04/20/2304/20/23 11

PHYS-575/CSI-655PHYS-575/CSI-655Introduction to Atmospheric Physics and ChemistryIntroduction to Atmospheric Physics and Chemistry

Lecture Notes #3 – Part 1: Lecture Notes #3 – Part 1: ThermodynamicsThermodynamics

1. Thermodynamics Review/Tutorial - Ideal Gas Law - Heat Capacity - 1st & 2nd Laws of Thermodynamics - Adiabatic Processes - Energy Transport2. Hydrostatic Equilibrium3. Adiabatic Lapse Rate – DRY4. Adiabatic Lapse Rate - WET5. Static Stability6. SLT and the Atmosphere

04/20/2304/20/23 22

What is Thermodynamics?What is Thermodynamics?

"Now, in the second law of thermodynamics...""Department of Entropy"

Thermodynamics is the study of heat and its transformation from a macroscopic point of view.

04/20/2304/20/23 33

1. Thermodynamics Tutorial1. Thermodynamics TutorialThermodynamics is the study of heat and its transformation to andfrom other sources of energy, from a macroscopic point of view.

Statistical Mechanics connects thermodynamics to the microscopic world through the statistical description of an ensemble of atoms ormolecules that constitute a macroscopic system.

The transfer of heat, in turn, is driven by differences in temperatureor potential differences associated with chemical reactions.

In the interest of crafting a brief tutorial for applications to the atmosphere, I haveglossed over some of the finer (but yet important) points of thermodynamics.

For more complete treatment:General: Fundamentals of Statistical and Thermal Physics (McGraw-Hill Series in Fundamentals of Physics) by Frederick Reif, 1965.Atmospheric: Atmospheric Thermodynamics, by C.F. Bohren and B.A. Albrecht, Oxford University Press, Oxford, 1998.

04/20/2304/20/23 44

Defining TemperatureDefining Temperature

Temperature is a measure of the mean kinetic energy of gas molecules.

The temperature of an ideal monatomic gas is related to the average kinetic energy of its atoms as they move. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold).

kTmv2

3

2

1 2

04/20/2304/20/23 55

Measure of TemperatureMeasure of Temperature

dvevdvvf cv 2)/(2)(

kTdvvfvm

vmmv2

3

22

1

2

1

0

222

Temperature of a measure of the mean kinetic energy of gas molecules.

m

kTc 2

04/20/2304/20/23 66

Temperature ScalesTemperature Scales

The The FahrenheitFahrenheit Scale:  Scale:  • we are most familiar with this one we are most familiar with this one • water freezes at 32 degrees F. water freezes at 32 degrees F. • water boils at 212 degrees F. water boils at 212 degrees F. • when we cool to the absolute lowest when we cool to the absolute lowest

temperature we reach -459 degrees temperature we reach -459 degrees (this is referred to as Absolute Zero) (this is referred to as Absolute Zero)

The The CelsiusCelsius Scale: Scale: • water freezes at 0 degrees C. water freezes at 0 degrees C. • water boils at 100 degrees C. water boils at 100 degrees C. • Absolute Zero is at -273 degrees C. Absolute Zero is at -273 degrees C.

The The Kelvin Kelvin ScaleScale • Absolute Zero is 0 K Absolute Zero is 0 K • a temperature change of 1 degree K is a temperature change of 1 degree K is

the same as a temperature change of 1 the same as a temperature change of 1 degree C. degree C.

• water freezes (or melts) at 273 K water freezes (or melts) at 273 K • water boils at 373 K water boils at 373 K

The Kelvin is scale is the more useful scale The Kelvin is scale is the more useful scale for our course since it refers to Absolute for our course since it refers to Absolute Zero in a direct way. Zero in a direct way.

Temperature is a measure of the random kinetic energy of atoms and/or molecules

<1/2 mv2> = 3/2 k T

04/20/2304/20/23 77

The Ideal Gas LawThe Ideal Gas LawLaboratory experiments show that the pressure, volume, and temperature ofany material may be related by an Equation of State (EOS). These variablesare known as State Variables. All atmospheric gasses follow an equation of state known as the Ideal Gas Law (IGL) to a very high degree of accuracy. We assume the IGL to be exact in atmospheric science.

The Ideal Gas Law may be written: pV = mRT where

p = pressureV = volumem = mass T = temperature (absolute Kelvin; K = oC + 273.15)R = gas constant

The gas constant R depends upon the particular gas under consideration.Since m/V = ρ (density of the gas), the IGL may be written:

p = ρRT

04/20/2304/20/23 88

We can also define α = 1/ρ, known as the specific volume,to write the IGL as:

pα = RTBoyle’s Law: For fixed temperature, the pressure of a gas is inverselyproportional to its volume, i.e., P ~ 1/V.

Additional forms of the Ideal Gas Law:

A mole (gram-molecular weight) of any substance is the molecular weight M of the substance expressed in grams. For example, the molecular weight of water is 18.015 gm, so 1 mole of water is 18.015 gm of water. The number of moles (N) in a mass m (in grams) of a substance is given by:

N = m/M

The number of molecules in 1 mole of any substance is a universalconstant called Avogadro’s number, NA.

NA = 6.022 x 1023 molecules per mole.

Ideal Gas Law (IGL) - continued

04/20/2304/20/23 99

The Ideal Gas Law for 1 mole of any gas can be written:

pV = R*T

Where R* is the universal gas constant = 8.3145 J K-1 mol-1.So for N moles of any gas, the IGL will be:

pV = NR*T

The gas constant for 1 molecule of any gas is also a universal constantknown as Boltzmann’s constant, k = 1.38 x 10-23 J K-1 molecule-1

So for a gas with n gas molecules per unit volume V, the IGL is then

p = nkT

Ideal Gas Law (IGL) – continued again

04/20/2304/20/23 1010

Ideal Gas Law - SummaryIdeal Gas Law - Summary

Ideal Gas Law: P = nkT = ρRT= RT/α

PV = R*T/MP = pressurem = mass per gas particlen = number density of gas particles ρ = mn = mass densityα = 1/ρ = specific volumeV = volume of one mole of gask = Boltzmann’s constantR = gas constant (gas specific) = R*/ΜM = molar massR* = universal gas constantT = temperature

04/20/2304/20/23 1111

Heat CapacityHeat Capacity

dT

dQC

The Heat Capacity of a material is a measure of its ability to absorb and retain heat.

More precisely, the Heat Capacity is the energy (dQ) required to increase the temperature of a unit volume of any substance from T to T+dT (in Kelvin)

The Heat Capacity depends upon the nature of the material and its temperature.

The Heat Capacity also depends upon exactly how the energy is added. If theheat is added to a gas at constant volume the heat capacity is lower than if theheat is added at constant pressure. The reason is that heat performs work if the volume changes.

04/20/2304/20/23 1212

Heat Capacity - continuedHeat Capacity - continued

Heat capacity is related to the ability of a substance to store energy. Energy can be stored in a variety of ways. For a gas, the most obviousway to store energy is in random kinetic energy of the gas molecules.

kTmv2

3

2

1 2

The 1/2mv2 is the kinetic energy of a molecule of mass m moving with avelocity v. There is ½ kT of energy “per degree of freedom” of the molecule.For a molecule moving in 3-dimensions, there are 3 degrees of freedomand thus the average kinetic energy is stored as 3/2kT.

If there are other ways for a molecule to store energy, then the heat capacitywill be higher.

Thus the Heat Capacity depends upon the phase of the substance.

04/20/2304/20/23 1313

Heat Capacity of WaterHeat Capacity of Water

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/gifs/HeatCapacity02.gif

The Heat Capacity of water makesan excellent example.

When frozen, water molecules do nothave translational kinetic energy andthus its heat capacity is low. Moleculescan only vibrate.

Thawing requires heat and thus isa portion of its heat capacity.

Upon thawing, water molecules canhave kinetic energy of translation andthe heat capacity increases with temperature.

Evaporation requires heat and thus increases the heat capacity.

04/20/2304/20/23 1414

Heat Capacity - continuedHeat Capacity - continued

As can be seen with water, the Heat Capacity is a function of only temperature.

Thus we define the Internal Energy, U, of a unit volume of material to be the measure of the amount of thermal energy stored in the material. The Internal Energy thus depends only upon temperature.

For a gas, the distribution of speeds is a strong function of temperature.

So the internal energy increases as the temperature increase.

U = ρCvT

If you add heat (dq) to a gas, you can cause the internal energy (U) toincrease and/or cause the gas to expand and do work on its environment.

04/20/2304/20/23 1515

Specific HeatsSpecific Heats

dT

dq

constvv dT

dqC

Suppose a small quantity of heat dq is added to a unit mass of material andthis causes the material to rise in temperature from T to T+dT.

Then is the specific heat of the material.

If the volume of the material is kept constant, then the specific heatat constant volume Cv is defined as:

However, if the volume of the material is kept constant, then dq = du(heat changes internal energy and does no work on the environment)and:

constvv dT

duC

For an ideal gas u depends only upon temperature (T), so Cv depends only upon T.

04/20/2304/20/23 1616

We can also define a specific heat at constant pressure Cp as:

constpp dT

dqc

But when heat is added to a parcel of gas at constant pressure, some energy can be used in expanding the gas.

So more heat must be added to a given mass of material at constant pressure to raise it to a given temperature than if the material was kept at constant volume.

Specific Heats - continued

04/20/2304/20/23 1717

The 3 Laws of ThermodynamicsThe 3 Laws of Thermodynamics

First Law:First Law: You can You can’’t win.t win. Second Law:Second Law: You can You can’’t break even.t break even. Third Law:Third Law: You can You can’’t get out of the game.t get out of the game.

04/20/2304/20/23 1818

The Three Laws of ThermodynamicsThe Three Laws of Thermodynamics

1) Conservation of Energy: Energy is neither created nor destroyed, it is merely converted from one form to another.

2) The Entropy of an isolated system increases when a system undergoes a spontaneous change.

3) The Entropy of all substances approaches zero as the temperature (in Kelvin) approaches zero. All substances have zero energy at absolute zero.

04/20/2304/20/23 1919

The First Law of Thermodynamics Conservation of Energy: Energy is neither created

nor destroyed, it is only changed from one form to another.

What is Energy?Forms of Energy:-- Gravitational Potential-- Kinetic Energy-- Chemical Energy-- Electromagnetic Energy-- Rest mass energy

For any system (e.g. a specific collection of matter), the changein energy of the system is equal to the energy transferred by workplus the energy transferred by heat.

Heat is the transfer of energy to or from a system associated with a temperature difference.Work is the transfer of energy to a system by the application of a force.

04/20/2304/20/23 2020

The first law of thermodynamics is the The first law of thermodynamics is the application of the application of the conservation of energyconservation of energy principle to thermodynamic processes:principle to thermodynamic processes:

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/firlaw.html

04/20/2304/20/23 2121

The First Law and WorkThe First Law and Work

http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1.html

04/20/2304/20/23 2222http://www.fas.org/irp/imint/docs/rst/Sect14/stability2.jpg

04/20/2304/20/23 2323

Useful Forms of the Useful Forms of the First Law of ThermodynamicsFirst Law of Thermodynamics

PdVdTCdU

PdVdQdU

dWdQdU

p

04/20/2304/20/23 2424

Second Law of ThermodynamicsSecond Law of Thermodynamics

Entropy is the heat added (or subtracted), ΔQ, to a system divided by its temperature in Kelvin (T). It is a measure of the disorderof a system; a measure of the unavailability of a system’s energy todo work; a measure of the disorder of the molecules in a system;a measure of the number of possible states of a system.

The Entropy of an isolated system increases when the system undergoes a spontaneous change.

dQ is the heat absorbed in an isothermal and reversible process.

T

QS

04/20/2304/20/23 2525

Implications of the Second LawImplications of the Second Law

It is impossible for any process (e.g., engine), working in It is impossible for any process (e.g., engine), working in a cycle, to completely convert surrounding heat to work.a cycle, to completely convert surrounding heat to work.

Dissipation will always occur.Dissipation will always occur. Entropy will always increase.Entropy will always increase.

The Second Law of Thermodynamics states that it is impossible to completely convert heat energy into mechanical energy. Another way to put that is to say that the level of entropy (or tendency toward randomness) in a closed system is always either constant or increasing.

The Second Law of Thermodynamics

04/20/2304/20/23 2626

Work and Heat DissipationWork and Heat Dissipation

No matter how efficient the system (engine) is, dissipation will always occur.This usually appears as heat released from the system to its surroundings

04/20/2304/20/23 2727

Work and HeatWork and Heat

http://physics.uoregon.edu/~courses/dlivelyb/ph161/heat_engine_schem.gif

04/20/2304/20/23 2828

Work and EfficiencyWork and Efficiency

The Second Law of Thermodynamics states that it is impossible for any heat engine to be 100 % efficient: No process is possible which results in the extraction of an amount of heat from a reservoir and its conversion to an equal amount of mechanical work.

04/20/2304/20/23 2929

Efficiency of AutomobilesEfficiency of Automobiles

04/20/2304/20/23 3030

The Second Law and Heat Dissipation The Second Law and Heat Dissipation for a Automobilefor a Automobile

04/20/2304/20/23 3131

The Atmosphere as an Engine with The Atmosphere as an Engine with Associated DissipationAssociated Dissipation

04/20/2304/20/23 3232

Atmospheric Circulation acts an Engine transferring Atmospheric Circulation acts an Engine transferring

heat from a hot region to a cold regionheat from a hot region to a cold region..

04/20/2304/20/23 3333

Energy Flow in the Biosphere as an Engine with Energy Flow in the Biosphere as an Engine with DissipationDissipation

Visible light contains most energy from the sun (per wavelength interval) and overlaps the region where the atmosphere is most transparent, and also is the region where most photosynthesis occurs in the biosphere.

04/20/2304/20/23 3434

Usable EnergyUsable Energy

http://trc.ucdavis.edu/biosci10v/bis10v/week2/2webimages/figure-06-03b.jpg

04/20/2304/20/23 3535

The Third Law of ThermodynamicsThe Third Law of Thermodynamics

The Entropy of all substances approaches zero as the temperature (in Kelvin) approaches zero. All substances have zero energy at absolute zero.

04/20/2304/20/23 3636

The Use of The Use of Thermodynamic Thermodynamic

DiagramsDiagrams

A pair of variables:(P, V) or (P,T) or (V,T) or…denote a state of the system.

A P-V diagram shows the possible states that the system can have.

dW = PdV = Force x Displacement

04/20/2304/20/23 3737

Parcel ConceptsParcel ConceptsBelow approximately 100 km altitude, Below approximately 100 km altitude, air is relatively well mixed. air is relatively well mixed.

Virtually all mixing is accomplished Virtually all mixing is accomplished by the exchange of air by the exchange of air ““parcelsparcels”” which which have horizontal dimensions ranging have horizontal dimensions ranging from mm to the scale of the Earth.from mm to the scale of the Earth.

An air An air ““parcelparcel”” of infinitesimal of infinitesimal dimensions is assumed to be:dimensions is assumed to be:

(1)(1) Thermally insulated from the Thermally insulated from the environment (no energy exchange)environment (no energy exchange)

(2)(2) Moving slowly so that kinetic energy Moving slowly so that kinetic energy of motion is much smaller than itof motion is much smaller than it’’s s total energy.total energy.

In reality, both of these assumptions are violated to some extent. But for small displacements over small time intervals they can be excellent approximations.

04/20/2304/20/23 3838

Thermodynamic Descriptions of the AtmosphereThermodynamic Descriptions of the Atmosphere

During any atmospheric process, the state of a parcel of atmospheric gas (P, V, T, S, etc) will change. The Laws of Thermodynamics determine exactly how these changes can occur. Phase Diagrams describe these changes in the state variables describing the gas.

04/20/2304/20/23 3939

Thermodynamic Descriptions of the AtmosphereThermodynamic Descriptions of the Atmosphere

During any atmospheric process, the state of a parcel of atmospheric gas (P, V, T, S, etc) will change.

Any pair of variables can be used to describe the state of the system: (P,V) or (T, S) or (P, S), etc.

04/20/2304/20/23 4040

Example: Adiabatic Process - No Energy In/OutExample: Adiabatic Process - No Energy In/Out

04/20/2304/20/23 4141

Parcel Concepts: Applications of the laws of Parcel Concepts: Applications of the laws of thermodynamics to air parcelsthermodynamics to air parcels

First Law of Thermodynamics: dQ = dU + PdV

Internal Energy: dU = CpdT

Second Law of Thermodynamics: dS = dQ/T

Adiabatic means dQ = 0. dQ implies dS = 0.

Thus an adiabatic process is also an isentropic process.

04/20/2304/20/23 4242

Energy TransportEnergy Transport

(1) (1) Conduction:Conduction: is the transfer is the transfer of energy by collisions of energy by collisions between particles (generally between particles (generally atoms or molecules). Also atoms or molecules). Also known as diffusive transport known as diffusive transport of energy.of energy.

(2) (2) Convection:Convection: is the motion of is the motion of a fluid caused by density a fluid caused by density gradients which are a result of gradients which are a result of temperature differences.temperature differences.

(3) (3) Radiation:Radiation: is the transport of is the transport of energy by photons.energy by photons.

There are three primary ways that energy is transported in planetary atmospheres.

04/20/2304/20/23 4343

Diffusion of Mass and HeatDiffusion of Mass and Heat

Diffusion can be driven by concentration gradients, temperature gradients,and pressure gradients. When diffusion is produced by temperature gradientsthis is known as thermal conduction and leads to the transfer of heat.

04/20/2304/20/23 4444

Thermal ConductionThermal Conduction is the transfer of heat by collisions is the transfer of heat by collisions between particlesbetween particles

QQ = heat flux (erg cm = heat flux (erg cm-2-2 s s-1-1))

dT/dzdT/dz = temperature gradient in z direction = temperature gradient in z direction

κκTT = thermal conductivity is a measure of a material= thermal conductivity is a measure of a material ’’s physical ability to s physical ability to conduct heat.conduct heat.

FickFick’’s First Law of Diffusions First Law of Diffusion

The rate of change of energy per unit volume is given by:The rate of change of energy per unit volume is given by:

FickFick’’s Second Law of Diffusions Second Law of Diffusion

dz

dTQ T

dz

dQ

dt

dU

04/20/2304/20/23 4545

Thermal ConductionThermal Conduction is the transfer of heat by collisions is the transfer of heat by collisions between particlesbetween particles

UU = internal energy = = internal energy = ρρCCPPTT, where, where

ρρ = mass density = mass density

CCPP = heat capacity at constant = heat capacity at constant

pressure (it can also occur at pressure (it can also occur at constant v)constant v)

TT = temperature = temperature

Where Where κκD D = thermal diffusivity = thermal diffusivity

= = κκTT//ρρCCPP

2

2

z

T

dt

dTC Tp

2

2

z

T

t

TD

Or

04/20/2304/20/23 4646

ConvectionConvectionConvection is the motion of a fluid caused by density gradients which result from temperature differences.

Examples:Boiling waterCloud formation Plate tectonics

04/20/2304/20/23 4747

Convection in the AtmosphereConvection in the AtmosphereConvection is the motion of a fluid caused by density gradients which result from temperature differences.

04/20/2304/20/23 4848

The Atmospheric General CirculationThe Atmospheric General Circulationis a Manifestation of Convective Currentsis a Manifestation of Convective Currents

04/20/2304/20/23 4949

Quantifying Convective Energy TransportQuantifying Convective Energy Transport

dz

dTQ eFirst Law of Diffusion

The rate of change of energy per unit volume is given by:

Second Law of Diffusion

Ke = Eddy Diffusion Coefficient

The Convective Energy Flux can be quantified analogous to the thermal conduction flux, if the thermal diffusion coefficient is replaced by an Eddy Diffusion Coefficient, Ke.

The key problem in convection and mixing is the choice of Ke. It is usuallydetermined by observations of tracer motions in the atmosphere.

2

2

z

T

t

Te

04/20/2304/20/23 5050

Energy Transport by RadiationEnergy Transport by Radiation

kc

c = 2.998 x 108 ms-1 speed of electromagnetic radiationλ = wavelength (wavenumber = k = 1/λ)

ν = frequency, such that: Energy:

h = Planck’s Constant

hE

04/20/2304/20/23 5151

Radiation: EmissionRadiation: Emission

Thermal Blackbody Emission

The Sun is a near-blackbody at 5770 K

The Spectrum of Solar Radiation

04/20/2304/20/23 5252

Radiation: Absorption and Emission by MatterRadiation: Absorption and Emission by Matter

Line emission and absorptionLine emission and absorption

Line wavelengths correspond to energy changes in absorbing/emitting atoms.

04/20/2304/20/23 5353

Atomic Structure: Photon Absorption & EmissionAtomic Structure: Photon Absorption & Emission

04/20/2304/20/23 5454

SpectroscopySpectroscopy Each element (and molecule) has a unique spectroscopic signature. Each element (and molecule) has a unique spectroscopic signature. This is due to their unique structure and energy level distributions.This is due to their unique structure and energy level distributions.

04/20/2304/20/23 5555

Atmospheric Absorption by MoleculesAtmospheric Absorption by Molecules

04/20/2304/20/23 5656

04/20/2304/20/23 5757

The Greenhouse EffectThe Greenhouse Effect:: A Consequence of A Consequence of Radiation Absorption and Emission by Radiation Absorption and Emission by

Greeenhouse Gases in the AtmosphereGreeenhouse Gases in the Atmosphere

04/20/2304/20/23 5858

The Greenhouse Effect and Global WarmingThe Greenhouse Effect and Global Warming

Sunlight brings energy into the climate system; most of it is absorbed by the oceans and land. 

THE GREENHOUSE EFFECT: Heat (infrared energy) radiates outward from the warmed surface of the Earth.

Some of the infrared energy is absorbed by greenhouse gases in the atmosphere, which emits the energy in all directions.

Some of this infrared energy further warms the Earth.

GLOBAL WARMING:Increasing concentrations of CO2 and other "greenhouse" gases trap more infrared energy in the atmosphere than occurs naturally. The additional heat further warms the atmosphere and Earth’s surface.

04/20/2304/20/23 5959

Vertical Structure of the EarthVertical Structure of the Earth’’s Atmosphere: s Atmosphere: Illustration of Heat Transport MechanismsIllustration of Heat Transport Mechanisms

Convection: The steepvertical temperature gradient produces unstable air parcels.

Radiation: Stratosphericozone (O3) absorb solar ultraviolet photons which cause local heating.

Conduction: Solar extreme ultraviolet (EUV) photons absorbed in the upper atmosphere deposit energy which is conducted downwards.

04/20/2304/20/23 6060

Applications of Thermodynamics to Applications of Thermodynamics to Atmospheric ProcessesAtmospheric Processes

What is a Storm?What is a Storm? Parcel ConceptsParcel Concepts Hydrostatic EquilibriumHydrostatic Equilibrium Vertical Temperature ProfileVertical Temperature Profile Adiabatic Lapse RateAdiabatic Lapse Rate Dry vs. Wet AtmosphereDry vs. Wet Atmosphere Static StabilityStatic Stability The Second Law of Thermodynamics and the The Second Law of Thermodynamics and the

AtmosphereAtmosphere

04/20/2304/20/23 6161

What is a Storm?What is a Storm?

http://www.noaanews.noaa.gov/stories2005/images/ivan091504-1515zb.jpg

1. Do all storms have the same cause?

2. Do all storms have the same ending?

3. Are there aspects that all storms have in common?

04/20/2304/20/23 6262

Storm Definitions:Storm Definitions: behave violently, as if in state of a great anger behave violently, as if in state of a great anger take by force; "Storm the fort" take by force; "Storm the fort" rain, hail, or snow hard and be very windy, often with rain, hail, or snow hard and be very windy, often with

thunder or lightning; "If it storms, we'll need shelter" thunder or lightning; "If it storms, we'll need shelter" a violent weather condition with winds 64-72 knots (11 a violent weather condition with winds 64-72 knots (11

on the Beaufort scale) and precipitation and thunder and on the Beaufort scale) and precipitation and thunder and lightning lightning

blow hard; "It was storming all night" blow hard; "It was storming all night" a violent commotion or disturbance; "the storms that had a violent commotion or disturbance; "the storms that had

characterized their relationship had died away"; "it was characterized their relationship had died away"; "it was only a tempest in a teapot" only a tempest in a teapot"

attack by storm; attack suddenly attack by storm; attack suddenly a direct and violent assault on a strongholda direct and violent assault on a stronghold

04/20/2304/20/23 6363

What Causes a Storm?What Causes a Storm?

http://systhread.net/pics/storm/jpegs-orig/10.jpg

04/20/2304/20/23 6464

How Long Can a Storm Last?How Long Can a Storm Last?

Jupiter’s Giant Red Spot

04/20/2304/20/23 6565

Baby Red Spots?Baby Red Spots?

04/20/2304/20/23 6666

Review: Parcel ConceptsReview: Parcel ConceptsBelow approximately 100 km altitude, Below approximately 100 km altitude, air is relatively well mixed. air is relatively well mixed.

Virtually all mixing is accomplished Virtually all mixing is accomplished by the exchange of air by the exchange of air ““parcelsparcels”” which which have horizontal dimensions ranging have horizontal dimensions ranging from mm to the scale of the Earth.from mm to the scale of the Earth.

An air An air ““parcelparcel”” of finite dimension is of finite dimension is assumed to be:assumed to be:

(1)(1) Thermally insulated from the Thermally insulated from the environment (no energy exchange)environment (no energy exchange)

(2)(2) Moving slowly so that kinetic energy Moving slowly so that kinetic energy of motion is much smaller than itof motion is much smaller than it’’s s total energy.total energy.

In reality, both of these assumptions are violated to some extent. But for small displacements over small time intervals they can be excellent approximations.

04/20/2304/20/23 6767

Ideal Gas Law - SummaryIdeal Gas Law - Summary

Ideal Gas Law: P = nkT = ρRT=RT/α PV = R*T/M

P = pressurem = mass per gas particlen = number density of gas particles ρ = mn = mass densityα = 1/ρ = specific volumeV = volume of one mole of gask = Boltzmann’s constantR = gas constant (gas specific) = R*/ΜM = molar massR* = universal gas constantT = temperature

How do you describe air Parcels?

04/20/2304/20/23 6868

Parcel Concepts: Applications of the laws of Parcel Concepts: Applications of the laws of thermodynamics to air parcelsthermodynamics to air parcels

constvpvp dT

dqC

,,

Internal Energy (kinetic energy):

Internal Energy (general): U = ρCpT

Internal Energy Change: dU = CpdT; Work=PdV

First Law of Thermodynamics: dQ = dU + PdV

Second Law of Thermodynamics: dS = dQ/T

Hydrostatic Equilibrium: dP = -ρgdz

Adiabatic means dQ = 0. dQ implies dS = 0.

Thus an adiabatic process is also an isentropic process.

Specific Heats (Constant P,V)

kTmv2

3

2

1 2

04/20/2304/20/23 6969

Hydrostatic EquilibriumHydrostatic Equilibrium

Hydrostatic Balance:ρg dAdz = P(z) dA – P(z+dz) dAP(z+dz) = P(z) + (dP/dz) dz

Thus dP/dz = -ρg

Air pressure at any height in theatmosphere is due to the force per unit area exerted by the weight of all of the air lying above that height.

The air is in hydrostatic balance if thenet upward force on a thin slab of airis equal to the net downward force onthe slab.

Thus the change in pressure between thetop and bottom of the thin slab is equalto the weight (dM g = ρdV g) of the slab per unit area, where dV = dAdz.

04/20/2304/20/23 7070

Hydrostatic EquilibriumHydrostatic Equilibrium

gdzdP

gkT

mPg

dz

dP

Hydrostatic Balance: dP = - ρgdz Ideal Gas Law: P = nkT = (ρ/m) kT = ρRT

P = pressurem = mass per gas particlen = number density of gas particlesρ = mn = mass densityk = Boltzmann constantR = gas constantT = temperature dz

kT

mg

P

dP

H

ZZ

P

Pdz

kT

mg

P

dP o

o

Z

Z

P

P oo

ln

mg

kTH

04/20/2304/20/23 7171

Hydrostatic EquilibriumHydrostatic Equilibrium

H

ZZ

P

P o

o

ln

This means that pressure falls off exponentially with altitude z.H = kT/mg is the Atmospheric Scale Height, and is also the equivalent thickness of the atmosphere for constant temperature.

Near the Earth’s surface, H ~ 7-8 km.

Hzzo

oezPzP /)()()(

04/20/2304/20/23 7272

Atmospheric Scale HeightAtmospheric Scale Height

Hzzo

oezPzP /)()()(

This means that pressure falls off exponentially with altitude z, with a e-folding “distance” in the vertical of H. For an isothermal atmosphere (T = constant), density would have the same functional form.

H is also the equivalent “thickness” of the atmosphere. If the entire atmosphere was compressed to sea level pressure (Po), then the atmosphere would extend to a height of H.

Near the Earth’s surface, H ~ 7-8 km.

H is also roughly the altitude to which an atom moving vertically can reach. This can be seen by equating an atom’s kinetic to gravitational potential energy:

½ mv2 = 3/2 kT = 3/2 mgH

mg

kTH where

04/20/2304/20/23 7373

Geopotential HeightGeopotential Height

z

gdzz0

)(

dpgdzd

The Geopotential, Φ, at any point in the Earth’s atmosphere is defined asthe work that must be done against the Earth’s gravitational field to raisea mass of 1 kg from sea level to that point. Units are J kg-1 or m2s-2.

The force (in newtons) acting on 1kg at height z above sea level is numericallyequal to g. The work (in joules) done in raising 1 kg from z to z+dz is:

From hydrostatic equilibrium (dp = -ρgdz) we get:

The Geopotential at height z is then:

gdzd

04/20/2304/20/23 7474

We can also define a quantity called theWe can also define a quantity called the Geopotential HeightGeopotential Height, , Z, as:Z, as:

z

gdzgg

zZ

00

1)(

Where go is the globally averaged acceleration due to gravity at the Earth’s surface.

Geopotential Height is used as the vertical coordinate in most atmospheric applications in which energy plays an important role (e.g., large scale motions). In the lower atmosphere, there is only a small difference between the physical height z and the geopotential height Z.

04/20/2304/20/23 7575

Thickness of Layer Between Pressure LevelsThickness of Layer Between Pressure Levels

RT

pg

z

p

p

dpRTgdzd

2

1

2

1

p

p p

dpRTd

2

1

12

p

p p

dpTR

1

2012

p

p p

dpT

g

RZZ

In meteorological applications it is not convenient to deal with density, as itis difficult to measure. Pressure makes a more convenient vertical variable.

Hydrostatic equilibrium:

Geopotential:

Integrate the geopotential between two pressure levels, p1 and p2.

implies:

From the definition of geopotential height:

This is known as the thickness of the layerbetween two pressure layers.

04/20/2304/20/23 7676

Scale Height for an Isothermal AtmosphereScale Height for an Isothermal Atmosphere

1

2012

p

p p

dp

g

RTZZ

1

2012

p

p p

dpT

g

RZZ

1

2012

p

p p

dpT

g

RZZ

If T = constant, OR if the meantemperature is used in this expression,then we get:

)/ln( 2112 ppHZZ

H

ZZpp

)(exp 12

12

Which can be integrated exactly:

Or, by raising e to the power of each side:Scale Height: H = RT/go

H ~ 7-8 km in lower atmosphere

Thickness:

04/20/2304/20/23 7777

Thickness and Heights of Constant Pressure SurfacesThickness and Heights of Constant Pressure Surfaces

H

ZZpp

)(exp 12

12

Note that there is a uniquerelationship between P & Z.

Thus pressure can (and is)used as a vertical coordinate.

Lines of constant pressure areknown as isobars.

Pressure decreases monotonically with height, thus pressure surfacesnever intersect.

Pressure is a principle driver of atmospheric motions, and thus characterizingthe variation of pressure can provide insight into atmospheric dynamics.

04/20/2304/20/23 7878

First Law of Thermodynamics, Once AgainFirst Law of Thermodynamics, Once Again

dQ = dU + PdV

Heat Flow (in/out) = change in internal energy + work done by parcel

Using the specific heat relationship we have

dU = Cv dT

So we can write: dQ = Cv dT + PdV

as equivalent statement of the First Law of Thermodynamics.

04/20/2304/20/23 7979

Derivation of the Dry Adiabatic Lapse Rate (DALR)Derivation of the Dry Adiabatic Lapse Rate (DALR)

Hydrostatic Equilibrium: dP = -ρg dz, or VdP = -g dz(where V = specific volume = 1/ρ)Ideal Gas Law: P = ρRT = RT/ VFirst Law of Thermodynamics: dQ = CvdT + P dVRewrite IGL PV = RTDifferentiate the IGL P dV + V dP = RdT = (Cp-Cv) dT(For an ideal gas Cp – Cv = R)Combine FLT & IGL dQ = Cv dT + (Cp-Cv) dT – V dP dQ = CpdT – V dPBut for an adiabatic process (no energy flow into or out of the parcel) dQ = 0

So CpdT = V dP = -g dz dT/dz = -g/Cp DALR

Note that the DALR doesn’t say anything about the actual value of T,but it provides a very strong constraint on how T varies with altitude.

04/20/2304/20/23 8080

Adiabatic Processes for ParcelsAdiabatic Processes for Parcels

If we slowly move a parcel of dry air vertically, such that there is no energyflow with its environment (dQ=0), then its temperature will change withaltitude following the Dry Adiabatic Lapse Rate (DALR).

pd C

g

dz

dTDALR

04/20/2304/20/23 8181

The Dry Adiabatic Lapse RateThe Dry Adiabatic Lapse Rate

04/20/2304/20/23 8282

Dry Adiabatic Lapse Rate and the PV DiagramDry Adiabatic Lapse Rate and the PV Diagram

The (P,V) curve in a thermodynamic diagram of vertical motion in the atmosphere following the DALR will be an Adiabat.

04/20/2304/20/23 8383

Dry Adiabatic Lapse RateDry Adiabatic Lapse Rate

For the Earth:DALR ~ -7-8 K/km

If we know the temperature ofthe atmosphere are any level,and we know that the heat fluxis zero, i.e. adiabatic, then wecan deduce the temperature atany other level.

04/20/2304/20/23 8484

Entropy and Potential TemperatureEntropy and Potential Temperature

First Law of Thermodynamics: dQ = dU + PdV = dU - VdPInternal Energy: dU = Cp dTSecond Law of Thermodynamics: dS = dQ/T

S = Entropy of the system

04/20/2304/20/23 8585

Entropy and Potential TemperatureEntropy and Potential Temperature

VdPdUPdVdUdWdUdQ

dPPRTdTCVdPdTCTdS pp )/(

P

dPR

T

dTCdS p

Where κ =R/Cp (approximately 2/7 for a diatomic gas)So is the constant of integration.

oopo P

PR

T

TCSS lnln

'ln op STpCS

FLT

SLT

Integration

Rewrite:

04/20/2304/20/23 8686

Adiabatic Processes: dQ = 0Adiabatic Processes: dQ = 0Constant Entropy Processes: dS = 0Constant Entropy Processes: dS = 0

P

dPR

T

dTC p

P

dPR

T

dTCdS p

oP

PT

p P

dPR

T

dTC

From the Second Law of Thermodynamics dS = dQ/T, an adiabaticprocess is thus an isentropic process.

dS=0

P

PR

TC o

p lnln

Integrate:

P

PT o

Potential Temperature

04/20/2304/20/23 8787

Potential TemperaturePotential Temperature

If we take a parcel of gas at If we take a parcel of gas at T,P and change it T,P and change it adiabatically to standard adiabatically to standard pressure Ppressure Poo, it will have a , it will have a temperature of temperature of θθ..

Why do airplanes needWhy do airplanes needair conditioners?air conditioners?

http://dewx.easternuswx.com/Figures/Figure_2.jpg

kmP(mbar)

Slanted blue lines are adiabats

P

PT o

04/20/2304/20/23 8888

Parcel Concepts: Key IdeasParcel Concepts: Key Ideas

constvv dT

dqC

kTmv2

3

2

1 2 Internal Energy (kinetic energy):

Internal Energy (general): U = ρ CpT

Internal Energy Change: dU = CpdT; Work=PdV

First Law of Thermodynamics: dQ = dU + PdV

Second Law of Thermodynamics: dS = dQ/T

Hydrostatic Equilibrium: dP = -ρ gdz

Specific Heats (at constant P)

Specific Heats (at constant V)

constpp dT

dqC