Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 389

Then, in auxiliary co-ordinates, the container surface obeys the equation

R 2
a 2 +

2 z 2
a 2 1

τ

e 2 =

1. (6.19)

−

We then adopt the prolate spheroidal co-ordinates, (ν,μ), in the auxiliary system.
The co-ordinate surfaces of the 'radial-like' co-ordinate ν are the prolate spheroids

R 2
k 2 ν

z 2
k 2 1 +

2 =

1, 1 <ν< ∞ , (6.20)

2

− ν

while the co-ordinate surfaces of the 'angular-like' co-ordinate μ are the hyperbol-
oids of two sheets,

R 2
k 2 1

z 2
k 2

2 − 2 =−

1, − 1 <μ<1. (6.21)

− μ μ

If the container surface is given by ν = ν 0 , by comparison of (6.20), for ν = ν 0 ,
with (6.19), we find that

1 , a 2 1

e 2

k 2 ν

−

a 2

2
0

k 2

2

= − = ν

0 .

(6.22)

2

τ

On substitution for τ from (6.15), the parameters ν 0 and k of the container surface
are found to be

1

a 1

e 2 σ

2

e 2

2

− − σ

ν 0 =

2 , k

=

1 .

(6.23)

e 2

2

1 − σ σ −

Solving (6.20) and (6.21) for R 2 and z 2 in terms of the prolate spheroidal co-

ordinates (ν,μ), we find

k 2 ν 1 1

2 , z 2

R 2

2

k 2

2 2

= − − μ = ν μ . (6.24)

We then have the co-ordinate relations

k ν 1 1

2 cosφ =

x =

2

− − μ R cosφ, (6.25)

k ν 1 1

2 sinφ = 2

y = − − μ R sinφ, (6.26)

k τνμ = τ z . z = (6.27)

The co-ordinate surfaces (6.20) and (6.21) may be shown to be orthogonal. Dif-
ferentiating (6.20) with respect to R ,forfixedν,gives

dz

2

dR =− ν

R
z .

(6.28)

ν

2

− 1