Fowler A. Mathematical Geoscience

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600 9 Magma Transport

view, particularly advanced by Courtillot (1999) is that mass extinctions are often

associated with episodes of massive extrusion of ﬂood basalts. In particular, the for-

mation of the Deccan Traps in India represents a stuttering outpouring of something

approaching a million cubic kilometres of basaltic lava over a period of perhaps hun-

dreds of thousands of years at about the same period, and the timing of other such

ﬂood basalt eruptions appears coeval with other extinctions in the geologic record.

Courtillot thus associates extinctions, including the dinosaur extinction, with the

basaltic ﬂoods; the occurrence of the meteorite is not denied, but is not considered

to be a necessary cause (though it would undoubtedly help).

The gigantic eruptions themselves are thought to be associated with the exis-

tence of ‘superplumes’, essentially a manifestation of convection in the Earth’s

mantle (Schubert et al. 2001; Bercovici 2009a, 2009b). It is generally thought that

‘hotspots’ like Hawaii and Iceland are associated with the existence of convective

upwellings (‘plumes’) from within the mantle, possibly originating from the core-

mantle boundary, and which are consequently associated with vulcanism. In this

view, the massive basaltic eruptions are associated with the sort of massive plumes

which one might expect if mantle convection were episodic (as appears to be the

case on Venus): see also Chap. 8. The plume hypothesis itself (Morgan 1971), seem-

ingly an innocuous consequence of convection, is itself a matter of controversy.

Magma Transport The description of the porous ﬂow of melt in the astheno-

sphere really starts with the work of Turcotte and Ahern (1978) and Ahern and

Turcotte (1979), who describe the essential physics of the process, and indeed solve

the problem for one-dimensional transport. While they refer to compaction, they do

not model it explicitly. Compaction itself is introduced by McKenzie (1984), Scott

and Stevenson (1984), and Fowler (1985b), more or less in the same way by each

author, and thereafter the subject swells with the work of Hewitt, Katz, Spiegel-

man, Ribe, Bercovici, and many others. For no particularly good reason, much of

this early literature ignored the melting term (which after all is what produces the

melt), in which case the model is in some sense conservative, and one-dimensional

solutions have solitary wave properties (Scott and Stevenson 1984), termed by them

DePaolo and Manga (2003) summarise the hypothesis, and on the same page Foulger and Natland

(2003) raise various objections. The root of the controversy is (as so often) a misunderstanding of

how models should be applied. The objections of Foulger and Natland are that (i) there is no tomo-

graphic evidence for plume tails; (ii) hot spots are not ﬁxed; (iii) there is no evidence for excessive

temperatures at hot spots. These objections are based on a naïve understanding of how convection

works. One should not expect to see plume tails tomographically, nor should hot spots be neces-

sarily ﬁxed, nor should there be large temperature excess in strongly variable viscosity convection.

The objections are not based on data, but on preconception. The debate has become somewhat

creationist in tone, with its own website (www.mantleplumes.org), and occasional inﬂammatory

pieces, such as that of McNutt (2006), with the bye-line: “At least one chain of hot-spot volcanoes

is not caused by a plume rising up from the core-mantle boundary, calling for a reexamination of

the plume hypothesis.” McNutt’s remarks were chastised by Hofmann and Hart (2007). Just as the

question for global warming sceptics is: how could CO

not cause increased temperatures? the

question to be asked here is really: how could there not be plumes if the mantle is convecting at

high Rayleigh number?

9.8 Notes and References 601

‘magma solitons’, or magmons. Pursuit of these mythical beasts occupied a good

deal of attention for a number of years, particularly as there is a nice laboratory ana-

logue (Scott et al. 1986), although one can show that inclusion of melting and also

physically relevant boundaries suggests that in practice such waves will not be seen

(Fowler 1990).

Just as for water ﬂow under glaciers (Chap. 10) or over erodible sediment

(Chap. 6), porous ﬂow in the asthenosphere is prone to channelise (Aharonov

et al. 1995; Nicolas 1986; Hewitt and Fowler 2009; Spiegelman et al. 2001;

Stevenson 1989), and it seems likely that an arterial network of melt channels will

be formed. The upwelling melt could freeze on to the lithosphere, form a sub-

lithospheric lake, or promote fracturing and consequent continued rise through the

lithosphere.

Magmafracture Fracture mechanics is really a sub-branch of the theory of (lin-

ear) elasticity. Classical books are those by Sneddon and Lowengrub (1969), Sih

(1973) and England (1971). A more recent book, which also deals with time-

dependent (dynamic) fracture is that by Freund (1990). Speciﬁc deﬁnitions of

the stress intensity factor are given by Yang (2008) and Freund (1990). Two-

dimensional crack problems often involve the methods of complex variables, for

which see Carrier et al. (1966), Muskhelishvili (1953), Gakhov (1990), and also

Noble (1988).

The application of magma-driven (non-equilibrium) fractures in the Earth’s litho-

sphere was properly initiated by Spence and Turcotte (1985) (earlier work had been

done by Weertman (1971)), and developed subsequently by Spence et al. (1987),

and later Lister and co-workers (e.g., Lister and Kerr 1991; Richardson et al. 1996;

Bolchover and Lister 1999). The second of these deals with melt channels in a vis-

cous, porous medium, which is related to Question 9.9, while the last raises the

interesting issue of side-wall solidiﬁcation.

Dykes and Diapirs The issue of solidiﬁcation haunts the question of transport of

magma through the lithosphere. Two mechanisms are possible; the passage of mag-

mafractures, as above, and the passage of diapirs: huge molten blobs of magma,

which ascend due to their buoyancy by viscously deforming the surrounding coun-

try rock. This was suggested by Marsh (1982), following an original idea of Grout

(1945). The problem has been further studied by Morris (1982), Weinberg and Pod-

ladchikov (1994) and Bittner and Schmeling (1995). There is no essential differ-

ence between the two mechanisms, they simply differ by virtue of the geometry and

the deformation mechanism of the surrounding rock. Marsh suggested velocities of

−9

−1

for a 3 km diapir, with a consequent Péclet number of O(1).

To calculate the solidiﬁcation time t

for a diapir of diameter d, we estimate the

heat ﬂux per unit area from the diapir as

kT

, where T is the temperature of the

diapir above its surroundings; thus ρLd

∼

kT

· d

· t

, where L is latent heat,

whence

∼

St, (9.182)

602 9 Magma Transport

where κ is the thermal diffusivity, and St is the Stefan number

St =

T

, (9.183)

with c

being the speciﬁc heat. Using L = 3 × 10

Jkg

−1

, c

= 10

Jkg

−1

T = 10

K, we have St ∼

,sot

is essentially the conductive time scale. The

distance which the diapir moves in this time is

∼Pe St d, (9.184)

where the Péclet number is

Pe =

. (9.185)

Thus if also Pe ∼O(1), then the diapir freezes.

In more detail, the diapir moves by softening a thin rind of the surrounding very

viscous country rock, and moving through almost regelatively (regelation refers to

the passage of a heavy object through a block of ice; cf. the discussion of glacier

sliding in Chap. 10). The precepts of strongly variable viscosity (cf. Chap. 8)tellus

that this rind has a thickness such that the temperature drop across it is of O(

∗

)

(assuming RT

 E

∗

), where E

∗

is the activation energy for creep, T

is the

freezing temperature of the diapir (or more strictly the solidus temperature of the

solid/liquid interface), and R is the gas constant. For values E

∗

= 500 kJ mol

−1

R =8.3Jmol

−1

, T

=1,200 K,

ε =

∗

∼0.02 (9.186)

is indeed small, and the thickness of the rind d

is thus

∼εd (9.187)

(there is an additional factor

−T

, where T

is the ambient temperature of the

surrounding rock, but we take this to be approximately one). The velocity of ascent

is thus found from balancing the viscous shear stress

in the rind (where η

the rind viscosity) with the buoyancy induced shear stress ρgd , where ρ is the

difference in density between solid and liquid. This leads to an expression for the

Péclet number which resembles a sort of Rayleigh number:

Pe =

ερgd

. (9.188)

Taking d =3km,ρ =500 kg m

−3

, η

=10

Pa s, we calculate Pe ∼2.7.

In order to progress a diapir to the surface, Marsh suggested repeated injections,

so that the passage way could be repeatedly warmed. This seems a little artiﬁcial,

9.8 Notes and References 603

and it is not necessary for basaltic magma, where the viscosity is sufﬁciently low

that cracks can propagate at sufﬁcient speed that freezing does not occur.

The problem particularly arises for granitic plutons, where huge batholiths have

been emplaced in the continental crust. The viscosity of rhyolite is much higher

than that of basalt, so that fractures would have to be much larger, ascent times

would be greater, and freezing more likely. Despite this, ascent via fracture has

been suggested (Clemens and Mawer 1992; Petford et al. 1994). On the other hand,

because the ascent distance d

depends on the cube of the size, 10 km diapirs could

rise 20–30 times their own size, and thus to the surface. Smaller diapirs would not

do so well, though.

Against the idea of large diapir ascent may be the choice of viscosity. The value

= 10

Pa s is a typical asthenospheric sort of value, perhaps associated with

basalts at 1,500 K. But the melting point of granites is much lower, perhaps around

1,000 K, which is suggestive of higher values of η

. If this were the case, it would

virtually rule out diapiric uprise via rind softening. An alternative possibility is that

alluded to above in passing, the analogy with regelation. Regelation refers to the

passage of a heavy object through a block of ice. Because increased pressure lowers

the melting point of ice, a thin layer of water forms below the object and is squirted

round by the pressure gradient to the rear, where it refreezes, thus enabling passage

of the solid through the ice (Nye 1967). The analogy here is not quite the same,

however. If a layer of hot basalt is ponded at the base of the continental crust, it

will melt the crust, forming a light silicic magma which ﬂoats on top of the basalt.

As the basalt cools, crystals will begin to grow at the interface with the rhyolite. As

the crystals continue to grow, they will most likely form a mush or slurry, and will

cause the interstitial liquid to become increasingly silicic. Eventually, the heavy mat

of crystals will become unstable and sink to the ﬂoor (or this may occur continu-

ously). Apparently this will continue until the temperature decreases to the (cotectic)

point where also the rhyolite is crystallising. From this point the upper silicic layer

is also being removed, and the whole system effectively moves upwards at a rate

controlled by recrystallisation, and creep of the country rock need not be involved.

In the process, the magma presumably becomes granitic through the incorporation

of the crustal rocks. The idea is essentially due to Huppert and Sparks (1988).

Alloy Solidiﬁcation There are a number of books and proceedings volumes

on solidiﬁcation in multi-component systems. Amongst these, one should mention

Flemings (1974), Kurz and Fisher (1998), Davis et al. (1992), Ehrhard et al. (2001)

and Loper (1987), the last three of which provide multidisciplinary comparisons be-

tween different ﬁelds. Much of the interest originated from the study of the forma-

tion of metal alloy castings in metallurgy, particularly the formation of the defects

known as ‘freckles’. Early on it was found that an experimentally viable analogue

to the industrially important alloys such as lead–tin was the aqueous ammonium

chloride solution. The group led by Herbert Huppert at Cambridge has pioneered

the exploration of this and other aqueous solutions in freezing conditions, and Hup-

pert in particular, together with his colleague Steve Sparks, has uncovered, or per-

haps created, a gold mine of experimentally driven insights into all sorts of geolog-

ical processes in magma chambers; see, for example, Huppert (1986, 1990, 2000).

604 9 Magma Transport

Meanwhile, Huppert’s student Grae Worster (see, e.g., Worster 1997, 2000) has led

the way in experimental and analytic investigations of the ammonium chloride and

other aqueous crystallising solutions, more recently extending the laboratory work

also to applications in sea ice growth and freezing colloidal suspensions.

Nucleation and Crystallisation The classical theory of nucleation and crystal

growth is due to Avrami (1939, 1940). Dowty (1980) gives a lucid discussion of

the thermodynamics of nucleation and crystal growth rate. The theory is used in

the context of magma chambers by Brandeis et al. (1984), Brandeis and Jaupart

(1986), Spohn et al. (1988) and Hort and Spohn (1991). Worster et al. (1990)also

use a simpliﬁed theory of non-equilibrium growth in their discussion of convective

processes in magma chambers.

Double-Diffusive Convection Salt ﬁngers were discovered by Melvin Stern

(1960), and rapidly became part of the investigative culture at Woods Hole, and in

the GFD summer program there. Turner (1974) gives a nice review of the basic phe-

nomena of double-diffusive convection. He, Huppert and Sparks are the principal

advocates of the importance of ﬂuid mechanical (including double-diffusive) pro-

cesses in magma chambers, and Huppert (2000) gives a thorough and wide-ranging

review of much of this work. Turner’s (1973) book is the classic reference. A nice

book is that edited by Brandt and Fernando (1995), which contains a number of

historical recollections as well as a wide variety of applications.

Layered Igneous Intrusions The classic work on the Skaergaard layered intru-

sion is the voluminous book by Wager and Brown (1968). Largely observational,

but associating the layers with occasional turbidity currents, it was not until the pa-

per by McBirney and Noyes (1979) that genuine dynamical concepts began to be

applied. Noyes was a physical chemist, who brought to the problem the relatively

recent (and shocking) discovery of pattern formation in the Belousov–Zhabotinskii

chemical reaction, which is maintained through the interaction of nonlinear kinetics

with diffusion. In their paper, they suggest that layering may occur in an analogous

way to Liesegang rings, ﬁrst reported in 1896 by Liesegang in photographic emul-

sions, where silver dichromate is precipitated in a series of bands. Similar features

are seen in orbicular granites. A model for Liesegang rings was proposed by Keller

and Rubinow (1981) (and is discussed further in Sect. D.11 of Appendix D), but

for layered intrusions, such a model would need to be extended to multiple crystal

growth, as outlined by Maaløe (1978). The book edited by Parsons (1987) contains

a wealth of informative articles on the Skaergaard and other intrusions, for example

in the article by Irvine (1987). Oscillatory crystallisation has also been suggested

as a cause of layering by Wang and Merino (1993), but the construction of their

mathematical model is not too clear.

Sparks et al. (1993) suggest an interesting mechanism, whereby the increasing

crystal content of a convecting magma causes it to slow down, leading to a criti-

9.8 Notes and References 605

cal state where the crystals suddenly fall out of the solution, following which the

newly particle-free magma convects more vigorously, and can again circulate the

next batch of crystals. While this reasonably leads to layering, it is less obvious how

chemically distinct layers can occur.

The explanation for the layering in the Rum intrusion as being due to successive

injections of ultrabasic magma below a more evolved residual magma was given

by Huppert and Sparks (1980). The classic description of the Rum intrusion is by

Brown (1956). A more recent review is by Emeleus (1987). Note that the large scale

(tens of metres) layers are themselves subject to ﬁne-scale layering.

Volcanoes The Encyclopedia of volcanoes (Sigurdsson 2000) is a voluminous

compendium of knowledge written by experts. Some 1,400 large pages in length, it

has the advantage of consisting of twenty page long articles, so that there is space

for exposition in some depth. Each article comes with an opening glossary, and the

reference lists are usefully succinct. Despite its size, it is relatively cheap. The book

by Francis and Oppenheimer (2004) is more discursive and (thus) very readable,

a book to read in bed, snugly situated between a popular science book and a dry

monograph. The book by Sparks et al. (1997) describes volcanic eruption plumes,

and relates them to mathematical models of the different ﬂows.

Two-Phase Flows Two-phase ﬂows, as we have described them for ﬂow in a

volcanic vent, have mostly been studied in industrial situations; two particular ap-

plications are in boilers and nuclear cooling systems. In either case a ﬂuid (water, for

example) ﬂows through a tube where it is externally heated sufﬁciently that it starts

to boil. As the void fraction (i.e., gas volume fraction) increases, the ﬂow passes

through a succession of different ﬂow régimes: bubbly, slug, churn and annular. In

bubbly ﬂow, small bubbles of gas form a dispersed phase in the continuous liquid

phase. Slug ﬂow consists of tube-wide slugs of ﬂuid interspersed with plugs of gas.

A liquid ﬁlm drains at the wall as the gas bubbles ascend. This relatively regular

régime becomes irregular at higher gas ﬂow, when it is called churn ﬂow, and at

higher gas ﬂow rates still, the gas forms a continuous core ﬂowing through a liquid

ﬁlm which coats the walls; this is annular ﬂow. The same régimes occur whether the

ﬂow is heated (e.g., steam–water) or not (e.g., air–water). Because the liquid always

wets the wall, even in annular ﬂow, boiling occurs at the internal gas–liquid inter-

faces, so that the liquid is mildly superheated (although nucleation will also occur

at the walls, particularly in bubbly ﬂow).

There are a number of good engineering books on two-phase ﬂow, for example

Whalley (1987), Collier and Thome (1996), Hewitt and Hall-Taylor (1970), Wallis

(1969), Bergles et al. (1981) and Butterworth and Hewitt (1977). Mathematicians

have tended to get hung up with details of the averaging process whereby the model

equations are formulated, rather than the business of solving them. Drew and Pass-

man (1999) gives an up-to-date account of this work. In the context of volcanic

ﬂows, two-phase ﬂow models have been used by Melnik (2000), Starostin et al.

(2005) and Bercovici and Michaut (2010), for example.

606 9 Magma Transport

9.9 Exercises

9.1 The equations of plane strain are given by

∂σ

∂x

∂τ

∂y

+ρf

=0,

∂τ

∂x

∂σ

∂y

+ρf

=0,

where σ

and σ

are the normal stresses, τ is the shear stress, x and y sub-

scripts denote partial derivatives, and (f

) represents a body force per unit

mass.

The elastic constitutive relations are

=2μu

+λ(u

=2μv

+λ(u

τ =μ(u

where (u, v) are the components of displacement.

Use complex variables z =x +iy, ¯z =x −iy, to show that the force balance

equations can be written in the form

∂

∂z

[σ

−σ

+2iτ]+

∂

∂ ¯z

(σ

+σ

) +

∂V

∂ ¯z

=0,

where we suppose the body force can be written in terms of the potential V ,

thus

ρf =

∂V

∂ ¯z

where f =f

+if

The complex displacement is deﬁned by D =u +iv. Show that

+σ

=2(λ +μ)(D

¯z

−σ

+2iτ =4μD

¯z

and deduce that D satisﬁes

∂

∂z

[4μD

¯z

∂

∂ ¯z



2(λ +μ)(D

¯z

)



∂V

∂ ¯z

=0,

and hence show that

4μD

+2(λ +μ)(D

¯z

) +V =

4(λ +2μ)

λ +μ



(z),

where Ω



(z) is an arbitrary (analytic) function of z.

9.9 Exercises 607

Assuming now that V is real, show by taking the complex conjugate of this

equation and eliminating (D

¯z

), that

4μD

μV

λ +2μ

=−2



(z) +2κΩ



(z),

where κ =

λ+3μ

λ+μ

Assuming that

∂X

∂z

=V , show that this equation can be integrated to obtain

2μD =−

μX

2(λ +2μ)

+(¯z −z)



(z) +κΩ(z) +φ(¯z),

where φ(¯z) is a second arbitrary analytic function of its argument.

9.2 A crack L of width h containing magma in a porous medium (the astheno-

sphere) is described by the equations

∂h

∂t

∂

∂x



12η



ρ

g +

∂Π

∂x



μk

(1 −ν)η

Π =

2π(1 −ν)

−



∂h

∂s

s −x

where x is distance upwards along the vertical crack; μ is the shear modulus,

ν is Poisson’s ratio, k is the permeability, η is the magma viscosity, ρ

the

difference between solid and magma densities, and g is the acceleration due to

gravity. The diffusive term represents the suction of melt into the crack from

the porous medium. By choosing suitable scales for h ∼h

and t ∼t

in terms

of x ∼l, show that the model can be written in the dimensionless form

∂h

∂t

∂

∂x





1 +δ

∂Π

∂x



Π =

2π

−



∂h

∂s

s −x

and deﬁne the dimensionless parameter δ. Show that δ is small if l l

, where

=(12k)

1/5



(1 −ν)ρ



3/5

and estimate l

if μ =2 ×10

Pa, ρ

=300 kg m

−3

, g =10 m s

−2

, ν =

k =10

−12

. Deduce that on an asthenospheric length scale l

∼

10 km, δ is

indeed small.

Now suppose that a crack is nucleated at the interface x = 0 between the

porous asthenosphere in x<0 and the impermeable lithosphere in x>0,

where k =0. Assuming δ =0, show that a similarity solution exists in x<0,

in which

h =

F(η)

1/4

,η=−

1/2

and ﬁnd the resultant equation for F .

608 9 Magma Transport

Show from this equation that F →0asη →∞, and by detailed consider-

ation of the asymptotic form of the solution, show that in fact we require

1/2

F →0asη →∞,

assuming the crack initially has ﬁnite (zero) volume.

Assuming that h is continuous at x =0, show that the solution for the crack

thickness in 0 <x<x

h =

√

1/4



√



+36F



1/2



1/2

where F

=F(0), and show that the crack front position is

√

Deduce that, if also h

is continuous at x = 0, the boundary condition for F

at η =0is

12F



+1 =0,

where F



(0).

Sketch or compute the resulting form of the solution, and comment on the

physical applicability of the results, in terms of magma velocity and transport

time through the lithosphere. See also Fowler and Scott (1996).

9.3 A crack L is nucleated in the asthenosphere, which is treated as a viscous,

porous medium of viscosity η

and permeability k. The pore pressure in the

medium is p and its far ﬁeld (magmastatic) value is p

−T − ρ

gx, where

− T − ρ

gx = P is the far ﬁeld (lithostatic) solid pressure, and x points

vertically upwards along the crack; ρ

and ρ

are the solid and melt densities,

respectively, while g is the acceleration due to gravity. The effective pressure

P − p at the crack takes a value N which is related to the rate of viscous

closure w of the crack by the relation

N =−

2η

−



∂w

∂s

s −x

(see Ng 1998, p. 55 ff.) Assuming that ∇

p = 0, use complex variable meth-

ods to show that the net inﬂux of melt Ω to the crack, deﬁned on L by

Ω =

∂p

∂y



where η

is the melt viscosity, is given by

Ω =



−2η

ρ

−x

)

1/2



where ρ

=ρ

−ρ

, and the crack is taken to be in −l<x<l.

9.9 Exercises 609

Show that the second term in the above expression is absent if it is assumed

that p →P in the far ﬁeld.

9.4 A vertical crack L of width h is nucleated in a viscous, porous asthenosphere

containing melt. The ﬂuid ﬂux upwards (in the x direction along the crack) is

q, and conservation of melt in the crack is described by

∂h

∂t

∂q

∂x

+Ω,

where m is the rate of melting of the crack walls, and Ω is the supply from the

porous rock via suction; ρ

is the melt density.

The ﬂuid ﬂux is given by the Poiseuille law

q =

12η



ρ

g +

∂N

∂x



where η

is the melt viscosity, g is gravity, ρ

=ρ

−ρ

is the difference

in density between solid and melt, and N is the effective pressure in the crack.

The wall melting, due to potential energy release, is determined from

mL =q



ρ

g +

∂N

∂x



where L is the latent heat.

The crack closes at a rate w, determined from the closure equation

N =−

2η

−



∂w

∂s

s −x

where η

is the solid viscosity, and the crack width is described by the kine-

matic condition

∂h

∂t

−w.

Finally the melt suction is given by

Ω =−

4kη

∂

∂x

where k is the permeability.

Suppose that x ∼l, and that N ρ

gl. In this case show that h satisﬁes

the equation

+A(h

)

=ABh

∂

∂x



−rABh



where you should deﬁne the parameters A, B, C and r, and ﬁnd typical val-

ues for these assuming ρ

= 3 ×10

kg m

−3

, ρ

= 2.5 ×10

kg m

−3

, g =

10 m s

−2

, η

=10

Pa s, L =3 ×10

Jkg

−1

, k =10

−12

, η

=10

Pa s.