Fowler A. Mathematical Geoscience

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380 6 Landscape Evolution

Hence show that the characteristic speeds satisfying det(λA −B) = 0are

λ =

and λ =∞, and thus that the system is a mixed hyperbolic/parabolic

system.

[To say that one root of the equation is inﬁnity means that the normal

quadratic one would expect has degenerated to a linear equation, because the

rank of A is only one. This is because of the presence of only one time deriva-

tive in the pair of equations.]

Show that if a term εq

is added to the ﬁrst equation in (∗) above, where ε

is small, then a second ﬁnite root of O(1/ε) occurs. [This root tends to inﬁnity

as ε →0.]

6.6 Suppose that in a model of hillslope erosion, sediment transport is given by

F = q

, where q is water ﬂux and S is surface slope, and α and β are

constant, and that uplift U and rainfall r are constant. Show that if x measures

distance from a ridge of a one-dimensional hillslope (i.e., whose elevation

depends only on x), and the equations for q and F are

=r, F

=U,

then the slope S = cx

for values of c and ν which you should determine.

Hence show that (geomorphically) concave slopes occur if α>1, and concave

slopes occur if α<1.

6.7 A tectonic province is uplifted rapidly, and thereafter erodes according to the

equation

∂s

∂t

=∇.



F ∇s



where S =|∇s| is the terrain slope. Suppose that F =qS, and that

∇.



q∇s



=−p,

where p represents precipitation.

If the topography varies only in the x direction, show that a similarity so-

lution can be found in the form s =

), and hence show that f(σ) =

B exp(−|σ |). What determines the constant B?

Find a comparable (cylindrically symmetric) similarity solution in two di-

mensions, and show in this case that

s =

exp



−



Determine B in terms of the initial uplifted volume V .

6.8 Suppose that in the stability equations



=−

qψ

σψ =[f



]



−(f

φ)



−

fψ

6.9 Exercises 381

subject to

φ =f



=0atx =0,

ψ =0atx =1,

we take k = 0, i.e., perturbations are in the x direction only. Show that φ = 0

and that ψ satisﬁes the equation

σψ =[f



]



with



=0atx =0,

ψ =0atx =1.

Allowing for the fact that ψ and σ may be complex, show that



|ψ|

dx =−



|ψ



dx,

and deduce that all eigenvalues σ are negative, i.e., one-dimensional perturba-

tions are stable. What does the assumption f

> 0 mean physically?

6.9 Suppose that φ satisﬁes the differential equation

Λφ





















with boundary conditions

φ =F









=0atx =0,



=0atx =1.

Show that if F =qS and S is constant, then Ψ =Sφ



/q satisﬁes the equation

ΛΨ =

[qΨ



]



with

qΨ



=0onx =0,

Ψ =0onx =1,

and then

φ =



qΨ

dx.

Suppose that q =x. Show that the solution for Ψ which satisﬁes both bound-

ary conditions is

Ψ =J



2k(−Λ)

1/2



382 6 Landscape Evolution

providing

Λ =−



0,n



where j

0,n

is the nth (positive) zero of the Bessel function J

Compare this result with that obtained from a WKB analysis assuming

q =0.

[The zeros of the zeroth order Bessel function satisfy j

0,n

∼ (n −

)π for

large n.]

6.10 The function A(ζ, p) is deﬁned by the integral

A(ζ, p) =

2πi



−p

ζt−

dt,

where the contour L goes from ∞e

4πi/3

to ∞e

2πi/3

in the complex t plane

(it is the contour L

in Fig. C.1). Use the method of steepest descents to ﬁnd

the asymptotic behaviour of A(ζ, p) as ζ →±∞.

6.11 Show that for γ , α, λ, A and B of O(1), and ε 1, the roots of the polynomial

εα

+ε(λ +B)m

−(λ −A)m −2γ(λ+B) =0

are given by

m =m

≈

2γ(λ+B)

A −λ

,m=m

≈±

iθ

√

−ν +···,

where

θ =



γ(A−λ)



1/2

,ν=γ(λ+B)



2α

A −λ



Hence show that if

φ(x)=U

−

(where one of the coefﬁcients U

can be chosen arbitrarily), then satisfaction

of the boundary conditions φ(0) =φ



(0) =φ



(1) =0 requires

√

εν

+···,U

=±

−

√

εν

+···

(if U

−

is chosen in this way), and

θ =θ

εν

+···,

where θ

=(r +

)π, for integral values of r.

Hence show that

λ ≈

A −

αθ

−εB



1 +

2γ



+···

1 +ε



1 +

2γ



+···

6.9 Exercises 383

Deduce that



(γ ) ≈

αθ

(1 −ε)

−

2ε(B +A)

and therefore that λ has a maximum as γ varies at

γ ≈



2ε(B +A)



1/2

6.12 Suppose that u is non-negative and of compact support, and that

1/3

=u +u

ξξ

Suppose that u ≈α[ξ

(τ ) −ξ]

near a margin ξ =ξ

(τ ) (where u =0), and

α and ν are positive. Show by balancing terms that possible choices for ν and

are

ν =3,

=3α

1/3

and

ν =1,

=0.

Show further in the latter case that if

u =α(ξ

−ξ)+β(ξ

−ξ)

5/3

+···,

then

≈

5β

3α

2/3

and deduce that both marginal advance or retreat are possible.

6.13 Suppose that u satisﬁes the equation

3/2



3/2



ξξ

on an interval Ω, with u = 0on∂Ω, and u ≥ 0att = 0. Explain why the

solution u will be non-negative.

Let ψ and λ be the ﬁrst eigenfunction and eigenvalue of the equation



+λψ =0,ψ=0on∂Ω,

(thus ψ is of one sign, let us say positive, in Ω: without loss of generality,

choose



ψdξ=1). Show that the positive quantity E(t) =



uψ dξ satis-

ﬁes the equation

E =(1 −λ)



3/2

ψdξ.

Use the measure dω =ψdξ and Jensen’s inequality

to deduce that if λ<1,

E ≥(1 −λ)E

3/2

, and therefore that the solution blows up in ﬁnite time if the

interval length |Ω|>π.

See (1.163).

384 6 Landscape Evolution

Equally, show that E decreases if λ>1(|Ω|<π), so that the solution is

apparently bounded.

Show that an exact blow-up solution of the form

1/2

B cos

αξ

−t

, |ξ|<

2α

u =0 otherwise, exists in Ω =(−L,L) providing L>π/2α, and if B =

and α =

6.14 The equations



2μ



+β)

3/2

−τ

∗

−h

3/2

1/2



and



2μh

3/2

1/2

where S

= (S

+ μ

)

1/2

, describe the long proﬁles of channel slope and

water ﬂux in the Smith–Bretherton model. Suppose that τ

∗

=0 and deﬁne

Γ =



1 +



3/2

Show that

lnQ =



Γ(S

+μ

)

1/2

−S

By integrating this, show that

Q ∝(S

+S)

Γ/(Γ

−1)

(Γ S

−S)

1/(Γ

−1)

[The substitution S =μ sec θ may help.]

6.15 Let Q =H

3/2

1/2

and Z satisfy the equation

∂Q

∂x

∂

∂Y



∂Z

∂Y



in 0 <Y <Y

, where S =S(x) and Y

is constant, and where

∂Z

∂Y

=0atY =0.

Suppose that H

3/2

Q(x)

cosξ , where ξ =

πY

Show that

∂Z

∂Y

−

∂H

∂Y

g(x),

Using the Cauchy–Schwarz inequality, the boundedness of E implies the boundedness of the

(Ω) norm. This is not the same as proving the boundedness of u (or equivalently, that of the

∞

(Ω) norm).

6.9 Exercises 385

where g(x) should be determined, and thus show that the characteristics of the

equation

αS −a



αS +S





=Sa

−



−

βH



when written in terms of x and ξ,are

sin ξ =exp



−



g(x)

S(x)



where x =σ when Y =Y

. Draw the shape of the characteristics in the (x, Y )

plane, assuming g>0.

Now suppose that

D =

∂

∂Y



βH

1/2

∂H

∂Y



;

show that

=−

πβ

and deduce that on a characteristic,

=α −b(x)a,

and determine b(x).

Show that if S is constant, then b(x) is constant, and deduce that

a =



1 −exp



−b(x −σ)



Illustrate the behaviour of the solution graphically in the two cases α ≶ α

where

Show that if also

Q is constant, then

σ =x −

lncosec ξ,

and deduce that

a =



1 −sin

b/α



Chapter 7

Groundwater Flow

Groundwater is water which is stored in the soil and rock beneath the surface of the

Earth. It forms a fundamental constituent reservoir of the hydrological system, and it

is important because of its massive and long lived storage capacity. It is the resource

which provides drinking and irrigation water for crops, and increasingly in recent

decades it has become an unwilling recipient of toxic industrial and agricultural

waste. For all these reasons, the movement of groundwater is an important subject

of study.

Soil consists of very small grains of organic and inorganic matter, ranging in size

from millimetres to microns. Differently sized particles have different names. Partic-

ularly, we distinguish clay particles (size <2 microns) from silt particles (2–60 mi-

crons) and sand (60 microns to 1 mm). Coarser particles still are termed gravel.

Viewed at the large scale, soil thus forms a continuum which is granular at the

small scale, and which contains a certain fraction of pore space, as shown in Fig. 7.1.

The volume fraction of the soil (or sediment, or rock) which is occupied by the pore

space (or void space, or voidage) is called the porosity, and is commonly denoted

by the symbol φ; sometimes other symbols are used, for example n, as in Chap. 5.

As we described in Chap. 6, soils are formed by the weathering of rocks, and

are speciﬁcally referred to as soils when they contain organic matter formed by the

rotting of plants and animals. There are two main types of rock: igneous, formed by

the crystallisation of molten lava, and sedimentary, formed by the cementation of

sediments under conditions of great temperature and pressure as they are buried at

depth.

Sedimentary rocks, such as sandstone, chalk, shale, thus have their porosity

built in, because of the pre-existing granular structure. With increasing pressure,

the grains are compacted, thus reducing their porosity, and eventually intergranular

cements bond the grains into a rock. Sediment compaction is described in Sect. 7.11.

Igneous rock tends to be porous also, for a different reason. It is typically the

case for any rock that it is fractured. Most simply, rock at the surface of the Earth

There are also metamorphic rocks, which form from pre-existing rocks through chemical changes

induced by burial at high temperatures and pressures; for example, marble is a metamorphic form

of limestone.

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

387

388 7 Groundwater Flow

Fig. 7.1 A granular porous

medium

is subjected to enormous tectonic stresses, which cause folding and fracturing of

rock. Thus, even if the rock matrix itself is not porous, there are commonly faults

and fractures within the rock which act as channels through which ﬂuids may ﬂow,

and which act on the large scale as an effective porosity. If the matrix is porous at

the grain scale also, then one refers to the rock as having a dual porosity, and the

corresponding ﬂow models are called double porosity models.

In the subsurface, whether it be soil, underlying regolith, a sedimentary basin,

or oceanic lithosphere, the pore space contains liquid. At sufﬁcient depth, the pore

space will be saturated with ﬂuid, normally water. At greater depths, other ﬂuids

may be present. For example, oil may be found in the pore space of the rocks of

sedimentary basins. In the near surface, both air and water will be present in the

pore space, and this (unsaturated) region is called the unsaturated zone, or the vadose

zone. The surface separating the two is called the piezometric surface, the phreatic

surface, or more simply the water table. Commonly it lies tens of metres below the

ground surface.

7.1 Darcy’s Law

Groundwater is fed by surface rainfall, and as with surface water it moves under a

pressure gradient driven by the slope of the piezometric surface. In order to char-

acterise the ﬂow of a liquid in a porous medium, we must therefore relate the ﬂow

rate to the pressure gradient. An idealised case is to consider that the pores consist

of uniform cylindrical tubes of radius a; initially we will suppose that these are all

aligned in one direction. If a is small enough that the ﬂow in the tubes is laminar

(this will be the case if the associated Reynolds number is 1000), then Poiseuille

ﬂow in each tube leads to a volume ﬂux in each tube of q =

πa

8μ

|∇p|, where μ is

7.1 Darcy’s Law 389

the liquid viscosity, and ∇p is the pressure gradient along the tube. A more realistic

porous medium is isotropic, which is to say that if the pores have this tubular shape,

the tubules will be arranged randomly, and form an interconnected network. How-

ever, between nodes of this network, Poiseuille ﬂow will still be appropriate, and an

appropriate generalisation is to suppose that the volume ﬂux vector is given by

q ≈−

μX

∇p, (7.1)

where the approximation takes account of small interactions at the nodes; the nu-

merical tortuosity factor X  1 takes some account of the arrangement of the pipes.

To relate this to macroscopic variables, and in particular the porosity φ, we ob-

serve that φ ∼ a

, where d

is a representative particle or grain size so that

q/d

∼−(

μX

)∇p. We deﬁne the volume ﬂux per unit area (having units of veloc-

ity) as the discharge u. Darcy’s law then relates this to an applied pressure gradient

by the relation

u =−

∇p, (7.2)

where k is an empirically determined parameter called the permeability, having units

of length squared. The discussion above suggests that we can write

k =

; (7.3)

the numerical factor X may typically be of the order of 10

To check whether the pore ﬂow is indeed laminar, we calculate the (particle)

Reynolds number for the porous ﬂow. If v is the (average) ﬂuid velocity in the pore

space, then

v =

. (7.4)

If a is the pore radius, then we deﬁne a particle Reynolds number based on grain

size as

2ρva

∼

ρ|u|d

√

, (7.5)

since φ ∼ a/d

. Suppose (7.3) gives the permeability, and we use the gravitational

pressure gradient ρg to deﬁne (via Darcy’s law) a velocity scale

; then

∼

3/2







∼10[d

]

, (7.6)

This scale is thus the hydraulic conductivity, deﬁned below in (7.9).