Fowler A. Mathematical Geoscience

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530 8 Mantle Convection

that of the chemical structure of the mantle. There are two aspects to this. The most

fundamental is that as depth increases in the mantle, there are a number of phase

changes which occur. The upper mantle below the lithosphere is largely thought to

consist of olivine, (Mg,Fe)

SiO

, and this undergoes a transformation to a spinel

phase between 400 and 500 km depth. A further transition occurs at 650 km, where

the spinel dissociates into a perovskite phase (Mg,Fe)SiO

and wüstite (Mg,Fe)O.

Depending on the exact composition, other phase changes may occur at different

pressures. In the mantle, the presence of these phase changes is detected seismi-

cally, and they are associated with density increases of several percent (Anderson

2007).

(Some) descending lithospheric slabs clearly sink (at least) to the vicinity of the

650 km seismic discontinuity, so a reasonable initial simplifying assumption may

be that of a two layer mantle, with the olivine in the upper mantle separated from

the perovskite lower mantle. The simplest consequence would seem to be that con-

vection might occur separately in two, or possibly more, layers. In fact, this idea

underlies the original concept of shallow mantle convection, and underpinned the

choice of depth scale in models such as that of Turcotte and Oxburgh (1967). Later

investigations raised doubts that the density jump would be sufﬁcient to prevent

whole-mantle convection, and many studies now assume this. The issue revolves

round the magnitudes of the relative density jump across a phase change boundary,

ρ

, the corresponding buoyancy term αT , and the slope of the Clapeyron curve

relating the phase change pressure to the temperature. At least for the descending

lithospheric slabs, we may take T ∼ 10

K, and thus αT ∼ 3 × 10

−2

.This

may be comparable to the density increase across the 650 km boundary. Numerical

studies (Christensen and Yuen 1984, 1985) suggest that at least some form of pene-

tration is likely, and this is consistent with ideas of hot spots associated with plumes

originating at the core–mantle boundary.

However, layered convection seems the easiest explanation for the inference,

from geochemical studies of different magmas at the surface of the Earth, that the

mantle consists of at least two, and possibly more, distinct reservoirs, which have

been chemically isolated for much of the Earth’s history (Tackley 2009). This in-

ference is based on the different trace elements present in the erupted magmas.

Mid-ocean ridge basalts (MORB) are depleted in the so-called incompatible trace

elements, while the continental crust is enriched in these same elements, which leads

us to suppose that the continents form as the residue from continual melting of the

MORB source region. Mass balance calculations suggest that this source only oc-

cupies around half the mantle (the estimates vary a good deal). The easiest vision

is to suppose that the phase change at 650 km causes a form of ‘leaky’ (see below)

layered convection, thus providing the separate mantle reservoirs.

However, this simple picture is increasingly complicated by further geochemi-

cal signatures. One such is that ocean island basalts (OIB, such as Hawaii) have

anomalous helium isotope ratios, suggesting that they originate from a primordial

reservoir. Most simply, this is construed to be the lower mantle, and that is consistent

with the idea that they come from mantle plumes, which presumably penetrate the

barrier at 650 km. Various other complications arise, and have led to various differ-

ent conceptualisations of how convection works in practice. It is important to note

8.8 Notes and References 531

that these discussions of geochemical signatures ignore the possible importance of

transport processes associated with eruptions.

Two further seismic zones warrant mention. The ﬁrst is a low velocity zone just

below the lithosphere. The easiest interpretation for this is that it represents the con-

vective overshoot in temperature which typically arises in high Rayleigh number

convection, possibly associated with the presence of partial melt. The other anoma-

lous region is the D



layer of some 200 km thickness above the core–mantle bound-

ary. There are several ways this layer can be construed. One is that it arises from

a phase change from perovskite to a ‘post-perovskite’ phase. Since the D



layer is

very variable in thickness, this may not be sufﬁcient. A dynamical interpretation is

that it represents the remnants of foundered subducted lithospheric slabs; this is at-

tractive to some parts of the geochemical story, as well as to seismological inference

that slabs do in some places penetrate to the lower mantle. Such penetration does not

argue against an essentially layered style of convection, since the negative buoyancy

associated with slabs is much larger than elsewhere in the mantle. Foundering slabs

also suggest (à la Howard) a nice explanation for the semi-regular massive ﬂood

basaltic eruptions associated with superplumes (Courtillot 1999; Yuen et al. 2007).

Lastly, the interface between the molten iron (or iron oxide) of the core and the

mantle is a surface of phase change with an associated phase diagram which must

describe the reactions which occur there. If we suppose that the primordial Earth

consists of a liquid iron Fe core surrounded by a perovskite (Mg,Fe)SiO

mantle,

then Knittle and Jeanloz (1991) found experimentally that reactions occurred in

which FeO and SiO

were formed:

(Mg

1−x

)SiO

→a

MgSiO

FeO

FeSi

SiO

; (8.310)

here a

are various stoichiometric coefﬁcients. As with all such reactions where the

solidus and liquidus temperatures depend on pressure and concentrations, it is possi-

ble or likely that a region of mixed phase may exist in the solid. Knittle and Jeanloz’s

suggestion is that such a putative region corresponds to the D



layer. This reaction

incidentally provides a nice reason why the liquid outer core of the Earth is an alloy

of iron with some lighter element. The core–mantle boundary reaction produces the

iron oxide which dissolves in the outer core, while the inner core solidiﬁes at its

expense, releasing oxygen-rich liquid which will tend to pond at the top of the core

(because the motion is so sluggish).

How does all this tie in to our notion of variable viscosity convection? For stag-

nant lid convection, the lid thickness is quite substantial, both in numerical compu-

tations and as predicted by (8.178) and (8.184), and the whole upper mantle might

well be in the stagnant lid, so that the phase change would have little effect. And

if, when subduction is initiated, the slabs do indeed penetrate the lower mantle in

places, then this will induce (locally) rapid convection in the lower mantle, as well

as deposition of the slab at the base, and the subsequent eruption of a superplume.

One objection to layered convection which has been made in the past is that ‘there

would be a strong thermal boundary layer at the base of the upper mantle’ (Tackley

2009). This is an unfounded inference based on constant viscosity convection, but

532 8 Mantle Convection

inappropriate for variable viscosity convection. Indeed, it seems that variable vis-

cosity convection with subduction can provide a picture which is fairly consistent

with most of the basic features which have been observed.

Howard’s 1966 Convection Paper Lou Howard’s vision of convection (Howard

1966), discussed in Sect. 8.7, was presented at the International Congress of Applied

Mechanics in Münich in 1964. For the most part it is a discursive review of earlier

work on turbulent convection by Malkus, Spiegel, and others, now rather dated. It

is only in the ﬁnal two pages that Howard advances his conceit of turbulent thermal

convection as consisting of long, quiescent phases where thermal boundary layers

grow from the heated and cooled surfaces into the interior, interrupted by rapid con-

vective eruption of plumes as the boundary layers become unstable. As a paradigm,

it is essentially identical to the concept of episodic Venusian convection.

Howard’s conception, together with the publication in 1963 of Lorenz’s famous

paper (Lorenz 1963), led to a very productive sequence of research ideas based

round the activity at the GFD summer program at the Woods Hole Oceanographic

Institution. In particular, Howard and Malkus invented their famous water wheel

(e.g., Matson 2007), which provides a bridge between the Lorenz equations and the

essentially similar behaviour in the Howard convection model.

Malkus’s experi-

ment is described by Sparrow (1982), and an attempt to draw an explicit compar-

ison between Howard’s rough description and a formal asymptotic solution of the

Boussinesq convection equations was made by Fowler (1992b).

8.9 Exercises

8.1 It is required to show that a vector potential A for an incompressible velocity

ﬁeld u in a domain D can be deﬁned so that

u =curl A, div A =0.

Suppose that ψ is chosen to satisfy

∇

ψ =−u in D,

with boundary condition

∇.ψ =0on∂D.

Using the Cartesian identity

∇

≡grad div−curl curl,

Indeed, one can show that the behaviour of the Lorenz equations in the limit of large ‘Rayleigh’

and Prandtl numbers r and σ is equivalent to Howard’s physical description (Fowler and McGuin-

ness 1982).

8.9 Exercises 533

show that ∇

(∇.ψ) =0inD, and deduce that ∇.ψ =0. Deduce that a suitable

vector potential is A =∇×ψ.

Show that if D = R

, the vorticity is ω = curlu and u → 0as|r|→∞,

then

u =

4π



curl ω(r



)dV(r



)

|r −r



Show also that a general representation of the velocity is of the form

u =curl ψj + curl curl χj,

and derive the form of the Boussinesq equations in terms of ψ,χ and T .

8.2 Explain how high Rayleigh number, variable viscosity convection can be used

to explain the styles of mantle convection on Venus and Earth, and use it to

explain and interpret the following observations on the Earth:

volcanism occurs at plate boundaries;

earthquakes occur at plate boundaries;

the continents drift relative to each other;

earthquakes occur regularly on the San Andreas fault;

oceanic trenches occur at convergent plate boundaries;

the oceans are relatively shallow at mid-ocean ridges;

black smokers occur at mid-ocean rises;

chains of islands occur in the Paciﬁc ocean (e.g., the Hawaiian islands);

ocean island basalts are distinct from mid-ocean ridge basalts.

8.3 Explain what is meant by post-glacial rebound, and how it can be used to infer

values for the mantle viscosity.

Explain how the mantle can behave like a ﬂuid even though it is solid.

The viscosity of mantle rock is measured to be of the form

η =

2Aτ

n−1

exp



∗

+pV

∗



where typical values are (with large error bars) A =10

MPa

−n

−1

, n =3.5,

∗

= 525 kJ mol

−1

, R = 8.3Jmol

−1

, and V

∗

= 17 cm

mol

−1

.Use

these values to infer likely values of the mantle viscosity at the base of the

lithosphere if T = 1500 K, τ = 10

Pa, and the depth is 100 km (assume

pressure is lithostatic, i.e. p = ρgz, where z is depth, g ≈ 10 m s

−2

, and

ρ ≈ 3.3 ×10

kg m

−3

). Is this value consistent with the post-glacial rebound

value of 10

Pa s? If not, how big would the error bars on E

∗

need to be to

make it consistent?

8.4 Write down the Boussinesq equations describing convection of a constant vis-

cosity ﬂuid heated from below (assume acceleration terms are negligible). Ex-

plain what the Boussinesq approximation means. What are suitable boundary

conditions for convection in the Earth’s mantle? By choosing suitable scales

for the variables, write the model in non-dimensional form, and deduce that the

534 8 Mantle Convection

ﬂow depends only on a single dimensionless parameter, the Rayleigh number,

and deﬁne what this is.

If the acceleration term had been included (thus, dimensionally, ρ

−∇p +···), show that the size of the inertial acceleration terms is given by

1/Pr, where Pr =ηc

/k is the Prandtl number. Assuming c

≈10

Jkg

−1

k ≈ 4Wm

−1

, and η ≈10

Pa s, estimate the importance of the inertia

term.

8.5 Write down a dimensionless set of equations describing Boussinesq convec-

tion of a high Prandtl number, constant viscosity ﬂuid in a horizontal layer of

ﬂuid, and deﬁne the Rayleigh number R.

Using suitable boundary conditions for the ﬂow describing convection in

the Earth’s mantle, show in detail that convection will occur if R>R

, where

you should deﬁne R

Use suitable values for the Earth’s mantle to show that R  R

(assume

α = 3 ×10

−5

−1

, T = 3000 K, ρ = 4 ×10

kg m

−3

, g = 10 m s

−2

, η =

Pa s, κ =10

−6

−1

, and that the depth to the core–mantle boundary is

3000 km).

8.6 The amplitudes of orthogonal sets of weakly nonlinear convective rolls are

described by the equations

A =

9π

A −



+cB



B =

9π

B −



+cA



where an overdot indicates differentiation with respect to the slow time vari-

able τ .

Show by rescaling the variables that the model can be reduced to the form

A = A −A



+cB



B =B −B



+cA



Find the steady states, and calculate their stability in the two cases c<1 and

c>1, and draw the consequent phase planes in each case.

8.7 The slowly varying amplitude of weakly nonlinear convective rolls satisﬁes

the scaled equation

=A −A

where T is a slow time scale and X is a long space scale. Write down the

equation satisﬁed by a travelling wave solution A =f(ξ), where ξ =X −cT .

Assuming c>0 and g =−f



, write down the equations satisﬁed by f

and g, and, by analysing the phase plane, show that there is always a solution

connecting (1, 0) to (0, 0).[It may help to consider the function E =

−

8.9 Exercises 535

Show that if c<2, the origin in the (f, g) plane is a stable focus, while if

c>2, it is a stable node. Deduce that for c<2, travelling wave solutions have

oscillatory tails in which A<0.

Now consider the slope s =

of the chord joining a point on the travelling

wave trajectory in the phase space to the origin. Show that initially s is posi-

tive and increasing. As long as 0 <f <1, show by consideration of

dξ

as a

function of s that s<

, providing c ≥2, and deduce that in this case f (and

thus A) remains positive as it approaches zero.

What is the implication for convection in a large pan, if the motion is initi-

ated locally?

8.8 Describe what is meant by continental drift, and how the theory of plate tec-

tonics and mantle convection can be used to explain it. How do mid-plate

volcanoes, such as Hawaii, ﬁt in with this theory?

The scaled Boussinesq equations for two-dimensional thermal convection

at inﬁnite Prandtl number and large Rayleigh number R in 0 <x<a,0<z<

1, can be written in the form

ω =−∇

ψ,

∇

ω =

−ψ

=δ

∇

where δ = R

−1/3

. Explain what is meant by the Boussinesq approximation,

and explain what the equations represent. Explain why suitable boundary con-

ditions for these equations which represent convection in a box with stress free

boundaries, as appropriate to convection in the Earth’s mantle, are given by

ψ =0,ω=0, on x =0,a, z=0, 1,

T =

on z =0,T=−

on z =1,T

=0onx =0,a.

Show that, if δ 1, there is an interior ‘core’ in which T ≈0, ∇

ψ =0.

By writing 1 −z =δZ, ψ =δΨ and ω =δΩ, show that Ψ ≈u

(x)Z, and

deduce that the temperature in the thermal boundary layer at the surface is

described by the approximate equation

−Zu



≈T

with

T =−

on Z =0,T→0asZ →∞.

The same technique can be used for the Fisher equation.

536 8 Mantle Convection

If u

is constant, ﬁnd a similarity solution, and show that the scaled surface

heat ﬂux q =∂T/∂Z|

Z=0

is given by

q =



πx

8.9 Suitable equations to describe mantle convection in a radial geometry are

∂(ru)

∂r

∂w

∂z

=0,

∂p

∂r

∂(rτ

)

∂r

∂τ

∂z

+τ

∂p

∂z

∂(rτ

)

∂r

∂τ

∂z

+ρ



1 −α(T −T

)



=2η

∂u

∂r

=2η

∂w

∂z

=η



∂u

∂z

∂w

∂r



η =

2Aτ

n−1

exp



∗



2τ

=2τ

+τ

+(τ

+τ

)

∂T

∂t

∂T

∂r

∂T

∂z

=κ∇

Here, r and z are radial and vertical coordinates, u and w the corresponding

velocity components, and τ

, τ

and τ

the radial normal deviatoric stress,

vertical normal deviatoric stress and shear stress, respectively.

Scale the equations by writing

p −ρ

gz, τ

,τ

∼

T ∼T

,η∼η

,t∼

, x ∼d, u ∼

and show that the resulting dimensionless equations depend on the parameters

∗

αρ

,Λ=

2Aη





n−1

exp



∗



,ε=

∗

(†)

8.9 Exercises 537

Now suppose that the dimensionless thickness of the lithosphere is ν.

Rescale the variables by writing

z ∼ν, u ∼Ra

∗3/5

ν, w ∼Ra

∗3/5

,t∼ν

η ∼



νεRa

∗1/5



,τ∼τ

∼ν

∗

,p∼νRa

∗

∼τ

∼ν

∗

and write down the rescaled equations in terms of the parameters ε, ν, and ω,

where

ω =νεRa

∗1/5

, (∗)

and we have deﬁned

Λ =Ra

∗3(n−1)/5

. (‡)

Show that this deﬁnes an asthenospheric viscosity scale η

= η

given



∗2

αρ

gdν



n−1

exp



∗



and that if

Ra =

αρ

then

=(νε)

1/5

Ra.

Now compare this to the scales used in Sect. 8.5, where a viscosity scale

was used, and parameters Ra

and Λ

deﬁned as above using η

instead of

. There we deﬁned

(νε)

=ε

n−1

n+1

. (∗∗)

With this deﬁnition, show that, using (∗) above,

=ε

n−1

n+1

Use Eqs. (†), (‡), (8.184) and (8.185) to show that

6(n−1)



n−1

n+1



2n+3

and deduce that if ν is deﬁned using (∗∗), then ω =1.

538 8 Mantle Convection

8.10 Two-dimensional convection of a temperature- and pressure-dependent New-

tonian viscous ﬂuid at inﬁnite Prandtl number is described by the dimension-

less equations

=τ

+τ

=τ

−τ

−R(1 −T),

=−2ηψ

=η(ψ

−ψ

=∇

=τ

+τ

η =exp



1 −T +μ(1 −z)

εT



where

R 1,ε1,μ∼1,

and the velocity is given in terms of the stream function by

u =(−ψ

,ψ

Deﬁne the isoviscous temperature

iso

=1 +μ(1 −z),

and put

T =T

iso

+εφ;

show that the model can be reduced to the form



∂

∂z

−

∂

∂x





η(ψ

−ψ

)



∂

∂x∂z

(ηψ

) =RεT

iso

−μψ

=ε



∇

φ +ψ

−ψ



Deduce that a possible structure for the ﬂow in the interior of the convec-

tion cell is a horizontal ﬂow with ψ = ψ(z), in which case the viscosity is

determined by solution of



∂

∂z

−

∂

∂x



(ψ



η) =RεT

iso

Chapter 9

Magma Transport

Rocks which are formed at the Earth’s surface are of two main types, igneous and

sedimentary. A third type of rock, metamorphic, is one which has been subject to

post-formation processes of thermal and chemical alteration, often due to the effects

of elevated temperature and pressure on burial.

Sedimentary rocks are formed where sediments accumulate, at the bottom of

lakes and oceans, and the different rock types are associated with different types

of sediments. Shale is formed from clay, sandstone from sand, limestone from car-

bonate muds, often comprised of skeletal fragments of small marine organisms. The

sediment grains form a rock when squeezed together, through the precipitation of

cementing phases such as calcite in the intergranular pore space.

Igneous rocks, on the other hand, form when molten magma crystallises. This

may occur deep within the Earth’s crust in magma chambers, or else at the Earth’s

surface, when magma is extruded in volcanic eruptions. Such eruptions occur in

many different ways, and the study of the dynamics of volcanic eruptions is a prob-

lem of interest in its own right. In this chapter, however, we will be more concerned

with the processes whereby magma is able to move from deep in the Earth’s mantle

to regions near or at the Earth’s surface.

All rocks in the Earth are silicates, that is to say the principal constituent is silica,

SiO

, and this typically provides more than half of the composition, usually com-

bined with other compounds. Silicate rocks are formed by reaction of silica with

various metal oxides, in particular the oxides of aluminium (Al

), iron (FeO and

), magnesium (MgO), calcium (CaO), sodium (Na

O) and potassium (K

O),

as well as water. Because all of these compounds occur in varying proportions, the

subject of geochemistry which describes the chemistry of terrestrial rocks is an ex-

tremely complex one. The presence of many different components in mantle rocks

means, for example, that phase diagrams to describe melting and solidiﬁcation are

extremely complicated, and the stability regions of different phases depends on the

geochemical composition as well as temperature and pressure. The subject of ig-

neous petrology has thus been principally concerned with the geochemical evolu-

tion of magmas, and it is only relatively recently that the corresponding physical

evolution has been considered. In this chapter we are primarily concerned with the

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

539