Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 462

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Radius

Figure 8.3 The generalised displacement potential for the axisymmetric mode
designated (4,1,0) in the notation of Greenspan (1969, p. 64). It is even in the
equatorial plane and has a period of 18.254 hours.

The programme EIGENS.FOR finds the eigenvectors and eigenfrequencies, with
input matres.dat containing the eleven matrices required for the computation and
produced by MAT.FOR. Also input is the file decomp.dat containing the decom-
pression factor. As well as searching for vanishing determinant of the coe cient

matrix as a function of frequency and iterating on eigenvectors as outlined,
EIGENS.FOR also produces the graphics files THREED and FLOWV as output.
THREED is a perspective plot of the generalised displacement potential χ, while
FLOWV is a plot of the flow vector field projected on the meridional plane and with
the azimuthal component shown in perspective orthogonal to the meridional plane.
As a first example of the computation, we show an axisymmetric solution of
the Poincare inertial wave equation known as the (4,1,0) mode in the notation
of Greenspan (1969, p. 64). Solutions of the Poincare inertial wave equation are
included in the code because it has analytical solutions as presented in Section 6.1,
and these can be used as a check on the numerical procedures. The programme
POINCARE.FOR calculates these analytical solutions and results for a variety of
modes are tabulated in Table 6.1. The generalised displacement potential for this
mode is shown as a perspective plot in Figure 8.3. The flow vector field for this
mode is shown projected on the meridional plane, and the azimuthal component of
the flow is shown in perspective orthogonal to the meridional plane in Figure 8.4.
Comparison with the analytical solution gives agreement with the eigenfunction
to three significant figures, while the eigenfrequency is reproduced to seven signi-
ficant figures, for the 3 3 finite element grid used in the calculation. Since the

eigenfrequency is at a stationary point of the functional, the error in the eigenfreq-
uency is proportional to the square of the error in the eigenfunction.

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