Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 387

p 1 is the perturbation in pressure, W is the centrifugal potential (5.2), and ρ 0 is
the uniform mass density. Equation (6.1) equates the acceleration in the rotating
frame (3.77) to the body force per unit mass, while equation (6.2) expresses the
incompressibility of the fluid, or more precisely, the assumption that the flow in
the uniform fluid is su ciently slow that its compressibility can be ignored.

The left side of equation (6.1) is linear in the displacement field. It is the vector

2 u

=− ω + 2 i ω Ω × u =−∇ χ, (6.4)

which gives

+ ω

2 u .

1
2 i ω

Ω × u = (6.5)

Then,

2 Ω

Ω · =− ω · u , (6.6)

and

2 ( Ω × u )

Ω × =− ω

+

2 i ω Ω ×

( Ω × u )

2 i ω ( Ω · u ) Ω −Ω

2 u 2 ( Ω × u )

=− ω +

2 u ,

1
2 i ω

2 u +

4 ( Ω

2 2

= − ω + ω · ) Ω + ω Ω (6.7)

on substitution for Ω × u from equation (6.5), and for Ω · u from equation (6.6).

Solving (6.7) for the displacement field gives

ω

.

1

2

u =−

2 ω 2

− 4 Ω ( Ω · ) + 2 i ω Ω × (6.8)

2

− Ω

ω 4

Potential χ then becomes a generalised displacement potential, giving the displace-
ment field in terms of its gradient, through

ω

×∇ χ .

1

2

u =

2 ω 2

∇ χ − 4 Ω ( Ω ·∇ χ) + 2 i ω Ω (6.9)

2

ω − 4 Ω

In order for this displacement field to satisfy the condition of incompressible
flow (6.2), its divergence must vanish. Using the vector identities,
∇· Ω ( Ω ·∇ χ) = ) 2 ( Ω ·∇ χ) ∇· Ω + Ω ·∇ ( Ω ·∇ χ) = ( Ω ·∇ χ, (6.10)

and

∇· ( Ω ×∇ χ ) =∇ χ · ( ∇× Ω ) − Ω ·∇×∇ χ =

0, (6.11)

on taking the divergence of expression (6.9), and setting it to zero, we find that

2 2

) 2 ω ∇ χ − 4 ( Ω ·∇ χ = 0, (6.12)