θ =1’ √ (2.33)

π The solution for w is found by substituting

0

Eq. (2.39) into Eq. (2.20) to obtain

The integral in Eq. (2.33) has a special name called

the error function: ∞

(2n + 1)π x

(’1)n

32 22

’ κn π t

w(x, t) = 3 cos e

· 2

4L

2 π (2n + 1) 3

’· 2 2L

erf(·) = √ e d· (2.34) n=0

π 0

(2.40)

The solution for θ, therefore, is

Substituting Eq. (2.40) into Eq. (2.38) and re-

θ = 1 ’ erf(·) = erfc(·) (2.35)

arranging for h gives

where erfc(·) is called the complementary error

∞

U L2 x2 (’1)n

32κ

function. Written in terms of the original vari- h(x, t) = 1’ 2 ’ 3 2

πL (2n + 1)3

2κ L

ables, Eq. (2.35) becomes n=0

(2n + 1)π x 22

’ κn π t

x — cos e (2.41)

2

h(x, t) = h 0 erf √

4L

(2.36) 2L

2 κt

Most solutions of the diffusion equation in in¬- Examples of Eq. (2.41) are plotted in Fig-

ure 2.3a for values of κt ranging from 1000 m2

nite and semi-in¬nite domains can be written in

to 2500 m2 . These results provide an estimate for

terms of error functions.

Series solutions and similarity solutions are the time scale necessary to achieve steady state.

If we assume κ = 1 m2 /kyr, then steady state is

the two basic types of analytical approaches to

solving the diffusion equation. In the following achieved after approximately 2.5 Myr of uplift for

sections we will apply these equations to speci¬c a hillslope 50 m in length. Interestingly, the time

problems that arise in geomorphic applications. scale required to develop steady state does not

depend on the uplift rate.

2.2.4 Transient approach to steady state Figure 2.3b presents the same results as Fig-

In Section 2.2.1 we found the steady-state diffu- ure 2.3a except that the hillslope length has been

sive hillslope to be given by Eq. (2.13). Here we normalized to 1. In this approach the results de-

solve the time-dependent diffusion equation that pend only on a single nondimensional parameter

κt/L 2 . This nondimensional framework is more

describes the transient approach to steady state.

We consider a diffusing hillslope of length L with ¬‚exible because we do not need to plot differ-

constant uplift rate U : ent solutions for hillslopes of different lengths.

While Figure 2.3a is speci¬c to hillslopes 50 m

‚h ‚ 2h

’κ 2 =U (2.37) in length, Figure 2.3b allows us to estimate the

‚t ‚x

time required to achieve steady state for any hill-

The boundary conditions for this problem are

slope length. If κ = 1 m2 /kyr and L = 100 m, for

given by Eqs. (2.13) and (2.12). The initial condi-

example, steady state is achieved (i.e. the relief

tion is h(x, 0) = 0. The constant uplift term on

has achieved 90% of the steady-state relief) after

the right side of Eq. (2.37) can be eliminated by

κt/L 2 = 1 or t = 10 Myr.

introducing a new variable w that quanti¬es the

In this chapter we will usually present results

difference from steady state. The variable w is de-

in terms of actual length scales in order to de-

¬ned by

velop intuition about the rates of landscape evo-

U2 lution. In practice, however, it is often best to

w(x, t) = h(x, t) ’ (L ’ x 2 ) (2.38)

2κ solve the problem in a fully nondimensional way

and then apply speci¬c values of κ, L , and t to

The evolution of w is governed by the diffusion

equation with the same boundary conditions scale the results appropriately.

38 THE DIFFUSION EQUATION

(a) 1.0 (a)

=

kt

=

kt

2500 m2

2250 m2

0.8

0.6

2hk

UL2 Q2a

0.4 Q2b

Q2c

Q3

0.2

= 250 m2

kt

0.0

0 10 20 30 40 50 (b)

x (m)

(b) 1.0

=

kt/L2 =

kt/L2

1.0

0.8

0.9

0.6

2hk

UL2

Q2a Q2b

0.4

0.2

kt/L2 = 0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

x/L

Fig 2.3 (a) Approach to steady state in a hillslope subject to

uniform uplift or base level drop for a hillslope 50 m in length. Q3

Q2c

(b) Same as (a) except that results are plotted in terms of

dimensionless length x /L . Fig 2.4 (a) Hanaupah Canyon alluvial fan, showing

mid-Pleistocene (Q2a) to Holocene (Q3) terraces. (b) Aerial

photographs of Q2a, Q2b, Q2c, and Q3 terraces, illustrating

the correlation between age and degree of roundedness of

2.2.5 Evolution of alluvial-fan terraces the terrace slope adjacent to gullies.

following fan-head entrenchment

In this section we consider the evolution of

initially-planar hillslope terraces subject to rapid aggrading mode, the ratio of sediment to water

base-level drop. This example provides an oppor- is high and sediment is deposited across the fan

tunity to use both series and similarity solu- (or the available accommodation space in the en-

tions within the context of a real-world hillslope- trenched zone of an older abandoned terrace) to

evolution application. form a low-relief deposit by repeated episodes of

First some background. In the Basin and channel avulsion (Figure 2.5a). When the ratio of

Range province of the western United States, sediment to water decreases, such as when cooler

many alluvial fans have a suite of terraces that and wetter climate conditions cause revegetation

rise like a ¬‚ight of stairs from the active channel. of hillslopes and anchoring of the available re-

Figure 2.4 illustrates the classic alluvial fan ter- golith, the sediment deposited during the pre-

races of Hanaupah Canyon in Death Valley, Cali- vious period of high sediment ¬‚ux is rapidly en-

fornia. These terrace suites are generally consid- trenched (Figure 2.5b). This entrenchment acts as

ered to be caused by variations in the sediment- a rapid base-level drop for the abandoned terrace

to-water ratio in channels draining the mountain and gullies quickly form at the distal end of the

front (Bull, 1979). When alluvial fans are in an terrace and migrate headward (Figure 2.5c). At

2.2 ANALYTIC METHODS AND APPLICATIONS 39

1.0

(a) 1 (b) 2

kt =

0.8

20 m2

A A 0.6

h

A h0 kt =

1280 m2

0.4

3(d) 4

(c) kt =

0.2

2560 m2

0.0

A A

A A 0 50

x(m)

Fig 2.6 Plots of Eq. (2.42) for κt values of 50, 100, 200 . . .

B

6400 m2 and a terrace width of 2L = 100 m. The early phase

of the model is characterized by penetration of erosion into

the scarp interior. After κt ≈ 1000 m2 no planar remnant

Fig 2.5 Schematic illustration of a cycle of aggradation and remains and the scarp pro¬le can be approximated by a sine

fan-head entrenchment. When alluvial fans are in an function with height decreasing linearly with time.

aggradational mode, the ratio of sediment to water is high

and sediment is deposited across the available

accommodation space to form a low-relief surface by once the gully has formed. This approach pro-

repeated episodes of channel avulsion (a). When the ratio of vides a relative-age dating and correlation tool for

sediment to water decreases, the sediment deposited during

fan terraces (Hsu and Pelletier, 2004). Series solu-

the previous high-sediment-¬‚ux period is rapidly entrenched

tions (Eq. (2.20)) are the most comprehensive ap-

leaving an abandoned terrace (b). This entrenchment acts as a

proach to this problem, but similarity solutions

rapid base-level drop for the abandoned terrace and gullies

can be used to model young terraces (those with

quickly form at the distal end of the terrace and migrate

planar remnants) with no loss of accuracy.

headward (c).

First we consider series solutions. Substituting

f (x) = h 0 into Eq. (2.20) gives

∞

any point along a line parallel to the mountain (’1)n

4h 0

h(x, t) =

front, a pro¬le can be plotted that crosses one π 2n + 1

n=0

or more gullies. In young terraces, gully pro¬les (2n + 1)π x κ(2n+1)2 π 2 t

’

— cos e (2.42)

4L 2

are characterized by narrow valleys and broad,

2L

planar terrace treads. In older pro¬les, gully ero-

sion penetrates farther into the terrace. Given Equation (2.42) can be used to model terrace gully

a suf¬ciently old terrace, hillslope erosion will pro¬les through their entire evolution from back-

completely penetrate the terrace tread and no wearing scarp to downwearing gully remnant.

planar remnant will be preserved. The effects of That two-phase evolution is illustrated in Figure

erosional beveling can be seen in the aerial or- 2.6. The early phase of the model is character-

thophotographs shown in Figure 2.4b. As the ter- ized by penetration of the backwearing scarp into

the terrace interior. After κt ≈ 250 m2 , no planar

race age decreases from mid-Pleistocene (Q2a) to

latest Pleistocene--early Holocene (Q3), more pla- remnant remains and the scarp pro¬le can be

nar terraces are preserved and the degree of ter- approximated as a sine function (or cosine func-

tion, depending on where x = 0 is de¬ned) with

race rounding at gullies decreases markedly.

Equations (2.36) and (2.20) can be used to a linearly-decreasing height over time.

model the evolution of gully pro¬les assuming The results of Figure 2.6 nicely illustrate the

that gullies form rapidly following entrenchment ¬ltering aspects of diffusion. A series solution

and that the gully ¬‚oor has a constant elevation is composed of sine functions multiplied by an

40 THE DIFFUSION EQUATION

relatively constant over these time periods. If we

1.0

assume a κ value of 1 m2 /kyr, the estimated age

Q2c

kt = 50 m2

Q2a of the Q2c terrace is 50 kyr and the Q2b terrace is

0.8

150 kyr. These values are reasonably close to the

kt > 240 m2

estimated ages of these terraces based on avail-

0.6

h able geochronology.

Q2b

h0 Normalization is appropriate in comparing

kt = 150 m2

0.4

diffusion-model predictions to natural landforms

because the relative shape of the hillslope does

0.2 not depend on the absolute height of the hill-

slope. In other words, the landform height can

0.0 Hanaupah Canyon, Death Valley, CA be scaled up or down and the relative hillslope

20

10 30

0 40 50

x (m) shape predicted by the diffusion model will re-

main the same.

Fig 2.7 Gully pro¬les of the Q2a, Q2b, and Q2c fan

In the ¬eld, the length of scarp backwearing

terraces at Hanaupah Canyon, Death Valley, California.

can be used as a tool to correlate terraces. The

Best-¬t morphologic ages to the diffusion pro¬le given by

length of erosional penetration into a scarp can

Eq. (2.42) are 240 m2 , 150 m2 , and 50 m2 , respectively. Only a √

be estimated as » = κt. This is a general result

minimum age can be determined for Q2a because the height

for all diffusion problems. This relationship im-

of the original terrace is unknown.

plies that Q2c- and Q2b-aged terraces wear back

by approximately 7 and 12 m, respectively, assum-

ing κ = 1 m2 /kyr.

exponential damping term that is a function of

wavelength. Diffusion acts to ¬lter the small- The similarity solution can be used to esti-

wavelength components of the topography over mate the age of the Q2c and Q2b terraces because

time through the exponential term in the series planar remnants are preserved on these terraces.

solution. For large values of t, all of the small As such, the terrace width can be considered to

wavelength (i.e. large n) terms will be negligible be semi-in¬nite. To solve the problem with the

and the only signi¬cant term will be the largest- similarity method, we consider a semi-in¬nite

wavelength sine function. This is a general re- terrace of height h 0 subject to rapid base-level

drop at h = L . The solution to this problem is

sult: all base-level-controlled hillslopes will ap-

proximate a sine function during their waning

L ’x

h(x, t) = h 0 erf √ (2.43)

stages of evolution, regardless of their initial

2 κt

conditions.

Equation (2.43) is identical to Eq. (2.42) for small

Figure 2.7 illustrates normalized surveyed

values of κt.

cross sections of the Q2a, Q2b, and Q2c terraces

in Hanaupah Canyon along with their best-¬t pro-

2.2.6 Evolution of alpine moraines

¬les to Eq. (2.42). The Q2c scarp is best repre-

sented by Eq. (2.42) with κt = 50 m2 , while Q2b is Advancing alpine glaciers bulldoze rock and de-

best represented by κt = 150 m2 . These κt values bris out along their margins. When the glacier

are sometimes referred to as morphologic ages. retreats, a terminal moraine is formed. Terminal

Only a minimum κt value can be determined for moraines are initially triangular in cross section.

Q2a because no planar remnant remains and we As such, the solution to the diffusion equation

do not know the height of the original terrace. with a triangular initial condition provides a sim-

Therefore, once the terrace pro¬le is normalized, ple model of moraine evolution.

any κt value greater than ≈ 250 m2 matches the In the simplest case of a symmetrical

observed pro¬le equally. These results suggest moraine, only half of the moraine need be con-

that the Q2b terrace on the Hanaupah Canyon sidered. The highest point of the moraine forms a

divide located at x = 0 (Figure 2.8a). At the base of

fan is approximately three times older than the

Q2c terrace, assuming that κ values have been the moraine two different boundary conditions

2.2 ANALYTIC METHODS AND APPLICATIONS 41

1.0 t=0 (a)

t = 20 kyr (Pinedale)

t = 40 kyr

0.8 t = 60 kyr

...

t = 140 kyr (Bull Lake)

0.6

h

h0

0.4

(c)

moraines

0.2

constant elevation at base of moraine

k = 1m2/kyr

0.0

1.0

(b)

t = 20 kyr (Pinedale) formerly

0.8

glaciated valley

t = 140 kyr (Bull Lake)

0.6

h

h0

0.4

0.2

no local base level control

k = 1 m2/kyr

0.0

0 40

20 30 50

10

x (m)

20 ka) moraine, for example, is rounded at the

Fig 2.8 (a) Diffusion model of moraine evolution plotted

top over a width of approximately 10 m. A Bull

for 20 kyr intervals up to t = 200 kyr, assuming a κ value of

1 m2 /kyr, with a ¬xed-elevation boundary condition at the Lake moraine (approx. 140 ka), in contrast, has a

slope base. (b) Same as (a), but with a ¬xed-elevation rounded crest of approximately 30 m in width as-

boundary condition at x = ∞. (c) Virtual oblique aerial suming a κ value of 1 m2 /yr.

photograph of a series of terminal moraines in Owens Valley, Second, we consider the case in which the

Inyo County, California.

moraine deposits sediment at its slope base (such

as when the moraine is surrounded by an allu-

can be considered. First we consider the case of vial piedmont). In this case, sediment shed from

a ¬xed elevation at a distance L from the divide. the moraine will be deposited at the base of the

As in previous examples, this boundary condition moraine and on the surrounding piedmont. The

is most appropriate for cases in which a stable boundary condition in this case is a ¬xed eleva-

tion of zero at x = ∞. For the case of an initially

channel is present at the slope base. Substituting

f (x) = h 0 (L ’ x) into Eq. (2.20) gives the solution triangular hillslope, the solution is

∞

8h 0 1

L ’x

h(x, t) = 2 2h 0

h(x, t) = (L ’ x)erf √ + (L + x)

π (2n + 1)2

L

n=0 4κt

(2n + 1)π x κ(2n+1)2 π 2 t

L +x

’ x

— cos e (2.44)

4L 2

— erf √ ’ 2xerf √

2L 4κt 4κt

κt (x’L )2 (x+L )2 x2

Figure 2.8a shows plots of Eq. (2.45) at 20 kyr e’ + e’ ’ 2e’ 4κt

+2 4κt 4κt

π

increments from t = 20 kyr to t = 200 kyr assum-

ing a κ value of 1 m2 /kyr. A Pinedale-age (approx. (2.45)

42 THE DIFFUSION EQUATION

Figure 2.8b illustrates pro¬les of Eq. (2.45) for and Anderson, 1979; Hanks and Andrews, 1989).

the same ages as in Figure 2.8a. For the case In this approach, the far-¬eld gradient (i.e. the

with a depositional slope base the erosion of the gradient of the initially unfaulted surface) must

moraine crest occurs at a slightly lower rate com- ¬rst be subtracted from the midpoint gradient

pared to the base with a ¬xed base level. The to take into account the effects of scarps cut into

main effect of the deposition is to widen the sloping surfaces. The plot of midpoint gradient

moraine as sediment progrades out on the ad- minus the far-¬eld gradient is compared to char-

jacent slope base. acteristic curves for the diffusion equation with

different parameter values (e.g. Bucknam and

2.2.7 Evolution of pluvial shoreline and Anderson, 1979; Hanks et al., 1984) to determine

fault scarps which parameters best match the observed data.

Both methods are potentially unreliable because

The study of pluvial shoreline and fault scarps

they reduce the information content of the scarp

has had great historical importance in hillslope

to the slope angle or gradient at a single point.

geomorphology. These landforms have been stud-