Moisture Variables on Skew T Log P Diagram

AOS 330 LAB 8

List of Variables

• Mixing ratio (w)• Saturation mixing ratio (ws)

• Specific Humidity (q)• Vapor pressure (ev)

• Saturation vapor pressure (es)

• Relative humidity (RH)• Dewpoint (Td)

• Dewpoint Depression• Virtual Temperature (Tv)

• Wet-Bulb Temperature (Tw)

€

ΔTd

Mixing Ratio (w or rv)

• Mixing ratio (rv) – a way to tell how much vapor there is relative to a mass of dry air

• It is conserved as long as there is no condensation or evaporation.

• Units : g kg-1

€

rv =ρ vρ d

=

evRvTpdRdT

=evpd

RdRv

=εevp− ev

€

ε ≡RdRv

= 0.622

Specific Humidity (q)

• Mass of water vapor per unit mass of moist air

• But mass of water vapor is very small compare to the total mass (~1-2% of the total mass)

• • , (ev<< p)

€

qv ≡ρ vρ

=ρ v

ρ d + ρ v=rv

1+ rv=

εevp− ev (1−ε)

≈εevp

€

rv =εevp− ev

≈εevp

€

rv =qv

1−qv

€

rv ≈ q ≈εevp

Vapor Pressure (ev)

• ev – partial pressure of vapor in (Pa)

• es – saturation vapor pressure over plane surface of pure water

Saturation Vapor Pressure (es)

• Vapor pressure ev - most directly determines whether water vapor is saturated or not.

• ev < es(T) subsaturated, evaporation

• ev = es(T) saturated

• ev > es(T) supersaturated, condensation

• es only depends on temperature.

• es(T) increases with increasing temperature.

Relative Humidity

• Subsaturated: RH < 100%• Saturated: RH = 100%• Supersaturated: RH > 100%• Depends on both vapor pressure ev and the air

temperature T

€

RH =eves(T)

€

S ≡ RH −100%

Dewpoint (Td)

• Consider the case ev < es(T), we could always reduce es(T) to ev by lowering the temperature.

• Dewpoint is the temperature at which moist air became saturated over a plane surface of pure water by cooling while holding ev constant.

• Only depends on the vapor pressure ev.

€

es(Td ) = ev

Saturation Mixing Ratio (ws)

• It is the mixing ratio for which air is saturated at specific T and P.

€

ws(T, p) =εes(T)

p− es(T)≈εes(T)

p

Saturation Mixing Ratio (ws)

• Depend on both temperature and pressure• In units of (g kg-1)• If we choose P to be 622 hPa, then€

ωs(T, p) ≈εes(T)

p

€

ε ≡RdRv

≈ 0.622

€

ωs(T,622hPa) ≈0.622es(T)

622= 0.001es(T)

€

ωs(T,622hPa)g

kg

⎡

⎣ ⎢

⎤

⎦ ⎥= es(T) hPa[ ]

To find saturation vapor pressure (es)

622hPa3 deg C

7.8 gkg-1

A temperature of 3 deg C at 622hPa is correspond to a saturated mixing ratio of 7.8 g kg-1. The saturation vapor pressure is ~ 7.8 hPa.

Dewpoint, Mixing Ratio, and Dewpoint Depression

€

ΔTd = T −Td

P

TTd

wsw

• To apply ideal gas law to mixture of air and vapor• Moist air equation of state :

Virtual Temperature (Tv)

€

p = pd + ev = (ρ dRd + ρ vRv )T

€

=ρRd (ρ dRd + ρ vRv

ρRd)T

€

=ρRd

1+Md

Mv

rv ⎛

⎝ ⎜

⎞

⎠ ⎟

1+ rvT

€

p = ρRdTv

€

Tv ≡ T(1+ 0.61rv ) ≈ T(1+ 0.61q)

€

p = ρRmT

€

Rm = Rd

1+Md

Mv

rv ⎛

⎝ ⎜

⎞

⎠ ⎟

1+ rv

€

Tv =

1+Md

Mv

rv ⎛

⎝ ⎜

⎞

⎠ ⎟

1+ rvT

Wet-Bulb Temperature (Tw)• It is the temperature to

which air is cooled by evaporation until saturation occurs.

• Assume that all of the latent heat of vaporization is supplied by the air

• Normand’s Rule: To find Tw, lift a parcel of

air adiabatically to its LCL, then follow moist adiabat back down to parcel’s original P

One Other Variable: Θ • Potential

Temperature Θ

• Temperature of the parcel if it were compressed or expanded dry adiabatically to 1000 hPa.

• Conserved in dry adiabatic process

Critical Levels on Thermodynamic Diagram

• The level at which a parcel lifted dry adiabatically will become saturated.

• Find the temperature and dewpoint of the parcel (at the same level, typically the surface). Follow the mixing ratio up from the dewpoint, and follow the dry adiabat from the temperature, where they intersect is the LCL.

Lifting Condensation Level (LCL)

Finding the LCL

References

• Petty, G (2008). A First Course in Atmospheric Thermodynamics, Sundog Publishing.

• Potter and Coleman, 2003a: Handbook of Weather, Climate and Water: Dynamics, Climate, Physical Meteorology, Weather Systems and Measurements, Wiley, 2003