A major problem in atmospheric energetic dynamics is addressed. In using a

particular mathematical derivation it is verified that 0.5% of the total potential

energy is available for conversion into kinetic energy

Camerlengo, Alejandro Livio

Saadon, Mohd Nasir

Universiti Putra Malaysia Institutional Repository

Pertanika J. Sci. & Techno!. 7(1): 33-37 (1999)ISSN: 0128-7680© Universiti Putra Malaysia PressThe Role of the Available Potential Energy in the AtmosphereAlejandro Livio Camerlengo and Mohd. Nasir SaadonFaculty of Applied Science and TechnologyUniversiti Putra Malaysia TerengganuMengabang Telipot21030 Kuala Terengganu, Malaysiae-mail: alex@upmt.edu.myReceived 9 May 1997ABSTRAKSuatu masalah utama (pokok) dalam dinamik tenaga atmosfera di bicarakan.Dengan menggunakan ungkapan tertentu formula matematik, didapati bahawa0.5% dari tenaga potensi keseluruhan yang tersedia unruh konversi berubahmenjadi tenaga kinetik.ABSTRACTA major problem in atmospheric energetic dynamics is addressed. In using aparticular mathematical derivation it is verified that 0.5% of the total potentialenergy is available for conversion into kinetic energy.Keywords: atmospheric energetics, available potential energy, kinetic energy,internal energyINTRODUCTIONStrengthening or weakening of atmospheric systems which form the weatherpattern is often measured in terms of the kinetic energy they possess. Underadiabatic motion, however, the total energy of the whole atmosphere remainsconstant.In general, the motion of the atmosphere is not adiabatic. In particular,this is true in the lower layer of the atmosphere.Friction is the only non-adiabatic process which ordinarily generates internalenergy and destroys kinetic energy (Lindzen 1967). The remaining non-adiabatic processes, including the release of latent heat, alter only the internalenergy directly. Thus, the only possible sources for the kinetic energy of thewhole atmosphere are both the potential energy and the internal energy.It has been shown that the net gain of kinetic energy of the wholeatmosphere occurs at the expense of both the kinetic energy and the internalenergy (Haurwitz 1941). Following Margules (1903), the sum of the potentialenergy and internal energy is known as total potential energy,The total potential energy is not a good measure of the amount of energyavailable for conversion into kinetic energy under adiabatic flow. A quantitythat measures the energy available for conversion into kinetic energy (KE),under adiabatic flow, is generally used. Such a quantity is known as the availablepotential energy (APE). The available potential energy of the atmosphere mayA.L. Camerlengo and Mohd. asir Saadonbe defined as the difference between the total potential energy and theminimum total potential energy, which may result from any adiabaticredistribution of mass.Following Lorenz (1967), the major properties of the APE are:(i) the sum of the APE and the KE is conserved under adiabatic flow,(ii) the APE is completely determined by the distribution of mass.(iii) the APE > 0 if the density stratification is not both horizontal andstatistically stable.THE PROBLEMBoth the atmospheric potential, P, and the internal, I, energy per unit mass ofthe whole atmosphere are defined as g z and cv T, respectively. Because P andI of a vertical column above a unit area bear the ratio:PII = (c - c) I cp v vand since an element of mass per unit area is :d ml dx dy = dpl g,it yields:PoP + I = (cp/g)fT dpoUpon substitution in equation (3) ofwhere Poo = 1000 hpa, it follows that:(1)(2)(3)(4)Upon integration by parts of equation (5), it yields:where the subindex b (t) represents the bottom (top). Therefore:(6)(7)34 PertanikaJ. Sci. & Techno!. Vo!. 7 No.1, 1999The Role of the Available Potential Energy in the AtmosphereUpon usage of equation (4), it follows that:(8)Because the pressure at the top of the atmosphere is zero, the second termin the RHS of equation (7) vanishes. Thus,eo ~P + I = [cp/g(k + I)Pook] fpk+ld8 + fp k+ld8o 00(9)DISCUSSIONThe average over an isentropic surface is denoted by a bar. It follows that:(10)The minimum total potential energy which may result from adiabaticrearrangement occurs whenever p = p everywhere. It is obtained by settingp = p in equation (9). Thus, the average potential energy per unit area of theearth's surface is:A = [cp/g(k + l)p,xn f(pk+l - pk+l~ 8oUpon expansion, by the binomial theorem, of p = P + p', it yields:(11)pk+l = (p + p,)k+l = pk+l + (1 + k)p' pk + [(I+k)/2](pfpk-1 + .... (12)The average of equation (12) is taken. Bearing in mind that p' = 0, ityields:pk+l = pk+! + [(k/2)(I+k)pk-l(p,/ + [(I+k) k (k-l)/31] pk-2 (p,)3 (13)Upon integration, it yields:f(pk+1 - pk+l)d e = [k (k+l)I2] fpk-'(p,)2d e + [(l+k) k (k-l)I3!] Jpk-2 (p'f de(14)It follows that:A = [cp/g(k+l)Pooknpk+l{[k(k+l)/2] (p'IX)2 - [(I+k) k (k-l)/3!] (p'IX)3 + ...}d8(15)where X = p. The power series (15) must converge if p' < X everywhere.However, the rapidity of its convergence depends upon typical values of p'IX.PertanikaJ. Sci. & Technol. Vol. 7 o. I, 1999 35A.L. Camerlengo and Mohd. asir SaadonAssume, for example, that on a particular isentropic surface p = 1000 hpaover half of the area. Over the other half of the area, p decreases linearly from1000 hpa to 300 hpa. In this case, p = 825 hpa, (p,2)/X2 = 0.075 and (P.3)/X3=- 0.019. The ratio of these last two quantities is approximately 0.06. It may beconcluded that this series is very well represented by its leading term. It followsnaturally that:A= [(k c p )!(2 g Pook)] f pk+l(p'IX)2 d8o(16)Therefore, A depends upon the variance of pressure over the isentropicsurfaces. This variance is related to the variance of 8 on an isobaric surface.while e and T represent the average values of 8 and T; and 8' and T'represent the departures of 8 and T from their respective average values (eand T).The function e (p) is approximately determined by p = P (e (p)), insuch a way that:p' "" p (8 - 8') - p(8) "" - 8' d pld 8 (17)Therefore, it follows that:A = [(k cp )/(2 g Pook)] j {pk+1 (E)'na pia 8)1(a 8/a x) X2l}d 8 (18)oBut d P = (ap;a8) (8·)2 d8 Thus, it yields:A = [(k c p)/(2 gpook)] I{pk-l (8,)2 /(-d 8;aX)}d pFrom the hydrostatic approximation, it follows that:where Yd = -d T/ d z Id = g/c p' Because of that fact that:8'/T' = 8/T,it follows naturally that:PoA = (1/2) J {To )(Yd-yt(T'/H)2 d powhere H = T.(19)(20)(21)(22)36 PertanikaJ. Sci. & Technol. Vol. 7 o. 1. 1999The Role of the Available Potential Energy in the AtmosphereCONCLUSIONEquation (22) is suitable for estimating the ratio A/(P + Y). This ratio equalsa suitable average value of (Yd 1',2)/[2(Yry)1'2]. If Y = 0.66 Yd and 1',2(15°C)2 are taken as typical values of the atmosphere, then it follows that:X/[p + Y] "" 1/200 (23)It is verified that less than 1% of the total potential energy is available forconversion into kinetic energy. As pointed out by Holton (1972), the atmosphereis a rather inefficient heat engine.ACKNOWLEDGEMENTThis study was supported by a short grant of Universiti Putra Malaysia. Theauthors gratefully acknowledge this support.REFERENCESHAUR\\1TZ, B. 1941. Dynamic Meteorolcgy. ew York: McGraw-HilI.HOLTON, J.R. 1972. An Introduction to Dynamic Meteorolcgy. ew York: Academic Press.LINDZEN, R.S. 1967. Planetary waves on beta-planes. Monthly Weather Review 95: 441-451.LORENZ, E.W. 1967. The Nature and Theory oj the General Circulation oj the Atmosphere. WorldMeteorological Organization.MARGULES, M. 1903. Uber die Energie der Sturme. Jahrb. Zentralanst. Meteor. Wien, p. 1-26. Translation in C. Abbe (1910). Smithsonian Inst. Meteorol. ColL 54: 533-595.PertanikaJ. Sci. & Technol. Vol. 7 o. 1, 1999 37

The Role of the Available Potential Energy in the Atmosphere

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