Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 411

and the scaled, flow-induced gravitational potential obeys

2 V 1

∇ =Γ ρ 1 . (6.151)

The dimensionless self-gravitation number is defined in Table 6.2.

From the numerical values in Table 6.2, to parts in 10 9 , the equation of mass
conservation (6.144) takes the dimensional form
∂ρ 1
∂ t +

Γ

v ·∇ ρ 0 + ρ 0 ∇· v = 0, (6.152)

with subscripted zero indicating near-equilibrium quantities, and subscripted ones
indicating perturbation quantities arising from the flow. Similarly, outside bound-
ary layers, to parts in 10 6 , the equation of motion (6.146) takes the dimensional
form

ρ 0 ∂ v v − ρ 1 g 0 r ) .

∂ t +

2 Ω × =−∇ p 1 + ρ 0 g 1 − Ω × ( Ω × (6.153)

The centrifugal acceleration Ω × ( Ω × r ) can be expressed as −∇ W with

1
2 Ω

2 r 2 sin 2

W =− θ, (6.154)

a function of radius r and co-latitude θ. Similarly, the near-equilibrium gravita-
tional acceleration, g 0 , can be expressed as V 0 , with V 0 the near-equilibrium

gravitational potential. Combining the two potentials, as customary, into the total
geopotential U 0 (5.1), the equation of motion becomes
ρ 0 ∂ v

−∇

v

∂ t +

2 Ω × =−∇ p 1 + ρ 0 g 1 − ρ 1 ∇ U 0 . (6.155)

It is also customary to combine the near-equilibrium gravitational acceleration
with the centrifugal acceleration to form the total near-equilibrium gravity vector,
denoted by g 0 , with

g 0 =−∇

U 0 . (6.156)

The near-equilibrium equation for pressure gradient, (6.145), takes the dimensional
form of the hydrostatic condition

∇ = ρ 0 g 0 .

p 0 (6.157)

ff ff

In the energy equation (6.149), thermal di usion e ects amount to only parts in

10 16 and viscous dissipation e ects amount to only a few parts in 10 9 at most, so

that the flow is highly isentropic and the dimensional form of the energy equation
becomes

ff

∂ s
∂ t +

Ds
Dt =

v ·∇ s = 0. (6.158)