Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 399

Tabl e 6 . 1 Some eigenperiods of the Poincare inertial wave
equation found by the programme POINCARE.

Period in Period in

m n σ sidereal hours hours

1 2 0.5013 23.9392 23.874

0 4 0.6556 18.3037 18.254

1 4 0.8546 14.0411 14.003

1 4 0.3068 39.1118 39.005

2 4 0.6171 19.4461 19.393

3 4 0.2510 47.8175 47.687

1 6 0.9312 12.8871 12.852

2 6 0.8224 14.5908 14.551

1 8 0.7946 15.1023 15.061

2 8 0.8974 13.3727 13.336

3 8 0.8126 14.7682 14.728

While the outer core is a contained, rotating fluid, the container is deformable,
the fluid is compressible, stratified and self-gravitating, and there is an inner body.
To consider these realistic properties in its dynamics, we examine the equations for
conservation of mass, momentum and energy, in addition to those for gravitation.
The outer core is also the seat of the Earth's main magnetic field and electro-
magnetic e ects must be considered. Fortunately, for motions with wavelengths

large compared to the electromagnetic skin depth, core oscillations can be treated
as a purely mechanical phenomenon (Crossley and Smylie, 1975; Acheson, 1975).
In this approximation, the induced field is roughly uB / L , where u is the typical
particle displacement, B is the magnetic field strength and L is a common length
scale for both the oscillation and the field. The Lorentz force density is then approx-
imately uB 2

ff

/μ 0 L 2 , withμ 0 the permeability of free space. For motions with periods
of a fraction of a day and longer, the Coriolis acceleration is usually dominant and
the Lorentz force density measured against it is B 2 L 2 /μ 0 ρ Ω ω, where ρ is the mass

density, ω is the angular frequency of the oscillation and is Earth's angular velo-

city of rotation. For a field strength of 10 − 2 T, length scale 10 3 km, mass density
10 4 kgm − 3 and an oscillation period of one day, the ratio amounts to only about
1.5

Ω

10 − 6 .
The outer core is a compressible, stratified fluid, and its motions are subject to
the law of mass conservation. Suppose a surface

×

S

, moving with the fluid, encloses

V

a volume , containing the total mass

V ρ( x i , t ) d

V , (6.86)