Geography Reference
In-Depth Information
4.5.1
The Barotropic (Rossby) Potential Vorticity Equation
For a homogeneous incompressible fluid, the continuity equation (2.31) simplifies
to
∇·
U
=
0 or, in Cartesian coordinates,
∂u
∂x +
∂v
∂y
∂w
∂z
=−
so that the vorticity equation (4.22b) may be written
f ) ∂w
∂z
D h
+
f )
=
+
(4.23)
Dt
Section 3.4 showed that the thermal wind in a barotropic fluid vanishes so that
the geostrophic wind is independent of height. Letting the vorticity in (4.23) be
approximated by the geostrophic vorticity ζ g and the wind by the geostrophic wind
(u g , v g ), we can integrate vertically from z 1 to z 2 to get
h D h ζ g +
f
ζ g +
f w ( z 2 )
w ( z 1 )
=
(4.24)
Dt
However, because w
Dz/Dt and h
h(x,y,t),
Dz 2
Dt
Dz 1
Dt =
D h h
Dt
w ( z 2 )
w ( z 1 )
=
(4.25)
Substituting from (4.25) into (4.24) we get
D h ζ g +
f
1
ζ g +
1
h
D h h
Dt
f
=
Dt
or
D h ln ζ g +
f
D h ln h
Dt
=
Dt
which implies that
ζ g +
D h
Dt
f
=
0
(4.26)
h
This is just the potential vorticity conservation theorem for a barotropic fluid,
which was first obtained by C. G. Rossby [see (4.13)]. The quantity conserved
following the motion in (4.26) is the Rossby potential vorticity .
 
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