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Comments |
A fluid property
Ψ
(x + t)
that satisfies
D
D
t
Ψ
=
0
,
where
D
D
t
=
∂
∂
t
+
u
.
∇
,
is said to be "materially conserved".
Ertel's theorem states that if Ψ is materially conserved, then
D
D
t
q
=
v
J
,
where q is as given above, and
J
=
(
∇
p
×
∇
v
)
.
∇
Ψ
The quantity q is called the potential vorticity. Since Ψ
can be any materially conserved quantity, the potential vorticity q represents a class of quantities.
If J should be equal to zero, then
D
D
t
q
=
0
and potential vorticity itself is materially conserved.
Ertel's theorem is important because:
i) it is a very general description of fluids, and leads to other circular and vorticity theorems,
ii) as seen above, potential vorticity can be conserved, which means Ertel's theorem takes on the status of a conservation law,
iii) it is the governing equation for an important class of motions, including planetary geostrophic and quasi-geostrophic motions.
It also has applications in oceanography.
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