---
title: "Magmatism at Subduction Zones"
book: "High School Earth and Space Sciences"
subject: earth
language: en
chapter: 22
exercises: 15
source: https://one-course.com/books/earth/2/en/chapter/22-magmatism-at-subduction-zones
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 22 — Magmatism at Subduction Zones

On 18 May 1980, at 8:32 in the morning, a magnitude 5.1 earthquake shook Mount St. Helens. At once the north flank of the volcano, which had been bulging outwards by about two metres a day, slid away: the largest debris avalanche in recorded history, about $2.5\,\mathrm{km}^{3}$ of rock. Freed of its load, the gas-rich magma inside blew out sideways through the scar, at no less than $480\,\mathrm{km}/\mathrm{h}$, flattening forests over $600\,\mathrm{km}^{2}$. Within fifteen minutes a column of ash stood more than $24\,\mathrm{km}$ high, and the eruption went on for nine hours. Sixty-one people were killed or never found.

![The ash column of Mount St. Helens on 18 May 1980, which rose more than 24\, km in less than fifteen minutes. Photograph Donald A. Swanson, USGS (public domain).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/img-3deeac75e3fe.jpg)

*The ash column of Mount St. Helens on 18 May 1980, which rose more than $24\,\mathrm{km}$ in less than fifteen minutes. Photograph Donald A. Swanson, USGS (public domain).*

Mount St. Helens is one of a line of volcanoes — Baker, Rainier, Hood, Shasta and others — that stands above the place where a small oceanic plate, the Juan de Fuca Plate, sinks beneath North America. This chapter asks why volcanoes are found there, why their eruptions are so often violent, and what becomes of the magma that never reaches the surface.

**You already know.**

At a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction), a plate of [oceanic lithosphere](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-oceanic-lithosphere) sinks back into the mantle beneath another plate; its earthquakes, the [Wadati–Benioff zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-benioff), trace the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) down to as deep as $670\,\mathrm{km}$ ([Chapter 15](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#ch-g11-subduction-zones)). The mantle is made of [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite); it is solid, but a rock that rises and is less squeezed may melt in part, by [decompression melting](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-partial-melting), as beneath a ridge ([Chapter 14](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#ch-g11-ocean-ridges)). [Andesite](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-rock-names) is a [volcanic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) between basalt and [rhyolite](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-rock-names), [diorite](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-rock-names) its [plutonic](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) equivalent, and a [pluton](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion) a large body of magma that cooled underground ([Chapter 17](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#ch-g11-igneous-rocks)). The [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) is thick and light, mostly granitic rock on top ([Chapter 19](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#ch-g12-continental-crust)). An explosive eruption blows the magma to pieces with its own gas (Book 1, grade 7).

## 22.1 Volcanic arcs

Most of the active volcanoes on land do not stand at random. Of the 1214 volcanoes that have erupted since the end of the last ice age and are listed in the catalogue of the Global Volcanism Program, 838 — more than two out of three — stand above a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction). Around the Pacific they make an almost unbroken ring, often called the Ring of Fire, and every line of them runs parallel to a trench, on the side of the plate that does not sink.

![The volcanoes active since the end of the last ice age, red where they stand above a subduction zone, with the trenches (black line, triangles on the plate that overrides). Data: Smithsonian Global Volcanism Program; plate boundaries of Bird (2003); land from Natural Earth.](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/fig-e990e90e3cd2.svg)

*The volcanoes active since the end of the last ice age, red where they stand above a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction), with the trenches (black line, triangles on the plate that overrides). Data: Smithsonian Global Volcanism Program; plate boundaries of Bird (2003); land from Natural Earth.*

**Definition 22.1 (Volcanic arc).**

A *volcanic arc* is a line of volcanoes on the overriding plate of a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction), parallel to the trench, standing above the place where the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) has reached a depth of about a hundred kilometres. It is an *island arc* when it rises from the sea floor, as in Japan or Indonesia, and a continental arc when it stands on a continent, as in the Andes or the Cascades.

The edge of the arc nearest the trench is called the **volcanic front**: on the trench side of it, there are no volcanoes at all. How deep is the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath it?

**Proposition 22.2 (The slab beneath the volcanic front).**

If the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is a plane that dips at an angle $\theta$ from the trench, and the volcanic front lies at a horizontal distance $d$ from the trench, then the top of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) lies beneath the front at a depth

$$
H = d \tan\theta .
$$

**Proof.** Take a vertical section across the trench. The [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab), the surface and the vertical line beneath the front make a right triangle, with the right angle beneath the front: the side along the surface is $d$, the vertical side is $H$, and the angle at the trench is $\theta$. So $\tan\theta = H/d$. ∎

A compilation of 839 volcanic centres along $33\,000\,\mathrm{km}$ of arcs measured $H$ beneath each of them: averaged over stretches of arc $500\,\mathrm{km}$ long, it ranges from 72 to $173\,\mathrm{km}$, with a mean of $105\,\mathrm{km}$. Arcs above steep [slabs](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) lie close to their trench, arcs above gently dipping [slabs](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) far from it, but the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath them is always about a hundred kilometres down.

**Method 22.3 (Finding the slab under a volcano).**

1. Take the earthquakes of a narrow band across the arc, and plot their depth against their distance across it, at true scale.
2. Keep the earthquakes deeper than about $70\,\mathrm{km}$ : shallower ones also occur in the overriding plate and on the fault between the two plates.
3. Draw the line, or read the depth, of the deep earthquakes beneath the volcanic front: that is the depth of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) there.
4. Check it with $H = d\tan\theta$ , measuring $d$ from the trench and the dip $\theta$ on the section.

![A section across north-east Japan at true scale: the earthquakes of magnitude 4.5 or more between 37.5 and 40.5 N, 1976–2025 (grey), the volcanoes of the same band (red triangles) and the line fitted to the earthquakes deeper than 70\, km (blue, dashed). Two out of five of the shallow events east of 141.5 E date from 2011, the year of the great earthquake of March. Data: USGS earthquake catalogue; Smithsonian Global Volcanism Program; plate boundaries of Bird (2003).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/fig-e5d46a3b00a2.svg)

*A section across north-east Japan at true scale: the earthquakes of magnitude 4.5 or more between $37.5{}^{\circ}$ and $40.5{}^{\circ}$ N, 1976–2025 (grey), the volcanoes of the same band (red triangles) and the line fitted to the earthquakes deeper than $70\,\mathrm{km}$ (blue, dashed). Two out of five of the shallow events east of $141.5{}^{\circ}$ E date from 2011, the year of the great earthquake of March. Data: USGS earthquake catalogue; Smithsonian Global Volcanism Program; plate boundaries of Bird (2003).*

**Example 22.4 (North-east Japan).**

In the band of the figure, the easternmost volcano is Iwate, at $141.0{}^{\circ}$ E, and the Japan Trench lies at $144.1{}^{\circ}$ E. One degree of longitude there is $111.2\cos39{}^{\circ} \approx 86.4\,\mathrm{km}$, so $d \approx 3.12 \times 86.4
\approx 270\,\mathrm{km}$. The deep earthquakes dip at about $20{}^{\circ}$, and

$$
H = 270 \times \tan20{}^{\circ} \approx 98\,\mathrm{km}.
$$

The earthquakes deeper than $70\,\mathrm{km}$ that lie beneath the front itself have a median depth of $98\,\mathrm{km}$: the two agree, close to the world average.

## 22.2 Water melts the mantle

Beneath an arc, the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is cold, and so is the [oceanic crust](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-oceanic-lithosphere) on top of it: it cannot melt. The magma must come from the hot mantle above the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab), the **mantle wedge**. But the wedge is not rising, so it cannot melt by decompression as it does beneath a ridge. Something else lowers its melting temperature: water.

The [oceanic crust](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-oceanic-lithosphere) is soaked in sea water near the ridge where it forms, and much of that water ends up locked inside its minerals, above all in serpentine, made from the olivine and pyroxene of the crust. As the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) sinks and warms, these minerals break down and give their water away. The water rises into the hot [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) of the wedge above.

**Definition 22.5 (Hydrous melting).**

*Hydrous melting* (or flux melting) is the [partial melting](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-partial-melting) of a rock caused by the addition of water, which lowers the temperature at which the rock begins to melt.

The temperature at which a rock begins to melt, at a given pressure, is its **solidus**. For dry [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) it is well known; with water it is much harder to measure, and published values still differ by 200 to $600\,{}^{\circ}\mathrm{C}$. The most recent experiments place it between 950 and $1000\,{}^{\circ}\mathrm{C}$ at $3\,\mathrm{GPa}$, about $95\,\mathrm{km}$ down, and just above $1000\,{}^{\circ}\mathrm{C}$ from 5 to $11\,\mathrm{GPa}$. Dry [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) at $3\,\mathrm{GPa}$ begins to melt only at about $1470\,{}^{\circ}\mathrm{C}$: water lowers the solidus by about $500\,{}^{\circ}\mathrm{C}$.

![The temperature at which mantle peridotite begins to melt, against depth: dry (orange) and saturated with water (blue, the two ranges measured so far). At the depth of the slab beneath arcs, about 100\, km, water lowers it by about 500\, C. Sources: Hirschmann (2000), with the PREM densities; Wang and others (2020); Kawamoto and Holloway (1997).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/fig-efdeb9a767c3.svg)

*The temperature at which mantle [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) begins to melt, against depth: dry (orange) and saturated with water (blue, the two ranges measured so far). At the depth of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath arcs, about $100\,\mathrm{km}$, water lowers it by about $500\,{}^{\circ}\mathrm{C}$. Sources: Hirschmann (2000), with the PREM densities; Wang and others (2020); Kawamoto and Holloway (1997).*

So the water does what heating alone could not: it makes part of the hot wedge melt. The magma, lighter than the solid mantle around it, rises towards the surface; some erupts, much more stays and cools underground. The place where the water is set free, and where the wedge above it is hot enough to melt once wet, is what fixes the volcanic front about a hundred kilometres above the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab).

![Magmatism at a subduction zone (schematic, not to scale). As the slab warms, its minerals give up their water, which rises into the hot mantle wedge and makes it melt in part. The magma rises: some of it erupts at the volcanoes, more of it cools in the crust as plutons.](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/fig-c0a04ce28919.svg)

*Magmatism at a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction) (schematic, not to scale). As the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) warms, its minerals give up their water, which rises into the hot mantle wedge and makes it melt in part. The magma rises: some of it erupts at the volcanoes, more of it cools in the crust as [plutons](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion).*

## 22.3 Andesites and granodiorites

The first melts of wet [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) are not basaltic. In experiments at $1\,\mathrm{GPa}$, water-saturated [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) gives melts of andesitic composition. Rising through the thick crust of the arc, magmas also change as they cool, from basaltic towards andesitic and rhyolitic. Arc magmas are therefore richer in silica than the basalts of ridges — andesitic magma holds 55 to $65\,\%$ of it — and they carry the water that made them.

Both properties make their eruptions violent. The more silica a magma holds, the more viscous it is, and the harder it is for its gas to escape; explosive eruptions are favoured by a high gas content and a high viscosity, as in andesitic and rhyolitic magmas. Almost all composite volcanoes stand above [subduction zones](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction): cones thousands of metres high and up to $20\,\mathrm{km}$ across, with slopes of 10 to 30 degrees, whose eruptions range from quiet lava flows to explosions. Beneath Mount St. Helens, a magma chamber about $1\,\mathrm{km}$ wide extends from about 6 to $14\,\mathrm{km}$ down.

Most arc magma never erupts. It cools slowly in the crust and crystallises into coarse-grained rocks.

**Definition 22.6 (Granodiorite).**

*Granodiorite* is a coarse-grained [plutonic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) like granite, but with more plagioclase than potassium feldspar: 20 to $60\,\%$ of its light minerals are quartz, and 65 to $90\,\%$ of its feldspar is plagioclase; it often holds biotite or hornblende.

![Left: andesite, a fine-grained volcanic rock of arcs. Right: granodiorite of the Sierra Nevada, with white plagioclase, grey quartz and black biotite and hornblende. Photographs James St. John (CC BY 2.0).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/img-8fb470cbd17d.jpg)

![Left: andesite, a fine-grained volcanic rock of arcs. Right: granodiorite of the Sierra Nevada, with white plagioclase, grey quartz and black biotite and hornblende. Photographs James St. John (CC BY 2.0).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/img-4143bbcfc346.jpg)

*Left: [andesite](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-rock-names), a fine-grained [volcanic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) of arcs. Right: [granodiorite](#def-g12-subduction-magmatism-granodiorite) of the Sierra Nevada, with white plagioclase, grey quartz and black biotite and hornblende. Photographs James St. John (CC BY 2.0).*

On average, the upper [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) holds about $67\,\%$ $\ce{SiO2}$: it is [felsic](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-felsic-mafic), like the granites that arcs make in quantity in their [batholiths](#def-g12-subduction-magmatism-batholith).

**Definition 22.7 (Batholith).**

A *batholith* is a large mass of [plutonic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) that covers more than $100\,\mathrm{km}^{2}$ where erosion has exposed it. It forms where many [plutons](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion), intruded one after another, join beneath the surface.

One of the largest in the world, the Coast [Plutonic](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) Complex, runs along the Pacific coast from the Vancouver region of western Canada to south-eastern Alaska. The Sierra Nevada [batholith](#def-g12-subduction-magmatism-batholith) of California formed above a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction); most of its granites in Yosemite crystallised in the Cretaceous, 105 to 85 million years ago. The rocks above them were worn away over millions of years, and the [plutons](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion) that cooled deep down now form the mountains.

![The Sierra Nevada, seen from the east: granite and granodiorite of the Sierra Nevada batholith, which crystallised deep down above a subduction zone in the Cretaceous. Photograph James St. John (CC BY 2.0).](https://one-course.com/images/onecourse/chapters/earth-2/g12-subduction-magmatism/img-6efda088997f.jpg)

*The Sierra Nevada, seen from the east: granite and [granodiorite](#def-g12-subduction-magmatism-granodiorite) of the Sierra Nevada [batholith](#def-g12-subduction-magmatism-batholith), which crystallised deep down above a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction) in the Cretaceous. Photograph James St. John (CC BY 2.0).*

## 22.4 How continents grow

**Definition 22.8 (Continental growth).**

*Continental growth* is the addition of new [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust), made from the mantle, mostly by the magmatism of arcs. The net growth of the continents is that addition minus the [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) that returns to the mantle.

Arcs are where the light rock of the continents is made. Estimates of the magma that arcs add today are about $1.5\,\mathrm{km}^{3}$ a year at island arcs and $1.0\,\mathrm{km}^{3}$ a year at continental arcs; averaged over the long term, about $2.7\,\mathrm{km}^{3}$ a year. But [subduction zones](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction) also take [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) back into the mantle: sediment carried down with the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) and pieces of the overriding plate scraped from beneath, about $3.2\,\mathrm{km}^{3}$ a year. The two are close, and the net growth of the continents over the last several hundred million years may have been nil, or even negative.

Continents also grow at their edges in another way. [Continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) is too light to be subducted, so when a plate carries a continent into a trench the two continents are welded together in a collision ([Chapter 20](https://one-course.com/books/earth/2/en/chapter/20-building-mountains#ch-g12-mountain-building)). Each continent has an old core, a [craton](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-craton), surrounded by belts of younger rock, the deformed [roots](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-root) of former mountain belts, added one after another at its edges ([Chapter 19](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#ch-g12-continental-crust)).

## 22.5 Exercises

**Exercise 22.1 ★.**

Give the word: (a) a line of volcanoes parallel to a trench, on the plate that does not sink; (b) melting caused by adding water to a rock; (c) a [plutonic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) with more plagioclase than potassium feldspar and more than $20\,\%$ quartz; (d) a mass of [plutonic rock](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-volcanic-plutonic) larger than $100\,\mathrm{km}^{2}$; (e) the addition of new [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) made from the mantle.

**Solution of Exercise 22.1.**

(a) a [volcanic arc](#def-g12-subduction-magmatism-arc); (b) [hydrous melting](#def-g12-subduction-magmatism-hydrous-melting) (flux melting); (c) a [granodiorite](#def-g12-subduction-magmatism-granodiorite); (d) a [batholith](#def-g12-subduction-magmatism-batholith); (e) [continental growth](#def-g12-subduction-magmatism-continental-growth).

**Exercise 22.2 ★.**

On 18 May 1980 the ash column of Mount St. Helens rose more than $24\,\mathrm{km}$ in less than 15 minutes. (1) What was its least mean rising speed, in $\mathrm{km}/\mathrm{h}$ and in $\mathrm{m}/\mathrm{s}$? (2) Compare it with the speed of the lateral blast, at least $480\,\mathrm{km}/\mathrm{h}$.

**Solution of Exercise 22.2.**

(1) At least $24/0.25 = 96\,\mathrm{km}/\mathrm{h}$, that is $96\,000/3600 \approx
27\,\mathrm{m}/\mathrm{s}$. (2) The blast was at least $480/96 = 5$ times faster.

**Exercise 22.3 ★.**

Of the 1214 volcanoes of the catalogue active since the end of the last ice age, 838 stand above a [subduction zone](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction). (1) What percentage is that? (2) On which side of a trench are these volcanoes? (3) Why are there none between the trench and the volcanic front?

**Solution of Exercise 22.3.**

(1) $838/1214 \approx 0.690$, about $69\,\%$. (2) On the side of the plate that overrides, the one that does not sink. (3) Between the trench and the front the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is not yet deep enough: it has not reached the depth, about a hundred kilometres, where its water is set free into a mantle hot enough to melt once wet.

**Exercise 22.4 ★.**

A coarse-grained rock holds $25\,\%$ quartz, $45\,\%$ plagioclase, $15\,\%$ potassium feldspar and $15\,\%$ biotite and hornblende. (1) What fraction of its feldspar is plagioclase? (2) Is it a [granodiorite](#def-g12-subduction-magmatism-granodiorite)?

**Solution of Exercise 22.4.**

(1) $45/(45 + 15) = 0.75$: $75\,\%$ of the feldspar is plagioclase. (2) Yes: it is coarse grained, holds more than $20\,\%$ quartz, and its plagioclase fraction lies between 65 and $90\,\%$.

**Exercise 22.5 ★.**

Give two reasons why the volcanoes of arcs erupt more explosively than those of ridges.

**Solution of Exercise 22.5.**

Arc magmas are richer in silica, so they are more viscous; and they carry the water that made them, so they are rich in gas. A high viscosity and a high gas content both favour explosive eruptions.

**Exercise 22.6 ★★.**

(1) A volcanic front lies $150\,\mathrm{km}$ from its trench, above a [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) that dips at $35{}^{\circ}$. How deep is the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath it? (2) Compare with the average of $105\,\mathrm{km}$.

**Solution of Exercise 22.6.**

(1) $H = 150 \times \tan35{}^{\circ} \approx 150 \times 0.700 \approx
105\,\mathrm{km}$. (2) It is the world average.

**Exercise 22.7 ★★.**

Take the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) to lie $105\,\mathrm{km}$ beneath every volcanic front. How far from the trench is the front when the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) dips at $20{}^{\circ}$, and when it dips at $45{}^{\circ}$? What does that tell you about where to look for the arc?

**Solution of Exercise 22.7.**

$d = H/\tan\theta$: $105/\tan20{}^{\circ} \approx 105/0.364 \approx
288\,\mathrm{km}$ at $20{}^{\circ}$, and $105/\tan45{}^{\circ} = 105\,\mathrm{km}$ at $45{}^{\circ}$. The gentler the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab), the farther from the trench the arc stands.

**Exercise 22.8 ★★.**

At $100\,\mathrm{km}$, the pressure is about $3.16\,\mathrm{GPa}$. (1) Using $T =
-5.104P^2 + 132.9P + 1120.7$ ($P$ in $\mathrm{GPa}$, $T$ in ${}^{\circ}\mathrm{C}$), compute the temperature at which dry [peridotite](https://one-course.com/books/earth/2/en/chapter/14-ocean-ridges-and-the-oceanic-lithosphere#def-g11-ocean-ridges-peridotite) begins to melt there. (2) With water, it begins to melt at about $1000\,{}^{\circ}\mathrm{C}$. By how much does water lower it? (3) The mantle wedge there is at, say, $1300\,{}^{\circ}\mathrm{C}$. Does it melt dry? Wet?

**Solution of Exercise 22.8.**

(1) $T = -5.104 \times 3.16^2 + 132.9 \times 3.16 + 1120.7 \approx -51.0 +
420.0 + 1120.7 \approx 1490\,{}^{\circ}\mathrm{C}$. (2) By about $1490 - 1000 =
490\,{}^{\circ}\mathrm{C}$. (3) Dry, no: $1300\,{}^{\circ}\mathrm{C}$ is below $1490\,{}^{\circ}\mathrm{C}$. Wet, yes: it is above $1000\,{}^{\circ}\mathrm{C}$.

**Exercise 22.9 ★★.**

The debris avalanche of Mount St. Helens moved about $2.5\,\mathrm{km}^{3}$ of rock. Arcs add about $2.7\,\mathrm{km}^{3}$ of new crust a year. (1) How many days of the world’s arc magmatism does the avalanche represent? (2) Why is the comparison only a matter of size?

**Solution of Exercise 22.9.**

(1) $2.5/2.7 \approx 0.93$ of a year, about 340 days. (2) The avalanche moved old rock that already existed; arcs make new rock from the mantle. Only the volumes are compared.

**Exercise 22.10 ★★.**

(1) How much new crust do arcs add in a million years at $2.7\,\mathrm{km}^{3}$ a year? (2) The [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust) holds about $9 \times 10^{9}\,\mathrm{km}^{3}$. How long would arcs alone take to make that volume, if none of it were lost? (3) Compare with the age of the Earth, 4.5 billion years.

**Solution of Exercise 22.10.**

(1) $2.7 \times 1\,000\,000 = 2.7 \times 10^{6}\,\mathrm{km}^{3}$. (2) $9 \times 10^{9}/2.7
\approx 3.3 \times 10^{9}$ years, 3.3 billion years. (3) It is less than the age of the Earth, so arcs could have made the whole [continental crust](https://one-course.com/books/earth/2/en/chapter/19-the-continental-crust#def-g12-continental-crust-continental-crust); but [subduction](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction) also takes crust back, so in fact much of the crust has been made and destroyed more than once.

**Exercise 22.11 ★★.**

(1) In the section across Japan, why are many earthquakes shallower than $70\,\mathrm{km}$ left out when the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is fitted? (2) The deep earthquakes lie $116\,\mathrm{km}$ down at $140.5{}^{\circ}$ E and $80\,\mathrm{km}$ at $141.5{}^{\circ}$ E, with $86.4\,\mathrm{km}$ per degree. Compute the dip of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab).

**Solution of Exercise 22.11.**

(1) Shallow earthquakes also occur in the overriding plate and on the fault between the two plates; only the deep ones lie in the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) alone. (2) $116 - 80 = 36\,\mathrm{km}$ over $86.4\,\mathrm{km}$: $\tan\theta = 36/86.4
\approx 0.417$, $\theta \approx 22.6{}^{\circ}$.

**Exercise 22.12 ★★★.**

Show that, for a front at distance $d$ from the trench above a [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) of dip $\theta$, the distance measured down the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) from the trench to the point beneath the front is $d/\cos\theta$. For Japan ($d = 270\,\mathrm{km}$, $\theta = 20{}^{\circ}$), how long is it? The Pacific Plate moves towards Japan at about $83\,\mathrm{mm}$ a year; taking that speed all along the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab), how long does a piece of sea floor take to travel from the trench to beneath the volcanoes?

**Solution of Exercise 22.12.**

In the right triangle of the proposition, the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is the hypotenuse $L$ and $d$ the side next to the angle $\theta$, so $\cos\theta = d/L$ and $L =
d/\cos\theta$. For Japan, $L = 270/\cos20{}^{\circ} \approx 270/0.940 \approx
287\,\mathrm{km}$. At $83\,\mathrm{mm}$ a year: $2.87 \times 10^{8}/83 \approx 3.5 \times 10^{6}$ years, about 3.5 million years.

**Exercise 22.13 ★★★.**

The magma chamber of Mount St. Helens is about $1\,\mathrm{km}$ wide and extends from about 6 to $14\,\mathrm{km}$ down. (1) Treating it as a vertical cylinder, estimate its volume. (2) Compare it with the $2.5\,\mathrm{km}^{3}$ of the avalanche. (3) Compute the [lithostatic pressure](https://one-course.com/books/earth/2/en/chapter/21-metamorphic-rocks#def-g12-metamorphic-rocks-lithostatic) at its top and bottom ($\rho =
2830\,\mathrm{kg}/\mathrm{m}^{3}$, $g = 9.81\,\mathrm{m}/\mathrm{s}^{2}$).

**Solution of Exercise 22.13.**

(1) $V = \pi r^2 h = \pi \times 0.5^2 \times 8 \approx 6.3\,\mathrm{km}^{3}$. (2) About $6.3/2.5 \approx 2.5$ times the volume of the avalanche. (3) $P = \rho g z$: at $6\,\mathrm{km}$, $2830 \times 9.81 \times 6000 \approx
1.67 \times 10^{8}\,\mathrm{Pa} = 0.17\,\mathrm{GPa}$; at $14\,\mathrm{km}$, about $3.89 \times 10^{8}\,\mathrm{Pa}$ = $0.39\,\mathrm{GPa}$.

**Exercise 22.14 ★★★.**

Arcs add about $2.7\,\mathrm{km}^{3}$ of crust a year, and [subduction zones](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-subduction) take back about $3.2\,\mathrm{km}^{3}$. (1) What is the net change per million years? (2) Over 500 million years, what fraction of a crust of $9 \times 10^{9}\,\mathrm{km}^{3}$ is that? (3) The authors call the net growth “effectively nil”. Why, given the uncertainty of each estimate?

**Solution of Exercise 22.14.**

(1) $2.7 - 3.2 = -0.5\,\mathrm{km}^{3}$ a year, a loss of $5 \times 10^{5}\,\mathrm{km}^{3}$ per million years. (2) $500 \times 5 \times 10^{5} = 2.5 \times 10^{8}\,\mathrm{km}^{3}$, that is $2.5 \times 10^{8}/9 \times 10^{9} \approx 2.8\,\%$ of the crust. (3) Each rate is an estimate, uncertain by much more than the difference of $0.5\,\mathrm{km}^{3}$ a year between them: within the errors, the two balance.

**Exercise 22.15 ★★★.**

A [batholith](#def-g12-subduction-magmatism-batholith) exposes $20\,000\,\mathrm{km}^{2}$ of [granodiorite](#def-g12-subduction-magmatism-granodiorite) whose [plutons](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion) solidified between 5 and $15\,\mathrm{km}$ down. (1) Estimate the volume of magma it represents, taking it $10\,\mathrm{km}$ thick. (2) At $1\,\mathrm{km}^{3}$ a year, the rate of all the world’s continental arcs together, how long would they take to make it? What does that tell you about the time over which a [batholith](#def-g12-subduction-magmatism-batholith) is built? (3) How much rock had to be eroded from above it before it could be seen?

**Solution of Exercise 22.15.**

(1) $20\,000 \times 10 = 2 \times 10^{5}\,\mathrm{km}^{3}$. (2) $2 \times 10^{5}/1 =
200\,000$ years for all the continental arcs of the world together. One arc receives only a small part of that magma, so a single [batholith](#def-g12-subduction-magmatism-batholith) takes far longer: it is built [pluton](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion) by [pluton](https://one-course.com/books/earth/2/en/chapter/17-igneous-rocks#def-g11-igneous-rocks-intrusion), over millions of years. (3) At least the $5\,\mathrm{km}$ of rock above its top.

**Problem 22.1.**

Weekend problem — Beneath the volcanoes of Japan

The volcanoes of north-east Japan between $37.5{}^{\circ}$ and $40.5{}^{\circ}$ N lie between $139.7{}^{\circ}$ E (Megata, the westernmost) and $141.0{}^{\circ}$ E (Iwate, the easternmost); the Japan Trench lies at $144.1{}^{\circ}$ E. The table gives the median depth of the earthquakes deeper than $70\,\mathrm{km}$ in three strips of longitude, 1976–2025. Take $111.2\,\mathrm{km}$ for one degree of latitude.

| longitude (${}^{\circ}$ E) | 140.5 | 141.0 | 141.5 |
| --- | --- | --- | --- |
| median depth ($\mathrm{km}$) | 116 | 98 | 80 |

**Part I — The volcanic front.**

1. Show that one degree of longitude at $39{}^{\circ}$ N is about $86.4\,\mathrm{km}$ .
2. How far is the volcanic front from the trench?
3. How wide is the arc, from Iwate to Megata?
4. On which plate do the volcanoes stand, and on which side of the trench?
5. Why are there no volcanoes between the trench and Iwate?

**Part II — The earthquake section.**

6. By how many kilometres do the median depths change from $140.5{}^{\circ}$ to $141.5{}^{\circ}$ E?
7. Over what horizontal distance?
8. Compute the dip of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) .
9. Continue the line from $141.0{}^{\circ}$ E, $98\,\mathrm{km}$ , up to the surface. Where does it reach it, and how far is that from the trench?
10. Suggest why the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is less steep near the trench than deeper down.

**Part III — The slab beneath the volcanoes.**

11. Compute $H = d\tan\theta$ with the distance of question 2 and a dip of $20{}^{\circ}$ .
12. Compare it with the median depth beneath the front.
13. Compare both with the world range, 72 to $173\,\mathrm{km}$ , and mean, $105\,\mathrm{km}$ .
14. How deep is the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath Megata, using the line of question 9?
15. Taking the crust of the arc to be about $35\,\mathrm{km}$ thick, is the top of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath the front in the crust or in the mantle? How far below the base of the crust is it?

**Part IV — Where the water comes from.**

16. Where did the water of the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) enter its rocks, and in which mineral is much of it held?
17. At $100\,\mathrm{km}$ the pressure is about $3.16\,\mathrm{GPa}$ . Compute the dry solidus there with $T = -5.104P^2 + 132.9P + 1120.7$ .
18. With water, the solidus is between 950 and $1000\,{}^{\circ}\mathrm{C}$ . By how much does water lower it?
19. Why can the cold [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) itself not be the source of the magma?
20. Sum up: how deep is the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath the volcanic front of north-east Japan?

**Solution of Problem 22.1.**

**1.** $111.2 \times \cos39{}^{\circ} \approx 111.2 \times 0.777 \approx
86.4\,\mathrm{km}$.

**2.** $(144.1 - 141.0) \times 86.4 \approx 3.1 \times 86.4 \approx
268\,\mathrm{km}$, about $270\,\mathrm{km}$.

**3.** $(141.0 - 139.7) \times 86.4 \approx 112\,\mathrm{km}$.

**4.** On the overriding plate, the one that does not sink, west of the trench: the Pacific Plate sinks beneath it.

**5.** There, the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is not yet deep enough to release its water into a mantle hot enough to melt.

**6.** $116 - 80 = 36\,\mathrm{km}$.

**7.** One degree of longitude, $86.4\,\mathrm{km}$.

**8.** $\tan\theta = 36/86.4 \approx 0.417$, so $\theta \approx
22.6{}^{\circ}$.

**9.** The line rises $36\,\mathrm{km}$ every $86.4\,\mathrm{km}$ eastwards, so it reaches the surface $98/0.417 \approx 235\,\mathrm{km}$ east of Iwate, at $141.0 + 235/86.4 \approx 143.7{}^{\circ}$ E: about $(144.1 - 143.7) \times 86.4
\approx 35\,\mathrm{km}$ west of the trench.

**10.** The [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is not a flat plane: it enters the trench almost horizontal and bends down more and more as it sinks.

**11.** $268 \times \tan20{}^{\circ} \approx 268 \times 0.364 \approx
98\,\mathrm{km}$.

**12.** They agree: $98\,\mathrm{km}$ both ways.

**13.** Both lie within the world range, a little shallower than the mean of $105\,\mathrm{km}$.

**14.** Megata lies $112\,\mathrm{km}$ west of Iwate, so the [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) beneath it is about $98 + 0.417 \times 112 \approx 145\,\mathrm{km}$ deep.

**15.** In the mantle, about $98 - 35 = 63\,\mathrm{km}$ below the base of the crust.

**16.** It entered near the ridge where the crust formed, from sea water; much of it is held in serpentine.

**17.** $T = -5.104 \times 3.16^2 + 132.9 \times 3.16 + 1120.7 \approx
1490\,{}^{\circ}\mathrm{C}$.

**18.** By $1490 - 1000 = 490$ to $1490 - 950 = 540\,{}^{\circ}\mathrm{C}$: about $500\,{}^{\circ}\mathrm{C}$.

**19.** The [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) is several hundred degrees colder than the mantle around it, too cold to melt even with water: it is the hot mantle wedge above it, once wetted, that melts.

**20.** The [slab](https://one-course.com/books/earth/2/en/chapter/15-subduction-zones#def-g11-subduction-zones-slab) lies about **$100\,\mathrm{km}$ beneath the volcanic front** of north-east Japan ($98\,\mathrm{km}$ by both methods).