Session 2, Unit 3
Atmospheric Thermodynamics
Ideal Gas Law
Various forms
m
PV  nRT 
RT
M
m PM
 
V
RT
R
P 
T
M
1
Where  

Hydrostatic Equation
Air density change with atmospheric
pressure
dP   gdz
dP
  g
dz
g dz   dP
First Law of Thermodynamics
For a body of unit mass
dq  dw  du
 dq=Differential increment of heat added to the body
 dw=Differential element of work done by the body
 du=Differential increase in internal energy of the body
dw  P d
dq  du  P d  cv dT  P d
Heat Capacity
At constant volume
 dq 
 du 
cv  



 dT  const  dT  const
For ideal gas cv 
du
dT
At constant pressure
 dq 
cp  

 dT  p const
Heat Capacity
Relationships
cv M  C v
cpM  Cp
C p  Cv  R
 
Cp
Cv
Concept of an Air Parcel
An air parcel of infinitesimal dimensions
that is assumed to be



Thermally insulated – adiabatic
Same pressure as the environmental air at
the same level – in hydrostatic equilibrium
Moving slowly – kinetic energy is a
negligible fraction of its total energy
Adiabatic Process
Reversible adiabatic process of air
dq  cv dT  P d
Add and subtract  dP : dq  cv dT  d ( P )   dP
Use ideal gas law : dq  (cv  R) dT   dP
dq  c p dT   dP
Adiabatic process dq  0
c p dT   dP  0
Combine with ideal gas law
dP C p dT

P
R T
Lapse Rate
Combine hydrostatic equation and ideal
gas law
dP
PM
  g   g
dz
RT
dP
gM

dz
P
RT
For adiabatic process
dP C p dT

P
R T
Lapse Rate
Therefore
gM
dT  
dz
Cp
dT/dz is Dry Adiabatic Lapse Rate (DALR)
Dry Adiabatic Lapse Rate

Dry adiabatic lapse rate (DALR)
dT
gM
9.81m / s 2  29 g / mol
kg
Pa  m  s 2




3
dz
Cp
kg
3.5  8.314m  Pa / mol  K 1000 g
o
o
o
K
C
F
C
 0.00978  9.78
 5.37
 10
m
km
1000 ft
km

Or on a unit mass basis
dT
g
9.81m / s 2


 9.8K / km
dz
c p 1004 J / kg  K

Or the expression in the textbook:
dT ( g / g c )    1 9.95 o C


 DALR


dz
R   
km
Lapse Rate
Effect of moisture
C  (1  w)C p, Air  wC p,WaterVapor
'
p
Because
C p  (1   )C pAir   CPWaterVapor
C p ,W aterVapor  C p , Air
C p'  C p

Wet adiabatic lapse rate < DALR
(temperature decreases slower as air parcel rises)
Condensation
Lapse Rate
Superadiabatic lapse rate (e.g., 12oC/km)
Subadiabatic lapse rate (e.g., 8oC/km)
Atmospheric lapse rate


Factors that change atmospheric temperature
profile
Standard atmosphere
(lapse rate ~ 6.49 oC/km or 3.56 oF/1000 ft)
Potential Temperature
Current state: T, P
Adiabatically change to: To, Po
 Po 
To  T  
P
  1 


  
Set Po = 1000 mb,
To is potential temperature 
If an air parcel is subject to only adiabatic
transformation,  remains constant
Potential temperature gradient
  dT 

 DALR

z  dz  actual
Session 2, Unit 4
Turbulence and Mixing
Air Pollution Climatology
Atmospheric Turbulence
Turbulent flows – irregular, random, and
cannot be accurately predicted
Eddies (or swirls) – Macroscopic random
fluctuations from the “average” flow

Thermal eddies
 Convection

Mechanical eddies
 Shear forces produced when air moves across a rough
surface
Lapse Rate and Stability
Neutral
Stable
Unstable
Richardson Number and
Stability
Stability parameter
g   
s 

T  z 
Richardson number



Stable
Neutral
Unstable
  
g

z 
Ri  
_ 2


du
T

dz




Stability Classification
Schemes
Pasquill-Gifford Stability Classification

Determined based on
 Surface wind
 Insolation

Six classes: A through F
Turner’s Stability Classification

Determined based on
 Wind speed
 Net radiation index


Seven classes
Feasible to computerize
Inversions
Definition
Types





Radiation inversion
Evaporation inversion
Advection inversion
Frontal inversion
Subsidence inversion
Fumigation
Planetary Boundary Layer
Turbulent layer created by a drag on
atmosphere by the earth’s surface
Also referred to as mixing height
Inversion may determine mixing height
Planetary Boundary Layer
Neutral conditions

Mixing height
u*
h 
f

Increased wind speed and surface
roughness cause higher h.
Planetary Boundary Layer
Unstable conditions

Mixing height


t0


2 H dt
t

h
dT  


 C p   DALR  dz  


1
2
Planetary Boundary Layer
Stable conditions

Mixing height
u*
h  0 .4
L
f
Surface Layer
Fluxes of momentum, heat, and
moisture remain constant
About lower 10% of mixing layer
Surface Layer
Monin-Obukhov length
L
C p Tu *3
kgH
Monin-Obukhov length and stability
classes
Surface Layer Wind Structure
Neutral air
u*  z 
u  ln  
ka  z0 
Surface Layer Wind Structure
Unstable and stable air
  z 
z 

ln    m  
 L 
  z0 
For unstable air
u
u *
ka
1  x 2 

1  x 
m  2 ln 
 ln 
  2arc tan( x) 

2
 2 
 2 
z

x  1  16 
L

For stable air
z
m  5 
 L
1
4
Friction Velocity
ka u
u* 
 z 
z

ln    mk u 
 z 0 u  ln  z   L z 
a
*
z 
 0
m
 L
Measurements of wind speed at
multiple levels can be used to
determine both u* and z0
Power Law for Wind Profile
Wind profile power law
u  z 
  
u m  zm 
Value of p
p
Estimation of Monin-Obukhov
Length
For unstable air
For stable air
z
Ri 
L
z
Ri

L 1  5 Ri
Bulk Richardson Number
 dT

 DALR 

gz  dz

Rb 
2
T 

u




Rb
Ri  2
p
2
Air Pollution Climatology
Meteorology vs. climatology
Meteorological measurements and
surveys
Pollution potential
-low level inversion frequency in US
Air Pollution Climatology
Mean maximum mixing height
determined by



Morning temperature sounding
Maximum daytime temperature
DALR
Stability wind rose