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
(ζ
+
f )
=
(ζ
+
(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
.