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An equation of state describes the relationship
among pressure, temperature, and density of any material. |
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All gases are found to follow approximately the
same equation of state, which is referred to as the “ideal gas law
(equation)”. |
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Atmospheric gases, whether considered
individually or as a mixture, obey the following ideal gas equation: |
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The ideal gas law can be applied to the
combination of atmospheric gases or to individual gases. |
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The value of gas constant for the particular gas
under consideration depends on its molecular weight: |
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Rgas = R* / Mgas |
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where R* = universal gas constant = 8314.3 J deg-1 kg-1 |
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The gas constant for dry atmospheric air is: |
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Rair
= R* / Mair = 8314.3/28.97 = 287 J deg-1 kg-1 |
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(Mair
@ 0.80*MN2 + 0.20*MO2 = 0.80*28 + 0.2*32 =
28.8) |
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The gas constant for water vapor is: |
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Rvapor
= R* / Mvapor = 8314.3/18.016 |
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= 461 J deg-1 kg-1 |
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Question: Calculate the density of water vapor
which exerts a pressure of 9 mb at 20°C. |
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Moist air has a lower apparent molecular weight
that dry air. |
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The gas constant for 1 kg of moist air is larger
than that for 1 kg of dry air. |
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But the
exact value of the gas constant of moist air would depend on the amount of
water vapor contained in the air. |
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It is inconvenient to calculate the gas constant
for moist air. |
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It is more convenient to retain the gas constant
of dry air and use a fictitious temperature in the ideal gas equation. |
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This fictitious temperature is called “virtual
temperature”. |
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This is the temperature that dry air must have
in order to has the same density as the moist air at the same pressure. |
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Since moist air is less dense that dry air, the
virtual temperature is always greater than the actual temperature. |
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Where
T: actual temperature |
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p: actual (total) pressure =
pd + e |
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pd: partial pressure exerted by dry air |
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e:
partial pressure exerted by water vapor |
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e = Rd/Rv
= 0.622 |
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The
hydrostatic equation tells us how quickly air pressure drops wit height. |
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èThe rate at which air pressure decreases with
height (DP/ Dz) is equal to the air density (r) times the
acceleration of gravity (g) |
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Since P= rRT (the ideal gas law), the hydrostatic
equation becomes: |
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dP = -P/RT x gdz |
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è
dP/P = -g/RT x dz |
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P
= Ps exp(-gz/RT) |
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P = Ps
exp(-z/H) |
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The atmospheric pressure decreases exponentially
with height |
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One way to measure how soon the air runs out in
the atmosphere is to calculate the scale height, which is about 10 km. |
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Over this vertical distance, air pressure and
density decrease by 37% of its surface values. |
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If pressure at the surface is 1 atmosphere, then
it is 0.37 atmospheres at a height of 10 km, 0.14 (0.37x0.37) at 20 km,
0.05 (0.37x0.37x0.37) at 30 km, and so on. |
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Different atmospheric gases have different
values of scale height. |
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The heavier the gas molecules weight (m) è the
smaller the scale height for that particular gas |
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The higher the temperature (T) è the more
energetic the air molecules è the larger the scale height |
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The larger the gravity (g) è air
molecules are closer to the surface è the smaller the scale height |
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H has a value of about 10km for the mixture of
gases in the atmosphere, but H has different values for individual gases. |
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The core of a hurricane is warmer than its
surroundings. |
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The intensity of the hurricane (as measured by
the depression of pressure surface) must decrease with height. |
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Thus, a warm core hurricane exhibits its
greatest intensity near the ground and diminish with increasing height
above ground. |
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Energy is the capacity to do work. |
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Heat is one form of energy. |
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Heat is one form of internal energy which is
associated with the random, disordered motion of molecules and atoms. |
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Internal kinetic/potential energy are different
from the macroscopic kinetic/potential energy. |
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Air
temperature is a measurement of the average internal kinetic energy of air
molecules. |
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Increase
in internal kinetic energy in the form of molecular motions are manifested
as increases in the temperature of the body. |
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This law states that (1) heat is a form of
energy that (2) its conversion into other forms of energy is such that
total energy is conserved. |
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The change in the internal energy of a system is
equal to the heat added to the system minus the work down by the system: |
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Therefore, when heat is added to a gas, there
will be some combination of an expansion of the gas (i.e. the work) and an
increase in its temperature (i.e. the increase in internal energy): |
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Heat and temperature are both related to the
internal kinetic energy of air molecules, and therefore can be related to
each other in the following way: |
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Add
(remove) heat to (from) the air parcel (diabatic processes) |
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(1)
Conduction: requires touching |
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(2)
Convection: Hot air rises |
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(3)
Advection: horizontal movement of air |
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(4)
Radiation: exchanging heat with space |
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(5)
Latent heating: changing the phase of water |
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Without adding (removing) heat to (from) the air
parcel |
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(1)
Adiabatic Process: Expanding and compressing air |
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Conduction is the process of heat transfer from
molecule to molecule. |
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This energy transfer process requires contact. |
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Air is a poor conductor. (with low thermal
conductivity) |
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Conduction is not an efficient mechanisms to
transfer heat in the atmosphere on large spatial scales. |
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Convection is heat transfer by mass motion of a
fluid (such as air or water). |
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Convection is produced when the heated fluid
moves away from the heat source and carries energy with it. |
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Convection is an efficient mechanism of heat
transfer for the atmosphere in some regions (such as the tropics) but is an
inefficient mechanism in other regions (such as the polar regions). |
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Advection is referred to the horizontal transport of heat in
the atmosphere. |
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Warm air advection occurs when warm air replaces
cold air. Cold air advection is the other way around. |
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This process is similar to the convection which
relies on the mass motion to carry heat from one region to the other. |
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Advection can be considered as one form of
convection. |
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Radiation is heat transfer by the emission of
electromagnetic waves which carry energy away from the emitting object. |
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The solar energy moves through empty space from
the Sun to the Earth and is the original energy source for Earth’s weather
and climate. |
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Latent heat is the heat released or absorbed per
unit mass when water changes phase. |
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Latent heating is an efficient way of
transferring energy globally and is an important energy source for Earth’s
weather and climate. |
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The latent heat of evaporation is a function of
water temperature, ranging from 540 cal per gram of water at 100°C to 600
cal per gram at 0°C. |
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It takes more energy to evaporate cold water
than evaporate the same amount of warmer water. |
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If a material changes its state (pressure,
volume, or temperature) without any heat being added to it or withdrawn
from it, the change is said to be adiabatic. |
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The adiabatic process often occurs when air
rises or descends and is an important process in the atmosphere. |
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Air
pressure decreases with elevation. |
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If a
helium balloon 1 m in diameter is released at sea level, it expands as it
floats upward because of the pressure decrease. The balloon would be 6.7 m
in diameter as a height of 40 km. |
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Air
molecules in the parcel (or the balloon) have to use their kinetic energy
to expand the parcel/balloon. |
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Therefore, the molecules lost energy and slow
down their motions |
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è The
temperature of the air parcel (or balloon) decreases with elevation. The
lost energy is used to increase the potential energy of air molecular. |
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Similarly when the air parcel descends, the
potential energy of air molecular is converted back to kinetic energy. |
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è Air
temperature rises. |
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Static stability is referred as to air’s
susceptibility to uplift. |
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The static stability of the atmosphere is
related to the vertical structure of atmospheric temperature. |
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To determine the static stability, we need to
compare the lapse rate of the atmosphere (environmental lapse rate) and the
dry (moist) adiabatic lapse rate of an dry (moist) air parcel. |
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The environmental lapse rate is referred to as
the rate at which the air temperature surrounding us would be changed if we
were to climb upward into the atmosphere. |
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This rate varies from time to time and from
place to place. |
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Ge = environmental lapse rate |
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Gd = day adiabatic lapse rate |
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Gm = moist lapse rate |
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Absolutely Stable |
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Ge < Gm |
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Absolutely Unstable |
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Ge > Gd |
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Conditionally Unstable |
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Gm < Ge < Gd |
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At the end of a sunny day, warm air near the
surface, cold air aloft. |
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In the early morning, cold air near the surface,
warm air aloft. |
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The later condition is called “inversion”, which
inhibits convection and can cause sever pollution in the morning. |
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The potential temperature of an air parcel is
defined as the the temperature the parcel would have if it were moved
adiabatically from its existing pressure and temperature to a standard
pressure P0 (generally taken as 1000mb). |
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In the atmosphere, air parcel often moves around
adiabatically. Therefore, its potential temperature remains constant
throughout the whole process. |
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Potential temperature is a conservative quantity
for adiabatic process in the atmosphere. |
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Potential temperature is an extremely useful
parameter in atmospheric thermodynamics. |
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Evaporation: the process whereby molecules break
free of the liquid volume. |
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Condensation: water vapor molecules randomly
collide with the water surface and bond with adjacent molecules. |
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On
average, 1 meter of water is evaporated from oceans to the atmosphere each
year. |
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The
global averaged precipitation is also about 1 meter per year. |
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Earth’s surface lost heat to the atmosphere when
water is evaporated from oceans to the atmosphere. |
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The evaporation of the 1m of water causes
Earth’s surface to lost 83 watts
per square meter, almost half of the sunlight that reaches the surface. |
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Without the evaporation process, the global
surface temperature would be 67°C instead of the actual 15°C. |
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by mass |
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by vapor pressure |
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Specific Humidity: How many grams of water vapor
in one kilogram of air (in unit of gm/kg). |
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Relative Humidity: The percentage of current
moisture content to the saturated moisture amount (in unit of %). |
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Clouds form when the relative humidity reaches
100%. |
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The
air’s content of moisture can be measured by the pressure exerted by the
water vapor in the air. |
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The total pressure inside an air parcel is equal
to the sum of pressures of the individual gases. |
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In the left figure, the total pressure of the
air parcel is equal to sum of vapor pressure plus the pressures exerted by
Nitrogen and Oxygen. |
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High vapor pressure indicates large numbers of
water vapor molecules. |
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Unit of vapor pressure is usually in mb. |
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Saturation vapor pressure describes how much
water vapor is needed to make the air saturated at any given temperature. |
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Saturation vapor pressure depends primarily on
the air temperature in the following way: |
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è |
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Saturation pressure
increases exponentially with air temperature. |
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Two ways: |
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Increase (inject more) water vapor to the air (Aà B). |
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Reduce the temperature of the air (A à C). |
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If a planet has a very high temperature that the
air can never reach a saturation point |
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Water vapor can be added into the atmosphere. |
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More water vapor traps more heat (a greenhouse
effect) |
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The planet’s temperature increases furthermore |
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Ever
more water evaporated into the atmosphere |
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More
greenhouse effect |
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More
warming |
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More
water vapor |
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….. |
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Dew point temperature is another measurement of
air moisture. |
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Dew point temperature is defined as the
temperature to which moist air must be cool to become saturated without
changing the pressure. |
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The close the dew point temperature is to the
air temperature, the closer the air is to saturation. |
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Notes