Vorticity
Vertical component of vorticity: i.e., the rotation about the local vertical


v u
ς  k    V or in cartesian coordinate s, ς 

x y
There are three types of vorticity used in geophysical fluid dynamics
Relative vorticity = ς
Planetary vorticity = f
Absolute vorticity = ςa = ς + f
If the wind is geostrophic, the geostrophic relative vorticity is given by
g 2
1 2
ςg = ∇ Φ = ∇ z, where Φ = gz is geopotenti al
f
f
Vorticity
y
+
∂v ∂u
ς=
∂x ∂y
+
u

0
y
v
0
x
x
ς  0 for cyclonic circulation
ς  0 for anticyclon ic circulation
Divergence

∇ • VH =
∂u ∂v
+
∂x ∂y
Divergence:
From the continuity equation -

 u v w 
1 dρ

   V  



ρ dt
 x y z 
For incompressible flow:
dρ
0
dt

 u v  w

   VH  


 x y  z
Most meteorologists simply use divergence when they mean the

horizontal divergence   VH
Divergence

u v
  VH 

 x y
∂u
?
∂x
u
0
x
v
0
y

∂u ∂v
∇ • VH =
+
> 0, divergence
∂x ∂y

∂u ∂v
∇ • VH =
+
< 0, convergenc e
∂x ∂y
Vorticity and Divergence
Why they are important?
What’s the unit?
s-1
What’s the order of the magnitude?
Relative Vorticity
~ 10 -5 s -1
Scale Analysis (Synoptic Scales)
∂v 10 ms -1 10 ms -1
=
=
= 10
6
∂x 1000km 10 m
-5
s -1
For severe weather , vorticity can go up to ~ 10 -3 s-1 .
Divergence , ~ 10 -6 s -1
Why?
For severe weather , divergence can go up to ~ 10 -3 s-1 .
Vorticity and Divergence (Synoptic Scale)
Relative Vorticity
~ 10 -5 s -1
∂v ∂u
ς=
∂x ∂y
∂v
when
>0
∂x
Divergence , ~ 10 -6 s -1

u v
  VH 

 x y
∂u
=> >0
∂y
Same signs for
both terms
∂u
when
>0
∂x
∂v
=>
<0
∂y
Divergence smaller because
∂u
∂v
and
are opposite signs! ! !
∂x
∂y
Vorticity in Nature Coordinates
v ∂v
ς= R ∂n
v
R
Curvature effect
n
V
S
Shear effect
The smaller the R (i.e., curved more, shorter waves), the stronger the
curvature term. R is positive for cyclonic circulation and negative for anticyclonic circulation.
Vorticity
?
curvature
Shear
Vorticity Advection
Ridge
Trough
Trough
V wind speed
L
dςa
 0,
dt
0
Relative
vorticity
advection
Planetary
vorticity
advection


∂ςa ∂ς ∂f ∂ς
df
=
+
=
= -V • ∇ςa = -V • ∇ς - v ,
∂t ∂t ∂t ∂t
dy
df
>0
dy
Vorticity Advection (shorter Rossby waves)
For short waves,
relative vorticity dominates
Ridge
Trough
Trough
ς min
V
ς max
ς max
L
Wavelength

 ∂ς
∂ς
=>
= - V • ∇ς = - V
(nature coordinate s)
∂t
∂s
Downstream of a trough or upsteam of a ridge
 ∂ς
∂ς
∂ς
V > 0,
< 0, =>
≈ -V
> 0 (PVA)
∂s
∂t
∂s
Vorticity Advection (shorter Rossby waves)
∂ς
>0
∂t
PVA
If one uses geostrophi c wind to approximat e real wind, then
g
1
ςg = ∇2Φ = ∇2z, where Φ = gz is the geopotenti al.
f
f
(assume wave solution)
g
2
∂ςg f ∂(∇ z)
g ∂z
∂z
=
∝> 0, =>
< 0, => z decreases with time.
∂t
∂t
f ∂t
∂t
PVA
Adiabatic
cooling
z decreases
with time
divergence
aloft
ascent
convergenc
e
Vorticity Advection (shorter Rossby waves)
For short waves,
relative vorticity dominates
Ridge
Trough
Trough
ς min
V
ς max
ς max
L

ς
ς

 V  ς  V
(nature coordinate s)
t
s
Down stream of a trough or upsteam of a ridge
V  0,
ς
ς
ς
 0, 
 -V
0
s
t
s
Positive Vorticity Advection, PVA.
Vorticity Advection (shorter Rossby waves)
For short waves,
relative vorticity dominates
Ridge
Trough
Trough
ς min
V
ς max
ς max
L
Waves propagate eastward!

ς
ς

 V  ς  V
(nature coordinate s)
t
s
Down stream of a trough or upsteam of a ridge
V  0,
ς
ς
ς
 0, 
 -V
0
s
t
s
Negative Vorticity Advection, NVA.
Vorticity Advection (shorter Rossby waves)
NVA
∂ς
<0
∂t
If one uses geostrophi c wind to approximat e, then
(assume wave solution)
g 2
∂ςg ∂ f ∇ z
g ∂z
=
∝< 0,
∂t
∂t
f ∂t
NVA
Adiabatic
warming
z increases
with time
∂z
=>
> 0, => z inceases with time.
∂t
convergence
aloft
decent
divergence
Vorticity Advection (shorter Rossby waves)
564
βL2
C = U- 2 ,
4π
Trough
df
where β =
,
dy
Wave propagation speed
570
576
U
C wave speed
Tropopause
L2
z
C
4
0
speed
2
level of
non-divergence
500-600 mb
Vertical Coupling
564
Trough
570
576
U
C wave speed
Tropopause
L2
z
C
4
level of
non-divergence
500-600 mb
2
0
speed
Will grow
Will grow
Low-level cyclone and 500mb heights
5520 5460
996
cold air mass
992
**
.
cold front
32
31
988
..
39
37
direction of
propagation
49
46
56
52
warm air mass
warm front
Upper and low-level coupled together –
upper level trough enhances the low-level cyclone development.
Low-level cyclone and 500mb heights
5520 5460
5520 5460
upper level trough is about to
catch low-level system.
upper level trough catches
low-level system – low level
system is decaying.