Some characteristics of the atmosphere during an adiabatic process

PROGRESS IN NATURAL SCIENCEVol. 16, No. 6, June 2006Some characteristics of the atmosphere during an adiabatic processGAO Ll2, LI Jianping' ** and REN Hongli'(1. LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 00029, China; 2. Graduate School of Chinese A-cademy of Sciences, Beiing 100049, China; 3. National Climate Center, Laboratory of Climate Studies, China Meteorological Adminis-tration, Beijing 100081, China)Received September 10, 2005Abstract Some important characteristics of the atmosphere during an adiabatic process are investigated, which include the invari-ability of atmnospheric entropy range and local surface potential temperature, the conservation of the atmospheric mass intervened betweenany isentropic surface and the ground, and the isentropic surface intersecting with the ground. The analysis shows that the atmosphericreference state (ARS) for investigation on available potentia! energy (APE) should be deined objectively as the state which could be BP-proached from the existing atmosphere by adiabatic adjustment, and be related to initial atmospheric state before adjustment. For the initialatmosphere state at any time, its corresponding ARS is dfferent from the one at another time. Based on the above mentioned conclusions,the reference state proposed by Lorenz cannot be obtained physcally, so a new conception, the conditional minimum total potential ener-gy, is put forward in order to obietively investigate atmospheric APE.Keywords: available potential energy, adiabatic, isentropic surface, atnospheric reference state, conditional minimum total po-tential energy.The studies on available potential energy (APE)In the work on theoretical derivation and calcula-always play an important role in atmospheric energet-tion of APE, to define a suitable atmospheric refer-ics. The terminology of APE was first introduced forence state (ARS) by redistributing atmospheric massthe general circulation by Margules'1I, and later rede-under thermodynamically reversible adiabatic processfined by Lorenzl2.31. Furthermore, approximate andis of importance. In Lorenz’s first ARS without to-exact formulas for calculating APE and its generationpography, the atmosphere is barotropic, horizontal, .were promoted, but those derivations are traditionallystably stratified, and in minimum TPEI2,3,5]. Fur-based on the assumptions that hydrostatic balancether researches on ARS have been done, such as theprevails, the atmosphere is stably stratified every-case with topography, etc. Actually, such distri-where, the latent energy does not contribute to thebutions of ARS have been directly designed physical-internal energy, and surface topography can be ig-ly， whereas some characteristics of atmosphere onnored. There have been many approximations devel-isentropic surfaces ( ISs ) under adiabatic conditionoped for handling those assumptions. For example,have not yet been mathematically investigated. More-Dutton and Johnson derived a“ more exact" equationover, how ARS can be obtained from an existing at-by eliminating the assumption of hydrostatic balmospheric state in terms of adiabatic adjustment, andance'”，and Lorenz developed moist available energywhether ARS designed physically can be approached,by considering the vapor processl5), and Taylor tookhave not been proved mathematically. Evidently,surface topography into accounto. Lorenz' s APEsolving these essential problems are very important forconcept has also been extended to study the variationthe comprehension of ARS and APE，which will beof APE in a limited region during the process of for-examined in the present paper.mation of storms'[7-10]In recent years, the theory of 1 Variation of local surface potential tem-APE has new development in many aspects'll- -131,perature during an adiabatic processand has also been applied widely to the studies of at-mospheric and oceanic energetics-14 -18] .APE is defined by Lorenz as the difference be-twee中国煤化ITPE) of an eisig* Supported by the National Science Fund for Distinguished Young Scholars (Grant.TYTHCNMHG°ic Research DevelopmentProgram of China (Grant No.2006CB400503)** To whom correspondence should be adressed. E -mail: gaoli@ mail. iap. ac. cnProgtess in Natural Science Vol.16 No.6 2006 www. tandf. co. uk/ journals645atmospheric state and some suitably defined referenceature is invariant .statel2.31. Usually the reference state represents theminimum total potential energy (MTPE), which canThis characteristic can be easily documented.be attained by redistributing atmospheric mass underGiven that the range of atmospheric potential temper-thermodynamically reversible adiabatic condition. Ac-ature is originally expressed as [0min(0)，0max(0)]cording to Lorenz’ s definition of ARS， the atmo-and written as [0min(t),0mx(t)] at any time t. Ifsphere is barotropic, borizontal, stably stratified, and0min(t)≠0min(0), 0max(t)≠0max(0), there will ex-in the state of MTPE. If there is no surface topogra-ist motion passed through isentropic surface duringphy, the local surface potential temperature of thethe process of adiabatic adjustment, which is evident-ARS is homogeneous and just identical to the mini-ly contradictory to the above- mentioned.mum of potential temperature before adiabatic adjust-Under adiabatic condition, lower boundary con-ment. Taylor put forward two particular ARSs withsurface topography consideredl6l. One has the mini-dition (2) can be written as 0s= 0. If there is nomum entropy due to taking surface topography intoboundary flowl19l(namely Vhs= 0, corresponding toARS directly, and the other has uniform surface pres-viscous lower boundary condition), or in more com-sure, undulate isentropic surfaces with topography,mon sense, there is no entropy advection on the sur-and the minimum enthalpy. We can see that for dif-face ( Vns*V0s=0), we can obtain that local changeferent purposes, the definition of ARS can be chosenof 0s is zero. So it can be obtained thatarbitrarily. Therefore, it is necessary to examineCharacteristic 2. During an adiabatic process, ifwhether the assumed ARS can be approached really inan existing atmospheric state by adiabatic adjust-there is no flow across the boundary, or there is noment. So far rigorous derivations have not been seenentropy advection on the surface, the surface poten-tial temperature will locally remain invariable.yet.This implies that local potential temperature onWe first give boundary conditions in sphericalisentropic coordinate (λ,q,0,t) (where λ, φ indi-the surface always keeps its original value in the exist-cate geographical longitude and latitude, respective-ing atmosphere. Under the condition of no boundaryly)， λ∈[0,2x]，φ∈[- π/2,π/2], θ is potentialflow and no surface entropy advection， local surfacepotential temperature does not vary with the processtemperature and t the time) as fllosl9,20;of atmospheric adiabatic adjustment.Upper boundary: when θ= 0r= const,Based on Characteristics 1 and 2, it is easy toθ= 0;(1)obtain two simple corollaries as follows:lower boundary:when θ= 0g(A,q,t),Corollary 1. During an adiabatic process, ifads.- us_ as+ sas,(2)there is no boundary flow, or there is no entropy ad-9tacosp aa ap’vection on the surface, the range of atmospheric en-where θ, 0s are potential temperature at the top oftropy or potential temperature in any vertical air col-atmosphere and on the surface of the earth, respec-umn will be constant.tively. ug， Us are zonal and meridional componentsof wind at the surface, respectively, and a is meanCorollary 2. During an adiabatic process, ifradius of the earth.vection on the surface, the position of intersectionAs we have known, under adiabatic condition,lines between isentropic surfaces and the ground willthe potential temperature of air parcel remains con-never change.stant, viz. dθ=0. Since vrtical θ velocity θ = d0/It is clear that the fore mentioned derivations candt= 0, ISs are substantial surfaces. During the probe reasonable with or without topography. Thiscess of adiabatic adjustment, substantial surfaces canneither vanish nor be created, and there is no motionshow中国煤化工f adiabatic adjust-which can pass through ISs. So we obtainmentC NMH Gat any place on thesurface remains nvariaole Irom the existing atmo-Characteristic 1. During an adiabatic process,sphere to its ARS. Therefore, after adjustment, thethe range of atmospheric entropy or potential temper-ISs intersected with ground are not always superpos-546www. tandf. co. uk/journals Progress in Natural Science Vol. 16 No.6 2006able with iso- geopotential surfaces, which indicatescurved surface σG= o1Uσ1s to the atmospheric topthat the previous definitions of ARS can be unobjec-with the upper boundary condition, hence it followstive.thatIn addition, because atmospheric state is diabati-d8do+192(边dθdoJo, 202ta0\at 1cally changing at every time, surface potential tem-i1sperature varies with time, which makes the local sur-._face potential temperature after adiabatic adjustmenta coso;(最(唱门。different at every time. This implies that ARS shouldbe different with respect to the existing atmosphere+l 二vcosq)]}> dθdostate at different time. In the following discussion,we can see that the invariability of local surface poten-I儿。acos;[(00」。tial temperature is quite important for the atmosphericadiabatic process due to the mass conservation restric-tion. .+21 vcosq,唱1]},} d0do501]2 Atmospheric mass conservation on isen-「的( 2)do-|[es(R20 )do = 0,(4)tropic surfaces during an adiabatic processisin which dσ=a2 cos φdλdφ is the integral elementBased on the invariability of local surface poten-in horizontal direction. Sincetial temperature under adiabatic condition，now some.1acharacteristics of the atmosphere on the ISs duringat o。- 2ta0g_ atthe process of adiabatic adjustment will be further dis-cussed. The atmospheric mass conservation is a basicwith the aid of atmospheric upper boundary condi-law as we know. For the characteristics of mass con-tion, the first two terms of Eq. (4) can be rewrittenservation, the previous work focused on the case thatasISs do not intersect with ground!2-4But the caserp°r旦边_dθdσ +[["3(2 d0daa0\ atthat ISs intersect with ground, which is quite essen-°is .tial to understanding the features of ISs, has not been=-[(2)do-[(2 dodocumented clearly.°1°1sIn the spherical isentropic coordinate, the mass__ 2do+ [(中Osdo. (5)continuity equation. may be expressed as:\ 201g。 at "「2|边__「. al边Lat\a01一g acosp Laxa01Thus, the integral of pressure tendency on the curvedsurface σG is obtained. For any variable A, there ex-1「auos器)], +是o明)=0,acosp Lapists Ado =[A。do+ |Asda, in which Ag, is(3)where p is pressure, u and V are zonal and meridionalthe value of variable A on the isentropic surface 01,components of wind, respectively. Here, the pressureand As the value of A on the ground.tendency on the top of atmosphere is ignored for con-venience, namely apg/at = 0[19,20].For the fourth term of Eq. (4), with the law oftaking derivative of variable- limit integral we can ob-Considering the general case that any isentropictainsurface θ1 intersects with ground, let 01 denote theoverground section of the isentropic surface θ1( name-acosplax("器明acosply, if(λ,p)∈σ1, θ≥θs(入, p)), and let σ1s denotethe ground section of the isentropic surface 01 ( name-l1sa0s((6)中国煤化工05 a0)。ly, if(λ,q)∈σ1s,θ≤θs(λ,p)), and let r denotea collection of intersection lines between isentropicYHCNMH(p9τ_ 1surface θ1 and ground ( namely the boundary I =acosp LapJ 0。UCOs aoav| =0。acospσ1∩σ1s). We can integrate Eq. (3) from the closed .Progress in Natural Science Vol.16 No.6 2006 www. tandf. co. uk/ journals547usads(边(7)the atmospheric mass over a closed curved surface σGa ap\aolgcan also maintain conservation.Substituting (5)- -(7) into Eq. (4)，we obtainTherefore, it can be seen that:)0s.do+J[( 2301g。 atdσCorollary 3. During an adiabatic process, if。there is no surface entropy advection, (i) the atmo-spheric mass between any two continuous isentropica cospsurfaces will maintain conservation; (ii) the atmo-spheric mass between any isentropic surface and+ La\Jovcospaodlσground will maintain conservation.us_ aθs. usads\( 边}As the above derivations have shown，for theJ \acosq af a aφ1\aogcase that there exists surface entropy advection,°1swhich means surface potential temperature varies[o.2)。do-|[0s(品。do=0,with time, Eq. (10) cannot be obtained from (9)Hence, the mass between any IS and ground can notin which, when (λ,φ)∈σ1,θw=θi; when (入,φ)maintain conservation if IS intersects with ground.∈σis, θw= 0g. With the lower boundary conditionControlled by the law of mass conservation, either no(2)，we can further obtainsurface entropy advection or the invariability of localsurface potential temperature is an essential physicalT 0do=-do.(8)ata0 1日,restriction for the adiabatic process in the atmo-sphere.For an atmospheric adiabatic process, the right-handside of Eq.(8) is equal to zero, thus3 Conditional minimum total potential ener-do=0.(9)eyAccording to Characteristic 3, the atmosphericIn Eq. (9)，whether the order between integral anddifferential operators can be exchanged will entirelymass enclosed with ISs and ground maintains conser-depend on the fact whether the boundary T=σ1∩σ1svation, but its distribution may change. So there ex-varies with time. As Corollary 2 has shown, whenists a minimum total potential energy (MTPE). Fur-there is no surface entropy advection, the positions ofthermore, for the initial atmospheric state at anythe intersection lines ( namely the boundary T) be-time, its corresponding ARS is different from the onetween ISs and ground will remain invariable. At thisat another time. Thus, such a MTPE should be con-moment, we can obtain the characteristics of atmo-ditional and depends on the initial distribution of at-spheric mass conservation over a closed curved surfacemospheric state at different time, which is called theσc by Eq. (9):conditional minimum total potentialenergy(CMTPE) here. During an adiabatic process, not on-2(J pdo)=0,(10)ly the potential temperature of air parcel is invariable,but also the mass maintains conservation. Restrictedwhere g is the acceleration of gravity. In particular,by the law of mass conservation, the positions of in-if σisis null set中, then the isentropic surface has notersection lines between ISs and ground are also fixedintersection with ground, and σ1=σG. Eq. (10) isand unchangeable with time. Hence， in Lorenz’ sstill proper. On the other hand, if σ1 is null set中,definition, the ideal ARS with horizontal ISs may beσs=σG, and Eq. (10) represents the conservation ofnever approached by adiabatic adjustment, and itsthe whole atmospheric mass. Thus it follows thatcorresponding TPE is just a lower limit of TPE a-mong different ARSs. which mav be called the abso-Characteristic 3. During an adiabatic process,lute中国煤化工atmospheric mass over the isentropic surfaces which，CNMH G,do not intersect with ground will maintain conserva-THAPE(2.3, the in.tion; when isentropic surfaces intersect with ground,tegrand is the difference between TPE and MTPE inunder the condition of no surface entropy advection,unit air column. In order to guarantee consistent isen-648www. tandf. co. uk/journals Progress in Natural Science Vol. 16 No.6 2006tropic surface, which means vertical integral limits at4 DuttonJ. A. and Johnson D. R. The theory of avilable potenialany column keep the same during the process of adia-energy and a variational approach to atmospheric energetics. Ad-vances in Geophys, 1967, 12: 333- 436.batic adjustment， Lorenz has made an extension5 Lorenz E. N. Available energy and the maintenance of a moist eir-( named Lorenz extension here): when 0<θg(λ,p),culation. Tellus, 1978, 30: 15- -31.p(A,p,0)= ps(λ,p). Thus, the pressure of every6 Taylor K. E. Formulas for calculating available potential energyover uneven topograpby. Tellus, 1979, 31: 236- -245 .spot under ground is substituted by corresponding7 Smith P. J. A computational study of the energetics of a limited re-surface pressure, which makes all of ISs closed. Asgion of the atmosphere. Tellus, 1969, 21(2); 193-201.we know, the ISs intersected with ground cannot8 SmithP. J. On the cntibution of a limited region to the globalreach horizontal states by adiabatic adjustment underenergy budget. Tellus, 1969, 21(2): 202- 207.9 JohnsonD. R. The available potential energy of storms. J. At-the restriction of mass conservation. So Lorenz exten-mos. Sei., 1970, 27: 727--741.sion is almost impossible to be applied physically, al-Edmon H. J. Jt. A reexamination of limnited-area available poten-though it is mathematically feasible. Actually this ex-tial energy budget equations. J. Atmos. Sci.， 1978, 35; 1655-tension is unnecessary, because local surface potential1659.1 BoerG. J. Zonal and eddy forms of the available potential energytemperature has invariability under adiabatic condi-equations in pressure coordinates. Tellus, 1975， 27(5): 433-tion. So any ARS, which cannot be approached by a-diabatic adjustment, is not objective.12 Siegmund P. The generation of available potenial energy, accord-.ing to Lorenz ' exact and approximate equations. Tellus, 1994,On the other hand, any atmospheric state with46A(5): 566-582.3 Shepherd G. A unified theory of available potential energy. At-different distribution of surface potential temperaturemos. Ocean., 1993, 31: 1- -26.should have its corresponding ARS approached by a-14 OortA. H., AscherS. C., Levitus s. et al. New estimnates ofdiabatic adjustment. Because the positions of intersec-the available potential energy in the world ocean. J. Geophys.tion lines between ISs and ground are fixed, theoreti-Res.1989, 94: 3187-3200.cally the ARS after adjustment exists uniquely. If theHuang R. Mixing and available potential energy in a boussinesq oMTPE is regarded as a criterion, the APE can be de-cean. J. Phys. Oeanogr., 1998, 28: 669- -678.6 Blanchard D. O. Asessing the vertical distribution of convectivetermined in terms of the difference between the TPEavailable potential energy. 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