Thermodynamics of moist convection Part I: heat engine and buoyancy. Olivier Pauluis (NYU) Porquerolles

Transcription

1 Thermodynamics of moist convection Part I: heat engine and buoyancy Olivier Pauluis (NYU) Porquerolles

2 Outline Introduction: the nature of the problem Entropy of moist air Heat engines and water vapor. Equation of state of moist air Idealized moist Rayleigh-Benard convection Buoyancy and entropy fluxes Isentropic circulation

3 Net cooling above ~-100W/m 2 Kinetic energy production and dissipation ~2-5W/m 2 Net heating at the surface ~100W/m 2 D " 4Wm #2 $ % " $ V 1m ~ 0.1ms #1 But kinetic energy production and dissipation are highly intermittent!

4 Different weather regime are associated with dramatically different amount of kinetic energy production.

5 1A. Thermodynamics 1st Law - energy conservation "U + W = Q 2nd Law what does it mean? "S = Q T + "S irr "S irr # 0

6 "S = Q T + "S irr "S irr # 0 S: entropy is a state variable. There are reversible processes. Entropy is conserved for reversible adiabatic transformations, i.e. two systems have the same entropy if they can be join by a sequence of reversible adiabatic transformation. But real transformations are irreversible and associated with a positive entropy production.

7 1B. Entropy of moist air Moist air can be treated as an ideal mixture of dry air, water vapor and liquid water. The entropy per unit mass of dry air S is then: S = s d + rs v + r l s l With S: entropy per unit mass of dry air; r: mixing ratio (water vapor concentration) r l : mixing ratio for condensed water r T =r+r l : mixing ratio for total water s d,s v.s l : specific entropy for dry air, water vapor and liquid water

8 The specific entropies are defined up to an additive constant: s d = C pd ln T T o " R d ln p d p o + s d 0 s v = C pv ln T T o " R v ln e e o + s v0 s l = C l ln T T o + s l 0 We cannot put all the integration constant to 0 because the entropy of water vapor and liquid water must be such that: s v " s l = L v T at saturation (e = e s (T) or H =1) Moist entropy S : set s l0 = s d0 = 0 " S = (C pd + r T C l )ln T + R d ln p $ d + r L v T 0 p 0 T # R ' & v lnh) % (

9 A brief note on adiabatic invariants: The thermodynamic properties of dry air can be described by 2 state variables, say entropy and pressure. As pressure is not invariant, any adiabatic invariant is function of entropy alone. df(s) = 0 for reversible adiabatic transformations. dt For moist air, we need at least three state variables, e.g. entropy, total water concentration, and pressure. Any function of entropy and total water content is an adiabatic invariant: df(s,r T ) dt = 0 for reversible adiabatic transformations.

10

11 2. Idealized heat engine Q out T out Q in T in Idealized problem: convection transport water vapor and energy upward from a warm/moist source to a dry/cold sink. Situation is analogous to shallow, non-precipitating convection.

12 2a. Carnot cycle " 3 1" 2 : isothermal expansion at T in 2 2 " 3 : adiabtic expansion with S 2 = S 3 3 " 4 : isothermal compression at T out 4 "1: adiabatic compression with S 2 = S 3 Total water content is constant through the entire cycle! Q out T T in p 2 Cold descent Warm ascent Q in T out T in T out 4 "S 3 S

13 Mechanical work is defined as W = "#(S,r T, p)dp Using the thermodynamic relationship TdS = dh "#dp we get: W = External heating "Q = dh #$dp = TdS Heating at the warm source: $ Q in = # "Q " TdS = (T in # T out )$S 2 # 1 + = TdS = Tin $S Efficieny " C = W Q in = T in # T out T in

14 2B. Steam cycle 1" 2 : isothermal moistening at T in 2 " 3: adiabtic expansion T T in " 4 : isothermal drying at T out 4 "1: adiabatic compression Heating is due solely to evaporation! Q out T out H 1 4 "S 3 S 3 dry descent moist ascent T out Q in = L"r T T in "r T r T

15 The expression TdS = dh "#dp is only valid for close transformations. To account for the addition or removal of mass, we need an additional term: TdS = dh "#dp " gdr T where g is the Gibbs free energy of water vapor per unit of mass: g = h " Ts = C pv (T " T 0 " ln T T 0 )+ R v T lnh # R v T lnh

16 Mechanical work: with W = " TdS + " gdr T = (T in # T out )$S + (g in # g out )$r T g in = R v T in lnh in g out = R v T out lnh out Surface heating: Entropy change: Q in = T in "S + g in "r T = L"r T $ "S = L # g ' in & )"r T % ( T in Efficieny " H = W Q in = T in # T out T in + R vt out L ln H in H out

17 Efficieny " H = T in # T out T in Carnot efficiency The efficiency depends on the state of the system!!! Saturated case: H=1 + R vt out L ln H in H out Efficieny " H,sat = W = T in # T out Q in T in General case: the relative humidity increases with height, i.e H out " H in # $ H % T in & T out T in Hence, a steam cycle produces at best as much mechanical work as a Carnot cycle additional term depending on saturation

18 Three regimes: " H " C = #T T The cycle is unsaturated at all time: efficiency is minimum. The cycle is partially saturated: efficiency increases with amount of water in the cycle. The cycle is saturated at all time: efficiency is maximum and given by the Carnot efficiency ~ 0.16" C H out =1 H in =1 r T,in Unsaturated cycle Partially saturated Fully saturated

19 2C. There is no free lunch Its rate of change is given by dg = sdt + "dp For a reversible isothermal process, we have: dg "#dp = 0 $ %g + W = 0 The amount of work that can be extracted is equal to the reduction in free energy! And it is only possible to increase the free energy if work is exerted on the system

20 water vapor transport penalty due to an increase in the free energy as water is transported upward ( g v,in " g v,out )# 0 w $ r $ T g v = R v T lnh Free energy increase with height in an unsaturated ascent Bur is constant whenever the air is saturated g out g in T out T in

21 2d. Mixed Carnot-steam cycle 1" 2 : isothermal heating and moistening at T in 2 " 3: adiabtic expansion T " 4 : isothermal cooling and drying at T out 4 "1: adiabatic compression Intermediary steps 5 and 6 such that cycle is a humidifier and is a Carnot cycle S

22 Latent and sensible heat flux: Q lat = L"r T Bowen ratio: Efficieny " = Q sen = T in "S + (g in # L)"r T B = Q sen Q lat B 1+ B " + 1 c 1+ B " H = T in # T out T in B R v T out L ln H in H out The efficiency of an atmospheric heat engine depends on both its degree of saturation and on the Bowen ratio.

23 small "T low H $ Work = "Q in = T in # T out & + 1 % T in 1+ B R v T out L ln H in H out ' ) Q in ( large "T low H large "T high H

24 3A. Equation of state for moist air Equation of state relates various thermodynamic properties (I.e state variable) of a fluid. We are particularly interested in expressed properties like specific volume in terms of adiabatic invariants and pressure. Moist air is treated as a mixture of dry air, water vapor and condensed water. Water vapor and dry air are treated as ideal gases. Liquid water is treated as incompressible and its volume is neglected.

25 Specific volume: We start from the ideal gas law PV = (N d + N v )R * T = ( N d m d ) R * T+ ( N v m v ) R * T m d m v Then, after some reorganization, we get: # 1 r & " = % R d T + R v ( T $ 1+ r T 1+ r T ' P = ( 1+ R v Rd r ) R T d 1+ r T P = R dt ) P At the end, we can express the specific volume as a (smooth) function of four state variables, i.e.: " = "( p,r,r,r T )

26 Thermodynamic equilibrium Condensed water can only be present if it is in thermodynamics equilibrium with water. I.e we have either e " e s (T) and r l = 0 or e = e s (T) and r l # 0 (unsaturated air) (saturated air) e : partial pressure of water vapor e s : saturation vapor pressure T : Temperature q l : concentration of condesate water

27 Phase transition and partial derivatives Thermodynamic equilibrium introduces a switch condition in the description of the state of moist air: # r l = 0 for r T < r s (T, p) $ % r T " r s (T, p) for r T < r s (T, p) This implies a discontinuity in the partial derivatives of the equation of state: # "r l & % ( $ "r T ' p,t * = 0 for r ) r T s (T, p) +, 1 for r T > r s (T, p) This applies not only to liquid water content, but to almost all states variables.

28 Equation of state for moist air

29 % Adiabatic Lapse Rate : " = -#g $T ( ' * & $p ) S,r T

30 3B. Boussinesq Approximation Start with the compressible N-S equation du dt = "#$p " gk "%$ 2 u d# dt "#$ u = 0 Assume an isentropic reference state Expand for small perturbation in pressure and density: du dt = "#P + Bk "$# 2 u # u = 0

31 Variations of density only enter through the buoyancy term in the vertical momentum equation: B = B(X 1...X n,z) = g "(X 1...X n, p ref (z)) #" ref (z) " ref (z) where the specific volume! " = "(X 1...X n, p) is a non-linear function of the pressure p and other state variables X 1 X n. The simplest choice is then to choose X 1 X n to be adiabatic invariants: dx i dt = "# 2 X i

32 What about the adiabatic invariants? We use the entropy S and the total water concentration q T as adiabatic invariants: The buoyancy is a non-linear function of the two invariants:

33 Piece-wise linear equation of state: Linearize the equation of state within the saturated and unsaturated regions: Introduce two new variables, the dry and moist buoyancies D and M:

34 For unsaturated parcels, While for saturated parcels,

35 We still need a condition for saturation: M " D # "N s 2 z The full system is then

36 z Buoyancy in a unsaturated adiabatic ascent B = D " N s 2 z B(z) Buoyancy in a saturated adiabatic ascent B = M Buoyancy

37 z B(z) z =z b B(D 0,M 0,z) Adiabatic ascent - Parcel becomes saturated at z =z b Buoyancy

38 Moist Rayleigh-Benard convection Cloud layer Subcloud layer Analog to the classic Rayleigh-Benard convection but now, both the dry and moist buoyancyes D and M must be specified at the upper and lower boundary

39 5 non-dimensional parameters 3 Parameters in the equations! Dry Rayleigh number! Moist Rayleigh number! Prandtl number And 2 are hidden in the buoyancy term:

40 Five-dimensional parameter space Dry Rayleigh number Moist Rayleigh number Prandtl number Condensation in Saturated Ascent Surface Saturation Deficit

41 2 Limiting cases: Unsaturated atmosphere: if the whole atmosphere is unsaturated - i.e when M 0 " D 0 " N s 2 H # 0 and M H " D H " N s 2 H # 0 This problem is equivalent to the Rayleigh-Benard problem with Ra = Ra D and Pr = Pr Saturated atmosphere: if the whole atmosphere is unsaturated - i.e when M 0 " D 0 # 0 and M H " D H # 0 This problem is equivalent to the Rayleigh-Benard problem with Ra = Ra M and Pr = Pr

42 Atmospheric moist convection

43 Case 1: stratocumulus regime

44 Buoyancy flux and cloud base "Clouds" "Cloud base"

45 Buoyancy flux and cloud base In Boussinesq system, generation of kinetic energy is given by the Integral of buoyancy flux: " t KE = # wbdz $ D In classic Rayleigh-Benard convection, we have " t B + " z wb = #" zz B so that the buoyancy flux is constant with height Enhanced buoyancy flux dry But it is not the case for stratiform convection

46 Buoyancy flux and cloud base B = max(m,d " N s 2 z) (not an adiabatic invariant anymore!) " t D + " z wd = #" zz D " t M + " z wm = #" zz M In unsaturated regions: wb = wd In saturated regions: wb = wm Cloud base of stratocumulus and water deficit increase

47 Mixing line Over long time-scale, the solution collapse toward a mixing line, I.e. the dry and moist buoyancy can be expressed as a function of a mixing fraction M = "M H +(1# ")M 0 D = "D H +(1# ")D 0

48 t T f = 2.5 t T f = 5.0 t T f = 7.5 t T f =10.0 t T f =12.5 t T f = 62.5

49 However, rate of collapse decrease with Rayleigh number

50 But the steady state distribution depends also on the Rayleigh number

51 Which has direct impact on the cloudbase/cloud fraction

52 Case 2: stratocumulus to cumulus

53 Changing cloud fraction Variation of M H and thus of CSA and Ra M

54 Stratocumulus to Cumulus CSA=0.17 CSA=0.35 CSA=0.53 Transition to Cumulus Cloud layer breaks up and disappears

55 Case 3: Conditional instability

56 Conditional instability

57 Conditional instability Stable stratification for unsaturated parcels But unstable stratification for saturated parcels Ra D < 0 Ra M > 0

58 Conditional instability Stable stratification for unsaturated parcels But unstable stratification for saturated parcels Ra D < 0 Ra M > 0

59 z Buoyancy in a unsaturated adiabatic ascent B = D " N s 2 z Mean buoyancy profile B(z) Buoyancy in a saturated adiabatic ascent B = M Buoyancy

60 Cloud water Vertical velocity

61 And convection becomes inceasingly intermittent at high Ra.

62 Simulations evolve toward a localized turbulent patch at high aspect ratio

63 Conclusion Atmosphere can be viewed as a heat engine that generates kinetic energy by transporting energy from warm to cold. Relative humidity is a key factor in determining how much work is produced by atmospheric circulation. This can be captured by a simple steam cycle. This behavior is related to the non-linear state equation associated with the different behavior between staurated and unsaturated air. A simplified piecewise linear equation of state can capture the main effect of phase transition on dynamics, and use to simulat idealized moist convection.

64 Latent and sensible heat flux: Q lat = L"r T Bowen ratio: Efficieny " = Q sen = T in "S + (g in # L)"r T B = Q sen Q lat B 1+ B " + 1 c 1+ B " H = T in # T out T in B R v T out L ln H in H out The efficiency of a mixed cycle depends on both relative humidity and Bowen ratio.

65 ( " 0 w # B # $ % ad " 0 w # S # & 'g v * ) 'z + G + -", 0 w # r # T Z Z Z Z CB CB CB CB T Z " ad "g v = "R v T lnh Z $ " #g v & % #z + G ' ) ( w " r " T w " S " w " B "

66 II. Buoyancy flux We consider now a convective layer. The generation of kinetic energy is given by the vertical integral of the buoyancy flux: dke dt B is the buoyancy = $ " 0 w # B # dz B(S,r T,z) = G " p #" 0 (z) " 0 (z)

67 A. Stratocumulus convection Linearize the buoyancy flux & " 0 w # B # $ %B ) ( + '%S * r T,p & " 0 w # S # + ( %B '%r T The partial derivatives can be rewritten using the Maxwell relationships: ) + * S,p " 0 w # r # T # % $ "B "S & ( ' r T,p # = ) 0 G "* & % ( $ "S ' r T,p # = ) 0 G "T & % ( $ "p ' r T,S # = + "T & % ( $ "z ' r T,S =, ad # "B & % ( $ "r T ' S,p # = ) 0 G "* d % $ "r T & ( ' S,p # + G = ) 0 G "g v % $ "p & ( ' r T,S # + G = + "g v % $ "z & ( ' r T,S + G

68 After integration: dke dt $z % 0 = " 0 w # B # dz = (T in & T out )" 0 w # S # + ( g v,in & g v,out )" 0 w # r # T & G$z"0 w # r # T Work done by a Carnot cycle water vapor penalty geopotential energy gained by water Work done by a mixed cycle

69 Thermodynamics of moist air Part 2: Global cosiderations Olivier Pauluis (NYU) Porquerolles - Sept. 2010

70 Outline Introduction: the nature of the problem Entropy of moist air Heat engines and water vapor. Equation of state of moist air Idealized moist Rayleigh-Benard convection!mixing in stratocumulus Buoyancy and entropy fluxes Isentropic circulation

71 Cold and dry Warm & moist height height X (z) w " X "(z) Adiabatic invariants (e.g. entropy, total water, equivalent potential temperature, M and D ) are well-mixed. And their flux is constant.

72 For non-invariants (e.g. liquid water, temperature, buoyancy) height height X (z) Y (z) Y (z)= F(X 1 (z), X 2 (z), p(z))!the vertical derivative changes abruptly at the cloud base

73 For non-invariants (e.g. liquid water, temperature, buoyancy) height height w " X "(z) w " Y "(z) w " Y " = #Y w " X " #Y 1 + w " X " 2 #X 1 #X 2!The vertical flux is discontinuous at cloud base!

74 Excess evaporation at cloud top Excess condensation at cloud base height height Condensation - evaporation w " Y "(z) Excess condensation at cloud base acts as a source (or sink) for non-conserved quantities (liquid water, buoyancy, etc )

75 What are your favorite invariants?

76 I understand it It makes it easy to compute buoyancy or density (e.g. M and D) It can be measured It can be conserved under specific diabatic transformation I can easily write its tendency equation under general (non-conservative) conditions

77 Buoyancy depends on location of cloud base " H " C = #T T ~ 0.16" C Efficiency of steam cycle depends on relative humidity Unsaturated cycle H out =1 H in =1 r T,in

78 Buoyancy flux In the Boussinesq approximation, the generation of kinetic energy is given by the vertical integral of the buoyancy flux: dke dt B is the buoyancy = $ " 0 w # B # dz B(S,r T,z) = G "(S,r T, p 0 (z)) #" 0 (z) " 0 (z) = G$ 0 (z) "(S,r T, p 0 (z)) #" 0 (z) [ ]

79 Linearize the buoyancy flux with & " 0 w # B # $ %B ) ( + '%S * r T,p & " 0 w # S # + ( %B '%r T The partial derivatives can be rewritten as # % $ "B "S & ( ' # "B & % ( $ "r T ' r T,p S,p # = ) 0 G "* & % ( $ "S ' # = ) 0 G "* & % ( $ "r T ' r T,p S,p ) + * S,p = ) G # 0 "* & % ( 1+ r T 0 $ "S ' = ) G # 0 "* d & % ( 1+ r T 0 $ "p ' " 0 w # r # T B(S,r T,z) = G" 0 (z) #(S,r T, p 0 (z)) $# 0 (z) [ ] r T,p r T,S + G 1+ r T 0 " d = " 1+ r T : specific volume per unit mass of DRY AIR

80 Maxwell relationships: TdS = dh "# d dp " gdr $ " d = #H ' $ & ), T = #H ' $ & ) and g = #H ' & ) % #p ( % S,r T #S ( p,r T %#r T ( S,p $ "# & d % "S $ & % "# d "r T ' $ ) = " 2 H ' $ & ) = "T ' & ) ( p,r T %"S"p( "p r T % ( ' ) ( p,s $ = " 2 H ' $ & ) = "g ' & ) %"r T "p( "p r T % ( S,r T S,r T

81 Maxwell relationships: # % $ "B "S & ( ' r T,p = ) 0G #"* & % ( 1+ r T 0 $ "S ' = 1 1+ r T 0 + ad r T,p = ) 0G #"T & % ( 1+ r T 0 $ "p ' r T,S # "B & % ( $ "r T ' S,p = ) G # 0 "g& % ( 1+ r T 0 $ "p' = ) G # 0 "* d & % ( 1+ r T 0 $ "p ' r T,S = + 1,# "g&.% ( 1+ r T 0 - $ "z ' + r T,S r T,S 1 1+ r T 0 G / + G1 0 + G 1+ r T 0

82 " B = 1 1+ r T0 # ad " S $ % + B " = # ad S " $ dg & ' dz + G ( ) * q " T 1 % dg 1+ r T 0 & ' dz + G ( ) * r " T " S : Total entropy perturbation per unit mass of dry air " " S = " S 1+ r T 0 : Total entropy perturbation per unit mass of moist air q T " = r T " 1+ r T 0 : Specific humidity perturbation (and not mixing ratio...) % w " B " = # ad w " S " $ dg & ' dz + G ( ) * w " q " T

83 After integration: dke dt $z % 0 = " 0 w # B # dz = (T in & T out )" 0 w # S # + ( g v,in & g v,out )" 0 w # q # T & G$z"0 w # q # T Work done by a Carnot cycle water vapor penalty geopotential energy gained by water Work done by a mixed Carnot-steam cycle

84 ( " 0 w # B # $ % ad " 0 w # S # & 'g v * ) 'z + G + -", 0 w # r # T Z Z Z Z CB CB CB CB T Z " ad "g v = "R v T lnh Z $ " #g v & % #z + G ' ) ( w " r " T w " S " w " B "

85 Global considerations

86 However, Columbus did not sail directly west from Spain. Rather, he went South to the Canaries islands. The prevailing winds in the Canaries blow from the East. This is what Columbus needed in order to sail West.

87 These easterly winds are present through over all subtropical oceans. They are now known as the Trade winds for the key role they played in the Transatlantic commerce. I think that the causes of the General Trade- Winds have not been fully explained by those who have wrote on the subject George Hadley (1735)

88 George Hadley (1735) Hadley explanation for the Trade winds: There is a global circulation, with air rising at the Equator, and subsiding over the Poles Conservation of (angular) momentum implies that air moving toward the equator acquires a easterly component.

89 Clouds and the Hadley circulation

90 Ferrel was the first to identify the role of rotation in atmospheric motions (the Coriolis effect). Also, using Maury s data, he identifies a reverse circulation associated with the westerly winds in the midlatitudes. Ferrel (1836)

91 Bjerknes (1921) Midlatitudes are however dominated by storms (aka synoptic scale eddies ). Understanding the interplay between the storm and global circulation is a key issue in modern meteorology.

92

93 The general circulation of the atmosphere The Earth s atmosphere receives most of its energy at the surface and in the Tropics. But it emits infra-red radiation rather uniformly. The circulation acts to transport energy from equator to Pole. Other important constraint related to angular momentum balance. latitude

94 Eulerian averaging Temperature (K) Zonal wind (m/s) Pressure(mb) latitude latitude Eulerian averaging: take the time and zonal average at fixed latitude and pressure (or height) F (", p) = 1 2#T T % 0 2# % 0 F($,", p,t)d$dt

95 Eulerian mean circulation meridional wind (m/s) Stream function wind (kg/s) Pressure(mb) latitude latitude Average velocity at constant pressure or height: v (", p) = 1 2#T The circulation can be diagnosed by computing the stream function: T % 0 p surf % p 2# v($,", p,t)d$dt dp "(#, p) = 2$v acos# g % 0

96 Ferrel cells Hadley cells Polar cell 45S Equator 45N Polar cell Eulerian-mean circulation exhibits the classic three-cell structure. But the Ferrel cell is a reverse circulation that transports energy toward the equator.

97 Circulation in isentropic coordinates (Dutton, Johnson, Held and Schneider) Rather than averaging in eulerian coordinates, one can average the circulation at constant value of the potential temperature!: F " (#,") = 1 2$T # F(%,#,",t)d%dt Motivation: the potential temperature is related to entropy and is conserved for reversible adiabatic transpormation in the absence of phase transition. T & 0 2$ & 0 " # ($,#) = ' 2% & # v # acos$ d# 0

98 Stream function on potential temperature surfaces " latitude The three cell structure disappears: there is a single Equator-to- Pole cell in each hemisphere. The circulation is direct (high entropy air flows poleward, low entropy air flow equatorward).

99 Why the circulation in eulerian and isentropic coordinates are in the opposite direction? v < 0 v > 0 longitude In the midlatitudes, the flow is highly turbulent: the meridional velocity alternates between poleward and equatorward at all levels.

100 In the stormtracks: Eulerian-mean circulation v p < 0 at low pressure v < 0 v > 0 Isobaric surface v p > 0 at high pressure longitude In the midlatitudes, the flow is highly turbulent: the meridional velocity alternates between poleward and equatorward at all levels. This idealized eddies is associated with a poleward flow at high pressure/low level, and equatorward flow at high level

101 In the stormtracks: Isentropic circulation " # v > 0 at high # v < 0 Cold Warm v > 0 Potential temperature surface " # v < 0 at low # longitude Thickness variations are such that the upper isentropic layer encompass larger fraction of the poleward flow. Such pattern also corresponds to a net poleward energy mass transport.

102 Isentropic flow and eddy mass transport " # v > 0 at high # v < 0 v > 0 Potential temperature surface " # v < 0 at low # longitude The mass flux on isentropic surfaces can be written as: Total mass flux " # v # = " # # v # + "# 'v' # Transport by mean circulation Eddy transport The mass transport by the eddies is in the opposite direction to the mean wind.

103 Parcel trajectories (eulerian) mean velocity Equator Midlatitudes Pole The potential temperature is more or less conserved, and the the circulation on isentropes do a better job at capturing the mean lagrangian trajectories of air parcels. In the midlatitudes, the parcels move on average in the opposite direction to the (eulerian) mean velocity.

104 What about moisture? How to define an isentropic surface in a moist atmosphere? Previous studies have used the potential temperature! as definition of entropy. The equivalent potential temperature! e is conserved for reversible adiabatic transformation, even when phase transition take place. Why not use! e then? Does it matter?

105 'Dry isentropes': " l = cst 'Moist isentropes': " e = cst Pressure(mb) " l " e latitude latitude Different definitions imply different isentropic surfaces.! e includes a contribution from the latent heat content, and has often a minimum in the middle of the atmosphere.

106 Stream function on dry isentropes Stream function on moist isentropes " l " e latitude latitude Same single cell structure But amplitude of the circulation differs!

107 DJF DJF " l " e JJA JJA " l " e latitude latitude

108 Mass transport 0n moist isentropes Annual mean In the Tropics: Transport on dry isentropes is larger Mass transport on dry isentropes JJA Mass transport (kg s -1 ) DJF latitude In the Midlatitudes: Transport on moist isentropes is larger

109 Why the difference in mass transport? Both! e and! l are conserved along adiabatic trajectories. Rather than isentropic surfaces, we can think of having a set of isentropic filaments -i.e. lines of constant value for both! e and! l. The poleward mass transport along such isentropic filaments is defined as: M(" l 0," e 0,#) = 1 2$T " e ( ( ( %v&(" l '" l 0 )&(" e '" e 0 ) d) dp g dt " l

110 Mass flux distribution at 40N - DJF Convection/precipitation Evaporation " e Radiative Cooling " e #" l ~ L C p q t " e < " l : impossible " l

111 In the Midlatitudes: DJF DJF " l " e 40N 40N Circulation on moist isentropes is larger than that on dry isentropes.

112 Mass flux distribution at 40N - DJF Poleward flow " e Equatorward flow " l Isentropic filaments that intercept the surface

113 Mass flux and stream function at 40N Stream function on dry isentropes: $ & 0 % & 0 " # l ($) = M(# l,# e )d# e d# l Stream function on moist isentropes: $ & 0 % & 0 " # e ($) = M(# l,# e )d# l d# e " e " e " l " l

114 Mass transport at 40N - DJF " e Isentropic filaments that intersect the surface Portion of the mass transport included in the circulation on moist isentropes but not to that on dry isentropes " l The additional mass transport on moist isentropes takes place filaments near the Earth s surface. The equivalent potential temperature corresponds to upper tropospheric value of the potential temperature. This corresponds to a poleward flow of warm, moist air near the surface that is ready to rise into the upper troposphere.

115 Why the circulation on moist isentropes is larger? Dry isentrope v < 0 v > 0 longitude

116 In the stormtracks: Circulation on moist isentropes Dry isentrope v < 0 v > 0 Moist isentrope Moist air moving poleward longitude Moist isentropes found in the upper troposphere also intersects the Earth s surface. Such situation corresponds to a poleward flow of warm, moist air near the surface.

117 Hadley circulation: DJF DJF " l " e 10N 10N Circulation on dry isentropes is larger than circulation on moist isentropes.

118 Mass flux distribution at 10N - DJF Equatorward flow Poleward flow

119 Mass flux distribution at 10N - DJF The poleward and equaotrward flows overlap on moist isentropes But they are well separated on dry isentropes

120 Dry isentropes Moist isentropes Pressure(mb) 0N 40N 40N latitude 0N latitude In the equatorial regions, the potential temperature increases uniformly with height, but not the equivalent potential temperature. The equivalent potenital temperature in equatorward and poleward flows of the Hadley cell are close to each other.

121 Quadrant II Quadrant I " e 305K Quadrant III 305K " l

122 Annual mean Quadrant I: Upper tropospheric flow Mass flux JJA DJF latitude Tropics: upper tropospheric and moist air go in opposite direction Quadrant II: Low level warm moist air Quadrant III: Low level dry air Midlatitudes: upper tropospheric and moist air go in the same direction

123 Circulation on dry isentropes The global circulation has two poleward components in the midlatitudes: an upper tropospheric branch of high! e -! l ; Moist branch: additional mass flow on moist isentropes an a lower branch of warm, most air with high! e - low! l, which ascent into the upper troposphere within the stormtracks. Mass transport is comparable in each branch.

124 Entropy transport (JK -1 s -1 ) Entropy transport : F S = Annual mean latitude p surf " 0 vs dp g Moist entropy transport Dry entropy transport Water transport F Sm = F sl + L v T 0 F q

125 Annual mean "#(K) "#(K) "#(K) JJA DJF latitude Gross Stratification: "# = F # "$ # ~Entropy (or potential temperature) per unit mass transported

126 In the tropics: "# e << "# l! l Vertical stratification of humidity " q z is in opposite direction to that for potential temperature " z # z!! e

127 In the midlatitudes: "# e ~ "# l In order to have enhanced mass transport but same stratification, the additional mass transport must take place at! l corresponding to lower tropospheric value, but! e corresponding to upper troposphere. It implies that horizontal variations of! e ( and of water vapor ) are comparable to its vertical variations.

128 Conclusions Heat engine and buoyancy flux computations yields the same answer. Entropy and buoyancy are closely tied: B " #$ S " ad Global mean circulation depends on the coordinate system - its is true also for any isentropic circulation. In the midlatitudes, the mass transport on moist isentropes is approximately twice as large as that on dry isentropes. The additional mass transport corresponds to a low-level, poleward flow of warm, moist air that ascends into the upper troposphere within the stormtracks.

129 Further reading Moist thermodynamics and convection: Iribarne JV, Godson WL, 1973: Atmospheric thermodynamics Emanuel, K.A., 1994: Atmospheric Convection. Oxford University Press, 580 pp. Stevens, B., 2005: Atmospheric moist convection, Annu. Rev. Earth Planet. Sci : Idealized moist Rayleigh-Benard convection: Pauluis 0. and J. Schumacher, 2010: Idealized moist Rayleigh-Benard convection with a piecewise linear equation of state. Comm. Math. Sci.8, Pauluis, O., 2008: Thermodynamic Consistency of the Anelastic Approximation for a Moist Atmosphere. J. Atmos. Sci. 65, Atmospheric heat engines: Pauluis, 0., 2010: Water vapor and mechanical work: a comparison of Carnot and steam cycles. To be published in J. Atmos. Sci. Goody, R., 2003: On the mechanical efficiency of deep, tropical convection. J. Atmos. Sci., 50,

130 Pauluis, O. and I.M. Held, 2002: Entropy budget of an atmosphere in radiative-convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci., 59,! Pauluis, O., V. Balaji and I.M. Held, 2000: Frictional dissipation in a precipitating atmosphere. J. Atmos. Sci., 57, Emanuel, K. A. and M. Bister, 1996: Moist convective velocity and buoyancy scales. J. Atmos. Sci., 53, Renno, N. and A. Ingersoll, 1996: Natural convection as a heat engine: A theory for CAPE. Global Isentropic Circulation: Pauluis, O., A. Czaja and R. Korty, 2010: The global atmospheric circulation in moist isentropic coordinates. To be published in J. Climate. Pauluis, O., A. Czaja and R. Korty, 2008: The global atmospheric circulation on moist isentropes. Science, 321, Held, I. M. and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, Johnson, D. R., 1989: The forcing and maintenance of global monsoonal circulations: An isen- tropic analysis. Advances in Geophysics, 31,