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
Thermodynamics
M. D. Eastin
Water Vapor in the Air
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
Thermodynamics
M. D. Eastin
Review of Clausius-Clapeyron Equation
Basic Idea:
p (mb)
• 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)
C
221000
Liquid
Solid
1013
6.11
T
Vapor
P
(mb)
Vapor
esw
100
374
T (ºC)
T
Liquid
Liquid
and
Vapor
Thermodynamics
0
Sections of the P-V and P-T diagrams for
which the Clausius-Clapeyron equation
is derived in the following slides
V
M. D. Eastin
Review of Clausius-Clapeyron Equation
Mathematical Representation:
p (mb)
• Application of the Carnot Cycle…
C
221000
dp s
l

dT TΔ
Liquid
Solid
1013
where:
T =
l =
dps =
dT =
Δα =
Temperature of the system
Latent heat for given phase change
Change in system pressure at saturation
Change in system temperature
Change in specific volumes between
the two phases
6.11
T
Vapor
0
de sw
lv

dT Tα v  α w 
Thermodynamics
de si
ls

dT Tα v  α i 
100
374
T (ºC)
dp wi
lf

dT Tα w  αi 
M. D. Eastin
Review of Clausius-Clapeyron Equation
Computing saturation vapor pressure for a given temperature:
Version #1: Assumes constant latent heat of vaporization (lv = constant)
Less accurate at extreme temperatures
 lv  1
1 

esw (mb)  6.11 exp 

 R v  273.15 T (K)
Version #2: Accounts for temperature dependence of the latent heat [lv(T)]
Most accurate across the widest range of temperatures


6808
esw (mb)  6.11 exp53.49 
 5.09lnT ( K )
T (K )


Thermodynamics
M. D. Eastin
Review of Systems
• Our atmosphere is a heterogeneous
closed system consisting of multiple
sub-systems
• We will now begin to account for the
entire system…
Dry Air
(gas)
pd, T, ρd, md, Rd
Liquid Water
pw, T, ρw, mw
Open sub-system
Water Vapor
e, T, ρv, mv, Rv
Open sub-system
Closed sub-system
Energy Exchange
Mass Exchange
Thermodynamics
Ice Water
pi, T, ρi, mi
Open sub-system
M. D. Eastin
Moist Air Parameters
Our Approach:
• Apply what we have learned thus far:
• Learn how to compute:
Equation of State
First Law of Thermodynamics
Second Law of Thermodynamics
Phase and Latent Heats of water
Clausius-Clapeyron Equation
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)
Thermodynamics
M. D. Eastin
Basic Moisture 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
Dry Air (N2 and O2)
Water Vapor (H2O)
pd  ρd R d T
e  ρ v R vT
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
Thermodynamics
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
M. D. Eastin
Basic Moisture Parameters
2. Mixing Ratio (w):
Definition: Mass of water vapor per unit mass of dry air:
mv ρv
w

md ρd
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:
pd  ρd R d T
e  ρ v R vT
Rd e
w
Rv p  e
How do we find “e”
from observations?
p  pd  e
Thermodynamics
M. D. Eastin
Basic Moisture Parameters
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)
 lv  1
1 
e  6.11 exp 
 
 R v  273.15 Td 
where:
e has units of mb
Td has units of K
Needed Information for Computation:
Rd e
w
Rv p  e
Thermodynamics
Observed variables:
Computed variables:
Physical Constants:
Units:
p, Td
e
Rd, Rv, lv
g/kg
M. D. Eastin
Basic Moisture Parameters
3. Saturation Mixing Ratio (w sw):
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)
w sw 
w sw
Thermodynamics
mv ρv

md ρd
R d esw

R v p  esw
How do we find “esw”
from observations?
M. D. Eastin
Basic Moisture Parameters
3. Saturation Mixing Ratio (w sw):
How do we find “esw”?
Our integrated Clausius-Clapeyron equation
Use T to find the saturation vapor pressure (esw)
esw
 lv  1
1 
 6.11 exp 
 
 R v  273.15 T 
where:
esw has units of mb
T has units of K
Needed Information for Computation:
w sw
Thermodynamics
R d esw

R v p  esw
Observed variables:
Computed variables:
Physical Constants:
Units:
p, T
esw
Rd, Rv, lv
g/kg
M. D. Eastin
Basic Moisture Parameters
4. Specific Humidity (q):
Definition: Mass of water vapor per unit mass of moist air:
m
ρ
q v  v
m
ρ
where:
m  md  mv
  d   v
It is closely related to mixing ratio (w):
w
q
1 w
q
w
1 q
Since both q << 1 and w << 1 in our atmosphere, we often assume
qw
Thermodynamics
M. D. Eastin
Basic Moisture Parameters
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
mv
r
m vsw
Using the Ideal Gas laws for dry and moist air:
e
r
e sw
Note:
Thermodynamics
How do we find “e” and “esw”
from observations?
w
r
w sw
M. D. Eastin
Basic Moisture Parameters
5. Relative Humidity (r):
Finding “e” and “esw”:
 lv  1
1 
e  6.11 exp 
 
 R v  273.15 Td 
esw
where:
 lv  1
1 
 6.11 exp 
 
 R v  273.15 T 
e and esw have units of mb
Td and T has units of K
Needed Information for Computation:
r
Thermodynamics
e
e sw
Observed variables:
Computed variables:
Physical Constants:
Units:
T d, T
e, esw
lv, Rv
%
M. D. Eastin
Skew-T Log-P Diagram
Pressure (200 mb)
Thermodynamics
M. D. Eastin
The Skew-T Log-P Diagram
w sw
R d esw

R v p  esw
w sw (p,T )
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
Thermodynamics
M. D. Eastin
Application: The Skew-T Diagram
Example:
Typical surface observations at the Charlotte-Douglas airport in March:
p = 1000 mb
T = 25ºC
Td = 16ºC
Find the following using a Skew-T Diagram:
Saturation Mixing Ratio (wsw)
Mixing Ratio (w)
Specific Humidity (q)
Relative Humidity (r)
Thermodynamics
M. D. Eastin
Application: The Skew-T Diagram
Given: p = 1000 mb
T = 25°C
Td =16°C
Saturation Mixing Ratio:
w sw
R d esw

R v p  esw
w sw (p,T )
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)
wsw = 22 g/kg
P = 1000 mb
Thermodynamics
T = 25°C
M. D. Eastin
Application: The Skew-T Diagram
Given: p = 1000 mb
T = 25°C
Td =16°C
Mixing Ratio:
Specific Humidity:
w
Rd e
Rv p  e
w (p,Td )
q
w
1 w
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)
w = 12.3 g/kg
q = 12.2 g/kg
P = 1000 mb
Thermodynamics
Td = 16°C
M. D. Eastin
Application: The Skew-T Diagram
Given: p = 1000 mb
T = 25°C
Td =16°C
Relative Humidity:
r
e
e sw
r
w
w sw
r (p,T ,Td )
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
r = 56%
w = 12.3 g/kg
wsw = 22 g/kg
P = 1000 mb
Thermodynamics
Td = 16°C
T = 25°C
M. D. Eastin
Moist Air Parameters during Processes
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)
Thermodynamics
M. D. Eastin
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
Temperature Cools: T1 → T2
• Mass of water vapor and
dry air are constant
• 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
Thermodynamics
Vapor pressure
• Isobaric transformation
esw(T)
esw1
Td
esw2
e
T2
T1
Temperature
M. D. Eastin
Moist Air Parameters during Processes
Isobaric Cooling: Dew Point Temperature (Td)
• 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)
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
Isobaric Cooling: Dew Point Temperature (Td)
T
Td 
 T Rv

1 ln r 
lv


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)
Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
T, r
----Rv, lv
K
M. D. Eastin
Application: The Skew-T Diagram
Given: p = 1000 mb
T = 25°C
r = 56%
Dew Point Temperature:
Td (p,T ,r)
r
w
w sw
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 = 56%
w = 12.3 g/kg
P = 1000 mb
Thermodynamics
Td = 16°C
wsw = 22 g/kg
T = 25°C
M. D. Eastin
Moist Air Parameters during Processes
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
lv
Tw  T  w  w sw 
cp
where:
wsw is the saturation mixing ratio at Tw
w is the mixing ratio at Td
Important
See your text for the full derivation…
Needed Information for Computation:
• Can not be mathematically solved for without iteration
• Easiest to solve for graphically on a Skew-T diagram
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
Adiabatic 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!
Thermodynamics
M. D. Eastin
Application: The Skew-T Diagram
Given:
p = 1000 mb
T = 25ºC
Td = 6ºC
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
Tw = 14ºC
P = 1000 mb
Thermodynamics
Td = 6°C
T = 25°C
M. D. Eastin
In Class Activity
Calculations:
Observations from this morning at CLT:
p = 1000 mb
T = 8.3ºC
Td = 2.8ºC
Compute: w, q, wsw, r
Skew-T Practice:
Observations from yesterday afternoon as CLT:
Graphically estimate:
p = 1000 mb
T = 13.5ºC
r = 32%
Td, Tw
Write your answers on a sheet of paper and turn in by the end of class…
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
 1000

θ m  T
 p 
Rd
(1 0.26q)
cp
Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
p, T, Td (or r)
e, w, q (also esw if using r)
cp, Rd, Rv, lv
K
M. D. Eastin
Moist Air Parameters during Processes
Adiabatic Expansion (or Compression): Moist Potential Temperature (θm)
Note: Since q << 1 in our atmosphere, the difference between the moist
potential temperature (θm) and the dry potential temperature (θ) is
extremely small
 1000

θ m  T
 p 
Therefore:
Rd
(1 0.26q)
cp
The two are essentially equal:
 1000

θ  T
 p 
Rd
cp
θm  θ
The moist potential temperature (θm) is rarely used in practice
Rather, the dry potential temperature (θ) is used
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
w
r
w sw
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
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
Reaching Saturation by Adiabatic Ascent:
Where is the typical Lifting Condensation Level (LCL)?
Cloud
Base
LCL
Rising unsaturated parcels cool to saturation
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
TLCL 
1
 55
 1   ln r 



 T  55   2840
See your text for the full derivation…
Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
T, r (or Td)
----- (e, esw if using Td)
----K
M. D. Eastin
Application: The Skew-T Diagram
Given:
p = 1000 mb
T = 25ºC
Td = 6ºC
Temperature of the Lifting Condensation Level (TLCL):
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. The TLCL is found at the intersection point of the two lines
6. The corresponding pressure pLCL also defines the LCL
TLCL = 2ºC
PLCL = 740 mb
P = 1000 mb
Thermodynamics
Td = 6°C
T = 25°C
M. D. Eastin
Moist Air Parameters during Processes
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
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
Saturated (Moist) Adiabatic Ascent:
Clouds with no precipitation
• Shallow
• No loss of condensed water
• Experience some entrainment
• Ascent is almost reversible
Thermodynamics
Which one occurs in reality?
Clouds with precipitation
• Shallow or Deep
• Loss of condensed water
• Experience some entrainment
• Ascent is almost pseudo-adiabatic
M. D. Eastin
Moist Air Parameters during Processes
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
 1000

θ e  TLCL 
 p 
R d (c p  w t c w )
where: w t  mv  mw  md


lv w
exp

 c p  w t c w TLCL 
Important
Needed Information for Computation:
• Difficult to compute for since mw is unknown
• Can be computed if mw is observed
(e.g. by radar) or estimated
Thermodynamics
Cannot be determined
on a Skew-T diagram
M. D. Eastin
Moist Air Parameters during Processes
Pseudo-Adiabatic 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 falls out of the parcel
• Dry adiabatically descends to 1000 mb
 1000

θ ep  T 
 p 
0.285 (1 0.28w)

 3376

exp w 1  0.81w 
 2.54
 TLCL


Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
p, T, Td, r
e, w, TLCL
Rd, Rv, lv
K
M. D. Eastin
Application: The Skew-T Diagram
Given:
p = 1000 mb
T = 25ºC
Td = 6ºC
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.
θep = 307 K
(34ºC + 273)
T = 20°C
P = 1000 mb
Thermodynamics
Td = 0°C
M. D. Eastin
Moist Air Parameters during Processes
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
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
Thermodynamics
M. D. Eastin
Moist Air Parameters during Processes
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
θw 
θ ep

 3376

exp w 1  0.81w 
 2.54
 θw


where: w is the mixing ratio at θw
Important
See your text for the full derivation…
Needed Information for Computation:
• Can not be mathematically solved for without iteration
• Easiest to solve for graphically on a Skew-T diagram
Thermodynamics
M. D. Eastin
Application: The Skew-T Diagram
Given:
p = 700 mb
T = 8ºC
Td = -11ºC
Wet-bulb Potential Temperature (θw):
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 1000 mb
6. The θw is the resulting temperature at 1000 mb
P = 700 mb
Td = -11°C
T = 8°C
θw = 287 K
P = 1000 mb
Thermodynamics
(14ºC + 273)
M. D. Eastin
Additional Parameters
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)
p  ρR d Tv
where:
p  pd  e
  d   v
Tv  (1 0.61q)T
w
q
1 w
R e
w d
Rv p  e
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…
Thermodynamics
M. D. Eastin
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)
Tv  (1 0.61q)T
See your text for the full derivation…
Advantage:
Simple way to account for variable moisture in an air parcel
Will use more in next chapter…
Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
p, T, Td (or r)
e, w, q
Rd, Rv, lv
K
Cannot be determined
on a Skew-T diagram
M. D. Eastin
Additional Parameters
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)
 1000

θ v  Tv 
 p 
Rd
cp
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…
Needed Information for Computation:
Observed variables:
Computed variables:
Physical Constants:
Units:
Thermodynamics
p, T, Td (or r)
e, w, q
cp, Rd, Rv, lv
K
Cannot be determined
on a Skew-T diagram
M. D. Eastin
Summary: Relationship of Parameters
Lots 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
TLCL  Td  Tw  T  Tv
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
w    m  v  e  ep
Thermodynamics
M. D. Eastin
Summary: The Skew-T Diagram
Given:
p = 800 mb
T = 8.5ºC
Td = -8.0ºC
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)
• Temperature at the LCL (TLCL)
• Pressure at the LCL (PLCL)
• Wet-bulb potential temperature (θw)
• Pseudo-adiabatic equivalent
potential temperature (θep)
Note: All parameter symbols are color-coded with their locations
PLCL
P = 800 mb
P = 1000 mb
Thermodynamics
TLCL
Td, w
Tw
θw
T, wsw
θ
θep
M. D. Eastin
Water Vapor in the Air
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
Thermodynamics
M. D. Eastin
References
Petty, 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
Thermodynamics
M. D. Eastin