Transcript 07. Vorticity and Circulation

FLUID ROTATION
Circulation and Vorticity
Circulation: A measure of
the rotation within a finite
element of a fluid

 
C   V  dl   V cos   dl
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 d

dt dt

 
 V  dl

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

u 
 v 
C   udx  vdy  u x   v  x y   u  y x  v y
y 
 x 


u 
 v 

C   udx  vdy  u x   v  x y   u  y x  v y
y 
 x 

 v u 
C    xy
 x y 
 C  v u
  
lim
   relative vorticity
xy  0
 xy  x y
The relative vorticity is the microscopic equivalent of macroscopic circulation
Consider an arbitrary large fluid
element, and divide it into small
squares.

u 
 v 
C A   udx vdy  u x   v  x y   u  y x  v y
y 
 x 


u 
 v 
CB   udx vdy  u x   v  x y   u  y x  v y
y 
 x 

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 
 v u 
  xy

x y 
Area 
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
V 
Use Taylor series expansion and denote velocity along the top leg as  V 
s

n


(negative because we are integrating counterclockwise)
Note that d (s) =  n
 
V 

C   V  dl  V s  n   V 
n s
n 

CALCULATE VORTICITY
 
V 

C   V  dl  V s  n   V 
n s
n 

V
C  Vs  V n  Vs 
ns
n
V
C  V n 
ns
n
 V
 C  Vn V ns



V


ns n ns
s n
ns 0 ns 
  lim

V V
  
Rs n
Shear
V V
  
Rs n
Flow curvature
V V
  
Rs n
Vorticity due to the earth’s rotation
Consider a still atmosphere:
 
V  R
R
Earth’s rotation
U  a cos
no motion
along this
direction

Ce   U  dl  U A dx A  U B dx B
Ce  a cos   a cos   d  a cos   a cos   d
Ce  a cos   a cos   d  a cos   a cos   d
after some algebra and trigonometry……
Ce  2 sin   2a2 cos sin  d  2 sin   A
Ce  f  A
Ce
 f
A0 A
lim
Earth' s vorticity  2 sin   f
Cartesian expression for vorticity
  w v   u w   v u 
3D relative vorticity vector    V  
 iˆ     ˆj    kˆ
 y z   z x   x y 

Vertical component of
vorticity vector (rotation in a
horizontal plane
 v u
 ˆ
ˆ
  k   k   V  
x y
Absolute vorticity (flow +
earth’s vorticity)


  kˆ a  kˆ   Va
Absolute vorticity
  
 v u 
   f
 x y 
The vorticity equation in height coordinates
du
1 p
 fv  
dt
 x
dv
1 p
 fu  
dt
 y
(1)
(2)
Expand total derivative
u
u
u
u
1 p
u
 v  w  fv  
 Fx
t
x
y
z
 x
Take
v
v
v
v
1 p
 u  v  w  fu  
 Fy
t
x
y
z
 y
(2) (1)

x
y
write relative vorticity
v u

x y
as 
 u v   w v w u 




f
f
1   p  p   Fx Fy 
  u  v  2 
  
u
v
w
   f     




t
x
y
z

x

y

x

z

y

z

x

y


x

y

y

x

x
y 

 


 
 u v   w v w u  1   p  p   Fy Fx 
d   f 
  2 
  
   f     




dt

x

y

x

z

y

z


x

y

y

x

x
y 

 


 
 u v   w v w u  1   p  p   Fy Fx 
d   f 
  2 
  
   f     




dt

x

y

x

z

y

z


x

y

y

x

x

y

 


 

Rate of change
of relative vorticity
Following parcel
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically
sheared flow
Pressure/density
solenoids
Gradients in force
Of friction
 u v   w v w u  1   p  p   Fy Fx 
d   f 
  2 
  
   f     




dt

x

y

x

z

y

z


x

y

y

x

x

y

 


 

Rate of change
of relative vorticity
Following parcel
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically
sheared flow
Pressure/density
solenoids
Gradients in force
Of friction
 u v   w v w u  1   p  p   Fy Fx 
d   f 
  2 
  
   f     




dt

x

y

x

z

y

z


x

y

y

x

x

y

 


 

Rate of change
of relative vorticity
Following parcel
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
 u v   w v w u  1   p  p   Fy Fx 
d   f 
  2 
  
   f     




dt

x

y

x

z

y

z


x

y

y

x

x

y

 


 

Rate of change
of relative vorticity
Following parcel
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically
sheared flow
Geostrophic wind = constant
N-S wind component
due to friction
 Fy 


 x 
Pressure/density
solenoids
Gradients in force
Of friction
The vorticity equation in pressure coordinates
du


 fv  Fx
dt
x
dv


 fu  Fy
dt
y
(1)
(2)
Expand total derivative
u
u
u
u

u
 v 

 fv  Fx
t
x
y
P
x
Take
  v u 
  v u
    u   
t  x y 
x  x y
v
v
v
v

u  v 

 fu  Fy
t
x
y
P
y
(2) (1)

x
y

  v u
f v   
y  x y


  v u 
f  
 

P  x y 

 v u
 u v   u  v    Fy Fx 
    f     

   x  y 

x

y

x

y

P

y

P

x


 
 

write relative vorticity
v u

x y
as 
 u v   u  v    Fy Fx 




 u   f   v   f   
   f     





t
x
y
P

x

y

P

y

P

x

x

y

 
 

The vorticity equation
 u v   u  v    Fx Fy 




 u   f   v   f   
   f     





t
x
y
P

x

y

P

y

P

x

x

y

 
 

Local rate of
change of relative
vorticity
Horizontal advection
of absolute vorticity
on a pressure surface
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically
sheared flow
Gradients in force
Of friction
Vertical advection
of relative vorticity
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