Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 378

mantle & crust axis

inner core axis

r

q

q i

f i - f

q

O

Figure 5.7 Spherical triangle formed by the three axes; namely, the mantle and
crust axis, the inner core axis and the axis of the radius vector r in the inner
core reference frame. By the law of cosines for spherical triangles cosθ =
cosθ i cosθ + sinθ i sinθ cos(φ i − φ ).

Again, because γ is only a first-order quantity,

2 φ r d ρ 0
dr 0

∇ ×

( r ρ( r )) f cosθ sinθ .

=

(5.247)

Following the same arguments as before,

∇ × ( r ρ( r ))

f i 2 iP − 2 (cosθ ) e − i φ ı r d ρ 0

dr 0 3 P 2 (cosθ ) e i φ

=− +

f 2 P − 2 (cosθ ) e − i φ j r d ρ o

dr 0

1

3 P 2 (cosθ ) e i φ

+ − + . (5.248)

In evaluating the torque integrals (5.219) in the inner core frame of reference, we
require the representation of P 2 (cosθ) in that frame. As shown in Figure 5.7, the
axis of the mantle and crust is taken to pass through co-latitude θ i and longitude φ i
in that frame. The radius vector r passes through co-latitude θ in the mantle and
crust frame and through co-latitude θ and longitude φ in the inner core frame of
reference. As shown in Figure 5.7 a spherical triangle is then formed with sides θ i
and θ enclosing the angle φ i − φ , with θ forming the third side. By the addition
formula (B.9), the required expression for P 2 (cosθ) is given as

2

1) m P 2 (cosθ i ) P − 2 (cosθ ) e im (φ i − φ ) P 2 (cosθ) = ( − . (5.249)

m =− 2