Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 394

In auxiliary cylindrical co-ordinates ( R ,φ, z ), the governing Poincare inertial wave
equation (6.49), taking the azimuthal dependence again as exp im φand multiplying
through by R 2 , becomes

R ∂χ
∂ R

2

R ∂
∂ R

R 2 ∂

χ
∂ z 2 −

m 2 − χ = 0. (6.64)

Di ff erentiation of (6.62) yields

1 1

η 1

2 2 2

− ξ − η − ξ

R ∂ξ

∂ξ
∂ z =− ∂ R =− ξ

,

k ξ

2 ,

(6.65)

2 2 2

ξ − η − η

while di ff erentiation of (6.63) yields

1 1

ξ 1

2 2 2

R ∂η

− ξ − η ∂η

∂ z =

− η

∂ R = η

,

2 .

(6.66)

k ξ ξ

2

− η

2 2

− η

Following the same procedure as before in the derivation of equation (6.42), we
find equation (6.64), in the auxiliary spheroidal co-ordinates (ξ,η), becomes
1

2 1 2

∂
∂ξ

1 1

− ξ − η ∂χ

∂ξ

∂
∂η

∂χ
∂η

2 2

m 2 − ξ − − η − χ = 0. (6.67)

ξ

2

− η

2

Again, the variables separate and χ is found to be the product of associated
Legendre functions in the co-ordinates ξ and η.
For a rigid container, the normal component of displacement must vanish. In
physical cylindrical co-ordinates,

R ∂χ φ im k ∂χ

∇ χ =

∂ R + R χ + ∂ z ,

(6.68)

where R ,

k are unit vectors in the directions of the co-ordinates ( R ,φ, z ), res-
pectively. Thus, from expression (6.9), the displacement in cylindrical co-ordinates
is proportional to the vector

φ ,

2

ω ∇ χ − 4 Ω ( Ω ·∇ χ) + 2 i ω Ω ×∇ χ

m R χ +

k ω

2 ∂χ

ω ∂χ

2 m R χ ∂χ

∂ R

φ i ω

R ω

2

=

∂ R +

+ 2 Ω + − 4 Ω

∂ z .

(6.69)

For the container illustrated in Figure 6.1, the vector

k dz
dR

R

t = + (6.70)

is tangent to the surface. Hence, the vector

k dR
dz

R

n = − (6.71)