Differential Solar Heating Hadley Cells of Rising and Falling Air.
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Moist air thermodynamics (Reading Text 3.5-‐3.7, p79-‐101)
Topics: • Variables that descript moist air • staDc stability for moist convecDon, saturated and pseudo-‐ adiabaDc lapse rates, equivalent potenDal temperature
• StaDc stability • The second law of thermodynamics for moist air, the Clausius-‐Clapeyron equaDon
Variables • Mixing raDo, w, and specific humidity, q:
– W=mv/md, mass of vapor vs. mass of dry air – q=mv/(mv+md)=w/(w+1) mass of vapor vs. total air mass
• Vapor pressure, e=[w/(w+ε)]p – Where recall ε=Mv/Md=Rd/Rv=0.622, p: air pressure Because:
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e =nv
nv + ndp =
mv
Mvmd
Md
+mv
Mv
p ×
Mv
mdMv
md
=
mv
mdMv
Md
+mv
md
p =w
ε + wp
• nv, nd: numbers of mole of vapor and dry air molecules, respecDvely. mv/md, mass of vapor v
• Mv and Md, molecular weight of vapor and dry air
• Virtual temperature Tv: – Recall from Sect. 4.1, that Tv=T/[1-‐e/p(1-‐ε)] (Eq. 3.16 in the text), we can express Tv ≈ T (1+0.61W) (see below)
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e =w
ε + wp ⇒ e
p=
wε + w
Tv = T 1
1− ep
(1−ε)= T 1
1− wε + w
(1−ε)= T ε + w
ε + w − w + wε
Tv= T1+ w /ε(1+ w)
= T 1+ ( 10.622
−1) w1+ w
%
& '
(
) * = T 1+ 0.61 w
1+ w%
& '
(
) *
Tv≈ T 1+ 0.61w( ) because w << 1, w +1 ≈w
QuesDon:
If the atmospheric consists 10 g per 1 kg of dry air and temperature is 27C at the sea-‐level, what is the mixing raDo (w), specific humidity (q), vapor pressure (e) and virtual temperature of this atmosphere?
• W=10g/1 kg=0.01 • q=(10g)/(1000g+10g)=0.0099=9.9g/kg • e=wP/(ε+w)=0.01X1000hPa/(0.622+0.01)=15.8Pha
• Tv≈T(1+0.61w)=(273+27)K(1+0.61X0.01)=301.8K=28.8C
• SaturaDon vapor pressure, es: the maximum vapor pressure with respect to a plane surface of pure water at T. It is a funcDon of T.
• Below 0°C, the saturaDon vapor pressure with respect to ice, esi for a plane ice surface is lower than that with respect of of water.
• Thus, atmosphere can be saturated for ice but not for water. Supper saturaDon e>esi can occur in real atmosphere (e.g., near tropopause).
• SaturaDon mixing raDo, ws=mvs/md=0.622es/(p-‐es)≈0.622es/p – SaturaDon mixing raDo is more commonly used in meteorology and climate than es.
– See p. 82 of the text for derivaDon.
• RelaDve humidity, RH=100w/ws=100e/es – RelaDve humidity is derived from dew point or frost point temperature, Td, and air temperature, T, which can be measured readily.
• Td: the temperature at which dew starts to form, when ws(Td)=w of the air, thus RH=100ws(Td)/ws(T) Thus, RH is determined by saturaDon mixing raDo of dew/frost point temperature vs. that of air temperature for the same pressure value.
Example:
• Meteorological measurements show that a) air T=30°C and Td=25°C; b) The next day, T remains the same, but Td reduces to 20°C. Calculate RH for both cases use the figure below:
a) es(T=30C)=40 hPa, es(Td=25C)=30 hPa, RHa=100*es(Td)/es(T)=75%
b) es(Td=20C)=25 hPa RHb=100*es(Td)/es(T) =25hPa/40hPa=62.5%
RelaDve humidity dropped about 12.5% in case b)
• Lii condensaDon level (LCL): – T in a rising air parcel would decrease follow the dry adiabaDc lapse
rate (9.8C/km). At the height where T reduces to the same value as Td, condensaDon occurs. This height is referred to as the liiing condensaDon level (LCL). It is also cloud base.
T
height
Td
LCL
• LCL in a skew T-‐ln p chart:
Figure 3.10 The liiing condensaDon level of a parcel of air at A, with pressure p, temperature T and dew point Td, is at C on the skew T − ln p chart.
Td T
• T in a rising air parcel would follow constant θ line (orange, dry adiabaDc line). The green isothermal lines indicate T and Td of the surface air (1000 hPa). The pressure level at which constant θ line cross the isothermal line for Td is the LCL for this air.
Saturation moist adiabatic and pseudoadiabatic lapse rates:
• As an rising air parcel above LCL, condensaDon occurs and the temperature change inside of the air parcel is determined by combined dry adiabaDc cooling and latent heaDng due to condensaDon (dT/dz=(αdp/dz-‐Ldw)/Cp, vs. dry adiabaDc dT/dz=(αdp/(Cpdz)). Thus, the rate of temperature decrease with height, i.e., the moist adiabaDc lapse rate (Γm), is less than that of dry adiabaDc lapse rate.
• Unlike the dry adiabaDc lapse rate (Γd=R/Cp=9.8K/km), Γm is not a constant. Γm depends on net amount of condensaDon.
• Two assumpDons about Γm: – SaturaDon moist adiabaDc lapse rate: All condensed water is retained in
the rising air parcel, and can be re-‐evaporated if T increases. It is a reversible process.
– PseudoadiabaDc lapse rate: All condensed water falls out the rising air parcel. Thus, re-‐evaporated is not possible. It is a irreversible process.
Saturation moist adiabatic vs. pseudoadiabatic lapse rates: • Saturated moist adiabatic lapse rate is greater than pseudoadiabatic
lapse rate (T decreases faster with height), because – In case of the former, condensed water can be re-evaporated which absorbs
latent heat. Thus, air parcel tends to be less buoyant and Tv (virtual temperature) decreases faster with height.
– Under pseudoadiabatic case, no re-evaporation of condensed water. The rising air parcel is more buoyant and Tv decreases slower with height.
• Moist lapse rate varies. Typically, Γm is about 4 K/km in the warm and humid lower troposphere, and ~ 6-‐7 K/km in the mid-‐troposphere, close to Γd in the upper troposphere. Why?
Example:
Meteorological measurements show that air T=30°C and Td=25°C, T=-‐5°C at 500 hPa.
a) Determine the LCL of an rising air parcel from the surface
b) Determine the lapse rate between the LCL and 500 hPa assuming i) all condensed liquid water falls out immediately; ii) only a half of the liquid fall out, the other half rising with the air parcel and re-‐evaporate immediately. Ignore the heat absorbed by the condensed water.
c) Height of 500 hPa is 5.5 km, 950hPa is 0.5 km. Latent heat of vaporizaDon, Lv=2.5X106 J kg-‐1. Surface pressure is 1000 hPa.
SoluDon:
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a. From the surface to LCL, ∂T∂z
= Γd = 9.80K/km, T −Td =∂T∂z
zLCL
zLCL =T −TdΓd
=(30− 25)K9.8K /km
≈ 0.51km
b. At LCL, es,LCL = es (25C) = 30hPa, ws = 0.622 es
p= 0.622 30hPa
950hPa= 0.020 = 20g /kg
At 500 hPa, es,500hPa(-5C) = 5 hPa, based on es - T relation shownin the figure.
ws,500hPa = 0.622 esp
= 0.622 5hPa500hPa
= 6.2×10−3 = 6.2g /kg
i) The total condensed water between LCL and 500 hPa, δes = (20− 6.2)g /kg = 13.8g /kg
dTdz
=1Cp
αdpdz
=αρg=g
− L dwdz
*
+
, , , ,
-
.
/ / / /
=gCp=Γd
−LCp
dwdz
= 9.8K /km − 2.5X106.Jkg−1
1004JK −1kg−1⋅
13.8×10−3
5km ×103m /km)
= (9.8− 6.9)K /km ≈ 2.9K /Kmii) If a half of the condensed water re- evaporates, the net condensed water is 6.9g/kg.dTdz
= (9.8− 3.45) /1004K /m ≈ 6.35K /Km Thus, the value of moist lapse rate depends on the amount of net condensation in the rising air parcel.
• Wet-‐bulb temperature, Tw: – In surface meteorological staDon, web-‐bulb temperature represents T measured by a glass bulb thermometer wrapped by wet clothe over which ambient air is drawn.
– Because evaporaDon of wet clothe also cool ambient T and increase w, thus Tw > Td (cool dry adiabaDcally unDl condensaDon occurs) <T, usually is the arithmeDc mean of T and Td.
Equivalent potenDal temperature, θe:
• When condensaDon occurs in a rising air parcel, the latent heat released by condensaDon, Lvdws, would warm the air parcel, thus against the dry adiabaDc cooling, as shown by the 2nd law of thermodynamics Lvdws=CpdT-‐αdp.
• The equivalent potenDal temperature, θe, is defined as the temperature would be if all the water vapor were condensed and rain out immediately (pseudoadiabaDc moist process), and all the potenDal energy were used to heat the air. Thus, it is higher than the potenDal temperature θ (only include internal and potential energy, not latent energy).
• θe is derived from the 2nd law of thermodynamics -‐Lvdws=CpdT-‐αdp assuming all the water vapor is condensed and rain out immediately.
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because θ = T pop
#
$ %
&
' (
R / c p, lnθ = lnT - R
Cpln p+
RCp
ln pocons tan t
Cpdθθ
= CpdTT
− R dpp
, because dqT
= CpdTT
− R dpp
dqT
= Cpdθθ
⇒ Lvdws
T= Cp
dθθ
⇒
dθθ
= −Lvdws
CpT≈ −d Lvws
CpT
#
$ % %
&
' ( ( beause 1
T2 <<1T
dθθθ e
θ
∫ = − d Lvws
CpT
#
$ % %
&
' ( ( 0
ws∫
θe ≈θ ⋅ exp Lvws
CpT
#
$ % %
&
' ( (
θe is defined as the equivalent potential temperature, represents the potential tempearture (θ ) would be if all the vapor were condensedand its latent heat were used to increase θ .
• Equivalent potenDal temperature, θe is a conserved variable for an pseudoadiabaDc adiabaDc process, i.e., there is no heat exchange between the system and the ambient, and 100% latent heat release due to condensaDon of water vapor is used to increase T.
• The moist adiabaDc process is not a true adiabaDc process, because of latent heat release. However, we can treat it as it were a adiabaDc process if we use θe concept.
• Θe is similar to θ for dry air.
Normand’s rule-‐finding Td and Tw using θ and θe lines
• Find LCL: – Find the θ line that intersects with
isothermal line that represents surface T.
– Find the ws line that intersects with Td at 1000 hPa.
– The intersecDon between θ and ws represents LCL.
• Finding Tw in Skew T-‐lnp chart: – From the LCL follow the constant θe
line down to the sea level (1000 hPa), you will find the wet bulb temperature, Tw at the surface.
• Q: Why does LCL is determined by the Td follow the Ws instead of isothermal line?
Effect of ascent followed and descent on air temperature and humidity:
• Once a convecDve air reaches LCL (point 2), it would ascent follow constant θe line unDl it reaches cloud top (point 3) and begin descending.
• The temperature of the descending air depends on the type of moist process during ascending process: – If it follows pseudoadiabaDc process, all the
condensed water falls out, no evaporaDon would occur during the descending, T follows dry adiabaDc process (constant θ line) back to surface (point 4, 900 hPa in this case) with T2 (see exercise 3.10 for quanDtaDve descripDon)
– If it follows saturated moist process, condensed water retained in the air parcel would re-‐evaporate, T during descending follows θe to LCL (point 2), then follow θ back to surface (point 1). Thus, T is reversal.
T2
LCL
T T2 1
1
2
2
3
3
4
4
In-‐class discussion-‐summary:
• Specific humidity is more commonly used in meteorological applicaDon. Do you expect the value of specific humidity more or less than that of mixing raDo? Why?
• How is RH determined by meteorological measurement? • What is the difference between the saturaDon moist adiabaDc
and pseudoadiabaDc moist assumpDons? Which assumpDon would lead to overesDmate buoyancy of the convecDng air (or strength of convecDon) and which assumpDon would underesDmate the buoyancy of convecDve air? Why?
• What determine value of the moist adiabaDc lapse rate? • What type of lapse rate does equivalent potenDal
temperature represent?