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 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.
16 θ≈θLvLpd exp (-L r /c T)
An unapproximated expression for θl is:
χ χ−γ⎡ ⎤ ⎛⎞pLrovL⎛⎞⎛⎞rrLL − θ≡L T1⎜⎟⎜⎟⎜⎟ − 1exp − ⎢ ⎥ ⎝⎠prr(crc)T⎝⎠⎝⎠ε+TT⎣⎢ pdTpv + ⎦⎥
where χ≡afRrRcdTv+ / ch pdTpv+ rc
γ ≡ rRTv/ ch c pdTpv+ rc
The liquid-water virtual potential temperature
The liquid-water virtual potential temperature θlv is defined by:
χ χ−11 χ− ⎡ ⎤ ⎛⎞pLro ⎛⎞⎛⎞⎛⎞rrLL r L − vL θ≡LvT1 v ⎜⎟⎜⎟⎜⎟⎜⎟ − 1 − 1exp − ⎢ ⎥ ⎝⎠p1rrr(crc)T⎝⎠⎝⎠⎝⎠+ε+TTT⎣⎢ pdTpv +⎦⎥
¾ Note that θLv = θv when rL = 0
17 Some notes
¾ θe, θL, and θLv are conserved in reversible adiabatic processes involving changes in state of unsaturated or cloudy air.
¾ θe, θL, and θLv are not functions of state - they depend on p, T, r and rL ¾ → Curves representing reversible, adiabatic processes cannot be plotted in an aerological diagram
¾ In a saturated process, r = r*(p, T)
The pseudo-equivalent potential temperature
¾ Approximation: neglect the liquid water content (set rL = 0).
¾ This makes the approximate forms of θe, θL, and θLv functions of state which can be plotted in an aerological diagram. ¾ Adiabatic processes where the liquid water is ignored are called pseudo-adiabatic processes. They are not reversible!
* ¾ The formula θe*≈θexp (Lvr /cpdT) is an approximation for the pseudo-equivalent potential temperature θep . A more accurate formula is: 0./(.) 2854 1− 0 28r F po I L F3376 IO θep = TG J expMrr (1081+− . )G 254 . JP H p K N H TLCL KQ
Temperature at the LCL
18 The Lifting Condensation Level Temperature
¾ TLCL is given accurately (within 0.1°C) by the empirical formula:
−1 1 ln(TT / ) T = L + K d O + 56 LCL M P NTd − 56 800 Q
TK and Td in degrees K
[Formula due to Bolton, MWR, 1980]
¾ If an air parcel is lifted pseudo-adiabatically to the high atmosphere until all the water vapour has condensed out,
θep = θ.
¾ The pseudo-equivalent potential temperature is the potential temperature that an air parcel would attain if raised pseudo-adiabatically to a level at which all the water vapour were condensed out.
¾ The isopleths of θep can be plotted on an aerological diagram. These are sometimes labelled by their temperature at 1000 mb which is called the wet-bulb
potential temperature, θw.
19 The adiabatic lapse rate
¾ Lift a parcel of unsaturated air adiabatically
¾ it expands and cools, conserving its θv
¾ its T decreases with height at the dry adiabatic lapse rate, Γd : ⎛⎞dT g 1+ r Γ=−d ⎜⎟ = ⎝⎠dzdq= 0 cpdpvpd 1+ r(c / c )
¾ Note that r is conserved, but r* decreases because e*(T) decreases more rapidly than p. ¾ Saturation occurs at the lifting condensation level (LCL) * when T = TLCL and r = r (TLCL, p).
Above the LCL, the rate of which its temperature falls, Γm, is less than Γd because condensation releases latent heat.
For reversible ascent:
⎛⎞dT g 1r+ T Γ≡−m ⎜⎟ = × dz c c ⎝⎠s pd 1r+ pv cpd ⎡ Lr ⎤ ⎢ 1+ v ⎥ ⎢ RTd ⎥ 2 ⎢ cL L(1r/)rv +ε ⎥ ⎢1r++L 2 ⎥ ⎣⎢ crcRT(crc)pd++ pv v pd pv ⎦⎥
When rT is small, the ratio Γm/Γd is only slightly less than unity, but when the atmosphere is very moist, it may be appreciably less than unity.
20 The moist static energy and related quantities
The first law gives
dq = dk −αddp
where dq is expressed per unit mass of dry air.
Adiabatic process (dq = 0) → dk −αddp = 0
αd = α(1 + rT) For a hydrostatic pressure change, αdp = −gdz.
Under these conditions:
dh≡ (cpd++++= r T c L )dT d(L v r) (1 r T )gdz 0
Some notes
If rT is conserved, we can integrate
hc=+()()tan.pdr TcT l+ L vr + 1+ r T gz= cons t
¾ The quantity h is called the moist static energy. ¾ h is conserved for adiabatic, saturated or unsaturated transformations in which mass is conserved and in which the pressure change is strictly hydrostatic. ¾ h is a measure of the total energy: (internal + latent + potential)
21 The dry static energy
¾ Define the dry static energy, hd.
¾ Put rT = r =>
h(crc)T(1r)gz.dpdL=+ ++
¾ This is conserved in hydrostatic unsaturated transformations.
¾ h and hd are very closely related to θe and θ.
The (virtual) liquid water static energy
¾ Define two forms of static energy related to θL and θLv. ¾ These are the liquid water static energy:
hcwpdTpvvLT= ()()+ r cTL− r + 1+ r gz and the virtual liquid water static energy
⎛⎞ε+rT LrvL hcLv=−+ pd ⎜⎟ Tρ gz ⎝⎠ε+rrTL − 1r + T
¾ hLv is almost precisely conserved following slow adiabatic displacements.
¾ If rL = 0, hLv = cpdTv + gz (just as θL reduces to θ).
22 Vertical profiles of dry conserved variables
θ , θv Dry static energy z (km) z z (km) z
deg K 105 J/kg
Vertical profiles of moist conserved variables
θ, pseudo θe Moist static energy z (km) z z (km) z
deg K 105 J/kg
23 The stability of the atmosphere
¾ Consider the vertical displacement of an air parcel from its equilibrium position ¾ Calculate the buoyancy force at its new position ¾ Consider first an infinitesimal displacement ξ; later we consider finite-amplitude displacements ¾ Parcel motion is governed by the vertical momentum equation d2ξ = b dt2
⎛⎞ρ−ρ⎛⎞ α−α b(ξ=− ) gpa = g p a is the buoyancy force ⎜⎟⎜⎟ ⎝⎠ραpa⎝⎠ per unit mass
Newton's law for an air parcel: d2ξ buoyancy force = dt2 unit mass
buoyancy force∂ b :b()ξ =ξ+ unit mass∂ z z0=
db2ξ∂ 2 − ξ=0 dt∂ z z0=
The motion equation for small displacements is:
d2ξ ∂b +=N2ξ 0 where N2 =− dt2 ∂z
24 The motion equation for small displacements is:
d2ξ +=N2ξ 0 dt2 ∂b where N2 =− ∂z
For an unsaturated displacement, θvp is conserved and we can write
⎛⎞⎛⎞TTvp− vaθ−θ vp va b(ξ= ) g⎜⎟⎜⎟ = g ⎝⎠⎝⎠Tvaθ va
Since θvp = constant ≅θva,
2 ∂bgθvp ∂θva ∂θ va Ng=− =2 ≅ ∂zzzθ∂va θ∂ va
Stability criteria
¾ Parcel displacement is:
• stable if ∂θva/∂z> 0
• unstable if ∂θva/∂z < 0
• neutrally-stable if ∂θva/∂z = 0
¾ A layer of air is stable, unstable, or neutrally-stable if these criteria are satisfied in the layer.
25 ¾ In a saturated (cloudy) layer of air, the appropriate conserved quantity is the moist entropy s (or the
equivalent potential temperature, θe) ¾ Must use the density temperature to calculate b.
¾ Replace αp in b by the moist entropy, s. ¾ In this case
2 1s⎡ ∂ ∂rT ⎤ NclnTg=Γ−Γ+⎢ mLm()⎥ 1r+∂T ⎣ z ∂ z⎦
¾ A layer of cloudy air is stable to infinitesimal parcel
displacements if s (or θe) increases upwards and the total water (rT) decreases upwards. It is unstable if θe decreases upwards and rT increases upwards.
Some notes
¾ The stability criterion does not tell us anything about the finite-amplitude instability of an unsaturated layer of air that leads to clouds. ¾ Parcel method okay, but must consider finite displacements of parcels originating from the unsaturated layer.
26 Potential Instability
¾ A layer of air may be stable if it remains dry, but unstable if lifted sufficiently to become saturated. ¾ Such a layer is referred to as potentially unstable.
¾ The criterion for instability is that dθe/dz < 0.
unstable lift saturated/cloudy
stable unsaturated
Conditional Instability
¾ The typical situation is that in which a displacement is stable provided the parcel remains unsaturated, but which ultimately becomes unstable if saturation occurs. ¾ This situation is referred to as conditional instability. ¾ To check for conditional instability, we examine the buoyancy of an initially-unsaturated parcel as a function of height as the parcel is lifted through the troposphere, assuming some thermodynamic process (e.g. reversible moist adiabatic ascent, or pseudo-adiabatic ascent). ¾ If there is some height at which the buoyancy is positive, we say that the displacement is conditionally-unstable.
27 ¾ If some parcels in an unsaturated atmosphere are conditionally-unstable, we say that the atmosphere is conditionally-unstable. ¾ Conditional instability is the mechanism responsible for the formation of deep cumulus clouds. ¾ Whether or not the instability is released depends on whether or not the parcel is lifted high enough. ¾ Put another way, the release of conditional instability requires a finite-amplitude trigger. ¾ The conventional way to investigate the presence of conditional instability is through the use of an aerological diagram.
200
LNB
300 )
b dry pseudo-
m (
adiabat adiabat
e
r
u
s
s e
r 500 p 10 g/kg
700 LFC 850 LCL 1000 20 oC 30 oC
28 Positive and Negative Area Convective Inhibition (CIN)
The positive area (PA)
p LFC 1 2 1 2 PA=−=2 uLNB2 u LFCz ch T vp − T va Rd dln p pLNB The negative area (NA) or convective inhibition (CIN)
p NA== CINparcel T − T R dln p ∫ ( vp va) d pLFC
Convective Available Potential Energy - CAPE
The convective available potential energy or CAPE is the net amount of energy that can be released by lifting the parcel from its original level to its LNB.
CAPE = PA − NA
We can define also the downdraught convective available potential energy (DCAPE)
po DCAPEid= zp R()ln Tρρap− T d p i
The integrated CAPE (ICAPE) is the vertical mass-weighted integral of CAPE for all parcels with CAPE in a column.
29 Pseudo θe Reversible θe z (km) z z (km) z
ms−1 ms−1
Buoyancy Liquid water z (km) z z (km) z
zL km zL km
30 reversible with ice
reversible Height (km) Height
pseudo-adiabatic
Buoyancy (oC)
Downdraught convective available potential energy (DCAPE)
po DCAPEid= zp R()ln Tρρap− T d p i
o Td T qw = 20 C
700 LCL 800 mb, T = 12.3oC Tw Td 850 T r* = 6 g/kg Tρp DCAPE ρa
1000
31 Buoyancy and θe
z Lifted parcel Environment ()ρ −ρ bg=− p o ρo
Tvp ρp To(z), ro(z), p(z), ρo(z)
(T− T ) or bg= vp vo To
Lifted parcel Environment (T− T ) bg= ρp vo To
Tρp ρp To(z), ro(z), p(z), ρo(z)
Below the LCL (Tρp= Tvp)
sgn (b) = sgn { Tp(1 + εrp) − To(1 + εro) = Tp − To + ε[Tprp − Toro]}
ε = 0.61 At the LCL (Tρp= Tvp)
* sgn (b) = sgn [Tp(1 + εr (p,Tp)) − To(1 + εro)]
* * * =sgn[Tp(1 + εr (p,Tp) − To(1 + εr (p,To)) + εTo(r (p,To) − ro)]
32 * * * sgn (b) = sgn [Tp(1 + εr (p,Tp) − To(1 + εr (p,To)) + εTo(r (p,To) − ro)]
Since T is a monotonic function of θ , ** v e b ∝ ()θ−θ+Δep eo
z small
parcel saturated * θθeo eo LFC LCL
moist adiabat (T θ r ) L L L (TL θe* r*) (Td rT) (rT = r* + rL) E F G p2
P = pLCL –p2 > 0
LCL or saturation level (TLCL , pLCL) D P = p –p < 0 r* - isopleth LCL 2 p1 B C A (T r) d (Tw θε rw) (T θ r) dry adiabat After Betts
33 The saturation point
¾ The point (TLCL , pLCL) is referred to as the saturation point. ¾ The saturation point of an air parcel is a conserved quantity in the absence of diabatic or mixing processes. ¾ One can plot the saturation points of parcels from a radiosonde sounding on an aerological diagram. ¾ The state of a parcel of air is characterized by its saturation point and the departure of the actual pressure from the saturation pressure, i.e. P = pLCL –p. ¾ If P > 0, the parcel is cloudy, if P < 0 it is unsaturated.
The End
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