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

 1. A large area of continent consisting 30 km of crust with an average density of 2.8 gm/cc is over 90 km thick material with density 3.1 gm/cc . It is covered with a 1.6 km thick layer of ice (density 0.9 gm/cc), and is in isostatic equilibrium. Then the ice melts. After equilibrium has been regained, by how much has the rock surface of the continent changed? (density of the asthenosphere is 3.2 gm/cc).

Solution:

  Block A: Thickness of the crust =30 km
               Density of the crust =2.8 gm/cc
               Thickness of the mantle = 90 km
               Density of the mantle =3.1 gm/cc 

Block B: Thickness of the ice =1.6 km
              Density of the ice =0.9 gm/cc
              Thickness of the crust =30 km
               Density of the crust =2.8 gm/cc
               Thickness of the mantle = 90 km
               Density of the mantle =3.1 gm/cc


$\bf \color{black} {g ( h_c\times\rho_c + h_m\times\rho_m +h_{asth}\times\rho_{asth})_{block A} =g (h_{ice}\times\rho_{ice}+ h_c\times\rho_c + h_m\times\rho_m)_{block B} }$

$\bf \color{black} { ( h_c\times\rho_c + h_m\times\rho_m +h_{asth}\times\rho_{asth})_{block A} = (h_{ice}\times\rho_{ice}+ h_c\times\rho_c + h_m\times\rho_m)_{block B} }$

$\bf \color{black} { (30\times2.8 + 90\times3.1 +h_{asth}\times3.2)_{block A} = (1.6\times0.9+ 30\times2.8 + 90\times3.1)_{block B} }$

$\bf \color{black} { (h_{asth}\times3.2)_{block A} = (1.6\times0.9)_{block B} }$

$\bf \color{black} { h_{asth} = \frac{1.6\times0.9}{3.2} }$

$\bf \color{black} { h_{asth} = 0.45 km}$


Therefore, $h_a$ =0.45 km ; that is, the weight of the ice caused the lithosphere to sink 0.45 km.

As 1.6 km of ice was added but the lithosphre sank 0.45 km , the top of the ice is only 1.15 km above the original continental surface.


Reference : Looking into the Earth by Alan E.Mussett