ATMS 500
2008 (Middle latitude) Synoptic Dynamic Meteorology
Prof. Bob Rauber
106 Atmospheric Science Building
PH: 333-2835
e-mail: rauber@atmos.uiuc.edu
Time: Tues/Thurs, 3:00 pm. -4:15 a.m.
Synoptic dynamics of extratropical cyclones
Our goal: Examine the fundamental dynamical constructs required to
examine the behavior of extratropical cyclones:
Martin, J. “Mid-latitude atmospheric dynamics” Wiley
-Kinematic properties of the horizontal wind field
-Fundamental and apparent forces
-Mass, Momentum and Energy
-Governing equations (Conservation of momentum, mass, energy, eqn of state)
-Equations of motion and their application
-Balanced flow
-Isentropic flow
-Circulation, vorticity, Potential vorticity
-Quasi-Geostrophic theory
-Ageostrophic flow
-QG diagnostic tools ( equation, Q vectors)
-Frontogenesis
-Vertical circulation about fronts
-Semi-geostrophic theory
-Potential vorticity diagnostics
Synoptic dynamics of extratropical cyclones
Potential additional topics
Extratropical Cyclones
-Wave dynamics
-Jetstream and Jetstreak dynamics
-Subtropical and Polar front jets
-Cyclogenesis and explosive cyclogenesis
-Cyclone lifecycle
-Pacific/Continental/East Coast cyclone structure and dynamics
-Fronts and frontal dynamics
-Occlusions
COURSE WEBSITE
http://www.atmos.uiuc.edu/courses/atmos500-fa08/
Posted on this website are:
1) All PowerPoint files I use in class
2) Any Papers I will review
3) Syllabus
GRADES
A. Mid-term exam (25% of grade)
B. Final exam (25% of grade)
C. Derivation Notebooks (26% of grade)
D. Homework (24% of grade)
In this course we will use Système Internationale (SI) units
Property
Length
Mass
Time
Temperature
Frequency
Force
Pressure
Energy
Power
Name
Meter
Kilogram
Second
Kelvin
Hertz
Newton
Pascal
Joule
Watt
Symbol
m
kg
s
K
Hz (s-1)
N (kg m s-2)
Pa (N m-2)
J (N m)
W (J s-1)
Review of basic mathematical principles
Constant: A quantity that has a single value
Scalar: A quantity that is described completely by it’s magnitude
Examples:
temperature, pressure, relative humidity, volume, snowfall
Vector: A quantity that requires more than one value to describe it completely
Examples:
position, wind, vorticity
Vector Calculus




A  Ax i  Ay j  Az k




B  Bx i  By j  Bz k
 
If: A  B
Ax  Bx
Then:
Ay  By
Az  Bz

The magnitude of A is given by:

A  Ax2  Ay2  Az2


1/ 2
Adding Vectors
 



A  B   Ax  Bx i  Ay  By  j   Az  Bz k
Adding vectors is commutative
Adding vectors is associative
   
A B  B  A



     
A B C  A B C

Subtracting Vectors
Simply adding the negative of the vector
 
  

A B  A  B
 



A  B   Ax  Bx i  Ay  By  j   Az  Bz k
Multiplying a Scalar and a Vector




FA  FAx i  FAy j  FAz k
Multiplying two Vectors to obtain a scalar (the scalar or dot product)
   
A  B  A B cos
 






A  B  Ax i  Ay j  Az k  Bx i  By j  Bz k



Carrying out multiplication gives nine terms
 
 
 
 
 
 
 
A  B  Ax Bx i  i   Ax B y i  j   Ax Bz i  k
 
 
 
 Ay Bx  j  i   Ay B y  j  j   Ay Bz j  k
 
 
 
 Az Bx k  i  Az B y k  j  Az Bz k  k




 
A  B  Ax Bx  Ay By  Az Bz
Commutative and distributive    
A B  B  A
properties of scalar product


      
A B  C  A B  A C
Multiplying two Vectors to obtain a vector (the curl or cross product)
Magnitude of cross product
  
i j k
 
A  B  Ax Ay Az
Bx By Bz
 
A  B  A B sin 
Where quantity on the
RHS of the equation is
called the determinant
 



A  B  Ay Bz  Az By i   Ax Bz  Az Bx  j  Ax By  Ay Bx k
Properties of the cross product
It is not commutative
   
A B  B  A

 
 
 
A  B  B  A
rather

  
  
It is not associative A  B  C  A  B  C


      
A B  C  A B  A C
Derivatives of vectors and scalars
 
 d

F
mV
dt

  dm

dV  dm
F m
V
 mA  V
dt
dt
dt


 
V  ui  vj  wk




dV du  dv  dw 
di
dj
dk

i
j
k u v  w
dt dt
dt
dt
dt
dt
dt
Special mathematical operator we will employ extensively
The del operator:
T 
T  T  T 
i
j
k
x
y
z
     
 i 
j k
x
y
z
Used to determine the gradient of a scalar quantity

       



  A   i 
j  k   Ax i  Ay j  Az k
y
z 
 x
  Ax Ay Az 

  A  


y
z 
 x

Used to determine the divergence of a vector field


       



  A   i 
j  k   Ax i  Ay j  Az k
Used to determine the rotation or
y
z 
 x
curl of a vector field
  
i
j k
     Az Ay   Az Ax   Ay Ax 
  j 

 A 
 i 



  k 
x y z
z   x
z   x
y 
 y
Ax Ay Az
Special mathematical operator we will employ extensively
2
2
2


F

F

F
2
  F    F   2  2  2 
y
z 
 x




A    Ax
 Ay
 Az
x
y
z
The Laplacian Operator
The advection operator
The Taylor Series Expansion
A continuous function can be represented about the point x = 0 by a power
series of the form

f ( x)   an x n  a0  a1 x  a2 x 2  ...  an x n
n 0
Provided certain conditions are true. These are:
1) The polynomial expression passes through the point (0, f(0))
2) f(x) is differentiable at x = 0
3) The first n derivatives of the polynomial match the first n derivatives of
f(x) at x = 0.
For these conditions to be met, we must chose the “a” coefficients properly

f ( x)   an x n  a0  a1 x  a2 x 2  ...  an x n
(1)
n 0
Let’s substitute x = 0 into the above equation
f (0)  a0
Take first derivative of (1) and substitute x = 0 into the result
f (0)  a1
f (0)
 a2
Take second derivative of (1) and substitute x = 0 into the result
2
f (0)
 a3
Take third derivative of (1) and substitute x = 0 into the result
6
Carrying out all derivatives
f n (0)
an 
n!
f (0) 2 f (0) 3
f n (0) n
f ( x)  f (0)  f (0) x 
x 
x  ... 
x
2!
3!
n!
f (0) 2 f (0) 3
f n (0) n
f ( x)  f (0)  f (0) x 
x 
x  ... 
x
2!
3!
n!
To determine the value of a function at a point x, near x0, this
function can be generalized to give
f ( x0 )
f n ( x0 )
2
x  x0   ...
x  x0 n
f ( x)  f ( x0 )  f ( x0 )x  x0  
2!
n!
We will use this function often, except that we will ignore the higher order terms
f ( x)  f ( x0 )  f ( x0 )x  x0 
This is equivalent to assuming that the function changes at most linearly
In the small region between x and x0
Centered difference approximation to derivates
Consider two points, x1 and x2
in the near vicinity of point x0
Use Taylor expansion to estimate value of f(x1) and f(x2)
f ( x0 )
f n ( x0 )
2
 x   ...
 x n
f ( x1 )  f ( x0  x)  f x0   f ( x0 ) x  
2!
n!
f ( x0 )
f n ( x0 )
2
x   ...
x n
f ( x2 )  f ( x0  x)  f x0   f ( x0 )x  
2!
n!
f ( x0 )
f n ( x0 )
2
 x   ...
 x n
f ( x1 )  f ( x0  x)  f x0   f ( x0 ) x  
2!
n!
f ( x0 )
f n ( x0 )
2
x   ...
x n
f ( x2 )  f ( x0  x)  f x0   f ( x0 )x  
2!
n!
Subtracting equations gives:
f ( x0 )
x 3  ...
f ( x0  x)  f ( x0  x)  2 f ( x0 )x   2
3!
x   ....
f x0  x   f x0  x 
f ( x0 ) 
 f ( x0 )
2x
6
2
Ignoring higher order terms
f ( x0 ) 
f x0  x   f x0  x 
2x
Estimate of first derivative
Adding equations gives:
f ( x0 ) 
f x0  x   2 f x0   f x0  x 
x 2
Estimate of second derivative
Temporal changes of a continuous variable
Q  Q( x, y, z, t )
expand differential:
 Q 
 Q 
 Q 
 Q 


dQ  
dx

dy

dz





 dt


 x  y , z ,t
 z  x, y ,t
 t  x , y , z
 y  x , z ,t
divide by dt
dQ  Q   Q  dx  Q  dy  Q  dz
  


  

dt  t   x  dt  y  dt  z  dt
Note that the position derivatives are the wind components
dx
u
dt
v
dy
dt
w
dz
dt
dQ  Q   Q  dx  Q  dy  Q  dz
  


  

dt  t   x  dt  y  dt  z  dt
dQ  Q   Q   Q 
 Q 
  w

  u
  v

dt  t   x   y 
 z 
We can write this in vector form:
dQ  Q  

  V  Q
dt  t 
Let’s let Q be temperature T and switch around the equation:
T dT 

 V  T
t
dt
T dT 

 V  T
t
dt
ADVECTION OF T
local change in temperature
At a point x, y, z
the change in temperature
following an air parcel
the import of temperature
To x,y,z by the flow

 V  T
is warm advection:
warm air is transported
toward cold air
ADVECTION ALWAYS IMPLIES A GRADIENT EXISTS IN THE PRESENCE
OF A WIND ORIENTED AT A NON-NORMAL ANGLE TO THE GRADIENT