1. Atmospheric Thermodynamics
Sections 1.2, 1.3, 1.5, and 1.6 of old AS
Chapter 3 of new AS (on line)
Pages 123 to 169 or UWC
(only covers some of the topics)
Chapters 2 and 3 of MSE
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2. The Florida State University
Topics In This Section
Properties of gases.
Ideal gas law (several versions)
The first law of thermodynamics
Changes in pressure and density with height
Adiabatic Processes
Water vapor in the Atmosphere
Static Stability
The Second Law of Thermodynamics and Entropy
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3. The Florida State University
Properties of Matter
Key properties can be divided into large-scale and small (atomic) scale properties.
Large scale properties can be further divided into
properties that do change when the material is subdivided:
Volume
Mass
Number of molecules
those that don’t change when the material is subdivided:
Density
Pressure
Temperature
Small scale feature are related to atomic structure.
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4. Pressure Cooker Example: Constant density; Changing T & P
P = Pressure T = Temperature  = Density C = Constant
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5. Ideal Gas Law (General Form)
We will discuss the general form of the ideal gas law, then the form that is almost universally used in meteorology
General for of ideal gas law
P v = n R* T
P = pressure [Pa]
v = volume [m3]
n = number of molecules
units of moles
1 mol = x1023 molecules
R* = universal gas constant = 8314 J mol-1 K-1
T = temperature [K]
There are many other equivalent forms of this law.
It applies when the density of the gas is sufficiently small that the electron fields of neighboring molecules are usually negligible.
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6. Avogadro’s Number And Hypothesis
The number of molecules per mole is Avogadro’s number
NA = x1023
Avogadro’s Hypothesis:
Gases containing the same number of molecules occupy the same volumes at the same pressure and temperature.
The universal constant for one molecule of gas is Boltzmann’s constant (k).
k = R* / NA
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7. Boyle’s Law and Charles’ Two Laws
If temperature is constant, then the chemistry version of the ideal gas law easily explains Boyle’s Law:
If the temperature of a fixed mass of gas is constant, the volume of the gas is inversely proportional to the pressure.
Pv = nR*T  Pv  constant
This law applies for isothermal conditions.
Charles’ First Law:
For a fixed mass of gas at a constant pressure, the volume of gas is directly proportional to its absolute temperature.
Pv = nR*T  v  T
Charles’ Second Law:
For a fixed mass of gas held within a fixed volume, the pressure of the gas is proportional to its absolute temperature.
Pv = nR*T  P  T
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8. Ideal Gas Law (Meteorology Form)
The general form is modified to apply to the lower atmosphere.
General form of ideal gas law
Pv = nR*T
Divide by volume
P = (n/v)R*T
Multiply n/v by the molecular weight of the mixture (M), and divide R by the same quantity
P = (M n / v) (R* / M) T
M n is the mass of mixture. Dividing mass by volume is density ()
Replace the universal R* with an lower atmosphere R
R = R* / M = JK-1kg-1
P =  R T
In most applications that you will see, we will use R rather than R*.
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9. Alternative Forms of Ideal Gas Law
The meteorology form of the ideal gas law can easily be modified to be in terms of specific volume () rather than density ()
 = 1 /  = volume / mass [m3kg-1]
P  = R T
For a gas containing no molecules per unit volume, the ideal gas law becomes
P = no k T
The meteorology version of R is dependent on a weighted mean of the molecular masses of the molecules that make up the gas. The value for the meteorology version of R is determined with the assumption that there is no water vapor in the air. It is usually written Rd.
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10. Ideal Gas Law for Moist Air
The value for the meteorological R on the last page was derived assuming dry air.
The concentration of water vapor is a highly variable quantity in the lower atmosphere.
R could be modified to account of this variability.
R becomes Rd( r), where r is the mixing ratio (the mass of water vapor divided by the mass of dry air).
The meteorological ideal gas law becomes
P =  Rd ( r) T
However, in practice, most meteorologists prefer to write this as
P =  Rd Tv, where Tv is the virtual temperature Tv = T( r)
CAUTION: T and Tv are in units of K, not C
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11. Derivation of Virtual Temperature
A Partial Pressure is the pressure due to one type of gas molecule. The partial pressures sum to the total pressure.
The partial pressure for dry air is
Pd = d Rd T, Rd = 287 J K-1 kg-1
d is the density of dry air.
The partial pressure for water vapor is
Pv = v Rv T, Rv = J K-1 kg-1
v is the density of water vapor.
In meteorological applications it is common to use the symbol e to indicate the partial pressure of water vapor.
e = v Rv T
The total pressure is P = Pd + e
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12. Derivation of Virtual Temperature
The density of moist air can be written as
 = (md + me) / V = d + v
Where d and v are determined assuming that only the dry air occupied the whole volume V, and that only the water vapor occupied the whole volume V.
Then
 = (P - e) / (Rd T) + e / (Rv T)
Where  = Rd / Rv = Mw / Md =
Then
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13. Derivation of Virtual Temperature
The ideal gas law can be used to show that e / P is equal to
Where r is the mixing ratio, the mass of water vapor divided by the mass of dry air, usually discussed in units of g water vapor per kg of dry air, but applied in kg water vapor / kg dry air.
Consider that 1/(1-x) can be approximated as 1+x, provided that x is small. Therefore Tv  T ( r )
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14. More on Virtual Temperature
Virtual temperature is a fictitious temperature rather than an actual temperature.
Virtual temperature is the temperature that dry air would have to achieve to have the same density as moist air at the same pressure as the dry air.
Since moist air is less dense than dry air, the virtual temperature is always equal to or greater than the actual temperature.
Even for very moist air, the virtual temperature is only a few degrees warmer than the actual temperature.
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15. Application of the Hydrostatic Equation
One of the key equations in Meteorology is the hydrostatic equation.
It describes the change in pressure with changes in height.
P/ z = - g
Integrate the hydrostatic equation from the height of interest to the top of the atmosphere (approximated as infinity).
This result indicates that the pressure at any height in the atmosphere is equal to the weight of the air above that height, within a column of unit area (that is, a column that has an area of 1 in the units of distance squared).
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16. How Rapidly Does Pressure Change With Increasing Height?
Consider both the Hydrostatic Equation and the ideal gas law.
P/ z = - g
P =  Rd Tv
Solve the ideal gas law for density, and replace density in the hydrostatic equation
 P/ z = -(P / Rd Tv) g
Solving this results in an approximately logarithmic change in pressure with changes in altitude (assuming Rd and Tv are constant – they are treated as mean values):
Recall that  P / P =  ln( P ), where ln means nature log
 ln(P) /  z = -g / Rd Tv, or  ln(P) = -(g / Rd Tv)  z
Integrate to get the Hypsometric equation
Note that P1 is the pressure at height z1
z2 – z1 = (Rd / g) Tv ln( P1 / P2 ) (Hypsometric Equation)
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17. More on the Hypsometric Equation
Recall that  P/ z = -(P / Rd Tv) g
 ln(P) /  z = -g / Rd Tv, or  ln(P) = -(g / Rd Tv)  z
Integrate to get the Hypsometric equation
Note that P1 is the pressure at height z1
z2 – z1 = (Rd / g) Tv ln(P1 / P2 ) (Hypsometric Equation)
ln(P(z)) = ln(Psfc) -(g / (Rd Tv) ) z
P(z) = Psfc exp(-g z / (Rd Tv) )
Approximate because Tv is not really a constant.
The rate of change of pressure with height decreases as the pressure decreases (i.e., as the height increases).
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Geopotential Height
For applications where energy is the key consideration (e.g., large scale circulations), it is often desirable to use geopotential height (Z) as a vertical coordinate.
The geopotential () at any point in the atmosphere is defined as the work that must be done against the planet’s gravitational attraction to raise a 1 kg weight from mean sea level to that point.
The value of gravitational attraction is not quite constant with height. Recall that the force of attraction between two masses is a function of the distance (r) between their centers of mass.
F = G mplanet mobject r-2
The geopotential height (Z) is
z (km) Z(km) g (ms-2)
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19. Changes in Temperature with Altitude
The mean surface pressure is 1013 mb
See that P/ z =
[P(z2) – P(z1)] / [z2 – z1] decreases as the pressure decreases.
If we assumed a constant density, equal to the mean surface value, then the thickness of the atmosphere would be about 10 km.
This can be shown by integrating the hydrostatic equation.
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20. Change in Density with Altitude
Again recall the ideal gas law
P =  Rd Tv
Solving for density:
 = P / (Rd Tv)
Recall that pressure decreases with height. Assume that the changes in Rd and Tv are relatively small.
Then, the density changes approximately as the pressure changes, with a decreasing rate of change as altitude increases.
 = Psfc exp(-g z / (Rd Tv) ) / (Rd Tv)
= sfc exp(-g z / (Rd Tv) )
The above equation cannot be used to calculate what the temperature profile would have to be for the density to be constant for all heights. Why?
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21. More on the Hypsometric Equation - A more precise version -
The hypsometric equation can also be derived from the starting point of geopotential height. Use the hydrostatic equation to substitute for dz.
Integrate between two pressure levels: P(1) and P( 2).
Divide by the gravitational attraction (go) at mean sea level.
This approach does not assume that the virtual temperature is constant, and also accounts for the change in gravitational acceleration with height.
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Scale Height
Scale height is the thickness of the atmosphere if it had constant virtual temperature (an isothermal atmosphere, with constant vapor pressure).
Then the virtual temperature (Tv) in the hypsometric equation can be taken outside the integral. The resulting equation is
Rearranging the equation
The scale height (H) is
How does the pressure change if the height is increased by the scale height?
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23. Scale Height Example of Application
Consider an approximately isothermal layer from Z=200km to Z=1400km, with a temperature of 2000K. If the ratio of the number density of oxygen atoms to the number density of hydrogen atoms is 105 at Z=200km, what is the value of this ratio at Z=1400km?
Solution: at these heights gases are distributed by the process of diffusion, rather than strong mixing of the troposphere. Applying the chemistry version of the ideal gas law to an isothermal layer indicates that the number density is proportional to the pressure.
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24. Scale Height Example of Application
Solution:
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25. Layer Thicknesses and Heights of Constant Pressure Surfaces
The Hypsometric Equation indicates that the vertical distance between two pressure layers is proportional to the average virtual temperature for the layer.
A vertical profile of observations of pressure, temperature, and humidity can be used to determine the heights of the pressure levels (or constant pressure surfaces).
Larger values if virtual temperature indicate larger distances between pressure levels.
Example: calculate the thickness of the 1000 to 500hPa layer, for average virtual temperatures of (a) Tv = 15C, and (b) Tv = -40C.
(a) Tv = 288 K  Z = 5846m; (b) Tv = 233 K  Z = 4730m
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26. Example: Adjustment to Sea Level Pressure
Surface pressures can be somewhat difficult to interpret. Changes in surface pressure with location can be due to
Weather related changes (front, highs, lows), or
Differences in the heights of the surface stations.
This problem is dealt with by adjusting surface pressure observations to mean sea level (Z=0).
Want to adjust pressures from Psfc to Pmsl.
Solve the above equation for Pmsl.
Near sea level this adjustment is approximately 1hPa/8m.
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27. Example: Aircraft Altimeter Calibration
Most (pre-GPS) aircraft do not routinely measure their height. The height above the surface is highly desirable for purposes of landing.
However, they can easily measure pressure and temperature. These observations, combined with surface observations at the airport, can be used to estimate the height relative to the surface.
Assume that the temperature changes linearly with height: T = To - z, where  is the change in temperature with height (lapse rate).
Combine the ideal gas law and the hydrostatic equation:
Integrate from the surface (Psfc(zsfc)) and height of the airplane (P(z))
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28. Example: Aircraft Altimeter Calibration
Integrate
If you are calculating height above ground, you can set zsfc = 0.
If, Tsfc = 300K, and =7K/km,
If (P/Psfc) = 0.90, z=0915m
If (P/Psfc) = 0.85, z=1403m
If (P/Psfc) = 0.70, z=3019m
If (P/Psfc) = 0.50, z=5672m
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29. First Law of Thermodynamics
The first law of thermodynamics describes the energy budget of a system (e.g., a parcel or air or a pile of rocks).
Energy balance implies that energy is neither added nor removed from the system.
Energy budget implies that energy can be added to or taken from the system.
The first law is applied to closed systems. Closed systems are those where the total amount of matter (total of gas, liquid, and solid) remains constant.
The first law of thermodynamics defines the change in internal energy (u) as equal to the difference of the heat transfer (q) into a system and the work (w) done by the system.
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30. First Law of Thermodynamics
Let q be the amount of energy that is added to the closed system.
A negative value of q indicates that energy is taken from the system.
Let w be the amount of external work done by the system. This is the work done by the closed system to modify the external environment.
For example, the closed system could expand or contract it’s volume.
Let u be the internal energy of the system (e.g., thermal energy). If u1 is the energy at one time, and u2 is the energy at a later time, then
u2 – u1 = q - w
Where q and w are the energy input and the external work done by the system (e.g., change in volume, PE, or KE) during this time period.
Written as a derivative this equation becomes
du = dq - dw
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31. Example: First Law of Thermodynamics
Consider a gas filled cylinder, with constant cross sectional area, and a piston at one end of the cylinder.
The volume of the cylinder is the cross section times the length from the fixed end to the inner face of the piston.
The pressure of the gas is inversely proportional to it’s volume.
P v = constant
The work done by changing the gas volume is force (P x area) times distance (x). Assume pressure is constant.
W = P * area * distance = P * change in v
dW = P dv
The work done by the close system is equal to pressure times the change in volume.
If this equation is applied to a unit mass (1kg), then
dw = Pd
Alternative form of 1st law: dq = du + Pd
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Joule’s Law
When gas expands without doing any external work, by expanding into a chamber that has been evacuated, and without taking in or giving out heat, the temperature of the gas does not change.
What does this mean in the context of the 1st law?
dw = 0 (does not do work)
dq = 0 (does not lose or receive energy; an adiabatic process)
du = dq - dw  du = 0
This implies that the internal energy is independent of the volume of gas, so long as no work is done in changing that volume.
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Specific Heat
Specific heat of a material is defined as dq/dT. However, the numerical value depends on how the material changes as it receives or loses heat.
Is pressure held constant?
Is volume held constant?
First consider the case where volume is held constant.
If v is constant, then the work done by the gas is zero, and dq = du.
The 1st law can then be written as
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Specific Heat
Consider the case where pressure is held constant as the gas changes volume.
In this case, work must be done by the gas to change the volume.
Volume increases as temperature increases, requiring energy from the gas,
Volume decreases as temperature decreases, which gives the gas more energy.
Modify the form of the 1st law:
Use the equation of state (ideal gas law) to replace d(P) = RdT
Therefore, Cp = Cv + R
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Enthalpy
Enthalpy (H) is a measure of energy stored as internal energy and work related to pressure and volume. Specific enthalpy (h) is the enthalpy per unit mass.
h = u + P
Since u, P, and  are functions of state, h is also a function of state.
Specific enthalpy can be written in another form through the following manipulations.
dh = du + d(P)
dh = du + Pd + dP
Recall that du = CvdT and that dq = CvdT + d(P) - dP, then
dq = dh - dP
dh = dq + dP
This is another version of the 1st law of thermodynamics
Recall that dq = CpdT - dP, therefore
dh = CpdT
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Dry Static Energy
Dry static energy (Eds) is equal to the enthalpy (H) plus the potential energy (PE).
Eds = h +  = CpT + gz
With some juggling of equations, it can be shown that
dq = dEds = d(CpT + gz)
The conservation of dry static energy can be used to derive the dry adiabatic lapse rate (dq = 0 indicates an adiabatic process).
dEds = d(CpT + gz) = 0
Therefore CpdT + gdz = 0
Recall that the lapse rate () is –dT/dz
Which can be solved to show that the dry adiabatic lapse rate (d) is
d = –dT/dz = g/Cp
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What is an Air Parcel?
It should be useful to understand the conditions that are assumed to apply to an air parcel.
(1) An air parcel is thermally insulated from its environment, so that its temperature changes adiabatically as it rises or sinks, always at exactly the same pressure as the environmental air at the same level, which is assumed to be in hydrostatic equilibrium.
(2) An air parcel is moving slowly enough that the macroscopic kinetic energy of the air parcel is a negligible fraction of its total energy.
In practice, one or more of these assumptions is almost always invalid.
What are some common examples of breakdowns in these assumptions?
Despite these problems, the simple model usually works very well.
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38. Potential Temperature
The potential temperature () of an air parcel is defined as the temperature that the air parcel would have if it were expanded or compressed adiabatically, from its existing pressure and temperature to a standard pressure (usually 1000 hPa).
Apply the the 1st law of thermodynamics:
dq = CpdT - dP
Since the process is adiabatic dq = 0.
The specific density can be substituted for through the ideal gas law: P  = R T
Resulting in
Integrate from upward from the surface (Tsfc= , Psfc) to the height of the parcel (T,P).
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39. Potential Temperature
Solving the integral results in
Multiply by R/Cp, then take the antilog.
Solve for the potential temperature (; Poisson’s equation)
If (P/Psfc) = 0.90 & T=288K   = 296.8K
If (P/Psfc) = 0.85 & T=283K   = 296.5K
If (P/Psfc) = 0.70 & T=268K   = 296.8K
If (P/Psfc) = 0.50 & T=243K   = 296.2K
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40. Thermodynamic Diagrams
There are several types of thermodynamic diagrams that can be extremely useful for weather analysis.
These charts usually have lines showing adiabats (lines of constant potential temperature), isotherms (lines of constant temperature), isobars (lines of constant pressure), and lines of constant humidity.
There is often a great deal of additional information.
The charts can be used to find dew point temperatures, heights of the cloud base, levels of free convection, limits of convection, and estimate the energy available through convection.
The adiabats are plotted by solving Poisson’s equation for temperature.
Each value of potential temperature () has a unique adiabat.
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41. Thermodynamic Diagrams Pseudoadiabatic Chart
Lines of constant temperature would be vertical lines on this plot.
Usually only the portion shaded in blue if printed for use with our atmosphere.
Most atmospheric sounds have an angle between the adiabat and the isotherm.
This allows a rather small range of angles on the plot.
A different type of plot, the skew T – ln P chart, allows for more resolution.
Figure from Wallace and Hobb’s Introduction to Atmospheric Sciences
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42. Thermodynamic Diagrams Skew T – ln P Chart
For a Skew T – ln P Chart, the
Ordinate (vertical axis) is –ln(P)
The minus sign puts smaller values above larger values
Abscissa (horizontal axis) is T – c ln(P)
Where c is a constant
Figure from Wallace and Hobb’s Introduction to Atmospheric Sciences
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43. Thermodynamic Diagrams Skew T – ln P Chart
Solving the formula for the abscissa (x) in term of y:
y = c-1x + c-1T
Shows that lines of constant temperature (T) have a slope of c-1.
This means that isotherms are parallel.
Since T is in the y-offset, the parallel lines are not overlapping!
The constant is usually chosen to make the angle between the isotherms and isobars about 45.
This angle results in the ‘skew’ part of the name.
The angle between isotherms and dry adiabats is approximately 90, which make variations in the atmospheric profile much more visually apparent.
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44. Evaporation Experiment
Humidity refers to water in the form of vapor.
Water vapor will exist anywhere that water exists next to a gas or vacuum.
If a barrier is removed between water and a vacuum (Fig. A), there will be zero pressure above the water.
Water vapor from the liquid will enter the vacuum
Eventually, an equilibrium will be achieved. On average, the number of vapor molecules leaving the liquid equals the number of molecules entering the liquid: the net vapor molecule flux is zero.
Figure from Meteorology by Danielson, Levin and Abrams
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Partial Pressures
The equilibrium water vapor pressure in the previous example, would not change if there were additional gases in the system, so long as the temperature did not change.
Each of these pressures (called partial pressures) can be determined from the ideal gas law (Chemistry version, or Meteorology version):
P = (n / v) R T or
P = r (R/Md) T
The sum of the partial pressures is the total pressure.
The vapor pressure is the partial pressure of water vapor.
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46. Equilibrium Vapor Pressure Variations
Equilibrium occurs (statistically speaking) when (on average) the number of molecules evaporating from the water equals the number condensing on the water.
The equilibrium vapor pressure is a strong function of temperature.
The equilibrium vapor pressure for cold air is small compared to the equilibrium vapor pressure for hot air
Figure from Meteorology by Danielson, Levin and Abrams
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47. Saturation Vapor Pressure
The Clausius-Clapeyron equation describes how the saturation vapor pressure changes with temperature.
Eq in MSE
where eo = kPa, To = 273K, and Rv = 461 JK-1kg-1 is the gas constant for water vapor.
L is either the latent heat of vaporization (Lv = 2.5x106 Jkg-1), or the latent heat of deposition (Ld = 2.83x106 Jkg-1), depending on whether or not we are describing equilibrium with a flat surface of water or ice.
Air that contains less water vapor than the saturation value is called unsaturated (e < es).
Figure from Meteorology by Danielson, Levin and Abrams
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48. Vapor Pressure Over Supercooled (Unfrozen) Droplets
Water droplets can exist at temperatures less than freezing.
These droplets are called supercooled.
The saturation vapor pressure over ice is less than the saturation value over water.
Note that all these values apply to a saturation over a flat surface.
Figures from Meteorology by Danielson, Levin and Abrams
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Mixing Ratio
Figure from Meteorology by Danielson, Levin and Abrams
Mixing ratio (r) is an air parcel’s mass of water vapor divided by its mass of dry air.
Where e = Rd/Rv = is the ratio of gas constants for dry air and water vapor.
The saturation mixing ratio is found when e = es.
Eq. 5.3 in MSE
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50. Specific Humidity vs. Mixing Ratio
The specific humidity (q) is the air parcel’s mass of water vapor divided by it’s total mass (dry air plus water vapor). (Page 99 of MSE).
The absolute humidity (rv) is the concentration of water vapor in air, times the molecular mass (with mass in kg).
It has the same units as density.
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51. Saturation Mixing Ratio Dependency on Temperature
The saturation mixing ratio is shown for the typical range of temperatures on the earth.
The air is saturated only along the curved line.
Unsaturated below the line.
Supersaturated above the line.
Figure from Meteorology by Danielson, Levin and Abrams
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52. Relative Humidity vs. Mixing Ratio
Relative humidity is the water vapor pressure divided by the saturation water vapor pressure: e / es = q / qs = r / rs = r / r s.
It can also be thought of as the mixing ratio divided by the saturation mixing ratio, or the specific humidity divided by the saturation specific humidity.
Relative humidity is usually expressed as a percentage.
Figure from Meteorology by Danielson, Levin and Abrams
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53. Relative Humidity vs. Temperature
The relative humidity is modified by changes in temperature, even when the amount of water vapor in the air is constant.
This dependency is due to the change in saturation value.
Why is the relative humidity usually greatest around sunrise?
What causes fog to form?
Figure from Meteorology by Danielson, Levin and Abrams
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A Hair Hygrometer
Hygrometers are instruments than measure humidity.
For centuries the key element of these instruments has been hair.
How many people have found that hair becomes less manageable in very humid air?
A human hair can change approximately 2.5% in length when RH changes from 0 to 100%.
Hair changes length depending on the relative humidity: increasing in length as the relative humidity increases.
Figure from Meteorology by Danielson, Levin and Abrams
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Example of Houston Air Temperature: Mixing Ratio and Dew Point Temperature
Consider the surface air in Houston
T = 30C, r = 14 g/kg
Note that the symbol r is often used instead of w in these graphics.
If more water vapor is added to the air, then the mixing ratio can increase until is reaches the saturation value, approximately 27 g/kg.
Saturation can also be achieved by cooling the air down to the dew point temperature (Td), approximately 19C
What might cause such changes?
Why should you be concerned about the veracity of observations when Td > T ?
Figure from Meteorology by Danielson, Levin and Abrams
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56. Finding the Actual and Saturation Mixing Ratios
Given a chart similar to the one on this page, it is easy to find the mixing ratio and the saturation mixing ratio given the dew point temperature and the air temperature.
These are relatively common observations.
The dew point temperature can be used to find the actual mixing ratio.
The temperature can be used to find the saturation mixing ratio.
Figure from Meteorology by Danielson, Levin and Abrams
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57. Calculating the Dew Point Temperature
The dew point temperature can be modeled, and then determined without the use of charts (page 100 in MSE).
Start from the Clausius-Clapeyron equation
Replace T with Td and es with e, where e is the observed vapor pressure, and Td is the temperature for which that amount of vapor would saturate the air.
Solving for Td
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58. Saturation Level or Lifting Condensation Level
(page 101 in MSE)
If a parcel of unsaturated air is lifted high enough, its temperature will drop to the point where it is saturated.
This height is the Lifting Condensation Level (LCL).
This is the height of the cloud base for convective clouds.
Convective clouds form due to rising air.
The LCL is a measure of the surface humidity: the lower the humidity the higher the LCL.
The height of the LCL can be estimated by
zLCL  a (T – Td)
where a = km/C
Why does the LCL not apply to stratiform (advective) clouds?
Figure from Atmospheric Science by Wallace and Hobbs
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A Dewpoint Hygrometer
Electronic/mechanical instruments can be used to measure the dewpoint.
They take in a sample of air, and reduce the temperature until condensation occurs.
These are becoming cheap and reliable for normal indoors conditions.
Other humidity sensors have hygroscopic (water absorbing) materials, and measure the electrical resistance of the material.
Figure from Meteorology by Danielson, Levin and Abrams
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60. The Sling Psychrometer and Wet Bulb Temperature
A sling psychrometer is a simple and still used instrument for measuring humidity.
It measures wet bulb temperature (Tw). Which is the temperature to which a moistened thermometer will cool when well ventilated.
A moistened cloth is wrapped around the bulb of a thermometer (called the web bulb thermometer), and then spun around and around to ventilate it.
Why does it cool?
What energy gaining process balances the cooling process?
Figure from Meteorology by Danielson, Levin and Abrams
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61. Wet Bulb Temperature (page 101 of MSE)
The difference between the ambient air temperature (T) and the wet bulb temperature (Tw) is due to cooling of the air around the thermometer’s bulb from the conversion of water to water vapor.
The energy that is used in this cooling comes from the air.
This energy exchange must be balanced (otherwise Tw would not be stable).
Cp (T – Tw) = -Lv (r – rw)
For people interested in a formulas for converting wet bulb temperatures to other humidity measures (e.g., mixing ratio), see pgs. 101 and 102 of MSE.
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62. Determining Values of Humidity Variables
Summary of humidity variables.
T  saturation humidity values
Td  actual humidity values
Note: the word diagram can be replaced with formula.
Bottom graphic
Given two of the observations (brown circles), various approaches (blue boxes) can be used to determine the variable in the other attached circle.
Veracity check: Td < Tw < T
Figure from Meteorology by Danielson, Levin and Abrams
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63. Temperature, Dewpoint and Humidity in Oregon
Figure from Meteorology by Danielson, Levin and Abrams
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64. Example: Mean July Dewpoints and Humidities
The example shows July dewpoint temperatures and noon-time relative humidities.
Low values occur in deserts.
Locations of high values are less consistent between plots.
Why?
Figure from Meteorology by Danielson, Levin and Abrams
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65. The Florida State University
The Heat Index
Perspiration is a mechanism by which human bodies regulate their temperature, to prevent overheating.
In dry conditions this can be effective.
When the air is near saturation, this cooling mechanism is not effective.
The Heat Index considers how humidity modifies the ease of evaporation, and consequently the apparent temperature.
The equation describing the heat index, in terms of temperature and relative humidity, is based human perceptions (not theory).
Figure from Meteorology by Danielson, Levin and Abrams
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66. Indoor Humidity Conditions in Winter
The outdoors specific humidity is approximately equal to the indoor value unless there is a substantial moisture source within the house.
Conditions outdoors: T = -4 and Td = -8C
Indoor conditions: T = 21 and Td = -8C
Results in an indoors RH of 13%.
Figure from Meteorology by Danielson, Levin and Abrams
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67. Cooling of Surface Air Late in the Day or Night
If the temperature cools, and the amount of water vapor remains constant (and non-zero), then the relative humidity will increase.
The cooling can continue below the dew point only if
The air becomes super saturated, or
Condensation occurs
Condensation adds energy to the air, warming it, slowing the decrease in temperature.
Figure from Meteorology by Danielson, Levin and Abrams
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68. Forecaster Rule of Thumb
The overnight minimum temperature is not likely to be below the dew point temperature.
Unless the moisture characteristics change.
For example, drier air might move into the area.
Unless storms occur and cause mixing of surface air with higher altitude air.
Or much colder air is blown into the region.
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69. The Florida State University
Dew and Frost
If the surface temperature decreases below the dew point of the air, then air very near the surface will also cool below the dew point.
Under these conditions, water vapor in the air will condense on the surface, until a new equilibrium is achieved.
The condensed water forms dew.
If the dew point temperature is below freezing, then it could more accurately be thought of as a frost point.
Ice on highways can be a major safety hazard!
It is also a hazard for many plants.
Forecasting occurrences of frost hopefully allows for appropriate preparations.
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70. The Florida State University
Adiabatic Expansion
As a parcel of air rises, it changes in pressure, temperature, and density.
For adiabatic expansion, there is no exchange of energy with the surrounding environment.
Work is done to expand the parcel, thereby reducing the temperature.
While the air parcel is unsaturated, the change in temperature follows the dry adiabatic lapse rate
Figure from Meteorology by Danielson, Levin and Abrams
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71. Adiabatic Expansion of an Air Parcel
Figure from Meteorology by Danielson, Levin and Abrams
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72. Temperature-Pressure Plotting Chart
Characteristics of five air parcels are shown.
Which parcels have the same pressure?
Which parcels have the same temperature?
Figure from Meteorology by Danielson, Levin and Abrams
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73. Dry Adiabatic Temperature Changes Chart
There are several versions of these charts.
Any can easily be used to see how the temperature changes with height or pressure.
These are also lines of constant potential temperature.
Figure from Meteorology by Danielson, Levin and Abrams
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74. Process of Dewpoint Changes
The dew point temperature is a function of pressure.
The mixing ratio and dew point change (decrease) as a parcel rises, and increase as a parcel falls.
The dew point temperature corresponding to any pressure can be found by following the green lines.
Figure from Meteorology by Danielson, Levin and Abrams
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75. Determining the Lifting Condensation Level (LCL)
The LCL can be found by finding the intersection of
the dry adiabat line (blue) corresponding to the parcel’s temperature and pressure, and
the constant mixing ratio line corresponding to the parcel’s mixing ratio and pressure.
Note that not all parcels will start at sea level, nor is the surface constant.
Figure from Meteorology by Danielson, Levin and Abrams
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76. Phases of Matter and Transitions of Phases of Matter
Figure from Meteorology by Danielson, Levin and Abrams
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77. Energy Associated with Phase Changes
Changing phase requires either energy input (melting, sublimation, and evaporation), or releases energy (condensation, deposition, and freezing).
H2O examples:
Latent heat of vaporization (evaporation): Lv  x 106 J kg-1
Latent heat of fusion: Lf  3.34 x 105 J kg-1
Ignoring other considerations, how much energy is required to change 1 gram of water to water vapor?
1g H20 x 1 kg x x 106 J kg-1 = 2.5 x 103 J =2258J
1000 g
How much energy is released when the 1g of water vapor condenses?
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78. The Moist Adiabatic Lapse Rate
Rising air that is saturated DOES NOT follow the dry adiabat.
Why?
Recall that we determined our dry adiabatic lapse rate by assuming the change in energy with height is zero.
d(energy) / dz = 0
Energy = thermal energy + potential energy = CpT + gz
- (dT / dz) = g / Cp  10C/km
The moist adiabatic lapse rate is different because of the input of energy from latent heat release.
- (dT / dz) = g / Cp - (Lv / Cp) dr/dz
Units check: [dT / dz] = K m [(Lv / Cp) dr/dz] = (J kg-1) (J-1 K kg) (kg kg-1 m-1) = K m-1
The moist lapse rate is variable: usually 4 to 7C/km
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79. Adiabatic Diagram Example: The Moist Adiabat
The change in temperature with height is slower following a moist adiabat.
On the type of diagrams we are using, the moist adiabat has a greater slope (rise divided by run) than the dry adiabat.
From any point on the plot, the moist adiabat will be clockwise of the dry adiabat.
Figure from Meteorology by Danielson, Levin and Abrams
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80. Three Types of Stability
The top example (A) is a stable system.
If the ball is moved, it returns to the original position.
The middle example (B) is an unstable system.
If the ball moves off the top of the dome, it will continue to move.
The bottom example (C) is meta-stable. It can change positions, and need not continue moving or return to the original position.
Figure from Meteorology by Danielson, Levin and Abrams
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81. The Florida State University
An Instrumented Tower
If observations are taken at many heights (or pressures), profiles of atmospheric variables can be plotted.
Profile means information on how the data changes with height.
This information provided describes how the environment changes with height.
Figure from Meteorology by Danielson, Levin and Abrams
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82. Environment Temperatures
Another example of environmental conditions is an office building.
Each room can be thought of as being the local environment. The elevator can move up through the environment.
The elevator is analogous to the parcel of air.
It moves up or down through the environment.
The environment does not change when the parcel moves through it.
Figure from Meteorology by Danielson, Levin and Abrams
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83. Example: Early Morning Air in Albuquerque, NM
A parcel starting at the surface rises along the dry adiabat.
Its temperature remains below the environmental temperature.
Its density remains greater than the density of the environmental air.
It returns to the altitude where it started.
Figure from Meteorology by Danielson, Levin and Abrams
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84. Example: Afternoon Temperatures in Albuquerque, NM
The surface temperature has warmed due to day time heating.
If the parcel rises it is warmer than the environmental air.
It will continue to rise until that condition is not met.
Where might it stop rising in this example?
Figure from Meteorology by Danielson, Levin and Abrams
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85. Stability of Saturated Air
If the air in this example was unsaturated, the parcel wound not rise.
The situation is stable.
If the air is saturated, and the parcel begins to rise, it will ‘follow’ the moist adiabat.
The parcel’s temperature will be above the environmental temperature.
The situation is unstable.
Therefore the stability is a function of the environmental temperature and the parcel’s humidity.
Figure from Meteorology by Danielson, Levin and Abrams
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86. The Florida State University
Stability Ranges
An environmental lapse rate that is greater than the moist adiabatic lapse rate is absolutely (always) unstable.
Smaller (more negative) Environmental Lapse rate
An environmental lapse rate that is less than the dry adiabatic lapse rate is absolutely (always) stable.
A lapse rate between the dry and moist adiabatic lapse rates is conditionally (sometimes) unstable.
Bigger Environmental Lapse rate
Figure from Meteorology by Danielson, Levin and Abrams
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87. Environment Lapse Rates
The top row of examples show environments that are made more stable by changes in the temperature profile.
The bottom examples show environments that are made less stable by temperature changes.
Figure from Meteorology by Danielson, Levin and Abrams
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88. Convective Cloud Formation
Convective clouds are formed by rising air.
The height of the cloud depends on the humidity and the environmental temperature profile.
It will grow taller so long as the top parcel remains less dense than the surrounding air (and a little farther).
When the rising air cannot rise any farther, it is forced to spread out and form an anvil.
Figure from Meteorology by Danielson, Levin and Abrams
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89. Warm and Cold Front Cloud Types
Figure from Meteorology by Danielson, Levin and Abrams
Different types of clouds are associated with warm and cold fronts.
Rapid rising of air occurs at the leading edge of a cold front.
This motion creates convective clouds.
A rapidly moving cold front can produce a wall of cumulonimbus clouds.
The air along a warm front moves more slowly, creating a much more varied progression of cloud types.
Watching the evolution of cloud types provides insights into the past and future weather.
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90. Lifting Destabilizes an Air Layer
When a layer of air is lifted, the air at the top of the layer cools more than the air at the bottom.
This change causes the air in the lifted layer to be less stable.
Lifting can occur for several reasons.
Suggestions?
Figure from Meteorology by Danielson, Levin and Abrams
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91. The Florida State University
Radiation Fog
Example of how night-time cooling can cause radiation fog, and later cause a stratus layer.
Why is it called radiation fog?
Figure from Meteorology by Danielson, Levin and Abrams
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92. The Florida State University
Advection Fog
Figure from Meteorology by Danielson, Levin and Abrams
Advection fog is caused when relatively warm moist air is blown (advected) over a cooler surface.
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93. Cloud Formation by Mixing
Clouds can also form when two unsaturated air parcels are mixed.
E.g., parcels with conditions corresponding to points A and C.
This mechanism is possible only because the saturation vapor pressure is a non-linear function of temperature, with a concave-up shape.
Figure from Meteorology by Danielson, Levin and Abrams
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94. Cloud Formation by Vertical Mixing
Figure from Meteorology by Danielson, Levin and Abrams
Vertical mixing can also cause the formation of clouds.
If environmental air that is highly stable becomes well-mixed (evenly distributing the temperature), the resulting air will have a dry adiabatic lapse rate and a much greater surface temperature and humidity.
If parcels are raised, they can form clouds
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95. Examples of Land Sea Breezes
Well known in Classical world
Ancient Greeks took advantage of land breezes to make exiting a port easier (with the wind).
Sea breeze: surface flow (2) is from the sea.
Land breeze: surface flow is (2) from the land.
Air rises (3) over the relatively warm surface, and falls (5) over the relatively cool surface.
These processes create a circulation.
Source:
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96. The Florida State University
Example: Sea Breeze
Source:
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97. Land and Sea Breeze Circulations
The average condition (Fig. A) shows uniform thickness over land and sea.
During the day, the temperature of the surface will increase more over land than over water.
Both increase (more details on the next slide).
The thermal expansion causes an adjustment of the pressure levels, and results in a pressure gradient that aids the return flow towards the cooler surface.
Figure from Meteorology by Danielson, Levin and Abrams
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98. Chinook Winds (or in Europe foehn)
Chinook winds occur when a cold high or ridge (cP air) is downwind of mountains, and warm air (mT or mP) is pushed over the mountains.
The maritime air increases its thermal energy by moist adiabatic ascent, and increases it’s temperature by dry adiabatic descent.
The difference in thermal energy is due to the release of latent heat.
A 27°C change occurred in two minutes, and >40°C changes can occur in hours.
A foot of snow can be evaporated or melted in hours,
Then frozen when the cold air returns.
Figure from Meteorology by Danielson, Levin and Abrams
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99. The Florida State University
Santa Anna Winds
Until around 1980, there was a Californian law that accepted a plea of temporary insanity during a Santa Anna event.
In extreme cases, Santa Anna winds are 100mph winds, of >100°F air, with very little humidity.
Santa Anna winds have the force of a tropical cyclone, dry out vegetation, and fan fires burning the dried vegetation.
Occur most frequently in Autumn, when there is a persistent high pressure system over the Great Basin.
Winds descending thousands of feet from the mountains to the coast, warming adiabaticly in the process.
Figure from Meteorology by Danielson, Levin and Abrams
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100. The Second Law of Thermodynamics
The 2nd law of thermodynamics describes what fraction of heat can be converted to do work.
Hurricanes can be described (albeit oversimplified) as a large application of the 2nd law of thermodynamics.
Originators of various forms of the 2nd law are Carnot, Clausius and Lord Kelvin.
The key application of the 2nd law is the Carnot cycle, which describes the work done as part of a cyclic process.
Since the process is cyclic, the starting conditions are equal to the ending conditions (for an integer number of cycles).
Therefore, the work done by the system must be equal to the energy input to the system.
A heat engine is a device that does work through the agency of heat.
The most common meteorological application is a hurricane.
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