1 MAY 1996 FANG AND TUNG 1241A Simple Model of Nonlinear Hadley Circulation with an ITCZ: Analytic and Numerical Solutions M~ FA~ ANO KA KIT TUNGDepartment of Applied Mathematics, University of Washington, Seattle, Washington(Manuscript received 5 April 1995, in final form 23 August 1995)ABSTRACT Simple analytic solutions are constructed for an axially symmetric, nonlinear, slightly viscous circulation ina Boussinesq atmosphere in the presence of intense convection at an intertropical convergence zone. The latitude-height extent of the Hadley circulation is obtained, as well as its streamfunction, zonal wind, and temperature distribution. Numerical solutions of the viscous primitive equations are also obtained to verify the analyticsolutions. The strength of the circulation is stronger than previous results based on dry models and is now closeto the observed value. The extent of the Hadley region is also quite realistic.1. Introduction From the late 1970s, a series of studies (Schneiderand Lindzen 1977; Schneider 1977; Held and Hou1980; Lindzen and Hou 1988; Hou and Lindzen 1992;Plumb and Hou 1992) advanced our understanding ofthe dynamics of the Hadley circulation in the absenceof eddies. In these studies, many features of the Hadleycirculation, such as its meridional extent, flat horizontaltemperature gradient, angular momentum conservation, and the intensification of the winter circulationcell have been explored. Schneider (1977) noted that the nonlinear advectionof angular momentum is important and the absoluteangular momentum is conserved if friction is negligible. Held and Hou (1980), Lindzen and Hou (1988),and Hou and Lindzen (1992) developed a simple approximate theory using the concept of angular momentum conservation. In these studies, all thermal forcingare idealized by a simple Newtonian cooling processes:Q = (TE - T)/r, and frictional processes are parameterized by viscous diffusion. Hereafter, we will refer tothese models as Newtonian cooling models (NC models). The physical interpretation of TE is the temperature at equilibrium in the absence of large-scale meridional circulation. In this formulation, Te can be viewedas the consequence of the local convective-radiativeequilibrium in the Tropics. When a large-scale circulation (the Hadley circulation) is present, the temper Corresponding author address: Dr. Ka Kit Tung, Department ofApplied Mathematics, University of Washington, Box 35420, 408Guggenheim Hall, Seattle, WA 98195.ature T in the NC models differs from its equilibriumstate Te in the region of the circulation. It is well known that moisture convergence and latent heat release in the intertropical convergence zone(ITCZ) play an important role in the dynamics of theHadley circulation (Riehl and Malkus 1958). Therehave been attempts to incorporate latent heating in theNC models by simply changing the specification of Tein an otherwise dry model. For example, Hou and Lindzen (1992) used a latitudinally concentrated T~ to include the effect of a concentrated latent heating in theITCZ. Although one could include some effects of moistconvection this way, for example, by defining Te to bethat obtained under convective-radiative equilibriuminstead of the value obtained under radiative equilibrium, there is a fundamental conceptual difference between a moist convecting and a dry atmosphere in theway temperature adjustment is accomplished, whichhas only recently been clarified (see Emanuel et al.1994; Yanai et al. 1973; Cane and Sarachik 1989).Since cumulus convective adjustment is probably amuch faster process, the local temperature in regionsof active convection should be nearly in convectiveradiative equilibrium--that is, T ~ Te. The effect ofthe large-scale circulation on T is of second order andaffects mainly the conditions for convection (e.g., theavailability of moisture convergence) and to a less extent the lower-level temperature in the subcloud layer(Emanuel et al. 1994). In other words, the temperatureprofile itself, rather than the net heating, is controlledby convection. Such a conceptual change in the waywe view the relationship between temperature and heating actually results in considerable simplification intheoretical models of Hadley circulation.c 1996 American Meteorological Society1242 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 9 Confirmation of this conceptual model can sometimes be found in numerical models incorporatingmoist convection. Satoh (1994) constructed a moreconsistent model by including the hydrological cycleand a radiation-convection scheme in his axisymmetric model. His results show that the vertical temperatureprofile in the strong rising region of the Hadley circulation is determined by local moist processes. It decreases from the sea surface temperature upward insuch a way that the profile follows the moist adiabaticlapse rate. Thiis vertical profile of temperature in themoist convective region is consistent with results fromone-dimensional convective models (Sarachik 1978;Satoh and Hayashi 1992). The temperature is almostflat latitudinally inside the Hadley circulation cell. Outside the circulation cell, the temperature depends on thesurface temperature and is determined by the local radiative equilibrium. Satoh's results also show that the upper-level poleward flow conserves angular momentum, and zonalwind and vertically averaged temperature are in cyclostrophic balance. Satoh's numerical work provides themotivation for our present attempt at constructing asimple analytic: solution to the problem. Satoh (1994)also tried to obtain some analytic results on some averaged quantities, such as the total poleward mass flux,and the extent of the Hadley circulation at the top andthe bottom of his model, making additional simplifyingassumptions not justified by his numerical results. Inaddition, the two-dimensional (meridional-height)structure of the circulation and the detail of the meridional velocity and the vertical velocity have not beenfully explored. Our analytic solutions are a result ofasymptotic deduction and provide a detailed meridional-height dist~ribution of the meridional circulation,temperature, zonal wind, and the separation curve between the circulation region and the equilibrium region.Our numerical :solutions illustrate how viscosity affectsthe extent of the Hadley circulation and the total massflux. In this paper, we propose a simple physically consistent model to describe the Hadley circulation in a convective axisymmetric atmosphere. In this model, theITCZ is idealized as an extremely narrow band centeredat one latitude, not necessarily symmetric about theequator. The vertical temperature profile in the ITCZ isdetermined by fhst moist convection (and so is providedby a one-dimensional cumulus convection model or simply given by a moist adiabat), and the net heating at theITCZ is determdned by the resulting large-scale circulation as part of the solution. This model provides analternative approach that differs from the earlier NCmodels, in which the latent heating is specified, and thetemperature and circulation are resulting solutions.Away from the ITCZ, the diabatic heating (cooling) inour model is still approximated by a Newtonian coolingform as in the NC models. Under the assumption ofsmall viscosity and long Newtonian cooling time awayfrom the ITCZ, angular momentum conservation, cyclostrophic balance, and hydrostatic balance can be shownto hold asymptotically away from the ITCZ. This modelturns out to be even simpler than the NC model, and ananalytic solution in the "nearly inviscid limit" is obtainable. In section 2, we describe the formulation of ourmodel. Some general features of the solution in thenearly inviscid limit are discussed in section 3. Analyticsolutions are obtained in section 4. The examples forboth equatorially symmetric and asymmetric equilibrium temperatures are shown in section 5. The solutionsare verified by numerical solutions presented in section6. Concluding remarks follow in section 7. Some cases.not given analytic solutions in section 3 are discussed inthe appendix.2. The modela. The primitive equations In this simple model, we consider a set of steady,axially symmetric primitive equations for a Boussinesqfluid on a sphere of radius a rotating with rate f~, confined between the bottom solid surface and a stress-freelid at height H. Let T be the temperature, To the constant of globally averaged temperature, Q the diabaticheating rate per unit mass, and (u, v, w) the velocityof the fluid in the longitudinal, latitudinal (4,), and thevertical (z) direction, respectively. The equations areOu v Ou Ou uv -- - -- tan4,~+~+W oz a - 2~v sin4, = ~,V2u (1)0v v 0v 0v u20--~ + -a ~ + w--Oz + --a tang>1 0~+ 2f~u sin4, = ~V2v - --- (2)a 04,10(v cos4,) Owa 04, + ~zz = 0 (3)0~ TOz - g To (4)o-7+5~+w +rd = ~v:r+ Z,,where Fd is the dry adiabatic lapse rate and Cp is thespecific heat of the gas at constant pressure. To calculate the steady-state solution analytically, we drop thetime derivative terms in Eqs. ( 1 ) - (5). The numericalsolutions presented in section 6 use time stepping andso need the full time-dependent equations.I MAY 1996 FANG AND TUNG 1243b. Heating Suppose that the diabatic heating can be approximated by the Newtonian cooling law as in Schneider(1977): Q TE-T Cp- r(tt) ' (6)where kt -= sinqb, Te is the equilibrium temperature andr is the relaxation time.~ Suppose a convection regionis centered at latitude ktl. Since the local convectionprocess is much faster than the large-scale circulation,we can incorporate this process in the form (6) by assuming r(kt) -' 0 as/~ ~ ktl so that the temperature at/.q approaches Te, the temperature at "convective-radiative equilibrium." This TE can be found using a detailed one-dimensional model incorporating latent heating and radiation. An important characteristic of Tefound by most such models is that its lapse rate is determined approximately by a moist adiabat; that is, T(l~l, z)-~ re(l~l, z) = TM(Z) ~ TMO -- I~mZ, (7)where Fm is the moist adiabatic lapse rate at/~. TheITCZ is considered as a very narrow band so that it isapproximated by a single latitude tq. Away from thislatitude, the diabatic heating is approximated by a slowNewtonian cooling process with 'r(/~) = r0, where r0is a long constant relaxation time (-20 days). Thesurface temperature at/~, TMo, is known to be determined mainly by the sea surface temperature, and to aminor extent by the large-scale circulation (see Emanuel et al. 1994). This latter dependence is ignored inpresent work and so TM0 = Ts(/~l), where Ts(tz) is thelocally specified surface temperature. Although we willuse the simple form (7) in our examples discussedlater, our model can alternatively incorporate a Te calculated using a more detailed one-dimensional moistmodel as input. The magnitude of the net heating Q and hence thevertical velocity at the ITCZ in this model depends onthe ratio ( Te - T) / r, as the numerator and denominatoreach goes to zero. If T -o Te faster than r -~ 0, therewill be no heating, as the temperature adjusts rapidlyto the local convective equilibrium. Otherwise therewill be finite or even infinite heating. We allow for thelatter possibility because T may approach TE slowerthan r -' 0 at/z~. The magnitude of the heating at theITCZ is to be determined as part of the solution by thelarge-scale circulation. We therefore write for ourmodel ~ More precisely, Te is the temperature at equilibrium obtained inthe absence of the large-scale meridional circulation. It can be obtained by a one-dimensional model that treats the atmosphere locally,column by column, to be in radiative-convective equilibrium, as in,for example, Sarachik (1978) and Satoh and Hayashi (1982). Q T-TE - -- + q(z)6(t~ - tz~), (8) C, %where q(z) is to be determined by mass conservationfor the large-scale circulation. The delta function distribution is necessary to allow for the possibility of finite heating integrated over the ITCZ. If q = 0, theITCZ will contribute no net heating. In any case, r(/~, z) - Te(/~, z), as tz --'/~, (9)and so is known without a knowledge of the large-scale(Hadley) circulation. The highly idealized delta function heating form (8) was also adopted by Schneider(1983) in his modeling of Martian great dust storms.But in his parameterization, q, the strength of heatsource, is related to dust/solar absorption and is specified, while in our model q(z) is part of the solution. Itis determined by the amount of sinking that occurs elsewhere in the circulation region.c. Boundary conditions The boundary conditions for the velocities are noslip conditions at the bottom and stress-free lid at thetop. The boundary is considered to be insulated so thatno heat flux crosses it. There is no poleward velocity-that is, v cosqb = 0--at the poles.d. Nondimensionalization As in Fang and Tung (1994), nondimensional variables are introduced by asterisks as follows: z* = z/H,r* = 2~r, (u*, v*) = (u, v)/U, w* ~ (w/U)(a/H),ig* = ~/2~2aU, V.2 = H2~72, and r gH RaT T* To 2QaU Ro To 'Here U is a typical zonal velocity, Ro --- U/(2~2a) isRossby number, and R~ ~ gH/(2~a)2. After the nondimensionalization process, the superscript asterisksfor the nondimensional variables are dropped withoutconfusion. The following nondimensional fundamentalsteady-state equations are obtained:Ro v~-~+W~zz-UVtanqb - sin~bv = EV2u (10)Rov + W~zz + tan~b 0~ +sinqbu=EV2v-~ (11) O(v cos4~) Ow 0q, + ~zz = o (12) 0~ ~ = r. (13) Oz1244JOURNAL OF THE ATMOSPHERIC SCIENCESVOL. 53, NO. 9At/~, the temperature is from (7) T(i~, z) = TM(Z) ~ T~o - 6mZ, (14)and away from/~, we have r = r0 and ( cgT OTo~ oz ) E T~-T Ro v~-7,+w~-+~aw =-~72T+~, (15) o' Towhere E = ~,/(2f~H2) is Ekman number, and rr = ~/g is Prandtl number. Additionally, 6a =- FaHRu/(ROT0) is the nondimensional dry adiabatic lapse rateand ~m ---- FmHRu/(RoTo) is the nondimensional moistlapse rate. The parameter regime relevant to the real atmosphere is E~ 1, r0>> 1, and Ero~ 1.For a dimensional Newtonian relaxation time of 20days, we find % ~ 250 >> 1.For a typical kSnematic viscosity coefficient of v = 5m2 s-l, we have E~ 10-44 1,where we have taken H = 15 km as the height of the"tropopause." The product yields Ero ~ 0.025 ~ 1.The condition, Ero ~ 1, defines the "nearly inviscidlimit" in the present work. The Prandtl number cr is taken to be order one orlarger. The value of the Rossby number, Ro, is mostlynot restricted2 i[n the present work, except that the caseof Ro < 80E: has been dealt with separately in Fangand Tung (1994), where exact solutions were found.The work presented here assumes that Ro > 80E2. Forthe terrestrial atmosphere Ro = 0.03 is small, but isstill larger than 80E2.e. Occurrence of Hadley circulation In general, a Hadley circulation will occur, with orwithout moist convection, whenever there is a meridional gradient of the equilibrium temperature Te at theequator. Only under very restricted conditions [E -- 0,cgrE/cgl~ = O, ~)2TE/O~2 ~ 0 at tz = 0 and Ro smallenough (see Fang and Tung 1994)] can the atmosphererelax to the radiative-convective equilibrium with (v,w) = (0, 0) over the globe. For the case E- 0+, OTs/0/~ = 0 but 02TE]OI~2 -: 0 at tz = 0, the equilibrium 2 In the boundary layer asymptotic analysis presented in section 5,the nature of the viscous and thermal diffusive boundary layers isdifferent dependent on whether Ro is small or not. Only the case ofsmall Ro is worked out in detail in that example.solution, although it exists, is unstable to symmetricinertial instability in the purely inviscid limit, and, inthe "nearly inviscid" limit (E small, but nonzero), itis incompatible with the presence of viscosity, no matter how small it is (Hide's theorem, see Held and Hou1980). So even in this case, a Hadley circulation exists,at least near the equator.3. Some general results There are some very simple but rather general resultsthat can be shown to be valid for axisymmetric circulations with or without convection. The most importantof these is the result on latitudinal temperature homogenization, which has previously be noted in numericalresults in dry (Held and Hou 1980; Lindzen and Hou1988) as well as moist (Satoh 1994) model atmosphere. These general results will first be derived here,along with conditions for their validity, as part of theleading order asymptotic, solution of the primitive equations. Specializing to our case in the presence of anITCZ will be done in section 4.a. Angular momentum conservation In the "nearly inviscid limit," defined by Er0 small, absolute angular momentum in the zonal direction is conserved along streamlines away from thin viscous boundary layers and the ITCZ. Away from the ITCZ, and for E -- 0 +, one can showfrom Eq. (12) that (v, w) are proportional to Q andtherefore are scaled by 1/to. We can rescale them byintroducing v = O/to, w = ~/ro. Thus, the angularmomentum Eq. (7) can be rewritten asRo O ~ + - ~-zz ~ UO tanq5 02u -sinqb0=Er00z2. (16)Thus, it can be seen that if Ero ~ 1, the absolute angular momentum is conserved along streamlines awayfrom viscous boundary layers; that is, OL OL ~+~~zz=0 for Ero~l, (17)where L --= cosOqb + 2 Rou cosqb is the absolute angularmomentum. In the current problem, there is a viscous boundarylayer near the lower boundary, and an "inner viscouslayer" near the meridional edge of the Hadley circulation region. In these regions, absolute angular momentum conservation may not hold. There are two possible solutions to Eq. (17) awayfrom these boundary layers: one is an equilibrium solution with no circulation [i.e., (v, w) =- 0], and theother is a statement 'of conservation of absolute angular1 M^- 1996 FANG AND TUNG 1245momentum conservation: L = L(~) where (v, w) * 0,with ffJ being the streamfunction for (v, w), v cos4'= -O~t/Oz, w = OkV/Ot~. In most relevant cases, theequilibrium solution (v, w) = 0 can be ruled out forone reason or the other (see the discussion at the endof section 2). In these cases we are then left with theonly solution L = L(~), (18)which, incidently, includes the relevant case of nonzerocirculation in some region in the Tropics, and no circulation in the extratropics near the poles. In the purely inviscid case, the functional form forL is nonunique, being determined presumably by theinitial conditions. In the slightly viscous (or sometimescalled the "nearly inviscid") case (E small but nonzero), absolute angular momentum, as well as otherconserved quantity, should be homogenized insideclosed streamlines at steady state, a result first statedby B atchelor (1956), and later expressed as Hide's theorem by Schneider (1977) and Held and Hou (1980).Since the real atmosphere is never in a steady state, itis not clear which limit is the more appropriate one.This issue will be left to a future paper on time-dependent Hadley circulations. Here we shall pursue theslightly viscous steady-state limit further. In such a limit, the solution to (18) in the Hadleycirculation region is L = Lo, a constant,or, stated in another way, ~2 _ ~s2 u = 2Ro(1 - 1~2)1/2 , (19)where/Zs is the stagnation point where u = 0, whichyields Lo = 1 - /~. At this stage we have not yet determined where/~s is.b. Cyclostrophic balance It is often assumed that geostrophy (or cyclostrophicbalance in the nonlinear case) holds for large-scalenearly inviscid flows away from the equator. To theseconditions one must add the additional constraint thatthe timescale of temperature adjustment must belong--that is, r0 >> 1. Consequently, cyclostrophic balance does not hold inside an ITCZ and in other regionsof intense convection, nor in a viscous boundary layer.It will not be invoked in the present work in theseregions. The meridional momentum equation ( 11 ) can be rewritten asr~- 0 ~-~ + v~ ~zz + R-u2 tan4' E 020 0/I> + sin4'u = . (20) r o Oz2 04'For r0 >> 1, (20) reduces to a statement about cyclostrophic balance: 0~ Rou2 tan4' + sin4'u = -- (21) 04,Remarkably, cyclostrophic balance holds up to a thinregion within ~ ~ Rx/-~T02 ~ 7 X 10-4of the equator for Ro ~ 0.03 and % ~ 250.c. Latitudinal temperature homogenization Combining cyclostrophic balance (21) with hydrostatic balance (13), we obtain the so-called thermalwind relation: OT O0z (R-u2 tan4, + u sin4,) = - ~. (22) In the Hadley circulation region, since u, given by(19), is a function of latitude only, the temperaturemust satisfy OT -- = 0. (23) 04, In the nearly inviscid limit: Er0 ~ 1, r0 >> 1, there is no horizontal temperature gradient in the Hadley cir culation region, away from viscous boundary layers. Thus, the effect of the Hadley circulation is to homogenize the temperature latitudinally within itscore. There may appear to be a technical difficulty in applying (23) across the equator because cyclostrophicbalance (21) does not hold there. However, since theregion where (21) may break down is thin meridionally, integrating (20) across the equator shows that ~,and hence T, should be approximately continuous.Hence (23) holds even across the equator.4. Analytic solution with an ITCZa. Temperature As pointed out before, the result on latitudinal ternperature homogenization, (21 ), was found by previousauthors in numerical solutions for both the dry andmoist models. In the dry NC models, the (latitudinallyconstant) temperature that the model attains in the Hadley circulation is determined by the so-called "equalarea rule" (Held and Hou 1980), so that the mass fluxin the subsidence region, where Q = (T~ - T)/ro isnegative, can balance the mass flux in the upwellingregion, where Q is positive. Consequently, T thus determined is roughly halfway between the maximum T~1246 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 9and the minimum Te tbund in the latitudes of the Hadley circulation region. In the current model, when an intensive convectiontakes place in an ITCZ located at/~, and where the"statistical equilibrium" (Emanuel et al. 1994) holds,the temperature there is locally determined by the moistprocesses. We have, from (7),T(/~, z) = Te(tz~, z) = T~t(z) ---- TMo -- 6rnZ.Combining this with the fact that T(/~, z) is independentof latitude inside the circulation region, we have, forthe whole region inside the core of the Hadley circulationT(/~, z) = T(/Zl, z) = Te(/~, z) = r~,(z) ------ rMo - 6mZ.(24) The temperature in the Hadley region is completely and explicitly determined. The physical consequence of (24) is that since theITCZ usually occurs near where the tropical sea surfacetemperature is the wannest (assuming that there isenough moisture convergence), the temperature attained in the core of the Hadley circulation has thesame, warmer value. Thus Tin the Tropics in the modelwith an ITCZ is usually wanner than the T in the dryNC models. Consequently, the region of subsidence iswider and downward mass flux is greater in the currentmodel than in previous dry NC models. This radiativelydetermined downward mass flux forces a low-levelmass convergence into the ITCZ, producing an upwelling as a result. This Hadley mass flux thus produced is usually stronger than in dry models. Contrary to common perceptions, the new conceptual model requires one to think of net heating and upwelling at the ITCZ not as a direct result of local latentheat release by the con?ection, but as a result of forcedascent by renaote radiative-cooling-induced subsidence. Of course, the magnitude of subsidence is afunction of the global temperature distribution in theHadley cell. Latent heat release raises the local temperature at ~, which is then homogenized by the Hadley circulation to latitudes away from tq. A corollary of (24) is that the vertical lapse rate in the Hadley circulation is approximately the same at all latitudes and follows approximately the moist adiabat appropriate for the latitude of the ITCZ. This is also consistent with the numerical result ofSatoh (1994), who found "the vertical temperatureprofile in the upward motion region of the Hadley circulation is determined by the moist process to be amoist adiabat, and the profiles in the downward motionhave the same moist adiabat."b. Geopotential If *-o >> 1, the cyclostrophic balance, (22), is valid,except possibly in the thin viscous boundary layers.Note that the left-hand side of (21) can be written as 0 2 OL Rou2 tan~b + u sin~b ~ ~ (Rou /2) u 2 0/~since L is constant within the circulation region. Sincethere is a strong vertical motion carrying the absoluteangular momentum all the way to the top in the ITCZ,we know the stagnation point,/~s, should be in the samelocation of the ITCZ, that is,/~s --- /~. Integrating thecyclostrophic balance equation (21) from/~ to/~ gives ~(/z, Z) = ~(z) - Rou2/2 (25)if the integration path is entirely within the circulationcell (angular momentum conserving region where OL/0tz = 0). Here, ~(z) is the geopotential profile at thelatitude/~, where u = 0--that is, ~(z) -- ~(/~, z).Equation (25) gives the relationship between the geopotentials at two points in a horizontal line within theangular momentum conserving region. If the wholevertical line/z = tz~ is in the circulation cell, the geopotential in the entire circulation region is given by(25).c. Meridional circulation In the circulation region, the vertical velocity at latitudes other than/~ can be obtained by combining (15)and (24) to yield T~-T~ ~ w- 6~r0Ro' t~:~tq. (26)[Note that since T is independent of tz, the v(OT/Oc~)term in (15) vanishes in the circulation region.] Here,6~ ~ 6~ - 6m is a nondimensional static stability. TheVertical velocity at /zx is determined by mass conservation. From (8), we know that w can also be writtenas T,- T Ro6,w(/~, z) = q(z)6(l~ -/z~) + -- (27) TOBased on the mass conservation requirement [derivablefrom ( 12)1 f~w(tz, z)d~ = 0, (28) 1we obtain, for the magnitude of net heating at the ITCZ, f_ q(z) = - T~. - T~ d/z. (29) 1 TOIt is seen that it is equal to the total downward massflux due to radiative cooling away from the ITCZ.1 MAY 1996 FANG AND TUNG 1247FIG. 1. Schematic structure of the Hadley circulation and its boundaries (see text).Thus, net heating in the ITCZ is not locally determinedbut is a result of the large-scale subsidence elsewherein the Hadley circulation region. Outside the circulationregion, there is no meridional circulation and T In the NC models (e.g., Held and Hou 1980; Lindzenand Hou 1988), mass is required to balance in the Hadley circulation in the absence of an ITCZ by the socalled equal-area rule (Held and Hou 1980; Lindzenand Hou 1988). This temperature is usually colder thanthe TM in our model. Consequently, the large-scaleHadley circulation is weaker than that in our model.The circulation strengths in the two models becomenearly equal to each other if in our model we changeTu to be the calculated temperature profile from Heldand Hou (1980) so that q(z) ~ 0 in (29).d. Equilibrium zonal wind As an example, we use the same Ts distribution for/~ ~/~1 as in Held and Hou (1980), Lindzen and Hou( 1988 ), and Satoh (1994): ....rs r00 An(/z - go)2 Az-- (30) To VH'where Av~ FH/To and F is the lapse rate of Ts. Thenondimensional form is Rn Ts=~oo[r0o-Au(g-U0)2- AvZ]. (31)The equilibrium zonal wind can be obtained from (22)withT= Tsandu=0atz=0:where R ~ RuAu can be considered as a thermalRossby number.3 It is noted that the equilibrium zonalwind does not exist in the region where0 ~< /~ ~< 8R/~oZ/(1 + 8Rz).Therefore, a Hadley circulation will at least occupy thisregion. The actual Hadley circulation region may belarger.e. Boundary of the circulation region Suppose in the nearly inviscid limit, the viscous effect is confined in narrow viscous boundary layers only(the scale of these boundary layers shrinks to zero ast/-- 0 +). The meridional plane is mainly divided intotwo regions: the angular momentum conserving regionand the equilibrium region, separated by a line z= z,(/~) or tz =/z,(z). There may be a thin boundarylayer (of thickness ~ E4-~o) around the separation lineconnecting all variables smoothly to each side. Forsmall but nonzero viscosity there is also a viscousboundary layer near the lower and upper boundaries.The circulation pattern is schematically depicted in Fig.1. The dark shaded area is the inviscid core of the circulation region where the zonal wind is given by (19)and the temperature is given by (24). The light shadesarea is the viscous boundary layer. Outside the shadedarea is the equilibrium region where T = Ts and thezonal wind is given by (32). Now, we shall determine the boundaries of the Hadley circulation region. From (13), we have- (I~, z,) = ~o(l~) + T(l~, z')dz'= ~o(l~) + rs(t~, z')clz',(33)where ~0(/~) is the geopotential at the lower surface.On the other hand, suppose the path from/~ to tz at z= z, is in the circulation region, then from (25) Ro U2'ee(~, z,) = ee,(z,) - ~(34)The continuity of the geopotential across the separationline leads to the following equation: fj * Ro u2'- o(/~) + Ts(/z, z')dz' = ~,(z,) - -~- (35)( 3 2) 3 The definition of R here is one-quarter of that in the definition of Held and Hou (1980).1248 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, NO. 9Note r(kq, z) = TM(z)and ~(z,) ~- ~[~(/2~, z,) = ~o(P~) + r~t(z')dz'.Substituting these expressions and (19) into (35), wehavefj* (rE- rt~)dz' = ~0(/2t) (/22 _ m2)2- ~0(/~) - 8 Ro(1 -/22) ' (36)Thus, if we know/2~ and the geopotential distributionat the bottom boundary, we can obtain the separationline from (36). The geopotential variation at the bottom boundary is due to the effect of viscosity. Beforewe discuss that, it should be pointed out that (36)should strictly be obtained with quantities evaluatedabove the bottom viscous boundary layer. In particular,the lower limit of the (I) integral in (36) should be d,the thickness of the boundary. However, since (TE- TM)dz' = (Te- T~t)dz' + O(E), (37)the difference is ignored. Similarly, the geopotentialson the right-hand side of (36) should also be evaluatedat z = d instead of z = 0, but the difference is generallyignored in boundary-type asymptotic calculation. Evaluating the meridional momentum equation ( 11 )at z -- 0 and applying the no-slip boundary conditions,we find O'go 0% I 0,~ = E b-ffz~ ~=o' (38) Outside the Hadley circulation region--that is, bt>/2;~ or/2 </~;, where/.g- and/2; are the northern andsouthern extent of the Hadley circulation region, respectively-the atmosphere is in radiative equilibriumexcept for the viscosity induced flow in the boundarylayers at the lower and upper boundaries. The viscousproblem in this region can be cast in the form of aCharney (1973) problem. Its analytic solution was obtained previous.ly by Fang and Tung (1994) to beE-- = =Oz~ cos4, Oz~ z=o \~-~/ (a + b) + e-X[(c + d) sink - (c - d) cosk] xe4x -- 2e2x sin2k - 1(39)where a = -e4x q- ex cos(k) + e3x cOS(K) -- e2x COS(2k) + e2x sin(2k) b = -e4x -t- 2e3x cos(k) - e2x cos(2X) + ex sin(k) + e3x sin(k) - e2x sin(2k) c = -e2x - e4x + ex cos(k) + e3x cos(k) - ex sin(k) + e3x sin(X) + e2x sin(2k) d = e2x - ex cos(X) - e3x cos(X) + e2x cos(2X) - ex sin(k) + e3x sin(k)and 3, -= x/I/zl/(2E), T~(/2) is the equilibrium surfacetemperature distribution [i.e., T,(/2) = T,(p, 0)]. For/z :v 0 and E small, we have k -, m. Equation(39) becomes~for E~ 1. (40)Equation (40) can be used together with (36) to determine the extent of the Hadley circulation for varioussmall values of E. Note that as E -0 0, 4~0 is a constantfor p :~ O.f Upwelling and heating at the ITCZ The vertical velocity in the circulation region isgiven by (27). Integrating the vertical velocity fromsouth pole to north pole, we have Ro6~ wdp = q(z) + - (41) --1 1 TOSince the temperature T differs from TE in the circulation region only, (TE- r)dlz = (TE- T~)dl~, (42) 1 ~where ~ (z) ~e ~e no~em and sou~em bound~esof the circulation region at height z. Thus, from (29),we have for th~ s~ength of ~e concentrated heating: 1 fuzz) q(z) = -~ (Te- T~)d~. (43) TO 4 The continuity of O~/O& near the edge of the Hadley region isinvoked. Equation (40) is thus assumed to hold at the edge as wellas outside.1 M^- 1996 FANG AND TUNG 12495. Some examplesa. Symmetric equilibrium temperature For an equatorially symmetric surface temperatureprofile, we assume, in this explicit example, that thevertical profile of the equilibrium temperature is a linear function of the height starting from a prescribedsurface temperature T,(/2) with a lapse rate F (-Fm)at each latitude (except at/20: Te(/2, z) = T~(I~) - Fz, /2 -~ /2~. (44)In the nondimensional form, we have Te = ~oo [Ts(/2) - Avz] /2 #: /2,. (45)At the ITCZ,/2~, the convective-radiative equilibriumtemperature, is given by (14): Rn Te = T~t(z) = ~oo (TMo - AMZ), (46)where surface temperature Ts is given by an equatorially symmetric distribution T~(/2) = Too - An/22~. (47) From (43), we know 1 q(0) = ~oo o_,, [ru0 - (Too - An/22~)]d/2.As for the ITCZ, we require q (z) ~> 0, otherwise therewill not be moisture convergence into the convectionregion. Considering Ts(/2~) = Too - Au/22~n and q(0)~> 0, we need 2n Tsto - T~(/21) > 2n +~ An/2~' (48)Thus, if Tt~o is given by Ts(/2~), to have an ITCZ located at/~ for a symmetric Te given by (44) the onlypossible location of the ITCZ is the latitude withwarmest surface temperaturefi This result is consistentwith studies of the statistical relationship between seasurface temperature and cumulus activity, which suggest that high SST is a necessary condition for the intensive cumulus activity (Graham and Barnet 1987). From (36), we know the boundary of the circulationregion is determined byR"-~ /22n~., q.. (15 - = 8 Ro(1 -/22) - [~o(/2~) - (bo(/2)], (49) ~ More precisely, the ITCZ should be located at the latitude withhighest virtual surface air temperature, which may not necessarily bethe latitude with highest surface temperature due to the high contentof moisture in the ITCZ.where 6 and 6,~ are nondimensional lapse rates corresponding to F and Fro, respectively. For mathematicalsimplicity, we take F = Fm and n = 1 as an example.(The difference between Fm and F will result in aslightly wider extent of circulation region and a slightenhancement of meridional circulation strength.) Forsimplicity, in the following calculation we choose U= gHAn/(2~2a), and consequently Ro = R. In thenearly inviscid limit, E-~ 0+,/I>0(/2) is a constant, andso (49) gives /22 . ! 8 Rzz,= 8R(1-/22)' or /2~ = + + 8Rz' (50)The Hadley circulation in this symmetric case has zerowidth at the bottom in the inviscid limit. Setting z = 1in (49) gives the extent of the Hadley circulation at thetop boundary 6: ~8R/27=+ l+8R(52)This formula is similar to that in Satoh (1994) withC - oo.7 From (44), we have the strength of the concentrated heating in the ITCZ 2 2( 8~zq(z) =--37-o/2~(z) = --37-o[, 1 ~- ~zJ (53)The vertical velocity and meridional velocity are An Anw = ~q(z)6(/2) RA~r-~/22, (54) hHv cos~b - q'(z) sgn(/2), (55) 2RA~where A, ---- (Fa - F,~)H/To and the streamfunction is- - An ~[/2,(Z)3_/23] if /2>0 3RA~r-~-~[--[/2,(Z)3 + /23] if /2 < 0. (56) 6 For n = 2 and in the "inviscid limit," it gives z, = l/[8R(1_/~2)], and the extent of the Hadley circulation at the top is given by ~t7 = _+~/1 ~/~R (51)It is found that there does not exist an angular momentum conservingcirculation region unless R > ~[8. The solution for R < I/8 was givenin Fang and Tung (1994). For R > ~/8 the circulation region is detached from the lower surface. It starts from z.(0) = l/(SR) ratherthan from the lower boundary. ? In Satoh (1994), if viscosity v is fixed and the difference step inz direction, dz -- 0, the friction coefficient C approaches infinity.1250JOURNAL OF THE ATMOSPHERIC SCIENCESVOL. 53, NO. 9The streamlines (outside viscous boundary layers) andthe circulation region are shown in Fig. 2a. Note thatthere is a thin viscous boundary layer near the top (notshown) where the poleward meridional flow occurs.The dimensional streamfunction ~d can be written as~d = poaHU~l~ = poa2 A~To ([/~.(z)3 - tz3] if /~ > 0 x3(Fa-Fm)roq[-[/~,(z)3+/23] if /~<0.The value of the parameters used are R = 0.03 and AH= 1/6, F = 6degkm-~,Fd= 9.8degkm-~,H= 15km, To = 300 deg, and r = 20 days. The total massflux as a dimensional variable is 1.12 x 10~- kg s-~ ineach hemisphere, which is more than twice as strongas the symmetric case obtained by Lindzen and Hou(1988) (4.6 x 109 kg s-~ ). It is found that the strengthof the circulation is proportional to horizontal temperature gradient (AHT0) and is inversely proportional tothe relaxation time and the static stability for a fixedthermal Rossby number R. Inside the Hadley circulation region, the temperature is given by T~ in (24) andthe zonal wind is given by angular momentum conserving wind u in (19). Outside the Hadley circulationregion, the temperature is given by the equilibrium temperature Te (46) and the zonal wind is given by theequilibrium zonal wind ue in (32). Figure 2b showsthe temperature distribution (solid lines) and the equilibrium temperature (dashed lines). The zonal wind isshown in Fig. 2c. Note the discontinuity of the temperature and zonal velocity across the lateral edge ofthe Hadley cell. Such a solution is, strictly speaking,valid only in the limit of E -~ 0. For E small but nonzero, there will be a viscous boundary layer near theinterface and there will be a viscous correction to theleading-order asymptotic solution.1 ) VISCOUS CORRECTIONS: EXTENT OF THE I'][ADLEY CELL The solution given in (50) for the location of theedge of the Hadley cell is valid in the limit E -~ 0. Forsmall but nonzero values of E it can be shown that theviscous correction to (50) is minor near the top, butcan be significant near the bottom, where the limitingwidth of the Hadley cell in (50) is zero. Substituting(38) and (39) into (49), with T,(/~) given by (47), wefind the modified location for various Es. This is depicted in Fig. 2; for various values of E. As expected,the width of the Hadley cell near the surface is sensitiveto E since it is very sensitive to the small value of ~0 (0)- q>o(qb.). [/~,. ~ (~o(0) - rI>o(/~))TM ~ E~/~ from(49).] In general, the width is nonzero. This fact isphysically significant because it is the convergence inthe lower viscous boundary layer that is supplying thenecessary moisture for convection in the ITCZ. 2) VISCOUS CORRECTIONS: BOUNDARY LAYER NEAR THE EDGE OF THE HADLEY CELL When E is small but nonzero, there is in general aviscous correction to the leading order solution whenever there is a sharp gradient. Because, as seen in Fig.2a, there is a discontinuity in temperature across thelateral edge of the circulation region, there exists a layernear the edge where E/aV2T is not negligible even forsmall E. An "inner friction layer" (actually "thermal"in the present case) is present near z = z.(/~), and asingular perturbation analysis needs to be performedhere. As in section 3a, we let (~, ~) = (v, w)/~o. Equation(15) is written asRo 0~-~+w~~z-Z +~a = Ero K72T + (re- T). (57)The thermal diffusion term, the first term on the fighthand side of (57), now needs to be retained even forsmall Ero, because of the high gradients of T acrossthe boundary layer. Since the interface tz =/~,(z) is astreamline, we expect the most rapid varying directionto be perpendicular to the streamline. It is now convenient to introduce the streamline coordinates, following Batchlor (1956). Let ~ be the streamfunction defined by 0 cos ~ = Ozand 0~Let ~ = const lines be everywhere perpendicular to thestreamlines ~ = const, such that (, ~) now form atwo-dimensional orthogonal coordinate system, Since I OT/O~[ >> OT/OO ] in the inner boundarylayer near the edge of the circulation region, we haveasymptotically, ~ OT- h2 cOT V~T = V'Vr ~ V~, V~,~ ~ 0~,~, (58)where h2 -~ 1~7~[2 -- (02 cos2~b q- (H/a)2~2).Upon introduction of the stretched boundary layercoordinate (~, r/) =- (~/ Er,~-~o/a, ~/),Eq. (57) now becomes ( OT ) 202T Ro tVvl~+td- =h ~--~+(r--r).(59)1 M^- 1996 FANG AND TUNG 12510.8 '~l,, 59o,,O'] ''0.20.1 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (~)~l Tn'"~/' ,;" ' ~18~ ' "~ ' X'~o.~01 / ~/ ~ / ~ ~ ~3.5 , , X , X, X I-1 ~.8 -0.6 -0.4 ~.2 0 0.2 0.4 0.6 0.8 13.5! 32.5'I f ............oP'" .... R,/ , , , -1 -0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 FIG. 2. Analytic inviscid solution for an equatorially symmetric surface temperature: (a) Stream function ~; (b) temperature T (solid lines) and equilibrium temperature TE(dashed lines) distribution; and (c) zonal wind u at variousaltitudes.(b)The left-hand side is small for Ro ~ 1 (which is validfor the terrestrial atmosphere), and soh2 -92T' - T' = 0, 0~2where T' is the viscous correction to the leading-ordersolution. In (60), h2 is taken to be independent of ~and is given by leading-order solution obtained earlierin the limit E -* 0. The solution to (60) on differentside is 0.5(TM-- TE) exp(-~/h) for ~ > 0T' -- 0.5(TM-- TE) exp(Mh) for ~ < 0, which satisfies the asymptotic matching condition: T-orE, ~-0o~; r~r,,,, ~--,-~;(60) and T=0.5(TM- TE), ~=0. The use of curvilinear coordinates turns out to be cumbersome for expressing the result of (61) in terms of physical coordinates. This process can be simplified by replacing the curvilinear coordinate (~, ~/) by a local linear coordinate (~ ', r/') defined by 1 r~' - [(~ - t~o) + k(z - Zo)]; 1(61) ~' -~ [k(/z - /z0) - (z - z0)], (62)1252 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 9 0.8 0.7 0.6 ~ - 0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 014 016 0'.8 FIG. 3.extent of ~ circulation is for E = 0 (solid), E = 2 X 10-~ (dashdo~ed), E = 3 x l0-~ (dashed), ~d E = 2.5 X 10-~ (outer so~d).where (~, zo) is the edge of the Hadley cell and k--- (dz.(l~)/dl~)l~o is the local slope of the edge. Thenew coordinate is identical to the old curvilinear coordinate at (/~0, Zo). Thus, except for an exponentiallysmall error, (60) can be approximated by 02T, h'2~ - r' = 0, (63) 0~,2where h'2 = ((O~'/Oz)2 + (H/a)2(O~c'/O/.t)2)l(uo.zo).The new solution is the same as (61), except with h'replacing h and ~' replacing ~. It is plotted in Fig. 4afor various values of E~o/a. It is seen that the jumpin T in the limiting solution E--' 0 (Fig. 2a) is replacedby a smooth transition from TM on one side to the equilibrium Te on the other side of the boundary layer. Thedegree of smoothness increases with E~o/a. This analytic solution will later be verified by the numericalsolution in section 6. Once the temperature is obtained with viscous correction, the corresponding zonal wind can be obtainedeasily using the thermal wind relationship (22). Onecan easily show, using the viscous meridional momentum equation ( 11 ), that the thermal wind relationshipholds even inside the inner boundary layer. This is because the leading-order solution for v is continuous, andso the viscous term EV2v is of order E. The nonlinearadvection of the meridional velocity is O( 1/to2) and sois also small. The zonal winds for various values of Eare plotted in ]gig. 4b.8 The asymptotic solution willalso be verified numerically in section 6. ~ Some interpolation across the equator is needed for the viscoussolutions.3) LOCATION OF THE ITCZ Any/~ * 0 in the case of symmetric surface temperature will lead to a negative q(z) from (47). It implies that for a symmetric equilibrium temperature distribution, the ITCZ should be located at the latitude ofwarmest sea surface temperature. However, when theviscous effect is taken into account, it could shiftslightly off the equator (we will discuss it in the nextsection). However, for a surface temperature without a distinctive maximum at the equator but with a broad peaknear the equator, as used by Numaguti (1993) for describing the distribution of SST in middle and westernPacific: 32.82.62.42.2 - 5-01.4-01.3-0:2 -0t. 1 ~ 011 0:2 0~.3 0~.4 0.5 (~) 2.5 ~ z = .~.// ~'~'" 5 -1 -0.8 '0.6 -0.4 -0.2 0 0.2 0.4 0.60.8 Fro. 4. Viscous co~ecfion to "inviscid li~t" at v~ous altitudes, with solid lines for "inviscid li~t"; dashed lines for e~ (Ero/a)m = 0.16; dash-dotted lines for e = 0.04: (a) Temperamre T; (b) Zonal wind u.I M^Y I996 FANG AND TUNG 1253Ts( it ) =~ (T0o- A,/--2) if I/~l <itvR, T (64)~oo( oo-A,it2) elsewhere,where/~v is the width of the broad peak region. We findthat the ITCZ still occurs over the warmest surface temperature, but now there is no preferred location for theITCZ within this broad peak region. There is a possibility that two ITCZs can coexist at both sides of theequator, at +/.q, and a pair of weak reversed circulationcells exist between -it~ and /~. Figure 5 shows thecontour of the meridional circulation for ~ = it~ = 0.1,F = 7 deg km-~,Fm = 6 deg km-~, and other physicalparameters the same as those used in Fig. 2.b. Asymmetric equilibrium temperature In the eastern Pacific region, the SST has a localminimum at the equator due to strong oceanic upwelling. The maximum of SST is usually located in theNorthern Hemisphere. As an example, when the surface temperature is asymmetric about the equator, weuse [same as in Lindzen and Hou (1988)], for it =~ it0, TE=~oo[T00-A,(it-it0)2- AvZ], (65)and gttT = Tu(z) = ~oo (TMo - At~z)(66)in the circulation region. For simplicity, we take Av= Z~. We now proceed to find the boundary of theHadley region. Suppose at latitude tz,, the top boundary is in the circulation region while the bottom boundary is not--that is,/~ < ~, </~,+ or it;- < it, < it[(the situation is depicted in Fig. 1). From (36) and(39), we have[(/~, - it0)2 - (/~ - ito)2]z, (~ _ itS)2 = ~o(it,) - ~o(it~) + 8R(1 - it) (67)and ~o(/~,) - (~0(/~1) = O(~/~). Here, we still chooseU = gHA,/(2f~a) and Ro = R. In the latitudes wherethere is cimulation above the bottom boundary--thatis, it~ < it, </~--(36) givesZ, = O;and(]~2 -- ]~12)2 1o.g0.80.70.60.50.40.30,20.1 ~'~, "', '1 ,~.o~/ / ,, ,-0.8 -0.6 -0.4 -..0.2 0 0.2 0.4 0.6 0.8FIG. 5. The stream lines in the "inviscid core" for a symmetricsurface temperature with a broad maximum from -0.1 to 0.1. Similar to the discussion for the symmetric temperature, if we set ~o(/~,) - ~o(/~) = 0 in the nearlyinviscid limit, the extent of circulation at the bottomboundary z, = 0 is given by/~ = -+/~. For I/~1 >(66) gives9: (it + I'Ll)2(]'~ -- ]'~1) (69) z, = 8R(1 - it2)(/~ + kq - 2kt0)The extent of the circulation at the top boundary isobtained by setting z, = 1 in (69): - 8R(1 -/~,2)(it, + itl - 2/~o) = 0. (70)As it1 -'* ito, the top extent has a simple formula + -kto - 48J~(1 + 8R) - 8R/zo~U;- = (71) l+8RThis is the same as obtained previously by Satoh(1994), who, however, did not obtain the location ofthe boundary at other heights. From (43), we also can obtain the strength of thenet heating in the ITCZ: 1q(z) = 3r'--~ [(it$- tz-)3 - (itT~ -/Zo)3 - 3(/~1 - itO)2(it; -- J[/,.~)], (72)~0 (it) = (I)0 (itl) 8 Ro (1- /.t2) . (68) case9 Theof/.qcase=/~o-f +/~' -=as/~-- --,needs0% to be considered as the limit of the1254 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 9 0.9 0.8 0.7 0.6 0.5 0'4~ 0.~ 0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1 -0.8 -.0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 6 5 4 3~ 2 1,/',..-'" ............... : \ / ...... Z' '-7,'7,~ .... ~"% ~ ~......=.=.,,.s,z~. ...... '2.' -018 -016 -0'.4 -012 ~ 012 014 016 018 FIG. 6. Analytic inviscid solution for an equatorial asymmetric surface temperature (t~o = 0.1,/~ = 0.11): (a) Streamfunction $; (b) temperature T (solid lines) and equilibriumtemperature T~ (dashed lines) distribution; and (c) zonal windu at various altitudes.(b)where/z~(z) m:e the locations of the edge of the circulation cell at altitude z in summer and winter hemisphere, and "plus" or "minus" signs corresponds to/z > /~ and/z < /z~, respectively. The location of theITCZ is constrained by the requirement: q(z) ~> O. Inparticular, it is :required that 1q(0) = 3r-~ [(~;' -/~-)3 - (-/~ - t~-)3 - 6/~l(/~l -- /~0):] ~> 0. (73)Assuming the mass is conserved at each 'circulationcell, we have the streamfunction:- +- = k[(/~ -/~o)3 - (/z -/~o)3 - 3(/~1 -/Zo)2(/~[ -/z)], (74)where k -= AM/(3RAsro). The streamlines and the circulation region are shown in Fig. 6a for /to = 0.1,Pt = 0.11. Other parameters are the same as in the Caseof symmetric surface temperature. The dimensionalmass flux of the circulation in the winter cell (3.14x 10l- kg s-l) is stronger than the asymmetric caseobtained in Lindzen and Hou (1988) for the same/~0(2.2 x 10x- kg s-X)~- and is also stronger than thatobtained in our symmetric case. Our value is very closeto the observational data obtained by Oort (1983).[Multiplying 2~r to our value leads to 19.73 x 10~-'kg s-l, compared to the observed values in solstices(19.7 x 10l- kg s-1 in DJF, 1963-1973, and 18.1x 10~- kg s-~ in JJA, 1963-1973).] The distributionof the temperature and the zonal wind are shown inFigs. 6b and 6c, respectively. Viscous corrections are 10 In Lindzen and Hou (1988), As = 1/8 = 0.125, while ours is0.19. If we use their value, it will lead to an even stronger mass flux.1 MAY 1996 FANG AND TUNG 1255not shown. These generally tend to smooth out only thediscontinuities in the limiting solution.6. Numerical calculation Numerical calculations have been performed forboth equatorially symmetric and asymmetric equilibrium temperatures. The model used is described byEqs. ( ! ) - (7), which is basically the same set of Boussinesq primitive equations as in Held and Hou (1980),except here the vertical temperature profile in the ITCZis specified. The parameters used are the same as in thecalculation of the analytic solutions in sections 5a and5b. For simplicity, we take F = F,~ and TM0 = Ts(/~)in the calculations. The steady solution is obtained byintegrating forward in time from a motionless and equilibrium temperature initial condition. At each time step,Eqs. ( 1 ) - (6) are solved for a uniform r0 in the wholedomain and the temperature in the 1TCZ is reset to itsequilibrium value after each time. The E -o 0 + case couldnot be done numerically since no steady solution can bereached via time stepping when E is below some value.The grid is a staggered grid with 50 points in the verticaland 180 points from the south pole to the north pole. Thenumerical scheme utilized is standard in all respects. For the symmetric equilibrium temperature given by(44) and (47), a series of calculations have been completed for E = 5 x 10-5, I x 10-4, 2 x 10-4, 4X l0-4, and 8 x 10-4. The geopotenfial at the lowersurface, ~0, is shown in Fig. 7. It confirms the scalingmentioned earlier that the magnitude of ~0(/~) is pro~/~Ertional to ~/-~. The maximum values of ff)o(tZ) versusare plotted in Fig. 8. It is seen that it follows almosta perfect straight line, confirming the earlier resultbased on asymptotic scaling. In the following calcula 0'045~~/~l I ? ~ \ 0.0~[ / \ . // ^ Xx, ' ~.~5 -1 -0.8 ~.6 ~.4 ~.2 0 0.2 0.4 0.6 0.8 FIG. 7. ~e geopotential at the lower surface for E = 5 X 10% 1X 10-4, 2 X 10-4, 4 x 10-~, and 8 X 10-a (the maximum valueincreases with E). 0,045 0.04 0.035 0.03~,.~0.025 0.02 0.015 0.01 O.I ~050,01 0.015 ~ 0.02 0.025Fla. 8. The maximum values of ~o(tt) vs ~/~.0.03tions, a "standard" Ekman number E is taken to be 4x 10-4 (t/= l0 m2 s-l). Figure 9a shows the numerically obtained streamfunction solution for E = 4 x 10-4, while Fig. 9b showsthe analytic streamfunction for small E. The analytic andnumerical solutions are in good agreement with respectto the magnitude and the extent of the circulation in the"inviscid core." Compared with the analytic solution inthe E = 0 case (Fig. 2), the strength of the circulationfor E = 4 X 10-4 is about 20% less since the inviscidcore extends up to z ~ 0.85 only. It is nevertheless stilltwice as strong as that obtained in Lindzen and Hou(1988). The latitudinal extent of the circulation near thebottom shrinks as E becomes small in the numerical solutions, as was founded earlier in our analytic solution.However, we could not obtain a numerical solution foran arbitrarily small E, The extent of the circulation atthe lower surface is very sensitive to E. The numericalresults also show that the poleward flow is confined ina narrow viscous boundary layer near the top. Figure 10a shows the zonal wind at various altitudes,the numerical solution for E = 4 x 10-4 is shown indotted lines and the analytic solution for E = 4 x 10-4is shown in solid lines, while the E = 0 analytic solutionis shown in dashed lines. The maximum jet speed ofthe numerical solution agrees very well with that of theanalytic solution for small E. The agreement deteriorates slightly with respect to the position of the jet maximum closer to the surface. This is probably due to thefact that the asymptotic parameter e -- ~ is not smallenough for the analytic solution to be accurate, whichwas obtained under the assumption that the bottom viscous boundary layer does not overlap with the innerfrictional layer. Figure 10b shows the distributions oftemperature for E = 4 x 10-4. The numerical solutionis shown in dotted lines. The analytic solution for1256 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 90.0.80.70.60.~10.30.20.1 10.9 ~ t ti-0.0~196 -,0 ~ , ,9~T--- ,-- , ~ , 0.6 0.7 0.6 0.5I~ 0.4 0.3 0.2 0.1 0 -1-1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 /~ (~) ~ ~ ~ , ~ , -018 -018 -014 -012 ; oi~o14o16o18 i~ (b)FIG. 9. The streaml."unctions for/~o = 0,/~] = 0, and E = 4 X 104: (a) Numerical result, and (b) analytic result.E = 4 x 10-4 is shown in solid lines, while that for E= 0 is shown in dashed lines. In the circulation region,the latitudinal temperature gradient is approximatelyzero, as in the analytic solutions. The slight discrepancynear the midlatimde is mainly due to the use of locallinear coordinate; transform. Figure 11 displays the numerical solution for anasymmetric equilibrium temperature given in section5.2 (/~0 = /~] = 0.1). Figure lla shows the streamfunction. Due to an intensified circulation in the wintercell, the viscous effect becomes less effective and thestrength of the circulation tends closer to the analyticsolution for E = 0 (10% less). Figure llb shows thezonal wind at the top and it is close to the E = 0 analyticsolution in the winter cell. However, in the summercell, the circulation cell becomes very weak (1/i2 asstrong as in the winter cell) and the viscosity has significant effect on the zonal wind. Figure 1 lc shows thedistributions of the temperature for the numerical solution (solid lines) and the E = 0 analytic solution(dashed lines). It is also shown that the horizontal temperature profile is flat in most of winter cell and it presents a sharp gradient between the winter cell and theequilibrium region. In the summer cell, the temperatureis very close to its equilibrium state. The sensitivity test for the location of the ITCZ hasbeen performed and the results are shown in the ap 4~ 15 Z~ .75 08 .2 ~- ' - ~ 0.? "' / ~ x ~.0.6 ~ .....o,s "~ ~ x ~.,-o.~1 I Y .... ~ X I 0.~ " " ~.8 ~,6 -0.4 ~.2 0 0.2 0,4 0.6 0.8 (~) Fzo. 10. Solution of zonal wind and the temperature for go = 0, ~= 0, ~d E = 4 X 10~: (a) Zon~ wind at v~ous altitudes, wheredotted lines ~e for the numefic~ result and solid lines for the an~yticsolutioa. ~e aaalytic soludoa for E = 0 is showa in dashed liaes;(b) temperature ~s~bu~ons of ~e analytic solutions (solid lines),numerical solution (dotted lines), and an~ytic solution for E = 0(dashed lines).1 MAY 1996 FANG AND TUNG 1257 I ' ' -~'-~-= = P-----'-'-'~2'- = = '~ ' ' .130.4 : :::: ~'~1~.~ o.~.o,[ / ,', ....%.~ "-1 !o.~ ~ ', ,, ,, ,, .. ~ ,o.,/ ~o;,,,,,, ,, / o.~o.4~ ~v ~ ~ ~ ~ /~. : ',,', ', 0.34 '1-1 -0.8 ~.6 ~.4 -0.2 0 0.2 0.4 FIG. 11. Numerical solution for it0 = 0.1,/~ = 0.1, and E= 10-4: (a) Contour of stream function; (b) zonal wind at thetop, where solid line is for the numerical result and dashedline for the analytic solution in the "inviscid limit"; and (c)temperature distributions of the numerical solution (solidlines) and the "inviscid limit" (dashed lines).pendix. Since the ITCZ is defined as the region wherea strong mass convergence occurs, the net heatingshould be positive. This gives a constraint on the specification of/~ in our model. Based on our numericalsolutions, it is found that although the location of theITCZ does not necessarily coincide with the locationof the warmest surface temperature, the two should notbe too far separated. In our calculation, since the resulting concentratedheating q(z) has a maximum near the top, the streamfunction also has a single maximum near the top. This issupported by the observation (Oort 1983). This is different from that obtained by Hou and Lindzen (1992). Our numerical calculation is repeated for a cyclostrophic model, that is, with the advection term in themeridional velocity equation ignored. The results showthat there is no significant difference and confirm theconclusion obtained asymptotically that the cyclostrophic balance is a valid approximation in the adoptedparameter regime (r0 >> 1).7. Concluding remarks The present work differs from previous papers onHadley circulations (Held and Hou 1980; Lindzen andHou 1988; Hou and Lindzen 1992) in that we assumedthat in regions of intense moist convection (the ITCZ),the temperature profile locally adjusts to the moist adiabat on a timescale much faster than the Newtoniancooling relaxation time adopted by other authors.Therefore, the temperature there is known and not tobe determined by the large-scale circulation. It is thenet heating that needs to be determined as part of thesolution. This approach is consistent with the concep1258 JOURNAL OF THE ATMOSPHERIC SCIENCES VOL. 53, No. 9tual model of Emanuel et al. (1994) and with the numerical result of Satoh (1994), which includes moistconvection. Away from the region of intense convection, temperature adjusts radiatively on a slower timescale. The large-scale Hadley circulation is then responsible for determining the temperature distributionwithin the Hadley cell away from the ITCZ. The zonalabsolute angular momentum in the inviscid core isfound to be constant, which is an extension of the conclusion given by Batchelor (1956) for the case ofsteady 2D flow with closed streamlines. Thus, there isno horizontal gradient of the temperature inside theHadley circulation cell at each height. The vertical temperature profile,' within the Hadley cell should followthe moist adiabat appropriate for the latitude of theITCZ. This result is asymptotically valid in the limit ofsmall viscosity (E -~ 0 +) and large radiative relaxationtime (r0 >> 1, dimensionally larger than a few hours).The strength of' the Hadley circulation found this wayis stronger than in the previous results and is now veryclose to the observed. The studies in Emanuel et al. (1994), Yanai et al.(1973), and Satoh and Hayashi (1992) suggest thatconvection determines the vertical lapse rate approximately as the moist adiabat. Although the large-scalecirculation does not alter this lapse rate in the Tropics,it does change the sea surface temperature and the subcloud-layer entropy in the context of a coupled oceanatmosphere system. The strong large-scale circulationwill reduce the magnitude of the surface temperaturein the Tropics. This process is through a convectivedowndraft of low-entropy air and an increased surfaceheat flux to the ocean. As a simple decoupled model,this paper ignores this feedback of the large-scale circulation to the sea surface temperature. It also does notpredict the location of the ITCZ. Nevertheless, ourwork suggests that, from mass balance considerationalone, the location of the ITCZ needs to be near thelatitude of the warmest tropical sea surface temperature. Otherwise, there is not enough subsidence in therest of the Hadley circulation region to support a risingbranch in the ITCZ. Another unrealistic assumption in this work is theimposition of a lid at the model troposphere. We havesince repeated the calculations without lid, and can report that there is no significant change in the resultsunless the height of the ITCZ exceeds the tropopause,since there is no penetration of the Hadley circulationacross the tropopause. (More precisely, w = 0 in thelower stratosphere if there is no horizontal gradient ofT~. In the presertce of TE gradient across the equatorabove the tropopause, there will be a circulation in thestratosphere. However, this is not an extension of thetropospheric Hadley circulation.) Acknowledgments. We wish to thank Prof. Ed Sarachik and Dr. Hu Yang for their valuable discussions.The helpful comments by Dr. Isaac Held, Prof. MankinMak, and another reviewer are gratefully acknowledged. This research is supported by NASA GrantNAG-l-1404. APPENDIX Numerical Solution for t~ :~ po Numerical calculations for the cases when the ITCZis not located at the latitude of the warmest surfacetemperature have also been performed. Figure A1 displays the solution for P0 = 0, tz~ -- 0.1. The contour ofthe streamfunction is shown in Fig. Ala. The strengths 10.90.5 ' [ f / -0.0~ t I I "+ i I I ~ ~ t I I I I I % t I ~ ~ I I I ~ '" ' -T// I I t ~ ~ I t ~ ~ ~ I t ~ x ~.( ~II~__L_I ,, ,, ,, I I I I I I ', ,, ,, ',[,, I I I t I I 0--13' I I I ~ ~ ~ ~ ~ ~ -_2~22;~99~ [ ~0'-7--~'~ , , , , 0.8 0.7 0.6[q 0.4 0.3 0.2 0.1 0 -1 -0.8 -0.6 --0.4-0.2 0 0.2 0.4 0.6 0.8 p 2.5 ' ' ' '~i ' ' ' i~' ' ' ;, ~! ) ' i!i i: : :'~ ~ :t ~.s I;:: /;'~ I~ :t 0.: ": .,, ,,.' ~x~ -0'51 --0'.8 -0'.6 -0'.4 -0'.2 ; 012 0'.4 0'.6 0'.8 /~ (b) F~G. A1. Numerical solution for /z0 = 0, /~, = 0.1: (a) Streamfunction and (b) zonal wind at the top, where solid line is for thenumerical result, dotted line and dashed line are for the angular momentum conserving zonal wind originated from ~ = 0, 0.1 respectively.1 MAy 1996 FANG AND TUNG 1259of two cells are equal but each is slightly weaker thanits counterpart in Fig. 9a. The circulation region as awhole is still almost symmetric about the equator butit is separated at the ITCZ, and the winter cell has alarger size. According to the discussion in section 3a,the concentrated heat source q(z) will be negative nearthe lower surface if/~ -~ /~0 in the inviscid limit. Butin the viscous case, q(z) is still positive and, therefore,physically permissible due to the low-level return flowin the viscous boundary layer. Figure Alb shows thenumerical solution for zonal velocities at the top insolid line. Also shown are the zonal winds calculatedfrom (19), which is the expression of angular momentum conservation for a parcel rising from Ps- The dotted 0.7 0.6 51 0.3.0.4 i 121 ].001 0.2 3,001 -' --0.8 -.0.6 -O.4 -O.2 0 0.2 0.4 0.6 0.8 (~) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fla. A2. Numerical solution for /~0 = 0, /~t = 0.3: (a) Streamfunction; (b) Zonal wind at different altitudes (z = 1, 0.75, 0.5, 0.25corresponding to the large value of the zonal wind to the small valuerespectively), and dotted line is for the angular momentum conserving zonal wind with/zt = 0. 0,9 ~ll t/~/-q'-~~lOB I ~ ~,~ *, x x~ -0330715 0.5 '~ ~, 0.4 ~.~71 ~ ~ ~ ~.~593 0.3 ~ 0.~ ~:, ~.~ -0.4~.~ 0 0.~ 0'.~0'.~ 0'.~ (~) 4.~ :~ , , , ~ ~: :~ i ~ ,: i i ~' / i ~; 2.5 .z - F 2 i E .-' i 6 /! .. 0 ".. i ,-"~ ~'~ ~.~ -0.~ ~.4 ~.~ 0 0.2 0.4 0'.~ 0~ FIG. A3. Numerical solution for ~o = 0.15, ~ = 0.1: (a) S~emfunction and (b) zonal wind at ~e top, where solid line is for ~enmehcal ~sult, dashed line ~d dotted line ~e for ~e angul~ momentum conse~ing zonal wind for ~ = 0.1, 0.15, respectively, ~ddash-dotted line is for ~e equilibrium zon~ wind.line is for/~s = 0 and the dashed line is for/~s = 0.1, Itis seen that the numerically calculated zonal wind liesin between these two simple curves, and, therefore, wemay conclude that the numerical solution for this casefollows an angular momentum conserving path with therising motion occurring in a wider region from 0 to 0.1.Because the circulation region is symmetric about theequator and the equilibrium temperature is symmetric,the zonal wind and the temperature are also symmetricabout the equator. When the ITCZ moves further poleward, the patternof the circulation changes. Figure A2 displays the solution for Po = 0, /~t = 0.3. The streamfunction is1260 JOURNAL OF THE ATMOSPHERIC SCIENCES Vow.. 53, No. 9O.E0.?0.~0.50.40.30.20.10-1543-1 ..~ .0.0 Q_.. ~ .... ~ ~..~.~___-~= ~ .~ t It'-j-~ ~ ~ ~[//.. ~,~. ~.01!7 ', ',',~'~',~.~ .~ ~.~ ~ [~1111~ ) ~x~, ~.C ~)~ it~x ~ ~ ~ ~ II t ~ ill ~t~ ~ s ~.01~ ~ 1~ I ~ I~1 I I [ / i [ ~ t I I I ~ . ~0~ t ~ x~ ~ ! ~ ~ ~ ~ t ~ ~ ~ t/ I ~ ['XLX;,,, * ~ ~ ~l I ~. , , , , , -0.6 ~.B ~.4 ~.2 0 0.2 0.4 0.6 0.8 (~) .... /ii ~~~,t,~ ~ t tf't I ~ ~.~~~'~'~'~~ ~ . I i ~'.~~'.~ ~.~ ~.~ o o~ o'.~ o'.~ o'.~o.g0.80.70.60.50.30.20.1 FIG. A4. Numerical solution for t~0 = 0.3, /~ = 0.1: (a)Stream function; (b) zonal wind at the top, where solid line isfor the numerical result, dashed line is for the angular momentum conserving zonal wind originated from/~ = 0.1, anddash-dotted line is for the equilibrium zonal wind; (c) temperature distributions of the numerical result (solid lines) andthe equilibrium state (dashed lines).(~)shown in Fig. A2a. It is found that the cell containingthe ITCZ has a large size and a strong circulation, whilethe whole circulation domain is still symmetric aboutthe equator. In this case, the concentrated heating qo(z)is negative and a sinking rather than rising motion occurs in the ITCZ. Thus, this case may not be physicallyrelevant to the :modeling of the latent heating in theITCZ. Figure A2b shows the zonal wind at various altitudes (z = 1, 0.75, 0.5, 0.25), and the dashed line isthe angular mo~nentum conserving zonal wind whenthe rising motion occurs at the equator in (19). It isfound that the zonal wind follows an angular momentum conserving path originated from tz0 rather than/~,and the sinking motion at t~ carries the angular momentum downward. The temperature in this case is stillsymmetric about the equator. For an asymmetric surfac,e, temperature, the circulation does not change if the ITCZ drifts slightly awayfrom the warmest place poleward or equatorward. Figure A3 displays the solution for/~o = 0.15,/~ = 0.1.Figure Ala shows the strearnfunction. A winter cellstarts from/~ and it is slightly stronger than its counterpart in Fig. lla since the difference between TE andT is large in the winter cell. The summer cell is veryweak and is viscosity dominated. Figure A3b showsthe zonal wind at the top. It is found that the solution(solid line) follows an angular momentum conservingpath where the rising motion occurs at the ITCZ [/~s= 0.1 in (19) ] (dashed line) rather than a path wherethe rising motion occurs at the warmest place [/~.= 0,15 in (19)] (dotted line). The region of thisangular momentum conserving zonal wind is from1 MAY 1996 FANG AND TUNG 1261/~0 = 0.15 to the whole winter cell. Poleward of po= 0.15, the zonal wind is less than its equilibrium value(dash-dotted line). This region starts from/~o and extends poleward to the latitude where the equilibriumzonal wind attains the maximum (/~ ~ 0.5). When/~0 and/~ are separated further, the wintercell and the summer cell are separated at the confluent point where u = 0 and v = 0. This point now isnot at the same latitude as the ITCZ nor the warmestplace. Figure A4 displays the solution for/~0 = 0.3,/~x = 0.1. The contour of the streamfunction is shownin Fig. A4a. The winter cell starts at the confluentpoint, which is poleward of/~0 = 0.3 (/~ m 0.37), andextends across the equator to the winter hemisphere.The strength is twice that in Fig. 14a. The summercell is poleward of the confluent point up to the latitude where the equilibrium zonal attains maximum(see Fig. A4b). Figure A4b shows the zonal wind atthe top. There is an interesting new feature in thatzonal wind is discontinuous inside the circulation region (at/~l = 0.1). It seems possible that there aremultiple angular momentum conservation regions.Figure A4c shows the contour of the temperature(solid lines) and the equilibrium temperature(dashed lines). The horizontal profile of the temperature has a slight curvature since the meridional circulation is very strong.REFERENCESBatchelor, G. K., 1956: On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech., 1, 177-190.Cane, M. A., and E. S. Sarachik, 1989: Course on Ocean-Atmo sphere Interactions in the Tropics. United Nations Development Organization, International Center for Science and Technology, 288 pp.Chamey, J. G., 1973: Planetary fluid dynamics. Dynamic Meteorol ogy, P. Morel, Ed., D. Reidel, 97-351.Emanuel, K. A., J. D. Neeling, and C. S. Bretherton, 1994: On large scale cimulafion in convecting atmosphere. Quart. J. Roy. Me teor. Soc., 120, 1111-1143.Fang, M., and K. K. Tung, 1994: Solution to the Chamey problem of viscous symmetric circulation. J. Atmos. Sci., 51, 1261 1272.Graham, N. E., and T. P. Bamet, 1987: Sea surface temperature, surface wind divergence, and convection over tropical oceans. Science, 238, 657-659.Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric cir culations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, 515-533.Hou, A. Y., and R. S. Lindzen, 1992: The influence of concentrated heating on the Hadley circulation. J. Atmos. Sci., 49, 1233 1241.Lindzen, R. S., and A. Y. Hou, 1988: Hadley circulation for zonally averaged heating centered off the equator. J. Atmos. Sci., 45, 2416-2427.Numaguti, A., 1993: Dynamics and energy balance of the Hadley circulation and the tropical precipitation zones: Significance of the distribution of evaporation. J. Amos. Sci., 50, 1874-1887.Oort, A. H., 1983: Global atmospheric circulation statistics, 1958 1973. NOAA Prof. Paper 14. U.S. Govt. Printing Office, 180 PP.Plumb, R. A., and A. Y. Hou, 1992: The response of a zonally sym metric atmosphere to subtropical thermal forcing: Threshold be havior. J. Atmos. Sci., 49, 1790-1799.Riehl, H., and J. S. Malkus, 1958: On the heat balance of the equa torial trough zone. Geophis., 6, 503-538.Sarachik, E. S., 1978: Tropical sea surface temperature: An interac tive one-dimensional atmosphere-ocean model. Dyn. Atmos. Oceans, 2, 455-469.Satoh, M., 1994: Hadley circulation in radiative-convective equilib rium in an axially symmetric atmosphere. J. Atmos. Sci., 1947-1968.--, and Y.-Y. Hayashi, 1992: Simple cumulus models in one-di mensional radiative convective equilibrium problems. J. Amos. Sci., 49, 1202-1220.Schneider, E. K., 1977: Axially symmetric steady-state models of the basic state for instability and climate studies. Part II: Nonlinear calculations. J. Amos. Sci., 34, 280-296.--, 1983: Martian great storms: Interpretive axially symmetric models. Icarus, 55, 302-331. , and R. S. Lindzen, 1977: Axially symmetric steady-state mod els of the basic state for instability and climate studies. Part I: Linear calculations. J. Amos. Sci., 34, 263-279. Stevens, D. E., 1983: On symmetric stability and instability of zonal mean flows near the equator. J. Atmos. Sci., 40, 882-893.Tung, K. K., 1986: Nongeostrophic theory of zonally averaged cir- culation. Part I: Formulation. J. Atmos. Sci., 43, 2600-2618. Yanai, M., S. Esbensen, and J.-H. Chu, 1973: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci., 30, 611-627.

## Abstract

Simple analytic solutions are constructed for an axially symmetric, nonlinear, slightly viscous circulation in a Boussinesq atmosphere in the presence of intense convection at an intertropical convergence zone. The latitude–height extent of the Hadley circulation is obtained, as well as its streamfunction, zonal wind, and temperature distribution. Numerical solutions of the viscous primitive equations are also obtained to verify the analytic solutions. The strength of the circulation is stronger than previous results based on dry models and is now close to the observed value. The extent of the Hadley region is also quite realistic.