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ESS55Prof. Jin-Yi Yu
Lecture 2: Atmospheric Thermodynamics
Ideal Gas Law (Equation of State)
Hydrostatic Balance
Heat and Temperature
Conduction, Convection, Radiation
Latent Heating
Adiabatic Process
Lapse Rate and Stability
ESS55Prof. Jin-Yi Yu
The Ideal Gas Law
An
equation of state
describes the relationship among pressure,
temperature, and density of
any material
All gases are found to follow approximately the same equationof state, which is referred to as the
“ideal gas law (equation)”
Atmospheric gases, whether considered individually or as amixture, obey the following ideal gas equation:
P =
R T
pressure
Density=m/V
temperature
(degree Kelvin)
gas constant
(its value depends on the gas considered)
ESS55Prof. Jin-Yi Yu
Gas Constant
The ideal gas law can be applied to the combination ofatmospheric gases or to individual gases.
The value of gas constant for the particular gas underconsideration depends on its molecular weight:
R
gas
= R* / M
gas
where R* = universal gas constant = 8314.3 J deg
kg
The gas constant for dry atmospheric air is:
R
air
= R* / M
air
= 8314.3/28.97 = 287 J deg
kg
(M
air
0.80*M
N
+ 0.20*M
O
The gas constant for water vapor is:R
vapor
= R* / M
vapor
= 461 J deg
kg
ESS55Prof. Jin-Yi Yu
Applications of the Gas law
Question:
Calculate the density of water vapor which exerts a
pressure of 9 mb at 20°C. Answer: Use the ideal gas law:
P
v
ρ
R
v
T
and
P
v
= 9 mb = 900 Pa (a SI unit)
R
v
= R* / M
v
= 461 J deg
kg
T = 273 + 20 (°C) = 293 K.
So we know the density of water vapor is:
ρ
= P
v
/ (R
v
T) = 900 / (461*293) = 6.67 x 10
kg m
(from
Atmospheric Sciences: An introductory Survey
)
ESS55Prof. Jin-Yi Yu
Virtual Temperature
Moist air has a lower apparent molecular weight that dry air.
Î
The gas constant for 1 kg of moist air is larger than that for 1 kg ofdry air.
Î
But the exact value of the gas constant of moist air would depend on the amount of water vapor contained in the air.
Î
It is inconvenient to calculate the gas constant for moist air.
It is more convenient to retain the gas constant of dry air and use afictitious temperature in the ideal gas equation.
Î
This fictitious temperature is called “virtual temperature”.
Î
This is the temperature that dry air must have in order to has the samedensity as the moist air at the same pressure.
Î
Since moist air is less dense that dry air, the virtual temperature isalways greater than the actual temperature.
ESS55Prof. Jin-Yi Yu
How to Calculate Virtual Temperature?
Where
T: actual temperaturep: actual (total) pressure = p
d
p
d
: partial pressure exerted by dry air e: partial pressure exerted by water vapor
ε
= R
d
/R
v
ESS55Prof. Jin-Yi Yu
Hydrostatic Balance in the Vertical
vertical pressure force = gravitational force
ρ
x (dz) x (dA) x g
dP = -
ρ
gdz
dP/dz = -
ρ
g
(from
Climate System Modeling
)
The hydrostatic balance !!
ESS55Prof. Jin-Yi Yu
What Does Hydrostatic Balance Tell Us?
The hydrostatic equation tells us how
quickly air pressure drops wit height.
Î
The rate at which air pressure decreases with
height (
P/
z) is equal to the air density (
times the acceleration of gravity (g)
ESS55Prof. Jin-Yi Yu
Warm Core Hurricane
The core of a hurricane is warmer than its surroundings.
The intensity of the hurricane (as measured by the depression ofpressure surface) must decrease with height.
Thus, a warm core hurricane exhibits its greatest intensity near theground and diminish with increasing height above ground.
(from
Understanding Weather & Climate and Atmospheric Sciences: An Intro. Survey)
Pressure Surface
Z
surface
tropopause
hurricane center
ESS55Prof. Jin-Yi Yu
Air Temperature Air Pressure
hydrostatic balance
Air Motion
geostrophic balance
thermal wind balance
Energy (Heat)
The first law of thermodynamics
ESS55Prof. Jin-Yi Yu
Heat and Energy
Energy is the capacity to dowork.
Heat is one form of energy.
Heat is one form of internalenergy which is associatedwith the random, disorderedmotion of molecules andatoms.
Internal kinetic/potentialenergy are different from themacroscopic kinetic/potentialenergy.
water
no macroscopic kinetic/potential energy
internal kinetic energy (related to temperature)
internal potential energy
(related to the phase)
ESS55Prof. Jin-Yi Yu
What Is Air Temperature?
Air temperature is a measurement of the average internal kinetic energy of air molecules.
Increase in internal kinetic energy in the form of molecular motions are manifested as increases inthe temperature of the body.
ESS55Prof. Jin-Yi Yu
The First Law of Thermodynamics
This law states that (1) heat is a form of energythat (2) its conversion into other forms of energyis such that total energy is conserved.
The change in the internal energy of a system isequal to the heat added to the system minus thework down by the system:
U = Q - W
change in internal energy(related to temperature)
Heat added to the system
Work done by the system
ESS55Prof. Jin-Yi Yu
Therefore, when heat isadded to a gas, there will besome combination of anexpansion of the gas (i.e. thework) and an increase in itstemperature (i.e. the increasein internal energy):
Heat added to the gas = work done by the gas + temp. increase of the gas
H =
p
∆α
C
v
T
volume change of the gas
specific heat at constant volume
(from
Atmospheric Sciences: An Intro. Survey)
ESS55Prof. Jin-Yi Yu
Heat and Temperature
Heat and temperature are both related to the internalkinetic energy of air molecules, and therefore can berelated to each other in the following way:
Q = cm
T
Heat added
Specific heat = the amount of heat per unit mass required
Mass to raise the temperature by one degree Celsius
Temperature changed
ESS55Prof. Jin-Yi Yu
Specific Heat
(from
Meteorology: Understanding the Atmosphere
)
ESS55Prof. Jin-Yi Yu
Radiation
Radiation is heat transfer by the emission ofelectromagnetic waves which carry energy awayfrom the emitting object.
The solar energy moves through empty spacefrom the Sun to the Earth and is the originalenergy source for Earth’s weather and climate.
ESS55Prof. Jin-Yi Yu
Latent Heating
Latent heat is the heat released or absorbed per unit mass whenwater changes phase.
Latent heating is an efficient way of transferring energyglobally and is an important energy source for Earth’s weatherand climate.
(from
Meteorology:
Understanding theAtmosphere
)
80 cal/gm
600 cal/gm
680 cal/gm
ESS55Prof. Jin-Yi Yu
Latent Heat of Evaporation
The latent heat of evaporation is a function ofwater temperature, ranging from 540 cal per gramof water at 100°C to 600 cal per gram at 0°C.
It takes more energy to evaporate cold water thanevaporate the same amount of warmer water.
ESS55Prof. Jin-Yi Yu
Adiabatic Process
If a material changes its state (pressure,volume, or temperature) without any heatbeing added to it or withdrawn from it, thechange is said to be adiabatic.
The adiabatic process often occurs when airrises or descends and is an important processin the atmosphere.
ESS55Prof. Jin-Yi Yu
Air Parcel Expands As It Rises…
Air pressure decreases
with elevation.
If a helium balloon 1 m in
diameter is released at sealevel, it expands as it floatsupward because of thepressure decrease. Theballoon would be 6.7 m indiameter as a height of 40km.
(from
The Blue Planet
)
ESS55Prof. Jin-Yi Yu
What Happens to the Temperature?
Air molecules in the parcel (or the balloon) have to use their kinetic energy to expand the parcel/balloon.
Therefore, the molecules lost energy and slow down theirmotions
Î
The temperature of the air parcel (or balloon) decreases with elevation. The lost energy is used to increase the potentialenergy of air molecular.
Similarly when the air parcel descends, the potential energy ofair molecular is converted back to kinetic energy. Î
Air temperature rises.
ESS55Prof. Jin-Yi Yu
Dry Adiabatic Lapse Rate
(from
Meteorology: Understanding the Atmosphere
)
ESS55Prof. Jin-Yi Yu
Moist Adiabatic Lapse Rate (from
Meteorology: Understanding the Atmosphere
)
ESS55Prof. Jin-Yi Yu
Absolutely Stable Atmosphere
(from
Meteorology Today)
ESS55Prof. Jin-Yi Yu
Absolutely Unstable Atmosphere
(from
Meteorology Today)
ESS55Prof. Jin-Yi Yu
Conditionally Unstable Atmosphere
(from
Meteorology Today)
ESS55Prof. Jin-Yi Yu
Day/Night Changes of Air Temperature
At the end of a sunny day, warm air near the surface, coldair aloft.
In the early morning, cold air near the surface, warm airaloft.
The later condition is called “inversion”, which inhibitsconvection and can cause sever pollution in the morning.
End of Day
Night
(from
Is the Temperature Rising?)
ESS55Prof. Jin-Yi Yu
Stability and Air Pollution
Neutral Atmosphere (Coning)Stable Atmosphere (Fanning)Unstable Atmosphere (Looping)Stable Aloft; Unstable Below (Fumigation)Unstable Aloft; Stable Below (Lofting)
(from
Is the Temperature Rising?)
ESS55Prof. Jin-Yi Yu
Potential Temperature (
The potential temperature of an air parcel is defined as the thetemperature the parcel would have if it were movedadiabatically from its existing pressure and temperature to astandard pressure P
0
(generally taken as 1000mb).
θ
= potential temperature
T = original temperatureP = original pressureP
0
= standard pressure = 1000 mb
R = gas constant = R
d
= 287 J deg
kg
C
p
= specific heat = 1004 J deg
kg
R/C
p
ESS55Prof. Jin-Yi Yu
Importance of Potential Temperature
In the atmosphere, air parcel often moves aroundadiabatically. Therefore, its potential temperatureremains constant throughout the whole process.
Potential temperature is a conservative quantity foradiabatic process in the atmosphere.
Potential temperature is an extremely useful parameterin atmospheric thermodynamics.
ESS55Prof. Jin-Yi Yu
Adiabatic Chart
The expression of potential temperature can be modified into:
**T = (constant ***
θ
) P
0.
(from
Atmospheric Sciences: An Intro. Survey)
(from
The Physics of the Atmospheres)
ESS55Prof. Jin-Yi Yu
Measuring Air Moisture
by mass
by vapor pressure
in unit of g/kgin unit of g/m
3
in unit of %
ESS55Prof. Jin-Yi Yu
Observed Specific Humidity
(from
Meteorology Today)
ESS55Prof. Jin-Yi Yu
Specific .vs. Relative Humidity
Specific Humidity: How many grams of water vapor in one kilogramof air (in unit of gm/kg).
Relative Humidity: The percentage of current moisture content to thesaturated moisture amount (in unit of %).
Clouds form when the relative humidity reaches 100%.
specific humidity
6 gm/kg
saturated
specific humidity
10 gm/kgsaturated
specific humidity
20 gm/kg
Relative humidity 6/20 x 100
%
=
%
Relative humidity 6/10 x 100
%
=
%
ESS55Prof. Jin-Yi Yu
Vapor Pressure
The air’s content of moisture can be measured by the pressure exertedby the water vapor in the air.
The total pressure inside an airparcel is equal to the sum ofpressures of the individual gases.
In the left figure, the total pressureof the air parcel is equal to sum ofvapor pressure plus the pressuresexerted by Nitrogen and Oxygen.
High vapor pressure indicates largenumbers of water vapor molecules.
Unit of vapor pressure is usually inmb.
(from
Meteorology Today)
ESS55Prof. Jin-Yi Yu
Saturation Vapor Pressure
Saturation vapor pressure describes howmuch water vapor is needed to make theair saturated at any given temperature.
Saturation vapor pressure dependsprimarily on the air temperature in thefollowing way:
Î
Saturation pressure increasesexponentially with air temperature.
TheClausius-ClapeyronEquation
L: latent heat of evaporation;
α
: specific volume of vapor and liquid
ESS55Prof. Jin-Yi Yu
How to Saturate the Air?
Two ways:
(1)
Increase (inject more) water vapor to the air (A
Æ
B).
(2)
Reduce the temperature of the air (A
Æ
C).
(from “
IS The Temperature Rising
”)
ESS55Prof. Jin-Yi Yu
“Runway” Greenhouse Effect
If a planet has a very high temperature that the air cannever reach a saturation point
Î
Water vapor can be added into the atmosphere.
Î
More water vapor traps more heat (a greenhouse effect)
Î
The planet’s temperature increases furthermore
Î
Ever more water evaporated into the atmosphere
Î
More greenhouse effect
Î
More warming
Î
More water vapor
Î
ESS55Prof. Jin-Yi Yu
Dew Point Temperature
Dew point temperature isanother measurement of airmoisture.
Dew point temperature isdefined as the temperature towhich moist air must be coolto become saturated withoutchanging the pressure.
The close the dew pointtemperature is to the airtemperature, the closer theair is to saturation.
(from
The Atmosphere
)
ESS55Prof. Jin-Yi Yu
An Example
ESS55Prof. Jin-Yi Yu
Applications of Adiabatic Chart
(from
Meteorology Today)
