The Vorticity Equation
To understand the processes that produce changes in vorticity, we
would like to derive an expression that includes the time derivative of
vorticity:
d ⎛ ∂v ∂u ⎞
⎜ − ⎟ =K
dt ⎜⎝ ∂x ∂y ⎟⎠
Recall that the momentum equations are of the form
du
=K
dt
dv
=K
dt
x-component momentum equation
y-component momentum equation
Thus we will begin our derivation by taking
∂
∂
[y-component momentum equation] −
[x-component momentum equation]
∂y
∂x
∂
∂
[y-component momentum equation] −
[x-component momentum equation] =
∂y
∂x
1 ∂p ⎤ ∂ ⎡ ∂u
1 ∂p ⎤
∂ ⎡ ∂v
∂v
∂v
∂v
∂u
∂u
∂u
+ u + v + w + fu = −
−
+u
+ v + w − fv = −
ρ ∂y ⎥⎦ ∂y ⎢⎣ ∂t
ρ ∂x ⎥⎦
∂x ⎢⎣ ∂t
∂x
∂y
∂z
∂x
∂y
∂z
1 ∂2 p
1 ⎛ ∂p ∂ρ ⎞
∂ 2 v ∂v ∂w
∂u
∂f
∂ ∂v
∂ 2 v ∂v ∂u
∂ 2 v ∂v ∂v
⎟
⎜
+u 2 +
+v
+
+w
+
+f
+u
=−
+
ρ ∂x∂y ρ 2 ⎜⎝ ∂y ∂x ⎟⎠
∂x ∂t
∂x
∂x ∂x
∂x∂y ∂y ∂x
∂x∂z ∂z ∂x
∂x
∂x
1 ∂2 p
1 ⎛ ∂p ∂ρ ⎞
∂ ∂u
∂ 2u ∂u ∂u
∂ 2u ∂u ∂v
∂ 2u ∂u ∂w
∂v
∂f
⎟
⎜
+
−f
−v
=−
+
+u
+
+v 2 +
+w
ρ ∂x∂y ρ 2 ⎜⎝ ∂x ∂y ⎟⎠
∂y ∂t
∂x∂y ∂x ∂y
∂y
∂y ∂y
∂y∂z ∂z ∂y
∂y
∂y
∂ ⎛ ∂v ∂u ⎞
∂ ⎛ ∂v ∂u ⎞
∂ ⎛ ∂v ∂u ⎞
∂ ⎛ ∂v ∂u ⎞ ⎛ ∂v ∂u ⎞⎛ ∂u ∂v ⎞
⎜ − ⎟ + u ⎜⎜ − ⎟⎟ + v ⎜⎜ − ⎟⎟ + w ⎜⎜ − ⎟⎟ + ⎜⎜ − ⎟⎟⎜⎜ + ⎟⎟ +
∂t ⎜⎝ ∂x ∂y ⎟⎠
∂x ⎝ ∂x ∂y ⎠
∂y ⎝ ∂x ∂y ⎠
∂z ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂y ⎠⎝ ∂x ∂y ⎠
⎛ ∂w ∂v ∂w ∂u ⎞
1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞
∂f
+ ⎜⎜
−
⎟⎟ + v ∂y = ρ 2 ⎜⎜ ∂y ∂x − ∂x ∂y ⎟⎟
x
z
y
z
∂
∂
∂
∂
⎝
⎠
⎝
⎠
⎛ ∂u ∂v ⎞
∂ζ
∂ζ
∂ζ
∂ζ
+u
+v
+w
+ ζ ⎜⎜ + ⎟⎟ +
∂t
∂x
∂y
∂z
⎝ ∂x ∂y ⎠
⎛ ∂u ∂v ⎞
f ⎜⎜ + ⎟⎟
⎝ ∂x ∂y ⎠
∂f
df ∂f
∂f
∂f
=
+u +v + w
dt ∂t
∂x
∂y
∂z
⎛ ∂u ∂v ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞
1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞
∂f
⎜
⎟
⎟⎟ + v
=
−
−
f ⎜⎜ + ⎟⎟ + ⎜⎜
∂y ρ 2 ⎜⎝ ∂y ∂x ∂x ∂y ⎟⎠
⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠
d
(ζ + f ) = −(ζ + f )⎛⎜⎜ ∂u + ∂v ⎞⎟⎟ − ⎛⎜⎜ ∂w ∂v − ∂w ∂u ⎞⎟⎟ + 12 ⎛⎜⎜ ∂p ∂ρ − ∂p ∂ρ ⎞⎟⎟
dt
⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠
vorticity equation
1
Terms In Vorticity Equation
d
(ζ + f ) = −(ζ + f )⎛⎜⎜ ∂u + ∂v ⎞⎟⎟ − ⎛⎜⎜ ∂w ∂v − ∂w ∂u ⎞⎟⎟ + 12 ⎛⎜⎜ ∂p ∂ρ − ∂p ∂ρ ⎞⎟⎟
dt
⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠
A
B
C
D
A: Rate of change of absolute vorticity following the fluid motion
B: Effect of horizontal velocity divergence on vorticity
C: Transfer of vorticity between horizontal and vertical components
(“twisting term” or “tilting term”)
D: Effects of baroclinicity (“solenoidal term”)
Term A: Rate of change of absolute vorticity following the fluid motion
∂ (ζ + f )
∂ (ζ + f )
∂ (ζ + f )
d (ζ + f ) ∂ (ζ + f )
=
+u
+v
+w
∂t
∂x
∂y
dt
∂z
local
tendency
of absolute
vorticity
horizontal
advection
of absolute
vorticity
vertical
advection
of absolute
vorticity
2
Term B: Effect of horizontal velocity divergence on vorticity
⎛ ∂u ∂v ⎞
− (ζ + f )⎜⎜ + ⎟⎟
⎝ ∂x ∂y ⎠
⎛ ∂u
∂v ⎞
⎛ ∂u
∂v ⎞
If ⎜⎜⎝ ∂x + ∂y ⎟⎟⎠ > 0 (divergence), then vorticity will decrease if absolute
vorticity is positive. Vorticity will increase if absolute vorticity is
negative.
If ⎜⎜⎝ ∂x + ∂y ⎟⎟⎠ < 0 (convergence), then vorticity will increase if absolute
vorticity is positive. Vorticity will decrease if absolute vorticity is
negative.
This mechanism is quite important for large-scale midlatitude systems.
Term C: Transfer of vorticity between horizontal and vertical components
(“twisting term” or “tilting term”)
⎛ ∂w ∂v ∂w ∂u ⎞
⎟⎟
− ⎜⎜
−
⎝ ∂x ∂z ∂y ∂z ⎠
∂v
>0
∂z
In this example, vertical shear of v-component
wind is producing shear vorticity about an eastwest axis. The orientation of the vorticity vector
is shown by the solid red arrow.
East-west variations in the vertical velocity twist
or tilt this “vortex tube” toward a more vertical
orientation, as indicated by the broken red
arrow. This gives the vorticity vector a
component in the z-direction, indicating a
transfer of vorticity from the horizontal to the
vertical.
∂w
<0
∂x
∂v ∂w
d (ζ + f )
<0→
>0
∂z ∂x
dt
3
Term D: Effects of baroclinicity (“solenoidal term”)
1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞
⎜
⎟
−
ρ 2 ⎜⎝ ∂y ∂x ∂x ∂y ⎟⎠
This term arises because of the horizontal
variations in density that occur in a baroclinic
atmosphere. In this example, even though the
pressure gradient is uniform, variations in
density produce small variations in the pressure
gradient force. The variations in acceleration
that result lead to the production of positive
vorticity.
p1
p2
p3
p4
ρ4
ρ3
p4 > p3 > p2 > p1
ρ4 > ρ3 > ρ2 > ρ1
ρ2
ρ1
∂p
∂ρ
d (ζ + f )
< 0;
<0→
>0
∂y
∂x
dt
4