Download AT620 Review for Midterm #1

AT620 Review for Midterm #1
Part 1: Chapters 1-4
Brenda Dolan
October 17, 2005
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
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 1: Overview of Cloud Dynamics
Tp
LWC
g/m3
Comments
3
10 min
0.5-1
ABL, shallow
5000
10
10 min
1.0-1.5
Cumulonimbus
10,000
15
10 min
2.5-4
Supercell
12,000
40
5 min
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
Variable
15
20 min
Cloud Type
H(m)
W(m/s)
Ordinary
cumulus
1500
Towering
cumulus
Stable
Orographic
clouds
Larger than Ordinary Cu,
more LW
More unstable air, weaker
capping inversion,
convergence0
Grow in very unstable
conditions
Can last 2-6 hours;
characteristic BWER
< 1.0
Air just ascends with
topography
Chapter 2
Basic Thermodynamics
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





pV  mRT
p   RT
pV  nR * T
p  RT / 
p  n0 kT
(For a unit mass)
(For molecules)
 Other relations





n=m/M
R=R*/M
k=R*/NA
Rd=287 J K-1 kg-1
Rv=461 J K-1 kg-1
(Dry gas constant)
(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
Tds=du+pdα
(In terms of enthalpy)
(In terms of entropy)
 “PV Work”
 dw=pdV
v2

v2
v1
v1
w
 pdV  nR *  Td(lnV )
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”
dq
 ds 
(Reversible process)
T
 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
 pVϒ=const.
 dw=RTdα/α
(Adiabatic legs)
(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):
Qc  Qh
work done


Qh
heat absorbed
 Example: Hurricane

Ts  Tp
Ts
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
F  U  TS
dF   pdV  SdT
 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
G  U  TS
G  F  PV  U  TS  PV
dG  Vdp  SdT
Chapter 2: Basic Thermodynamics
 Free Energy
 Spontaneous Processes:
 A process in which the system tends to equilibrium
 ∆F,G<0
 ∆F,G=0
 ∆F,G>0
(Spontaneous process)
(Equilibrium process)
(Forbidden process)
 Chemical potential
 Change in internal energy per mole of substance when
material is added or taken away from the system

 U 
 
  n  S,V

 G 

  n  T , p
(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
Llv
s s 
T
v
l
dQ  L * m
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.
dgl  S l dT  V l de
dgv  S v dT  V v de
dgv  dg l
 AND, T,e are same, so equate and rearrange:


de
Llv

dT T (V v  V l )
Llv
ln e  
 const.
RT
(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
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)
BTE  nLV (  L  v )
 Surface energy (Area)
SE  A LV
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:

G  nLV ( v   L )  A LV
 G  nL
(chemical potential)
 e
4
 R3kT ln    4 R2 LV(for spherical drop)
3
 es 
 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.
2 LV
R* 
 e
nL kT ln  
 es 
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.
16 3
G* 
LV
 
3  nL kT ln e e 
s 

2
1) e/es < 1
all R
2) e/es > 1
sub-saturated
∆G>0 for
super-saturated
∆G + or –
Chapter 3: Nucleation of Liquid Droplets
 Relative humidity (e/es) above a pure water droplet of a
known radius:
 2 LV 
e
 exp 

es
 nL kTr 
 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
2 LV
 Critical Radius: R* 
 e
nL kT ln  
 es 
 Catalyst just increases the chance of random formation of a larger
drop (R* does not change)
 Critical energy barrier:
G* 
3
16 LV
 
3  nL kT ln e e 
s 

 Where f(m)=(2+m)(1-m)2/4
2
f (m)
Chapter 3: Nucleation of Liquid Droplets
 Nucleation on Insoluble Particles (but wettable)
 Curved, insoluble surface
 Critical Radius:
R* 
2 LV
 e
nL kT ln  
 es 
 Catalyst just increases the chance of random formation of a larger
drop (R* does not change)
 Critical energy barrier:
G* 
3
16 LV
 
3  nL kT ln e e 
s 

2
f (m, x)
 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
eA 
nA
eAO
nA  nB
 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
 2  
e
ims M 0
 exp 
1



3
es
n
kTr
 L
  M 4 / 3 r   ms





Chapter 3: Nucleation of Liquid Droplets
 Nucleation on water-soluble particles
 Saturation ratio using the molal osmotic coefficient, assuming a
dilute solution
e
 A B
A B 
 exp   3   1    3 
r r 
es
r r 
 where
A
2
nL kT
and
B
3 ms M w
nL kT
 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 of the
Atmosphere
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
 d ln  d lnT   Ra  d ln p  0 Poisson’s Equation
C 
 pa 
 θl,i: Ice-liquid water potential temperature
 Conserved for wet adiabatic (liquid and ice transformations)
di lnil  d ln 
Llv
L
di rl  iv di ri  0
c paT
c paT
 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:
 Rm Ra 
Llv
Lli
1
d ln  

d
ln
p

dr

d
r

[Q(R)  Q(D)]
v
i i

c pmT
c pmT
c pm
 c pm c pa 
 Neglecting heat stored in condensed water, Q(R) adn
Q(D):
d ln  
Llv
L
drv  il di ri  0
c paT
c paT
 Assuming: drv+dirl+diri=0, we can write
d ln 
Llv
L
di rl  iv di ri
c paT
c paT
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
Chapter 5
Atmospheric Aerosols
To be continued Wednesday…