Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 453

1.0
0.87

0

30

0.5 60

1 90

0 a / b

Radius r

Figure 8.1 The domain in the ( r , z ) plane broken into a 3 × 3 finite element grid,

=

equispaced in radius and co-latitude θ, with z cosθ.

yielding the expansion

1

σ

z 2

N 2 N 2

1

2

z 2 =−

2 σ

1

+

2 σ

1

− +··· . (8.36)

2 2 2

ζ

1 −

σ − σ −

2

For example, a period of 4 sidereal hours gives σ = 9, and for a weak departure

/ σ 1 = from neutral stratification we might have N 2 0.01, giving N 2

2

0.00125.
Thus, neglect of the second term in the expansion would lead to an error of slightly
over one part in a thousand. To this level of approximation, we can multiply the
functional by σ

= −

1 to make it an eighth-degree polynomial in σ. Similarly,
continuity of normal displacement becomes a quadratic in σ.
We can then define a finite element grid for support functions in the ( r , z ) plane.
The domain of the problem is shown in Figure 8.1 as a 3

σ

2 2

−

× 3 finite element grid,

equispaced in radius and co-latitude.

8.3 Finite element support functions
A particular finite element R ij occupies the interval ( i , i + 1) in radius r and the

interval ( j , j + 1) in the cosine of co-latitude z , as illustrated in Figure 8.2.

( i , j + 1) ( i + 1, j + 1)

R ij

Z

( i , j )

( i + 1, j )

r

Figure 8.2 The rectangular finite element R ij and its four nodes.