Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 414

With substitution from (6.156), dividing by ρ 0 and di ff erentiating with respect

to time, the equation of motion (6.155) becomes

1

2 v
∂ t 2 +

∂ ∂ v

∂ t =−∇ ρ 0 ∂ p 1

∂ V 1
∂ t

1
ρ

0 ∇ ρ 0 ∂ p 1

ρ 0 ∂ρ 1

1

2 Ω ×

∂ t −

−

∂ t + ∂ t g 0 ,

(6.170)

2

where V 1 is the decrease in gravitational potential associated with the flow, giving
g 1 = ∇ ects of flow pressure

fluctuations amount to only parts in 10 3 , and depend inversely on the square of
the period, and to this degree of approximation, ∂ p 1 /∂ t may be neglected along
with the second term on the right side of the equation of motion. The third term on
the right side of the equation of motion, on substitution from the equation of state
(6.168), becomes

V 1 (3.12). Again, at periods of several hours the e ff

ρ 0 g 0

p 0

ρ 0 ∂ρ 1

1 1 1

α

∂ t g 0 =

− v ·∇ ρ 0 +

2 v

·∇

ρ 0 g 0 v

g 0

1 g 0

g 0

· d ρ 0

dr +

ρ 0
α

=

2 v

·

g 0 1

g 0 ,

2

1
α

α d ρ 0

dr

=

2 v

· + (6.171)

ρ 0 g 0

taking gravity to be in the direction opposite to radius, and substituting from the
hydrostatic condition (6.157). Again, neglecting the e ect of flow pressure, and

substituting from the hydrostatic condition, equation (6.169) yields

ff

2

v · g 0 =− α ∇· v . (6.172)

Using this relation and expression (6.166) for the stability factor β, the equation of
motion is transformed to

∂χ
∂ t

2 v
∂

∂ ∂ v

∂ t =−∇

2 t +

2 Ω × − β g 0 ∇· v , (6.173)

with

∂χ
∂ t = ρ 0 ∂ p 1

1 ∂ V 1

∂ t . ∂ t −

(6.174)

At frequencies below seismic frequencies, flow pressure fluctuations have little

e ect on density compared to transport through the stratified density field or

through the hydrostatic pressure field. At periods of several hours, departures from
these subseismic conditions amount to only parts in 10 3 , and decrease inversely as
the square of the period.

ff