In Lithosphere and Asthenosphere you learned that the rigid lithosphere rests on hot, slowly flowing asthenosphere. Over thousands of years that rock behaves like a very thick fluid, so the crust floats on it. Floating explains a surprising fact: the highest mountains have the deepest roots.
Why it works
An object floats when it pushes aside its own weight of fluid (a formula we can solve for any letter): $\rho_{\text{object}} V_{\text{total}} = \rho_{\text{fluid}} V_{\text{under}}$. Ice ($0.917\,\text{g/cm}^3$) in seawater ($1.025$) floats about $89\%$ under. Crust ($2.7$) floats on mantle ($3.3$) the same way. Now compare two columns of the same width. Deep below both, at the bottom of the deepest root, the weight above each square meter must be the same; if it weren't, the soft mantle would flow until it was. A column with a mountain of height $h$ on top carries extra crust, so it must swap dense mantle for light crust below: a root $r$. Balancing the extra weight on top against the buoyancy gained below, $\rho_c h = (\rho_m - \rho_c)\,r$.
Key idea
Crust floats on the mantle in isostatic balance. A range of height $h$ sits on a root $r = h\dfrac{\rho_c}{\rho_m - \rho_c}$. With crust $2.7$ and mantle $3.3\,\text{g/cm}^3$, $r = 4.5h$: most of a mountain range is underground.
Worked examples
Worked example: The root of a range
A range stands 4 km above the plains. Crust $2.7$, mantle $3.3\,\text{g/cm}^3$. How deep is its root?
- $r = 4 \times \dfrac{2.7}{3.3 - 2.7} = 4 \times \dfrac{2.7}{0.6} = 4 \times 4.5$.
- $r = 18\,\text{km}$. If normal crust there is 35 km thick, the crust under the peaks is $4 + 35 + 18 = 57\,\text{km}$ thick, close to what seismic waves find under the Andes.
Answer: $18\,\text{km}$.
Worked example: Land pushed down by ice
An ice sheet 3 km thick ($\rho_{\text{ice}} = 0.917$) rests on the land for thousands of years. How far does the land sink into the mantle ($3.3$)?
- At balance, the ice's weight equals the weight of mantle pushed aside: $0.917 \times 3 = 3.3\,d$.
- $d = \dfrac{2.751}{3.3} \approx 0.83\,\text{km}$.
- When the ice melted at the end of the last ice age, the land began to rise back. Around Hudson Bay and the Baltic Sea it is still rising, up to about a centimeter a year.
Answer: About $0.83\,\text{km}$, roughly 830 m.
Common mistake
It is tempting to picture mountains sitting on a flat base of solid rock that holds them up. If that were true, all that extra rock would tug hard on things nearby. In the 1850s, surveyors in India found that the Himalayas pull plumb lines sideways far less than their mass should: the extra rock above is balanced by light crust replacing dense mantle below. Check with the iceberg: you never see an iceberg standing tall on a flat bottom; the higher it rises, the deeper it reaches.