Fowler A. Mathematical Geoscience

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580 9 Magma Transport

seems generally not to be the case; instead, magma accumulates in crustal magma

chambers, typically at depths of 10–20 kilometres below the surface. The question

then arises as to why this should be the case.

9.6.1 The Formation of Magma Chambers

There are various different types of magma chambers. The simplest are the dyke and

the sill. The dyke we have already discussed; it is the conduit through which magma

ascends through the lithosphere. It may supply a crustal magma chamber, or it may

provide the vent from a magma chamber to an erupting volcano. Particularly in the

latter case, the dyke may freeze after the eruption, and the solidiﬁed dyke may later

be exposed at the surface by erosion.

A sill is like a dyke on its side; it is a ﬂat tabular body which is presumably

formed when an upwelling dyke reaches an unconformity, where the rock density

decreases, and the propagating fracture ﬁnds it easier to propagate sideways rather

than upwards. Again, such formations are commonly exhibited at the surface fol-

lowing erosive removal of the overlying crust.

The term laccolith refers to a magma chamber which is initiated as a sill, but in

which the upwards buoyancy is sufﬁcient to uplift the overlying crust, forming a

gigantic blister. It may be that this is the principal way in which magma chambers

form. It is certainly the case that upwelling magma from the deep mantle typically

feeds crustal magma chambers, and these are large, long-lived bodies, which can

supply volcanic eruptions for millions of years before the magma chamber ﬁnally

crystallises. The dynamical processes involved in the creation of magma chambers

have not apparently been studied.

An anomalous and extreme example of the magma chamber is the batholith,

which refers to very large (hundreds of kilometres in horizontal dimension) intru-

sions of granite. Their mechanism of formation has been something of a problem to

explain. The reason is that, although granites are clearly formed as igneous rocks,

they are extremely silicic, and thus extremely viscous. They are too viscous to as-

cend via magmafracture (the velocity would be so small that they would freeze), and

the other suggested ascent mechanism, as large, buoyant diapirs, may not be realis-

tic. Another possible mechanism for the formation of granite batholiths is that they

form at the base of the crust, when a mantle plume melts the crust (a process called

anatexis), forming a hybrid, viscous magma. The consequent crystallisation forms

the batholith, which is subsequently exposed at the surface after relentless erosion.

Erosion of a millimetre a year allows 30 kilometres of erosion in 30 million years,

which is the right kind of rate to promote the production of surface batholiths.

9.6.2 Nucleation and Crystallisation

The emplacement and subsequent crystallisation of a magma chamber is a little

more complicated than the solidiﬁcation of a single component material, as for ex-

ample in the freezing of water. For pure liquids, we should expect a rind of crystals

9.6 Crystallisation in Magma Chambers 581

to grow inwards from the walls as cooling proceeds. In a multi-component liquid, a

good deal more goes on, and we can use our understanding of the solidiﬁcation of

castings in metal alloys to explain this.

The basics may be understood by reference to a two component mixture. The

thermodynamic phase diagram (like those extensively discussed in Sect. 9.3) causes

the crystallisation of one substance at the expense of the other, and thus the crys-

tallisation process releases ﬂuid of different composition to the bulk, and thus of

different density. The release of this ﬂuid at the walls of the chamber will therefore

cause compositional convection to occur. The form of this compositional convection

may be compromised by the thermal convection which will occur due to cooling at

the walls, and is discussed in the following Sect. 9.6.3.

For most substances, the thermal diffusivity is much larger than the composi-

tional diffusivity, κ D, and this causes crystal growth to occur in a mixed phase

region, a mush, where solid and liquid coexist. In the laboratory, the mush is typ-

ically formed of dendritic crystals, because growth preferentially occurs on pre-

existing crystal surface. In a magma chamber, growth is so slow that this may not

be true.

In fact, in a casting, convective currents can allow the splintering of small crystal

fragments, which can then circulate in the bulk liquid, forming seeds for the growth

of so-called equiaxed crystals. In the ﬁnal casting, one typically ﬁnds an outer den-

dritic or columnar rind surrounding an equiaxed core.

The principal difference between a crystallising magma chamber and a metallur-

gical casting lies in the chemical complexity of the magma, and a consequence of

this is that the actual processes of crystal nucleation and growth are extremely slow,

and in particular, the common laboratory assumption of thermodynamic equilibrium

is not applicable. Crystal growth requires an undercooling.

The classical theory of non-equilibrium crystal growth is due to Avrami, and uses

a concept known as the ﬁctive volume. Suppose that the crystal nucleation rate per

unit volume in a liquid is I , and the crystal surface growth rate (i.e., the surface

velocity of advance) is Y , both of these being functions of temperature (and thus

time, as the medium cools). A crystal nucleated at time t =τ has a volume

v =

4π





Y(θ)dθ



(9.135)

at time t (assuming they are spheres: other crystal shapes will simply modify the

pre-factor in (9.135)). In the time interval (τ, τ + dτ) there are I(τ)dτ crystals

generated per unit volume, and therefore the ﬁctive crystal volume per unit volume,

i.e., the ﬁctive crystal volume fraction, φ



,is



4π



I(τ)





Y(θ)dθ



dτ, (9.136)

whence we ﬁnd

∂φ



∂t

=4π



I(τ)Y(t)





Y(θ)dθ



dτ. (9.137)

582 9 Magma Transport

The crux of the matter lies in the observation that this ﬁctive volume fraction

of crystals represents the quantity of crystals that would be obtained if we allow

nucleation and growth of crystals within as well as outside crystals. This is clearly

incorrect, but a simple adjustment allows computation of the true crystal volume

fraction φ. Assuming the crystals are nucleated randomly in space and time, then

only that proportion of the change in ﬁctive volume δφ



in a small time interval δt

which occurs in the intercrystal liquid will contribute to a change in the true crystal

fraction. Thus, apparently,

δφ =(1 −φ)δφ



, (9.138)

and therefore the evolution equation for the crystal fraction is

∂φ

∂t

=4π(1 −φ)Y(t)



I(τ)





Y(θ)dθ



dτ. (9.139)

The simplest solution of this equation occurs at constant temperature and liquid

composition, when it is reasonable to take I and Y as constant. With φ =0att =0,

the solution is

φ =1 −exp



−



. (9.140)

The time scale for solidiﬁcation is t ∼ (

)

1/4

. The function given by (9.140)is

sigmoidal, and approaches one fairly promptly at t ≈(

1.8

)

1/4

. However, an inaccu-

racy of the model is that φ only tends to one as t →∞. It is fairly obvious that when

φ approaches one, the intercrystal liquid will be conﬁned to small tetrahedral pock-

ets, and the speciﬁc interfacial surface area per unit volume will be σ ∼(1 −φ)

2/3

Thus for φ ≈1,

∂φ

∂t

=σY ∼(1 −φ)

2/3

Y, (9.141)

and thus

φ ≈1 −kY

−t)

(9.142)

for some ﬁnite t

The difference between (9.140) and (9.142) lies in the way in which speciﬁc

surface area is deﬁned. In fact, (9.139) can be written in the form

∂φ

∂t

=(1 −φ)σ



Y, (9.143)

where σ



is the ﬁctive speciﬁc surface area. Since, evidently,

∂φ

∂t

=σY, it is clear

that the critical assumption (9.138) is equivalent to

σ =(1 −φ)σ



, (9.144)

and this tacitly assumes that surface area has the same distribution as volume. This

may be true for small enough φ, but it cannot be true as φ →1. Despite this, (9.139)

is commonly used.

9.6 Crystallisation in Magma Chambers 583

Undercooling The rates of nucleation I and growth Y depend on the degree of

undercooling below the liquidus temperature T

. The formation of a solid crystal of

volume V is associated with a free energy change of magnitude

G =σA−

VG

, (9.145)

where V

is the liquid molar volume, G

is the bulk free energy difference be-

tween solid and liquid, σ is the interfacial surface energy, and A is the crystal sur-

face area. As with propagating fractures, there is a penalty for creating new sur-

face energy, which is offset by the reduction in bulk free energy in forming the

solid nucleus. However, since A ∝ r

and V ∝ r

, where r is the radius of an as-

sumed spherical nucleus, we can see that G increases for small r to a maximum

at r =r

2σV

G

, before decreasing for larger r. There is thus an energy barrier to

nucleation: nuclei larger than r

will continue to grow, but those smaller than r

will shrink, and the creation of sufﬁciently large nuclei is associated with thermal

ﬂuctuations. In keeping with the precepts of statistical mechanics, the rate at which

such large ﬂuctuations occur is of Arrhenius form, with an activation energy given

by G

=G(r

), and this suggests that the nucleation rate will be of the form

I =K exp



−

G



. (9.146)

The activation energy G

is itself deﬁned by

G

16πσ

3G

, (9.147)

and the free energy change is related to the undercooling T =T

−T . In a single

component material, we have

G

≈S T , (9.148)

where S (> 0) is the entropy drop on crystallisation. We thus see that we may

assume that G

∝

T

. As a consequence, the nucleation rate increases rapidly

with undercooling, and in practice, there is a critical undercooling T

such that

nucleation is effectively absent for T

∼

T

At larger undercooling, other factors become important. In particular, the rate

of transport of molecules (by diffusion) to the nuclear interface is itself thermally

activated, and this has the effect of making the pre-factor K have an Arrhenius

dependence also. If the activation energy for diffusion is H , then we may write

the nucleation rate in the form

I =K



exp



−

H



exp



−



RT (T )



, (9.149)

584 9 Magma Transport

Fig. 9.17 Nucleation rate

(arbitrary units) given by

(9.149)usingvalues

=1200 K, K



=10

H

=2 ×10



=10

where



16πσ

3(S)

. (9.150)

Figure 9.17 shows a typical form for the nucleation rate, which is consistent with

experimentally measured rates. In particular, peak nucleation rates often occur at un-

dercoolings of the order of hundreds of degrees Kelvin, and critical undercoolings

T

can be of the order of tens of degrees. Experimental measurements normally

give a quenched nucleation density of typical order > 10

−3

, indicating that nu-

cleation is relatively easy in silicate melts. The critical undercooling is associated

with the very large values of



, and in fact it is appropriate to deﬁne the undercool-

ingsothatG



=RT

(T

)

(see Question 9.9).

Interfacial Growth Similar sorts of ideas describe the rate of crystallisation at a

pre-existing crystal interface. Essentially, crystal growth is an adsorption–desorption

process, with rates of both being thermally activated. We write

Y =R

−R

, (9.151)

where R

and R

are interfacial attachment and detachment rates. Attachment is

thermally activated in a similar way to the transport involved in nucleation, and we

write



exp



−

G



, (9.152)

where G

is an appropriate activation energy for diffusive transport. Most simply,

would be given by a similar expression, but for detachment, there is the addi-

tional energy barrier to overcome associated with the change in bulk free energy.

Thus we put



exp



−

G



exp



−

G



, (9.153)

9.6 Crystallisation in Magma Chambers 585

Fig. 9.18 Growth rate

(arbitrary units) given by

(9.154)usingvalues

=1200 K, K



=10

G

=2 ×10

S

where G

isgivenby(9.148) as before, and the pre-multiplicative constant K



must be the same if Y =0 when T =T

. This leads us to the growth rate in the form

Y =K



exp



−

G



1 −exp



−

ST



. (9.154)

The growth rate is similar to the nucleation rate, in the sense that there is a peak

at a certain undercooling, but there is no critical undercooling. Figure 9.18 shows a

representative illustration. As for nucleation, growth rates of silicates typically peak

at undercoolings of O(100) K, and the typical peak growth rates are of the order

of 10

−6

–10

−8

−1

, with the lower values at lower liquidus temperatures. This

indicates peak growth rates of the order of a metre a year for silicate melts, and

suggests that crystallisation will normally occur via the slow growth of crystals in a

crystal slurry.

The basic problem to study is then the crystallisation of an initial emplacement of

magma, by analogy with the simplest Stefan problem of interfacial growth. We con-

sider one-dimensional growth of a crystal mush from an initially emplaced magma

in x>0 next to cold country rock in x<0. In the magma, the temperature will

satisfy the heat conduction equation

∂(ρc

∂t

∂

∂x



∂T

∂x



+LS, (9.155)

where

ρc

=ρ

φ +ρ

(1 −φ), (9.156)

and ρ

are solid and liquid densities, c

and c

are solid and liquid speciﬁc heats

at constant pressure,

k =φk

+(1 −φ)k

(9.157)

is the thermal conductivity, k

and k

being those of solid and liquid, respectively,

and L is the latent heat per unit mass. S is the rate of formation of crystal, given by

586 9 Magma Transport

(9.139):

∂φ

∂t

=4π(1 −φ)Y(t)



I(τ)





Y(s)ds



dτ. (9.158)

We assume the solid and liquid densities, thermal conductivities and speciﬁc

heats are constant and equal, denoted ρ, k and c

, respectively, and we deﬁne the

thermal diffusivity to be κ =

ρc

. Then we have

∂T

∂t

=κ

∂

∂x

∂φ

∂t

∂φ

∂t

=4πN(1 −φ)Z

∂Z

∂t

=Y,

(9.159)

where Z =



Y(s)ds, and we have assumed that all the nucleation occurs instantly

at t =0, creating N nuclei per unit volume. We have

φ =Z =0att =0, (9.160)

and thus

1 −φ =exp



−

4πNZ



, (9.161)

whence the model reduces to

∂T

∂t

=κ

∂

∂x

4πNZ

Y exp



−

4πNZ



∂Z

∂t

=Y,

(9.162)

and the growth rate Y is a given function of T . Suitable boundary and initial condi-

tions are that

T =T

∞

,Z=0att =0,

T =T

at x =0,

T =T

∞

as x →∞.

(9.163)

If T

is the liquidus, we deﬁne the undercooling

θ =T

−T, (9.164)

9.6 Crystallisation in Magma Chambers 587

and we suppose Y is a function of θ . It is convenient to non-dimensionalise the

model by choosing scales

θ ∼T

,x∼d, t ∼

≡[t],

Z ∼



4πN



1/3

≡[Z],Y∼

[Z]

[t]

(9.165)

where we suppose the peak growth rate is Y

is at an undercooling of T

. Sup-

posing that T

∞

−T

∼T

, it is also natural to scale Y ∼Y

, and we can enable

this by choosing the length scale d so that [Z]=[t]Y

, i.e.,

d =





1/2

(4πN)

1/6

. (9.166)

The dimensionless model then becomes

=θ

−StZ

Y exp



−



=Y,

(9.167)

with initial and boundary conditions

θ =θ

∞

,Z=0att =0,

θ =θ

at x =0,

θ =θ

∞

as x →∞,

(9.168)

where the Stefan number is

St =

T

(9.169)

and

−T

T

,θ

∞

−T

∞

T

. (9.170)

We use typical values L = 3×10

Jkg

−1

, c

=10

Jkg

−1

, κ =10

−6

−1

T

= 100 K, N = 10

−3

(based on an eventual crystal size of 1 mm),

= 10

−7

−1

, to ﬁnd d ∼ 6.6cm,[t]∼1 hour, St ≈ 3. The crystals grow

in a fairly thin skin.

The key to understanding the solution lies in the observation that the width of

the skin, d ∼ 10 cm, is much less than the macroscopic dimension d

of a magma

chamber, so that on the large scale, the skin appears as an interface, and we might

expect the usual interfacial Stefan condition to emerge in the limit d d

And indeed, this is the case. If we write (9.167) in the dimensionless form

=θ

−Stφ

(9.171)

588 9 Magma Transport

(noting that φ = 1 −exp(−

) is dimensionless), then integration on the macro-

scopic scale of the integral form



(θ +St φ)dx =[θ

]

(9.172)

from s− to s+ (where these denote the limits as x → s from x<s and x>s,

respectively, on the macroscale, and s is the ‘location’ of the skin (e.g., where

φ =

))

gives the usual dimensionless Stefan condition

St ˙s =[θ

]

−

(9.173)

for the solution of the macroscale temperature ﬁeld. Note that this assumes that θ is

continuous across the interface.

The skin acts as a ‘shock’ over which φ changes rapidly, and we thus expect that

solutions of (9.167) will tend to travelling waves which accommodate this transition

(and that the front speed will be given by (9.173)). We put

ξ =x −ct, (9.174)

and look for solutions in which θ and Z are functions of ξ . These functions then

satisfy the ordinary differential equations

−cθ



=θ



−StZ

Y exp



−



−cZ



=Y(θ),

(9.175)

together with the (matching) boundary conditions



→g

,Z→0asξ →∞,



→g

−

as ξ →−∞,

(9.176)

where g

and g

−

denote the limiting macroscopic values of θ

and θ

s−

For simplicity, suppose that the emplaced magma is at the liquidus, so that θ →0

as ξ →∞, and g

=0. Suppose also that Y(θ)=θ . Then (9.175) has a ﬁrst integral



=−cZ



+St



1 −exp



−



, (9.177)

using the fact that Z → 0asξ →∞. The solutions can be analysed in the (Z, W )

phase plane, where



=W,



=−cW +St



1 −exp



−



(9.178)

More precisely, take A =s −η, B = s +η, and then let η →0withη d.

9.6 Crystallisation in Magma Chambers 589

Fig. 9.19 Phase portrait of

(9.178)forvaluesSt =3,

c =1

We are interested in trajectories which approach (0, 0) from Z>0, W<0, and

there is a unique such trajectory, which is the stable separatrix to the saddle point at

the origin. Figure 9.19 shows the phase portrait.

It is clear that as ξ →−∞, Z →∞, and thus W



∼ St − cW , and W grows

exponentially. This is not apparently consistent with (9.173), which requires in the

present case c =

−

. Since g

−

=θ



=−cZ



=−cW



at ξ =−∞, we see that this is

only consistent with (9.178)ifc is small. In fact (see Question 9.9), this is precisely

the case when the travelling wave solution should be valid, since c must be small in

order that it match to the outer solution near the wall of the chamber.

9.6.3 Double-Diffusive Convection

Crystals in magma chambers can hardly grow in a stagnant medium, however, since

convective processes will inevitably occur. In particular, because magmas are multi-

component ﬂuids, such convection is likely to be double-diffusive, and we now dis-

cuss this subject. Double-diffusive convection refers to the buoyant motions driven

by density differences of a ﬂuid which are induced by two competing substances.

Most commonly, these are temperature (T ) and composition (c), and their effect on

the density is represented by the relation

ρ =ρ



1 −α(T −T

) +βc



, (9.179)

where conventionally we think of increasing composition causing an increased den-

sity, as is the case in the common laboratory case of salt dissolved in water.

There are now two Rayleigh numbers: ﬁrst, there is the conventional thermal

Rayleigh number

Ra =

αρ

gT d

ηκ

, (9.180)

where d is the depth of the ﬂuid layer, T the prescribed temperature excess (base

over top), η the viscosity, and κ the thermal diffusivity. In addition, there is a com-