the morning. The evening behavior of M3results from an underestimate of the magnitude of the1615 14 13m 9 7MEAN SURFACE TEMPERATURE T~M~ OF DAY (HOURS)Do. 6d. As in Fig. 6a except z = 625 m.downward heat flux. In the morning the heat flux calculated using (1) is negative (toward the surface) whilethe real heat flux is upward, resulting in an excess ofheat at the surface. This is illustrated in Figs. 7 and 8where the surface energy budget is depicted for M2 andM3, respectively, for'z = 625 m. The most strikingdifference occurs around 0900 LST. In M2 the sensibleheat flux is already positive at 0800 LST while in M3it is still negative at 0900 LST. From 0800 to 1000LST, the net radiative flux is increasing rapidly from18 to 213 W m-2, causing a faster warming in M3 thanin M2, reflected in the sharper peak of the ground flux'curve in Fig. 8. During the evening the sensible heatflux is larger in magnitude than the ground flux in M2while the reverse is true in M3. We can conclude fromthis comparison that the inversion model proposed hereis better able to simulate the diurnal cycle of surfacetemperature than is a diagnostic formulation suchas (1).5. Conclusion We have presented here a schematic representationof the nocturnal inversion and its morning breakupthat has been used to simul~ttc successfully the diurnalvariation of the surface temperature in the Wangaradata. The model was designed for use in ACMs as areplacement for the traditional surface layer relationship between the heat flux and the vertical temperaturegradient during the nocturnal and morning phases ofthe diurnal cycle. Such a model can be used with virtually any vertical resolution, and thus eliminates thenecessity of resolving the boundary layer structure inACMs.450 MONTHLY WEATHER REVIEW VOLUME 114 The advantage is obtained without sacrificing thesurface processes, which may be as sophisticated as one ~o0wants them to be. Most processes taking place at theresolution of the ACM, such as horizontal and vertical 20advection and diffusion, radiation, convection and eventhe warming and cooling produced by the surface heat 200flux are naturally superimposed on the model structure.Over oceans, the present scheme is not needed since '~ 15othe surface temperature is usually assumed known and ~the diurnal cycle is much weaker. The formulation of ~ '" 100Delage (1985) may be used in this case. mThe pararneterization assumes a potential temper- ~ oature profile in (z/h)ue which was shown to be more ~appropriate than a linear profile. This profile does notimpose a constraint on the ACM since it is used only 0below the first interior level. The required NBL thickness was found to be largely independent of u. or L, -oa result that may not easily be generalized to othersites. However, a 10% error in the effective NBL height -! oo owill result in a 10% error (of opposite sign) in the potential temperature difference between the surface andthe first ACM level, and it is conceivable that he couldbe varied with location to remove these biases.The feasibility of implementing such a scheme inan ACM has already been demonstrated. The operational spectral model at the Canadian MeteorologicalCentre in Montreal has used an earlier version of thisinversion model for many years. When the parametersspecifying the surface energy budget are properly set,useful forecasts of surface temperature can be produceddirectly by the model (Delage and d'Amours, 1979).More studies of this type are underway with an ACMto establish the "state of the art" in. surface temperature300200~ 150'"' 100-5O SURFACE ENERGY BUDGETz-'=625 ml .' ~ -1000 ..... , ..... ,, .......... 6 12 18 TIME OF DAY (HOURS) FIG. 7. Mean surface energy budget for M2 at z = 625 m as afunction of time of day. R. (dotted line), G (broken line), ocpQo (solidline) and .oL, Eo (dashed line). SURFACE ENERGY BUDGETiz--62 ml " : : : '. 6 12 18 24 TIME OF DAY (HOURS) FiG. 8. As in Fig. 7 except for M3.representation, to indicate the problem areas and topropose improvements. Acknowledgments. I wish to thank Dr. Robert Benoitwho reviewed my manuscript, Dr. Claude Girard- forhis suggestions on the numerical algorithms and Dr.Herschel Mitchell who helped me with the text. APPENDIX Solution for the Morning Phase If in (14) Im is taken at t - 1, (16) can be rewrittenas: Fehj_ Ooj)e-( I ,-,\3 = tn I Imt 3(0~300t-- Ozt 'Jc Ozj -- Ooj -- '~-Z J 'The 'expression for Imt is substituted into (24) and theresulting cubic equation is solved for Oot. The solutionis O0t ~ a + (s + r)1/3 -- (S -- r)u3with a = O~t - Ozi + 0o.~ - [mt-l/2z, [ /~q'l''e s: r2+k3/1 , b r = ~ (e - ad), b = 3(Ozj - Ooj)2/F2hj, d = Ig/o~pCp, e =Imt-I + At F.'-' + lg 0o'-'. pCp WpCpFEnRU^RY 1986 YVES DELAGE 451 REFERENCESAndrt, J. C., and L. Mahrt, 1982: The nocturnal surface inversion and influence of clear-air radiative cooling. J. ~ltmos. Sci., 39, 864-878.Arya, S. P. S., 1981: Parameterizing the height of the stable atmo spheric boundary layer. J. ,4ppl. Meteor., 20, 1192-1202.Bhumralkar, C. M., 1975: Numerical experiments on the computation of ground surface temperature in an atmospheric general cir culation model. J. Appl. Meteor., 14, 1246-1258.Clarke, R. H., A. J. Dyer, R. R. Brook, D. G. Reid and A. J. Troup, 1971: The Wangara experiment: boundary layer data. Tech. pap. no. 19, Division of Meteorological Physics, CSIRO, 362 PP.Deardorff, J. W., 1972: Parameterization of the planetary boundary layer for use in general circulation models. Mon. Wea. Rev., 100, 93-106.Delage, Y., 1974: A numerical study of the nocturnal atmospheric boundary layer. Quart. J. Roy. Meteor. Soc., 100, 351-364. ,1985: Surface turbulent flux formulation in stable conditions for atmospheric circulation models. Mon. Wea. Rev. 113, 89 98. , and R. D'Amours, 1979: Forecasting surface temperature with the CMC operational model. Fourth Conf on Numerical Weather Prediction, Silver Spring, Amer. Meteor. Soc., 298-301.Dyer, A. J., and E. F. Bradley, 1982: An alternative analysis of flux gradient relationships at the 1976 ITCE. Boundary-Layer Me teorol., 22, 3-19.Louis, J.-F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Boundary-Layer Meteorol., 17, 187-202.Garratt, J. R., and R. A. Brost, 1981: Radiative cooling effects within and above the nocturnal boundary layer. J. Atmos. Sci., 38, 2730-2746.Mailhot, J., and R. Benoit, 1982: A finite-element model of the at mospheric boundary layer suitable for use with numerical weather prediction models. J. Atmos. Sci., 39, 2249-2266.Nieuwstadt, F. T. M., and H. Tennekes, 1981: A rate equation for the nocturnal boundary-layer height. J. Atmos. Sci., 38, 1418 1428.Stull, R. B., 1983: A heat-flux history length scale for the nocturnal boundary layer. Tellus, 35A, 219-230.Suarez, M. J., A. Arakawa and D. A. Randall, 1983: The parame terization of the planetary boundary layer in the UCLA general circulation model: Formulation and results. Mort. Wea. Rev., 111, 2224-2243.Wetzel, P. J., 1982: Toward parameterization of the stable boundary layer. J. ,4ppl. Meteor., 21, 7-13.Yamada, T., 1979: Prediction of the nocturnal surface inversion height. J. Appl. Meteor., 18, 526-531.

## Abstract

A parameterization of the surface nocturnal inversion of temperature is proposed to enable atmospheric circulation models to handle the surface energy budget without having to resolve the boundary layer. The scheme allows a wide range of vertical resolutions for the host model. The principle is to replace the traditional instantaneous flux-profile relationship in the surface layer by a time integrated heat conservation equation linking the surface heat flux to the amplitude and thickness of the temperature inversion. The model is able to reproduce successfully the mean diurnal cycle of the Wangara data, using the observed vertical profiles to simulate atmospheric models with various resolutions. Unbiased surface temperature results are obtained from runs in which the information from the host model is taken at a height ranging from 40 to 625 m above the surface.