Fowler A. Mathematical Geoscience

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330 5 Dunes

is given by

K(k) =

(ν)exp



−

iπν sgnk



2π|k|

for real values of k.

Now suppose that φ satisﬁes the evolution equation

∂

∂ξ



+αK ∗φ

−φ



=0,

where f ∗g denotes the Fourier convolution of f and g and α is small. Show

that the steady state φ = 0 is linearly unstable, and ﬁnd the wave number of

the maximum growth rate.

When this equation is solved numerically, coarsening occurs, with the

wavelength of the bedforms increasing with time. Show that if ξ ∼ L  1,

the equation can be approximated by Burgers’ equation with small diffusivity.

Hence explain the way in which coarsening occurs.

5.10 Suppose that

ψ(t) =

∞



r=0

=(1 −ξ

−1/2

(1 −ξ

−1/2

and it is desired to calculate the coefﬁcients f

numerically. By consideration

of the power series for ψ

(or otherwise!) show that an iterative recipe for f

=1,



n+1

−ξ

n+1

−ξ



−

n−1



s=1

n−s

5.11 A shear ﬂow is described by the dimensional equation

τ =ρ



ν +κ



∂u

∂z





∂u

∂z

where ν is the kinematic viscosity. Show that a suitable dimensionless form is

τ =



ε +κ



∂u

∂z





∂u

∂z

where ε =

, and Re is the Reynolds number. Use the method of strained

coordinates

(i.e., write z = s + εz

(s) +···, u ∼ u

(s) + εu

(s) +··· to

show that

u ≈



z +s



where s

2κ

in order to suppress higher singularities in u

See, for example, Van Dyke (1975).

Chapter 6

Landscape Evolution

Landscape is one of the most obvious features of the environment we live in. Rolling

countryside, rugged cliffs, wind-swept mountains; the scenery which surrounds us

is formed by topography, and the shapes which the Earth’s surface is moulded into

forms the subject of geomorphology.

Topography is created by tectonic processes, and removed by erosion. The most

striking example on the planet today is the continuing formation of the Himalayas,

as India continues to crash into Asia after its separation from Antarctica during the

break up of Gondwanaland over a 100 million years ago. This continental collision

buckles the Earth’s crust and forms mountain belts; in the case of the Himalayas,

the mountains rise to a height above sea level of nearly nine kilometres, and they

are still rising.

Creation of mountains creates hillslope, and precipitation and its subsequent

runoff provides an erosive mechanism. At low elevations, this is through a vari-

ety of actions of running water: slow processes such as rainsplash and sheetwash,

which carry surﬁcial sediments towards a developed stream system, in which more

rapid evacuation of sediments occurs. At higher elevations, precipitation as snow

may form glaciers, which similarly carve the landscape and erode it. Typical rates

of tectonic uplift and erosion are comparable, and very slow in stable, vegetated

landscapes: a hundred metres per million years (100 m Ma

−1

) is a typical rate for

each. In unvegetated landscapes such as badlands, erosion and gully formation can

be much more rapid.

6.1 Weathering

A necessary pre-condition for sediment removal is the break up of the basement

rock into pieces (boulders, cobbles, gravel, silt, clay: sediments of varying grain

The formation of the Himalayan mountains is actually more complicated than that which would

arise from a simple buckling mechanism. Geological investigations indicate that the Himalayas

are formed by a backﬂow of partially molten rock driven by its own buoyancy; the partial melting

occurs in the subducting Indian crust due to extremely high levels of uranium concentration.

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

DOI 10.1007/978-0-85729-721-1_6, © Springer-Verlag London Limited 2011

331

332 6 Landscape Evolution

sizes), and this is achieved through a variety of mechanisms. Mechanical weather-

ing refers to the physical disruption of rock. For example, joints and fractures are

natural consequences of rock formation; igneous rocks may have shrinkage cracks

as they cool, sedimentary rocks have bedding planes and faults formed through fail-

ure in tectonic compression. When exposed at the surface, these are subjected to

frost action and other thermally induced stresses, which break up the rock, and form

an unconsolidated regolith.

Chemical weathering refers to a variety of chemical effects which have similar

effects. Over long time scales, rain water will react with rocks. Feldspars break

down into clays (very ﬁne grained particles) in this way, and carbonate rocks are

dissolved by weakly acidic rainwater. This is particularly noticeable in limestone,

where dissolution of the rocks can lead to the formation of spectacular underground

cave systems as found in karst regions. Chemical weathering can also form sediment

by attacking intergranular cements.

The eventual result of weathering is the rotting of rock to a decayed state called

saprolite. Further evolution of the regolith, for example, by the effects of vegetation

or through transport, leads to the formation of soils. It is the sediments of the regolith

which are subject to erosion through ﬂuvial or glacial transport.

6.2 The Erosional Cycle

Our purpose in this chapter is to describe how the processes of tectonic uplift and

subsequent erosion can lead to the formation of the topographic patterns such as that

shown in Fig. 6.1, in which we see a typically dendritic pattern of stream channels

draining a catchment area.

The framework in which we do this is through a model which describes the evo-

lution of hillslope through the balance of the two processes, uplift and erosion. More

or less, this model will be

∂s

∂t

=U −v

, (6.1)

where s is the vertical height of the topography, U is the rate of uplift, and v

the rate of abrasion, or erosion, of the hillslope. Our particular approach will be to

study ﬁrst the steady state of this equation, in which uplift balances erosion, and

then to seek conditions under which a uniform slope is unstable, and under which

the typical pattern of a drainage system can develop.

However, it should be pointed out that this assumption is rather an idealisation,

possible but not necessary. Another framework is the cycle of erosion described in

the latter part of the nineteenth century by W. M. Davis, thus called the Davisian

cycle of erosion.

Davis’s idea was that uplift was a relatively sudden thing, so that topography

evolved from an initial elevated stage. In a young landscape, slopes are steep and

stream erosion is strong, but as the landscape ages, slopes become gentler and ero-

sion is less dramatic. The next orogenic event initiates the next cycle, and so on.

6.3 River Networks 333

Fig. 6.1 Hillslope topography. Photograph courtesy of Gary Parker

From the modelling point of view, this framework differs from the notion of con-

tinual uplift and erosion only in detail, and there is no serious problem in using it as a

starting point. The mathematical difﬁculty it raises is that instability and stream for-

mation then occur on a slowly developing background state, so that an appropriate

frozen time hypothesis is required in order to determine stability criteria. Because

of the disparity of time scales between stream formation and hillslope erosion, this

is in fact hardly an issue.

6.3 River Networks

River networks are (approximate) fractals.

That this is so may be inferred from a

number of power-law relationships satisﬁed by river systems. One such is Hack’s

law, which relates the length l of a stream to its drainage basin area A,asl ∼A

0.6

Other power-law relationships are exhibited by quantities associated with the

Horton–Strahler ordering system, for example between the number of streams and

their order, which is an algorithmic measure of the stream size and importance

within the network.

A fractal set is one whose dimension is non-integral. Curves have dimension one, areas have

dimension two. A way of characterising the dimension of a set is to count the number of boxes N

of size ε required to cover the set. If N ∼ε

−D

as ε →0, then the set has fractal dimension D.This

is consistent with our intuitive sense of dimension, but also allows the calculation of non-integer

dimension for such exotica as the Koch snowﬂake, the Sierpinski gasket, and so on. Fractal sets

typically exhibit power-law relationships in their description.

334 6 Landscape Evolution

Fig. 6.2 Geometry of

overland ﬂow

A question of fundamental philosophical signiﬁcance is to explain how fractal

sets can arise as the response of a continuous, deterministic system to constant forc-

ing. In the study of chaotic (time-dependent) systems, the answer to this is known.

Strange attractors have fractal dimension, and this is because the systems can be

characterised by the action of (Poincaré return) maps, which generate the Cantor

sets of the attractor by a continued process of splitting and mixing. It is less obvious

how such a process might work in generating spatially fractal sets, and this appears

to remain as a challenge for nonlinear dynamicists.

6.4 Denudation Models

The basic structure of erosional theories for channel formation is typically that of a

shallow water (St. Venant) model of overland ﬂow, coupled to an Exner type equa-

tion describing hillslope erosion. If we denote the vertical coordinate as z, and the

position of the free water surface and hillslope as η and s, respectively (see Fig. 6.2),

then a St. Venant model can be written as

+∇.(hu) =r,

+(u.∇)u =−g∇η −

f |u|u

(6.2)

These equations represent conservation of mass and momentum, and u is the mean

horizontal (or along slope) velocity. The source term r represents rainfall. The water

depth h is given by

h =η −s. (6.3)

The Exner equation to model the evolution of the hillslope s is usually taken in

the form

(1 −φ)s

+∇.q

=[−ρ

+ρ

]



+ρ

(1 −φ)U



, (6.4)

wherein φ is the bed porosity, ρ

is the sediment density, q

represents bedload

transport, and v

and v

, if present, represent erosion and deposition of sediments

(measured as velocities: see (5.44)); U would represent tectonic uplift, also mea-

sured as a velocity. The erosion and deposition terms arise when sediment can be

carried as suspended load, which is assisted by rapid ﬂow and small particles, and

thus may be relevant in stormﬂow conditions. In this case the mean concentration c

of suspended sediment is given by

(hc)

+∇.(hcu) =ρ

−ρ

. (6.5)

and v

have to be prescribed, as discussed in Chap. 5.

6.4 Denudation Models 335

A possible point of confusion which arises when we study models of this type is

that some of them include down-cutting or abrasion of the landscape, while others

do not, allowing sediment to move from place to place without actually being exca-

vated: do we use (6.2)or(6.4) to evolve the hillslope? The source of this confusion

lies in the way in which the bedload transport is described. Conventionally, there

is transport but no bedload density, and this masks a conceptual awkwardness in

the models. At the outset, we address this awkwardness by modifying (6.4)inthe

following way.

We suppose that s describes the interface between stationary bed and moving

bedload, and we deﬁne a bedload density ρ

and bedload transport q

, so that

+∇.q

=−ρ

+ρ

(1 −φ)v

, (6.6)

where v

is the abrasion or entrainment rate of the bed, measured as a velocity. The

bedload density has units of mass per unit area, and is conveniently written in the

form

=ρ

(1 −φ)a, (6.7)

where a is a length, and can be interpreted as the depth of the active deforming

bedload layer. Finally, the Exner equation is

(1 −φ)s

=−ρ

(1 −φ)v



+ρ

(1 −φ)U



. (6.8)

The earlier model is regained if ρ

→0(ora → 0), which is consistent with the idea

of bedload being a surface density. Otherwise, v

needs to be prescribed, in which

case it is natural to suppose it is a function of active depth a; it is this quantity v

which is really the down-cutting or excavation rate. It remains to be seen how small

a actually has to be in order that it be negligible; we will come back to this later.

Note that if a>0, then (6.3) must be modiﬁed to read

η =s +a +h. (6.9)

6.4.1 Sediment Transport

These equations, or generalisations of them, can be used to study some forms of

morphological instability (dunes, braiding, meandering, etc.), and this formed the

subject of Chap. 5. They also allow a description of the erosional instability mecha-

nism which is responsible for the formation of channels. The efﬁcacy of this depends

on the chosen forms for bedload transport and bed erosion terms. The basic insta-

bility mechanism is less subtle than the stress shift analysed in Chap. 5, and is due

simply to the fact that a locally increased depth h causes increased ﬂow u, and thus

increased erosion, which in turn allows a further increase in h.

The constituents of this mechanism involve prescription of the sediment trans-

port terms, as described in Chap. 5. Both erosion rate v

and bedload transport q

depend on stress delivered by the water ﬂow. This stress is taken to be

τ =fρ

|u|u. (6.10)

336 6 Landscape Evolution

Erosion and bedload are increasing functions of τ, except that allowance needs to

be made for the local bed slope (it is easier to move particles downhill than uphill).

As discussed in Chap. 5 (see the similar discussion at (5.109)), for a particle of

diameter D

at the bed, the streamﬂow exerts a force of approximately τ D

on it,

and it is this force which causes motion. On a slope, there is an additional force

due to gravity, approximately −ρ

∇s, where ρ

is the density difference

between particle and ﬂuid. Thus the net stress causing motion is actually

=τ −ρ

∇s. (6.11)

This slope effect will turn out to have a (crucial) stabilising effect in the equation

for s.

If the stress is below a critical yield stress, then no motion occurs, and both v

and q

will be zero. The relevant dimensionless quantity is the Shields stress,

μ =

ρ

, (6.12)

and sediment transport or erosion occurs if μ>μ

≈ 0.05. The critical Shields

stress depends somewhat on particle size (through the particle Reynolds number),

but can reasonably be approximated as a constant. Denoting the dimensional yield

stress as

=μ

ρ

, (6.13)

then typical assumptions for bedload transport are the Meyer-Peter and Müller rela-

tionship



1/2

ρ



(τ

−τ

)

3/2

, (6.14)

where Meyer-Peter and Müller chose values of K = 8 and μ

= 0.047. For bed

erosion, the Van Rijn relationship is similar:

∝(τ

−τ

)

3/2

. (6.15)

6.4.2 Non-dimensionalisation

The variables of the model are water depth h, water velocity u, water surface η,

suspended load c, active bedload depth a, bed surface s, effective shear stress τ

and bedload transport q

, which is assumed to be a function of τ

, such as that given

by (6.14). The describing equations of the model are (6.2), (6.9), (6.5), (6.6), (6.7),

(6.8) and (6.11).

First we scale the constitutive transport terms. These are the erosional, deposi-

tional, and abrasional velocities v

, v

, and v

, and the bedload transport q

.The

erosional and depositional velocities are scaled as in Chap. 5, thus

E, ρ

cD, (6.16)

where v

is a typical erosion rate and v

is the particle settling velocity, and E and

D are dimensionless erosion and deposition rates of O(1), which are functions of

6.4 Denudation Models 337

mean ﬂow velocity. In addition, we write

A, q

, (6.17)

where U

is a typical value of the uplift rate U , and q

is determined by a relation

such as that of Meyer-Peter and Müller. Since we suppose that v

is a function of

active depth a, the choice v

∼U actually provides a scale for a.

We choose scales for the other eight variables h, u, η, s, c, τ

,aswellasx and t,

by balancing water ﬂux with rainfall, gravitational acceleration with friction, erosion

rate with deposition rate, hillslope elevation with water surface elevation, bedload

transport with abrasion rate, and hillslope rate of change with both abrasion rate and

tectonic uplift (if this is present). If we suppose that d is a suitable hillslope height

scale and l is a suitable horizontal length scale, then we speciﬁcally choose

η, s ∼d, x ∼l, t ∼[t]=

,τ

∼fρ

[u]

u ∼[u]=





1/3

,c∼

,h∼[h]=l





1/3

(6.18)

where we take r

as a typical precipitation rate.

There appears as yet to be nothing to constrain the length and depth scales l

and d. The ﬁrst of these might be determined by the implied tectonic setting. The

simplest conceptual idea is the continuing uplift of an island (or a mountain belt),

with sea level ﬁxed at prescribed boundaries, and this determines a natural length

scale l, the scale of the island. The other scale is ﬁxed by the balance of uplift rate

with hillslope denudation, which requires

l =

(1 −φ)U

. (6.19)

This appears to determine l again, but in fact it determines d through the dependence

of q

on τ

and thus [u]. For example, if we take q

∝ τ

3/2

∝[u]

, then from

(6.18), we have q

∝d, and thus d ∝U

Using the scaled variables in the model equations (6.2), (6.3), (6.5), (6.6) and

(6.8), we obtain the dimensionless set

δεh

+∇.(hu) =r,

δF



δεu

+(u.∇)u



=−∇η −

|u|u

η =s +δh +δνa,

δε(hc)

+∇.(hcu) =γ(E−cD),

δνa

+∇.Q

=−α(E −cD) +A,

=−A +U,

=|u|u −β∇s,

(6.20)

More precisely,

∼(

ρ

(1−φ)

1/2

)

∼

, so high mountains are (in this theory) a consequence

of high uplift rate and low rainfall, which makes intuitive sense.

338 6 Landscape Evolution

and we suppose for the moment that

=f(τ

)N, (6.21)

where

N =

, (6.22)

and f is no longer used for the friction factor.

The parameters are deﬁned by

ε =

,δ=

[h]

[u]

g[h]

,β=

ρ

[h]

γ =

,ν=

[a]

[h]

,α=

(1 −φ)U

(6.23)

[a] being the scale for the active depth a. The deﬁnitions of ε and Froude number

F here correspond to those used in Chap. 5.

It is easy to get a sense of the size of the various parameters in the model, simply

from observation. We have seen in Chap. 5 that ε 1, and this is conﬁrmed in the

present case; if, for example, we take U

∼ 10

−3

−1

(1 km per million years)

and r

∼1my

−1

, then ε ∼10

−3

The important parameter δ is the ratio of stream depth to hillslope height, and

is small. An estimate for [h] based on values r

∼ 1my

−1

, L ∼ 10

m, f ∼ 0.1,

g ∼ 10 m s

−2

, d ∼ 10

m, is [h]∼2 cm, thus δ ∼ 10

−5

, and even in stormﬂow

conditions, a transient depth of as much as a metre yields δ ∼ 10

−3

. Note that for

this choice of d and U

, the (erosional) time scale is a million years.

The velocity scale corresponding to the above depth is 0.2 m s

−1

, and from this

we calculate that the Froude number is about 0.4. This is quite vigorous, indicative

of the implicit assumption of relatively bare (and thus smooth) ground. Generally,

overland ﬂow will have quite small Froude number, except in stormﬂow conditions;

even then, the Froude number does not greatly exceed unity.

The erosional parameters are γ , α and ν. Since bedload transport typically occurs

over an active depth of at most a few grains, we can safely assume that ν  1.

The particle settling velocity is given by (5.8), which dimensionally gives v

∼

]

−1

if D

=[D

] mm. Thus v

is typically quite large, and consequently γ

is large; a typical estimate is (for particle grain size 1 mm) γ ∼ 10

. Estimates for

are about 10

−2

, since typical suspended loads ρ

are of order 10 g l

−1

and for U

∼10

−3

−1

,wehaveα ∼ 10

1.

Approximately, β ≈D

/[h] 1; we will thus generally take β =O(1).

In summary then, we estimate

ε, δ 1,

,ν,β 1,

γ, α 1.

(6.24)

It is an unfortunate feature of applied mathematics that there are not enough letters in the Roman

and Greek alphabets, even allowing for capitals, overhats, tildes, asterisks, subscripts and super-

scripts. Duplication is inevitable, and here, apologetically, we use f for the dimensionless sediment

transport function, having just made use of it as the friction factor.

6.5 Channel-Forming Instability 339

6.4.3 The Issue of Time Scale

The sceptical reader will at this point enjoy some doubts about these ‘typical’ val-

ues. A ﬁrst obvious point is that for a supposed typical overland ﬂow depth of 2 cm,

the turbulent friction law (6.10) may not be appropriate. This is, in fact, easily mod-

iﬁed by allowing a laminar component of stress at low Reynolds number, but this

is a cosmetic component which does not alter the structure of the model (see also

Question 6.2).

A more subtle concern is illustrated by the size of the Froude number. The fact

of the matter is that it does not rain all the time, but in occasional showers and

storms, and furthermore, we might expect that erosion and bedload transport would

hardly occur, except in the most severe storms. Since our erosional time scale is

[t]=d/U ∼ 10

y, it seems likely that the principal erosion forming events are

such extremely rare storm-induced ﬂoods, and that during these, the Froude number

may be O(1) or higher. This raises the concern that the estimation of the parameters

on the basis of typical ‘average’ values may be inappropriate.

For example, consider a landscape where is there is an extreme storm for a day

every 100 years, and drought at other times (and ignore the fact that this would

actually be a desert). In this scenario, water ﬂow and erosion only occur during the

storm, and the intervening century is irrelevant for the purpose of calculating water

ﬂux and erosion. It then becomes appropriate to use a different time scale (and thus

also water depth and velocity scales) to describe the erosive effect of the storm. The

evolution of the topography over longer tectonic time scales then occurs through the

aggregation of these short time scale discrete erosional events. It is not obvious that

averaging in this way (i.e., solve the short time erosional problem and then average

over time) yields the same result as the approach we have taken (i.e., average over

time and then solve the erosional problem), but we shall proceed on this basis, partly

because it may well be appropriate in certain (e.g., temperate) environments, and

partly because this is the approach which has implicitly been taken in the literature

of the subject. Further consideration of this point is consigned to the exercises.

6.5 Channel-Forming Instability

The basic problem in understanding the development of river networks is the forma-

tion of a channel from a uniform overland ﬂow. To see how this can happen, we will

begin by considering a simpliﬁed version of the model equations (6.20). We assume

δ 1, ε  1, F

 1, ν  1, and ignore the corresponding terms in the equations.

The last of these (neglect of the term δν in (6.20)

) corresponds to the neglect of the

bedload density ρ

, even in the stormﬂow case where a signiﬁcant active layer may

be mobilised.

We also make the important assumption of supposing that sediments are coarse,

and are transported only as bedload, so that v

=0 and thus also c =0. This avoids

the issue of dealing with the large parameters γ and α (see also Question 6.4). With