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Exam: 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

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.

Chapter 1: Overview of Cloud Dynamics

- 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)

Chapter 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.

Chapter 2: Basic Thermodynamics

- 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)

Chapter 2: Basic Thermodynamics

- 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

Chapter 2: Basic Thermodynamics

- 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)

Chapter 2: Basic Thermodynamics

- 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 processes
- There is no change in entropy for a reversible process

Chapter 2: Basic Thermodynamics

- 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

Chapter 2: Basic Thermodynamics

- Carnot Cycle

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

Chapter 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)

Chapter 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

Chapter 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)

Chapter 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”

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)

Chapter 3: Nucleation of Liquid Droplets

- 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.

Chapter 3: Nucleation of Liquid Droplets

- Critical energy barrier
- The energy that must be overcome by fluctuations in the system in order to produce a critically-sized embryo.

- 1) e/es < 1 sub-saturated ∆G>0 for all R
- 2) e/es > 1 super-saturated ∆G + or –

Chapter 3: Nucleation of Liquid Droplets

- Relative humidity (e/es) above a pure water droplet of a known radius:
- Supersaturation
- S=(1-e/es)*100

Chapter 3: Nucleation of Liquid Droplets

- 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

Chapter 3: Nucleation of Liquid Droplets

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

Chapter 3: Nucleation of Liquid Droplets

- 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

Chapter 3: Nucleation of Liquid Droplets

- 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

Chapter 3: Nucleation of Liquid Droplets

- Köhler curves:
- Stable equilibrium: droplet will evaporate or grow back to original radius
- Haze 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.

Chapter 3: Nucleation of Liquid Droplets

- Köhler curves:

Chapter 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:

Chapter 4: Bulk Thermodynamics

- 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.

Chapter 4: Bulk Thermodynamics

- 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

Chapter 4: Bulk Thermodynamics

- 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

Chapter 4: Bulk Thermodynamics

- 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 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

Chapter 4: Bulk Thermodynamics

- Two diagrams meet these requirements almost perfectly
- Tephigram
- Skew T-log p

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