at620 review for midterm #1 part 1: chapters 1-4 brenda dolan october 17, 2005 part 1: chapters 1-4...

AT620 Review for Midterm #1

AT620 Review for Midterm #1

Part 1: Chapters 1-4Brenda Dolan

October 17, 2005

Part 1: Chapters 1-4Brenda Dolan

October 17, 2005

Exam: 21 October 2005Exam: 21 October 2005

Exam is closed book You may bring a calculator You will have 2 hours to complete the exam (8-10am)

Bring your own paper

Exam is closed book You may bring a calculator You will have 2 hours to complete the exam (8-10am)

Bring your own paper

Chapter 1 Overview of Cloud Dynamics

Chapter 1 Overview of Cloud Dynamics

Chapter 1: Overview of Cloud Dynamics


Cloud Dynamics: The study of the evolution of clouds including their formation and dissipation mechanisms, cloud air motions and the forces creating those motions. Cloud dynamics is a macroscopic view of clouds from an ensemble perspective.

Cloud Microphysics: the detailed examination of individual cloud particle physics. This is more a microscopic understanding of clouds.

Convective Clouds: “Wet chemical reactors”—transforming particles and gases into acid precipitation. Important vertical transport of heat, moisture, gases, aerosols and momentum from the Earth’s surface to the low, middle, and upper troposphere, and even the lower stratosphere.

Cloud Dynamics: The study of the evolution of clouds including their formation and dissipation mechanisms, cloud air motions and the forces creating those motions. Cloud dynamics is a macroscopic view of clouds from an ensemble perspective.

Cloud Microphysics: the detailed examination of individual cloud particle physics. This is more a microscopic understanding of clouds.

Convective Clouds: “Wet chemical reactors”—transforming particles and gases into acid precipitation. Important vertical transport of heat, moisture, gases, aerosols and momentum from the Earth’s surface to the low, middle, and upper troposphere, and even the lower stratosphere.



Layer Clouds: Important radiative properties for climate and the global heat budget. Much larger coverage.

Cumulus clouds: primarily buoyancy-driven clouds. An ascending parcel cools adiabatically, increasing the relative humidity, and once the RH is ~100%, hygroscopic aerosol particles take on water vapor and form cloud droplets.

Lagrangian time scale (Tp): the time it takes a parcel of air to enter the base of a cloud and exit the top.

Total lifetime of the cloud (TL)

Layer Clouds: Important radiative properties for climate and the global heat budget. Much larger coverage.

Cumulus clouds: primarily buoyancy-driven clouds. An ascending parcel cools adiabatically, increasing the relative humidity, and once the RH is ~100%, hygroscopic aerosol particles take on water vapor and form cloud droplets.

Lagrangian time scale (Tp): the time it takes a parcel of air to enter the base of a cloud and exit the top.

Total lifetime of the cloud (TL)



Cloud Type H(m) W(m/s) Tp LWC g/m3

Comments

Ordinary cumulus

1500 3 10 min 0.5-1 ABL, shallow

Towering cumulus

5000 10 10 min 1.0-1.5 Larger than Ordinary Cu, more LWMore unstable air, weaker capping inversion, convergence0

Cumulonimbus 10,000 15 10 min 2.5-4 Grow in very unstable conditions

Supercell 12,000 40 5 min Can last 2-6 hours; characteristic BWER

Fog 100 0.01 3 hr .10 Radiation, frontal, advection, ice/snow

Stratocumulus 1000 0.1 3 hr 0.05-0.25 BL clouds driven by radiative cooling at top

Stable Orographic clouds

Variable 15 20 min < 1.0 Air just ascends with topography

Chapter 2 Basic Thermodynamics

Chapter 2 Basic Thermodynamics

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics

Isothermal Process: A change in state occurring at constant temperature.

Adiabatic Process: A change in state occurring without the transfer of thermal energy between the system and its surroundings.

Cyclic Process: A change occurring when the system (although not necessarily its surroundings) is returned to its initial state.

Isothermal Process: A change in state occurring at constant temperature.

Adiabatic Process: A change in state occurring without the transfer of thermal energy between the system and its surroundings.

Cyclic Process: A change occurring when the system (although not necessarily its surroundings) is returned to its initial state.


Equation of State: Ideal gas law (For a unit mass) (For molecules)

Other relations n=m/M R=R*/M k=R*/NA

Rd=287 J K-1 kg-1 (Dry gas constant) Rv=461 J K-1 kg-1 (Water vapor gas constant)

Equation of State: Ideal gas law (For a unit mass) (For molecules)

Other relations n=m/M R=R*/M k=R*/NA

Rd=287 J K-1 kg-1 (Dry gas constant) Rv=461 J K-1 kg-1 (Water vapor gas constant)

pV =mRTp =ρRT

pV =nR* Tp =RT / α

p =n0kT


First Law of Thermodynamics Conservation of energy: The amount of internal energy

in a system is equal to the heat added to the system minus the work done by the system. dq=du+dw dq=du+pdα dq=CvdT+pdα dq=CpdT-αdp dq=dh-αdp (In terms of enthalpy) Tds=du+pdα (In terms of entropy)

“PV Work” dw=pdV

First Law of Thermodynamics Conservation of energy: The amount of internal energy

in a system is equal to the heat added to the system minus the work done by the system. dq=du+dw dq=du+pdα dq=CvdT+pdα dq=CpdT-αdp dq=dh-αdp (In terms of enthalpy) Tds=du+pdα (In terms of entropy)

“PV Work” dw=pdV

w = pdV =nR* Td(lnV)v1

v2

∫v1

v2

∫


Internal energy du=dq-dw du=dw (Adiabatic process) du=cvdT (specific internal energy) Joule’s law: Internal energy of an ideal gas is a function of

temperature only. (this comes from the fact that ideal gas molecules are not attracted to one another)

Enthalpy The amount of energy in a system capable of doing

mechanical work. h=u+pα dh=αdp dh=cpdT (For constant p or adiabatic process)

Internal energy du=dq-dw du=dw (Adiabatic process) du=cvdT (specific internal energy) Joule’s law: Internal energy of an ideal gas is a function of

temperature only. (this comes from the fact that ideal gas molecules are not attracted to one another)

Enthalpy The amount of energy in a system capable of doing

mechanical work. h=u+pα dh=αdp dh=cpdT (For constant p or adiabatic process)


• 2nd Law of Thermodynamics: 1) Thermal energy flows from warmer to colder (thermal energy

will not spontaneously flow from a colder to a warmer object)

2) The entropy of the universe is constantly increasing

Entropy “The amount of disorder in a system”

(Reversible process)ds=0 for adiabatic processesThere is no change in entropy for a reversible process

• 2nd Law of Thermodynamics: 1) Thermal energy flows from warmer to colder (thermal energy

will not spontaneously flow from a colder to a warmer object)

2) The entropy of the universe is constantly increasing

Entropy “The amount of disorder in a system”

(Reversible process)ds=0 for adiabatic processesThere is no change in entropy for a reversible process

ds =dqT


Carnot Cycle A series of state changes of working substance in which its

volume changes and it does external work Work done by Ttwo adiabatic and two isothermal legs

dw=-du=-cvdT (Adiabatic legs) pVϒ=const. dw=RTdα/α (Isothermal legs)

Initial and final states are the same du=0 for the system net heat absorbed=work done by working substance (dq=dw)

Efficiency (η):

Example: Hurricane

Carnot Cycle A series of state changes of working substance in which its

volume changes and it does external work Work done by Ttwo adiabatic and two isothermal legs

dw=-du=-cvdT (Adiabatic legs) pVϒ=const. dw=RTdα/α (Isothermal legs)

Initial and final states are the same du=0 for the system net heat absorbed=work done by working substance (dq=dw)

Efficiency (η):

Example: Hurricane

η =Qc − Qh

Qh

=work done

heat absorbed

η =Ts − Tp

Ts


Carnot Cycle Carnot Cycle

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics Free Energy

Helmholz Free Energy (or Helmholz function) Sets upper limit on the amount of non-pV work possible at

constant T, V (it is free energy since its decrease represents the maximum energy that can be freed in a process and made available for work)

Transitions can only take place to a state with a lower free energy

Gibbs Free Energy (or Gibbs function) Sets upper limit on the amount of non-pV work possible at

constant T, P (it is free energy since its decrease represents the maximum energy that can be freed in a process and made available for work)


Free Energy Helmholz Free Energy (or Helmholz function)

Sets upper limit on the amount of non-pV work possible at constant T, V (it is free energy since its decrease represents the maximum energy that can be freed in a process and made available for work)


Gibbs Free Energy (or Gibbs function) Sets upper limit on the amount of non-pV work possible at

constant T, P (it is free energy since its decrease represents the maximum energy that can be freed in a process and made available for work)


F =U −TSdF =−pdV −SdT

G =U −TSG =F + PV =U −TS+ PV

dG =Vdp−SdT

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics Free Energy

Spontaneous Processes: A process in which the system tends to equilibrium

∆F,G<0 (Spontaneous process) ∆F,G=0 (Equilibrium process) ∆F,G>0 (Forbidden process)

Chemical potential Change in internal energy per mole of substance when

material is added or taken away from the system

(Gibbs free energy)

Free Energy Spontaneous Processes:

A process in which the system tends to equilibrium ∆F,G<0 (Spontaneous process) ∆F,G=0 (Equilibrium process) ∆F,G>0 (Forbidden process)

Chemical potential Change in internal energy per mole of substance when

material is added or taken away from the system

(Gibbs free energy)

μ =δU

δn⎛⎝⎜

⎞⎠⎟

S ,V

μ =δG

δn⎛⎝⎜

⎞⎠⎟

T , p

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics Phase changes

For a system consisting of phases, to be in equilibrium it must be in thermal, mechanical, and chemical equilibrium: T1=T2=...=TΦ

p1=p2=...=pΦ

µ1=µ2=...=µΦ

Phase transition equilibrium gl=gv

∆g=0 For a substance in stable equilibrium between different phases,

the specific Gibbs energy of those phases are equal. Latent Heat

Amount of heat absorbed or given off (released) during a phase change

Phase changes For a system consisting of phases, to be in equilibrium it must

be in thermal, mechanical, and chemical equilibrium: T1=T2=...=TΦ

p1=p2=...=pΦ

µ1=µ2=...=µΦ

Phase transition equilibrium gl=gv

∆g=0 For a substance in stable equilibrium between different phases,

the specific Gibbs energy of those phases are equal. Latent Heat

Amount of heat absorbed or given off (released) during a phase change

dQ =L * m

sv −sl =Llv

T

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics Clausius-Clapeyron equation

Describes the variation in (vapor) pressure with temperature for a system consisting of two phases in equilibrium at a pressure and temperature.

AND, T,e are same, so equate and rearrange:

(Integrated form)

Clausius-Clapeyron equation Describes the variation in (vapor) pressure with

temperature for a system consisting of two phases in equilibrium at a pressure and temperature.

AND, T,e are same, so equate and rearrange:

(Integrated form)

dgl =−SldT +V ldedgv =−SvdT +Vvdedgv =dgl

de

dT=

Llv

T(Vv −V l )

ln e =−Llv

RT+ const.

Chapter 2: Basic ThermodynamicsChapter 2: Basic Thermodynamics Surface Tension, σ

Inward pull of molecules. It requires work to move a molecule from center to the outside (kind of like PE) Work must be done to create a curved surface (σdΩ) Treat interface as its own “phase”

Surface Tension, σ Inward pull of molecules. It requires work to move a

molecule from center to the outside (kind of like PE) Work must be done to create a curved surface (σdΩ) Treat interface as its own “phase”

Chapter 3 Nucleation of Liquid

Droplets

Chapter 3 Nucleation of Liquid

Droplets

Chapter 3: Nucleation of Liquid Droplets


Homogeneous (Spontaneous) Nucleation Random clustering of drops (chance aggregation of vapor

molecules) through thermal kinetic energy collisions Does not occur in the atmosphere because it requires very

high supersaturations (12%) Two forms of energy involved in process:

Bulk thermodynamic energy (volume)

Surface energy (Area)

Homogeneous (Spontaneous) Nucleation Random clustering of drops (chance aggregation of vapor

molecules) through thermal kinetic energy collisions Does not occur in the atmosphere because it requires very

high supersaturations (12%) Two forms of energy involved in process:

Bulk thermodynamic energy (volume)

Surface energy (Area)

BTE =nLV(μL −μv)

SE =Aσ LV



The total energy change associated with the spontaneous formation of a droplet of volume V and surface area A is:

(chemical potential)

(for spherical drop)

Critical radius R* (Kelvin’s Equation) The radius at which a drop is in unstable equilibrium. If it gains

one molecule, it will continue to grow. If one molecule leaves it will continue to evaporate.

The total energy change associated with the spontaneous formation of a droplet of volume V and surface area A is:

(chemical potential)

(for spherical drop)

Critical radius R* (Kelvin’s Equation) The radius at which a drop is in unstable equilibrium. If it gains

one molecule, it will continue to grow. If one molecule leaves it will continue to evaporate.

ΔG = −nLV (μ v − μ L ) + Aσ LV

ΔG = −nL

4

3π R3kT ln

e

es

⎛

⎝⎜⎞

⎠⎟+ 4π R2σ LV

R* =2σ LV

nLkT lnees

⎛

⎝⎜⎞

⎠⎟



Critical energy barrier The energy that must be overcome by fluctuations in

the system in order to produce a critically-sized embryo.

Critical energy barrier The energy that must be overcome by fluctuations in

the system in order to produce a critically-sized embryo.

ΔG* =16πσ LV

3

3 nLkT ln ees

( )⎡⎣⎢

⎤⎦⎥

2

1) e/es < 1 sub-saturated∆G>0 for all R

2) e/es > 1 super-saturated∆G + or –



Relative humidity (e/es) above a pure water droplet of a known radius:

Supersaturation S=(1-e/es)*100

Relative humidity (e/es) above a pure water droplet of a known radius:

Supersaturation S=(1-e/es)*100

e

es

=exp2σ LV

nLkTr⎡

⎣⎢

⎤

⎦⎥



Nucleation on Insoluble Particles (but wettable) Flat, insoluble surface

Φ, Contact angle between substrate surface and the tangent line to the droplet surface (wettable surface, Φ =0º, non-wettable surface Φ =180º)

Add a new term to dG Critical Radius:

Catalyst just increases the chance of random formation of a larger drop (R* does not change)

Critical energy barrier:

Where f(m)=(2+m)(1-m)2/4

Nucleation on Insoluble Particles (but wettable) Flat, insoluble surface

Φ, Contact angle between substrate surface and the tangent line to the droplet surface (wettable surface, Φ =0º, non-wettable surface Φ =180º)

Add a new term to dG Critical Radius:



Where f(m)=(2+m)(1-m)2/4

R* =2σ LV

nLkT lnees

⎛

⎝⎜⎞

⎠⎟

ΔG* =16πσ LV

3

3 nLkT ln ees

( )⎡⎣⎢

⎤⎦⎥

2 f (m)



Nucleation on Insoluble Particles (but wettable) Curved, insoluble surface

Critical Radius:



where f(m)=(2+m)(1-m)2/4 and x=r/r* (ratio of radii of dry particle radius to critical droplet radius

2 things play a role in determining saturation ratio: size of nucleating particle wettability

Nucleation on Insoluble Particles (but wettable) Curved, insoluble surface

Critical Radius:



where f(m)=(2+m)(1-m)2/4 and x=r/r* (ratio of radii of dry particle radius to critical droplet radius

2 things play a role in determining saturation ratio: size of nucleating particle wettability

R* =2σ LV

nLkT lnees

⎛

⎝⎜⎞

⎠⎟

ΔG* =16πσ LV

3

3 nLkT ln ees

( )⎡⎣⎢

⎤⎦⎥

2 f (m, x)



Nucleation on water-soluble particles Raoult’s Law

The vapor pressure of component A above the solution is less than the vapor pressure of component A in pure form by the factor

The presence of a solute B (e.g. salt) lowers the energy barrier associated with nucleation

Saturation ratio for a solution drop: curvature + solution terms

Nucleation on water-soluble particles Raoult’s Law

The vapor pressure of component A above the solution is less than the vapor pressure of component A in pure form by the factor

The presence of a solute B (e.g. salt) lowers the energy barrier associated with nucleation

Saturation ratio for a solution drop: curvature + solution terms

eA =nA

nA +nB

eAO

e

es

=exp2σ

nLkTr⎡

⎣⎢

⎤

⎦⎥ 1+

imsM0

M 4 / 3πr3ρ −ms( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥



Nucleation on water-soluble particles Saturation ratio using the molal osmotic coefficient, assuming a

dilute solution

where and

Saturation ratio depends on salt properties (Van’t Hoff factor and molecular weight) and radius of particle

Nucleation on water-soluble particles Saturation ratio using the molal osmotic coefficient, assuming a

dilute solution

where and

Saturation ratio depends on salt properties (Van’t Hoff factor and molecular weight) and radius of particle

e

es

=expAr

−Br3

⎡⎣⎢

⎤⎦⎥≈1+

Ar

−Br3

⎛⎝⎜

⎞⎠⎟

A =2σnLkT

B =3γmsMw

nLkT



Köhler curves: Stable equilibrium: droplet will evaporate or grow

back to original radiusHaze droplets: very small particles; equilibrium less

than supersaturation, and they can deliquesce (take on water vapor)

Unstable equilibrium: an evaporating drop will grow back to it’s original size and a growing droplet will continue to grow.

Köhler curves: Stable equilibrium: droplet will evaporate or grow

back to original radiusHaze droplets: very small particles; equilibrium less

than supersaturation, and they can deliquesce (take on water vapor)

Unstable equilibrium: an evaporating drop will grow back to it’s original size and a growing droplet will continue to grow.



Köhler curves: Köhler curves:

Chapter 4 Bulk Thermodynamics of the

Atmosphere

Chapter 4 Bulk Thermodynamics of the

Atmosphere

Chapter 4: Bulk ThermodynamicsChapter 4: Bulk Thermodynamics

1st Law of Thermodynamics under moist conditions, and for a cloud-free atmosphere

Most generally, 1st law for open thermodynamic multi-phase system:

Neglecting radiation and molecular dissipation in a cloud free atmosphere the first law becomes:

1st Law of Thermodynamics under moist conditions, and for a cloud-free atmosphere

Most generally, 1st law for open thermodynamic multi-phase system:

Neglecting radiation and molecular dissipation in a cloud free atmosphere the first law becomes:


We also introduced thermodynamic variables which are conserved for adiabatic motions: θ: Potential temperature

Conserved for dry, isentropic motions Poisson’s Equation

θl,i: Ice-liquid water potential temperature Conserved for wet adiabatic (liquid and ice transformations)

Reduces to θ in the absence of cloud or precip.

We also introduced thermodynamic variables which are conserved for adiabatic motions: θ: Potential temperature

Conserved for dry, isentropic motions Poisson’s Equation

θl,i: Ice-liquid water potential temperature Conserved for wet adiabatic (liquid and ice transformations)

Reduces to θ in the absence of cloud or precip.

d lnθ =dlnT −Ra

Cpa

⎛

⎝⎜

⎞

⎠⎟dlnp=0

di lnθil =dlnθ −Llv

cpaTdirl −

Liv

cpaTdiri =0


θe: Equivalent potential temperature Useful in diagnostic studies as a tracer of air

parcel motions

Conserved during moist and dry adiabatic processes

θeiv is conservative over phase changes but not if precipitation fluxes exist

θe: Equivalent potential temperature Useful in diagnostic studies as a tracer of air

parcel motions

Conserved during moist and dry adiabatic processes

θeiv is conservative over phase changes but not if precipitation fluxes exist


The first law for a moist atmosphere can be written as follows, assuming that the gas constants and heat capacities do not vary with temperature:

Neglecting heat stored in condensed water, Q(R) adn Q(D):

Assuming: drv+dirl+diri=0, we can write

The first law for a moist atmosphere can be written as follows, assuming that the gas constants and heat capacities do not vary with temperature:

Neglecting heat stored in condensed water, Q(R) adn Q(D):

Assuming: drv+dirl+diri=0, we can write

d lnθ =Rm

cpm

−Ra

cpa

⎛

⎝⎜

⎞

⎠⎟dlnp−

Llv

cpmTdrv +

Lli

cpmTdiri +

1cpm

[Q(R) +Q(D)]

d lnθ =−Llv

cpaTdrv +

Lil

cpaTdiri =0

d lnθ =Llv

cpaTdirl +

Liv

cpaTdiri


Wet-Bulb temperature, Tw

Temperature that results from evaporating water at constant pressure from a wet bulb

Wet-bulb potential temperature, θw

Determined graphically

Conserved during moist and dry adiabatic processes (as is θe)

Energy Variables Dry static energy (s)

Moist static energy (h)

Wet-Bulb temperature, Tw

Temperature that results from evaporating water at constant pressure from a wet bulb

Wet-bulb potential temperature, θw

Determined graphically

Conserved during moist and dry adiabatic processes (as is θe)

Energy Variables Dry static energy (s)

Moist static energy (h)

Chapter 4: Bulk ThermodynamicsChapter 4: Bulk Thermodynamics Purpose of Thermodynamic diagrams

Provide graphical display of lines representing major kinds of processes to which air may be subject

Isobaric, isothermal, dry adiabatic and pseudoadiabatic processes

Three desirable characteristics

Area enclosed by lines representing any cyclic process be proportional to the change in energy or the work done during the process(in fact, designation thermodynamic diagram is reserved for only those in which area is proportional to work or energy)

As many as possible of the fundamental lines be straight

The angle between the isotherms and the dry adiabats shall be as large as possible (90º)

makes it easier to detect variations in slope

Purpose of Thermodynamic diagrams

Provide graphical display of lines representing major kinds of processes to which air may be subject

Isobaric, isothermal, dry adiabatic and pseudoadiabatic processes

Three desirable characteristics

Area enclosed by lines representing any cyclic process be proportional to the change in energy or the work done during the process(in fact, designation thermodynamic diagram is reserved for only those in which area is proportional to work or energy)

As many as possible of the fundamental lines be straight

The angle between the isotherms and the dry adiabats shall be as large as possible (90º)

makes it easier to detect variations in slope

Chapter 4: Bulk ThermodynamicsChapter 4: Bulk Thermodynamics Two diagrams meet these requirements almost perfectly

Tephigram

Skew T-log p

Two diagrams meet these requirements almost perfectly

Tephigram

Skew T-log p

Chapter 5Atmospheric Aerosols

To be continued Wednesday…

Chapter 5Atmospheric Aerosols

To be continued Wednesday…

at620 review for midterm #1 part 1: chapters 1-4 brenda dolan october 17, 2005 part 1: chapters 1-4...

Documents

cloud microphysics

cloud air motions

cumulus clouds

form cloud droplets

convective clouds

evolution of clouds

macroscopic view of

buoyancydriven clouds