Sources and Sinks of Available Potential Energy in a Moist Atmosphere. Olivier Pauluis 1. Courant Institute of Mathematical Sciences

Transcription

1 Sources an Sinks of Available Potential Energy in a Moist Atmosphere Olivier Pauluis 1 Courant Institute of Mathematical Sciences New York University Submitte to the Journal of the Atmospheric Sciences April 29th, Corresponing author aress: O. Pauluis, Warren Weaver Hall, 251 Mercer street, New York City, NY pauluis@cims.nyu.eu.

2 Abstract A general expression for the sources an sinks of available potential energy (APE) in a moist atmosphere is erive. The available potential energy here is efine as the ifference between the total static energy of the atmosphere an that of a reference state that minimizes the total static energy after a sequence of reversible aiabatic transformations. The central result presente here is that small reorganizations of the reference state o not change its total static energy. This makes it possible to etermine the rate of change of APE even in the absence of an analytic formula for the reference, as it is the case in a moist atmosphere. The effects of changing the total water content of an air parcel are also consiere in orer to evaluate the change of APE ue to precipitation, evaporation an iffusion of water vapor. These results are then iscusse in the context of the APE prouction for an iealize raiative convective equilibrium.

3 1. Introuction The maintenance of the general circulation requires that kinetic energy be continuously generate to balance frictional issipation. In this respect, the atmosphere behaves as a heat engine that prouces mechanical work by transporting heat from warm to col regions. However, the amount of kinetic energy prouce is limite by the thermoynamic properties of moist convection. Pauluis et al. (2000) an Pauluis an Hel (2002a,b) have shown that irreversibilities associate with the Earth s hyrological cycle severely reuce the amount of kinetic energy generate by the atmospheric circulation. Unerstaning the thermoynamic constraints on kinetic energy prouction is a key element in builing a theory of the general circulation of the atmosphere. Early iscussions of the maintenance of the general circulation were base on the notion of Available Potential Energy (APE) rather than entropy. Lorenz (1955) efines the APE as the ifference in total static energy (internal plus geopotential) between the current state of the atmosphere an that of an iealize reference state, efine as the state that minimizes the static energy of the atmosphere after a sequence of reversible aiabatic transformations. The APE is thus a property of the entire atmosphere, in contrast to entropy, which is well-efine for iniviual air parcels. By construction, the sum of APE an total kinetic energy is conserve through reversible aiabatic transformations. The atmospheric circulation can be escribe as a heat engine in which external heat sources act as a net source of APE, which is then converte into kinetic energy by the mostly reversible, aiabatic motions before being issipate through frictional issipation.

4 It shoul be note that this escription of the atmospheric circulation in terms of APE has been limite to ry circulations. Moisture, if it is taken into consieration at all, only enters as a prescribe external heat source, accounting for the latent heat release associate with phase transitions of water. From a conceptual point of view, the latent heat release is not an external process, but correspons to an internal energy conversion, an the actual external heating occurs when water evaporates at the Earth s surface. The main limitation of the ry framework lies in that it requires a priori knowlege of the istribution of latent heat release in orer to etermine the source of APE. Given the new unerstaning of the importance of latent heat transport an moist processes for the entropy prouction, the ry framework seems inaequate to properly aress the maintenance of the general circulation in the Earth s atmosphere. Applications of APE to moist circulations have however been limite. Lorenz (1977) an Ranal an Wang (1992) show that the concept of APE, efine as the static energy ifference between the current an reference states, can be applie to a moist atmosphere as long as the transformations leaing to the reference state inclue reversible phase transitions. The funamental ifficulty here is that the reference state cannot be erive analytically, but must be etermine from an iterative minimization proceure. In the absence of an analytic expression, it becomes impossible to take a time-erivative require to compute the evolution of APE. It shoul be stresse here that, while a number of textbooks iscuss the general circulation in terms of APE, none provies an expression for the sources an sinks of APE in a moist atmosphere. They rely instea on a formulation that is only vali in the context of a ry atmosphere.

5 The issue of the sources an sinks of APE has receive little attention so far, to the exception of the recent stuies Bannon (2004, 2005), which erive the effects of moisture in the context of various formulations for the available energy. Bannon (2004) obtains a general formula for the sources an sinks of available energy, but this expression requires a precise knowlege of the evolution of the reference state. The purpose of the current paper is primarily to erive an analytic expression for the rate of change of the Lorenz APE. The main result here is the rate of change of APE associate with changes in the reference pressure cancels out. This makes it possible to obtain an expression for the sources an sinks of APE even in the absence of an analytic formula for the pressure in the reference state. This expression is equally vali for ry an moist atmospheres, an requires only the knowlege of the reference state, not its evolution. The iniviual contributions of various processes are analyze. In particular, an external heat source generates an amount of APE given by the net energy input multiplie by a potential efficiency. This potential efficiency is equal to that of a perfect heat engine acting between the parcel s temperature an its temperature in the reference state. The efficiency for evaporation is slightly lower, mainly ue to a water loaing contribution. Aiabatic processes can also act as source or sink of APE. It is foun that iffusion of sensible an latent heat can result in a reuction of the APE that can be comparable to the surface energy source. In aition, both precipitation an reevaporation can act as sources of APE. This latter point is in stark contrast with a ry atmosphere where aiabatic processes can only increase the APE through a reverse conversion of kinetic energy.

6 2. Time tenency of the Available Potential Energy The total energy in the atmosphere is given by the integral E = " u + gz + 1 V 2m (1) 2 where u is the internal energy, g the gravitational acceleration, Z the geopotential height an V the three-imensional velocity. The integral in (1) is taken over all the mass of the atmosphere, with m the mass of an air parcel. The first law of thermoynamics implies that in the absence of external heat sources an exchanges with the lower bounary, the total energy is conserve. Consier an atmosphere in an initial state A that is transforme, after a set of aiabatic processes, into a final state B. Then the net amount of kinetic energy generate by the successive transformations is equal to the net loss in static energy: " $ 1 2 V 2 m = #" $ u + gzm. (2) Equation (2) is a statement of energy conservation for an isolate system. It oes not provie any constraint on how much energy can be converte, which epens on the sequence of transformations consiere. For any atmospheric state, it is possible to efine a reference state as the state that minimizes the total static energy after a set of reversible aiabatic transformation. Because of (2), this is also the state that woul maximize the kinetic energy in the atmosphere. The Available Potential Energy (APE) is efine as the ifference in total static energy between the atmosphere in its current state an in the reference state: APE = # (u " u ref ) + g(z " Z ref )m, (3)

7 where u ref an Z ref enote the internal energy an geopotential height of a parcel in its reference state. As entropy s an total water content q t are conserve in reversible aiabatic transformation, the reference state is a function of the joint istribution of entropy s an humiity q t. A more etaile escription of the properties of the reference state can be foun in Lorenz (1977) an Ranall an Wang (1992). A convenient simplification arises when the atmosphere is in hyrostatic balance. Inee, in this case the vertical integral of the static energy is equal to the vertical integral of the enthalpy. Furthermore, Lorenz (1955) shows that the reference state is always hyrostatic. Therefore, the APE of a hyrostatic atmosphere is given by the ifference in the total enthalpy APE = # h " h ref m (4) where h an h ref are the enthalpy in the current an reference states. We are intereste here in etermining the rate of changes of APE. For simplicity, one can assume that the current state is in hyrostatic balance. In this case, the rate of change of APE is given by the ifference between the rate of change of the total enthalpy of the atmosphere an of the reference state: t APE = h " t m # h " ref m. (5) t In a non-hyrostatic atmosphere, aitional contributions arise from the fact that the total static energy is not equal to the total enthalpy, primarily relate to elastic processes. For a full iscussion of these aitional contributions, the reaer is referre to Bannon (2004). In a multiple component system like moist air, it is useful to istinguish between open an close transformations, epening on whether a given parcel exchanges mass with its surrouning. Diffusion of heat, raiative heating, evaporation of clou water, an

8 expansion an compression are examples of close transformations, as the total amount of water is conserve. In contrast, evaporation at the Earth s surface, iffusion of water vapor, an precipitation are open transformations, in which the amount of water in an air parcel varies, an require specific treatment Close transformations. Consier first the rate of change of APE for close transformations only. The rate of change of enthalpy is relate to the rates of change of entropy, specific humiity an pressure: h t = Q + " p t = T s q + #µ t t + " p t. (6) Here, q is the specific humiity, Q is the external heating rate, "µ = µ v # µ l = R v ln H is the ifference of chemical potential between water vapor an liqui water, with H the relative humiity, an α is the specific volume. The time erivative t here refers to the Lagrangian erivative following a parcel s trajectory. At this point, only close transformations are consiere, so the total water q T is constant. Similarly, the enthalpy change in the reference state is given by h ref t (t 0 ) = T ref s t + "µ ref q t + t ( p ref (t 0 +t ) $ #p). (7) p ref (t 0 ) As before, the subscript ref inicates a quantity evaluate in the reference state. The thir term on the right-han sie correspons to the pressure work as a parcel reference

9 pressure varies. Note that if p ref is a smooth function of time, this integral is equal to " ref p ref t. The integral notation is kept here, as p ref can be iscontinuous. Hence, combining (6) an (7) into (5) yiels t APE = # (T " T ref ) s t m + # ($µ " $µ ref ) q t m p ref (t 0 +t ) + % p t m " # # t ( # %p) m. p ref (t 0 ) (8) This equation is similar to that erive by Bannon (2004). The first two terms on the right-han sie inclue the generation of APE by external heating, but also the effects of iffusion of heat an phase transition. They will be iscusse in greater etail in Section 4. The thir term on the right-han sie can be written as # " p t m = $p # $t xyz + = t r # V % r pxyz # " &1 gzxyz & # p% r Vxyz = ' t &W = & t KE (9) after replacing the integral over the mass by a volume integral, with"m = xyz. Equation (9) inicates that the contribution of the total pressure change is equal to ifference between the increase in geopotential energy " an the work performe by t the atmosphere on itself W. This term thus accounts for the conversion of APE to kinetic energy " KE. The fourth term of the right-han sie of (8) is the change in APE t associate with a reorering of the reference state. As there is no analytic expression for p ref, the changes in the reference pressure woul have to be obtaine numerically. This woul make the computation of the sources an sinks of APE extremely ifficult.

10 However, it is now shown now that the change in APE associate with a reorering of the reference state always vanishes, making the computation of p ref t unnecessary. Consier two atmospheric reference states A an C escribe by given istributions of entropy s A (m) an s C (m), an water vapor q A (m) an q C (m). The pressure istributions in these reference states are p A (m), an p C (m), while h A (m) an h C (m) are the enthalpy istributions. Here, m is use as a parcel ientifier. The enthalpy ifference between the two states is given by C # (h C " h A )m = # ( # Ts)m + # ( # $µq)m + # ( # %p)m, (10) A C A C A with C " inicating an integral following a thermoynamic path between the two A reference states. The transformation between A an C can follow an infinite number of such paths. One possibility is to efine an intermeiary state B which has the same pressure istribution as A, but the same entropy an moisture istribution as C. The transformation between A an C can thus be subivie in a change of entropy an moisture at constant pressure (leg A to B), followe by a reversible aiabatic pressure ajustment (leg B to C). In this case, we have: B # (h B " h A )m = # ( # Ts)m + # ( # $µq)m (11) A C B A # (h C " h B )m = # ( # $p)m (12) By construction, C is the reference state for B, it minimizes the total static energy, an the integral B

11 C # ( # "p)m $ 0 (13) B is negative. Inee, if it were positive, B woul have a lower total enthalpy than C. Similarly, the state D is constructe as to have the same pressure istribution as C an the same entropy an total humiity as A. Hence, we have D # (h D " h C )m = # ( # Ts)m + # ( # $µq)m (14) As A is the reference state for B, we have Aing (11-12) an (14-15) yiels C D C A # (h A " h D )m = # ( # $p)m (15) A D # ( # "p)m $ 0. (16) D $ C A ' $ B D ' # & # "p + # "p) m = * # & # (Ts + +µq) + # (Ts + +µq) ) m. (17) % ( % ( B D A The constraints (13) an (16) imply that the left-han sie must always be negative. Instea of the cycle A-B-C-D, one can subivie the path between A an C into N+1 intervals by selecting a set of intermeiary reference states A 1,A 2, A n. As before, construct B i as the state with the same pressure istribution as A i an the same entropy an total water istributions as in A i+1. Similarly, the states D i hare constructe with the same pressure istribution as A i+1, an same entropy an total water istributions as A i. We can now construct the cycle A=A 0, B 0,A 1, B N, C=A N+1,D N+1, A N, D 1,A=A 0. The equivalent of equation (17) inclues now the integrals taken over 2N+2 segments: C

12 # N $ A i+1 ' N $ A i ' N $ B i ' * & # "p ) m + # * & # "p ) m = + # * & # (Ts +,µq) ) m i= 0% ( i= 0% ( i= 0% ( B i D i+1 A i N $ D i+1 ' + # * & # (Ts +,µq) ) m i= 0% ( A +1i (18) By increasing the number of subintervals N, the terms on the right-han sie converge towar the integral following a path of reference state: Lim N "# i= 0 N & B i ) N & D i+1 ), ( % (Ts + $µq) + = -Lim N "#, ( % (Ts + $µq) + ' A i * i= 0' A i+1 *, (19) C % = (T ref s + $µ ref q) where the subscript ref is use to emphasize the fact that an integral is taken along a path of reference states. In the limit of N " #, the right-han sie of (18) vanishes, an we have A lim N "# N & A i+1 ) N & A i ) %, ( % $p + m = -lim N "# %, ( % $p + m (20) i= 0' * i= 0' * B i D i+1 As A 0,A 1, A n+1 correspon to reference states, the left-han sie is always negative for finite value of N, an the right-han sie is positive. Therefore, the limit converges towar 0: lim N "# N & A i+1 ) N & A i ) B %, ( % $p + m = lim N "# %, ( % $p + m = % % $ ref p ref m = 0 (21) i= 0' * i= 0' * A B i D i+1 where the integral is taken following a path of reference states. This result is a irect consequence of the fact that the reference state minimizes the total static energy of the atmosphere for reversible aiabatic transformations. The cancellation of the pressure term in (21) is very similar to the constraine extremum theorem in calculus. Given two smooth functions F(X) an G(X), with X a n-imensional

13 vector, so that F has an extremum at X 0 uner the constraint that G(X) = 0, then the irectional erivative of F in any irection allowe by the constraint G(X)=0 must vanish: "F "G & (X 0 )x i = 0 for #x $ % N such that& k (X 0 )x i = 0. "x i "x i i In the context of the previous iscussion, one can interpret the function F as the total enthalpy of the atmosphere, while the set of constraints G k correspons to the requirement that the entropy an total water of each parcel is conserve. For a finite set of parcels, the extremum theorem implies that F the total enthalpy remain unchange for an infinitesimal aiabatic reistribution of the parcels near a reference state X 0 that minimizes F. The erivation of equation (21) can be viewe as a specialize proof of the extremum principle for an infinite number of parcels an in which the erivatives are expresse in terms of thermoynamic variables. Equations (21) implies in particular that the reference pressure term in (5) vanishes p ref (t 0 +t ) # # t ( # " ref p) m = 0, p ref (t 0 ) so that the sources an sinks of APE can be written as: t APE = # (T " T ) s ref t m + # ($µ " $µ ref ) q t m " t i KE. (22) The first two terms on the left-han sie are the sources an sinks of APE associate with the phase transition of water an with the external forcing, while the thir terms is the conversion of APE into kinetic energy. Before iscussing these terms in greater etail in section 3, we nee also to aress the impact of changing the total water content of a parcel.

14 2.2. Open transformations Open transformations here mean transformations in which the total water in a parcel is change. It is only necessary to consier the aition or removal of a single phase, which in practice can be either precee or followe by a phase transition. Consier the aition of a quantity q l of liqui water to a given parcel. The enthalpy of the parcel changes by h = h l q l = Ts " µ l q l (23) where h l =C l (T T 0 ) is the specific enthalpy of liqui water, s=s l q l is the entropy change in the parcel, s l = C l ln T T o is the specific entropy of liqui water, an T 0 is an arbitrary reference temperature. Note that the change in parcel entropy can be written as the sum of an entropy change an a chemical potential contribution, similar to the formula for a close transformation. The effect of the mass change is not limite to the parcel itself. Inee, for the atmosphere to remain hyrostatic, the pressure below the parcel must increase m by p = gq l, with A the horizontal area of the parcel. For an aiabatic atmosphere, A the compression is associate with an enthalpy increase "p in all the unerlying parcels. Integrate over the entire column, this is equal to the geopotential energy of the water gz p q l m. Combine with (23), this implies that the total enthalpy change in the atmospheric column is equal to the sum of the enthalpy an geopotential energy of the liqui water.

15 The change in APE is obtaine by taking the ifference in the integrate enthalpy between the current an reference states: # APE = C l (T " T ref " T ref ln T & % ) + g(z " Z ref )( mq l. (24) $ T ref ' Note that the contribution of the term gz ref q l m arises from integrating the changes in the reference pressure: # " ref p ref = gz ref q l m (25) This oes not contraict the result from the previous section. Inee, the reference state is etermine by minimizing the total static energy through a succession of reversible aiabatic transformations while conserving the total water. As this exclues open transformations, the extremum theorem oes not apply to the change of total enthalpy resulting from the aition or removal of water. It shoul be stresse here that the expression (24) is vali for the rate of change of APE by the aition of conense water in a hyrostatic atmosphere. Without this unerlying hyrostatic balance, there woul be an aitional contribution associate with elastic effects (see Bannon 2004). 3. Sources an Sinks of APE Equations (22) an (24) can be use to compute the impact of various processes on the APE. It shoul be note here that while the APE itself is a property of the istribution of entropy an water vapor of the whole atmosphere, the sources an sinks of APE can be compute for specific parcels, inepenently of the transformations taking place in the rest of the atmosphere.

16 This section iscusses the APE contribution of six main atmospheric processes: external heat source, surface evaporation, iffusion of heat an of water vapor, precipitation, an re-evaporation of conense water. The APE sources an sinks are also estimate for an iealize raiative-convective equilibrium. This example is meant to be illustrative, an complete calculation of the APE sources an sinks woul be performe in a subsequent paper External Heating An external heat source Q, such as surface sensible heat flux or raiation, is associate with an entropy change of. Hence, the net source of APE is equal to " "t APE ext = Q T # T ref $ m. (26) T This is equal to the heating rate multiplie by a potential efficiency. This efficiency is the same as that of a perfect heat engine acting between the parcel s temperature an its reference state. The potential efficiency is negative when the reference temperature is higher than air parcel. In such case, heating woul reuce the APE, while cooling woul increase it. For heating at the surface, the potential efficiency can be extremely large, up to 1/3 for an air parcel at T~300K an with a reference temperature T ref ~200K. The potential efficiency is however highly sensitive to the parcel s properties. In particular, heating in a parcel whose reference level remains near the surface woul generate little or no APE. Tropospheric cooling can also prouce APE if the cooling occurs in parcels whose

17 reference level is lower than their current locations. Frictional issipation is associate with a net irreversible entropy prouction equal to D, with D the amount of kinetic T energy issipate, so that its effect on APE is the same as a heat source with Q=D. As the potential efficiency is always less than 1, frictional heating always prouces less APE than the amount of kinetic energy issipate Surface Evaporation The evaporation at the Earth s surface can be treate as a net aition of conense water, followe by an external heating an phase transition. " "t APE T # T = $ L ref v E T q t m + $ R vt ref (lnh # ln H ref ) q t m + $ C l (T # T ref # T ref ln T ) q T ref t m + $ g(z # Z ) q ref t m (27) The first term on the right-han sie is similar to the contribution of an external in source in (26), an is equal to the net latent heat source multiplie by the potential efficiency. The secon term accounts for the variation of chemical potential for water vapor. For surface evaporation, the reference relative humiity will always be higher than the current humiity, an the secon term will always be negative. Note that if the parcel is saturate in the reference state,lnh ref = 0, so that the loss of APE is proportional to the irreversible entropy prouction ue to evaporation: R v T ref (lnh " lnh ref ) = "T ref #S irr (28) with "S irr the irreversible entropy prouction associate with the evaporation of water in unsaturate air.

18 The thir term on the right-han sie of (27) is associate with the increase heat capacity of the air parcel. A Taylor expansion yiels C l (T " T ref " T ref ln T ) # 1 T ref 2 C (T " T ref ) 2 l. (29) T ref This correspons to the work of a perfect heat engine transporting a quantity of heat C l (T " T ref ) over a temperature ifference 1 2(T " T ref ). The fourth term correspons to the work require to lift the aitional water from the surface to its reference level. Consier for example evaporation in an air parcel at the surface, with T = 300K, H = 0.8, Z = 0, an whose reference state is in the upper troposphere, with T ref = 200K, H ref = 1, Z ref = 15,000m. The contribution of the latent heating is T " T ref T L v # 750kJ /kg, the impact of the chemical potential is R v T ref ln H H ref " #20kJ /kg, the contribution of the specific heat of liqui water amounts to C l (T " T ref " T ref ln T T ref ) # 80kJ /kg, an the water loaing is g(z " Z ref ) #145kJ /kg. A imensional analysis inicates that the latent heat term ominates (27), though the combine contribution of the other terms cannot be entirely neglecte. Furthermore, as for external heating, the net effect of the evaporation is highly sensitive to a parcel reference state. In particular, evaporation into a stable parcel - whose reference state is the same as its current state - woul not change the APE Diffusion of heat Diffusion of heat can be viewe as a combination of a heat source an a heat sink in ifferent parcels. The iffusion of a quantity of heat Q from a parcel A to a parcel B can be obtaine through (26):

19 " "t APE = Q( T ref,a # T ref,b ) $ T # T ref,a ref,b Q (30) iff,sen T A T B T where it has been assume that the parcel temperatures are similar T A " T B " T. In a ry atmosphere, the reference temperature of an air parcel is etermine by its potential temperature. If iffusion takes place between parcel s at the same pressure, the term T ref,a " T ref,b T is proportional to the buoyancy ifference between the air parcels. This implies that, outsie very specific circumstances, iffusion of heat will be limite to exchange between air parcels with similar reference temperatures. In a moist atmosphere, the reference temperature of parcels at same temperature (or virtual temperature) an pressure can vary significantly because of water vapor. In particular, it is possible for a moist air parcel, with a reference level in the upper troposphere, can have the same buoyancy as a comparatively ry parcel with a reference level near the Earth s surface. This greatly enhances the impact of iffusion on APE in a moist atmosphere Diffusion of water vapor The APE changes ue to iffusion of water vapor can be obtaine by applying equation (27) to a balance water vapor source an sink. The main contribution is associate with the latent heat transport t APE iff,v $ " T ref,a # T ' ref,b & ) L v qm " T # T ref,a ref,b % T A ( T T B L v qm. (31) This main contribution is equal to the APE changes associate with a sensible heat flux of the same amplitue as the latent heat transport. As the iffusive flux of latent heat is usually an orer of magnitue larger than sensible heat, the iffusion of water vapor has

20 the potential to be the largest sink of APE in the atmosphere. For example, the iffusion of 1 g of water vapor from an unstable air parcel with T ref,a " 200K to a stable parcel with T ref,b " 200K woul result in a estruction of APE of approximately 500 Joules (accounting for all the terms, not only the latent heat contribution) Precipitation Falling precipitation can be treate by transferring conense water from one air parcel to another one, until it reaches the groun. Expression (24) inicates that the removal of conense water will affect the APE both through its effect on the heat capacity (first term on the right-han sie of 24) an geopotential (the secon term). For precipitation falling out of a convective upraft (assuming thus T > T ref,z < Z ref ), the reuction in heat capacity reuces the APE, while the reuction in water loaing increases the APE. The positive contribution of the water loaing term can be seen as the result of the enhance buoyancy ue to a reuction of water loaing. Its effect is equal to the ifference in geopotential energy between the remove water s current an reference states. A comparison between the terms in (24) inicates that the water loaing is the ominant effect, so that the removal of precipitating water correspons to a net source of APE for the atmosphere. This is a remarkable aspect of moist atmospheres: an aiabatic process can increase the APE without affecting the kinetic energy of the atmosphere. The role of the water loaing on APE can be better unerstoo by looking at the impact of aing water to a parcel at the Earth s surface, lifting the parcel to higher level, an removing the water. If the ascent is reversible, its reference level is unchange, an the net effect of water loaing is a reuction of APE, with APE = g(z in " Z out )qm.

21 This is the amount of work require to lift the water to the level at which it is remove. If the precipitation falls through air at rest, this is also the amount of kinetic energy issipate in the microscopic shear zones surrouning the falling precipitation, as iscusse in Pauluis et al. (2000) Re-evaporation by The change in APE associate with a re-evaporation of conense water is given APE = R v T ref (ln H H ref )q (32) For an air parcel with a reference level lower than the current level, so H > H ref, reevaporation woul increase the APE. Re-evaporation of precipitation in a ownraft increases the APE. In contrast, if the reference level is higher than the current level, for example in the case of re-evaporation in the subclou layer, then re-evaporation ecreases the APE. It is useful to compare the re-evaporation an iffusion of water vapor. Consier a rising upraft in an unsaturate environment. Re-evaporation can result as some conense water rifts from the upraft into the environment. A similar effect can be achieve if water vapor iffuses irectly from the upraft to the environment. From a thermoynamic point of view, both the re-evaporation an iffusion are irreversible, an are associate with exactly the same entropy prouction. Yet, from the point of view of APE, iffusion of water vapor woul be associate with a large estruction of APE (assuming that the reference temperature of the upraft is lower than the environment in Eq. 32), while the re-evaporation coul potentially act as a source. The ifference arises

22 from the fact that the iffusion reuces the upwar heat transport by the upraft, while the re-evaporation can increase the net upwar heat transport by generating a col ownraft. This example inicates that the prouction or estruction of APE is not irectly relate to the issipative nature of the process Iealize Raiative-Convective equilibrium To provie a quantitative example, the sources an sinks of APE for iealize raiative-convective equilibrium are consiere. This example consiers a quasi-steay atmosphere, heate at the surface with a surface latent heat flux of 90 Wm -2, an a sensible hear flux of 10Wm -2. This is balance by a net tropospheric cooling of 100Wm -2. The surface temperature is 300K, an the tropopause temperature is 200K. These numbers are chosen to be illustrative of tropical conitions. The reference state can be constructe by lifting the unstable parcels from the bounary layer to the tropopause, while the rest of the atmosphere, - the stable parcels is only compresse uring the ascent of the unstable parcels. The portion of the atmosphere that rises up to the upper troposphere cannot be etermine a priori, but it is reasonable to assume here that it correspons to the mass of the subclou layer, approximately 50mb. For the surface flux, one can assume that the surface energy fluxes are transferre across a thin surface layer to unstable air parcels. In this case, the effective efficiency for the sensible heat flux is 1/3, an the source of APE is t APE sen =3.3Wm -2. Using the same value as the example in Section 3.2 for the latent heat flux, the source of APE associate with the latent heat flux to be t APE =23.8Wm -2. The correspons to a E

23 lower efficiency (APE prouction per unit of energy) for the latent heat flux than for the sensible heat flux. This is primarily ue to the water loaing in (27). The effect of the raiative cooling epens on its vertical structure an on the portion of the atmosphere that is unstable. If the mass of unstable parcels correspons to an atmospheric layer of 50mb, the reference temperature of a stable parcel is higher by approximately 5K than its current temperature. For a uniformly istribute raiative cooling, the energy loss in the stable parcels is 95 Wm -2, which, translates into a source of APE of approximately 2Wm -2. The rest of the raiative cooling, 5Wm -2 acts on unstable parcel, with T~300K an T ref ~200K, an removes about 1.7Wm -2 of APE. The net contribution of the raiative cooling is small, less than t APE ra <0.5Wm -2. There is a near cancellation between the cooling of the stable an unstable parcels, which results from the choice of a uniform cooling rate an the fact that the mean temperature of the reference state is usually close to the mean atmospheric temperature. The contribution of internal processes requires further assumption, an shoul thus be accepte as tentative until further stuy. While precipitation is a issipative mechanism, it is shown in section 3.5 that it acts as a source of APE, when precipitation forms in parcels that are lower than their reference level. In our example, assuming that the precipitation falls on average from 5000m yiels a source of APE given by t APE ~Pg(Z-Z ref )~3.6Wm-2, with P the precipitation rate. Re-evaporation can act as prec well as a source or as a sink of APE epening on where it occurs. A imensional scaling of (32) yiels a value of t APE ~PR v T ~3Wm -2. The sign of the contribution is re"ev

24 unknown, an this value most likely overestimates the impact of re-evaporation. The impact of iffusion can be obtaine if one consiers that a fraction ε of the surface fluxes is iffuse from unstable parcels to stable parcels. In this case, the APE estroye by iffusion of heat an water vapor woul be given by t APE iff " #$ t APE sfc, with t APE = sfc t APE + sen t APE the contribution of the surface fluxes. E The APE prouction ue to the external heat sources an sinks is close to t APE sfc " 28Wm #2. This is extremely large both in comparison to the observe atmospheric issipation of 2 4 Wm -2 (see for example Peixoto an Oort, 1993) or results from numerical simulations of raiative-convective equilibrium such as Pauluis an Hel (2002a) where the kinetic energy prouction an issipation in convective motions is about 1Wm -2. This ifference cannot be explaine by the contribution of re-evaporation or precipitation (the latter is a source of APE in any case). The only significant sink of APE is iffusion of heat an water vapor. The net APE source ue to the surface heat flux an iffusion is approximately t APE + sfc t APE iff " Q surf T surf # T trop T surf (1#$), (33) where Q surf is the total surface energy flux, an T surf an T trop are the surface an tropopause temperatures respectively. (The contribution of evaporation has been approximate as that of an external heat source.) The ifference between the large APE source ue to surface flux, an the weak observe issipation can only be reconcile if,

25 after iffusion is taken into account, only a small fraction of the surface fluxes is use to increase the energy of stable parcels, which implies " ~ Discussion In this paper, an analytic formula for the sources an sinks of APE has been erive that is equally vali for ry an moist atmospheres. A key element of this erivation lies in proving that the contribution ue to the reorganization of air parcels in the reference state vanishes. This is a irect consequence of choosing a reference state that minimizes the total static energy of the atmosphere, an is therefore vali only in the context of Lorenz APE framework. It is worth stressing that espite the wiesprea use of APE to iscuss the maintenance of the atmospheric circulation, this is the first time that an analytic expression for the prouction an issipation of APE in a moist atmosphere has been explicitly erive. The expression for the APE prouction is such that the contribution of iniviual processes can be ientifie, inepenently of the transformations taking place in other parts of the atmosphere. This has been use here to iscuss the contribution of external heat sources, iffusion of water vapor, surface evaporation, precipitation an iffusion of water vapor. The prouction of APE by an external heat source is given by equation (26), an is equal to the net heat source multiplie by a potential efficiency. The potential efficiency is equal to the efficiency of a Carnot cycle with an energy source at the current parcel temperature an an energy sink at the parcel temperature in its reference state. For an energy source at the surface, the same energy flux will generate more APE if it occurs in a parcel with low reference temperature than in an air parcel with high reference

26 temperature. This inicates that the generation of APE is maximize for heating in unstable air parcels, i.e. parcels whose reference level is situate in the upper troposphere, while heating stable parcels, i.e. whose reference level is situate near the surface, has little impact on the APE. A comparison between ry an moist processes inicates that the ifferences are relate primarily to the effects of the water loaing, the heat capacity of liqui water, an to the chemical potential ifference between water vapor an liqui water. These contributions are in general small when an energy source or transport is involve. The ominant contribution of a latent heat flux is the same as that of a corresponing sensible heat flux. The primary ifference between ry an moist atmosphere is not in the expression for the APE sources an sinks, but lies in the fact that, in a moist atmosphere, parcels with same specific volume an pressure can have a very ifferent static energy an thus reference temperature. For this reason, one expects iffusion to play a major role in the APE buget. Estimates of the APE prouction for an iealize example have been consiere. For typical conitions, the surface energy flux woul prouce a very large source of APE if all the heating occurs in unstable air parcels, close to 30Wm -2 for a surface flux of 100Wm -2. This APE source is much larger than typical estimate of the work prouce an issipate on average by atmospheric motion. It is argue here that this implies that a large fraction of the surface energy flux must ens up after iffusion in stable parcels. If only a small fraction 1-ε of the surface flux heats up unstable parcels, the generation of APE can be approximate by t APE " Q surf T surf # T trop T surf (1#$). (34)

27 The net generation of APE ue to the surface energy flux an iffusion is strongly controlle by how much mixing takes place between stable an unstable air in the bounary layer. The same surface flux woul prouce very little APE in subsience regions, where there is little to no unstable air an " is close to 1, but can prouce tremenous amount of APE in convectively active regions such as a hurricane eyewall.

28 References: Bannon, P.R., 2004: Lagrangian Available Energetics an Parcel Instabilities. J. Atmos. Sci., 61, Bannon, P.R., 2005: Eulerian Available Energetics in Moist Atmosphere. Submitte to J. Atmos. Sci. Lorenz, E. N., 1955: Available potential energy an the maintenance of the general circulation. Tellus, 7, Lorenz, E. N., 1978: Available energy an the maintenance of a moist circulation. Tellus, 30, Pauluis, O. an I.M. Hel, 2002a: Entropy buget of an atmosphere in raiativeconvective equilibrium. Part I: Maximum work an frictional issipation. J. Atmos. Sci., 59, Pauluis, O. an I.M. Hel, 2002b: Entropy buget of an atmosphere in raiativeconvective equilibrium. Part II: Latent heat transport an moist processes. J. Atmos. Sci., 59, Pauluis, O., V. Balaji an I.M. Hel, 2000: Frictional issipation in a precipitating atmosphere. J. Atmos. Sci., 57, Peixoto, J.P. an A.H. Oort, 1993: Physics of Climate. American Institute of Physics, 564 pp. Ranall, D. A., an J. Wang, 1992: The moist available energy of a conitionally unstable atmosphere. J. Atmos. Sci., 49,

29 List of Table: Table 1: Sources an Sinks of Available Potential Energy in an iealize raiativeconvective equilibrium. See text for etails.

30 Process APE contribution Surface sensible heat flux t APE sen " 3.2Wm #2 Surface evaporation t APE E " 23.8Wm #2 Raiative cooling Precipitation t APE ra t APE prec " 0.5Wm #2 " 3.6Wm #2 Re-evaporation "3Wm "2 < t APE re-ev < 3Wm "2 Diffusion t APE iff % " #$ t APE + sen t APE ( ' * & E ) Table 1: Sources an Sinks of Available Potential Energy in an iealize raiativeconvective equilibrium. See text for etails.

31