Chapter 2 Thermodynamics of moist air Thermodynamics of moist and cloudy air ¾ Literature: • Wallace and Hobbs (1977, Chapter 2) • Iribarne and Godson (1973) • Houze (1993) • Emanuel (1994) • Smith (1997, NATO Chapter 2) 1 The equation of state for moist unsaturated air ¾ The state of a sample of moist air is characterized by its: • Pressure, p • Absolute temperature, T • Density, ρ (or specific volume α = 1/ρ), and • Some measure of its moisture content, e.g. • The water vapour mixing ratio, r, defined as the mass of water vapour in the sample per unit mass of dry air. ¾ These quantities are connected by the equation of state. Equation of state: specific gas constant for dry air pRTα = dd(/)/()11+ r ε + ε ≈ RT (.) 1061+ r ε = Rd/Rv = 0.662 specific gas constant water vapour r is normally expressed in g/kg, but must be in kg/kg in any formula! 2 Other moisture variables ¾ The vapour pressure, e = rp/(ε + r) which is the partial pressure of water vapour. ¾ The relative humidity, RH = 100 × e/e*(T). • e* = e*(T) is the saturation vapour pressure - the maximum vapour that an air parcel can hold without condensation taking place ¾ The specific humidity, q = r/(1 + r), which is the mass of water vapour per unit mass of moist air. More moisture variables ¾ The dew-point temperature, Td, which is the temperature at which an air parcel first becomes saturated as it is cooled isobarically. ¾ The wet-bulb temperature, Tw, which is the temperature at which an air parcel becomes saturated when it is cooled isobarically by evaporating water into it. The latent heat of evaporation is extracted from the air parcel. 3 The vertical distribution of r and RH obtained from a radiosonde sounding on a humid summers day in central Europe. Aerological (or thermodynamic) diagrams p T = constant ln p . (p, α) . (p, α) T = constant α 4 Aerological diagram with plotted sounding T = constant ln p . (p, Τd) (p, Τ) The effect of the water vapour on density is often taken into account in the equation of state through the definition of the virtual temperature: TTv = (/)/()(.)111061+ r ε + ε ≈ T + r Then, the density of a sample of moist air is characterized by its pressure and its virtual temperature, i.e. p ρ= RTv Moist air (r > 0) has a larger virtual temperature than dry air (r = 0) => the presence of moisture decreases the density of air --- important when considering the buoyancy of an air parcel! 5 The equation of state for cloudy air Consider cloudy air as a single, heterogeneous system specific volume = (total volume)/(total mass) α=VVV + + / M+ M+ MM+ ()ali( d vli) total mixing ratio of Divide by Md water substance αα=+d ()11rrll(/α αd )+ ii (/α αd )/( + rT ) RT 1 RTpe+ 1 RT1+ r / ε α = d = dd = d prdT1+ p prdT1+ p 1+ rT ε = Rd/Rv = 0.622 defines the density temperature for cloudy air: specific heat of Tρ =T(1+r/ε)/(1 + rT) water vapour The first law of thermodynamics The first law of thermodynamics for a sample of moist air may be written alternatively as dq= c′v dT + pdα, incremental changes in = cdT′p − α dp, temperature, pressure and specific volume the heat supplied per unit mass to the air sample specific heats of the air sample c′′vvd=+ c (1 0.94r) c ppd =+ c (1 0.85r) −1 −1 −1 −1 The quantities cvd ( ≈ 1410 J kg K ) and cpd ( ≈ 1870 J kg K ) are the corresponding values for dry air. 6 Adiabatic processes An adiabatic process is one in which there is zero heat input (dq = 0); in particular, heat generated by frictional dissipation is ignored. dTln= ( Rcdp′′ /p ) ln RR′ = d (/)/()11+ r ε + r If the process is also reversible and unsaturated, r is a constant. The equation can be integrated exactly (ignoring the small temperature dependence of c'p ) to give: a constant ln T= (R′ / c′p )ln p+ ln A Define A such that, when p equals some standard pressure, po, usually taken to be 1000 mb, T = θ. The potential temperature The quantity θ is called the potential temperature and is given by: R′ R 1+ r / ε d κ′ p cp′ p c p θ = TF ooI = TF I pd 1+ rcpv/ c pd ≈ TF o I HG p KJ HG p KJ HG p KJ κ′ = κ(.)1024r− The variation in κ is < 1 % and is usually ignored κ = Rcdpd/ We define the virtual potential temperature, θv by κ F po I θvv= T G J H p K take the value of κ for dry air. 7 Enthalpy The first law of thermodynamics can be expressed as dq=+α−α−α d(u p ) dp = dk dp k = u + pα is called the specific enthalpy. The enthalpy is a measure of heat content at constant pressure. For an ideal gas, k = cpT. Water substance and phase changes The latent heat associated with a phase transition of a substance is defined as the difference between the heat contents, or enthalpies, of the two phases; i.e. subscripts refer to Li,ii = kii -ki, the two phases The dependence of Li,ii on T and p may be obtained by differentiation using k = u + pα (for details see e.g. E94, p114). It turns out that: Li,ii = Li,iio + (cpii -cpi)(T − 273.16 K) Li,iio is the latent heat at the so-called triple point (T = 273.16 K = 0.01oC, and e = 6.112 mb) where all three phases of water substance are in equilibrium. e is the water vapour pressure 8 The Clausius-Clapeyron equation The pressure and temperature at which two phases are in equilibrium are governed by the Clausius-Clapeyron equation: ⎛ dp ⎞ Liii, ⎜ ⎟ = . ⎝ dT ⎠ii, i T(ααii− i ) For liquid-vapour equilibrium, αl << αv, and using the ideal gas law for vapour, we obtain an equation for the saturation vapour pressure, e*(T): de **Lev = 2 dT RTv Lv = the latent heat of vaporization. de **Lev = 2 dT RTv This equation may be integrated for e*(T). A more accurate empirical formula is (see E94, p117): lneTT *= 53 . 67957− 6743 . 769 /− 4 . 8451 ln e* in mb and T in K A corresponding expression for ice-vapour equilibrium is: lneTT *= 23 . 33086− 6111 . 72784 /+ 0 . 15215 ln These formulae are used to calculate the water vapour content of a sample of air. If the air sample is unsaturated, the dew point temperature (or ice point temperature) must be used. 9 Moist enthalpy ¾ Define the moist enthalpy k of a sample of cloudy air by the formula: Mdk = Mdkd + Mvkv + MLkL ¾ Then k = kd + rkv + rLkL , expressed per unit mass of dry air. ¾ But Lv(T) = kv -kL , => k = kd + Lvr + rTkL ¾ rT = r + rL. c is the specific heat ¾ Since kd = cpdT and kL = cLT, L of liquid water k = (cpd + rTcL)T + Lvr The moist enthalpy is conserved in an isobaric process as long as dq = 0 and drT = 0. Entropy An excellent reference is Chapter 4 of the book: C. F. Bohren & B. A. Albrecht Atmospheric Thermodynamics Oxford University Press 10 The moist entropy and related quantities ¾ In analogy with the definition of k, we define the total specific entropy per unit mass of dry air as: s = sd + rsv + rLsL ¾ sd , sv , and sL are the specific entropies of dry air, water vapour and liquid water, respectively. ¾ The equality of the Gibbs free energy in phase equilibrium between water vapour and liquid water leads to the expression: * LTssvv=−()L the saturation specific entropy of water vapour Some algebra ss=+dvL rsrs + L Lvr * ss=+d rsTL + +rs()vv − s * T LTssvv=−()L Substitute scTL = L ln scdpd=−ln TR d ln p d scvpv= ln TRe− v ln ** scvpv= ln TRe− v ln L r sc=+( rcTRp )ln − ln +−v rR ln( RH ) pdT L d d T v 11 Conservation of entropy L r sc=+( rcTRp )ln − ln +−v rR ln( RH ) pdT L d d T v ¾ s is conserved under reversible moist adiabatic transformations ¾ It is not conserved when •dq≠ 0: radiative heating or cooling, or conduction •dr≠ 0: by external evaporation, precipitation loss • by evaporation of rain into unsaturated air Equivalent potential temperature Put sc=+( pdTLe rclnRlnp) θ− d o This defines the equivalent potential temperature, θe R/(cdpdTL+ rc) −+rR /(c r c ) ⎡ ⎤ ⎛⎞pLrovvpdTL θ=e T(RH)exp⎜⎟ ⎢ ⎥ ⎝⎠p(crc)TdpdTL⎣⎢ + ⎦⎥ 12 Entropy ¾ Entropy s is defined by a process in which heat and water substance are added to air which is kept just saturated by reversible evaporation from a planar water surface as its temperature is raised from a reference temperature and partial pressure pd to the state (T, pd). ¾ s is conserved under reversible moist adiabatic transformations. (T , p ) ref d (T, pd) Tref T Equivalent potential temperature R/(cdpdTL+ rc) −+rR /(c r c ) ⎡ ⎤ ⎛⎞pLrovvpdTL θ=e T(RH)exp⎜⎟ ⎢ ⎥ ⎝⎠p(crc)TdpdTL⎣⎢ + ⎦⎥ = p = 1 = 1 ¾ For dry air (r = 0, rL = 0, rT = 0), θe reduces to θ. ¾ Note that θe is not a state variable → isopleths of constant θe cannot be plotted on an aerological diagram. 13 Differential form for s ¾ The differential form of s for (saturated) cloudy air (RH = 1, rL ≥ 0) is * dT dpdv⎛⎞ L r ds=+ (cpd r T c pv ) − R d + d ⎜⎟ Tpd ⎝⎠ T * ¾ An alternative form is, using 0 = dr + drL * dT de dpd F LrvLI ds=+() cpd r T c pv −rR T v* −R d −d T e pd H T K Alternative forms ¾ Betts showed that the two forms can be written more symmetrically as: * ds dθe dT Lv * dp ==+−cpm * cpm dq Rm 1+ rT θe T T p ds dθl dT Lv dp ==−−cpm cpm dqlm R 1+ rT θl T T p * (1 + rT) cpm = (cpd + r cpv +rLcl) (1 + r ) R = (R +R) where T m d v * * q = r /(1 + rT) ql = rL /(1 + rT) 14 The equations * ds dθe dT Lv * dp ==+−cpm * cpm dq Rm 1+ rT θe T T p ds dθl dT Lv dp ==−−cpm cpm dqlm R 1+ rT θl T T p define ¾ the saturation equivalent potential temperature, θe* ¾ the liquid-water potential temperature, θl Approximate forms of θe* and θl Let cpm ≈ cpd, Rm ≈ Rd then * ds dθe dT Lv * dp ==+−cpm * cpm dq Rm 1+ rT θe T T p * dθe dLθ v * cpd * =+cpd dr and θe θ T ds dθl dT Lv dp ==−−cpm cpm dqlm R 1+ rT θl T T p ddθL θ Lv ccpdpdL=− dr θθL T 15 Assume that * * (Lv/T)dr ≈ d(Lvr /T) and (Lv/T)drL ≈ d(LvrL/T) * Then dθe dLθ v * cpd * =+cpd dr θe θ T ⎛⎞Lr* θ≈θ* exp v e ⎜⎟ ⎝⎠cTpd R/(cdpdTL+ rc) ⎡ ⎤ c/f * ⎛⎞pLrov θ=e Texp⎜⎟ ⎢ ⎥ ⎝⎠pdpdTL⎣⎢(c+ r c )T ⎦⎥ dθL dθ Lv ccpdpdL=− dr θθL T θLvLpd≈θ exp (-L r /c T) Note that when rL = 0, θL = θ → the liquid water potential temperature is the potential temperature attained by the evaporation of all liquid water in an air parcel through reversible moist descent.