---
title: "Lithosphere and Asthenosphere"
book: "High School Earth and Space Sciences"
subject: earth
language: en
chapter: 12
exercises: 15
source: https://one-course.com/books/earth/2/en/chapter/12-lithosphere-and-asthenosphere
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 12 — Lithosphere and Asthenosphere

Nine tenths of an iceberg lie under water: the ice floats because it is a little less dense than the sea, and it sinks into the water until the water it pushes aside weighs as much as the whole iceberg. Mountain ranges behave in much the same way. Under the highest of them the crust is nearly twice as thick as usual, and most of that extra crust lies hidden underneath, like the ice under the waterline. But an iceberg floats in a liquid, and the mantle beneath a mountain range is solid rock. This chapter shows how a solid can behave like a fluid on which the outer shell of the Earth floats, and what that shell is.

**You already know.**

The Earth has a crust, a mantle and a core (Book 1, grade 6). The [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho), at the base of the crust, lies about $41\,\mathrm{km}$ down under the continents on average and a few kilometres below the ocean floor; seismic speeds and densities are known at every depth from a [reference Earth model](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-reference-model) ([Chapter 11](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#ch-g11-seismic-structure)). The rock gets hotter with depth, by about 25 to $30\,{}^{\circ}\mathrm{C}$ per kilometre near the surface (Book 1, grade 6). A body floating in a liquid pushes aside a weight of liquid equal to its own weight, and the pressure at the bottom of a column of material is the weight of the column divided by its area (the physics books).

![An iceberg seen from just above the water: the ice under the surface, which shows through the clear sea, is far bigger than the part in the air. Photograph: NOAA (public domain).](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/img-00ef2ceb2fff.jpg)

*An iceberg seen from just above the water: the ice under the surface, which shows through the clear sea, is far bigger than the part in the air. Photograph: NOAA (public domain).*

## 12.1 Two ways to layer the Earth

The crust, the mantle and the core are layers of different *materials*: the crust is made of rocks rich in silicon, aluminium and oxygen, the mantle of a denser rock, the core of metal. But the outer part of the Earth can also be divided by how its rock *behaves* when forces act on it for a very long time.

**Definition 12.1 (Lithosphere; asthenosphere).**

The *lithosphere* is the rigid outer shell of the Earth: the crust and the uppermost, coldest part of the mantle, which together break into the tectonic plates and do not flow. The *asthenosphere* is the layer of the mantle beneath the lithosphere, hotter and weaker, which flows slowly over long spans of time, like a very stiff paste.

The boundary between the two lies about 70 to $100\,\mathrm{km}$ down under the oceans and most continents, and deeper under the oldest parts of the continents; across the United States, for instance, the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is from about $40\,\mathrm{km}$ thick in the west to more than $200\,\mathrm{km}$ thick in the middle of the continent. The base of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is not the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho): most of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is mantle rock.

![The upper 400\, km of the Earth under a continent, to scale. Left: the layers by material. Right: the layers by behaviour. The rigid lithosphere contains the whole crust and the top of the mantle; the asthenosphere has no sharp lower limit.](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/fig-a5537349cc18.svg)

*The upper $400\,\mathrm{km}$ of the Earth under a continent, to scale. Left: the layers by material. Right: the layers by behaviour. The rigid [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) contains the whole crust and the top of the mantle; the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) has no sharp lower limit.*

## 12.2 The low-velocity zone

Seismic waves show the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere). Almost everywhere in the Earth, seismic speeds increase with depth, because the rock is more and more squeezed. Under the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) the opposite happens.

**Definition 12.2 (Low-velocity zone).**

The *low-velocity zone* is the layer of the upper mantle, below the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere), in which seismic waves travel more slowly than in the rock above and below it.

In the [reference Earth model](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-reference-model), the S speed falls slowly from the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) down to $220\,\mathrm{km}$, then jumps up. Seismic studies all over the world find a sharp drop of speed somewhere between 70 and $100\,\mathrm{km}$: the top of the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere). The rock there is hot enough to be close to melting, and may contain a very little melt; it may also hold more water than the rock above. Rock close to melting is weak: it slows seismic waves, and it can flow.

![The S speed in the upper 400\, km of the reference Earth model. Below the crust, the speed decreases slowly with depth down to 220\, km (shaded from 80\, km), then increases again with sudden steps.](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/fig-566402c95e96.svg)

*The S speed in the upper $400\,\mathrm{km}$ of the [reference Earth model](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-reference-model). Below the crust, the speed decreases slowly with depth down to $220\,\mathrm{km}$ (shaded from $80\,\mathrm{km}$), then increases again with sudden steps.*

## 12.3 The geotherm

**Definition 12.3 (Geotherm; geothermal gradient).**

A *geotherm* is the curve of temperature against depth beneath a place. The *geothermal gradient* is the rise of temperature per unit of depth, in degrees per kilometre.

Boreholes measure the [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) only in the first few kilometres, where the temperature rises by 25 to $30\,{}^{\circ}\mathrm{C}$ per kilometre (Book 1, grade 6). Deeper, it is computed from the heat that flows out through the surface and from the properties of the rocks; [Chapter 24](https://one-course.com/books/earth/2/en/chapter/24-the-earths-internal-heat#ch-g12-internal-heat) shows where that heat comes from. Two facts shape the result. Part of the heat is made inside the upper crust itself, so the gradient is steepest near the surface and smaller below the crust. And in the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere), which flows, the slow motion of the rock carries the heat and evens out the temperature: it rises only about $0.45\,{}^{\circ}\mathrm{C}$ per kilometre. The base of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is placed where the steep [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) meets the gentle one of the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere), at about $1300\,{}^{\circ}\mathrm{C}$.

![A model geotherm under a continent (blue) and the temperature at which mantle rock (peridotite, without water) begins to melt (orange). The geotherm is computed from the mean heat flow of the continents and measured rock properties; in the asthenosphere (grey) it rises slowly. The dashed line shows how the lithosphere’s steeper rise would continue. The two curves come closest, about 140\, C apart, near the base of the lithosphere.](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/fig-c4686ba3d7b8.svg)

*A model [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) under a continent (blue) and the temperature at which mantle rock (peridotite, without water) begins to melt (orange). The [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) is computed from the mean heat flow of the continents and measured rock properties; in the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) (grey) it rises slowly. The dashed line shows how the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere)’s steeper rise would continue. The two curves come closest, about $140\,{}^{\circ}\mathrm{C}$ apart, near the base of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere).*

The melting temperature of the mantle rock rises with depth, because the rock is more and more squeezed (Book 1, grade 6): about $1120\,{}^{\circ}\mathrm{C}$ at the surface, nearly $1500\,{}^{\circ}\mathrm{C}$ at $100\,\mathrm{km}$. In the model, the [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) comes closest to it just below the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere), about $140\,{}^{\circ}\mathrm{C}$ short of it. A little water, or a slightly hotter mantle, closes the gap: in the west of the United States, where the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is thin, a little of the mantle beneath it is partly molten.

## 12.4 Isostasy

### 12.4.1 Floating on a solid

Over the few seconds of an earthquake wave, the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is a solid. Over long spans of time, it flows. A load placed on the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) — a mountain range, an ice sheet — slowly pushes the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) aside until the load floats, like a block of wood on water.

**Definition 12.4 (Isostasy; compensation depth).**

*Isostasy* is the balance in which the outer layers of the Earth float on the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere): at a depth in the mantle called the *compensation depth*, every column of rock of the same cross-section, from the surface down to that depth, has the same weight, whatever the relief above it.

![Airy’s model of isostasy (not to scale). The crust is everywhere of the same density; where it stands high, it is thick and has a root that reaches deep into the denser mantle. Under the ocean, the crust is thin. All three columns weigh the same down to the compensation depth.](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/fig-243c0478b315.svg)

*Airy’s model of [isostasy](#def-g11-lithosphere-asthenosphere-isostasy) (not to scale). The crust is everywhere of the same density; where it stands high, it is thick and has a root that reaches deep into the denser mantle. Under the ocean, the crust is thin. All three columns weigh the same down to the [compensation depth](#def-g11-lithosphere-asthenosphere-isostasy).*

### 12.4.2 The root of a mountain range

**Proposition 12.5 (Airy isostasy).**

Let the crust have a uniform density $\rho_c$, and the mantle beneath it a density $\rho_m > \rho_c$. If a region stands a height $h$ above another one and both are in isostatic balance, its crust is thicker by

$$
\Delta H = h\,\frac{\rho_m}{\rho_m - \rho_c},
$$

of which $h$ stands above the other region and a root $r = h\,\rho_c/(\rho_m - \rho_c)$ reaches below its crust.

**Proof.** Take two columns of the same area $S$, the lowland with a crust of thickness $T$ and the range, whose crust stands $h$ higher and reaches $r$ deeper, so that its thickness is $T + h + r$. Down to a [compensation depth](#def-g11-lithosphere-asthenosphere-isostasy) below both [Mohos](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho), the range column has $h + r$ more crust; the lowland column has, instead, mantle over the thickness $r$. With $g$ the acceleration of gravity, the two weights are equal when

$$
\rho_c\,(h + r)\,S\,g = \rho_m\,r\,S\,g,
  \qquad\text{that is}\qquad
  r = h\,\frac{\rho_c}{\rho_m - \rho_c}.
$$

The crust is thicker by $h + r = h\,\bigl(1 + \rho_c/(\rho_m - \rho_c)\bigr)
= h\,\rho_m/(\rho_m - \rho_c)$. ∎

With the mean density of the continental crust, $\rho_c =
2830\,\mathrm{kg}/\mathrm{m}^{3}$, and that of the mantle just below the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho), $\rho_m =
3380\,\mathrm{kg}/\mathrm{m}^{3}$, the root is $2830/550 \approx 5.1$ times the height, and the crust is thicker by about $6.1$ times the height: a range $3\,\mathrm{km}$ above its surroundings has a root of about $15\,\mathrm{km}$.

**Method 12.6 (Computing a root).**

1. Find the height $h$ of the region above a reference region whose crust thickness is known.
2. Compute $\rho_m/(\rho_m - \rho_c)$ with the densities of crust and mantle.
3. The crust is thicker than the reference crust by $h\,\rho_m/(\rho_m -  \rho_c)$ ; the root alone is that minus $h$ .

**Example 12.7 (A plateau 2 km2\,\mathrm{km}2km high).**

A plateau stands $2\,\mathrm{km}$ above a lowland whose crust is $37\,\mathrm{km}$ thick. Its crust is thicker by $2 \times 3380/550 \approx 12.3\,\mathrm{km}$: about $49\,\mathrm{km}$ in all, with a root of about $10\,\mathrm{km}$.

### 12.4.3 Why the continents stand high

**Definition 12.8 (Hypsometric curve).**

The *hypsometric curve* of the Earth gives, for each elevation, the fraction of the Earth’s surface that lies higher than it. Its slope shows how much of the surface lies at each elevation.

If the surface had been shaped by chance, most of it would lie near one average level. It does not.

**Proposition 12.9 (Two levels).**

The elevations of the Earth’s surface cluster around two levels: one within a few hundred metres of sea level, on the continents, and one about 4 to $5\,\mathrm{km}$ below sea level, on the deep ocean floor. Land covers $29\,\%$ of the surface; more than half of the surface lies deeper than $3\,\mathrm{km}$ below sea level, and only $3\,\%$ rises higher than $2\,\mathrm{km}$.

![Share of the Earth’s surface in each band of 500\, m of elevation (from ETOPO1, at one degree; over Antarctica and Greenland, the top of the ice). The two peaks are the continents and the deep ocean floor.](https://one-course.com/images/onecourse/chapters/earth-2/g11-lithosphere-asthenosphere/fig-c325bb522e54.svg)

*Share of the Earth’s surface in each band of $500\,\mathrm{m}$ of elevation (from ETOPO1, at one degree; over Antarctica and Greenland, the top of the ice). The two peaks are the continents and the deep ocean floor.*

[Isostasy](#def-g11-lithosphere-asthenosphere-isostasy) explains the two levels. The continents are made of a thick crust, about $41\,\mathrm{km}$ on average, of light rock; the ocean floor of a thin crust of denser rock, only about $5\,\mathrm{km}$ thick. Both float on the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere): the thick, light continental crust floats high, the thin, dense oceanic crust low, and the oceans fill the low ground.

## 12.5 A flexible lid

[Isostasy](#def-g11-lithosphere-asthenosphere-isostasy) needs the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) to flow. The [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) above it does not flow; it bends. A load narrower than the thickness of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere), such as a single volcanic island, is carried by the stiffness of the plate, like a weight on a plank, and the plate sags around it. A load much wider than that thickness, such as a whole mountain range or an ice sheet, sinks until it floats. When the load is removed, the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) slowly rises again ([Chapter 23](https://one-course.com/books/earth/2/en/chapter/23-the-destruction-of-mountains#ch-g12-destruction-of-mountains)).

## 12.6 Exercises

**Exercise 12.1 ★.**

Define the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) and the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere). Which of the following belong to the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere): the oceanic crust, the continental crust, the uppermost mantle, the outer core?

**Solution of Exercise 12.1.**

The [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is the rigid outer shell of the Earth, made of the crust and the uppermost mantle; the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) is the hotter, weaker mantle beneath it, which flows slowly over long times. The oceanic crust, the continental crust and the uppermost mantle belong to the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere); the outer core does not.

**Exercise 12.2 ★.**

An iceberg of density $910\,\mathrm{kg}/\mathrm{m}^{3}$ floats in sea water of density $1025\,\mathrm{kg}/\mathrm{m}^{3}$. What fraction of its volume lies under water? If an iceberg with vertical sides stands $20\,\mathrm{m}$ above the water, what is its total height?

**Solution of Exercise 12.2.**

The iceberg sinks until it pushes aside its own weight of water: the fraction under water is $910/1025 \approx 0.888$, about $89\,\%$. The fraction above is about $11.2\,\%$, so the total height is $20/0.112 \approx 180\,\mathrm{m}$, of which about $160\,\mathrm{m}$ lie under water.

**Exercise 12.3 ★.**

The ground is at $15\,{}^{\circ}\mathrm{C}$ and the [geothermal gradient](#def-g11-lithosphere-asthenosphere-geotherm) is $26\,{}^{\circ}\mathrm{C}$ per kilometre. What is the temperature at the bottom of a borehole $3\,\mathrm{km}$ deep? Why can the same gradient not be used down to $100\,\mathrm{km}$?

**Solution of Exercise 12.3.**

$15 + 26 \times 3 = 93\,{}^{\circ}\mathrm{C}$. At $100\,\mathrm{km}$ the same gradient would give more than $2600\,{}^{\circ}\mathrm{C}$, far above the melting temperature of mantle rock; but [S waves](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-body-waves) cross the upper mantle, which is therefore solid. The gradient must become much smaller with depth.

**Exercise 12.4 ★.**

From the figure of the elevations of the Earth’s surface, give the two most common elevation bands and the share of the surface in each.

**Solution of Exercise 12.4.**

The band from 0 to $500\,\mathrm{m}$ above sea level, with about $16\,\%$ of the surface, and the band from 4000 to $4500\,\mathrm{m}$ below sea level, with about $11\,\%$ (the band from 4500 to $5000\,\mathrm{m}$ below is almost as large).

**Exercise 12.5 ★.**

With the formula $T = -5.104\,P^2 + 132.899\,P + 1120.661$ (temperature in ${}^{\circ}\mathrm{C}$, pressure $P$ in gigapascals), compute the temperature at which dry mantle rock begins to melt at the surface ($P = 0$) and at $3\,\mathrm{GPa}$, about $95\,\mathrm{km}$ down.

**Solution of Exercise 12.5.**

At $P = 0$: $1121\,{}^{\circ}\mathrm{C}$. At $3\,\mathrm{GPa}$: $-5.104 \times 9 +
132.899 \times 3 + 1120.661 \approx -45.9 + 398.7 + 1120.7 \approx
1473\,{}^{\circ}\mathrm{C}$.

**Exercise 12.6 ★★.**

On the figure of the model [geotherm](#def-g11-lithosphere-asthenosphere-geotherm), read the temperature at 50, 150 and $250\,\mathrm{km}$. In which layer, [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) or [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere), is each depth? Between which two of these depths does the temperature rise most?

**Solution of Exercise 12.6.**

About $730\,{}^{\circ}\mathrm{C}$ at $50\,\mathrm{km}$ ([lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere)), about $1370\,{}^{\circ}\mathrm{C}$ at $150\,\mathrm{km}$ and $1410\,{}^{\circ}\mathrm{C}$ at $250\,\mathrm{km}$ ([asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere)). The temperature rises most between 50 and $150\,\mathrm{km}$, by more than $600\,{}^{\circ}\mathrm{C}$, because most of that interval lies in the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere); between 150 and $250\,\mathrm{km}$ it rises by only about $45\,{}^{\circ}\mathrm{C}$.

**Exercise 12.7 ★★.**

A mountain range stands $3\,\mathrm{km}$ above a lowland; take $\rho_c =
2830\,\mathrm{kg}/\mathrm{m}^{3}$ and $\rho_m = 3380\,\mathrm{kg}/\mathrm{m}^{3}$. Compute its root and the extra thickness of its crust.

**Solution of Exercise 12.7.**

Root: $3 \times 2830/550 \approx 15.4\,\mathrm{km}$. Extra thickness: $3 \times 3380/550 \approx 18.4\,\mathrm{km}$, that is the $3\,\mathrm{km}$ of the range plus its root.

**Exercise 12.8 ★★.**

A region has a crust $55\,\mathrm{km}$ thick. With the densities of [Exercise 12.7](#exo-g11-lithosphere-asthenosphere-7), how high does it stand above a region whose crust is $41.1\,\mathrm{km}$ thick?

**Solution of Exercise 12.8.**

The crust is thicker by $55 - 41.1 = 13.9\,\mathrm{km}$, and $\Delta H = h
\times 3380/550$, so $h = 13.9 \times 550/3380 \approx 2.3\,\mathrm{km}$.

**Exercise 12.9 ★★.**

In the reference model, the S speed is $4.470\,\mathrm{km}/\mathrm{s}$ at $80\,\mathrm{km}$, $4.419\,\mathrm{km}/\mathrm{s}$ just above $220\,\mathrm{km}$ and $4.644\,\mathrm{km}/\mathrm{s}$ just below. Compute, in per cent, the decrease between 80 and $220\,\mathrm{km}$ and the jump at $220\,\mathrm{km}$. Why is a decrease of speed with depth surprising?

**Solution of Exercise 12.9.**

Decrease: $(4.470 - 4.419)/4.470 \approx 1.1\,\%$. Jump: $(4.644 - 4.419)/4.419 \approx 5.1\,\%$. With depth the rock is more and more squeezed, which makes seismic waves faster; a decrease means that something else wins: the rock is hot and close to melting, and weaker.

**Exercise 12.10 ★★.**

In the model [geotherm](#def-g11-lithosphere-asthenosphere-geotherm), the temperature is $610\,{}^{\circ}\mathrm{C}$ at the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho), $40\,\mathrm{km}$ down, and $1345\,{}^{\circ}\mathrm{C}$ at the base of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere), $100\,\mathrm{km}$ down. (1) Compute the mean gradient between the two. (2) Compare with the gradient at the surface, $26\,{}^{\circ}\mathrm{C}$ per kilometre, and explain the difference.

**Solution of Exercise 12.10.**

(1) $(1345 - 610)/60 \approx 12\,{}^{\circ}\mathrm{C}$ per kilometre. (2) That is about half the gradient at the surface. Part of the heat is made inside the upper crust; that heat must also cross the top layers to escape, which makes the gradient steeper near the surface. Below the crust only the heat coming from deeper down crosses the rock, and the gradient is smaller.

**Exercise 12.11 ★★.**

In the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) the temperature is taken as $1300 + 0.45\,z$ (in ${}^{\circ}\mathrm{C}$, $z$ in kilometres). Compute it at 200 and $400\,\mathrm{km}$. By how much does it rise between the two? Compare with the rise through the first $100\,\mathrm{km}$, from $15\,{}^{\circ}\mathrm{C}$ at the surface to $1345\,{}^{\circ}\mathrm{C}$ at the base of the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere).

**Solution of Exercise 12.11.**

At $200\,\mathrm{km}$: $1300 + 90 = 1390\,{}^{\circ}\mathrm{C}$; at $400\,\mathrm{km}$: $1300 + 180 = 1480\,{}^{\circ}\mathrm{C}$. The rise is $90\,{}^{\circ}\mathrm{C}$ over $200\,\mathrm{km}$. Through the [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) the temperature rises by $1345 - 15 = 1330\,{}^{\circ}\mathrm{C}$ over only $100\,\mathrm{km}$: about thirty times more per kilometre.

**Exercise 12.12 ★★★.**

Below the upper crust, the model [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) is close to $T \approx 118 +
G\,z$ (in ${}^{\circ}\mathrm{C}$, $z$ in kilometres), where the gradient $G$ depends on the heat coming from deeper down. (1) For $G =
12.3\,{}^{\circ}\mathrm{C}$ per kilometre, as computed for the east of the United States, find the depth at which it meets $1300 + 0.45\,z$. (2) Do the same for $G = 21.5\,{}^{\circ}\mathrm{C}$ per kilometre, as computed for the Basin and Range province of the west. (3) Compare with the measured thicknesses, from about $40\,\mathrm{km}$ in the west to more than $200\,\mathrm{km}$ in the middle of the continent.

**Solution of Exercise 12.12.**

(1) $118 + 12.3\,z = 1300 + 0.45\,z$ gives $z = 1182/11.85 \approx
100\,\mathrm{km}$. (2) $118 + 21.5\,z = 1300 + 0.45\,z$ gives $z = 1182/21.05
\approx 56\,\mathrm{km}$. (3) The hotter west has a much thinner [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere), as measured (about $40\,\mathrm{km}$ at the thinnest); the very thick [lithosphere](#def-g11-lithosphere-asthenosphere-lithosphere) of the middle of the continent needs an even smaller gradient.

**Exercise 12.13 ★★★.**

(1) Show that, in Airy’s model, a region whose crust is $\Delta H$ thicker than a reference region stands higher by $\Delta H\,(\rho_m - \rho_c)/\rho_m$. (2) What fraction of any extra thickness of crust appears as height? (3) Compare with the fraction of an iceberg that stands above the water.

**Solution of Exercise 12.13.**

(1) By the proposition, $\Delta H = h\,\rho_m/(\rho_m - \rho_c)$, so $h =
\Delta H\,(\rho_m - \rho_c)/\rho_m$. (2) $550/3380 \approx 0.16$: about one sixth of any extra crust shows as height, five sixths as root. (3) For the iceberg, the fraction above the water is $(1025 - 910)/1025 \approx 0.11$: the same rule, (density of the fluid $-$ density of the floating body)/density of the fluid.

**Exercise 12.14 ★★★.**

At $100\,\mathrm{km}$ the dry mantle rock begins to melt at $1489\,{}^{\circ}\mathrm{C}$, and the model [geotherm](#def-g11-lithosphere-asthenosphere-geotherm) is at $1345\,{}^{\circ}\mathrm{C}$. (1) By how many degrees would water have to lower the melting temperature for the rock to start melting there? (2) Why would even a very small fraction of melt make the rock weak and slow down seismic waves?

**Solution of Exercise 12.14.**

(1) $1489 - 1345 = 144\,{}^{\circ}\mathrm{C}$. (2) A little melt gathers in thin films between the grains of the rock. The grains are then no longer firmly locked together: the rock resists shear less, so [S waves](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-body-waves) slow down, and it can deform and flow far more easily.

**Exercise 12.15 ★★★.**

Only $3\,\%$ of the Earth’s surface rises above $2\,\mathrm{km}$, while $53\,\%$ lies more than $3\,\mathrm{km}$ below sea level. (1) If all the land were planed flat and its rock dumped into the oceans, would the surface end up near sea level? Use [isostasy](#def-g11-lithosphere-asthenosphere-isostasy) to explain your answer. (2) What does the existence of two levels tell us about the crust?

**Solution of Exercise 12.15.**

(1) No. Planing the land would make the continental crust a little thinner, but it would still be thick and light, and would still float much higher than the thin, dense crust of the ocean floor; the oceans would still fill the basins between the continents. (2) There are two kinds of crust, of different thickness and density: a thick, light continental crust and a thin, dense oceanic crust.

## 12.7 Problem: Why Tibet Stands Five Kilometres High

**Problem 12.1.**

Weekend problem — weighing a plateau

The Tibetan plateau is the largest high region on Earth. Between latitudes $31{}^{\circ}$ and $35{}^{\circ}$ North and longitudes $80{}^{\circ}$ and $95{}^{\circ}$ East, its mean elevation is $5010\,\mathrm{m}$; the mean elevation of all the land outside Antarctica is $670\,\mathrm{m}$. The continental crust is $41.1\,\mathrm{km}$ thick on average and has a mean density $\rho_c = 2830\,\mathrm{kg}/\mathrm{m}^{3}$; the mantle just below the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) has the density $\rho_m = 3380\,\mathrm{kg}/\mathrm{m}^{3}$.

**Part I — The plateau and an average continent.**

1. How much higher than the average continent does the plateau stand?
2. By what percentage is the crust less dense than the mantle beneath it?
3. A block of crust floating freely on mantle rock would sink until what fraction of its thickness lay below the mantle’s surface? Compare with the iceberg ( $910$ in $1025$ ).
4. The mantle is solid. Over which spans of time does it behave as a solid, and over which can it behave like the water around an iceberg?
5. Why must the [compensation depth](#def-g11-lithosphere-asthenosphere-isostasy) be chosen below the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) of both columns?

**Part II — Airy’s balance for two columns.**

6. Draw the two columns, the average continent and the plateau, from their surfaces down to a depth of $70\,\mathrm{km}$ below sea level, with the plateau’s crust thicker by $\Delta H$ .
7. Write that the two columns weigh the same, and deduce $\Delta H$ in terms of the height difference $h$ and of the densities.
8. Compute $\rho_m/(\rho_m - \rho_c)$ .
9. Compute the root of the plateau below the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) of the average continent.
10. How deep below sea level is the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) of the average continent, and how deep is that of the plateau?

**Part III — The thickness of the crust.**

11. Compute the thickness of the crust under the plateau.
12. Check the balance: compute, for each column, the mass of rock above a square metre down to $70\,\mathrm{km}$ below sea level, in tonnes.
13. How thick would the crust be under a lowland at sea level?
14. How many times thicker than the average continental crust is the crust under Tibet?
15. If the mantle were a little lighter, $\rho_m = 3300\,\mathrm{kg}/\mathrm{m}^{3}$ , what thickness would the model give? Is the result very sensitive to the densities?

**Part IV — Comparison with the seismic measurements.**

16. Seismic waves converted at the [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) place it about $70\,\mathrm{km}$ down beneath northern Tibet. Compare with question 11.
17. Just north of the plateau, the Qaidam basin stands at a mean elevation of $3230\,\mathrm{m}$ , and its [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) lies about $50\,\mathrm{km}$ down. What crust thickness does Airy’s model predict there?
18. The Qaidam basin is filled with several kilometres of sediments, lighter than average crust. Does that make the real crust thinner or thicker than the model’s? Explain.
19. What property of the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) makes such a balance possible?
20. Conclude: how thick is the crust under Tibet, and why does the plateau stand five kilometres high?

**Solution of Problem 12.1.**

**1.** $5.01 - 0.67 = 4.34\,\mathrm{km}$.

**2.** $(3380 - 2830)/3380 \approx 16\,\%$.

**3.** The fraction $2830/3380 \approx 0.84$ would lie below; the iceberg sinks a little deeper, $910/1025 \approx 0.89$, because ice is closer in density to water than crust is to mantle.

**4.** For the seconds of a seismic wave, or the time of an earthquake, the mantle is a solid; only over very long times does the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) flow and let the crust float.

**5.** Below the deepest [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho), both columns are made of the same mantle rock at the same depths, and they weigh the same there; all the differences lie above that depth and must be included.

**6.** The average continent: crust from $0.67\,\mathrm{km}$ above to $40.4\,\mathrm{km}$ below sea level, then mantle down to $70\,\mathrm{km}$. The plateau: crust from $5.01\,\mathrm{km}$ above sea level down to a [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) deeper by the root, then mantle down to $70\,\mathrm{km}$.

**7.** The plateau column has $h + r$ more crust, the continent column has mantle over the thickness $r$ instead: $\rho_c\,(h + r) = \rho_m\,r$, so $r = h\,\rho_c/(\rho_m - \rho_c)$ and $\Delta H = h + r =
h\,\rho_m/(\rho_m - \rho_c)$.

**8.** $3380/550 \approx 6.15$.

**9.** $\Delta H = 4.34 \times 6.15 \approx 26.7\,\mathrm{km}$; the root is $26.7 - 4.34 \approx 22.3\,\mathrm{km}$.

**10.** The average continent’s [Moho](https://one-course.com/books/earth/2/en/chapter/11-seismic-waves-reveal-the-interior#def-g11-seismic-structure-moho) lies $41.1 - 0.67 \approx
40.4\,\mathrm{km}$ below sea level; the plateau’s lies $40.4 + 22.3 \approx
62.8\,\mathrm{km}$ below sea level.

**11.** $41.1 + 26.7 \approx 67.8\,\mathrm{km}$.

**12.** One kilometre of rock above a square metre has a mass of $\rho \times 1000\,\mathrm{m}^{3}$: $2830$ tonnes of crust or $3380$ tonnes of mantle. Continent: $41.1 \times 2830 + (70 - 40.43) \times 3380
\approx 116\,300 + 99\,900 \approx 216\,300$ tonnes. Plateau: $67.77 \times 2830 + (70 - 62.76) \times 3380 \approx 191\,800 +
24\,500 \approx 216\,300$ tonnes. The two are equal.

**13.** $41.1 - 0.67 \times 6.15 \approx 37.0\,\mathrm{km}$.

**14.** $67.8/41.1 \approx 1.65$ times.

**15.** $3300/470 \approx 7.02$, $\Delta H \approx 4.34 \times 7.02
\approx 30.5\,\mathrm{km}$, and a crust of about $71.6\,\mathrm{km}$. A change of about $2\,\%$ in the mantle’s density changes the thickness by about $4\,\mathrm{km}$: the result depends noticeably on the densities, but stays close to $70\,\mathrm{km}$.

**16.** The model gives about $68\,\mathrm{km}$, the seismic waves about $70\,\mathrm{km}$: they agree within a few per cent.

**17.** $h = 3.23 - 0.67 = 2.56\,\mathrm{km}$, $\Delta H = 2.56 \times
6.15 \approx 15.7\,\mathrm{km}$, and a crust of about $56.8\,\mathrm{km}$, some $7\,\mathrm{km}$ more than measured.

**18.** Thinner. A column whose upper part is lighter than average weighs less for the same thickness, so it floats higher: a lighter crust needs a smaller root to stand at a given height. The real crust can therefore be thinner than the model’s, as measured.

**19.** Over long times the [asthenosphere](#def-g11-lithosphere-asthenosphere-lithosphere) flows: it moves aside under a heavy column and comes back under a light one, until the columns balance.

**20.** **The crust under Tibet is about $70\,\mathrm{km}$ thick**, much more than the $41\,\mathrm{km}$ of an average continent: the plateau stands five kilometres high because its thick, light crust floats on the denser mantle, with a root of more than $20\,\mathrm{km}$.