GATE-2018 (24) :Wyllie – Time average equation

1) A 2.0km thick, and isostatically uncompensated, elevated land mass of density 2.7 gm/cc is associated with a 14km root at the crust-mantle boundary. where the mantle is denser by 0.3 gm/cc than the lower crust. what would be the thickness of the root, when the land mass is isostatically compensated.

A) 18.0 km

B) 16.2 km

C) 14.4 km

D) 13.6 km

Solution:

From Airy-Heiskanen model assumes local isostatic compensation, i.e., the root-zone of a mountain lies directly under it.

Airy-Heiskanen model hydrostatic equilibrium the  pressures are equal at A and B, 

Therefore, Pressure at A= pressure at B from above figure.

Uncompensated elevated land mass (h)= 2km 

Uncompensated root = 14 km 

crustal density =2.7 gm/cc

mantle density = 3.0 gm/cc (2.7+0.3) from above problem

compensated root=?

 $(h\times\rho_c )+(t\times\rho_c)+(uncompensated root\times\rho_c) = (t\times\rho_c) + (compensated root \times \rho_m)$

$(2\times2.7 )+(t\times\rho_c)+(14\times2.7) = (t\times\rho_c) + (r_0 \times 3.0)$

$(5.4 )+(t\times\rho_c)+(37.8) = (t\times\rho_c) + (r_0 \times 3.0)$

$5.4 +37.8 =(r_0 \times 3.0)$

$(r_0 \times 3.0)=5.4 +37.8 $

$r_0 =\frac{43.2}{3}$

$r_0 =14.4 km$

Reference: Fundamentals of Geophysics -Lowrie