lecture 4: thermodynamics review, part 2 · pdf filedry atmospheric thermodynamics moist...

Dry atmospheric thermodynamicsMoist atmospheric thermodynamics

Radiative–convective equilibrium

Lecture 4:Thermodynamics Review, Part 2

Jonathon S. Wright

[email protected]

14 March 2017



Dry atmospheric thermodynamicsStabilityBuoyancyDry convection and stable inversions

Moist atmospheric thermodynamicsWater vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation




90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800

700

600

500

400

300

200

150

100

75

50

Pre

ssur

e [h

Pa]

200

210

210

220

220

220

230

230

240

240

250

250

260

260270

280290

Zonal mean temperature

190

200

210

220

230

240

250

260

270

280

290

300

Tem

pera

ture

[K]

data from JRA-55

Small thermal inertiaI specific heat of the atmosphere one

fourth of specific heat for the ocean

I mass three orders of magnitude less

I adjusts rapidly to changes insurface temperature

Thermodynamic importanceI radiative and convective ventilation

of the climate system

I energy transport and redistribution

Water vaporI specific heat and gas constant

increase with increasing humidity

http://jra.kishou.go.jp/



StabilityBuoyancyDry convection and stable inversions

200 225 250 275 300 325 350 375 400Temperature [K]

100

200

300

400

500

600

700

800

900

1000

Pre

ssur

e [h

Pa]

Temperature (T)Potential temperature (θ)

data from JRA-55

(pold, θold)

(pnew, θold) (pnew, θnew)T = θ(p0p

)Rd/cp

ρparcel =pnew

Rdθold

(p0pnew

)Rd/cp

ρenv = pnew

Rdθnew(

p0pnew

)Rd/cp

StabilityI the stability of a dry, well-mixed

atmosphere depends on the verticalgradient of potential temperature





N2 = gθ∂θ∂z

Brunt–Vaisala frequency

N =

√(gθ∂θ∂z

)Buoyancy

I the Brunt–Vaisala frequency for theatmosphere can be defined in termsof potential temperature

I buoyancy oscillations exist if θincreases with height (stable)

I convective instability results if θdecreases with height (unstable)




T1 T2

Temperature

z1

z2

Hei

ght

T(z)

θ1 θ2

Potential temperature

z1

z2

θ(z)

Dry convectionI solar radiation heats the surface,

which then warms the atmosphereabove it (sensible heating)

I as the air warms, its potentialtemperature increases and itbecomes less dense than the airabove it

I this air rises until its density equalsthat of its environment

I the depth and intensity of thisconvection depend on the lapserate and the surface warming




Temperature

Hei

ght

T(z)

Temperature

Hei

ght

T(z)

nighttime cooling

daytime T profile

adiabatic descent

adiabatic ascent

inversion

Nighttime inversionsI surface radiative cooling can reverse

the sensible heat flux so that it isdirected toward the surface

I the air immediately above thesurface cools, so that it is muchdenser than the air above it

I this process creates a stableinversion layer at the surface

Subtropical inversionsI air warmed adiabatically during

descent from the upper tropospheremay be warmer than air at the topof the boundary layer



Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800700

600

500

400

300

200

150

100

Pre

ssur

e [h

Pa]

2

468101214 16

Zonal mean specific humidity

0

2

4

6

8

10

12

14

16

18

20

Spe

cific

hum

idity

[g k

g−1 ]

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800700

600

500

400

300

200

150

100

Pre

ssur

e [h

Pa]

1010

2020

30 30

30

30

40

40

40

50

50

50

60

6060

70

70

7070

80

808080 90

Zonal mean relative humidity

0

10

20

30

40

50

60

70

80

90

100

Rel

ativ

e hu

mid

ity [%

]

data from JRA-55

ε =RdRv

≈ 0.622

Water vaporI vapor pressure: e = ρvRvT

I saturation vapor pressure:

e∗ ≈ exp

(53.68 − 6743.77

T− 4.85 lnT

)I mass mixing ratio:

r =ρvρd

= εe

p− e

I specific humidity:

q =ρv

ρv + ρd= ε

e

p− (1 − ε)e=

r

1 + r

I relative humidity: RH = e/e∗





80 60 40 20 0 20 40Temperature [°C]

10-2

10-1

100

101

102

103

104

105

Vap

or p

ress

ure

[Pa] Triple point: 0.01°C, 611.2 Pa

water vapor

liquid

ice

supercooledliquid

SaturationClausius–Clapeyron equation:

de∗

dT=

1

T

Lv

ρ−1v − ρ−1

c

≈ LvRvT 2

e∗

Empirical approximations:

e∗ ≈ exp

(53.68 − 6743.77

T− 4.85 lnT

)e# ≈ exp

(23.33 − 6111.73

T− 0.15 lnT

)

Relative humidity can be defined with respect to liquid water (RH = e/e∗) or ice (RHi = e/e#)




Water vapor modifies the specific heatThe specific heat of water vapor (cpv ≈ 1870 J K−1 kg−1) is larger than the specific heat of dry air(cpd ≈ 1004 J K−1 kg−1). The specific heat is the energy required to change the temperature by 1 K:

cpm ≡(∂Q

∂T

)p

=cpdmd + cpvmv

md +mv

=cpd + cpv(mv/md)

1 + (mv/md)

=cpd + cpvr

1 + r

= cpd1 + (cpv/cpd)r

1 + r

≈ cpd

[1 + r

(cpvcpd

− 1

)]≈ cpd(1 + 0.86r)




Water vapor modifies densityThe total density of a parcel containing water vapor with a mass mixing ratio of r is:

ρ = ρd + ρv =pdRdT

+e

RvT

=pdRdT

(1 + r)

=p

RdT

pdε(pd + e)

(1 + r)

=p

RdT

1 + r

1 + r/ε

=p

RmT, Rm ≡ Rd

1 + r/ε

1 + r

ε ≈ 0.622 < 1, so Rm > Rd: density decreases with increasing water vapor




Water vapor modifies densityThe dependence of density on water vapor is often expressed using virtual temperature, which is thetemperature that the parcel would have if it had the same density and pressure but no water vapor:

RdTv = RmT

Tv =RmRd

T =1 + r/ε

1 + rT

≈ T (1 + 0.608r)

r > 0, so Tv ≥ T . The virtual potential temperature

θv ≡ Tv

(p

p0

) Rdcpd

.

is directly related to density and is therefore a useful measure of stability in unsaturated moist air.




200 225 250 275 300 325 350 375 400Temperature [K]

100

200

300

400

500

600

700

800

900

1000

Pre

ssur

e [h

Pa]

Temperature (T)Potential temperature (θ)Virtual potential temperature (θv)

θv mainly deviates from θ at low levels,where mixing ratios are high

data from JRA-55





Phase changesSuppose daytime sensible heating results in dry convection that lifts moist near-surface air. This airwill cool adiabatically. If it reaches saturation (i.e., RH ≥ 1), then condensation will occur. Conversely,some portion of the precipitation falling through air with RH < 1 will evaporate. Both of theseprocesses involve water changing phase.

I Phase changes are diabatic processes

I Neither θ nor θv are conserved during processes in which water changes phase

I Condensation releases latent heat: θ increases

I Evaporation requires latent heat: θ decreases




Equivalent potential temperatureIt is useful to define a new quantity that is conserved for moist adiabatic processes (i.e., reversiblephase changes). The specific entropy of an air parcel containing water vapor and liquid water can beexpressed as the sum of the specific entropies for dry air, water vapor and liquid water:

s = sd + rsv + rlsl

= sd + (r + rl)sl + r(sv − sl)

= sd + rtsl + rLvT

+ r(sv − s∗v)

= [cpd lnT −Rd ln pd] + [rtcl lnT ] +rLvT

+ [r (cpv lnT −Rv ln e− cpv lnT +Rv ln e∗)]

= (cpd + rtcl) lnT −Rd ln pd +rLvT

− rRv ln(e/e∗)

where we have used an entropy form of the Clausius–Clapeyron equation (Lv = T (s∗v − sl)) under the

assumption that water vapor and liquid water are in equilibrium.




Equivalent potential temperatureDifferentiating and setting ds = 0:

(cpd + rtcl)d(lnT ) = Rdd(ln pd) + rRvd(

lne

e∗

)− r

LvT

which allows us to define the equivalent potential temperature as

θe = T

(pdp0

)−Rd/(cpd + rT cl) ( e

e∗

)−rRv/cpexp

(Lvr

(cpd + rtcl)T

)

I if the air is completely dry (r = 0), θe reduces to θ

I both θ and θe are conserved for dry adiabatic processes

I θe is conserved for moist adiabatic processes but θ is not

I both θe and θ change due to radiative heating/cooling and and sensible heat fluxes at the surface

I θe is also affected by latent heat fluxes at the surface




200 225 250 275 300 325 350 375 400Temperature [K]

100

200

300

400

500

600

700

800

900

1000

Pre

ssur

e [h

Pa]

Temperature (T)Potential temperature (θ)Virtual potential temperature (θv)Equivalent potential temperature (θe)

conditional instability:∂θe∂z < 0

data from JRA-55





Condensed water also modifies densityTv accounts for the dependence of density on water vapor, but does not account for the presence ofsuspended liquid water or ice

ρ =md +mv +ml +mi

Va + Vl + Vi

=1 + r + rl + ri

ρ−1d + rvρ

−1v + rlρ

−1l + riρ

−1i

≈ pdRdT

(1 + rt) =p

RdT

pdε(pd + e)

(1 + rt)

=p

RdT

1 + rt1 + r/ε

We can then define the density temperature Tρ, for which p = ρRdTρ in the presence of condensate:

Tρ ≡ T1 + r/ε

1 + rt

Unlike Tv, Tρ may be either smaller or larger than T , depending on the value of rt



figure from Manabe and Strickler, 1967

surface

Pure radiative equilibrium

no vertical mixing

http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281964%29021%3C0361%3ATEOTAW%3E2.0.CO%3B2




surface

Add dry convection

convection when Γ exceeds dry adiabatic Γ





surface

Add ‘moist’ convectionconvection when Γ exceeds global mean Γ




4 3 2 1 0 1 2 3 4

Diabatic heating [K d−1 ]

200

400

600

800

Pre

ssure

[hPa]

Sum

Turbulence

LW radiation

SW radiation

Convection

200 225 250 275 300 325 350 375 400Temperature [K]

200

400

600

800

Pre

ssure

[hPa]

T

θd

generated using CliMT

Radiative–convective equilibriumI distributions of T and θ are realistic

I diabatic heat budget shows balancebetween radiation (LW + SW) andconvection

http://people.su.se/~rcaba/climt/



90°S 60°S 30°S 0° 30°N 60°N 90°N1000

900800700600500400300200100

Pre

ssur

e [h

Pa]

280K280K300K

340K

380K

(a) Total heating rate

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

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Pa]

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380K

(b) Radiative heating rate

90°S 60°S 30°S 0° 30°N 60°N 90°N1000

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Pa]

280K280K300K

340K

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(c) Convective heating rate

3 2 1 0 1 2 3

Heating Rate [K d−1]

data from JRA-55

Diabatic heatingI includes radiative heating and

cooling, latent heating and cooling,and other processes

I diabatic heating rates are severalKelvins per day in much of theatmosphere – why aren’t thetropics constantly heating up?

I the tropical troposphere is inapproximate radiative–convectiveequilibrium

I the stratosphere is approximately inradiative equilibrium


lecture 4: thermodynamics review, part 2 · pdf filedry atmospheric thermodynamics moist...

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