Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 386

6

Rotating fluids and the outer core

The fluid outer core is bounded by the shell (mantle and crust) on the outside, and
by the solid inner core on the inside. It is thus a contained, rotating fluid.
One of the earliest studies of the dynamics of contained, rotating fluids was that
of Poincare (1885). In these early models the container was assumed rigid, the fluid
incompressible, uniform and not self-gravitating, with no inner body. This subject
developed rapidly, both theoretically (Greenspan, 1969) and through laboratory
experiments (Aldridge and Toomre, 1969). The governing equation is the inertial
wave equation. The first suggestion that it might have an analogue in the dynamics
of the Earth's liquid core appears to be due to Pekeris and Accad (1972). Of course,
in the real Earth the container is deformable, the fluid is compressible, stratified and
self-gravitating, and there is an inner body.
We begin with the derivation and study of the inertial wave equation, and then
proceed to a scale analysis of long-period motions in the real Earth, finishing with
the subseismic condition and the decompression factor.

6.1 The inertial wave equation
For a mode with time dependence exp i ω t , for small oscillations, the vector dis-
placement field u obeys the equations

2 u

− ω + 2 i ω Ω × u =−∇ χ, (6.1)

∇· u = 0, (6.2)

where ω is the angular frequency of oscillation, and Ω is the vector rotation rate of
the reference frame, taken to be Earth's mean rotation rate around a fixed spatial
direction. Here, χ is a reduced pressure potential given by

p 1
ρ 0 +

χ = W . (6.3)