Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 408

argument as expressed by (5.111). With substitution from (5.7) and (5.113) we see
that a near equilibrium is possible if the quantity

2

Ω /2π G ρ is small. In Earth's core

this dimensionless number is only about 10 − 3 .
The outer core is a fluid stratified in both density and temperature. Taking the
density stratification Δ ρ to be the result of a mean gravity ¯g 0 acting on a mean

density ¯ρ 0 , a flow characterised by velocity scale U would be expected to modify
near-equilibrium quantities by the ratio of the Coriolis acceleration to the relative
action of gravity on the density stratification, or
Ω

U
Δ ρ ¯g 0 /¯ρ 0 .

(6.137)

Ω − 1 as the time scale and L as the length scale, the velocity scale is U Using = L Ω ,

and the ratio (6.137) may be expressed as

2 L
Δ ρ ¯g 0 /¯ρ 0 ,

Ω

(6.138)

with = L denoting the Rossby number (the ratio of the momentum advection

to the Coriolis acceleration). If

U / Ω

T 0 is the mean temperature of the outer core and

Δ T characterises its stratification, the stratified density and temperature fields may

be written

Δ ρ ¯g 0 /¯ρ 0 ρ 1

2 L

+ Ω ρ = ¯ρ 0 +Δ ρ ρ e ( r ) , (6.139)

T 0 +Δ T T e ( r )

Δ ρ ¯g 0 /¯ρ 0 T 1

2 L

+ Ω

T = , (6.140)

where ρ e ( r )and T e ( r ) are dimensionless functions of radius giving the near-
equilibrium variation of density and temperature. ρ 1 and T 1 are the correspond-
ing flow-induced dimensionless quantities. The pressure and gravitational potential
fields may be similarly cast in dimensionless forms as

¯ρ 0 ¯g 0 L p e ( r ) p 1

2 L
¯g 0

Ω

p = + , (6.141)

¯g 0 L V e ( r ) V 1

2 L
¯g 0

Ω

V = + , (6.142)

where p e ( r )and V e ( r ) are dimensionless functions of radius giving their near-
equilibrium variations and p 1 and V 1 are their dimensionless flow-induced per-
turbations. For scaled values of velocity v , time t and gradient ∇ ,wehavethat

∂
∂ t =Ω

∂
∂ t , and

1

U v ,

L ∇ .

v = ∇= (6.143)