Geology Reference
In-Depth Information
Z
a
a
b
R
c
Figure 6.1 Cross-section of the container surface, where
a
is the equatorial radius,
b
is the polar radius and
c
is the focal distance.
the
inertial wave equation
of Poincare. In a Cartesian co-ordinate system (
x
,y,
z
)
with the
z
-axis aligned with the rotation axis, the Poincare equation becomes
1
∂
2
2
2
∂
∂
x
2
+
∂
χ
χ
∂y
1
σ
χ
∂
z
2
=
2
+
−
0,
(6.13)
2
with the angular frequency expressed by the dimensionless Coriolis frequency,
σ
=
2
.
(6.14)
Ω
For modes with periods less than 12 sidereal hours, the square of the Coriolis
frequency satisfies σ
2
> 1. In this case, the Poincare inertial wave equation can
be transformed to Laplace's equation (Bryan, 1889) in new auxiliary co-ordinates
(
x
,y,
z
) in which the
z
-axis is stretched by the factor 1/τ, where 0 <τ<1, giving
z
=
τ
z
, with
1
σ
2
τ
=
1
−
2
.
(6.15)
If the container is an ellipsoid of revolution, or an oblate spheroid, then in phys-
ical cylindrical co-ordinates (
R
,φ,
z
), with
R
the cylindrical radius and φ the east
longitude, as illustrated in Figure 6.1, its surface is described by the equation
R
2
a
2
+
z
2
b
2
=
R
2
c
2
csc
2
z
2
c
2
cot
2
α
+
α
=
1.
(6.16)
The eccentricity,
e
, of the cross-section of the container surface is defined as
e
=
c
/
a
. Since
a
2
b
2
c
2
=
+
,
(6.17)
we have
a
2
1
e
2
.
b
2
=
−
(6.18)
Search WWH ::
Custom Search