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MET 61 Introduction to Meteorology - Lecture 4
“Heat in the atmosphere”
Dr. Eugene Cordero
San Jose State University
W&H: Chap 3, Pg 74-84
Ahrens: Chapter 4
Class Outline:
• Sensible and latent heat
• First law of thermodynamics
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Thermodynamic Diagram
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Green
Dry Adiabats
Red
Moist Adiabats
Yellow
Saturation
Mixing
Ratio
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Sensible heat
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• Is related to the energy exchange that can be
measured by a temperature change:
Q H
 C p T
m air
Cp=specific heat at constant pressure
Cp=Cpd(1+0.84w); w=water vapor mixing
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Latent Heat
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• Latent heat is the energy exchanged during a
phase change (I.e. liquid to vapor etc.)
• Heat exchanged without a temperature change.
Q E
L
m water
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Latent Heat
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• The heat energy required to change water from
one state to another (e.g. water from a vapor to
a solid).
• Latent heat of evaporation:
Heat lost by environment
• Latent heat of condensation
Heat given to environment
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1st law of thermodynamics
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• Relates the change in internal energy with the
heat added and the work done by the body
du  dq  dw
U, q and w are defined as per unit mass
du- internal energy (proportional to motion of molecules)
dq- increment of heat added
dw- work done by the body
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1st law of thermo
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• Work: related to a force acting on an object to
cause a displacement.
• Basically, the temperature of a parcel changes
when heat is added (dq) or when work is done
(dw)
• Alternate form:
dq  c p dT 
dp

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Water Vapor
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The amount of water vapor present in the air can be expressed in a
variety of ways:
Mixing Ratio:
Where mv is the mass of water vapor; md is the mass of dry air:
Units for w are typically given in: (g of water vapor/ kg of air)
Typical values; 1-5 g/kg midlatitudes and up to 20 g/kg in the tropics
If there is no condensation or evaporation, then the mixing ratio
of an air parcel is a conserved quantity.
Note: symbol r is also commonly used for the mixing ratio
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Saturation Vapor Pressure
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The saturation vapor pressure is defined as the maximum amount of
water vapor necessary to keep moist air in equilibrium with a surface
of pure water or ice.
Consider a box containing air and water. If the box is initially dry, the
water will evaporate and the water vapor pressure in the air will
increase.
Eventually, an equilibrium will be reached where evaporation and
condensation are equal. At this point, the air is ‘saturated’ and the
vapor pressure, e=es.
The saturation vapor pressure: es=es(T)
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Saturation Vapor Pressure
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The Clausius-Clapeyron equation describes the relationship between
the saturation vapor pressure and temperature. You may see the
derivation of this expression later in the course or further courses. For
now, the relationship is:
Where e0=6.11 hPa, T0 = 273°K and L represents the latent heat of
vaporization (Lv== 2.453 × 106 J/kg) or the latent heat of deposition
(Ld=2.8 x 106 J/kg).
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The Clausius-Clapeyron equation
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Example problem: What is the saturation vapor pressure
when the temperature is 30° C?
Answer: Convert temperature to Kelvins, 30° C = 303 K
ln(Es/6.11) = (2.453×106 J/kg/461 J/kg)(1/273 - 1/303)
Es = (e1.92)(6.11) = 42.1 hPa
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Saturation Mixing Ratio
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The saturation mixing ratio, with respect to water is defined as the
ratio of the mass of saturated water vapor, mvs, to the mass of dry air,
md .
We can express the saturation mixing ratio in terms of pressure as:
Where vs' is the partial density of water vapor required to saturate air
and d' is the partial density of dry air.
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Saturation Mixing Ratio
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The above equation can be further simplified as:
For typical values of temperatures in the Earth’s atmosphere, p>>es,
and thus
Thus, ws=ws(T,P)
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Relative Humidity
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The relative humidity is generally the ratio of the amount of water
vapor in the air compared to the maximum amount of water the air can
hold.
actual vapor pressure
RH 
saturation vapor pressure
water vapo r content
RH 
water vapo r capacity
RH is a function of temperature:
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Dew Point
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The dew point temperature, Td, is the temperature air must be cooled
at constant pressure for the air to become saturated. Or, this is the
temperature where the actual mixing ratio and the saturation mixing
ratio are equal.
Dewpoint is a good measure of the amount of water in the air.
Td ~ 20°C uncomfortable
Td > 24 °C very uncomfortable!
The Clausius-Clapeyron can be used to determine the dew point. If
you know the saturation vapor pressure, es, and Td=T and solve for
Td, giving:
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Lifting Condensation Level
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The lifting condensation level (LCL) defines the level where a moist
parcel lifted adiabatically will become saturated.
Because dry air must be lifted further to reach the LCL than moist air,
the LCL height serves as another measure of the amount of water
vapor in the air.
During the lifting of a parcel, one often assumes that:
- the mixing ratio, w, of the parcel is conserved.
- the potential temperature of the parcel is conserved.
At the LCL, the mixing ratio, w and the saturation mixing ratio, ws are
equal.
One often uses a pseudoadiabatic chart to find the LCL. However,
there is a relationship:
LCL=125m (T-Td) in ºC
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Thermodynamic Diagram
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Green
Dry Adiabats
Red
Moist Adiabats
Yellow
Saturation
Mixing
Ratio
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Activity 3 (Due Feb 14th)
1. Show that for adiabatic motions,
increases in temperatures are
accompanied by decreases in
geopotential.
2. Derive an expression for the dry adiabatic
lapse rate.
3. Plot out the vertical distribution of
potential temperature between the surface
and 10hPa.
4. Exercise 3.42
5. Exercise 3.46
6. Exercise 3.47
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MET 61 Introduction to Meteorology
Questions
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1. Calculate the density of water vapor which
exerts a pressure of 9 mb at 20°C.
2. Determine the virtual temperature of moist air
at 30 °C which has a mixing ratio of 20 g/kg.
3. Air at 1000hPa and 18 °C has a mixing ratio of
6g/kg. What is the relative humidity and dew
point?
4. In (3), determine the LCL using the given
relationship between dew point and
temperature.
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