Surface wash at the semi-arid break in slope


Kirby, A.V.T.; Kirby, M.J.

Zeitschrift für Geomorphologie, Supplementbände 21: 151-176

1974


The break in slope in semi-arid areas is considered to represent neither a break in process nor a break in form but a local concentration of widespread slope concavities. The rapidly changing slope gradient within a narrow zone can be produced by the gradual transition of dominant process from gravity forces to surface wash acting together on material of 2-50 mm diameter. Field measurements of surface wash on unchannelled slopes in Arizona gave a mean transport rate for coarse material (> 1 mm) of 4-5 cm3/cm. yr. This is comparable to rates obtained for fine material on unchannelled slopes over 500-1500 years from archaelogical mounds in a semi-arid area of southern Mexico. For the coarse material the distances travelled increased with increasing slope gradient, amount of unvegetated surface and storm rainfall, and decreased with grain size and relative surface roughness. On natural slopes these factors interact to make overall sediment transport almost independent of gradient. Surface wash in shifting braided channels was found to be 100 times more effective in transporting fine material than unchannelled wash. Measured slope profiles across the break in slope show an overall correlation between slope and grain size which is independent of the sharpness of the concavity. Modelling of the break in slope as an equilibrium form produced by gradually changing combinations of gravitational and hydraulic tractive stresses provides an acceptable envelope curve to the empirical data. The sharpening up of the concavity at the break in slope is thus considered to develop through this transitional combination of present surface processes superimposed on a broadly concave wash slope.

II
Z.
Geomorph.
N.
F.
Suppl.
Bd.
21
151-176
Berlin
Stuttgart
Dezember
1974
Surface
wash
at
the
semi
-arid
break
in
slope
by
ANNE
KIRKBY
and
M.
J.
KIRKBY,
Leeds
with
12
Figures
and
3
Tables
Zusammenfassung.
Der
„Hangknick"
semiarider
Gebiete
wird
weder
als
das
Ergebnis
einer
Anderung
im
Prozell
node
einer
solchen
der
Form,
sondern
als
ortliche
Konzentration
sich
ca
ber
weite
Bereiche
erstreckender
konkaver
Hangteile
angesehen.
Die
sdmelle
Anderung
des
Hangwinkels
kann
als
Ergebnis
des
allmahlichen
Obergangs
von
vorherrschendem
Sdiwerkrafttransport
zur
Abspiilung
bei
einem
Material
von
2-50
mm
Durchmesser
angesehen
werden.
Messungen
der
Abspiilung
auf
unzerschnittenen
Hangen
in
Arizona
ergaben
fur
grobes
Material
(caber
1
mm)
eine
Transportrate
von
4-5
ms
pro
Jahr.
Dieser
Wert
ist
mit
Erosions-
raten
vergleidibar,
die
fiir
einen
Zeitraum
von
500
bis
1500
Jahren
an
vorgeschichtlichen
„Mounds"
im
semi-ariden
Teil
Sildmexikos
gemessen
wurden.
Bei
dem
groben
Material
nahm
der
zuriickgelegte
Weg
mit
wadisender
Hangneigung
und
in
Abhangigkeiten
von
der
nicht
durch
Vegetation
bedeckten
Flache
und
von
Starkregen
zu,
nahm
dagegen
aber
mit
zunehmender
Par-
tikelgroBe
und
zunehmender
Rauhigkeit
der
Oberflache
ab.
An
natiirlichen
Hangen
wirken
all
diese
Faktoren
miteinander
und
machen
den
Gesamtsedimenttransport
nahezu
unabhangig
vom
Gefalle.
Abspiilung
in
anastomosierenden
Rinnen
hat
sich
fur
den
Transport
von
FeinmateriaI
als
hundert
mal
wirkungsvoller
herausgestellt
als
nicht
an
Rinnen
gebundene
Abspillung.
An
den
gemessenen
Hangprofilen
zeigt
side
eine
allgemeine
Korrelation
von
Hangneigung
und
PartikelgroIe,
die
unabhingig
von
der
Stharfe
der
Konkavitat
ist.
Wird
der
Gefallsbruch
im
Modell
als
eine
Gleichgewichtsform
angesehen,
die
sich
aus
der
allmahlich
andernden
Kom-
bination
von
schwerkraftbedingter
und
hydraulischer
Sdileppkraft
ergibt,
resultiert
daraus
eine
annehmbare
Hiillkurve
fur
die
empirischen
Daten.
Die
Zunahme
der
Konkavitat
im
Bereich
des
„Hangknidis"
wird
also
als
das
Ergebnis
dieser
sich
andernden
Kombination
rezenter
Ober-
fl
achenprozesse
angesehen,
die
auf
einem
im
wesentlichen
konkaven
Hang
wirken.
Summary.
The
break
in
slope
in
semi
-arid
areas
is
considered
to
represent
neither
a
break
in
process
nor
a
break
in
form
but
a
local
concentration
of
widespread
slope
concavities.
The
rapidly
changing
slope
gradient
within
a
narrow
zone
can
be
produced
by
the
gradual
transition
of
dominant
process
from
gravity
forces
to
surface
wash
acting
together
on
material
of
2-50
mm
diameter.
152
A.
V.
T.
and
M.
J.
KIRKBY
Field
measurements
of
surface
wash
on
unchannelled
slopes
in
Arizona
gave
a
mean
transport
rate
for
coarse
material
(>
1
mm)
of
4-5
cm
3
/cm.
yr.
This
is
comparable
to
rates
obtained
for
fine
material
on
unchannelled
slopes
over
500-1500
years
from
ardiaelogical
mounds
in
a
semi
-arid
area
of
southern
Mexico.
For
the
coarse
material
the
distances
travelled
increased
with
increasing
slope
gradient,
amount
of
unvegetated
surface
and
storm
rainfall,
and
decreased
with
grain
size
and
relative
surface
roughness.
On
natural
slopes
these
factors
interact
to
make
overall
sediment
transport
almost
independent
of
gradient.
Surface
wash
in
shifting
braided
channels
was
found
to
be
100
times
more
effective
in
transporting
fine
material
than
unchannell-
ed
wash.
Measured
slope
profiles
across
the
break
in
slope
show
an
overall
correlation
between
slope
and
grain
size
which
is
independent
of
the
sharpness
of
the
concavity.
Modelling
of
the
break
in
slope
as
an
equilibrium
form
produced
by
a
gradually
changing
combination
of
gravi-
tational
and
hydraulic
tractive
stresses
provides
an
acceptable
envelope
curve
to
the
empirical
data.
The
sharpening
up
of
the
concavity
at
the
break
in
slope
is
thus
considered
to
develop
through
this
transitional
combination
of
present
surface
processes
superimposed
on
a
broadly
concave
wash
slope.
Resume.
La
rupture
de
pente
des
versants
en
region
semiaride
est
consideree
comme
ne
representant
ni
une
rupture
dans
la
forme,
ni
une
rupture
dans
les
processus,
mais
une
concen-
tration
locale
de
concavites
de
pente
etendues.
Le
rapide
diangement
dans
la
valeur
de
la
pence
dans
un
espace
etroit
peut
etre
produit
par
la
transition
graduelle
de
processus
dominants,
depuis
les
forces
de
gravite
jusqu'au
ruissellement
diffus,
agissant
ensemble
sur
du
materiel
ayant
de
2
a
50
mm
de
diametre.
Des
mesures
sur
le
terrain
du
ruissellement
diffus
sur
des
pentes
non
ravines
en
Arizona
ont
donne
une
vitesse
moyenne
de
transport
pour
le
material
grossier
(>
1
mm)
de
4
a
5
cm3/cm.
an.
Ceci
est
comparable
aux
valeurs
obtenues
pour
des
materiels
fins
sur
pentes
non
ravinees
pendant
500
a
1500
ans,
d'apres
des
tumuli
archeologiques
dans
le
climat
semi-aride
du
Sud
du
Mexique.
Pour
les
materiaux
grossiers,
les
distances
de
transport
croissent
avec
le
gradient
de
pente,
l'importance
de
la
surface
non
couverte
de
vegetation
et
les
pluies
d'orage,
et
decrois-
sent
avec
la
grosseur
du
grain
et
la
rugosite
relative
de
la
surface.
Sur
des
pentes
naturelles,
ces
facteurs
combinent
leurs
effets
pour
rendre
partout
le
transport
de
sediment
presque
indepen-
dant
de
la
pente.
Le
nettoyage
de
la
surface
dans
des
chenaux
amastomoses
qui
se
deplacent
fut
trouve
100
fois
plus
effectif
pour
transporter
le
materiel
que
le
ruissellement
diffus.
Des
profils
de
pente
mesures
a
travers
la
rupture
de
pente
montrent
une
correlation
gene-
ralisee
entre
la
pente
et
la
grosseur
du
grain,
et
qui
est
independante
de
la
brutalite
du
change-
ment
de
pente.
Un
modele
de
la
rupture
de
pence
consideree
comme
une
forme
d'equilibre
pro-
cluite
par
une
combinaison
graduellement
changeante
de
tensions
de
gravitation
et
de
traction
hydraulique
fournit
une
courbe
enveloppe
acceptable
des
donnees
empiriques.
L'accentuation
de
la
concavite
a
la
rupture
de
pente
est
ainsi
consideree
comme
se
developpant
grace
a
la
com-
binaison
transitoire
de
processus
de
surface
actuels
agissant
sur
une
pente
du
ruissellement
largement
concave.
Introduction
In
semi
-arid
areas
mountains
meet
the
plains
below
in
a
visible
discontinuity
of
gradient
termed
the
break
in
slope.
In
detail
the
gradient
is
not
discontinuous
but
changes
sharply
from
a
steep
(greater
than
15°)
to
a
gentle
(less
than
5°)
angle
within
a
short
distance.
The
contrast
between
the
angles
of
the
mountain
slope
and
the
pediment
or
plain
below
is
heightened
by
differences
in
the
size
of
mate
-
rial
on
them
so
that
a
steep
pile
of
large
boulders
appears
to
abut
on
to
a
fine,
Surface
wash
at
the
semi
-arid
break
in
slope
153
s
andy
plain.
In
humid
landscapes
changes
in
gradient
in
slope
profiles
are
more
continuous
and
less
visible.
The
sharpness
of
a
break
in
slope
is
generally
absent.
The
origin
and
development
of
the
semi
-arid
break
in
slope
is
the
concern
of
this
paper.
It
is
a
slope
form
which
has
been
attributed
to
widely
differing
processes
from
spring
sapping
(PEEL
1941)
to
lateral
planation
(
JoHNsoN
1932)
to
subsurface
weathering
(MABBUTT
1955).
As
a
general
feature
in
the
landscape
the
break
in
slope
is
characteristic
of
semi
-arid
and
arid
areas.
Aridity
would there-
fore
seem
to
play
a
key
role
in
its
formation
and
is
the
starting
point
for
most
explanations
of
its
origin.
From
this
point,
theories
tend
to
diverge
along
one
of
three
paths:
1.
that
its
origin
lies
not
in
surface
processes
but
in
subsurface
weathering
and
differential
erosion
related
to
a
priori
differences
in
rock
resistance
(through
lithology,
jointing,
faulting
etc.).
This
approach
has
been
widely
argued
by
KING
(1950)
for
Africa
and
elsewhere
and
supported
by
work
in
south-west
Africa
by
MABBUTT
(1952,
1955).
This
mechanism
is
though
to
be
most
appropriate
in
areas
where
sparse
vegetation
achieved
in
association
with
moderate
rainfalls
and
high
temperatures.
PI
2.
that
the
break
in
landscape
form
must
indicate
a
spatial
break
in
process
(best
stated
by
BRYAN
1922).
The
nature
of
the
break
in
process
advocated
has
reflected
the
characteristics
of
the
detailed
landforms
in
the
different
areas
studied.
For
example,
JOHNSON
(1932)
argued
for
pediments
formed
as
coalescing
rock
fans
in
the
southwestern
United
States
of
America
that
the
break
in
slope
is
trimmed
-up
by
marginal
streams
flowing
parallel
to
the
mountain
front
(also
RAHN
1966).
Alternatively,
where
pediments
are
obviously
not
rock
fans
but
alluvial
in
original,
the
break
in
slope
has
been
ascribed
to
a
spatial
disjuncture
between
erosion
and
deposition.
Even
where
both
the
mountain
slope
and
pediment
are
regarded
as
zones
of
sedi-
ment
transport,
a
real
break
in
the
slope
processes
operating
has
been
envisaged.
For
example,
between
weathering
and
removal
largely
by
gravity
on
the
slope
above,
and
sheetfloods
and
rill
wash
on
the
pediment
below.
Thus,
above
the
break
in
slope
erosion
is
limited
by
weathering
while
below
it
the
processes
are
limited
by
their
ability
to
transport
material.
3.
that
the
break
in
slope
results
from
temporal
changes
in
process
at
that
$oint,
from
erosion
to
deposition
during
local
or
regional
alternations
of
down
-
cutting
and
infilling.
These
alternations
in
process
may
be
attributed
to
climatic
change, geological
movements,
human
intervention,
or
to
a
redistribution
of
erosional
and
deposi-
tional
areas
on
the
plain
(DENNY
1967).
Each
of
these
theories
tends
to
focus
on
features
of
the
landscape
associated
with
the
break
in
slope
that
are
less
universal
than
the
fact
that
it
is
essentially
a
semi
-arid
feature.
Thus,
marginal
streams
are
found
but
they
are
highly
localised;
some
pediments
are
formed
in
rock
with
only
the
thinnest
veneer
of
unconsoli-
dated
material
while
others
are
self
-evidently
large-scale
depositional
features;
well
defined
breaks
in
slope
are
commonly
associated
with
granite
but
not
always
so.
Each
model
or
hypothesis
is
therefore
likely
to
have
greater
or
lesser
generality.
For
example,
it
has
been
clearly
shown
that
in
many
parts
of
Africa
154
A.
V.
T.
and
M.
J.
KIRKBY
and
Australia
breaks
in
slope
are
widely
associated
with
preferential
weathering
phenomena
(Rux-roN
1958,
TWIDALE
1967).
The
work
on
which
this
paper
is
based
took
place
mainly
in
southern
Arizona,
USA.
In
detail
therefore
the
landforms
and
processes
described
are
applicable
only
to
a
small,
and
highly
individualistic
semi
-arid
area.
Our
approach
to
the
problem
of
the
origin
and
development
of
the
break
in
slope
may
have
more
general
validity
that
we
are
dealing
not
with
a
break
in
form
but
the
local
concentration
of
what
is
a
widespread
slope
concavity.
This
local
con-
centration
of
concavity
does
not
require
either
spatial
or
temporal
discontinuity
of
process
to
form
it
but
can
be
achieved
by
a
continuous
gradation
of
surface
processes
presently
operating
on
the
mountain
front
and
pediment.
Field
investigations
in
southern
Arizona
were
concentrated
around
Tucson
and
the
Sacaton
Mountains
to
the
northwest
(fig.
1).
The
area
receives
from
80
to
350
mm
of
rainfall
per
annum,
falling
mainly
in
two
seasons;
a
summer
one
usually
well
defined
and
emphasised
by
dramatic
differences
in
grain
size
above
and
below.
Vegetation
and
human
occupance
is
sparse
and
most
of
the
areas
studied
showed
no
signs
of
human
use
other
than
extensive
animal
grazing
and
the
occasional
visit
by
tourists.
The
Sacaton
site
is
located
in
the
Gila
River
Indian
Reservation.
The
erosional
history
of
the
area
has
been
described
most
graphically
by
BRYAN
(1922),
and
in
greater
detail
by
TUAN
(1959)
and
others.
Yuma
1
Z
0
Phoenix
Sacaton
Mts.West
Mohawk
Mts.
El
Crater
Mts.
Ajois
Sacaton
Mts.
East
(S)
Picacho
Peak
AREA
SHOWN
IN
Fig.2
r
Sta.Catalina
Mts.
-
-
-
Black
Mts.
LA•
TUCSON
Sierrita
Mts
A
A
Whetstone
Mts.
Ste
Rita
Mts.
Fig.
1.
Southern
Arizona.
Location
map
showing
detailed
study
area
of
Figure
2,
and
location
of
slope
profiles
in
associated
15'
topographic
map
sheets.
1
Surface
wash
at
the
semi
-arid
break
in
slope
155
Measurements
of
wash
processes
a)
Field
areas
and
methods
Current
measurements
of
sediment
transport
by
surface
wash
processes
have
been
made
for
both
fine
(<
1
mm)
and
coarse
(>
1
mm)
material
forming
the
pedi-
ment
surface.
In
addition,
rates
of
movement
for
fine
material
over
longer
periods
of
time
(500-1500
years)
have
been
obtained
from
measurements
of
the
changes
in
form
of
dated
archaeological
mounds
in
southern
Mexico
(KIRKBY
&
KIRKBY
1974).
Fine
material
on
the
surface
was
measured
by
erosion
-pins
using
a
method
devised
by
the
US
Geological
Survey
(LEOPOLD,
EMMETT
&
MYRICK
1966).
Coarse
material
transport
rates
were
obtained
by
erosion
-lines
in
a
method
described
below.
0
Rai
ngauge
Measurement
site
100
Rainfall
in
mm.
0
2
MILES
14
0
48
0
Tucson
14.8
Cortaro
,iNurser
es
0
0
,...--
8.26
0
194.
Mt.Lemmon
A
°Mt.
Lemmon
Inn
CATALI
NA
11
0
1
1ill
li
D
C
Tucson
Camp
Exptl
Farm
O
Sabina
Canyon
144
150
12
"- 11
44
i
E
/
0
.
6
,
MOUNTAINS
152
\\
120
0
otA.••
.
.
TUCSON
218
0
Tucson
W
B
Airport
odw
Tucson
Mag.0b.S.
84
0
Lazy
H
Ranch
rrrr\
\
A‘
\\
1.
1
10\
3215V-
TANQUE
VERDE
MTS.
Wi
ill.,
IA
.1
I
I
Pig.
2.
Detailed
study
area
near
Tucson,
Arizona,
showing
measurement
sites
(lettered),
and
total
rainfall
during
the
July
—August
1964
measurement
period.
156
A.
V.
T.
and
M.
J.
KIRKBY
The
movement
of
both
coarse
and
fine
sediments
were
measured
in
the
area
around
Tucson,
Arizona.
Erosion
-line
sites
for
coarse
material
were
set
up
in
twelve
locations
(Fig.
2:
sites
C
-F
and
H
-N)
and
monitored
over
a
two
-month
period
during
the
summer
rainstorm
season
(
July
—August
1964).
During
this
time
different
sites
received
between
80
mm
and
220
mm
of
rain.
The
movement
of
material
was
measured
for
individual
rainstorms
at
each
site
at
a
total
of
70
storms
and
sites
in
all.
Fine
material
movement
was
recorded
at
one
of
the
above
sites;
in
the
Sacaton
Mountains
(location
E
in
Fig.
1),
where
erosion
-pins
were
installed
in
addition
to
erosion
-lines.
The
erosion
-pins
have
provided
transport
rates
over
longer
periods
of
time;
for
six
months
(January
to
July
1964)
and
12
months
(August
1964
—August
1965).
b)
Coarse
material
i)
Erosion
-lines
The
erosion
-line
method
of
surface
slope
transport
is
very
simple.
The
installa-
tions
at
a
single
field
site
consist
of
two
(and
up
to
five)
thin
lines
of
enamel
paint
sprayed
across
the
slope
surface
roughly
parallel
to
the
contour
and
to
each
other;
and
a
crest
-stage
rain
gauge.
Each
line
is
between
15
and
60
m
long
and
has
an
average
width
of
18
mm
below
which
it
was
not
possible
to
spray.
One
line
is
located
above
the
visual
break
in
slope
(on
10
to
20°
gradient)
and
the
other
below
it
(on
2-15°
gradient).
The
crest
-stage
rain
gauge
used
consisted
of
two
open
vessels;
the
smaller
one
inverted
inside
the
larger
and
containing
powdered
cork.
Rain
is
caught
in
the
outside
container
and
its
level
recorded
by
the
height
of
the
powdered
cork
left
on
the
inside
surface
of
the
inverted
inner
vessel
which
remains
at
the
highest
water
level
after
the
rain
has
evaporated.
The
inner
container
protects
its
contents
from rain
splash.
Once
the
painted
erosion
-lines
and
the
rain
gauge
have
been
installed
and
the
number
of
particles
of
each
size
class
per
unit
length
of
each
line
has
been
measured,
the
site
is
left
until
rain
falls
on
it.
In
the
Tucson
area,
summer
rain
occurs
as
localised
storms
so
that
it
was
possible
to
identify
slope
movement
recorded
with
individual
storm
rainfalls
on
each
site.
Measurement
of
surface
transport
was
made
by
counting
the
number
of
particles
of
each
given
size
that
had
moved
for
specified
distances
(in
units
of
inches).
For
this
study
the
size
classes
used
were
1-2
mm;
4-8
mm;
8-25
mm;
25-50
mm;
50-100
mm
etc.
The
count
is
continued
along
the
line
until
about
300
particle
movements
have
been
recorded
(100
to
1500),
or
the
end
of
the
line
is
reached.
Particle
sizes
are
deter-
mined
by
a
scale
and
distances
measured
with
a
steel
tape.
After
measurement
has
been
completed,
each
line
is
resprayed
in
a
different
colour
so
that
move-
ments
due
to
individual
storms
could
be
separated
and
errors
due
to
non
-recovery
of
particles
would
not
be
compounded.
The
amount
of
rainfall
is
recorded
and
the
rain
gauge
set
again
for
the
next
storm.
Limitations
to
the
method
are
that
movements
of
less
than
one
inch
are
not
recorded
owing
to
the
minimum
width
of
line
that
can
be
sprayed
on
the
surface.
Where
the
erosion
line
crosses
even
a
small
wash
or
line
of
concentrated
flow,
recovery
of
the
painted
particles
is
too
unreliable
for
accurate
transport
measure-
Surface
wash
at
the
semi
-arid
break
in
slope
157
m
ent
and
these
sections
of
the
line
were
ignored.
Similiarly,
particle
recovery
declines
with
increasing
distance
moved
so
that
the
method
is
less
reliable
for
longer
distances
of
travel
and
thus
for
longer
periods
of
time.
The
transport
m
easured
will
be
a
minimum
due
to
non
-recovery
of
particles
and
also
to
turning
o
ver
of
particles
so
that
their
unpainted
faces
are
uppermost.
It
is
not
expected
however
that
actual
transport
rates
are
more
than
double
the
observed
rates.
For
the
twelve
erosion
-line
sites
installed
around
Tucson,
daily
inspections
were
made
after
each
rainstorm
and
as
far
as
possible
during
storms
so
that
the
processes
could
be
actually
observed
and
filmed.
In
the
two
-month
period
a
total
of
70
line
measurements
were
obtained.
The
total
mass
transport
was
found
by
multiplying
the
distance
moved
by
the
cross-sectional
area
and
summing
for
all
particles.
The
rate
is
expressed
in
cubic
centimetres
of
material
crossing
each
l
r
centimetre
length
of
line.
Comparison
of
the
number
of
particles
in
each
size
class
on
the
line
initially
with
the
number
moving
provides
the
mean
distance
moved
for
each
particle
size.
Correction
for
the
number
of
particles
on
the
line
in
com-
parison
with
the
area
occupied
by
the
line
enables
a
grain
size
analysis
to
be
calculated
for
the
surface
material
at
that
point.
o
ii)
Observations
uring
rainstorms
coarse
surface
material
appears
to
move
in
two
distinct
ways
depending
on
the
amount
of
water
covering
the
pediment
surface.
When
there
is
no
free
surface
water,
raindrops
split
up
on
impact
and
smaller
droplets
are
thro
\\
n
back
up
into
the
air
carrying
with
them
small
particles
of
surface
material
(ELLisoN
1945).
Particles
up
to
5
mm
diameter
were
seen
to
be
transported
in
this
way
for
distances
up
to
150
mm
at
a
time.
This
rainsplash
transport
is
inhibited
by
(a)
any
vegetation
and
(b)
free
water
on
the
surface
of
5
mm
or
more
in
depth.
The
presence
of
free
water
on
the
surface
enabled
movement
by
hydraulic
action
to
take
place
which
is
less
inhibited
by
vegetation.
During
the
course
of
the
rainy
season
therefore
the
effect
of
the
rain
in
increasing
the
grass
cover
is
to
reduce
movement
by
rainsplash
much
more
than
movement
by
hydrau-
lic
action.
Neither
rainsplash
nor
hydraulic
action
is
effective
in
moving
stones
larger
than
50
mm
in
diameter
even
on
relatively
steep
slopes
except
by
undermining
th
e
downslope
side
of
boulders
through
removal
of
the
fine
material
supporting
em.
The
free
-flowing
water
did
not
flow
in
a
uniform
sheet
although
at
high
infall
intensities
each
particle
of
material
was
covered
by
a
thin
film
of
water
over
its
upper
surface
held
on
by
surface
tension.
Surface
water
which
was
free
-
flowing
was
always
unevenly
distributed
spatially
and
moving
in
concentrated
flows
which
however
were
unstable
in
their
form
and
location
so
that
any
part
of
the
pediment
surface
might
become
the
temporary
location
of
a
short-lived
con-
centrated
flow.
This
flow
takes
place
in
ill-defined
and
discontinuous
rills
whose
depths
are
little
greater
than
the
irregularities
found
everywhere
on
the
uncon-
solidated
surface.
The
rills
have
no
definite
banks
or
channel
form
that
can
be
Seen
after
the
flow
has
disappeared
and
at
very
high
flows
they
coalesce
laterally
(as
described
in
detail
by
EMMETT
1970).
158
A.
V.
T.
and
M.
J.
KIRKBY
Rainsplash
transport
and
concentrated
flow
thus
both
occur
on
slopes
that
remain
unchannelled.
The
rate
of
transport
is
therefore
more
or
less
independent
of
distance
from
the
divide
since
the
surface
water
has
not
travelled
any
appre-
ciable
distance
during
its
period
of
transporting
-work;
and
only
in
continuous
channels
is
there
any
systematic
increase
in
discharge
with
increasing
distance
from
the
divide.
By
concentrating
on
surface
flows
between
channels
only,
we
can
expect
little
dependence
of
total
transporting
capacity
on
distance
from
divide
or
on
other
measures
of
discharge.
The
effect
of
vegetation
in
preventing
rainsplash
is
visually
very
striking,
in
that
erosion
-lines
beneath
overhanging
shrubs
and
grass
are
almost
totally
protected,
even
though
the
cover
provided
appears
to
be
far
from
complete.
Qualitative
evidence
showing
the
long-term
result
of
this
protection
may
be
seen
in
the
low
(10-30
cm
high)
mounds
beneath
every
perennial
shrub.
Erosion
-
lines
painted
round
the
edge
of
these
mounds
recorded
net
inward
movement
up
the
slope
of
the
mound.
This
effect
appeared
to
be
due
to
driving
rain,
which
was
able
to
penetrate
beneath
the
shrubs
from
the
outside,
but not
from
the
inside.
Although
sediment
movement
by
wind
and
burrowing
animals
is
also
recognised
as
important
in
the
formation
of
vegetation
mounds,
it
is
evident
that
rainsplash
is
a
major
factor.
A
sequence
of
erosion
-lines
at
different
elevations
in
Oaxaca,
Mexico
shows
the
effect
of
vegetation
more
directly.
At
a
rainfall
of
600
mm
under
a
sparse
cactus-scub
vegetation,
bare
and
naturally
vegetated
surfaces
showed
no
appreciable
difference
in
erosion
-line
destruction.
Under
a
scrub
-oak
cover
and
700
mm
rainfall,
bare
areas
showed
a
3.8
times
increase
in
erosion
rates;
and
under
oak
-pine
forest
with
900
mm
rainfall
a
45
times
increase
in
erosion
100
50
0
=
20
u
E
010
>.
3
5
O
L'S
rtl
r
o
,
2
1
001
01
1
10
30
50
70
°/0
of
particles
moving
at
least
giving
distance
(probability
scale)
1
1 1
1
6
0
slope
17
0
slope
1-2mm
0
o
2-4mm
x
0
4-8mm
A
o
0
x
o•t
c>
A
0.
1 1 1 1
1
Fig.
3.
Cumulative
frequency
distribution
for
distances
moved
by
small
stones
on
and
17°
slopes
at
site
E,
following
a
16
mm
rain-
storm.
Surface
wash
at
the
semi
-arid
break
in
slope
159
was
recorded
(FLANNERY
et
al.,
1967).
On
the
basis
of
these
observations,
all
me
asurements
of
erosion
rate
have
been
referred
to
the
length
of
unvegetated
erosion
-line.
iii)
Analysis
of
results
Measurements
of
travel
distance
showed
movements
of
over
50
cm
in
a
single
storm
for
material
up
to
4
mm
in
dameter,
and
lesser
movements
for
larger
stones.
Short
distances
of
travel
could
not
be
measured
reliably
due
to
the
finite
(18
mm)
width
of
the
erosion
line.
Referring
the
numbers
moving
to
the
total
number
originally
present
in
the
line,
the
distributions
approximate
well
to
either
inverse
exponential
or
log
-normal
distributions.
An
example
from
site
E
after
a
16
mm
storm
is
shown
in
Fig.
3,
referred
to
a
log
-normal
distribution.
The
inverse
exponential
distribution;
(1)
p
(travel
distance
>
x)
=
e
x
17
(
is
perhaps
theoretically
preferable,
corresponding
to
a
Poisson
process
in
which
the
probability
of
coming
to
rest
is
constant
with
distance.
200
100
50
20
Fig.
4.
Total
distances
moved
in
July—
a.
00
2
August
1964
for
stones
grouped
in
size
74
and
slope
categories.
Comparative
data
1
E
for
20-50
mm
markers
on
fine-grained
a
shale
slopes
and
laboratory
experiments
0.5
on
uniform
grain
-size
slopes
for
1-2
mm
material
are
shown
in
an
arbitrary
relative
position.
0.2
0.03
0.05
01
0.2
0.5
1.0
_
4
7,
Itschrift
fur
Geomorphologie
N.
F.
Suppl.
Bd.
21
-
0
-
+
0
cc
,
x
Sine
of
slope
angle
Arizona
data
for
July-Aug.1964
/o
1-2mm
stones
A
4-8
mm
stones
/x
2-4mm
stones
/+
8-25mm
stones
Schumm(1964)
data
showing
annual
rates
,'•
20-50mm
markers
Lab
data
for
stones
forming
surfaces
of
uniform
grain
size
(relative
values)
.
A
1
-
1-2mm
stones
160
A.
V.
T.
and
M.
J.
KIRKBY
The
distances
of
travel
showed
a
strong
dependence
on
both
grain
size
(negative)
and
ground
slope
(positive),
though
this
was
to
some
extent
masked
by
local
effects.
Overall
trends
have
been
obtained
by
summing
the
mean
distances
of
travel
throughout
the
period
of
measurement,
and
plotting
averages
for
each
grain
size
and
slope
category.
The
results
given
in
Fig.
4,
show
a
dependence
on
grain
size,
and
a
dependence
on
gradient
which
may
be
somewhat
less
for
finer
material.
Data
for
20-50
mm
markers
on
fine-grained
shale
slopes
(ScHumisi
1964),
and
for
laboratory
experiments
of
rainsplash
on
surfaces
with
uniform
grain
size
(1-2
mm)
are
also
shown.
The
absolute
values
of
movement
are
not
comparable
but
in
each
case
the
rate
of
increase
with
gradient
is
much
greater
than
that
for
grain
size,
approximating
to
a
sine
-square
law.
It
is
suggested
that
on
natural
slopes,
where
mean
grain
size
increases
rapidly
with
slope,
small
stones
are
more
and
more
impeded
by
the
roughness
of
the
surface.
This
roughness
factor
partly
compensates
for
the
increased
gradient
in
accordance
with
Le
Chatelier's
principle.
In
contrast,
ScHumm's
data
refer
to
artificial
markers
on
uniformly
fine-grained
slopes,
and
the
laboratory
data
are
for
constant
grain
-
size
with
changing
gradient.
In
both
cases
roughness
remains
constant
as
gradient
changes.
This
point
is
taken
up
again
below
in
the
context
of
the
observed
variation
of
grain
size
with
gradient
on
natural
slopes
in
Arizona.
Turning
from
distances
of
travel
to
total
rates
of
sediment
transport,
it
is
clear
that
there
is
no
similar
clear
dependence
on
slope
gradient,
because
the
increase
in
slope
is
roughly
counterbalanced
by
the
reduced
mobility
of
the
coarse
debris
on
the
steeper
slope.
Instead,
transport
rates
for
the
several
erosion
lines
100
50-
-
E
E
a
a
-
ID
-
E
OH
I/
.
•I
•N
•1
•N
•1
•M
•0
•K
E.
H••0
H.
81111
•0
m
•I
•1
•K
•L
005
01
02
05
•1
2
.5
2
Transport
Rate
in
cm
3
/cm
of
unvegetated
slope.
Fig.
5.
Transport
rate
in
a
storm
as
a
function
of
storm
rainfall.
Rates
are
referred
to
unit
length
of
unvegetated
slope.
Rates
for
different
slope
gradients
show
no
significant
difference
and
have
been
averaged.
The
best
-fit
line
is
S
=
6.3
X
10-
5
r23.
(
Surface
wash
at
the
semi
-arid
break
in
slope
161
at
each
site
(on
different
slopes)
have
been
averaged
to
give
an
average
transport
rate
per
unit
length
of
unvegetated
line
which
has
been
plotted
against
total
storm
rainfall
(Fig.
5).
The
best
-fit
relationship,
(2)
S
=
6.3
X
10
-5
r
2
'
5
,
where
r
=
storm
rainfall
in
mm
and
S
=
transport
rate
in
cm
3
/cm
of
unvegetated
slope,
shows
an
appreciable
scatter,
which
is
perhaps
not
surprising
since
storm
rainfall
is
acting
as
a
surrogate
for
both
rainfall
and
intensity
factors.
The
results
of
this
correlation
have
been
used
in
two
ways.
First
to
produce
an
'effective
rainfall'
on
the
occasions
when
more
than
one
rain
elapsed
between
site
visits.
Total
effective
rainfall
was
thus
calculated
as:
(
3
)
4
(r.
2
.
5
)
10.4
Second,
the
rainfall
correlation
could
be
combined
with
frequency
distribu-
tions
of
daily
rainfall
(since
it
is
rare
to
find
more
than
one
storm
on
a
day)
to
extrapolate
to
meaningful
annual
totals
on
unvegetated
soil.
This
exercise
has
been
carried
out
for
Tucson
and
Needles,
Arizona
in
Fig.
6.
In
Tucson,
with
a
mean
annual
rainfall
of
325
mm,
the
total
transport
is
4.2
cm
3
/cm.yr,
the
geo-
Recurrence
intervals
in
years
for
Tucson
095
0;1
02
0;5
1
0
2
192050
Recurrence
intervals
in
year
for
Needles
0;1
0.2
0.5
1.0
2
5
10
50
20
100
100
100
Frequency
density
Need
ies.
c%)
(days/year)
10
10
10
01
1
0
0.1
001
001
10
0
01
.
10
Rate
of
debris
transport
(cm
3
/cm
day)
Total
debris
transport
density
at
given
rate
(cm
3
/cm
year)
Fig.
6.
Frequency
distribution
for
daily
rainsplash
transport
at
Tucson
and
Needles,
Arizona
(solid
lines);
and
distribution
of
total
debris
transport
at
a
given
rate,
showing
geomorphically
dominant
event
sizes.
Curves
computed
from
weather
bureau
daily
rainfall
data,
combined
with
the
best
-fit
relationship
of
Figure
5.
162
A.
V.
T.
and
M.
J.
KIRKBY
morphically
dominant
event
having
a
recurrence
interval
of
about
1
year.
In
Needles,
with
a
rainfall
of
125
mm,
the
total
transport
is
0.5
cm
3
/cm.yr,
and
the
dominant
event
has
a
RI
of
about
0.5
years
(Fig.
6).
A
number
of
such
calculations
suggests
an
empirical
relationship
between
annual
sediment
transport
and
annual
rainfall,
for
sites
with
a
comparable
pattern
of
rainfall
intensities.
On
unvegetated
ground,
(4)
S
=
1.2
R
2
.
2
,
where
R
is
annual
precipitation
in
mm.
On
natural
sites
the
effect
of
increasing
rainfall
is
counterbalanced
by
an
increasing
density
of
vegetation
cover.
This
produces
a
peaked
curve
for
sediment
transport
under
natural
vegetation,
similar
in
form
to
the
well-known
curve
of
LANGBEIN
&
ScHumm
(1958).
Using
generalised
values
for
vegetation
cover
in
the
Southern
United
States
the
peak
for
rainsplash
and
unconcentrated
wash
appears
to
correspond
to
a
rainfall
of
300-400
mm
(KIRKBY
1969).
c)
Fine
material
Data
on
the
movement
of
fine
material
have
been
obtained
from
the
Eastern
Sacaton
Mountains,
Arizona
and
from
the
Valley
of
Oaxaca,
in
southern
Mexico.
In
the
Sacaton
Mountains,
in
addition
to
an
erosion
-line
site,
a
Vigil
site
was
installed
as
part
of
the
Vigil
Network
(LEoPoLD
1962)
in
order
to
measure
the
movement
of
fine
material
and
to
monitor
cut
and
fill
in
the
washes
as
well
as
the
interfluves.
This
was
particularly
important
as
the
erosion
-line
method
did
not
permit
measurement
of
sediment
transport
in
the
washes.
In
the
Valley
of
Oaxaca
transportation
rates
of
fine
material
were
obtained
for
unchannelled
flow
by
the
measurement of
changes
in
the
profile
form
of
dated
archaeological
mounds.
Together
the
two
sets
of
data
enable
us
to
compare
unchannelled
transport
rates
with
sediment
movement
within
channels
for
fine
material.
Com-
parison
of
the
erosion
-line
data
with
the
archaeological
mound
data
also
provides
a
comparison
of
unchannelled
(interfluve)
transport
rates
for
coarse
and
fine
material.
i)
Sacaton
Mountains,
Arizona
The
Vigil
site
established
in
the
Sacatons
in
February
1964
(site
S
in
Fig.
1)
included
three
lines
of
erosion
-pins
which
were
resurveyed
in
August
1964
and
again
in
August
1965.
The
eastern
Sacatons
are
composed
of
granite
and
possess
a
sharply
defined
break
in
slope
with
a
smooth
pediment
profile
below
(Fig.
7).
The
pediment
surface
is
undissected
and
drained
by
a
shifting
braided
network
of
superficial
washes
less
than
a
meter
deep
and
commonly
incised
only
a
few
centimeters.
These
washes
begin
about
50
metres
below
the
break
in
slope;
that
is,
in
the
area
within
which
the
measurements
were
taken.
Generally,
therefore
local
relief
across
the
pediment
is
only
0.2
to
1.0
metres
near
the
Vigil
Site
(Fig.
7
b
—d).
Vegetation
on
the
pediment
is
sparse,
consisting
mainly
of
widely
scattered
palo
verde
and
mesquite
trees
and
saguaro
cactus.
Vegetation
mounds
Surface
wash
at
the
semi
-arid
break
in
slope
163
a)
Long
Profile
with
2
x
V.E.
37°
\
100'1.
95•
15
7i
60*/.
15
.
/.
15•/.
11.9
6.2°46°
so
'
0
(b)
Line
®
20
x
YE.
(c)
Line
c
20
x
V.E.
(d)
Line
®
20
x
V.E.
10*/.
2.8°
2
2V.
0*/.
1*/*
2.3°
2.1°
1.6°
®
11°
20m1
0
*/.
19°
®
100m
2M
1
100m
Fig.
7.
a.
Long
profile
showing
the
break
in
slope
and
pediment
surface
of
the
East
Sacaton
Mtns
at
the
Vigil
Site
(S
in
Fig.
1).
Points
1,
3
and
5
are
reference
points
in
(b),
(c),
(d)
below
and
Fig.
9.
or
b.
-d.
Cross
sections
of
pediment
surface
along
erosion
pin
lines.
Pins
are
located
alternately
in
and
between
channels
as
shown.
Elevation
differences
between
(b),
(c)
and
(d)
are
shown
to
scale.
are
common.
Median
grain
sizes
of
the
pediment
surface
are
lower
(0.5-0.8
mm)
than
the
smallest
particles
recorded
on
the
erosion
-lines
so
that
the
results
obtained
here
largely
reflect
the
movement
of
fine
material.
The
three
lines
of
erosion
-pins
were
put
in
at
60
m,
250
m
and
340
m
distance
downslope
from
the
break
in
slope
and
each
pin
was
installed
on
an
interfluve
and
a
wash
area
alternately.
Details
of
the
lines
and
the
local
situations
are
given
in
Table
1.
Each
pin
consists
of
a
25
cm
long
carriage
bolt
with
a
diameter
of
10
mm
and
a
loosely
fitting
washer.
The
pin
is
inserted
vertically
into
the
ground
leaving
the
washer
flush
with
the
ground
surface.
When
erosion
occurs
the
washer
is
lowered
to
the
new
ground
level
and
will
remain
there
even
if
there
is
subse-
quent
deposition.
This
simple
device
can
therefore
record
both
cut
and
fill
(but
Surface
wash
at
the
semi
-arid
break
in
slope
165
erosion
leads
to
a
general
lowering
of
the
pediment
surface
(Table
3).
The
infilling
of
washes
in
the
upper
pediment
where
one
might
expect
downcutting
to
be
most
active
is
a
function
of
the
highly
transitory
nature
of
the
washes
them-
selves.
Downcutting
is
achieved
almost
instantaneously
during
the
course
of
a
storm
and
because
there
is
no
valley
development
leading
to
the
concentration
of
future
erosion
within
the
channel,
subsequent
downcutting
is
unlikely
to
strike
the
same
spot
twice.
Thus
a
wash
of
a
few
centimetres
deep
once
formed
is
likely
to
be
subject
mainly
to
net
deposition
during
ensuing
rainstorms.
Many
of
the
inter
-wash
erosion
-pins
showed
no
movement
during
the
period
of
measurement
so
that
the
average
figures
are
dominated
by
a
few
large
values,
mainly
of
cut.
The
data
do
not
allow
any
conclusions
about
the
seasonality
of
cut
and
fill
since
the
difference
between
the
periods
recorded
is
largely
a
function
of
one
high
value
of
deposition
(9
mm)
at
a
single
point.
ii)
Valley
of
Oaxaca,
Mexico
In
the
semi
-arid
highland
valley
of
Oaxaca
with
a
rainfall
of
600
mm
per
year,
mean
rates
of
unchannelled
transport
of
fine
material
were
obtained
for
longer
periods
of
time
(500
to
1500
years)
using
dated
archaeological
house
mounds.
The
mounds
are
of
the
order
of
30
to
70
metres
across
and
1
metre
high
in
their
present
form
and
are
entirely
composed
of
fine
material
(silt
to
fine
sand)
inter-
mixed
with
broken
pottery
fragments.
Mounds
abandoned
at
different
times
between
1500
and
500
years
BP
show
a
progressive
lowering
and
spreading
out
over
time
from
an
initial
talus
about
10
metres
across
formed
by
the
collapse
of
a
mud
-brick
house.
.4
On
the
basis
of
a
simple
mathematical
model
in
which
the
transport
rate
is
assumed
proportional
to
the
slope
it
can
be
shown
that
the
mound
profile
appro-
'
ximates
to
a
normal
curve,
whose
variance
increases
linearly
with
time
and
is
independent
of
the
plan
shape
of
the
mound
(KIRKBY
&
KIRKBY,
1974);
that
is,
(
5
)
a
2
=
2
Dt,
where
the
transport
rate
(S)
is
(6)
S
=
D.s
Pig.
8
shows
the
variance
over
time
for
eight
house
mounds
measured
in
which
the
slope
of
the
line
gives
a
long
term
average
for
D
of
1070
cm'/cm
yr.
If
«e
translate
this
rate
to
the
rainfall
(on
the
basis
of
R
2
.
2
Equation
(4)
above)
and
slope
(2°)
applicable
for
silt
-fine
sand
in
Arizona,
a
value
of
11
6
1070
sin
(6r)
2.2
=
5.4
cma/cm
yr
is
obtained.
d)
Implications
of
measurements
The
transport
rates
for
unchannelled
surface
wash
(rainsplash
and
concentrated
flow)
are
similar
for
the
movement
of
both
coarse
and
fine
material
(4.2
cms/cm
166
500-
t
300
-
"E
c4
200
b
1CO
A.
V.
T.
and
M.
J.
KIRKBY
VARIATION
OF
MOUND
DISPERSION
RATE
OVER
TIME
San
Jose
Zaachila
Macuilsochitt
Zaachilo
Mitta
'B
San
Lorenzo
Mina
San
Jose
.
B
.
0
500
1000
1500
Time
since
abandonment
(
years
BPI
Fig.
8.
Variation
of
housemound
variance
over
time
for
a
series
of
dated
mounds
in
the
Oaxaca
Valley,
Mexico.
The
diffusivity,
D,
is
equal
to
the
constant
of
pro-
portionality
relating
transport
rate
to
slope
in
a
simple
model.
yr
and
5.4
ce/cm
yr
respectively).
Big
differences
in
rate
are
related
not
to
sedi-
ment
size
but
to
the
channelled-unchannelled
nature
of
the
surface
wash
flows.
For
the
Sacaton
Mountains,
if
the
measured
lowering
rates
are
extrapolated
back
to
the
divide
(which is
290
metres
above
the
highest
line
at
X),
overall
transport
rates
are
445
cm
3
/cm
yr
for
60
metres
below
the
break
in
slope
and
802
cma/cm
yr
at
340
metres.
These
rates
for
channelled
wash
are
about
a
hundred
times
greater
than
the
rates
for
unchannelled
surface
wash.
Thus
most
of
the
sediment
is
transported
in
channel
processes.
This
can
be
seen
in
the
difference
between
the
overall
transport
rates
for
the
pediment
surface
and
the
much
lower
rates
for
the
interfluve
areas
discussed
above.
It
can
also
be
seen
from
the
density
of
shifting
braided
channels
on
the
pediment
surface;
the
amount
of
cut
and
fill
recorded
in
them
and
their
spatial
development
over
a
six
month
period
(Fig.
9).
In
the
short
term
at
least,
braided
channel
networks
can
be
associated
with
degradation
of
the
pediment
surface.
The
major
conclusion
from
these
field
measurements
of
surface
wash
pro-
cesses
is
that
on
natural
slopes,
rates
of
sediment
transport
by
unchannelled
pro-
cesses
are
more
or
less
independent
of
slope
gradient.
Average
rates
are
about
5
crn
3
/cm
yr
for
areas
in
Arizona
with
250
mm
rainfall.
This
independence
of
transport
rate
on
gradient
is
despite
a
strong
dependence
of
particle
travel
distance
on
grain
size
and
slope.
Vegetation
cover
is
of
paramount
importance
in
its
effect on
transport
rates.
Break
in
slope
forms
In
addition
to
the
measurements
of
wash
processes
described
above,
the
form
of
the
break
in
slope
was
surveyed
along
12
profiles
in
southern
Arizona
(Fig.
11
and
for
another
22
slope
profiles
in
other
parts
of
the
western
United
States.
Slope
profiles
were
surveyed
with
Abney
clinometer
and
tape.
Grain
size
analyses
Surface
wash
at
the
semi
-arid
break
in
slope
...-
t
I
,...
3'nick
absent
This
section
2-64
now
absent
8-64.
or
/s
/
Vi
3"nick
en
i
c
\2"
-
-
.
...."
4
.
k
2-64-
8-64.
,
0
i
4
..
ii
nick
11
/4
4
N
(..
"....
4•
1
"
2
..nicks
I t
il4
0,
t
%
..,..
"
...
1 ,
../
...).
4nick
2-64
absent
8-64
a
5
Metres
jo
167
POINT
3
Fig.
9.
Changes
in
the
braided
channel
system
on
a
30
X
30
m
quadrat
near
reference
point
3
(Fig.
7).
Black
areas
show
changes
from
February
(2-64)
to
August
(8-64)
1964.
were
made
along
the
profile
by
stretching
the
tape
across
the
surface
and
summing
the
proportion
of
the
tape
length
intercepted
by
stones
in
each
size
category
(25-50;
50
-
100;
100-200
mm
etc.).
Thus,
for
example,
a
stone
of
between
50-100
mm
diameter
which
intercepts
the
line
for
25
mm
distance,
will
add
25
mm
to
the
50-100
mm
category.
Theoretically
this
simple
technique
provides
a
grain
size
distribution
which
is
comparable
to
a
sieve
analysis
of
the
surface
coarse
material.
168
A.
V.
T.
and
M.
J.
KIRKBY
Comparison
of
the
form
of
slope
profiles
for
different
climatic
and
topo-
graphic
conditions
throughout
the
western
United
States
including
southern
Arizona
show
that
sharp
breaks
in
slope
arise
in
two
distinct
contexts.
1.
In
semi
arid
areas
where
present
annual
rainfall
is
less
than
375
mm
(in
US
conditions).
Near
this
boundary
value
of
rainfall,
it
is
the
presence
or
absence
of
vege-
tation
which
is
most
clearly
related
to
and
diagnostic
of)
the
development
of
typically
humid
or
arid
slope
forms.
In
turn
vegetation
cover
is
of
course
related
to
other
factors
such
as
lithology
and
aspect.
Where
a
visible
break
in
slope
occurs,
it
is
always
associated
with
a
progressive
decrease
of
grain
size
with
slope
gradient
down
to,
and
across
the
plain
below.
2.
In
mountain
regions
where
talus
slopes
have
been
actively
accumulating.
These
talus
slopes
form
at
an
angle
distinctly
below
their
static
angle
of
repose,
because
their
stability
depends
on
the
ability
to
bring
moving
stones
to
rest.
They
have
a
concave
profile
in
their
lower
portions
over
which
surface
material
becomes
coarser
in
the
downslope
direction.
Beyond
the
foot
of
the
talus
spread
itself,
grain
size
is
determined
by
independent
factors.
60%
7
4,
33°
'•
98%
50
0
1
Horizontal
and
vertical
scale
3X
45%
Picacho
Peak,
basalt
17°
10°
45
.
/.
11°
Santa
Catalina
Mts
.
schist
13
.
/.
Summit
12°
32
0
W
Sacton
Nits:
granite
24°
30°
-,
.
20%
tgiy.
1;
7
r-
70
Fig.
10.
Representative
profiles
across
breaks
in
slope
on
basalt,
schist
and
granite,
showing
percentage
of
material
>
25
mm
(above
profile)
and
slope
angle
in
degrees
(below
profile).
The
talus
section
on
the
upper
profile
is
an
active
section
with
increasing
grain
size
downslope.
For
site
locations
see
Fig.
1.
The
size
grading
of
surface
material
in
the
region
of
the
break
in
slope
is
very
diagnostic
of
accumulation
forms (increasing
grain
size
downslope)
or
ero-
sional
forms
(decreasing
grain
size
downslope).
Where
the
two
contexts
(in
which
breaks
in
slope
are
found)
occur
together
below
semi
-arid
cliffs
there
is
charac-
teristically
first
an
increase
and
then
a
decrease
of
material
size
downslope.
The
limiting
case
is
where
the
slope
has
formed
as
a
scree
but
is
now
under-
going
weathering
in
situ,
and
boulder
material
is
removed
only
after
it
has
been
reduced
to
fine
grain
sizes.
This
situation
corresponds
closely
to
the
conditions
on
the
semi
-arid
mountain
slopes
we
are
considering.
It
is
described
by
the
mathe-
29%
12*
100m.
35
.
/.
10°
14
.
/.
S
.
/.
12°
1
11
ma
tical
model
of
BAKKER
&
LE
HEUX
(1952)
in
which
the
resulting
form
is
shown
to
be
a
straight
boulder
-veneered
slope
at
the
angle
of
rest
of
the
boulders.
The
simple
model
BAKKER
&
LE
HEUX
describe
is
complicated
by
three
considerations:
1.
the
failure
of
scree
slopes
to
attain
the
theoretical
angle
of
rest
because
of
the
dynamics
of
slope
movement;
2.
the
widespread
concavity
of
natural
scree
profiles;
and
3.
the
reworking
of
scree
debris
by
other
slope
processes,
particularly
creep,
which
further
reduce
slope
angle
and
ultimately
bring
it
close
to
the
angle
of
sliding
friction
of
the
debris.
The
combined
effect
of
these
factors
(discussed
in
KIRKBY
&
STATHAM
1974)
is
to
modify
slightly
the
simple
model
of
a
boulder
-controlled
slope
as
described
by
BRYAN
(1922).
They
affirm
however
that
BRYAN'S
boulder
-control-
led
slope
is
commonly
the
initial
form,
which
becomes
modified
by
surface
wash
processes
in
a
semi
-arid
environment.
Moreover,
these
factors
provide
a
boulder
-
controlled
slope
that
is
initially
concave
as
the
mountain
mass
retreats
through
rockf
all.
For
the
slope
forms
measured
in
southern
Arizona,
the
surface
material
was
always
found
to
decrease
in
size
downslope
as
progressively
lower
gradients
were
reached.
Typical
slope
profiles
and
associated
grain
sizes
are
shown
in
Fig.
10
and
7.
From
the
data
two
generalisations
can
be
made;
10
0-5
0-2
rn
,%
01
t
0-05
5
602
001
Surface
wash
at
the
semi
-arid
break
in
slope
169
s
,
t`
O
rid
s:4
10
0
0
1.
O
0
S
O
O
O
0
0
0
O
O
00
O
0 0
O
o
o
0
0
O
2
5
50
100
200
500
10
20
1000
Grain
size,
d
8
4n
mm.
Fig.
11.
Empirical
relationship
between
slope
and
grain
size
for
Arizona
breaks
in
slope.
The
broken
line
is
a
best
fit
relationship.
The
solid
line
is
a
theoretical
envelope
curve
(equation
[12]).
170
A.
V.
T.
and
M.
J.
KIRKBY
1.
there
is
an
overall
relationship
between
grain
size
of
surface
material
and
slope
angle
which
is
independent
of
lithology.
The
generalised
relationship
for
the
grain
-size
slope
curve
shown
in
figure
11
is
(7)
sin
?,
=
0.04
(d84)
0.44
Sharp
breaks
in
slope
are
associated
with
a
scarcity
of
certain
(middle
-range)
grain
sizes
in
the
field.
2.
When
the
form
of
the
break
in
slope
is
compared
for
several
lithologies
the
differences
are
not
due
to
variation
from
the
above
slope,
grain
-size
relation-
ship
but
to
different
widths
of
the
break
in
slope
zone.
The
width
of
this
zone
(from
a
15°
to
gradient)
is
between
20
to
80
metres
for
granite
and
from
250-750
metres
for
schists
and
basalts
(except
under
condi-
tions
of
active
cliff
retreat
where
breaks
in
slope
are
partly
accumulative
and
may
be
very
sharp).
Thus
schists
and
basalts
have
a
break
in
slope
zone
which
is
about
ten
times
as
wide
as
that
in
granite
(Fig.
10).
The
sharpness
(or
narrow-
ness)
of
the
break
in
slope
in
granite
is
associated
with
the
widely
described
discontinuous
breakdown
of
material,
which
goes
directly
from
joint
-block
boulders
to
constituent
crystals.
In
other
lithologies,
such
as
the
schists
and
basalts
studied
here,
continuous
breakdown
provides
intermediate
sizes
of
material.
Analysis
of
the
form
of
the
break
in
slope
in
southern
Arizona
thus
indicates
that
slope
profiles
are
everywhere
concave
and
that
this
concavity
can
be
initiated
by
processes
of
mountain
retreat.
It
is
everywhere
associated
with
decreasing
grain
size
on
lower
slopes
in
a
relationship
which
is
found
to
be
constant
for
different
lithologies.
The
form
of
the
break
in
slope
does
vary
in
different
rock
types
but
it
is
a
difference
in
the
width
of
the
break
in
slope
that
provides
the
visible
distinctions.
The
theoretical
relationship
between
grain
size
and
slope
The
field
evidence
demonstrates
that
wash
processes
are
effective
around
the
break
in
slope,
both
in
and
between
channels.
It
also
shows
that
sediment
trans-
port
on
interfluves
is
essentially
constant
as
grain
size
and
slope
angle
change
together
across
the
break
in
slope.
This
is
equivalent
to
an
assumption
of
'grade',
which
is
associated
with
a
quasi
-equilibrium
form.
If
equilibrium
can
be
assumed
overall,
then
sediment
transport
must
be
more
or
less
constant
for
both
channels
and
interfluves.
A
similar
argument
applies
to
roughly
constant
water
discharge,
provided
that
there
are
no
gross
changes
in
soil
storage
capacity
or
spring
heads
at
the
break.
Adapting
GILBERT'S
(1909)
argument
on
the
convexity
of
divides,
it
can
be
argued
that,
within
a
narrow
break
in
slope
zone,
the
proportional
change
in
discharge
must
be
insignificant,
so
that
the
major
contribution
to
any
change
in
sediment
transport
must
result
from
differences
in
the
more
rapidly
changing
slope
and
grain
-size.
These
arguments
converge
to
suggest
that
a
solution
of
the
flow
and
sediment
discharge
equations
for
constant
water
and
sediment
discharge
i
Surface
wash
at
the
semi
-arid
break
in
slope
171
provides
an
approximation
to
the
hydraulic
equilibrium
around
the
break
in
slope.
The
choice
of
appropriate
equations
for
water
flow,
and,
even
more
sediment
transport,
presents
difficulties
since
they
are
not
adequately
tested
for
the
thin
flow
and
high
roughness
conditions
of
overland
flow,
although
the
work
of
EMMETT
(1970)
encourages
the
view
that
flow
is
turbulent,
and
behaves
qualita-
tively
like
river
flows.
The
flow
equation:
(8)
q =
(2
ef)
vt. r
3,2. sah
is
therefore
adopted.
The
roughness
factor
defined
by
this
equation
is
given
empirically
by
WOLMAN
(1955)
in
the
form:
(
9
)
1
Vf
=
4.07
log
io
(r/d
84
)
+
2.0
where
q
is
the
water
discharge
per
unit
width,
r
is
the
hydraulic
radius
of
the
flow,
d
84
is
the
grain
size
than
which
84
0
/0
is
finer,
and
s
is
the
sine
slope.
The
latter
relationship
becomes
physically
meaningless
when
r
<
d
84
,
because
the
stones
are
then
sticking
out
of
the
water.
A
limiting
upper
value
of
the
rough-
ness
is
therefore
taken
at
this
value,
of
1
yr
2.0.
The
choice
of
a
suitable
sediment
transport
equation
is
more
problematical,
but
the
MEYER-PETER
&
MULLER
(1948)
equation
has
been
used
for
its
simplicity
and
dimensional
correctness:
(10)
S
=
(gAd
3
)
1
/
2
(rs/Ad
0.047)'/'
where
S
is
the
sediment
transport
per
unit
width,
A
is
the
ratio
of
submerged
sediment
to
water
densities
(1.65),
and
d
is
the
grain
size
being
carried.
The
term
rs/Ad,
the
dimensionless
tractive
stress,
needs
correcting
for
high
slopes
to
allow
for
the
reduction
of
tractive
stress
required
to
move
material
on
slopes
close
to
their
angles
of
rest.
To
do
this,
the
slope
term
must
be
interpreted,
in
the
sediment
transport
equation
only,
as:
sin
sin
99
(11)
s
=
.
sm
(99
—19)
where
is
the
slope
angle,
and
99
the
angle
of
friction.
A
second
problem
concerns
the
choice
of
an
appropriate
value
of
grain
size,
d,
since
this
should
represent
the
grain
size
actually
carried.
In
this
analysis,
the
chief
concern
is
with
movement
of
the
roughness
elements
of
the
slope,
so
that
d
is
provisionally
identified
with
d
e4
above.
172
A.
V.
T.
and
M.
J.
KIRKBY
These
equations
can
now
be
solved
to
eliminate
the
hydraulic
radius,
r,
and
express
grain
size
as
a
unique
function
of
slope,
for
the
given
water
and
sediment
discharges.
It
will
be
appreciated
that
evidence
from
slope
profiles
does
not
specify
either
S
or
q;
and
that
these
values
may
vary
from
one
break
in
slope
to
another.
With
these
reservations,
equations
(8)
to
(11)
can
be
solved
most
simply
in
terms
of
the
parameter
u
=
r/d
84
as
follows:
sin
i9
sin
q)
(12)
a/d
84
(u
0.077)
=
a/11
u
(4.71og
10
u
2)
2
/
3
sin"
V.
sin
(9,
0)
SA)"
1/2)
16
where
a
=
((8
g
and
=
(2/2
g
1/2)2/3
A
log
—log
plot
of
the
two
right-hand
expressions
against
sin
7,,
for
a
series
of
values
of
the
parameter
u,
gives
intersections
which
define
the
slope
grain
-size
relationship.
For
u
<
1,
the
roughness
term
is
replaced
by
the
constant
value
of
2.0
inside
the
bracket.
It
is
argued
that
these
theoretical
curves
should
be
fitted
not
as
a
bestfit
to
the
empirical
data,
but
as
an
envelope
curve
which
shows
the
maximum
slope
or
minimum
grain
size
corresponding
to
a
fixed
value
of
the
other
variable.
If
a
boulder
weathers
from
bedrock
to
form
a
new
joint
block
on
a
slope
where
it
is
too
large
to
be
transported,
then
it
will
remain
on
the
slope
until
it
weathers
down
to
a
size
which
can
be
carried.
Such
oversize
blocks
will
therefore
form
a
persistent
feature
which
will
commonly
be
represented
in
spatial
samples
of
the
landscape.
In
contrast,
an
undersized
block
will
be
carried
away
in
the
first
storm
and
deposited
on
the
hydraulically
appropriate
slope.
It
will
therefore
not
be
spatially
represented.
An
envelope
curve
of
this
type
has
been
fitted
to
the
data
of
Fig.
11,
the
dog
-leg
representing
the
change
-over
in
roughness
at
r
=
d
84
.
Although
the
fitting
of
such
a
curve
is
not
critical
where
S
and
q
can
be
chosen
freely,
the
values
give
at
least
an
order
of
magnitude
for
the
dominant
flow
conditions.
They
are:
q
=
5.1
cms/cm.s
S
=
0.09
cm
3
/cm.s
Suspended
sediment
concentration
=
17,500
mg,/1
=
35
0
.
On
a
slope,
this
flow
gives
a
depth
of
4.0
mm
and
a
velocity
of
12.7
cm/s.
On
a
30
°
slope,
the
flow
depth
is
5.1
mm,
at
a
velocity
of
10.0
cm/s.
If
the
curve
in
Fig.
11
is
continued
to
finer
grain
sizes,
it
has
a
minimum
(slope)
at
2.1
°
for
a
grain
size
of
about
0.1
mm.
It
is
therefore
expected
that
surfaces
for
which
d
8
4
<
0.1
mm,
will
not
be
represented
near
the
break
in
slope
because
this
material
is
readily
carried
away
on
all
slopes
under
Arizona
conditions.
1
I
Surface
wash
at
the
semi
-arid
break
in
slope
g.
12.
Empirical
correlation
bet
-
en
distance
of
travel
by
rain
-
ash
and
unconcentrated
wash
),
and
slope
(s),
grain
size
(d),
and
face
roughness
(d
84
).
Distances
totals
for
period
of
measure
-
t,
initially
grouped
by
slope
d
grain
size
as
in
Fig.
4.
173
o
1-2mm
2
-4mm
8
4-8mm
+
8
-25mm
100
-
6
10
-4
10
3
to
2
0
)
.1.6
c:i
8
) "
Imm
units)
If
the
overall
trend
of
grain
size
in
Fig.
11
is
combined
with
the
mean
stances
of
travel
of
Fig.
4,
the
relative
roughness
of
the
surface
to
stones
of
fferent
grain
sizes
can
be
interpreted
as
in
fig.
12,
in
which
the
mean
distance
f
travel,
13)
x
=
5,600
(s/d)
1.6(
did84)o.4.
is
improved
empirical
relationship
approaches
the
theoretical
exponent
for
ope
of
2.0
for
slopes
of
fixed
grain
-size
(constant
dId
84
),
but
gives
a
reduced
lope
exponent
of
(1.6-0.4/0.44)
=
0.7
for
natural
slopes.
It
discounts,
however,
e
varying
exponents
of
slope
for
different
grain
sizes
shown
in
Fig.
4.
The
extension
of
this
argument
to
total
transport
rates
on
slopes
of
naturally
arying
grain
-size,
using
the
overall
best
-fit
lines
in
fig.
11
and
12
again,
leads
to:
14)
S=
id
a
s1.8
d-
0.8
a
s
(1.6
-0.8/o.44)
a
0.24
f
this
slope
expression
is
embedded
in
a
slope
wash
model
in
which
overland
flow
erosion
is
proportional
to
the
square
of
the
distance
from
the
divide,
then
on
a
uniform
grain
-size
surface:
(15)
S
a
x
2
.5
1
-
6
;
whereas
on
a
varying
grain
-size
surface:
(16)
S
a
x
2
sO.24
174
A.
V.
T.
and
M.
J.
KIRKBY
Using
the
method
of
characteristic
forms
(KIRKBY,
1971),
the
approximate
slope
profiles
are
respectively:
(17)
y
o
y
a
x
0
.
4
and
y
a
x
-3
where
x
is
horizontal
distance
from
the
divide
and
y
is
elevation
(=
y
o
at
the
divide).
The
very
much
greater
concavity
of
the
latter
curve
gives
a
crude
measure
of
the
extent
to
which
observed
grain
size
differences
are
sufficient
to
account
for
the
sharpness
of
the
break
in
slope.
The
instability
of
the
latter
solution
near
the
divide
prevents
an
exact
comparison
on
this
basis.
This
type
of
model
shows
that
although
wash
slopes
are
concave
in
any
case,
the
progressive
decrease
in
grain
-size
with
gradient
will
produce
a
greatly
enhanced
concavity,
in
a
some-
what
more
exaggerated
form
than
that
described
by
HACK
(1957)
for
river
long
profiles.
Conclusions
The
theoretical
arguments
and
empirical
evidence
together
suggest
that
the
break
in
slope
is
formed
as
follows:
1.
The
break
in
slope
is
formed
in
a
zone
where
there
is
a
gradual
transition
of
process
between
a
mountain
slope
on
which
talus
processes
operate
and
mate-
rial
is
released
mainly
as
joint
blocks,
at
a
rate
which
is
controlled
by
the
rate
of
weathering;
and
a
wash
slope
on
which
removal
is
limited
by
the
transportation
capacity
of
the
wash.
Both
of
these
processes
are
characterised
by
the
formation
of
concave
slopes.
This
general
tendency
to
concavity
provides
the
basic
slope
profile
context
within
which
the
form
of
the
break
in
slope
is
detailed.
2.
Material
on
the
mountain
slope
weathers
in
situ
and
the
fine
breakdown
products
are
readily
removed,
giving
the
conditions
under
which
a
bedrock
slope,
at
the
angle
of
rest
of
its
veneering
boulder
layer,
can
be
formed.
Although
in
detail
such
slopes,
like
all
screes,
are
concave
in
profile
and
at
less
than
the
static
angle
of
repose,
this
slope
is
essentially
the
'boulder
-controlled
slope'
described
by
BRYAN
(1922,
p.
43).
3.
Joint
blocks
weather
in
situ
until
they
are
small
enough
for
hydraulic
action
to
move
them
with
the
assistance
of
gravity
forces.
Such
reduced
boulder
-
cores,
and
the
fine
material
weathered
from
them
will
be
transported
rapidly
to
a
slope
where
it
just
cannot
be
carried
the
minimum
slope
in
Fig.
11.
4.
Lower
angle
slopes
therefore
receive
increasing
amounts
of
fine
debris
from
up
-slope,
producing
at
least
a
transportation
layer
of
debris.
Even
where
the
low
angle
slopes
are
progressively
downwearing,
as
is
assumed
in
classic
pediment
models,
initial
weathering
of
fresh
joint
blocks
will
take
place
within
this
trans-
portation
layer,
so
that
material
will
already
be
partly
comminuted
by
weather-
ing
by
the
time
it
is
exposed
at
the
surface.
These
effects
provide
a
positive
feed-
back
which
will
accentuate
any
initial
concavity
produced
by
wash
or
scree
processes.
The
initial
pre
-disposition
to
concavity
is
therefore
an
essential
part
of
the
process
by
which
sharp
breaks
in
slope
are
developed.
5.
Within
the
break
in
slope
zone,
changes
in
discharge
are
normally
quite
Surface
wash
at
the
semi
-arid
break
in
slope
175
sufficient
to
account
for
the
degree
of
local
concavity.
Its
form
thus
relies
p
r
incipally
on
the
hydraulic
equilibrium
between
gradient
and
grain
-size.
Within
the
break
in
slope,
there
is
a
continuous
transition;
from
scree
processes
in
which
no
hydraulic
force
is
needed
to
initiate
movement,
to
wash
processes
in
which
the
direct
gravitational
component
may
be
neglected.
The
combination
of
hydrau-
lic
and
gravitational
forces
is
thought
to
be
critical
to
an
analysis
of
the
processes
acting.
Given
that
a
continuous
transition
in
process
exists
there
is
no
a
priori
reason
for
assuming
that
the
corresponding
transitional
form
is
very
narrow,
so
that
simple
assumptions
about
a
break
in
process
at
this
point
are
untenable.
6.
Where
bedrock
materials
weather
to
fine
material
rather
than
to
a
range
of
smaller
material,
as
occurs
with
granites
and
some
sandstones,
the
absent
grain
-
sizes
are
matched
by
a
scarcity
of
corresponding
slope
gradients
because
no
material
is
deposited
at
these
angles,
and
breaks
in
slope
are
unusually
sharp.
Though
instructive
and
well
-documented,
this
is
not
an
essential
part
of
the
argument.
This
decription
of
the
development
of
mountain
slopes
in
Arizona
comes
close
to
that
of
KIRK
BRYAN
in
1922:
"The
talus
on
granite
slopes
commonly
consists
of
a
layer
of
boulders
only,
and
in
many
places
scattered
boulders
and
patches
of
boulders
between
protube-
rant
knobs
of
the
bedrock
seem
to
determine
the
angle
of
the
slope
.
..
in
a
general
way
the
size
of
the
boulders
is
proportional
to
the
grade
of
the
slope."
(p.
45)
"In
many
ranges
slopes
that
average
25°-30°
rise
directly
from
the
plain
to
the
crest
of
the
mountains.
The
factors
which
produced
the
mountain
slopes
must
then
differ
radically
from
those
which
produced
the
plain."
(p.
38)
Where
we
would
disagree
with
BRYAN
is
in
his
inference
that
the
break
in
slope
represents
a
disjuncture
of
process
at
that
point.
From
the
evidence
pre-
sented
here
we
would
conclude
that
a
continuous
transition
of
processes
is
suffi-
cient
to
create
the
strong
concavity.
Thus
the
break
in
slope
is
something
of
a
misnomer,
for
viewed
in
the
context
of
the
total
slope
transport
system,
from
divide
to
stream,
it
occupies
a
central
place.
Acknowledgements
The
major
part
of
this
work
was
carried
out
in
Arizona
in
1964
with
the
support
of
the
US
Geological
Survey
when
MJK
was
a
visiting
research
scientist
with
the
Water
Resources
Division
in
Washington,
DC
and
AK
was
a
graduate
student
in
the
Department
of
Geography
at
The
Johns
Hopkins
University,
Baltimore.
Our
thanks
are
due
to
LUNA
B.
LEOPOLD
and
Tom
MADDOCK
of
the
USGS
and
to
M.
GORDON
WOLMAN
of
Johns
Hopkins
for
their
helpful
advice
and
interest.
Professor
L.
J.
BATTAN
of
the
University
of
Arizona
at
Tucson
provided
the
valuable
assistance
of
his
weather
radar
system
in
giving
us
advance
warning
of
thunderstorms
so
that
we
could
pursue
them.
The
work
in
Mexico
in
1966-1970
was
supported
by
the
Smithsonian
Institution
and
the
US
NatiOnal
Science
Foundation
as
part
of
the
Oaxaca
Archaeological
Project
and
we
gratefully
acknowledge
their
generous
support.
Fig.
3,
4
and
6
are
reproduced
from
M.
A.
CARSON
&
M.
J.
KIRKBY
1972.
Hillslope
Form
and
Process
with
permission
of
the
Cambridge
University
Press.
Zeitschrift
fur
Geomorphologic
N.
F.
Suppl.
Bd.
21
176
A.
V.
T.
and
M.
J.
KIRKBY
References
BAKKER,
J.
P.
8r
J.
W.
N.
LE
HEUX
(1952):
A
remarkable
new
geomorphological
law.
-
Kon.
Nederl.
Akad.
van
Wet.,
Ser.
B,
55:
399-410
and
554-571.
BRYAN,
K.
(1922):
Erosion
and
sedimentation
in
the
Papago
Country,
Arizona.
-
US
Geolog.
Surv.
Bull.
730
B.
DENNY,
C.
S.
(1967):
Fans
and
pediments.
-
Am.
J.
of
Sci.
265:
81-105.
ELLISON,
W.
D.
(1945):
Some
effects
of
randrops
and
surface
fl
ow
on
soil
erosion
and
infiltra-
tion.
-
Transac.
of
Am.
Geophys.
Union,
26:
415-429.
EMMETT,
W. W.
(1970):
The
hydraulics
of
overlands
flow
on
hillslopes.
-
US
Geolog.
Surv.
Professional
Pap.,
662-A.
FLANNERY,
K.
V.,
A.
V.
T.
KIRKBY,
M.
J.
KIRKBY
&
A
W.
WILLIAMS
(1967):
Farming
systems
and
political
growth
in
Ancient
Oaxaca.
-
Science,
158:
445-454.
GILBERT,
G.
K.
(1909):
The
convexity
of
hilltops.
-
J.
of
Geology,
17:
344-351.
JormisoN,
D.
W.
(1932):
Rock
fans
of
arid
regions.
-
Am.
J.
of
Sc.
'
223:
389-416.
KING,
L.
C.
(1950):
A
study
of
the
world's
plainlands.
-
Quart.
J.
of
the
Geolog.
Soc.
106:
101-133.
KIRKBY,
A.
V.
T.
&
M.
J.
KIRKBY
(1974,
in
press):
The
implications
of
geomorphic
processes
for
archaeological
reconnaissance
survey
in
semiarid
areas.
In
Sediments
in
Archaeology
(Ed.
Davidson
and
Shaddey).
-
Duckworth.
KIRKBY,
M.
J.
(1969):
Erosion
by
water
on
hillslopes.
In
Water,
Earth
and
Man
(Ed.
R.
J.
CHORLEY).
-
p.229-238,
London
(Methuen).
-
(1971):
Hillslope
process
-response
models
based
on
the
continuity
equation.
-
Inst.
of
Brit.
Geographers,
Spec.
Publ.
No.
3:
15-30.
KIRKBY,
M.
J.
&
I.
STATHAM
(1974):
Surface
stone
movement
and
scree
formation.
J.
of
Geology
(in
press).
LANGBEIN,
W.
B.
8c
S.
A.
SCHUMM
(1958):
Yield
of
sediment
in
relation
to
mean
annual
preci-
pitation.
-
Transact.
of
the
Am.
Geophys.
Union,
39:
1076-1084.
LEOPOLD,
L.
B.
(1962):
The
Vigil
Network.
-
Bull.
of
Intern.
Assoc.
for
Sci.
Hydrology,
7:
5-9.
LEOPOLD,
L.
B.,
W. W.
EMMETT
&
R.
W.
MYRICK
(1966):
Channel
and
hillslope
processes
in
a
semi
arid
area,
New
Mexico.
-
US
Geolog.
Surv.
Professional
Pap.,
352-G.
MEYER-PETER,
E.
&
R.
MULLER
(1948):
Formulas
for
bed
-load
transport.
-
Proc.
of
the
3rd
Meeting
of
Intern.
Assoc.
for
Hydraulics
Res.,
Stockholm.
PEEL,
R.
F.
(1941):
Denudation
Landforms
in
the
Central
Libyan
Desert.
-
J.
of
Geomorph.,
5:
3-23.
MABBUTT,
J.
A.
(1952):
A
study
of
granite
relief
from
South-West
Africa.
-
Geolog.
Mag.
89:
87-96.
-
(1955):
Pediment
Landforms
in
Little
Namaqualand.
-
Geogr.
J.
121:
77-85.
RAHN,
P.
H.
(1966):
Inselbergs
and
nickpoints
in
southwestern
Arizona.
-
Z.
fur
Geomorph.
10:
215-225.
RUXTON,
B.
P.
(1958):
Weathering
and
subsurface
erosion
in
granite
at
the
Piedmont
angle,
Balos,
Sudan.
-
Geolog.
Mag.
95:
353-377.
SCHUMM,
S.
A.
(1964):
Seasonal
variations
of
erosion
rates
and
processes
on
hillslopes
in
western
Colorado.
-
Z.
fur
Geomorph.,
Suppl.-Bd.
5:
215-238.
TUAN,
YI
Fu
(1959):
Pediments
in
southeastern
Arizona.
-
Univ.
of
Calif.,
Publ.
in
Geo-
graphy,
13.
TWIDALE,
C.
R.
(1967):
Origin
of
the
piedmont
angle
as
e
videnced
in
South
Australia.
-
J.
of
Geology,
75:
393-411.
WOLMAN,
M.
G.
(1955):
The
natural
channel
of
Brandywine
Creek,
Pennsylvania.
-
US
Geolog.
Surv.
Professional
Pap.
271.
Address
of
authors:
Dr.
ANNE
V.
T.
KIRKBY,
Dept.
of
Geography,
University
College,
Gower
Street,
London/
W
C
IE
6
PT,
England;
Prof.
M.
J.
KIRKBY,
Dept.
of
Geography,
University
of
Leeds,
Leeds
LS29
JT,
England.