Fowler A. Mathematical Geoscience

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610 9 Magma Transport

By choosing l =

√

C and deﬁning suitable scales for t and h, show that the

model can be written in the dimensionless form

+(h

)

xxt

+ε



−r







where

ε =

2ρ



kη



1/2

Show that ε  1, and deduce that h is approximately described by the modi-

ﬁed BBM equation

+3h

xxt

9.5 A pod of melt in a viscous porous medium is described by the scaled, gener-

alised BBM equation

+3h

xxt

in which we assume that h → 0asx →±∞. Show that



∞

−∞

hdx = M is

conserved, and that travelling wave solutions exist of the form h =φ(η) with

η =x −ct, providing

2

=φ

−

,φ(±∞) =0.

Deduce that φ =

√

2c sechη, and thus that c =

2π

Show also that if one only requires h to be bounded at inﬁnity, then there is

a one parameter family of periodic solutions for each c>0.

9.6 A magma-ﬁlled crack L of width h propagates through the lithosphere, and is

described by the dimensionless equations

∂h

∂t

∂

∂x





1 +δ

∂Π

∂x



=0,

Π =

2π

−



∂h

∂s

s −x

where Π is the underpressure in the crack, which is assumed to lie in −l<x<

l, and x points vertically upwards. Essentially, we view x =0 as the base of

the lithosphere, so that the physical crack is in x>0, with its virtual extension

in x<0 representing conditions of symmetry about x =0.

Show that exact similarity solutions can exist in terms of a similarity vari-

able η = xt

1/3

, and write down the equations for the similarity functions H

and P corresponding to h and Π , respectively. Explain why such a similarity

solution is unlikely to represent a physically appropriate solution.

The equation is closely related to that studied by Benjamin et al. (1972) as a model for long waves

in shallow water. We may infer that the present equation for h is well-posed.

9.9 Exercises 611

Show that if δ is small and is ignored, then a similarity solution exists for

input ﬂuxes q|

x=0+

= h

= t

, and show that the appropriate similarity vari-

able is η =

, and ﬁnd β in terms of ν. Find the equation satisﬁed by H(η),

the appropriately rescaled deﬁnition of h. By writing H =η

1/2

G and τ =lnη,

show that G



(τ ) =F(G), where you should deﬁne the function F . Show that

a solution is possible in which G ∼

1/2

for large η, and thus we can sup-

pose H = 1atx = 0. In this case, show that if 0 <β<1, then H ∼



η →∞, while if β>1, H ∼ H

+(η

−η)

1/2

as η →η

for some ﬁnite η

Deduce that a physically meaningful solution of the model, in which a ﬁnite

crack grows, only exists if ν>0, and give an expression for the crack front as

a function of time.

9.7 The width h of a crack in an elastic medium and its underpressure Θ are

determined near the crack tip by the dimensionless equations, on [0, ∞],

=1 −

Θ =

−



∞

∂h

∂ξ

dξ

ξ −X

subject to

h ∼2λ

√

X as X →0,

h →1asX →∞.

Use complex variable methods to invert the integral equation, and by integrat-

ing the result, ﬁnd an integral expression for h. Using integration by parts,

derive the nonlinear integral equation for h,

h(X) =



∞

K(x,ξ)



−1



dξ,

where

K(x,ξ) =(x −ξ)ln



√

x +

√

x −

√



−2



xξ.

Deduce that if h ∼2λ

√

X as X →0, then

λ =



∞





−1



dξ.

Show also that

h ∼1 +

2πX

as X →∞,

and deduce that the fracture has a bulbous nose. Sketch the expected form of

the shape (see also Spence et al. 1987).

612 9 Magma Transport

9.8 According to Muskhelishvili (1953, pp. 73 ff.), the behaviour of the Cauchy

integral

Φ(z) =

2πi



−l

φ(t)dt

t −z

near the end z =l is given by

Φ ∼−

∗

cosecγπ

2i(z −l)

φ ∼

∗

(l −x)

as x →l with 0 <γ <1 (and φ

∗

is constant).

Show also that if

Ψ(z)=

2πi



−l

ψ(t)dt

t −z

and ψ(l) =0, then



(z) ∼

2πi



−l



(t) dt

t −z

+O(1)

as z →l.

As x →l+ on y =0, we have σ

−iτ ∼2Ω



, where

Ω ∼

2πi



−l

2iμ

1 +κ

h(s) ds

s −z

in which κ =

λ+3μ

λ+μ

, and also ν =

2(λ+μ)

. Show that if h ∼ c(l − x)

1/2

x →l−, then

(x, 0) ∼

√

2π(x −l)

as x →l+,

where

K =

μc

√

2(1 −ν)

9.9 The crystal nucleation rate of a liquid is given by

I =K



exp



−

H



exp



−



RT (T )



where K



is a rate constant, G



and H are activation energies, T is temper-

ature, and T = T

−T is the undercooling below the liquidus temperature

. By writing

T =T

−(T

)θ,

9.9 Exercises 613

show that I can be written in the form

I =k



exp



−

δθ

ε(1 −δθ)



exp



−

(1 −δθ)



provided we choose

T







1/2

and deﬁne the parameters k



, δ and ε.

Use the values T

= 1200 K, K



= 10

H

= 2 ×10



= 10

Kto

ﬁnd typical values of T

, ε, δ and k



, and hence show that δ ε, and thus

that

I ∼k



exp



−



if θ ∼O(1),

I ∼k



−Θ

, if Θ ≡

δθ

∼O(1).

Write down a uniform approximation for I .WhyisT

a measure of the

critical undercooling?

9.10 The supercooling θ below the liquidus in a magma chamber satisﬁes the equa-

tion

=θ

−St φ

where

φ =1 −exp



−



=H(θ)Y(θ),

and H(θ) is the Heaviside step function. The initial and boundary conditions

are

θ =θ

∞

≥0,Z=0att =0,

θ =θ

< 0atx =0,

=0atx =

Assume that θ

,θ

∞

, St ∼O(1), ε 1. By writing x =

, t =

, Z =

show that a suitable outer approximation for θ and z is given by

=θ

=H(θ)Y(θ),

in 0 <X<S(T)and S(T ) < X < 1, where the outer solution satisﬁes θ =

z =0onX =S. Show that the outer solution for z in X<Sis

z =



−1

(X)



θ(X,T



)



614 9 Magma Transport

where S

−1

is the inverse function of S. What is the outer solution for z in

X>S?

For T close to (and greater than) S

−1

(X), show that

z ≈



T −S

−1

(X)



X=S−

and deduce that a suitable rescaling near X =S is obtained by writing

X =S(T ) +εξ, θ =εΘ, z =ε

and that the inner problem can be written, to leading order, as

0 =Θ

ξξ

+St

Sφ

−

= H(Θ)y(Θ),

where we write Y(εΘ)=εy(Θ), supposing that y ∼O(1).

Show that the matching conditions for Θ and Z are

Θ ∼g

−

ξ, Z ∼

X=S−

as ξ →−∞,

Θ ∼g

ξ, as ξ →∞,

(∗)

where g

=θ

X=S±

[To compute the condition on Z, you will need to show that for 0 <T −

−1

(X) 1,

T −S

−1

(X) ≈−

εξ

Hence show that S satisﬁes the Stefan condition

S =[θ

]

−

Suppose that y(Θ) =Θ. Show that Z satisﬁes

−

ξξ

−

+St

S exp



−



and explain why, if g

> 0, appropriate boundary conditions are

Z =Z

=0atξ =0.

Show further that the solution for Z satisﬁes the second condition in (∗) au-

tomatically (remember that θ[S(T ),T ]≡0). What are appropriate conditions

if g

=0?

Sketch or compute the solution for Z.

9.9 Exercises 615

9.11 Dimensionless equations describing double-diffusive convection are given by

∇.u = 0,



+(u.∇)u



=−∇p +∇

u +Ra T

k −Rs c

+u.∇T =∇

+u.∇c =

∇

where u is the ﬂuid velocity, p is pressure, T is temperature, c is concentration,

and the parameters are the Rayleigh number Ra, the solutal Rayleigh number

Rs, the Prandtl number Pr and the Lewis number Le.

Assuming a two-dimensional ﬂow, so that there is a stream function ψ

satisfying

u =(−ψ

,ψ

suppose that the ﬂow is driven by prescribed temperature and salinity differ-

ences between the upper and lower surfaces of a stress-free box, so that the

boundary conditions take the form

ψ =∇

ψ =0atz =0, 1,

T =c =0atz =1,

T =c =1atz =0.

Write down the steady state in which u = 0, and let C and θ denote per-

turbations to c and T , respectively. Assuming C,θ,ψ  1, write down the

linearised equations and boundary conditions for the ﬂow. By seeking solu-

tions proportional to exp(ikx +σt)sin mπz, show that



σ +K



σ +K





σ +





(Rs −Ra)σ

+Rs −



=0,

where

Show that this can be written in the form

+aσ

+bσ +c = 0,

where

a = K



Pr +1 +



b =K



Pr +



Pr(Rs −Ra),

616 9 Magma Transport

c =

Pr +k



Rs −



Deduce that direct instability (σ>0) occurs if

Ra −Le Rs >R

27π

and oscillatory instability occurs if

Ra >

(Pr +λ)Rs

1 +Pr

(1 +λ)(Pr +λ)

where λ =

Chapter 10

Glaciers and Ice Sheets

Glaciers are huge and slow moving rivers of ice which exist in various parts of the

world: Alaska, the Rockies, the Alps, Spitzbergen, China, for example. They drain

areas in which snow accumulates, much as rivers drain catchment areas where rain

falls. Glaciers also ﬂow in the same basic way that rivers do. Although glacier ice is

solid, it can deform by the slow creep of dislocations within the lattice of ice crystals

which form the fabric of the ice. Thus, glacier ice effectively behaves like a viscous

material, with, however, a very large viscosity: a typical value of ice viscosity is

6 bar year (in the metre–bar–year system of units!). Since 1 bar = 10

Pa, 1 year

≈3 ×10

s, this is a viscosity of some 2 ×10

Pa s, about 10

times that of wa-

ter. As a consequence of their enormous viscosity, glaciers move slowly—a typical

velocity would be in the range 10–100 m y

−1

(metres per year), certainly measur-

able but hardly dramatic. More awesome are the dimensions of glaciers. Depths

of hundreds of metres are typical, widths of kilometres, lengths of tens of kilome-

tres. Thus glaciers can have an important effect on the human environment in their

vicinity. They are also indirect monitors of climate; for example, many lithographs

of Swiss glaciers show that they have been receding since the nineteenth century,

a phenomenon thought to be due to the termination of the ‘Little Ice Age’ which

lasted from about 1500 to about 1900.

Where glaciers are the glacial equivalent of rivers, i.e. channelled ﬂow, ice sheets

are the equivalent of droplets, but altogether on a grander scale.

When an entire

continent, or at least a substantial portion thereof, has a polar climate, then snow

accumulates on the uplands, is compressed to form ice, and ﬂows out to cover the

continent, much as a drop of ﬂuid on a table will spread under the action of gravity.

However, whereas droplets can reach a steady state through the contractile effect of

surface tension, this is not relevant to large ice sheets. In them, equilibrium can be

maintained through a balance between accumulation in the centre and ablation at

the margins. This can occur either through melting of the ice in the warmer climate

Ice caps are smaller scale sheet ﬂows, such as the Vatnajökull ice cap in Iceland, whose horizontal

dimension is about 100 km.

A. Fowler, Mathematical Geoscience, Interdisciplinary Applied Mathematics 36,

617

618 10 Glaciers and Ice Sheets

at the ice margin, or through calving of icebergs. (Indeed, the same balance of ac-

cumulation at higher elevations with ablation at lower elevations is responsible for

the normal quasi-steady proﬁle of valley glaciers.)

There are two major ice sheets on the Earth, namely those in Antarctica and

Greenland (the Arctic is an ocean, and its ice is sea ice, rarely more than three

metres thick). They are on the order of thousands of kilometres in extent, and kilo-

metres deep (up to four for Antarctica). They are thus, in fact, shallow ﬂows, a fact

which greatly facilitates the solution of mathematical models for their ﬂow. Possibly

more famous are the ice sheets which covered much of North America (the Lauren-

tide ice sheet) and northern Europe (the Fennoscandian ice sheet) during the last ice

age. Throughout the Pleistocene era (that is, the last two million years), there have

been a succession of ice ages, each lasting a typical period of around 90,000 years,

during which global ice sheet volume gradually increases, interspersed with shorter

(10,000 year) interglacials, when the ice sheets rapidly retreat. The last ice age ﬁn-

ished some ten thousand years ago, so that it would be no surprise if another were

to start now. Perhaps the Little Ice Age was indeed the start of ice sheet build-up,

only to be interrupted by the Industrial Revolution and the resultant global warming:

nobody knows.

Further back in Earth’s geologic history, there is evidence for dramatic, large

scale glaciation in the Carboniferous (c. 300 My (million years) ago), Ordovician

(c. 500 My ago), Neoproterozoic (c. 600–800 My ago) and Huronian (c. 2,500 My

ago) periods. In the Neoproterozoic glaciation, it seems that the whole landmass

of the Earth may have been glaciated, leading to the concept of ‘snowball Earth’.

It was following the shrinkage of the global ice sheet that the explosion of life on

Earth started.

While the motion of ice sheets and glaciers can be understood by means of vis-

cous theory, there are some notable complications which can occur. Chief among

these is that ice can reach the melting point at the glacier bed, due to frictional heat-

ing or geothermal heat input, in which case water is produced, and the ice can slide.

Thus, unlike an ordinary viscous ﬂuid, slip can occur at the base, and this is deter-

mined by a sliding law which relates basal shear stress τ to sliding velocity u

and

also, normally, the effective pressure N = p

− p

, where p

and p

are ice and

water pressures. The determination of p

further requires a description of the sub-

glacial hydrology, and thus the dynamics of ice is intricately coupled to other phys-

ical processes: as we shall see, this complexity leads to some exotic phenomena.

10.1 Dynamic Phenomena

10.1.1 Waves on Glaciers

Just as on rivers, gravity waves will propagate on glaciers. Because the ﬂow is very

slow, they only propagate one way (downstream), and at speeds comparable to the

surface speed (but slightly faster). These waves are known as surface waves, as they

Snowball Earth was discussed in Chap. 2.

10.1 Dynamic Phenomena 619

Fig. 10.1 Changes of mean

surface elevation of Mer de

Glace, France, along four

cross-proﬁles over a period of

9 years. The broken line

corresponds to a wave

velocity of 800 m/a.

Reproduced from Lliboutry

(1958a), by permission of the

International Association of

Hydrological Sciences

are evidenced by undulations of the surface: an example is shown in Fig. 10.1.They

are examples of kinematic waves, driven by the dependence of ice ﬂux on glacier

depth.

A more exotic kind of wave is the ‘seasonal wave’. This has no obvious counter-

part in other ﬂuid ﬂows. It consists of (sizeable) perturbations in the surface velocity

ﬁeld which propagate down-glacier at speeds in the order of 20–150 times the sur-

face speed. There is no signiﬁcant surface perturbation, and these waves must in

fact be caused by variations of the basal sliding speed due to annual ﬂuctuations in

the basal water pressure. Although well known and reported at the turn of the cen-

tury, little attention has been paid to these waves in recent years. Figure 10.2 shows

measurements of Hodge on Nisqually Glacier which indicates the rapid passage of

a velocity wave downstream.

Mention should also be made of wave ogives, although we will not deal with

them here. They are bands (also known as Forbes bands) which propagate below

ice-falls, and are due to the annual ablation cycle.

10.1.2 Surges

Perhaps the most spectacular form of wave motion is the glacier surge. Surges are

large scale relaxation oscillations of the whole length of a glacier. They are roughly