Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 413

density is d ρ 0 / dr . If the actual lapse rate exceeds the adiabatic lapse rate, the

displaced fluid particle will experience a restoring force; if not, it will be subject to
a force in the direction of its displacement. Hence, the stability or otherwise of the
density stratification turns on the sign of the stability factor

−

2

α d ρ 0

dr .

β =

+

1 (6.166)

ρ 0 g 0

If β<0, the density stratification is stable; if β>0, the density stratification is
unstable. The ratio of the actual lapse rate for density to the adiabatic lapse rate,
givenby1

− β, is referred to as the stratification parameter. If the stratification is
stable, the displaced fluid particle will experience a restoring force δ r βρ 0 g

2 per
unit volume. It will, therefore, oscillate around its equilibrium position at angular
frequency ω v with

2

0 /α

g 0
α

2

2

ω

v =− β

. (6.167)

ω v is called the Brunt-Vaisala frequency. For an unstable density stratification,
ω v becomes purely imaginary and w v can be used as a stability parameter with
w

2 2

v =− ω v .

At long periods, flow pressure fluctuations have only a small e ect on the dens-

ity. In the scaled equation of state (6.164), terms on the right have been arranged in
order of decreasing magnitude. They represent the rate of density decrease due to
transport along the near-equilibrium density gradient (of order D R / F R ), the e

ff

ect
of adiabatic compression by the near-equilibrium pressure field (of order C / F R ),
the e

ff

ects of dilatation-compression by the flow pressure fluctuations (of order C ),
and non-linear terms in the flow variables (of order ). In comparison to the first
term, taken as unity, the others have magnitudes of roughly 0.41, 2.8

ff

10 − 4 ,and ×

10 − 8 . Neglect of the smallest term is part of the linearisation for slow flows or
small amplitude oscillations. The e

×

1.3

ff

ects of flow pressure fluctuations on the dens-
ity are small, and depend inversely on the square of the period. At periods of several
hours they amount to only parts in 10 3 . Neglecting this term leads to the subseis-
mic approximation (Smylie and Rochester, 1981). With these approximations, the
equation of state (6.164) takes the dimensional form

∂ρ 1 1

α

∂ t =−

v ·∇ ρ 0 +

2 v

·∇

p 0 . (6.168)

Neglecting only the non-linear term on the right of (6.163), (of order 10 − 9
compared to the largest term on the right), it takes the dimensional form

∂ p 1
∂ t .

2

ρ 0 α ∇· v =− v ·∇ p 0 − (6.169)