Vorticity
Vorticity is the microscopic measure of spin and rotation in a fluid.
Vorticity is defined as the curl of the velocity:
V
r
×
∇
Wind direction varies
→
clockwise spin
Wind speed varies
→
clockwise spin
Absolute vorticity (inertial reference frame):
Relative vorticity (relative to rotating earth):
a
a
V
r
r
×
∇
≡
ω
V
r
r
×
∇
≡
ω

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w
v
u
z
y
x
k
ˆ
j
ˆ
i
ˆ
V
∂
∂
∂
∂
∂
∂
=
×
∇
r
Expansion of relative vorticity into Cartesian components:
For large scale dynamics, the vertical component of vorticity is
most important. The vertical components of absolute and
relative vorticity in vector notation are:
relative vorticity
k
ˆ
y
u
x
v
j
ˆ
x
w
z
u
i
ˆ
z
v
y
w
V
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
=
×
∇
r
absolute vorticity
(
)
V
k
ˆ
r
×
∇
⋅
=
ζ
(
)
a
V
k
ˆ
r
×
∇
⋅
=
η
From now on,
vorticity implies the
vertical component
(unless otherwise
stated.)
(
)
f
sin
V
k
ˆ
e
=
=
×
∇
⋅
φ
Ω
2
r
The absolute vorticity is equal to the relative vorticity plus the
earth’s vorticity. Since the earth’s vorticity is
For large scale circulations, a typical magnitude for vorticity is
y
u
x
v
∂
∂
−
∂
∂
=
ζ
1
5
10
−
−
=
≈
s
L
U
ζ
then
and
f
f
y
u
x
v
+
=
+
∂
∂
−
∂
∂
=
ζ
η