Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 470

7

5

3

0.2645 0.2650 0.2655 0.2660 0.2665

Frequency (cycles per hour)

Figure 8.15 Axial translational mode resonance. The fitted central period is
3.7656

±

0.0015 hours.

8

6

4

2

0.248 0.249 0.250 0.251

Frequency (cycles per hour)

Figure 8.16 Prograde equatorial translational mode resonance. It is just to the left
of the S 6 solar heating tide at exactly a 4-hour period. A double resonance is fitted
with central period for the prograde mode recovered as 4.0150 ± 0.0010 hours.

where s is the unit vector in the orthometric direction (opposite to gravity).
Taking the scalar product of the equation of motion with f u ∗ , where u ∗ is the

complex conjugate of the displacement vector u , and integrating throughout the
outer core, we find that

−

2 2 d

f u ∗ · ω f | u | V− 2 i ω ( Ω × u ) d V

V V

f u ∗ ·∇ χ d

2

v u ∗ ·

= V+ f ω s ( s · u ) d V . (8.91)

V V

Breaking the displacement vector into its real and imaginary parts,

u = u R + i u I , (8.92)