Geology Reference
In-Depth Information
8
Variational methods and core modes
In the subseismic description of core dynamics, presented in the previous chapter,
the changes in density caused by adiabatic compression or expansion due to trans-
port through the hydrostatic pressure field are included, but those arising from flow
pressure fluctuations are ignored in comparison. While this leads to the subseismic
wave equation governing long-period core oscillations, the geophysicist's favourite
analytical tool, the representation of solutions by expansion in spherical harmonics,
is poorly convergent for such modes, due to tight Coriolis coupling between har-
monics of like azimuthal number but di
ering zonal number (Johnson and Smylie,
1977). Instead, we use local polynomial basis functions to represent the general-
ised displacement potential. We show that solutions of the governing subseismic
wave equation and boundary conditions have either purely even or purely odd
symmetry across the equatorial plane. The basis functions then take the forms
developed in Section 1.6.3. Solutions to the subseismic wave equation are found
through the development of a variational principle, which includes the continuity
of the normal component of displacement as a
natural boundary condition
of the
problem (Smylie
et al.
, 1992). The remaining elasto-gravitational boundary con-
ditions are incorporated through the load Love numbers described in the previous
chapter.
ff
8.1 A subseismic variational principle
A variational principle for the subseismic wave equation, (7.24), will involve sta-
tionarity of the functional with respect to variations in the generalised displace-
ment potential, χ, subject to boundary conditions. The vector displacement field
is given by expression (7.16), entirely in terms of the gradient of the generalised
displacement potential. As we have seen, substitution of this expression in the sub-
seismic form of the equation of continuity, (6.182), leads to the subseismic wave
equation.
Search WWH ::
Custom Search