Transcript AOSS 321, Fall 2006 Earth Systems Dynamics 10/9/2006

AOSS 321, Winter 2009
Earth System Dynamics
Lecture 4
1/20/2009
Christiane Jablonowski
[email protected]
734-763-6238
Eric Hetland
[email protected]
734-615-3177
Today’s lecture
We will discuss
• Divergence / Curl / Laplace operator
• Spherical coordinates
• Vorticity / relative vorticity
• Divergence of the wind field
• Taylor expansion
Divergence
(Cartesian coordinates)
• Divergence of a vector A:
  
 
x  Ax 
    Ax Ay Az

A =
 Ay 


y
z
y    x
   Az 
 
z 
• Resulting scalar quantity, use of partial derivatives

Curl
(Cartesian coordinates)
• Curl of a vector A:
Az Ay 
  



 
y z 



x
A

x
 
    Ax Az 

A=
 Ay 

x 
y     z
   Az  Ay Ax 

 


z 
y 
 x
• Resulting vector quantity, use of partial derivatives


Laplace operator
(Cartesian coordinates)
• The Laplacian is the divergence of a gradient
   T 
   
x  x  2
2
2


T

T

T

T
2




  T  =  T 

 2 2 2
y
z
y  y  x
   T 
   
z  z 
• Resulting scalar quantity, use of second-order
partial derivatives
Spherical coordinates
• Longitude (E-W) defined from [0, 2]: 
• Latitude (N-S), defined from [-/2, /2]: 
• Height (vertical, radial outward):
r
r
v

Spherical velocity components
• Spherical velocity components:
D
u  r cos
Dt
D
vr
Dt
Dr D(a  z) Dz
w


Dt
Dt
Dt
• r is the distance to the center of the earth, which is
related to z by r = a + z

• a is the radius of the earth (a=6371 km), z is the
distance from the Earth’s surface
Unit vectors in spherical coordinates
• Unit vectors in longitudinal, latitudinal and vertical
direction:
sin 


i  cos 


 0 
cos sin  


j  sin  sin  


 cos 
cos cos 


k  sin  cos 


 sin  
• Unit vectors in the spherical coordinate system
depend on the location (, )
• As in the Cartesian system: They are orthogonal
and normalized

Velocity vector in spherical coordinates
• Use the unit vectors to write down a vector in
spherical coordinates, e.g. the velocity vector
v  ui  vj  wk
• In a rotating system (Earth) the position of a point
(and therefore the unit vectors) depend on time
• The Earth rotates as a solid body with the Earth’s

angular speed of rotation  = 7.292  10-5 s-1
   t,
  0,
z  0
i  

i
Dia Di
• Leads to
    


t  t 
Dt D
Time derivative of the spherical velocity
vector
• Spherical unit vectors i , j ,k depend on time
• This means that a time derivative of the velocity
vector in spherical coordinates is more complicated
in comparison to the Cartesian system:



Dv D

ui  vj  wk
Dt Dt
Du
Dv
Dw
Di
Dj
Dk

i
j
ku
v
w
Dt
Dt
Dt
Dt
Dt
Dt
Spherical coordinates , , r
• Transformations from spherical coordinates to
Cartesian coordinates:
x  r cos cos
y  r sin  cos
z  r sin 

Divergence of the wind field
u 
 
• Consider the divergence of the wind vector v  v 
 
  
w 
 
x  u 
    u v w

v =
 v   
y    x y z
   w 
 
z 
•
•
•
•
 •
The divergence is a scalar quantity
If  v > 0 (positive), we call it divergence
If  v < 0 (negative), we call it convergence
If  v = 0 the flow is nondivergent
Very important in atmospheric dynamics
Divergence of the wind field (2D)
• Consider the divergence of the horizontal wind
vector
u
v h   
v 


•
•
•
•
  
x  u u v
  v h =      
  v  x y
y 
The divergence is a scalar quantity
If  v > 0 (positive), we call it divergence
If  v < 0 (negative), we call it convergence

If  v = 0 the flow is nondivergent
Unit 3, frame 17: http://www.atmos.washington.edu/2005Q1/101/CD/MAIN3.swf
Vorticity, relative vorticity
u 
 
• Vorticity of geophysical flows: curl of v  v 
 
(wind vector)   
w 
w v 
y  z 
 

x  u  


u

w
 
    v =   v    
y    z x 
   w  v u 
 
  
z 
x y 
• relative vorticity: vertical component of 
defined as   k    v  u

rel
x y
• Very important in atmospheric
dynamics
Relative vorticity
• Vertical component of 
 rel
v u
 k   

x y

• Relative vorticity is
 rel  0 (positive) for counterclockwise rotation
– 
–  rel  0 (negative) for clockwise rotation

–  rel  0 for irrotational flows
 • Great web page: http://my.Meteoblue.Com/my/

Real weather situations
500 hPa rel. vorticity and mean SLP
sea level
pressure
Positive rel.
vorticity,
counterclockwise rotation,
in NH: low
pressure
system
Divergent / convergent flow fields
1 
2 x 
• Compute the divergence of the wind vector v h  1 
 y 
2 
• Compute the relative vorticity
• Draw the flow field (wind vectors)
y

x
Rotational flow fields
• Compute the rel. vorticity of the wind vector
• Compute the divergence
• Draw the flow field (wind vectors)
y

x
 1 
 2 y 
v h  
1 
 x 
 2 
Taylor series expansion
• It is sometimes convenient to estimate the value of a
continuous function f(x) about a point x = x0 with a
power series of the form:
df (x 0 )
1 d 2 f (x 0 )
2
f (x)  f (x 0 ) 
(x  x 0 ) 
(x

x
)
0
2
dx
2! dx
1 d n f (x 0 )
n
 ...
(x

x
)
0
n
n! dx
df (x 0 )
 f (x 0 ) 
(x  x 0 )
dx
• In the last approximation, we neglected the higher
order terms