Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 342

Ignoring radiative transfer, the energy equation for static equilibrium is

∇· ( k ∇ T ) =− H , (5.106)

where k is the thermal conductivity and H is the rate of heat production per unit
volume. Taking the curl of (5.104) yields

∇ ρ ×∇ U = 0, (5.107)

showing that, in equilibrium, the surfaces of equal density coincide with equipo-
tentials, which by (5.104) are also isobaric surfaces. Equation (5.105) then implies
that the material on such surfaces has the same temperature, and is in the same ther-
modynamic state, provided that the composition and phase are assumed uniform,
and implies that we may take ρ, p , T , k and H all to be functions of U alone. Thus,
k ∇ T = k dT

dU ∇ U

= f ( U ) ∇ U , (5.108)

where f ( U ) is a function of U . Substitution into (5.106) yields

2 U + f ( U )

∇· ( f ( U ) ∇ U ) = f ( U ) ∇ ∇ U ·∇ U

=− H , (5.109)

2 U from the governing

the prime indicating di ff erentiation. With replacement of ∇

equation (5.5) for U , and with recognition of ∇ U as the negative of the gravity

vector, (5.109) becomes
f ( U ) 4π G ρ −

2 2 f ( U )

2 Ω + g =− H . (5.110)

In equilibrium, H depends on U alone, and therefore the left side of (5.110) must
depend only on U . From the analysis in Section 5.2, it is clear that, in general, g

2

cient f ( U ) must therefore

varies over any equipotential in a rotating Earth. Its coe

vanish, so f ( U ) is constant, giving

1

2
3 m ¯ρ/ρ

H = C ρ − , (5.111)

the constant C being

3 f ( U )
¯ρ

GM
d 3 ,

C =− (5.112)

and ¯ρ being the mean density of the Earth,

3 M
4π d 3 .

¯ρ = (5.113)

It is reasonable to expect the volume rate of heat generation to depend on the mass
density. Since m is only about 1 / 290 for the Earth, we can expect (5.111) to be