THERMODYNAMICS 1 /43

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1 THERMODYNAMICS 1

Atmosphere A multi-component Multi-Phase System The gas phase atmospheric cons1tuents; major gases; fixed propor1ons by volume (dry air) Nitrogen (N 2 ) 78,08 % Oxygen (O 2 ) 29,05 % Argon (Ar) 0,934

2 Atmosphere A multi-component Multi-Phase System The gas phase atmospheric cons1tuents; major gases; fixed propor1ons by volume (dry air) Nitrogen (N 2 ) 78,08 % Oxygen (O 2 ) 29,05 % Argon (Ar) 0,934 % Variable vapors and minor gases: Carbon dioxide (CO 2 ) Neon, Helium, Nitrous Oxide, Ozon, Methan, Sulfur compounds, organics Water vapor (0 4 % by volume) Because of its proclivity to change phase and the manner in which these phase changes affect the local temperature on the one hand, and foster diverse interac1ons with radiant energy on the other, water indelibly marks mo1ons in the lower atmosphere on all 1me and spa1al scales. 2

3 Notation The basic theromdynamic proper1es of the atmosphere depend on its component parts. These are defined in terms of their mass, such that for an equilibrium system four cons1tuents of the moist atmosphere can be defined: d dry air v water vapor l liquid water i ice Total mass of the system: m=m d +m v +m l +m i To describe the amount of ma[er we use: Specific mass: q x =m x /m It is useful in a thermodynamic descrip1on for an equilibrium system (the normalizing mass is invariant as long as the basic flow describes the mo1on of the dry air) Mixing ra1o: r x =m x /m d It is helpful when considering the possibility that different components of a mass element have their own velocity 3

4 We dis1nguish between: equilibrium condensed phases associated with clouds, which evolve with the thermodynamic state in a more or less reversible way, and larger hydrometeors which evolve in irreversible way Larger hydrometeors, like rain-drops and most forms of ice, develop through irreversible microphysical processes such as the collision and coalescence of water droplets. These are more difficult to approximate as an equilibrium phase. Because they are larger, non-equilibrium phases of water in the atmosphere are also more dilute and shortlived The present lecture (Thermodynamics) focuses on reversible thermodynamic processes. 4

5 Thermodynamic systems in local equilibrium are iden1fied with mass elements, some1mes referred to as air- or fluid-parcels. Formally the concept gains validity for mass elements small enough that the volume they occupy encompasses a scale much smaller than the scale over which thermodynamic proper1es vary, but much larger than the mean-free path. Diffusion rapidly homogenizes the atmosphere on scales smaller than the Kolmogorov lenght scale, η=(ν 3 /ε) 1/4 ν - viscosity of the atmosphere, ε - turbulent disipaaon rate In vigorous cumulus clouds: ε=0.05 m 2 /s 3, ν= m 2 /s η=0.5 mm; this is several thousand 1mes larger than the mean free path of an air molecule, making the concept of an air parcel a useful one. 5

6 6 Typically the specific condensate mass within a cloud is less than 1 g kg -1 ; much less than the specific mass of water vapor What is the volume frac1on of the water condensate? The dilutness of condensate can come into conflict with the concept of an equilibrium thermodynamic system, as on the Kolmogorov scale, the condensate is not con1nuously distributed d l d l 3 l 3 d d d l l d l 10 m cm 1 V V kg g 1 m m ; cm g 1 ; m kg 1 m m V V = = = = = = ρ ρ ρ ρ

7 In presence of the condensate (droplets, ice) that is not con1nously distributed the concept of air parcel should be enlarged. One ofen imagines an ir parcel as being on the scale of arround 1 m 3. Strictly speeking volumes of air this large cannot be thought of in terms of a single temperature, but the error of this approxima1on is typically much less than those associated with other approxima1ons invoked in the descrip1on of such systems. 7

8 Subscripts notation d dry air v water vapor l liquid water i solid water (ice) c condensate t total water (irrespec1ve of phase) s saturated state, or process e equivalent (all condensate) reference state l! liquid-free (all vapor) reference state 8

Equation of State Taking an air-parcel to be comprised of an ideal mixture of ideal gases, perhaps in the presence of condensate, the equa1on of state is that for an ideal gas of variable composi1on,

9 Equation of State Taking an air-parcel to be comprised of an ideal mixture of ideal gases, perhaps in the presence of condensate, the equa1on of state is that for an ideal gas of variable composi1on, such that: p = p d + p v = m R d d V + m v R v V T R d = 287JK 1 kg 1 R v = 461JK 1 kg 1 Defining the density of the gaseous/vapor mixture as ρ=(m d +m v )/V allows one to formulate the equa1on of state as: d v l i c t s e dry air water vapor liquid water solid water (ice) condensate total water (irrespec1ve of phase) saturated state, or process equivalent (all condensate) reference state p = ρrt R = q d R d + q v R v = 1 q v q c = R d 1+ R v 1 q v q c R d ( ) R d + q v R v To avoid dealing with a state dependent gas constant it is customary to define a density temperature, such that: p = ρ R d T ρ T ρ = T ( 1+ εʹq v q c ) m = m d + m v + m c 1= q d + q v + q c q d =1 q v q c ε ʹ = R v R d l! liquid-free (all vapor) reference state / m ε = R d R v

10 Density temperature, virtual temperature T ρ = T ( 1+ εʹq v q c ) T v = T ( 1+ εʹq v ) = T ( q v ) In the absence of water the density temperature is the air temperature, otherwise can be interpreted as the temperature of a dry air parcel having the same density and pressure as the given air parcel. In the literature the density temperature is ofen called the virtual temperature. 10

11 Buoyancy Given the pressure, the density temperature determines the density and thus is important to the concept of the buoyancy. The buoyancy (b) of a fluid parcel can be measured by the extent to which its density differs from a background or reference density. Assume that locally the density is given in terms of a devia1on from such a reference state density: ρ = ρ 0 + ρʹ b g ρʹ Tʹ g ρ 0 T 0 Rʹ = g R 0 ʹ T ρ T ρ 0 p = ρrt The approxima1on of the buoyancy in terms of the density temperature follows from the assump1on that the rela1ve change in pressure is small compared to the rela1ve change in density. 11

12 Enthalpy Entropy Gibbs free energy Thermodynamic functions of state 12

13 Enthalpy For an atmospheric system it proves useful to use temperature and pressure to describe the state of the system. The First Law becomes (h specific enthalpy): q = dh υdp h = q d h d + q v h v + q l h l + q i h i ( ) p c p = h T h = h 0 + ( q d c p,d + q v c p,v + q l c l + q i c i )T Limi1ng our considera1ons to ideal gases implies that enthalpy depends ONLY on temperature. For water and ice the subscript p for specific heat is omi[ed because water and ice are non-compressible. Physically only enthalpy differences are relevant. 13

14 Enthalpy Vaporisa1on enthalpy (latent heat): L υ = h υ h l It proves useful to rewrite the expression for the enthalpy in terms of the phase-change enthalpies, so that it only refers to the reference enthalpy of one of the phases : using L v to subs1tute h v in expressiion: h = q d h d + q υ h υ + q l h l h e = c e T + q v L v c e = c d + q t ( c l c d ) = q d c d + q t c l using L v to subs1tute h l h l = c l T q l L v c l = c d + q t + ( c υ c d ) = q d c d + q t c υ The subscripts e and l serves as a reminder of which reference state has been adopted. Both expressions of enthalpy are simillar if: ( c l c e )T = q t L υ = q t ( c υ c l )T 14

Entropy The Second Law postulates the existence of an entropy state func1on, S, defined by the property that in equilibrium the state of the system is that which maximizes the entropy func1on.

15 Entropy The Second Law postulates the existence of an entropy state func1on, S, defined by the property that in equilibrium the state of the system is that which maximizes the entropy func1on. Such a func1on has the property that q Tds Tds = dh υdp As an extensive state func1on the entropy, like the enthalpy, can be decomposed into its cons1tuent parts:: s = q d s d + q υ s υ + q l s l + q i s i 15

16 Dry air and water vapor entropy Dry air (ideal gas): dh = Tds +υdp dh = c d dt, υ = R d T p d ds = c d d lnt R d d ln p d s d = s d,0 + c d ln T T 0 ( ) R d ln( p d p 0 ) s d,0 is the reference entropy of dry air at the temperature T 0 and pressure p 0. Water vapor (ideal gas): s υ = s υ,0 + c υ ln( T T 0 ) R υ ln( p υ p 0 ) It is assumed that the specific heats are constant between T and T 0. 16

17 Entropy for condensed phases The condensate is assumed to be ideal so that changes in pressure do not contribute to entropy. s l = s l,0 + c l ln( T T 0 ) 17

18 The general expression for the composite entropy with respect to the equivalent reference state s e = q d s d + q υ s υ + q l s l ( ) = q d s d + q t s l + q υ s υ s l = q d s d,0 + q t s l,0 + q d c d ln T q d R d ln p d + q t c l ln T + q υ ( s υ s l ) T 0 p 0 T 0 q l = q t q υ s e = s e,0 + c e ln T R e ln p d + q υ ( s υ s l ) T 0 p 0 ( ) ( c l c d ) s e,0 = q d s d,0 + q t s l,0 = s d,0 + q t s υ,0 s d,0 c e = q d c d + q t c l = c d + q t R e = q d R d s e,0 is determined by the amount of water in the system and the reference state temperature and pressure denoted by T 0 and p 0 respec1vely 18

19 Entropy for the liquid-free reference state s l = q d s d + q υ s υ + q l s l ( ) = q d s d + q t s υ q l s υ s l = q d s d,0 + q t s υ,0 + q d c d ln T T 0 q d R d ln p d p 0 + q t c υ ln T T 0 qr υt ln p υ p 0 q l ( s υ s l ) q υ = q t q l s l = s l,0 + c e ln T T 0 q d R d ln p d p 0 q t R υ ln p υ p 0 q l s l,0 = s d,0 + q t s υ,0 s d,0 ( ) ( c υ c d ) c l = q d c d + q t c υ = c d + q t ( s υ s l ) 19

Gibbs free energy For a closed isobaric and isothermal system it follows from equa1on reversible system 0 = d ( H TS) Which introduces the Gibbs free energy, or Gibbs poten1al as G = H TS i.e. the energy available to do work in an isothermal and isobaric system.

20 Gibbs free energy For a closed isobaric and isothermal system it follows from equa1on reversible system 0 = d ( H TS) Which introduces the Gibbs free energy, or Gibbs poten1al as G = H TS i.e. the energy available to do work in an isothermal and isobaric system. TdS = dh Vdp that for a From this defini1on it follows that the difference in the Gibbs energy of two cons1tuents is related to the differencies in their enthalpies and entropies: g υ g l = h υ h l T(s υ s l ) From the postulates of thermodynamics, whereby in equilibrium H and T adopt values that maximize S, it follows that the Gibbs free energy of a system in equilibrium is a minimum. In equilibrium the specific Gibbs energy of each phase must be equal, otherwise a redistribu1on of the mass between the phases could lower the total Gibbs energy. 20

21 The Clausius-Clapeyron Equation g = h Ts dg = dh sdt Tds = υdp sdt Tds = dh υdp dg υ = υ υ dp s υ dt dg l = υ l dp s l dt dg υ = dg l dp = s υ s l υ υ υ l dt Clapeyron equa1on υ υ = R υ T p υ υ >> υ l L υ T = s υ s l Saturated water vapor pressure is ofen denoted by e s d(lne s ) = L v R υ T 2 dt Clausius-Clapeyron equa1on 21

22 Satura:on vapor pressure over liquid (solid line) and ice (dashed line). Colored circles and lines show vapor pressure in the atmosphere, binned according to temperature for different pressure levels (900 hpa, black; 700 hpa, blue, 500 hpa, orange, 300 hpa, red). At T = 0C the satura1on vapor pressure is Pa. At T = 30C the satura1on vapor pressure over liquid water is 50.8 Pa as compared to 38.0 Pa over ice at the same temperature. Satura1on with respect to liquid for T < 0C is relevant because supercooled water is ofen present in the atmosphere, with homogeneous nuclea1on of ice par1cles first occurring at about T = -38C. 22

23 The Clausius-Clapeyron equation The Clausius-Clapeyron equa1on very effec1vely delimits the distribu1on of water throughout the atmosphere because: The atmosphere sits atop a reservoir of water, which endeavors to bring the air above it into satura1on, but how much moisture can be maintained in an air parcel is strongly constrained by the satura1on value. If the amount of moisture exceeds the satura1on value it condenses, and condensate is effec1vely removed by precipita1on by the system. Hence the satura1on specific humidity limits the amount of water in the atmosphere. Because the Clausius-Clapeyron equa1on so strongly controls the distribu1on of water in the atmosphere, if one had to single out a par1cular equa1on as being the most important for the func1oning of Earth s climate, it would be this equa1on. 23

24 Potential temperatures 24

25 Potential temperatures The first and second laws dictate how temperature changes between a given state, and a reference state defined by its pressure p ϑ, and phase distribu1on, given by the triplet {p ϑ,q v,q l }. The temperature in this reference state is called the poten1al temperature (denoted by θ) as it measures the temperature the system would have to have in the reference state for the entropy of this state to be iden1cal to that of the given state. Poten1al temperatures are invariant under an isentropic process, but their proper1es and absolute values depend on the choice of the reference. The poten1al temperature, rather than the sensible temperature, is ofen preferred as a state variable because it is invariant for reversible transforma1ons of the air parcel. The poten1al temperature provides a convenient way to compare air parcels in different parts of the atmosphere, where for instance the pressure or humidity may vary. 25

26 Equivalent potential temperature θ e Reference state {p ϑ,q v,q l }= {10 5 Pa,0,q t } All the vapor is condensed into liquid at p ϑ =1000 hpa. For historical reasons this par1cular reference state is called an equivalent state and denoted by subscript e. For the equivalent reference state we equate the equa1on for s e with the value of T 0 chosen so that s e,0 =s e. s e = s e,0 + c e ln T R e ln p d + q υ ( s υ s l ) θ e p θ c e lnθ e = c e lnt R e ln p d + q υ ( s υ s l ) p θ 26

27 Equivalent potential temperature.. We express the pressure of the dry air in terms of the total pressure and the specific humidity p d = R d Tρ d p = RTρ R = q d R d + q υ R υ p d = p q R d d = p R e R R We express the entropy difference (s v -s l ) rela1ve to the vapor entropy in satura1on s υ s l = s υ s s + s s s l = s υ s s + L υ T s υ s s = c υ ln T θ e R υ ln p υ p θ c υ ln T θ e + R υ ln p s p θ = R υ ln p υ p s = R υ lnϕ φ rela1ve humidity 27

28 Equivalent potential temperature.. c e lnθ e = c e lnt R e ln p p θ R e R q υ R υ lnϕ + q L υ υ T θ e = T p θ p R e ce Ωe exp q L υ υ c e T Ω = R e R e R e ce ϕ q υr υ c e R e = q d R d c e = c d + q t ( c l c d ) R = R d + q t ( R υ R d ) For q t =0 the equivalent poten1al temperature gets a simpler form Ω e is a factor that is very near unity, and (because q t <<1) depends only very weakly on the thermodynamic state. π θ = T p R d c d 28

29 Liquid water potential temperature The counterpart to the equivalent poten1al temperature is the liquid-water poten1al temperature, which is the temperature an air parcel would have if were reversibly brought to the {p ϑ,q v,0} reference state. For the liquid-free reference state we equate the equa1on for s l with the value of T 0 chosen so that s l,0 =s l. θ l = T p θ p R l cl Ωl exp q L l υ R Ω l = c l T q d R d q d R d cl R q υ R υ q t R υ cl R l = R d + q t ( R υ R d ) c l = c d + q t ( c υ c d ) R = R d + q t ( R υ R d ) 29

30 Approximate forms: θ e, θ l Many of the nuances that moisture brings to thermodynamic descrip1ons can be neglected when considering small perturba1ons about a given Enthalpies h e and h l both describe the enthalpy of moist liquid-vapor system, hence in general h e =h l. However if, as is common, one assumes that: c e c d = c p c l c d = c p Large differences between h l and h e become apparent. These differences are however differences only in the absolute sense. In saturated case qʹv = qʹs = qlʹ perturba1ons in general are similar. It is customary to define many of the moist thermodynamic variables in an approximate form apprropriate to the considera1on of small perturba1ons. L θe = θ exp c v p L θl = θ exp c q T v p v ql T where θ π = T p R d c p 30

31 Saturated equivalent potential temperature L θ s θ exp c θ s is only a func1on of temperature and pressure, so it measures the thermal structure of the atmosphere θ s is constant following a saturated pseudo-adiabat. The word pseudo arises because formally the process corresponding to constant θ s is similar to a reversible adiabat, but the removal of condensate upon condensa1on, as implied by the use of qs instead of qt impies a loss of condensate enthalpy by the system, hence it is not truly adiaba1c. The difference between θ s and θ e measures the subsatura1on, as θ e < θ s. v p q T s 31

32 Moist adiabatic lapse rate Γ s dt dz θe = γγ d γ c R 1+ q s β υ T d R c p L 1+ q s β υ T c p T β T lnq s lnt = L υ R υ T lnq s ln p 5400K T c p = q d c d + q υ c υ Γ d = g c d 32

Non-dimensional lapse rate γ pressure: 1 000 hpa the air ini1ally saturated at 300 K solid curve pseudo-adiabat

33 Non-dimensional lapse rate γ pressure: hpa the air ini1ally saturated at 300 K solid curve pseudo-adiabat dashed curve adiaba1c lapse rate Γ s dt dz θe = γγ d γ c R 1+ q s β υ T d R c p L 1+ q s β υ T c p T β T 5400K T 33

34 Non-dimensional lapse rate γ pressure: 800 hpa the air ini1ally saturated at 300 K solid curve pseudo-adiabat dashed curve adiaba1c lapse rate Γ s dt dz θe = γγ d γ c R 1+ q s β υ T d R c p L 1+ q s β υ T c p T β T 5400K T 34

35 Water condensated in pseudoadiabatic process Equa1on for enthalpy of a composite system for adiaba1c process. dh = q +υdp q = 0, dh l = c l dt L υ dq l q l dl υ c p = q d c d + q υ c υ + q l c l 0 = c p dt L υ dq l υdp c p dt L υ dq l υdp = 0 L υ dq l = c p dt + gdz dq l = c p L υ dt dz + g c p υdp = gdz dz Γ = g d, Γ s = dt c p dz dq l = c p L υ ( Γ d Γ s )dz LWC = q d l ρ c p ( LWC) = ( Γ Γ )dz L v ρ d s 35

36 Rate of condensation d ( LWC) = c w ( T, p)dz c w (T, p) = c p L υ ρ Γ d ( 1 γ ) c p L υ ρ Γ d kg / m 4 = g / m 4 c w = g / m 4 36

37 p=1000 hpa 37

38 p=800 hpa 38

39 EUCREX Pawlowska et al., Atmos. Res

40 Pawlowska et al., Atmos. Res

41 Pawlowska et al., Atmos. Res

42 ACE-2 Brenguier, Pawlowska, Schueller, JGR

43 EUCAARI - IMPACT Jarecka et al.,, ACP.,

2σ e s (r,t) = e s (T)exp( rr v ρ l T ) = exp( ) 2σ R v ρ l Tln(e/e s (T)) e s (f H2 O,r,T) = f H2 O

2σ e s (r,t) = e s (T)exp( rr v ρ l T ) = exp( ) 2σ R v ρ l Tln(e/e s (T)) e s (f H2 O,r,T) = f H2 O Formulas/Constants, Physics/Oceanography 4510/5510 B Atmospheric Physics II N A = 6.02 10 23 molecules/mole (Avogadro s number) 1 mb = 100 Pa 1 Pa = 1 N/m 2 Γ d = 9.8 o C/km (dry adiabatic lapse rate)