Just above the core-mantle boundary

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Rays and travel times: from arrivals to a layered planet

Problem

Just above the core-mantle boundary, at radius

r = 3630\,\mathrm{km}

, a reference model of the lowermost mantle gives density

\rho = 5500\,\mathrm{kg\,m^{-3}}

, bulk modulus

K = 655\,\mathrm{GPa}

, shear modulus

\mu = 293\,\mathrm{GPa}

, and local gravity

g = 10.4\,\mathrm{m\,s^{-2}}

. ((a)) Using the relations between wave speed and elastic moduli, find

v_P

and

v_S

at this radius. ((b)) Hence find the seismic parameter

\Phi = v_P^2-\tfrac43v_S^2

, and confirm algebraically that it equals

K/\rho

for these two speeds. ((c)) Use the Adams-Williamson equation to estimate the increase in density, in

\mathrm{kg\,m^{-3}}

, over the lowermost

200\,\mathrm{km}

of the mantle just above the core. ((d)) The Earth's measured moment of inertia factor is

C/Ma^2 = 0.3307

, well below the value

0.4

for a uniform sphere, showing that mass is concentrated towards the centre. Is the density increase found in part (c), spread over the lowermost

200\,\mathrm{km}

of mantle, a large or a small contribution to that overall concentration, given that the mantle occupies less than half the Earth's radius above a much denser core?

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