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Sources and sinks of entropy in the atmosphere

Sources and sinks of entropy in the atmosphere 1. Introduction Johnson [ ] showed the importance of getting the entropy budget right in climate models. In particular, he pointed out potential problems arising from the numerical generation of entropy in these models. He noted that such extra entropy generation may be responsible for the cold bias that most climate models show. This numerical entropy source is hard to avoid with the normal choice of dependent variables in such models. Peixoto et al . [ ], Goody [ ], and Warner [ ] made estimates of the natural generation of entropy in the atmosphere, while Pauluis [ ], Pauluis et al . [ ], and Pauluis and Held [ ] evaluated the production of entropy in a numerical model of convection. Bister and Emanuel [ ] showed that entropy generation by dissipative heating is important in determining the maximum potential intensity of tropical cyclones. As Emanuel et al . [ ] and Goody [ ] point out, thinking of latent heat release as a source of heat external to the atmosphere, as is done by Peixoto et al . [ ], is probably not a good idea, as the generation of latent heat is a process tightly integrated with atmospheric motions. This implies that the use of moist entropy, where latent heating is an internal process, is preferable to the use of dry entropy, where latent heating constitutes an external entropy source. Consistent with this view, the term “entropy” is here taken to mean “moist entropy.” The specific entropy of air parcels (often in the guise of equivalent potential temperature) is generally considered to be conserved in the absence of external heat sources. However, entropy can be created by real irreversible generation processes as well as by spurious numerical processes in models. It also can be transported by molecular and eddy diffusive processes. Textbook examples of irreversible entropy generation are the rapid expansion and compression of an ideal gas. Such violent processes are generally only important in the atmosphere when shock waves occur, a phenomenon not associated with meteorological flows. Important irreversible processes generating entropy in the atmosphere include mechanical dissipation in turbulence, heating due to work done by falling hydrometeors, evaporation of precipitation in dry air, and mixing of dry and moist air, among others. The form of the entropy source in the atmosphere depends on what is included as part of an “atmospheric parcel.” This issue becomes rather complex when various forms of condensate and precipitation exist in the atmosphere. Pauluis [ ] and Pauluis and Held [ ] include all water substance whether in the form of vapor, advected condensate (small particles), or precipitating condensate (large particles). We propose instead to include dry air plus water vapor and advected condensate (known as “total cloud water”) as part of the parcel and consider precipitation to be external. This makes it easier to implement numerically the entropy governing equation, as the advection of falling precipitation does not enter. Of course, interactions between the air parcel and the precipitation must then be considered. One advantage of this approach is that the entropy inherent in precipitation that falls to the surface is lost to the atmosphere and should not be counted as part of the atmospheric entropy, though of course it is part of the overall entropy of the earth system. As noted above, if the problem of numerical generation of entropy by numerical models is not overcome, then careful evaluation of the natural sources of entropy is for naught. One solution to this problem is to write the numerical governing equations in a form that explicitly conserves entropy in the absence of nonconservative physical processes. This can be achieved by making the entropy and total cloud water primary dependent variables with the governing equations for them written in flux form. One disadvantage of this is that the entropy and total cloud water must be inverted for temperature and water vapor mixing ratio. The iterative calculations needed to accomplish this can be computationally intensive, but this problem can be minimized by generating look‐up tables at the beginning of the computation. Another important issue is the simultaneous conservation of entropy and energy. Obtaining correct budgets for both constitutes a difficult challenge. If this is impossible, then a decision has to be made as to which quantity it is more important to conserve. The purpose of this paper is to lay the groundwork for numerical models of the atmosphere that treat the entropy accurately. Aside from taking entropy as a primary dependent variable as discussed above, the sources of entropy in the atmosphere must be accurately evaluated. Pauluis [ ] and Pauluis and Held [ ] have taken initial steps in this direction. We use a somewhat more general technique for evaluating these sources to extend their results. Certain classical textbooks in the area of nonequilibrium thermodynamics are not clear in their presentation of irreversible entropy generation. In particular, the texts by De Groot [ ], De Groot and Mazur [ ], and Yourgrau et al . [ ] can mislead the unwary [see Landau and Lifschitz , ; Prigogine , for alternate treatments]. We attempt to clarify these derivations, as they are pertinent to this paper. Section 2 presents the atmospheric governing equations for the case in which entropy is a primary dependent variable. Care is taken to account for the fact the atmosphere is an open system with respect to water substance. In section 3, we present an accurate definition of entropy that takes into account the existence of the ice phase. This definition is a generalization of that presented by Emanuel [ ]. Section 4 presents the derivation of entropy sources due to heating, phase changes, diffusion, and the sources and sinks of water vapor. Section 5 compares our results with those of others and the conclusions are presented in section 6. 2. Governing Equations The specific entropy s is defined here as the entropy of a parcel divided by the mass of dry air in the parcel, which may be written s = s D + r V s V + r L s L + r I s I where s D is the specific entropy of dry air, s V , s L , and s I are the specific entropies of vapor, advected liquid water, and advected ice condensate, while r V , r L , and r I are the mixing ratios of these quantities. Since the system for which the entropy is defined consists of components that move with the dry air component, the governing equation takes the simple form d s d t = X E where d ()/ dt represents the usual material derivative of fluid dynamics and X E represents the reversible and irreversible sources of entropy per unit mass of dry air. Defining specific entropy relative to the mass of dry air is consistent with writing the mass continuity equation for dry air alone, thus avoiding the necessity of including water substance source terms in this equation: d ρ D d t + ρ D ∇ · v = 0 where ρ D is the density of dry air and v is its velocity, assumed to be the same as that of the water vapor and advected condensate. The momentum equation for the dry air component of the atmosphere in this formulation is d v d t + ∇ p − ∇ · ( 2 K D ) ρ D ( 1 + r T ) + g ( 1 + r T + r R 1 + r T ) k + 2 Ω × v = 0 where r T = r V + r L + r I is the total cloud water mixing ratio, r R is the mixing ratio of precipitation, p is the total pressure, K is the dynamic eddy mixing coefficient (i.e., it includes the density), k is the vertical unit vector, g is the acceleration of gravity, Ω is the rotation vector of the earth, and D i j = 1 2 ( ∂ v i ∂ x j + ∂ v j ∂ x i ) is the strain rate. (This, of course, assumes that the turbulence can be represented by a flux‐gradient relationship with an eddy viscosity, however derived. There are many schemes of varying complexity to determine K , which we do not go into here.) The precipitation is assumed to be falling at its terminal fall velocity with respect to the air. Transient situations in which this is not true are assumed to be unimportant. It is sufficient to focus on the motion of the dry air component, as the vapor and advected condensate are assumed to move with the dry air. The total cloud water mixing ratio obeys the equation d r T d t = 1 ρ D ∇ · ( K ∇ r T ) + E − P where E is the evaporation rate of precipitation per unit mass of dry air and P is the formation rate of precipitation. The corresponding equation for precipitation is d r P d t = 1 ρ D ∇ · ( ρ D w T k + K ∇ r P ) + P − E where only a single category of precipitation with terminal fall speed w T is included for simplicity. The flux forms of equations , , , and can be obtained in the standard manner by multiplying each by ρ D and invoking the mass continuity equation . 3. Entropy Definition The specific entropy for dry air in terms of temperature T and dry air partial pressure p D is s D = C P D ln ( T / T F ) − R D ln ( p D / p R ) + s R D where C PD is the specific heat of dry air at constant pressure, the gas constant for air is R D = R / m D , where m D is the molecular weight of dry air and R is the universal gas constant, and s RD is the constant reference entropy for dry air. The reference temperature has been taken to be the freezing point T F . The reference pressure for dry air p R is taken to be 1000 hPa. For vapor, we have a similar equation s V = C P V ln ( T / T F ) − R V ln ( p V / e S F ) + s R V where p V is the partial pressure of vapor, C PV is the specific heat of water vapor at constant pressure, R V = R / m V is the gas constant for water vapor where m V is the molecular weight of water, and s RV is the constant reference entropy for water vapor. The reference pressure for water vapor is e SF , the saturation vapor pressure at freezing. For condensate, assumed to be incompressible, the specific entropy is s C = C C ln ( T / T F ) + s R C where C C is its specific heat and s RC is the constant reference entropy for the condensate. This equation applies to both ice ( C → I ) and liquid water ( C → L ) with appropriate choice of constants. Following convention, the reference entropies for dry air and liquid condensate are set to zero, s R D = s R L = 0 . Equilibrium between a saturated atmosphere and liquid condensate then requires that s R V = L L ( T F ) T F where L L ( T ) is the latent heat of condensation. For ice, the reference entropy in equation is negative as the result of equilibrium between liquid water and ice at freezing, s R C = s R I = − L F T F where L F is the latent heat of freezing at the freezing point. Putting these results together and defining separate liquid and ice condensate components results in a seamless expression for the specific entropy valid both above and below freezing, s = ( C P D + r V C P V + r L C L + r I C I ) ln ( T / T F ) − R D ln ( p D / p R ) − r V R V ln ( p V / e S F ) + L L ( T F ) r V − L F r I T F where C L and C I are the respective specific heats of liquid and ice, assumed constant. (Note that the assumption of constancy for C I is not a good approximation. However, the vapor pressure is so low below freezing that this approximation is unlikely to have significant consequences in most circumstances.) Since this is an equilibrium equation, r L = 0 below the freezing point and r I = 0 above the freezing point. Furthermore, r V equals the saturation mixing ratio over liquid if r L is nonzero, and that for ice if r I is nonzero. One could rewrite the vapor pressure for water vapor as the saturation vapor pressure times the relative humidity, as is done by Emanuel [ ]. This has some conceptual advantages, but equation is simpler to calculate numerically. For later reference, we present the final equations for the specific entropies of vapor, liquid, and ice: s V = C P V ln ( T / T F ) − R V ln ( p V / e S F ) + L L ( T F ) / T F s L = C L ln ( T / T F ) s I = C I ln ( T / T F ) − L F / T F We also need the chemical potentials μ of liquid and ice condensate. Using the enthalpy equation h C = C C ( T − T F ) − L C ( T F ) + C P V T F and the general relation μ = h − s T , we find the chemical potentials of liquid and ice to be μ L = C L ( T − T F ) − C L T ln ( T / T F ) − L L ( T F ) + C P V T F and μ I = C I ( T − T F ) − C I T ln ( T / T F ) + T L F / T F − L I ( T F ) + C P V T F Recognizing that L F = L I ( T F ) − L L ( T F ) , we easily verify that μ L = μ I at T = T F , as required by liquid‐ice equilibrium at freezing. 4. Entropy Sources and Sinks We now consider the form of the specific entropy source term X E . By “source,” we mean all nonadvective contributions to the Eulerian entropy tendency, including (1) nonadvective transport, which moves entropy around but is not associated with a net integrated source; (2) entropy entering and exiting the atmosphere due to radiation, surface fluxes, and interaction with precipitation, which is considered to be outside the atmospheric system; and (3) the irreversible generation of entropy. The bases for the understanding of entropy sources and sinks within a parcel are the Gibbs equation and the open system version of the first law of thermodynamics [ Landau and Lifschitz , ; Prigogine , ]. The Gibbs equation comes from the assumption that the entropy S of a parcel is a function of its internal energy E , its volume V , and the masses of the individual components M i , resulting in d S = ( ∂ S ∂ E ) d E + ( ∂ S ∂ V ) d V + ∑ i ( ∂ S ∂ M i ) d M i = d E T + p d V T − ∑ i μ i d M i T where the thermodynamic definitions of temperature T , pressure p , and the chemical potential of each component μ i are used. For internal transformations such as phase changes, the total mass of the atmospheric parcel does not change. However, for transfers of mass into and out of the system, it does. We therefore separate mass changes into two parts, d M i = d N i + d O i with the dN i referring to internal transformations and the dO i indicating masses entering and exiting the system. The sum of the dN i is zero, but this condition does not hold for the dO i . The open system version of the first law is d E = d Q − d W + ∑ i e i d O i where the heat added to the test parcel is dQ , the work done by it is dW , and the specific internal energies of the various components are e i . The existence of the last term in equation simply indicates that mass entering or exiting the system carries energy with it. Substituting equations and into equation results in T d S = d Q − d W + p d V − ∑ i μ i d N i + ∑ i ( e i − μ i ) d O i We assume that the volume change for an open system d V = d V X + d V M is either a result of expansion and compression of the atmospheric parcel at constant mass ( dV X ) or due to the addition or removal of mass from the parcel at constant pressure and temperature ( dV M ). The work done by the parcel is therefore d W = p d V X , since adding mass to the parcel without changing its temperature and pressure changes the parcel's volume, but does no work. This is because no actual expansion occurs; in essence we are simply redefining the boundary of the (open) parcel to include the added mass. For this process, p d V M = ∑ i d ( p i V M ) = ∑ i d ( M i R i T ) = ∑ i R i T d O i = ∑ i ( h i − e i ) d O i where the ideal gas law is used, with p i being the partial pressure of the i th component, h i = e i + R i T is its specific enthalpy, and where it is recognized that d M i = d O i in this case. (This also works for the condensed components, which can be considered to be ideal gases with very massive molecules, resulting in vanishingly small partial pressure and gas constant.) Further insight comes from noting that d W = p d V X = p d V − p d V M = p d V − ∑ i ( h i − e i ) d O i . Recalling that the chemical potential can be written in terms of the specific enthalpy h i , the specific entropy s i , and the temperature as μ i = h i − s i T , substitution of equation into equation yields d S = 1 T ( d Q − ∑ i μ i d N i ) + ∑ i s i d O i . Therefore, the entropy of the parcel can change by adding heat, by undergoing nonequilibrium phase transformations or chemical reactions, and by adding or subtracting mass. For multiple phases in equilibrium, the chemical potentials μ i are all equal, meaning that the second term on the right side of equation vanishes in equilibrium, since ∑ d N i = 0 . Dividing equation by M D dt , the mass of dry air in the parcel and the time differential results in an expression for the entropy source term X E : X E = q T − 1 T ∑ i μ i ( d r i d t ) N + ∑ i s i ( d r i d t ) O where q is the heating rate per unit mass of dry air, r i is the mixing ratio of the i th component, the subscript N indicates an internal phase change or chemical transformation, and the subscript O indicates the addition or removal of a component from the parcel. We split X E into three parts corresponding to the three terms in equation , X E = X Q + X N + X O , and discuss each below. 4.1. Internal Phase Transformations Since the atmospheric parcel considered here does not include precipitation, internal phase transitions are those between vapor, advected liquid, and advected ice. Advected vapor and condensate tend to be very close to equilibrium. However, small liquid cloud droplets can be lifted significant distances above the freezing level before freezing occurs. We neglect this effect for advected condensate, as we assume that all phase transitions within the atmospheric system (i.e., excluding precipitation) occur in equilibrium, and therefore generate no entropy. It seems likely in any case that the entropy generation due to this effect is minor compared to other sources of entropy. Thus, we set X N = 0. Since precipitation is outside of the atmospheric system, nonequilibrium effects in the evaporation and melting of precipitation are treated differently. These effects are described below. 4.2. Entropy Sources Due to Heating The entropy source due to heating is X Q = ∑ i X Q i = ∑ i q i T i where the q i are the heating rates per unit mass of dry air from various mechanisms and the T i are the temperatures at which the respective heat sources act. The heating has five main components, heating due to radiation q R , large‐scale heat conduction q C , heat conduction on the microscale near precipitation particles q M , viscous dissipation of kinetic energy q V , and transfer of sensible heat to and from precipitation particles q P . The corresponding entropy source terms are X Q = X Q R + X Q C + X Q M + X Q V + X Q P . Evaluation of q R and X QR require a radiative transfer model and are not considered further here. The term X QM due to microscale heat conduction is discussed later in conjunction with the microscale vapor diffusion term X OM . 4.2.1. Large‐Scale Heat Conduction For heat conduction, ρ D q C equals minus the divergence of the molecular heat flux ρ D q C = ∇ · ( κ ∇ T ) where κ is the thermal conductivity of air. Thus, the entropy source due to heat conduction is X Q C = ∇ · ( κ ∇ T ) ρ D T , where we have assumed that heat from this source is added to the atmosphere at the ambient atmospheric temperature. Equation can be written ρ D X Q C = ∇ · ( κ ln T ) + κ | ∇ T | 2 T 2 , where the entropy source per unit volume ρ D X Q C consists of minus the divergence of a molecular entropy flux − κ ∇ T and a positive definite quantity proportional to the temperature gradient squared. The first term represents loss less molecular transport of entropy, while the second represents the irreversible generation of entropy in molecular conduction. Heat conduction in laminar flows in the atmosphere provides a negligible source of entropy. However, turbulence sends variance in temperature T down‐scale where the corresponding temperature variance is dissipated by heat conduction. We thus replace the molecular heat flux − κ ∇ T by the turbulent heat flux − C P D K ∇ T , where K is the dynamic eddy mixing coefficient. Thus, in the presence of turbulence, X Q C = C P D ∇ · ( K ∇ T ) ρ D T . This has the same properties as the molecular case, resulting in an irreversible entropy source plus the divergence of the eddy flux of entropy. It would be tempting to replace ∇ T by ∇ θ for the eddy flux case, where θ is the potential temperature, since the potential temperature is conserved in eddy motions on scales much greater than the inner scale. However, eddy viscosity differs from molecular viscosity in that some of the kinetic energy is transformed into potential energy and vice versa. This energy does not ultimately contribute to the irreversible generation of entropy. Furthermore, replacement of ∇ T by ∇ θ results in a form for equation that cannot be reduced to a positive definite source plus the divergence of a flux, indicating that this replacement is inappropriate. That said, equation constitutes no more than a consistent educated guess for the form of the entropy source in the case of turbulence representable by an eddy viscosity. A deeper analysis is needed to determine whether this guess is correct. 4.2.2. Viscous Dissipation Heating due to viscous dissipation in the atmosphere has two sources, the end point of the turbulent energy cascade and dissipation of work done on the atmosphere by falling precipitation. The source of kinetic energy per unit volume due to the turbulent energy cascade in an essentially incompressible fluid is v · ∇ · ( K D ) = ∇ · ( K v · D ) − 2 K | D | 2 where v is the velocity, D is the strain rate given by equation , and K D is the Reynolds stress. The first term on the right side of this equation simply moves kinetic energy around in the fluid without dissipating it. However, minus the second term is the heat source due to viscous dissipation, which should appear as a heat source in the entropy source equation. The entropy source due to turbulent dissipation and falling precipitation together takes the form X Q V = 2 K | D | 2 + g ρ D r P w T ρ D T [ Landau and Lifschitz , , p. 54; Pauluis et al ., ] where g is the acceleration of gravity, r P is the mixing ratio of precipitation, and w T is the terminal fall speed of hydrometeors. Note that both terms in this equation represent irreversible entropy generation and are positive definite. 4.2.3. Heat Transfer to Precipitation We now consider the transfer of heat to the atmosphere that is required to keep the precipitation at the wet bulb temperature T W as it warms while falling into warmer air, possibly melting at the freezing level. For simplicity, we assume that melting occurs instantaneously as precipitation particles cross the freezing level. The heat flow toward hydrometeors associated with evaporation is treated separately. The precipitation‐related heating term takes the form q P = r P ( v z − w T ) [ − C C ∂ T W ∂ z + L F δ ( z − z F ) ] where v z is the vertical component of air velocity, the condensate specific heat C C is that appropriate to ice or liquid depending on the temperature, and z F is the elevation where T W = T F . The corresponding atmospheric entropy source term is X Q P = q p T W . In most cases q P , and hence X QP , are negative, since heat is flowing from the atmosphere into the precipitation as it falls into warmer air. However, if a hydrometeor is being carried upward in an updraft, then q p could be positive as the hydrometeor cools. Since precipitation is external to the atmospheric system, this term results in the transfer of entropy in and out (mostly out) of the atmosphere as precipitation warms in its fall to the surface. 4.3. Mass Sources and Sinks Mass sources and sinks of vapor and condensate come from three processes, large‐scale diffusion, the formation and evaporation of precipitation, and microscale diffusion near evaporating hydrometeors: X O = X O D + X O P + X O M . 4.3.1. Large‐Scale Diffusive Source of Entropy The molecular diffusion of condensate particles (i.e., Brownian motion) is much weaker than the diffusion of vapor, due to the large masses of these particles. However, eddy motions produce eddy diffusive fluxes of vapor and advected condensate that are quite similar. Thus, from equation , we have X O D = ∑ i s i ( d r i d t ) O = 1 ρ D ∑ V , L , I s i ∇ · ( K ∇ r i ) , where ( d r i / d t ) O is the eddy diffusive time tendency of the i th component and the sum is over vapor, advected liquid, and advected ice. Note that diffusion of the dry air component does not occur since the parcel is considered to move with the dry air. As with the case of heat conduction, the diffusive flow of vapor away from an evaporating hydrometeor is considered separately. Equation may be written ρ D X O D = ∇ · ∑ V , L , I ( K s i ∇ r i ) − K ∑ V , L , I ∇ s i · ∇ r i . The first term on the right in this equation is minus the divergence of the entropy flux due to diffusion of the components of water substance. The second term should, in principle, be a positive definite entropy source. For the water vapor component, the gradient of entropy is ∇ s V = C P V ∇ ln T − R V ∇ ln p D − R V ∇ ln r V where r V ∝ p V / p D is used. In the original derivation of the Gibbs equation (see equation ), the pressure and temperature are assumed to be approximately constant, whence ∇ T , ∇ p D ≈ 0 . In this approximation, − ∇ s V · ∇ r V ≈ R V ∇ ln r V · ∇ r V , which is positive definite. For liquid and ice, R i → 0, and the entropy generation by condensate mixing is approximately zero. Thus, making approximations consistently results in positive‐definite entropy generation due to mixing of water substance components. In a more accurate formulation, water vapor fluxes proportional to the gradients in T and p D would need to be taken into account. 4.3.2. Formation and Evaporation of Precipitation The evaporation of precipitation in an unsaturated environment is a nonequilibrium process. However, a microscopic view of this process reveals that the evaporation from the hydrometeor into the thin layer adjacent to its surface is essentially reversible, since this layer is very close to saturation, with a temperature nearly equal to the temperature of the hydrometeor. The true irreversible entropy source in this case arises from a combination of molecular heat conduction toward the hydrometeor and water vapor diffusion away from it. In what follows, we assume that the hydrometeor takes on the wet bulb temperature T W of the atmosphere, i.e., it acts analogously to a wet bulb thermometer. Transients in temperature due to the fact that a hydrometeor is generally encountering increasing atmospheric temperatures as it falls are ignored. This simplifying assumption is valid for small hydrometeors, such as those occurring in stratiform rain, and poor for very large particles such as hail. In principle T W could be replaced by the actual temperature of the precipitation, as obtained, for instance, from a full energy budget of the falling particles. Precipitation formation occurs formally in the present context by the aggregation of advected condensate or its accretion onto existing precipitation. In both cases, advected condensate leaves the atmospheric system and the entropy sink due to this process is s C ( T ) P , where P is the formation rate of precipitation mass per unit mass of dry air. Evaporation is the reverse process, but it occurs at the wet bulb temperature at the surface of hydrometeor, with an entropy source due to the transfer of condensate back into the atmospheric system equal to s C ( T W ) E , where E is the evaporation rate of precipitation mass per unit mass of dry air. The specific entropy of condensate is used rather than that of vapor, because we formally account for the evaporation after the condensate to be evaporated leaves the hydrometeor and enters the atmospheric system, thus cooling the atmosphere rather than the condensate. Since the evaporation then occurs in the microlayer next to the drop where the air is saturated, it takes place under nearly equilibrium conditions, thus producing no additional entropy. Putting the precipitation generation and evaporation terms together, the entropy source due to the mass of advected condensate transferred to and from precipitation is given by X O P = s C ( T W ) E − s C ( T ) P . This is another term representing the flow of entropy between the atmosphere and precipitation as it forms and evaporates. 4.3.3. Microscale Diffusion Near Hydrometeors We now consider the small‐scale diffusion of heat toward an evaporating hydrometeor and the diffusion of vapor away from it. Both processes can be significant sources of entropy. The entropy generation from the diffusive flow of vapor from the saturated microlayer next to evaporating hydrometeors to the free atmosphere is X O M = E [ s V ( T , p V ) − s V ( T W , p S ) ] = E [ C P V ln ( T T W ) − R V ln ( p V p S ( T W ) ) ] where p V is the vapor pressure in the free atmosphere and p S ( T W ) is the saturation vapor pressure at the wet bulb temperature. An additional source of entropy in this case comes from the microscale heat conduction toward the hydrometeors needed to balance the evaporative cooling adjacent to the surface of the drop. This takes the form X Q M = E L C ( T W ) ( 1 T W − 1 T ) since the heat flow is down the temperature gradient from the ambient temperature T to the surface temperature of the drop T W . The latent heat of evaporation L C becomes the liquid form above the freezing level and the ice form below. Equations and may be combined into a single, simplified equation. Invoking the minor approximation that ln ( T / T W ) = ( T − T W ) / T and using the Clausius‐Clapeyron equation p S ( T ) = e S F ( T F T ) ( C C − C P V ) / R V exp ( L C ( T F ) R V T F − L C ( T ) R V T ) where e SF is the saturation vapor pressure at freezing, the result is X O M + X Q M = E [ − R V ln ( p V p S ( T ) ) + C P V ln ( T T W ) ] . This represents irreversible entropy generation and is positive‐definite since p V ≤ p S ( T ) and T W ≤ T . 4.4. Heat and Moisture Fluxes From the Ocean Surface The analysis of the entropy created by sensible and latent heat fluxes from the ocean surface differs from that for evaporating precipitation in that the surface itself is assumed to provide the heat driving the evaporation. There is thus no flow of heat to the surface needed to evaporate the surface condensate. The evaporation produces a source of water vapor in a thin layer of the atmosphere adjacent to the surface with entropy per unit mass equal to s V [ T S , p S ( T S ) ] , where T S is the temperature of the surface, p S ( T S ) is the saturation vapor pressure at the surface temperature, and T and p V are the temperature and water vapor pressure in the boundary layer. In analogy with the diffusion of water vapor away from evaporating precipitation, the entropy generated per unit mass as the vapor diffuses into the free boundary layer is s V ( T , p V ) − s V [ T S , p S ( T S ) ] . Summing these results in a net entropy generation per unit mass due to evaporation of s V ( T , p V ) . Assuming an evaporation rate per unit area per unit time of F V and a sensible heat flux out of the ocean surface of F S , the combined entropy source due to surface heat and moisture fluxes is F E = F V s V ( T , p V ) + F S / T . Land surfaces are not necessarily saturated and they require a slightly different treatment. 5. Comparison With Other Results 5.1. Pauluis and Held Pauluis [ ] and Pauluis and Held [ ] derived equations for the irreversible generation of entropy in a moist atmosphere. Most of their results had to do with generally neglected heat sources such as turbulent dissipation, dissipative work done by falling raindrops, and the divergence of the heat flux due to molecular conduction. However, they also showed that the diffusion of water vapor in the atmosphere is a significant source of irreversible entropy generation. Their expression in our notation for the irreversible generation of entropy by the diffusive transfer of a mass of water vapor dO from parcel 1 to parcel 2 is d S i r r = R V d O ln ( p V 1 / p V 2 ) where R V is the gas constant for water vapor and p V 1 and p V 2 are the vapor pressures of water vapor in parcel 1 and parcel 2. The change in entropy in a parcel according to our formulation is given by equation . Neglecting heating and phase changes, this equation reduces to d S = s V d O for the diffusive transfer of a mass dO of water vapor into a parcel. For two parcels in which transfer is from parcel 1 to parcel 2, the net irreversible entropy source is d S i r r = d S 2 + d S 1 = s V 2 d O 2 + s V 1 d O 1 = ( s V 2 − s V 1 ) d O = [ C P V ln ( T 2 / T 1 ) + R V ln ( p V 1 / p V 2 ) ] d O where d O ≡ d O 2 = − d O 1 , T 1 and T 2 are the temperatures in the two parcels, and C PV is the specific heat of water vapor at constant pressure. Note that our result is equivalent to that of Pauluis and Held if the temperatures of the two parcels are the same. However, since our results do not take into account all of the effects of temperature gradients in this case, our results are not necessarily more accurate than those of Pauluis and Held. There is also a difference between the Pauluis‐Held results and our equation for the generation of entropy by evaporating precipitation. In addition to − R V ln [ p V / p S ( T ) ] in the Pauluis and Held papers, we have the term C P V ln ( T / T W ) . However, this is much smaller than the first term since typically ln ( T / T W ) ≪ 1 whereas − ln [ p V / p S ( T ) ] can become quite large for low relative humidities. The difference probably has to do with our assumption that the droplet temperatures are equal to the wet bulb temperature rather than the actual temperature. Pauluis [ ] and Pauluis and Held [ ] have no entropy source corresponding to water substance entering and exiting the atmospheric system, as represented by equation , as they consider precipitation to be part of the system. The corresponding physical effects appear as precipitation falls to the surface. 5.2. Classical Textbooks Landau and Lifschitz [ ] and Prigogine [ ] explicitly take into account the fact that isothermal diffusion transfers the enthalpy as well as the mass of the diffused component (see p. 221). Without this enthalpy transfer, the specific entropy s i in equation would be replaced by − μ i / T . The irreversible generation of entropy by diffusion given by equation would be unaffected, since − μ V / T differs from s V by only an additive constant that would cancel in the calculation of dS irr . However, the entropy source due to precipitation evaporation given in equation (approximating T W by T ) would be given in terms of the chemical potential of the condensate μ C rather than its specific entropy, X O P = − μ C ( E − P ) T ( incorrect ) , which is seriously incorrect. (The author was first alerted to this problem by the disastrous effect of equation on a numerical model of convection!) Certain well‐known texts [e.g., De Groot , ; De Groot and Mazur , ; Yourgrau et al ., ] do not make it clear that the enthalpy transport associated with mass transport into a parcel (by diffusive or other mechanisms) must be included in the calculation of the irreversible generation of entropy. In the case of mass transfer by diffusion of an ideal gas, the effects of the error on irreversible entropy generation vanish by accident, as noted above. However, if the specific enthalpy of the substance being diffused does not vary linearly with absolute temperature, the effects of the error reappear. Furthermore, if material actually enters or exits the system under consideration (as with the conversion of total cloud water to and from precipitation in our case), the results are disastrously wrong, as shown above. 6. Conclusions Summarizing our analysis, the total atmospheric entropy source (exclusive of radiative effects) is X E = 1 ρ D [ C P D ∇ · ( K ∇ T ) T + ∑ V , L , I s i ∇ · ( K ∇ r i ) ] + 2 K | D | 2 + g ρ D r P w T ρ D T + r P ( v z − w T ) T W [ − C C ∂ T W ∂ z + L F δ ( z − z F ) ] + E s C ( T W ) − P s C ( T ) + E [ − R V ln ( p V p S ( T ) ) + C P V ln ( T T W ) ] . Line by line, we have: 1. molecular processes of heat conduction and diffusion of water substance, augmented by the down‐scale transfer of variance to molecular scales by turbulence; 2. production of heat by viscous dissipation augmented by turbulence and the work done by falling precipitation on the atmosphere; 3. heat transfer to falling precipitation required to drive its temperature toward the wet bulb temperature of the air; also included is the heat transfer required to melt frozen precipitation as it descends through the freezing level; 4. transfer of entropy to and from the atmospheric microlayer surrounding hydrometeors by the formation and evaporation of precipitation; 5. entropy source associated with the microscale diffusion of water vapor away from and the microscale heat conduction toward evaporating hydrometeors. In addition, entropy can enter the atmosphere from surface fluxes given by equation . Equation represents the total nonadvective source of entropy, including irreversible entropy generation, flow of entropy to and from precipitation and the surface, and nonadvective transports. It is conventional to separate these quantities in textbooks, but from the point of view of numerical simulations, the total source (which may be negative in places) is what is needed. The results as presented are also specific to the case in which turbulent fluxes are represented by an eddy viscosity, though no restriction is placed on how this eddy viscosity is determined. Our results extend the theoretical development of Pauluis [ ] and Pauluis and Held [ ] in a number of respects. In particular, the difference between the atmospheric temperature and that of precipitation particles is accounted for in the present work and the most important effects of the ice phase are included. It remains to incorporate these results in a model that exhibits no numerical generation of entropy. This is feasible, with certain other compromises, given the development in section 2. The model of Raymond and Zeng [ ] conserves the integrated equivalent potential temperature explicitly, so only a small modification of that model is required to achieve that goal. Acknowledgments Thanks are due to Kerry Emanuel and Olivier Pauluis for penetrating reviews. Thanks also to Sharon Sessions for useful discussions. This work was supported by National Science Foundation grant 1021049. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Advances in Modeling Earth Systems Wiley

Sources and sinks of entropy in the atmosphere

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Wiley
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©2013. American Geophysical Union. All Rights Reserved.
ISSN
1942-2466
eISSN
1942-2466
DOI
10.1002/jame.20050
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Abstract

1. Introduction Johnson [ ] showed the importance of getting the entropy budget right in climate models. In particular, he pointed out potential problems arising from the numerical generation of entropy in these models. He noted that such extra entropy generation may be responsible for the cold bias that most climate models show. This numerical entropy source is hard to avoid with the normal choice of dependent variables in such models. Peixoto et al . [ ], Goody [ ], and Warner [ ] made estimates of the natural generation of entropy in the atmosphere, while Pauluis [ ], Pauluis et al . [ ], and Pauluis and Held [ ] evaluated the production of entropy in a numerical model of convection. Bister and Emanuel [ ] showed that entropy generation by dissipative heating is important in determining the maximum potential intensity of tropical cyclones. As Emanuel et al . [ ] and Goody [ ] point out, thinking of latent heat release as a source of heat external to the atmosphere, as is done by Peixoto et al . [ ], is probably not a good idea, as the generation of latent heat is a process tightly integrated with atmospheric motions. This implies that the use of moist entropy, where latent heating is an internal process, is preferable to the use of dry entropy, where latent heating constitutes an external entropy source. Consistent with this view, the term “entropy” is here taken to mean “moist entropy.” The specific entropy of air parcels (often in the guise of equivalent potential temperature) is generally considered to be conserved in the absence of external heat sources. However, entropy can be created by real irreversible generation processes as well as by spurious numerical processes in models. It also can be transported by molecular and eddy diffusive processes. Textbook examples of irreversible entropy generation are the rapid expansion and compression of an ideal gas. Such violent processes are generally only important in the atmosphere when shock waves occur, a phenomenon not associated with meteorological flows. Important irreversible processes generating entropy in the atmosphere include mechanical dissipation in turbulence, heating due to work done by falling hydrometeors, evaporation of precipitation in dry air, and mixing of dry and moist air, among others. The form of the entropy source in the atmosphere depends on what is included as part of an “atmospheric parcel.” This issue becomes rather complex when various forms of condensate and precipitation exist in the atmosphere. Pauluis [ ] and Pauluis and Held [ ] include all water substance whether in the form of vapor, advected condensate (small particles), or precipitating condensate (large particles). We propose instead to include dry air plus water vapor and advected condensate (known as “total cloud water”) as part of the parcel and consider precipitation to be external. This makes it easier to implement numerically the entropy governing equation, as the advection of falling precipitation does not enter. Of course, interactions between the air parcel and the precipitation must then be considered. One advantage of this approach is that the entropy inherent in precipitation that falls to the surface is lost to the atmosphere and should not be counted as part of the atmospheric entropy, though of course it is part of the overall entropy of the earth system. As noted above, if the problem of numerical generation of entropy by numerical models is not overcome, then careful evaluation of the natural sources of entropy is for naught. One solution to this problem is to write the numerical governing equations in a form that explicitly conserves entropy in the absence of nonconservative physical processes. This can be achieved by making the entropy and total cloud water primary dependent variables with the governing equations for them written in flux form. One disadvantage of this is that the entropy and total cloud water must be inverted for temperature and water vapor mixing ratio. The iterative calculations needed to accomplish this can be computationally intensive, but this problem can be minimized by generating look‐up tables at the beginning of the computation. Another important issue is the simultaneous conservation of entropy and energy. Obtaining correct budgets for both constitutes a difficult challenge. If this is impossible, then a decision has to be made as to which quantity it is more important to conserve. The purpose of this paper is to lay the groundwork for numerical models of the atmosphere that treat the entropy accurately. Aside from taking entropy as a primary dependent variable as discussed above, the sources of entropy in the atmosphere must be accurately evaluated. Pauluis [ ] and Pauluis and Held [ ] have taken initial steps in this direction. We use a somewhat more general technique for evaluating these sources to extend their results. Certain classical textbooks in the area of nonequilibrium thermodynamics are not clear in their presentation of irreversible entropy generation. In particular, the texts by De Groot [ ], De Groot and Mazur [ ], and Yourgrau et al . [ ] can mislead the unwary [see Landau and Lifschitz , ; Prigogine , for alternate treatments]. We attempt to clarify these derivations, as they are pertinent to this paper. Section 2 presents the atmospheric governing equations for the case in which entropy is a primary dependent variable. Care is taken to account for the fact the atmosphere is an open system with respect to water substance. In section 3, we present an accurate definition of entropy that takes into account the existence of the ice phase. This definition is a generalization of that presented by Emanuel [ ]. Section 4 presents the derivation of entropy sources due to heating, phase changes, diffusion, and the sources and sinks of water vapor. Section 5 compares our results with those of others and the conclusions are presented in section 6. 2. Governing Equations The specific entropy s is defined here as the entropy of a parcel divided by the mass of dry air in the parcel, which may be written s = s D + r V s V + r L s L + r I s I where s D is the specific entropy of dry air, s V , s L , and s I are the specific entropies of vapor, advected liquid water, and advected ice condensate, while r V , r L , and r I are the mixing ratios of these quantities. Since the system for which the entropy is defined consists of components that move with the dry air component, the governing equation takes the simple form d s d t = X E where d ()/ dt represents the usual material derivative of fluid dynamics and X E represents the reversible and irreversible sources of entropy per unit mass of dry air. Defining specific entropy relative to the mass of dry air is consistent with writing the mass continuity equation for dry air alone, thus avoiding the necessity of including water substance source terms in this equation: d ρ D d t + ρ D ∇ · v = 0 where ρ D is the density of dry air and v is its velocity, assumed to be the same as that of the water vapor and advected condensate. The momentum equation for the dry air component of the atmosphere in this formulation is d v d t + ∇ p − ∇ · ( 2 K D ) ρ D ( 1 + r T ) + g ( 1 + r T + r R 1 + r T ) k + 2 Ω × v = 0 where r T = r V + r L + r I is the total cloud water mixing ratio, r R is the mixing ratio of precipitation, p is the total pressure, K is the dynamic eddy mixing coefficient (i.e., it includes the density), k is the vertical unit vector, g is the acceleration of gravity, Ω is the rotation vector of the earth, and D i j = 1 2 ( ∂ v i ∂ x j + ∂ v j ∂ x i ) is the strain rate. (This, of course, assumes that the turbulence can be represented by a flux‐gradient relationship with an eddy viscosity, however derived. There are many schemes of varying complexity to determine K , which we do not go into here.) The precipitation is assumed to be falling at its terminal fall velocity with respect to the air. Transient situations in which this is not true are assumed to be unimportant. It is sufficient to focus on the motion of the dry air component, as the vapor and advected condensate are assumed to move with the dry air. The total cloud water mixing ratio obeys the equation d r T d t = 1 ρ D ∇ · ( K ∇ r T ) + E − P where E is the evaporation rate of precipitation per unit mass of dry air and P is the formation rate of precipitation. The corresponding equation for precipitation is d r P d t = 1 ρ D ∇ · ( ρ D w T k + K ∇ r P ) + P − E where only a single category of precipitation with terminal fall speed w T is included for simplicity. The flux forms of equations , , , and can be obtained in the standard manner by multiplying each by ρ D and invoking the mass continuity equation . 3. Entropy Definition The specific entropy for dry air in terms of temperature T and dry air partial pressure p D is s D = C P D ln ( T / T F ) − R D ln ( p D / p R ) + s R D where C PD is the specific heat of dry air at constant pressure, the gas constant for air is R D = R / m D , where m D is the molecular weight of dry air and R is the universal gas constant, and s RD is the constant reference entropy for dry air. The reference temperature has been taken to be the freezing point T F . The reference pressure for dry air p R is taken to be 1000 hPa. For vapor, we have a similar equation s V = C P V ln ( T / T F ) − R V ln ( p V / e S F ) + s R V where p V is the partial pressure of vapor, C PV is the specific heat of water vapor at constant pressure, R V = R / m V is the gas constant for water vapor where m V is the molecular weight of water, and s RV is the constant reference entropy for water vapor. The reference pressure for water vapor is e SF , the saturation vapor pressure at freezing. For condensate, assumed to be incompressible, the specific entropy is s C = C C ln ( T / T F ) + s R C where C C is its specific heat and s RC is the constant reference entropy for the condensate. This equation applies to both ice ( C → I ) and liquid water ( C → L ) with appropriate choice of constants. Following convention, the reference entropies for dry air and liquid condensate are set to zero, s R D = s R L = 0 . Equilibrium between a saturated atmosphere and liquid condensate then requires that s R V = L L ( T F ) T F where L L ( T ) is the latent heat of condensation. For ice, the reference entropy in equation is negative as the result of equilibrium between liquid water and ice at freezing, s R C = s R I = − L F T F where L F is the latent heat of freezing at the freezing point. Putting these results together and defining separate liquid and ice condensate components results in a seamless expression for the specific entropy valid both above and below freezing, s = ( C P D + r V C P V + r L C L + r I C I ) ln ( T / T F ) − R D ln ( p D / p R ) − r V R V ln ( p V / e S F ) + L L ( T F ) r V − L F r I T F where C L and C I are the respective specific heats of liquid and ice, assumed constant. (Note that the assumption of constancy for C I is not a good approximation. However, the vapor pressure is so low below freezing that this approximation is unlikely to have significant consequences in most circumstances.) Since this is an equilibrium equation, r L = 0 below the freezing point and r I = 0 above the freezing point. Furthermore, r V equals the saturation mixing ratio over liquid if r L is nonzero, and that for ice if r I is nonzero. One could rewrite the vapor pressure for water vapor as the saturation vapor pressure times the relative humidity, as is done by Emanuel [ ]. This has some conceptual advantages, but equation is simpler to calculate numerically. For later reference, we present the final equations for the specific entropies of vapor, liquid, and ice: s V = C P V ln ( T / T F ) − R V ln ( p V / e S F ) + L L ( T F ) / T F s L = C L ln ( T / T F ) s I = C I ln ( T / T F ) − L F / T F We also need the chemical potentials μ of liquid and ice condensate. Using the enthalpy equation h C = C C ( T − T F ) − L C ( T F ) + C P V T F and the general relation μ = h − s T , we find the chemical potentials of liquid and ice to be μ L = C L ( T − T F ) − C L T ln ( T / T F ) − L L ( T F ) + C P V T F and μ I = C I ( T − T F ) − C I T ln ( T / T F ) + T L F / T F − L I ( T F ) + C P V T F Recognizing that L F = L I ( T F ) − L L ( T F ) , we easily verify that μ L = μ I at T = T F , as required by liquid‐ice equilibrium at freezing. 4. Entropy Sources and Sinks We now consider the form of the specific entropy source term X E . By “source,” we mean all nonadvective contributions to the Eulerian entropy tendency, including (1) nonadvective transport, which moves entropy around but is not associated with a net integrated source; (2) entropy entering and exiting the atmosphere due to radiation, surface fluxes, and interaction with precipitation, which is considered to be outside the atmospheric system; and (3) the irreversible generation of entropy. The bases for the understanding of entropy sources and sinks within a parcel are the Gibbs equation and the open system version of the first law of thermodynamics [ Landau and Lifschitz , ; Prigogine , ]. The Gibbs equation comes from the assumption that the entropy S of a parcel is a function of its internal energy E , its volume V , and the masses of the individual components M i , resulting in d S = ( ∂ S ∂ E ) d E + ( ∂ S ∂ V ) d V + ∑ i ( ∂ S ∂ M i ) d M i = d E T + p d V T − ∑ i μ i d M i T where the thermodynamic definitions of temperature T , pressure p , and the chemical potential of each component μ i are used. For internal transformations such as phase changes, the total mass of the atmospheric parcel does not change. However, for transfers of mass into and out of the system, it does. We therefore separate mass changes into two parts, d M i = d N i + d O i with the dN i referring to internal transformations and the dO i indicating masses entering and exiting the system. The sum of the dN i is zero, but this condition does not hold for the dO i . The open system version of the first law is d E = d Q − d W + ∑ i e i d O i where the heat added to the test parcel is dQ , the work done by it is dW , and the specific internal energies of the various components are e i . The existence of the last term in equation simply indicates that mass entering or exiting the system carries energy with it. Substituting equations and into equation results in T d S = d Q − d W + p d V − ∑ i μ i d N i + ∑ i ( e i − μ i ) d O i We assume that the volume change for an open system d V = d V X + d V M is either a result of expansion and compression of the atmospheric parcel at constant mass ( dV X ) or due to the addition or removal of mass from the parcel at constant pressure and temperature ( dV M ). The work done by the parcel is therefore d W = p d V X , since adding mass to the parcel without changing its temperature and pressure changes the parcel's volume, but does no work. This is because no actual expansion occurs; in essence we are simply redefining the boundary of the (open) parcel to include the added mass. For this process, p d V M = ∑ i d ( p i V M ) = ∑ i d ( M i R i T ) = ∑ i R i T d O i = ∑ i ( h i − e i ) d O i where the ideal gas law is used, with p i being the partial pressure of the i th component, h i = e i + R i T is its specific enthalpy, and where it is recognized that d M i = d O i in this case. (This also works for the condensed components, which can be considered to be ideal gases with very massive molecules, resulting in vanishingly small partial pressure and gas constant.) Further insight comes from noting that d W = p d V X = p d V − p d V M = p d V − ∑ i ( h i − e i ) d O i . Recalling that the chemical potential can be written in terms of the specific enthalpy h i , the specific entropy s i , and the temperature as μ i = h i − s i T , substitution of equation into equation yields d S = 1 T ( d Q − ∑ i μ i d N i ) + ∑ i s i d O i . Therefore, the entropy of the parcel can change by adding heat, by undergoing nonequilibrium phase transformations or chemical reactions, and by adding or subtracting mass. For multiple phases in equilibrium, the chemical potentials μ i are all equal, meaning that the second term on the right side of equation vanishes in equilibrium, since ∑ d N i = 0 . Dividing equation by M D dt , the mass of dry air in the parcel and the time differential results in an expression for the entropy source term X E : X E = q T − 1 T ∑ i μ i ( d r i d t ) N + ∑ i s i ( d r i d t ) O where q is the heating rate per unit mass of dry air, r i is the mixing ratio of the i th component, the subscript N indicates an internal phase change or chemical transformation, and the subscript O indicates the addition or removal of a component from the parcel. We split X E into three parts corresponding to the three terms in equation , X E = X Q + X N + X O , and discuss each below. 4.1. Internal Phase Transformations Since the atmospheric parcel considered here does not include precipitation, internal phase transitions are those between vapor, advected liquid, and advected ice. Advected vapor and condensate tend to be very close to equilibrium. However, small liquid cloud droplets can be lifted significant distances above the freezing level before freezing occurs. We neglect this effect for advected condensate, as we assume that all phase transitions within the atmospheric system (i.e., excluding precipitation) occur in equilibrium, and therefore generate no entropy. It seems likely in any case that the entropy generation due to this effect is minor compared to other sources of entropy. Thus, we set X N = 0. Since precipitation is outside of the atmospheric system, nonequilibrium effects in the evaporation and melting of precipitation are treated differently. These effects are described below. 4.2. Entropy Sources Due to Heating The entropy source due to heating is X Q = ∑ i X Q i = ∑ i q i T i where the q i are the heating rates per unit mass of dry air from various mechanisms and the T i are the temperatures at which the respective heat sources act. The heating has five main components, heating due to radiation q R , large‐scale heat conduction q C , heat conduction on the microscale near precipitation particles q M , viscous dissipation of kinetic energy q V , and transfer of sensible heat to and from precipitation particles q P . The corresponding entropy source terms are X Q = X Q R + X Q C + X Q M + X Q V + X Q P . Evaluation of q R and X QR require a radiative transfer model and are not considered further here. The term X QM due to microscale heat conduction is discussed later in conjunction with the microscale vapor diffusion term X OM . 4.2.1. Large‐Scale Heat Conduction For heat conduction, ρ D q C equals minus the divergence of the molecular heat flux ρ D q C = ∇ · ( κ ∇ T ) where κ is the thermal conductivity of air. Thus, the entropy source due to heat conduction is X Q C = ∇ · ( κ ∇ T ) ρ D T , where we have assumed that heat from this source is added to the atmosphere at the ambient atmospheric temperature. Equation can be written ρ D X Q C = ∇ · ( κ ln T ) + κ | ∇ T | 2 T 2 , where the entropy source per unit volume ρ D X Q C consists of minus the divergence of a molecular entropy flux − κ ∇ T and a positive definite quantity proportional to the temperature gradient squared. The first term represents loss less molecular transport of entropy, while the second represents the irreversible generation of entropy in molecular conduction. Heat conduction in laminar flows in the atmosphere provides a negligible source of entropy. However, turbulence sends variance in temperature T down‐scale where the corresponding temperature variance is dissipated by heat conduction. We thus replace the molecular heat flux − κ ∇ T by the turbulent heat flux − C P D K ∇ T , where K is the dynamic eddy mixing coefficient. Thus, in the presence of turbulence, X Q C = C P D ∇ · ( K ∇ T ) ρ D T . This has the same properties as the molecular case, resulting in an irreversible entropy source plus the divergence of the eddy flux of entropy. It would be tempting to replace ∇ T by ∇ θ for the eddy flux case, where θ is the potential temperature, since the potential temperature is conserved in eddy motions on scales much greater than the inner scale. However, eddy viscosity differs from molecular viscosity in that some of the kinetic energy is transformed into potential energy and vice versa. This energy does not ultimately contribute to the irreversible generation of entropy. Furthermore, replacement of ∇ T by ∇ θ results in a form for equation that cannot be reduced to a positive definite source plus the divergence of a flux, indicating that this replacement is inappropriate. That said, equation constitutes no more than a consistent educated guess for the form of the entropy source in the case of turbulence representable by an eddy viscosity. A deeper analysis is needed to determine whether this guess is correct. 4.2.2. Viscous Dissipation Heating due to viscous dissipation in the atmosphere has two sources, the end point of the turbulent energy cascade and dissipation of work done on the atmosphere by falling precipitation. The source of kinetic energy per unit volume due to the turbulent energy cascade in an essentially incompressible fluid is v · ∇ · ( K D ) = ∇ · ( K v · D ) − 2 K | D | 2 where v is the velocity, D is the strain rate given by equation , and K D is the Reynolds stress. The first term on the right side of this equation simply moves kinetic energy around in the fluid without dissipating it. However, minus the second term is the heat source due to viscous dissipation, which should appear as a heat source in the entropy source equation. The entropy source due to turbulent dissipation and falling precipitation together takes the form X Q V = 2 K | D | 2 + g ρ D r P w T ρ D T [ Landau and Lifschitz , , p. 54; Pauluis et al ., ] where g is the acceleration of gravity, r P is the mixing ratio of precipitation, and w T is the terminal fall speed of hydrometeors. Note that both terms in this equation represent irreversible entropy generation and are positive definite. 4.2.3. Heat Transfer to Precipitation We now consider the transfer of heat to the atmosphere that is required to keep the precipitation at the wet bulb temperature T W as it warms while falling into warmer air, possibly melting at the freezing level. For simplicity, we assume that melting occurs instantaneously as precipitation particles cross the freezing level. The heat flow toward hydrometeors associated with evaporation is treated separately. The precipitation‐related heating term takes the form q P = r P ( v z − w T ) [ − C C ∂ T W ∂ z + L F δ ( z − z F ) ] where v z is the vertical component of air velocity, the condensate specific heat C C is that appropriate to ice or liquid depending on the temperature, and z F is the elevation where T W = T F . The corresponding atmospheric entropy source term is X Q P = q p T W . In most cases q P , and hence X QP , are negative, since heat is flowing from the atmosphere into the precipitation as it falls into warmer air. However, if a hydrometeor is being carried upward in an updraft, then q p could be positive as the hydrometeor cools. Since precipitation is external to the atmospheric system, this term results in the transfer of entropy in and out (mostly out) of the atmosphere as precipitation warms in its fall to the surface. 4.3. Mass Sources and Sinks Mass sources and sinks of vapor and condensate come from three processes, large‐scale diffusion, the formation and evaporation of precipitation, and microscale diffusion near evaporating hydrometeors: X O = X O D + X O P + X O M . 4.3.1. Large‐Scale Diffusive Source of Entropy The molecular diffusion of condensate particles (i.e., Brownian motion) is much weaker than the diffusion of vapor, due to the large masses of these particles. However, eddy motions produce eddy diffusive fluxes of vapor and advected condensate that are quite similar. Thus, from equation , we have X O D = ∑ i s i ( d r i d t ) O = 1 ρ D ∑ V , L , I s i ∇ · ( K ∇ r i ) , where ( d r i / d t ) O is the eddy diffusive time tendency of the i th component and the sum is over vapor, advected liquid, and advected ice. Note that diffusion of the dry air component does not occur since the parcel is considered to move with the dry air. As with the case of heat conduction, the diffusive flow of vapor away from an evaporating hydrometeor is considered separately. Equation may be written ρ D X O D = ∇ · ∑ V , L , I ( K s i ∇ r i ) − K ∑ V , L , I ∇ s i · ∇ r i . The first term on the right in this equation is minus the divergence of the entropy flux due to diffusion of the components of water substance. The second term should, in principle, be a positive definite entropy source. For the water vapor component, the gradient of entropy is ∇ s V = C P V ∇ ln T − R V ∇ ln p D − R V ∇ ln r V where r V ∝ p V / p D is used. In the original derivation of the Gibbs equation (see equation ), the pressure and temperature are assumed to be approximately constant, whence ∇ T , ∇ p D ≈ 0 . In this approximation, − ∇ s V · ∇ r V ≈ R V ∇ ln r V · ∇ r V , which is positive definite. For liquid and ice, R i → 0, and the entropy generation by condensate mixing is approximately zero. Thus, making approximations consistently results in positive‐definite entropy generation due to mixing of water substance components. In a more accurate formulation, water vapor fluxes proportional to the gradients in T and p D would need to be taken into account. 4.3.2. Formation and Evaporation of Precipitation The evaporation of precipitation in an unsaturated environment is a nonequilibrium process. However, a microscopic view of this process reveals that the evaporation from the hydrometeor into the thin layer adjacent to its surface is essentially reversible, since this layer is very close to saturation, with a temperature nearly equal to the temperature of the hydrometeor. The true irreversible entropy source in this case arises from a combination of molecular heat conduction toward the hydrometeor and water vapor diffusion away from it. In what follows, we assume that the hydrometeor takes on the wet bulb temperature T W of the atmosphere, i.e., it acts analogously to a wet bulb thermometer. Transients in temperature due to the fact that a hydrometeor is generally encountering increasing atmospheric temperatures as it falls are ignored. This simplifying assumption is valid for small hydrometeors, such as those occurring in stratiform rain, and poor for very large particles such as hail. In principle T W could be replaced by the actual temperature of the precipitation, as obtained, for instance, from a full energy budget of the falling particles. Precipitation formation occurs formally in the present context by the aggregation of advected condensate or its accretion onto existing precipitation. In both cases, advected condensate leaves the atmospheric system and the entropy sink due to this process is s C ( T ) P , where P is the formation rate of precipitation mass per unit mass of dry air. Evaporation is the reverse process, but it occurs at the wet bulb temperature at the surface of hydrometeor, with an entropy source due to the transfer of condensate back into the atmospheric system equal to s C ( T W ) E , where E is the evaporation rate of precipitation mass per unit mass of dry air. The specific entropy of condensate is used rather than that of vapor, because we formally account for the evaporation after the condensate to be evaporated leaves the hydrometeor and enters the atmospheric system, thus cooling the atmosphere rather than the condensate. Since the evaporation then occurs in the microlayer next to the drop where the air is saturated, it takes place under nearly equilibrium conditions, thus producing no additional entropy. Putting the precipitation generation and evaporation terms together, the entropy source due to the mass of advected condensate transferred to and from precipitation is given by X O P = s C ( T W ) E − s C ( T ) P . This is another term representing the flow of entropy between the atmosphere and precipitation as it forms and evaporates. 4.3.3. Microscale Diffusion Near Hydrometeors We now consider the small‐scale diffusion of heat toward an evaporating hydrometeor and the diffusion of vapor away from it. Both processes can be significant sources of entropy. The entropy generation from the diffusive flow of vapor from the saturated microlayer next to evaporating hydrometeors to the free atmosphere is X O M = E [ s V ( T , p V ) − s V ( T W , p S ) ] = E [ C P V ln ( T T W ) − R V ln ( p V p S ( T W ) ) ] where p V is the vapor pressure in the free atmosphere and p S ( T W ) is the saturation vapor pressure at the wet bulb temperature. An additional source of entropy in this case comes from the microscale heat conduction toward the hydrometeors needed to balance the evaporative cooling adjacent to the surface of the drop. This takes the form X Q M = E L C ( T W ) ( 1 T W − 1 T ) since the heat flow is down the temperature gradient from the ambient temperature T to the surface temperature of the drop T W . The latent heat of evaporation L C becomes the liquid form above the freezing level and the ice form below. Equations and may be combined into a single, simplified equation. Invoking the minor approximation that ln ( T / T W ) = ( T − T W ) / T and using the Clausius‐Clapeyron equation p S ( T ) = e S F ( T F T ) ( C C − C P V ) / R V exp ( L C ( T F ) R V T F − L C ( T ) R V T ) where e SF is the saturation vapor pressure at freezing, the result is X O M + X Q M = E [ − R V ln ( p V p S ( T ) ) + C P V ln ( T T W ) ] . This represents irreversible entropy generation and is positive‐definite since p V ≤ p S ( T ) and T W ≤ T . 4.4. Heat and Moisture Fluxes From the Ocean Surface The analysis of the entropy created by sensible and latent heat fluxes from the ocean surface differs from that for evaporating precipitation in that the surface itself is assumed to provide the heat driving the evaporation. There is thus no flow of heat to the surface needed to evaporate the surface condensate. The evaporation produces a source of water vapor in a thin layer of the atmosphere adjacent to the surface with entropy per unit mass equal to s V [ T S , p S ( T S ) ] , where T S is the temperature of the surface, p S ( T S ) is the saturation vapor pressure at the surface temperature, and T and p V are the temperature and water vapor pressure in the boundary layer. In analogy with the diffusion of water vapor away from evaporating precipitation, the entropy generated per unit mass as the vapor diffuses into the free boundary layer is s V ( T , p V ) − s V [ T S , p S ( T S ) ] . Summing these results in a net entropy generation per unit mass due to evaporation of s V ( T , p V ) . Assuming an evaporation rate per unit area per unit time of F V and a sensible heat flux out of the ocean surface of F S , the combined entropy source due to surface heat and moisture fluxes is F E = F V s V ( T , p V ) + F S / T . Land surfaces are not necessarily saturated and they require a slightly different treatment. 5. Comparison With Other Results 5.1. Pauluis and Held Pauluis [ ] and Pauluis and Held [ ] derived equations for the irreversible generation of entropy in a moist atmosphere. Most of their results had to do with generally neglected heat sources such as turbulent dissipation, dissipative work done by falling raindrops, and the divergence of the heat flux due to molecular conduction. However, they also showed that the diffusion of water vapor in the atmosphere is a significant source of irreversible entropy generation. Their expression in our notation for the irreversible generation of entropy by the diffusive transfer of a mass of water vapor dO from parcel 1 to parcel 2 is d S i r r = R V d O ln ( p V 1 / p V 2 ) where R V is the gas constant for water vapor and p V 1 and p V 2 are the vapor pressures of water vapor in parcel 1 and parcel 2. The change in entropy in a parcel according to our formulation is given by equation . Neglecting heating and phase changes, this equation reduces to d S = s V d O for the diffusive transfer of a mass dO of water vapor into a parcel. For two parcels in which transfer is from parcel 1 to parcel 2, the net irreversible entropy source is d S i r r = d S 2 + d S 1 = s V 2 d O 2 + s V 1 d O 1 = ( s V 2 − s V 1 ) d O = [ C P V ln ( T 2 / T 1 ) + R V ln ( p V 1 / p V 2 ) ] d O where d O ≡ d O 2 = − d O 1 , T 1 and T 2 are the temperatures in the two parcels, and C PV is the specific heat of water vapor at constant pressure. Note that our result is equivalent to that of Pauluis and Held if the temperatures of the two parcels are the same. However, since our results do not take into account all of the effects of temperature gradients in this case, our results are not necessarily more accurate than those of Pauluis and Held. There is also a difference between the Pauluis‐Held results and our equation for the generation of entropy by evaporating precipitation. In addition to − R V ln [ p V / p S ( T ) ] in the Pauluis and Held papers, we have the term C P V ln ( T / T W ) . However, this is much smaller than the first term since typically ln ( T / T W ) ≪ 1 whereas − ln [ p V / p S ( T ) ] can become quite large for low relative humidities. The difference probably has to do with our assumption that the droplet temperatures are equal to the wet bulb temperature rather than the actual temperature. Pauluis [ ] and Pauluis and Held [ ] have no entropy source corresponding to water substance entering and exiting the atmospheric system, as represented by equation , as they consider precipitation to be part of the system. The corresponding physical effects appear as precipitation falls to the surface. 5.2. Classical Textbooks Landau and Lifschitz [ ] and Prigogine [ ] explicitly take into account the fact that isothermal diffusion transfers the enthalpy as well as the mass of the diffused component (see p. 221). Without this enthalpy transfer, the specific entropy s i in equation would be replaced by − μ i / T . The irreversible generation of entropy by diffusion given by equation would be unaffected, since − μ V / T differs from s V by only an additive constant that would cancel in the calculation of dS irr . However, the entropy source due to precipitation evaporation given in equation (approximating T W by T ) would be given in terms of the chemical potential of the condensate μ C rather than its specific entropy, X O P = − μ C ( E − P ) T ( incorrect ) , which is seriously incorrect. (The author was first alerted to this problem by the disastrous effect of equation on a numerical model of convection!) Certain well‐known texts [e.g., De Groot , ; De Groot and Mazur , ; Yourgrau et al ., ] do not make it clear that the enthalpy transport associated with mass transport into a parcel (by diffusive or other mechanisms) must be included in the calculation of the irreversible generation of entropy. In the case of mass transfer by diffusion of an ideal gas, the effects of the error on irreversible entropy generation vanish by accident, as noted above. However, if the specific enthalpy of the substance being diffused does not vary linearly with absolute temperature, the effects of the error reappear. Furthermore, if material actually enters or exits the system under consideration (as with the conversion of total cloud water to and from precipitation in our case), the results are disastrously wrong, as shown above. 6. Conclusions Summarizing our analysis, the total atmospheric entropy source (exclusive of radiative effects) is X E = 1 ρ D [ C P D ∇ · ( K ∇ T ) T + ∑ V , L , I s i ∇ · ( K ∇ r i ) ] + 2 K | D | 2 + g ρ D r P w T ρ D T + r P ( v z − w T ) T W [ − C C ∂ T W ∂ z + L F δ ( z − z F ) ] + E s C ( T W ) − P s C ( T ) + E [ − R V ln ( p V p S ( T ) ) + C P V ln ( T T W ) ] . Line by line, we have: 1. molecular processes of heat conduction and diffusion of water substance, augmented by the down‐scale transfer of variance to molecular scales by turbulence; 2. production of heat by viscous dissipation augmented by turbulence and the work done by falling precipitation on the atmosphere; 3. heat transfer to falling precipitation required to drive its temperature toward the wet bulb temperature of the air; also included is the heat transfer required to melt frozen precipitation as it descends through the freezing level; 4. transfer of entropy to and from the atmospheric microlayer surrounding hydrometeors by the formation and evaporation of precipitation; 5. entropy source associated with the microscale diffusion of water vapor away from and the microscale heat conduction toward evaporating hydrometeors. In addition, entropy can enter the atmosphere from surface fluxes given by equation . Equation represents the total nonadvective source of entropy, including irreversible entropy generation, flow of entropy to and from precipitation and the surface, and nonadvective transports. It is conventional to separate these quantities in textbooks, but from the point of view of numerical simulations, the total source (which may be negative in places) is what is needed. The results as presented are also specific to the case in which turbulent fluxes are represented by an eddy viscosity, though no restriction is placed on how this eddy viscosity is determined. Our results extend the theoretical development of Pauluis [ ] and Pauluis and Held [ ] in a number of respects. In particular, the difference between the atmospheric temperature and that of precipitation particles is accounted for in the present work and the most important effects of the ice phase are included. It remains to incorporate these results in a model that exhibits no numerical generation of entropy. This is feasible, with certain other compromises, given the development in section 2. The model of Raymond and Zeng [ ] conserves the integrated equivalent potential temperature explicitly, so only a small modification of that model is required to achieve that goal. Acknowledgments Thanks are due to Kerry Emanuel and Olivier Pauluis for penetrating reviews. Thanks also to Sharon Sessions for useful discussions. This work was supported by National Science Foundation grant 1021049.

Journal

Journal of Advances in Modeling Earth SystemsWiley

Published: Dec 1, 2013

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