Circulation: A measure of the rotation within a finite element of a fluid
C
 
V

 d l

 
V cos
  d l
 Arbitrary blob of fluid rotating in a horizontal plane

In meteorology, changes in circulation are associated with changes in the intensity of weather systems. We can calculate changes in circulation by taking the time derivative of the circulation: dC dt
 d dt


V

 d l
 
Circulation is a macroscopic measure of rotation of a fluid and is a seldom used quantity in synoptic meteorology and atmospheric dynamics.

Calculate the circulation within a small fluid element with area
 x
 y
C
  udx
 vdy

   x
 v

 v
 x
 x
 y


 u

 u
 y
 y


 x

   y

C
  udx
 vdy

   x
 v

 v
 x
 x
 y


 u

 u
 y
 y


 x

   y
C



 v
 x

 u
 y


 x
 y
 x
 y

0


C
 x
 y



 v
 x

 u
 y
   relative vorticity
The relative vorticity is the microscopic equivalent of macroscopic circulation

Consider an arbitrary large fluid element, and divide it into small squares.
C
A
  udx
 vdy
  x
 v

 v
 x
 x
 y


 u

 u
 y
 y


 x

   y
C
B
  udx
 vdy

   x
 v

 v
 x
 x
 y


 u

 u
 y
 y


 x

   y
Sum circulations: common side cancels
Make infinitesimal boxes: each is a point measure of vorticity and all common sides cancel

Consider an arbitrary large fluid element, and divide it into small squares.
Fill area with infinitesimal boxes: each is a point measure of vorticity and all common sides cancel so that:
C
   udx
 vdy

 
Area


 v
 x

 u
 y


 x
 y
The circulation within the area is the area integral of the vorticity

Understanding vorticity: A natural coordinate viewpoint
Natural coordinates: s direction is parallel to flow, positive in direction of flow n direction is perpendicular to flow, positive to left of flow

CALCULATE CIRCULATION
Note that only the curved sides of this box will contribute to the circulation, since the wind velocity is zero on the sides in the n direction
Denote the distance along the top leg as
 s
Denote the distance along the bottom leg as
 s + d(
 s)
Denote the velocity along the bottom leg as V
Use Taylor series expansion and denote velocity along the top leg as
(negative because we are integrating counterclockwise)

V


V
 n
 s
Note that d (
 s) =
  n
C
 
V

 d l


V
  s
  n


V


V
 n
 n
 s

CALCULATE VORTICITY
C
 
V

 d l


V
  s
  n


V


V
 n
 n
 s
C

V
 s

V
 n

V
 s


V
 n
 n
 s
C

V
 n


V
 n
 n
 s
 
 n

 s

0
C
 n
 s

V
 n
 n
 s


V
 n
 n
 s
 n
 s

V
 
 s


V
 n
 
V
R s


V
 n

 
V
R s


V
 n

 
V
R s


V
 n

Vorticity due to the earth’s rotation
Consider a still atmosphere:
 
V
 
R
U
  a cos

R
Earth’s rotation no motion along this direction
C e
 
U
 d l


U
A
 
A

U
B
 
B
C e


 a cos
       a cos
      d
 

 a cos
       a cos
      d


C e


 a cos
       a cos
      d
 

 a cos
       a cos
      d
 after some algebra and trigonometry……
C e

2
 sin
 
2 a
2
 cos
 sin
  d
 
2
 sin
 
A
C e
 f

A
A lim

0
C e
A
 f
Earth ' s vorticity

2
 sin
  f

Cartesian expression for vorticity
3D relative vorticity vector


  
V




 w
 y

 v
 z

 i
ˆ 
 u
 z

 w
 x
ˆ j



 v
 x

 u
 y

 k
ˆ
Vertical component of vorticity vector (rotation in a horizontal plane
Absolute vorticity (flow + earth’s vorticity)
Absolute vorticity
  k
 

 k
  
V


 v
 x

 u
 y
  k
 
 a
 k
  

V a
 


 v
 x

 u
 y


 f

du
 dt fv
 
1

 p
 x
(1) dv
 dt
Expand total derivative fu
 
1

 p
 y
(2)
 u
 t
 u
 u
 x
 v
 u
 y
 w
 u
 z
 fv
 
1

 p
 x

F x
Take

( 2 )
 x

 v
 t

( 1 )
 y
 u
 v
 x
 v
 v
 y
 w
 v
 z
 fu
 
1

 p
 y

F y write relative vorticity
 v
 x

 u
 y as

 
 t
 u
 
 x
 v
 
 y
 w
 
 z

   f



 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z


 u
 f
 x
 v
 f
 y

1

2


 
 x
 p
 y

 
 y
 p
 x






F
 x x


F y
 y

 d
   f dt


 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z



1

2


 
 x
 p
 y

 
 y
 p
 x






F y
 x


F x
 y



d
   f dt
Rate of change of relative vorticity
Following parcel


 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z



1

2


 
 x
 p
 y

 
 y
 p
 x






F y
 x


F x
 y


Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically sheared flow
Pressure/density solenoids
Gradients in force
Of friction

d
   f dt
Rate of change of relative vorticity
Following parcel


 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z



1

2


 
 x
 p
 y

 
 y
 p
 x






F y
 x


F x
 y


Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically sheared flow
Pressure/density solenoids
Gradients in force
Of friction

d
   f dt
Rate of change of relative vorticity
Following parcel


 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z



1

2


 
 x
 p
 y

 
 y
 p
 x






F y
 x


F x
 y


Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically sheared flow
Pressure/density solenoids
Gradients in force
Of friction
Solenoid: field loop that converts potential energy to kinetic energy
F
 ma
PGF
 a m m (or

) large acceleration small m (or

) small acceleration large
Cold advection pattern geostrophic wind

d
   f dt
Rate of change of relative vorticity
Following parcel


 u
 x

 v
 y





 w
 x
 v
 z

 w
 y
 u
 z



1

2


 
 x
 p
 y

 
 y
 p
 x






F y
 x


F x
 y


Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically sheared flow
Pressure/density solenoids
Gradients in force
Of friction
Geostrophic wind = constant
N-S wind component due to friction



F y
 x



du dt
 
 
 x
 fv

F x
(1) dv dt
 
 
 y
 fu

F y
(2)
Expand total derivative
 u
 t
 u
 u
 x
 v
 u
 y
 
 u

P
 
 
 x
 fv

F x
 v
 t
 u
 v
 x
 v
 v
 y
 
 v

P
 
 
 y
 fu

F y

 t


 v
 x

 u
 y


  u

 x


 v
 x

 u
 y
Take

( 2 )
 x


( 1 )
 y
 f


 v

 y


 v
 x

 u
 y
 f


 


P


 v
 x

 u
 y





 v
 x

 u
 y
 f




 u
 x

 v
 y





 u

P
 
 y

 v

P
 
 x






F y
 x


F x
 y

 write relative vorticity
 v
 x

 u
 y as

 
 t
  u

 x

  f

 v

 y

  f

 
 

P


  f
 

 u
 x

 v
 y





 u

P
 
 y

 v

P
 
 x






F y
 x


F
 y x



 
 t
  u

 x

  f

 v

 y

  f

 
 

P


  f
 

 u
 x

 v
 y





 u

P
 
 y

 v

P
 
 x






F
 x x


F y
 y


Local rate of change of relative vorticity
Horizontal advection of absolute vorticity on a pressure surface
Vertical advection of relative vorticity
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically sheared flow
Gradients in force
Of friction
In English: Horizontal relative vorticity is increased at a point if
1) positive vorticity is advected to the point along the pressure surface,
2) or advected vertically to the point,
3) if air rotating about the point undergoes convergence (like a skater twirling up),
4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion
5) if the force of friction varies in the horizontal.
Solenoid terms disappear in pressure coordinates: we will work in P coordinate from now on