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Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Lecture 4:Thermodynamics Review, Part 2
Jonathon S. Wright
14 March 2017
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Dry atmospheric thermodynamicsStabilityBuoyancyDry convection and stable inversions
Moist atmospheric thermodynamicsWater vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
Radiative–convective equilibrium
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
StabilityBuoyancyDry convection and stable inversions
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)
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
StabilityBuoyancyDry convection and stable inversions
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
StabilityBuoyancyDry convection and stable inversions
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
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∗
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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#)
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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)
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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.
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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.
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
Water vaporThermodynamic effects of water vaporThermodynamic effects of condensation and evaporation
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
figure from Manabe and Strickler, 1967
surface
Pure radiative equilibrium
no vertical mixing
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
figure from Manabe and Strickler, 1967
surface
Add dry convection
convection when Γ exceeds dry adiabatic Γ
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
figure from Manabe and Strickler, 1967
surface
Add ‘moist’ convectionconvection when Γ exceeds global mean Γ
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
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
Dry atmospheric thermodynamicsMoist atmospheric thermodynamics
Radiative–convective equilibrium
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
900800700600500400300200100
Pre
ssur
e [h
Pa]
280K280K300K
340K
380K
(b) Radiative heating rate
90°S 60°S 30°S 0° 30°N 60°N 90°N1000
900800700600500400300200100
Pre
ssur
e [h
Pa]
280K280K300K
340K
380K
(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