Water Vapor in the Air

Thermodynamics M. D. Eastin

Water Vapor in the Air

How do we compute the dewpoint temperature, the relative humidity, or the temperature of a rising air parcel inside a cloud?

Here we investigate parameters that describe water in our atmosphere


Outline:

Review of the Clausius-Clapeyron Equation Review of our Atmosphere as a System

Basic parameters that describe moist air Definitions Application: Use of Skew-T Diagrams

Parameters that describe atmospheric processes for moist air Isobaric Cooling Adiabatic – Isobaric processes Adiabatic expansion (or compression)

Unsaturated Saturated

Application: Use of Skew-T Diagrams

Additional useful parameters Summary



Basic Idea:

• Provides the mathematical relationship (i.e., the equation) that describes any equilibrium state of water as a function of temperature and pressure.

• Accounts for phase changes at each equilibrium state (each temperature)

Review of Clausius-Clapeyron Equation

Sublimatio

n

Fus

ion

Vap

oriz

atio

n

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

V

P(mb)

Vapor

Liquid

Liquidand

Vapor

T

esw

Sections of the P-V and P-T diagrams for which the Clausius-Clapeyron equation is derived in the following slides


Mathematical Representation:

• Application of the Carnot Cycle…

where: T = Temperature of the system l = Latent heat for given phase change dps= Change in system pressure at saturation dT = Change in system temperature Δα = Change in specific volumes between

the two phases

wv

vsw

ααTdT

de

l

Sublimatio

n

Fus

ion

Vap

oriz

atio

n

T

C

T (ºC)

p (mb)

3741000

6.11

1013

221000

Liquid

Vapor

Solid

iv

ssi

ααTdT

de

l

iw

fwi

ααTdT

dp

l

TΔdT

dps l



Computing saturation vapor pressure for a given temperature:

Version #1: Assumes constant latent heat of vaporization (lv = constant) Less accurate at extreme temperatures

Version #2: Accounts for temperature dependence of the latent heat [lv(T)] Most accurate across the widest range of temperatures

T(K)

1

273.15

1

Rexp11.6(mb)e

v

vsw

l

)(ln09.5

)(

680849.53exp11.6(mb)esw KT

KT



• Our atmosphere is a heterogeneous closed system consisting of multiple sub-systems

• We will now begin to account for the entire system…

Review of Systems

Water Vapor

e, T, ρv, mv, Rv

Open sub-system

Ice Waterpi, T, ρi, mi

Open sub-system

Dry Air(gas)

pd, T, ρd, md, Rd

Closed sub-system

Liquid Waterpw, T, ρw, mw

Open sub-system

Energy Exchange

Mass Exchange


Our Approach:

• Apply what we have learned thus far: Equation of StateFirst Law of

ThermodynamicsSecond Law of

ThermodynamicsPhase and Latent

Heats of waterClausius-

Clapeyron Equation

• Learn how to compute: Basic parameters that describe moist air

Each parameter using standard observations and/or thermodynamic diagrams (Skew-Ts)

What do we regularly observe? Total Pressure (p)Temperature (T)

Dewpoint Temperature (Td) or

Relative Humidity (r)

Moist Air Parameters


1. Equations of State for Dry Air and Water Vapor:

• Water vapor in our atmosphere behaves like an Ideal Gas • Ideal Gas → equilibrium state between Pressure, Volume, and Temperature

• Recall: Water vapor has its own Ideal Gas Law

Basic Moisture Parameters

TRρp ddd TRρe vvDry Air (N2 and O2) Water Vapor (H2O)

pd = Partial pressure of dry air

ρd = Density of dry air

T = Temperature of dry air

Rd = Gas constant for dry air ( Based on the mean molecular weights ) ( of the constituents in dry air ) = 287 J / kg K

e = Partial pressure of water vapor (called vapor pressure)

ρv = Density of water vapor (called vapor density)

T = Temperature of water vapor

Rv = Gas constant for water vapor ( Based on the mean molecular weights ) ( of the constituents in water vapor ) = 461 J / kg K


2. Mixing Ratio (w):

Definition: Mass of water vapor per unit mass of dry air:

We can use the Equation of States for dry air and water vapor with Dalton’s Law of partial pressures to place mixing ratio into variables we either observe or can calculate from observations:

How do we find “e” from observations?

d

v

d

v

ρ

ρ

m

mw

ep

e

R

Rw

v

d

TRρp ddd

TRρe vv

ep p d



2. Mixing Ratio (w):

How do we find “e”? Our integrated Clausius-Clapeyron equation

Use Td in place of T to find the vapor pressure (e)

where: e has units of mbTd has units of K

Needed Information for Computation:

Observed variables: p, Td

Computed variables: ePhysical Constants: Rd, Rv, lv

Units: g/kg

dv

v

T

1

273.15

1

Rexp11.6e

l

ep

e

R

Rw

v

d



3. Saturation Mixing Ratio (wsw):

Definition: Mass of water vapor per unit mass of dry air at saturation

Can be interpreted as the amount of water vapor an air parcel would contain at a given temperature and pressure if the

parcel was at saturation (with respect to liquid water)

How do we find “esw” from observations?

sw

sw

v

dsw ep

e

R

Rw

d

v

d

vsw ρ

ρ

m

mw



3. Saturation Mixing Ratio (wsw):

How do we find “esw”? Our integrated Clausius-Clapeyron equation

Use T to find the saturation vapor pressure (esw)

where: esw has units of mbT has units of K


Observed variables: p, TComputed variables: esw

Physical Constants: Rd, Rv, lv

Units: g/kgsw

sw

v

dsw ep

e

R

Rw

T

1

273.15

1

Rexp11.6e

v

vsw

l



4. Specific Humidity (q):

Definition: Mass of water vapor per unit mass of moist air:

where:

It is closely related to mixing ratio (w):

Since both q << 1 and w << 1 in our atmosphere, we often assume

ρ

ρ

m

mq vv vd mmm

vd

w1

wq

q1

qw

wq



5. Relative Humidity (r):

Definition: The ratio (or percentage) of water vapor mass in a moist air parcel to the water vapor mass the parcel would have if it was saturated with respect to liquid water

Using the Ideal Gas laws for dry and moist air:

Note:

vsw

v

m

mr

swe

er How do we find “e” and “esw”

from observations?

sww

wr



5. Relative Humidity (r):

Finding “e” and “esw”:

where: e and esw have units of mbTd and T has units of K


Observed variables: Td, TComputed variables: e, esw

Physical Constants: lv, Rv

Units: %

T

1

273.15

1

Rexp11.6e

v

vsw

l

dv

v

T

1

273.15

1

Rexp11.6e

l

swe

er



Skew-T Log-P Diagram

Isotherm

(T=-1

0ºC)

Satu

ratio

n M

ixin

g R

atio

(10

g/kg

)

Pressure (200 mb)

Dry Adiabat (283K)

Pseudo-A

diabat (283K)


The Skew-T Log-P Diagram

The lines of constant saturation mixing ratio are also skewed toward the upper left

These lines are always dashed and straight, but may vary in color

Our Version:

Pink dashed Lines

sw

sw

v

dsw ep

e

R

Rw

T)(p,w sw


Example:

Typical surface observations at the Charlotte-Douglas airport in March:

p = 1000 mbT = 25ºCTd = 16ºC

Find the following using a Skew-T Diagram:

Saturation Mixing Ratio (wsw)Mixing Ratio (w)Specific Humidity (q)Relative Humidity (r)

Application: The Skew-T Diagram


Given: p = 1000 mb Saturation Mixing Ratio: T = 26°C Td =18°C

1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for wsw from the saturation mixing ratio line

that corresponds to (p, T)

T = 26°CP = 1000 mb

wsw = 22 g/kg


sw

sw

v

dsw ep

e

R

Rw

T)(p,w sw


Given: p = 1000 mb Mixing Ratio: T = 26°C Specific Humidity: Td =18°C

1. Place a large dot at the location that corresponds to (p, Td) 2. Obtain value for w from the saturation mixing ratio line

that corresponds to (p, Td) 3. Compute q using the w value → 0.0123 / (1 + 0.0123)

Td = 18°CP = 1000 mb

w = 12.3 g/kg

ep

e

R

Rw

v

d

)T(p,w d

w1

wq

q = 12.2 g/kg



Given: p = 1000 mb Relative Humidity: T = 26°C Td =18°C

1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T) 3. Obtain value for w and wsw from the saturation mixing ratio

lines that corresponds to Td and T, respectively 4. Compute r → 0.0123 / 0.022

)TT,(p,r dswe

er

sww

wr


Td = 18°CP = 1000 mb

T = 26°C

wsw = 22 g/kg

r = 56%

w = 12.3 g/kg


Our Approach:

• Examine the following: Isobaric processes (occurring at the surface)Processes involving ascent → Unsaturated

→ Saturated

• Learn how to compute: Parameters that are conserved during typical atmospheric processes (isobaric, adiabatic)

Each parameter using standard observations and/or thermodynamic diagrams (Skew-Ts)

What do we regularly observe? Total Pressure (p)Temperature (T)

Dewpoint Temperature (Td) or

Relative Humidity (r)

Moist Air Parameters during Processes


Isobaric Cooling: Dew Point Temperature (Td)

Definition: Temperature at which saturation (with respect to liquid water) is reached when an unsaturated moist air parcel is cooled at constant pressure

• Parcel is a closed system

• Mass of water vapor and dry air are constant

• Isobaric transformation

• Total pressure (p) constant• Vapor pressure (e) constant • Mixing ratio (w) constant

• Saturation vapor pressure (esw) and saturation mixing ratio (wsw) change since they are both functions of the temperature


Temperature

T2 T1

esw1

Va

po

r p

res

su

re

Td

Temperature Cools: T1 → T2

esw2 e

esw(T)



• Such a process regularly occurs

• Radiational cooling near surface• Often occurs at night (no solar heating)• Can occur at ground level (dew) or through a layer (fog)




Obtained by integrating the Clausius-Clapeyron equation between our initial [esw = esw(T1), T = T1] and final [esw = e, T = T2] states, solving for T2, and setting T1 = T, e/esw = r, and T2 = Td (see your text)


Observed variables: T, rComputed variables: -----

Physical Constants: Rv, lv

Units: K


rlnTR

-1

TT

v

d

vl


Given: p = 1000 mb Dew Point Temperature: T = 26°C r = 56%

1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for wsw from the saturation mixing ratio line that

corresponds to (p, T) 3. Compute w using r and wsw → 0.56(0.022) 4. The Td value is the temperature at (p, w)

r)T,(p,Tdsww

wr


Td = 18°CP = 1000 mb

T = 26°C

wsw = 22 g/kg

r = 56%

w = 12.3 g/kg


Adiabatic Isobaric Process: Wet-Bulb Temperature (Tw)

Definition: Temperature at which saturation (with respect to liquid water) is reached when an unsaturated moist air parcel is cooled by the evaporation of liquid water

where: wsw is the saturation mixing ratio at Tw

w is the mixing ratio at Td

See your text for the full derivation…


• Can not be mathematically solved for without iteration• Easiest to solve for graphically on a Skew-T diagram


swp

w wwc

TT vl

Important


Moist Air Parameters during ProcessesAdiabatic Isobaric Process: Wet-Bulb Temperature (Tw)

• Such a process regularly occurs

• Evaporational cooling occurs near the surface during light rain• The temperature often feels colder when its raining → It is!


Wet-bulb Temperature (Tw):

1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T)

3. Draw a line from (p, Td) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. From the intersection point of the two lines, draw another line downward along a pseudo-adiabat to the original pressure (p) 6. The Tw is the resulting temperature at that pressure


Td = 6°CP = 1000 mb

T = 26°C

Tw = 15ºC

Given:

p = 1000 mb T = 26ºC Td = 6ºC


In Class ActivityCalculations:

Observations from this morning at CLT: p = 1000 mbT = 8.3ºCTd = 2.8ºC

Compute: w, q, wsw, r

Skew-T Practice:

Observations from yesterday afternoon at CLT: p = 1000 mbT = 13.5ºCr = 32%

Graphically estimate: Td, Tw

Write your answers on a sheet of paper and turn in by the end of class…


Adiabatic Expansion (or Compression): Moist Potential Temperature (θm)

Definition: Temperature an unsaturated moist air parcel would have if it were to expand or compress from (p, T) to the 1000 mb level


Observed variables: p, T, Td (or r) Computed variables: e, w, q (also esw if using r)

Physical Constants: cp, Rd, Rv, lv

Units: K


0.26q)(1c

R

m

p

d

p

1000Tθ


Adiabatic Expansion (or Compression): Moist Potential Temperature (θm)

Note: Since q << 1 in our atmosphere, the difference between the moistpotential temperature (θm) and the dry potential temperature (θ) isextremely small

Therefore: The two are essentially equal:

The moist potential temperature (θm) is rarely used in practice Rather, the dry potential temperature (θ) is used


0.26q)(1c

R

m

p

d

p

1000Tθ

θθm

p

d

c

R

p

1000Tθ


Reaching Saturation by Adiabatic Ascent:

• An unsaturated air parcel that rises adiabatically will cool via expansion• During the parcel’s ascent the following occurs:

• Potential temperature remains constant• Moisture content (w or q) remains constant• Saturation vapor pressure (esw) decreases• Saturation mixing ratio (wsw) decreases• Relative humidity (r) increases

Eventually:

Relative humidity will reach 100% and saturation occurs Condensation must take place to maintain the equilibrium

Lifting Condensation Level (LCL):

Definition: Level were an ascending unsaturated moist air parcel first achieves saturation due to adiabatic cooling and condensation begins to occur


sww

wr


Reaching Saturation by Adiabatic Ascent:

Where is the typical Lifting Condensation Level (LCL)?


LCL CloudBase

Rising unsaturated parcels cool to saturation


Temperature at the Lifting Condensation Level (TLCL):

Definition: Temperature at which an ascending unsaturated moist air parcel first achieves saturation due to adiabatic cooling and condensation begins to occur



Observed variables: T, r (or Td)Computed variables: ----- (e, esw if using Td)

Physical Constants: -----Units: K


55

2840rln

55T1

1TLCL


Temperature of the Lifting Condensation Level (TLCL):


3. Draw a line from (p, Td) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. The TLCL is found at the intersection point of the two lines 6. The corresponding pressure pLCL also defines the LCL


Td = 6°CP = 1000 mb

T = 26°C

TLCL = 2ºC

Given:

p = 1000 mb T = 26ºC Td = 6ºC

PLCL = 745 mb


Saturated (Moist) Adiabatic Ascent:

Once saturation is achieved (at the LCL), further ascent produces additional cooling (adiabatic expansion) and condensation must occur Cloud drops begin to form!

Two Extreme Possibilities:

1. Condensation Remains

All liquid water stays with the rising air parcel Implies no precipitation

• Closed system → no mass exchanged with environment• Adiabatic → no heat exchanged with environment• Reversible process → if the parcel descends, drops evaporate• Implies no entrainment mixing



Saturated (Moist) Adiabatic Ascent:

Once saturation is achieved (at the LCL), further ascent produces additional cooling (adiabatic expansion) and condensation must occur Cloud drops begin to form!

Two Extreme Possibilities:

2. Condensation is Removed

All condensed water falls out of rising air parcel Parcel always consists of only dry air and water vapor Implies heavy precipitation and no cloud drops

• Open system → Condensed water mass removed from system → Irreversible process

• Pseudo-adiabatic → No heat exchanged with environment → No dry air mass exchanged → No water vapor exchanged

• Implies no entrainment mixing



Saturated (Moist) Adiabatic Ascent: Which one occurs in reality?


Clouds with no precipitation

• Shallow• No loss of condensed water• Experience some entrainment• Ascent is almost reversible

Clouds with precipitation

• Shallow or Deep• Loss of condensed water• Experience some entrainment• Ascent is almost pseudo-adiabatic


Reversible Equivalent Potential Temperature (θe):

Definition: Temperature an unsaturated moist parcel would have if it:• Dry adiabatically ascends to saturation (to its LCL)• Moist adiabatically ascends until all water vapor was condensed and retained within the parcel• Dry adiabatically descends to 1000 mb

where:


• Difficult to compute for since mw is unknown• Can be computed if mw is observed (e.g. by radar) or estimated


LCLwtp

v

)cw(cR

LCLe Tcwc

wlexp

p

1000Tθ

wtpd

dwvt mmmw Important

Cannot be determinedon a Skew-T diagram


Pseudo-Adiabatic Equivalent Potential Temperature (θep):

Definition: Temperature an unsaturated moist parcel would have if it:• Dry adiabatically ascends to saturation (to its LCL)• Moist adiabatically ascends until all water vapor was condensed and falls out of the parcel• Dry adiabatically descends to 1000 mb


Observed variables: p, T, Td, rComputed variables: e, w, TLCL

Physical Constants: Rd, Rv, lv

Units: K


2.54T

3376w0.811wexp

p

1000Tθ

LCL

0.28w)(10.285

ep


Pseudo-Adiabatic Equivalent Potential Temperature (θep):

1. Place large dots at the locations that correspond to (p, Td) and (p, T) 2. Draw a line from (p, Td) upward along a saturation mixing ratio line

3. Draw a line from (p, T) upward along a dry adiabat 4. From the intersection point of the two lines, draw another line upward along a pseudo-adiabat until it parallels the dry adiabats 5. From this “parallel point” (where all vapor has been condensed) draw a line downward along a dry adiabat to 1000 mb. 6. The θep is the resulting temperature at 1000 mb.


Td = 2°CP = 1000 mb

T = 22°C

θep = 307 K(34ºC + 273)

Given:

p = 1000 mb T = 22ºC Td = 2ºC


Saturated (Moist) Adiabatic Descent:

A descending saturated air parcel will warm (adiabatic compression) The amount of temperature increase will depend on whether condensed water is present in the parcel

Two possible scenarios;

1. Parcel does not contain condensed water

• The parcel immediately become unsaturated• Dry adiabatic descent occurs• Potential temperature (θ) remains constant• Mixing ratio (w) remains constant

• Similar to the final leg of determining θep on the Skew-T diagram



Saturated (Moist) Adiabatic Descent:

A descending saturated air parcel will warm (adiabatic compression) The amount of temperature increase will depend on whether condensed water is present in the parcel

Two possible scenarios;

2. Parcel does contain condensed water

• Initial descent warms air to a unsaturated state• Produces an unstable state for the condensed water drops• Some water drops evaporate → cools the air parcels

→ moistens the air parcel → brings parcel back to

saturation• Subsequent descent requires additional droplet evaporation in order to maintain the saturated state

Saturated descent can occur as long as condensed water is present Once all the condensed water evaporates → dry-adiabatic descent



Wet-Bulb Potential Temperature (θw):

Definition: Temperature a saturated moist air parcel that contains condensed water would have if it descends adiabatically to 1000 mb

where: w is the mixing ratio at θw



• Can not be mathematically solved for without iteration• Easiest to solve for graphically on a Skew-T diagram


2.54θ

3376w0.811wexp

θθ

w

epw

Important


Wet-bulb Potential Temperature (θw):


3. Draw a line from (p, Td) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. From the intersection point of the two lines, draw another line downward along a pseudo-adiabat to 1000 mb 6. The θw is the resulting temperature at 1000 mb


Td = -11°C

P = 1000 mb

T = 9°C

Given:

p = 700 mb T = 9ºC Td = -11ºC

P = 700 mb

θw = 288 K(15ºC + 273)


Equation of State for Moist Air:

Obtained by combining the Equations of State for both dry air and water vapor with the mixing ratio and specific humidity (see your text)

where:

Advantage: Defines total density (combinations of dry air and water vapor) Used to more easily define the total density gradients that

determine atmospheric stability (or parcel buoyancy)

Will use more in next chapter…

vdTρRp

vd

ep p d

T0.61q)(1Tv

w1

wq

ep

e

R

Rw

v

d

Additional Parameters


Virtual Temperature (Tv):

Definition: The temperature a moist air parcel would have if the parcel contained no water vapor (i.e. vapor was replaced by dry air)


Advantage: Simple way to account for variable moisture in an air parcel Will use more in next chapter…


Observed variables: p, T, Td (or r)Computed variables: e, w, qPhysical Constants: Rd, Rv, lv

Units: K

T0.61q)(1Tv




Virtual Potential Temperature (θv)

Definition: Temperature a moist air parcel would have if it were to expand or compress from (p, Tv) to the 1000 mb level, and the parcel

contained no water vapor (i.e. vapor was replaced by dry air)

Advantage: Similar to θ and θm but accounts for variable moisture in a parcel Used to define atmospheric stability Will use more in next chapter…


Observed variables: p, T, Td (or r) Computed variables: e, w, q

Physical Constants: cp, Rd, Rv, lv

Units: K

p

d

c

R

vv p

1000Tθ




Summary: Relationship of ParametersLots of Temperatures!

• Each temperature defines the state of an air parcel at a single location• Differences result from → Whether moisture is included

→ Type of process involved

Lots of Potential Temperatures!

• Each potential temperature defines the state of an air parcel at 1000 mb• Differences result from → Whether moisture is included

→ Type of process involved

vwdLCL TTTTT

epe vmw


Can be used to estimate (or simplify the computation of):

• Mixing ratio (w)• Saturation mixing ratio (wsw)• Relative humidity (r)• Specific humidity (q)• Potential temperature (θ)• Wet-bulb temperature (Tw)

Note: All parameter symbols are color-coded with their locations

Summary: The Skew-T Diagram

Td, w

P = 1000 mb

T, wsw

Given:

p = 800 mb T = 9.5ºC Td = -8.0ºC

P = 800 mb

θw

TLCL

θepθ

Tw

PLCL

• Temperature at the LCL (TLCL)• Pressure at the LCL (PLCL)• Wet-bulb potential temperature (θw)• Pseudo-adiabatic equivalent potential temperature (θep)


Review:

• Review of the Clausius-Clapeyron Equation• Review of our Atmosphere as a System

• Basic parameters that describe moist air• Definitions• Application: Use of Skew-T Diagrams

• Parameters that describe atmospheric processes for moist air• Isobaric Cooling• Adiabatic – Isobaric processes• Adiabatic expansion (or compression)

• Unsaturated• Saturated

• Application: Use of Skew-T Diagrams

• Additional useful parameters• Summary



ReferencesPetty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.

Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp. Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.

Also (from course website):

NWSTC Skew-T Log-P Diagram and Sounding Analysis, National Weather Service, 2000

The Use of the Skew-T Log-P Diagram in Analysis and Forecasting, Air Weather Service, 1990

Documents

Water Vapor in the Air