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GEOLOGICAL
NOTES
Qualitative
Chaos
in
Geomorphic
Systems,
With
an
Example
from
Wetland
Response
to
Sea
Level
Rise
Jonathan
D. Phillips
Department
of
Geography
and
Planning,
East
Carolina
University,
Greenville, NC
27858-4353
ABSTRACT
The
spatial
and
temporal complexity of
earth
surface
processes
and
landforms
and
the
presence
of
deterministic
chaos
in
many
fundamental
physical
processes
provide
reasons
to
suspect
chaos in
geomorphic
systems.
A
method
is
presented
to
assess
the
likelihood
of
chaotic
behavior
in
a
geomorphic system.
The method requires identification
of
the fundamental
system
components,
their positive,
negative,
or
negligible
influences
on
each
other,
and
the
relative strength or
magnitudes
of
these
links.
Based
on
this information,
the
method
can
classify
geomorphic
systems
as
stable
and
nonchaotic,
unstable
and potentially
chaotic,
or
unstable
and
generally chaotic.
Positive,
self-enhancing
feedback
is
a
key
diagnostic
of
the likelihood
of
chaotic
behavior.
A
sample
application
of
the
method
to the
problem
of
coastal
marsh
response
to
sea
level
rise
is
provided, which
shows
the
marsh
to
be
unstable.
If
changes
in vegetation
cover
are
partly dependent
on
vegetation
density,
the
system
is
generally
chaotic
if
marsh
vegetation
exhibits
self-
enhancing
feedback
(for
example,
seed
source
effects)
and
potentially
chaotic
if
vegetation
exhibits
self-limiting
feedback
(competitive
effects).
The
attractors
controlling
the
chaotic
dynamics
represent
states
of
pronounced ero-
sion/drowning or
accretion/expansion.
Introduction
Geomorphologists
must
often
deal
with
complex
systems
with
numerous
interrelationships among
system
components.
In
many
cases,
the qualitative
relationships among
system
components
can be
identified,
but
the
exact
form
of
the
governing
equations
or
the
magnitudes
of
the
interactions are
unknown.
Such
systems
may be
termed
partially
specified.
May
(1973),
Levins
(1974),
and Jeffries
(1974)
showed
how
standard
techniques
in linear
mathematical
stability
theory
could
be
applied
to
linear
or
nonlinear
partially
specified
systems in
ecology
and
population biology.
Slingerland
(1981)
developed
the qualitative
analysis of
complex
sys-
tems
in
a
geomorphological
context,
and
Scheideg-
ger
(1983)
showed that
the
conceptual
framework
and
analytical
method were applicable
to
a
broad
range
of
geomorphological
problems.
Qualitative
analysis of
complex,
partially specified
systems
has
been
applied
to
several geomorphic
problems
(Phil-
lips and
Steila
1984;
Phillips
1987a,
1990;
Schei-
degger
1987;
Slingerland
and
Snow
1988). The
qualitative
analyses
are
generalizations
of
linear
perturbation
theory,
which
has
been
widely used
1
Manuscript received October 6, 1991;
accepted
Decem-
ber
19, 1991.
in
geomorphology
(e.g.,
Callander
1969;
Smith
and
Bretherton
1972; Loewenherz 1991).
The
purpose
of
this
paper
is
to
extend
the quali-
tative
analysis
of
partially
specified
geomorphic
systems
to
evaluate
the
likelihood
of
deterministic
chaos.
Deterministic
chaos
is
complex,
random-
like
behavior
arising
from
the
dynamics of
deter-
ministic nonlinear
systems.
While
there
is
no for-
mal
physical
or
mathematical
definition
of
chaos,
one
universally
recognized
distinguishing
feature
of
a
chaotic
system
is
that it
exhibits sensitive de-
pendence on initial
conditions
and
increasing
di-
vergence
over
time.
Chaos
has
been shown
to exist
in
some
funda-
mental physical
processes,
such
as
turbulent
flows
and
atmospheric
dynamics,
which suggests
that
earth surface
features
affected by
these phenomena
may also
exhibit
chaos
(Culling
1987, 1988;
Hug-
gett
1988;
Malanson
et
al.
1990).
Further,
it
has
been suggested that
complex,
apparently
random
spatial
and temporal
patterns
in
topography, strati-
graphic
sequences,
and
soil
properties
may
arise
from
deterministic
nonlinear
dynamics and
may
thus
represent
spatial
manifestations
of
chaotic
landscape
processes
(Culling
1987,
1988;
Slinger-
land
1989;
Turcotte
1990;
Malanson
et
al. 1990).
IThe
Journal
of Geology,
1992,
volume
100,
p.
365-374]
©
1992
by
The
University
of Chicago.
All
rights
reserved.
0022-1376/92/10003-006$1.00
365
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GEOLOGICAL
NOTES
Phillips (1992a,
1992b)
has
specifically
addressed
this
issue
for
geomorphic
systems
and
for
surface
runoff
phenomena
and
showed
that
a
wide
variety
of
these
systems are
potentially
chaotic.
Chaos
in
a geomorphic system
would
have
pro-
found
implications.
Because
chaotic
systems
are
highly sensitive
to
initial
conditions,
and
slightly
different
initial states
lead to
divergence
in
behav-
ior
later,
chaotic behavior
makes
long-term
deter-
ministic
prediction
impossible
in
many cases,
as
detailed
knowledge
of
initial
conditions
is
typi-
cally not
available.
Principles
of
chaos may
chal-
lenge
the concept
of
equifinality,
whereby
initially
different
landforms
converge
over
time
to
similar
final
forms
(Culling
1987),
particularly where equi-
finality
has been asserted
as
a
result of
a
nonlinear
system
being
treated
as
though it were
linear.
On
the
other hand,
the
presence
of
chaos
would
sug-
gest
that
some
of
the
complex,
apparently
random
spatial
and
temporal
patterns
observed
in earth
sur-
face
processes
and landforms
may
have relatively
simple
deterministic
explanations.
Finally,
the
presence
of chaos
has
the
potential
to reconcile
the
simultaneous
occurrence
of
short range
complex-
ity
and
pseudo-randomness
and long
range
order
and
pattern
apparent
in
the
landscape.
Note
in
this
regard
that
chaotic behavior
does not
neces-
sarily
inhibit
short-term
deterministic
prediction
or
long-term
probabilistic
prediction.
Chaos
in
Partially
Specified
Systems
Brief
introductions
to
the
basic
concepts
of
chaos
theory
in
geological
contexts are
given by
Keilis-
Borok
(1990),
Turcotte
(1990),
and
Slingerland
(1989).
Texts
on
chaos
theory
and
nonlinear
dy-
namical
systems
are
numerous
and include
Baker
and
Gollub
(1990), Thompson
and
Stewart
(1986),
Wiggins
(1990),
Rasband
(1990) and
Schuster
(1988).
Puccia
and
Levins
(1985)
provide
compre-
hensive
treatment
of
the
qualitative
modeling
of
complex systems.
The
behavior
of a
geomorphic system
with
n
components
can
be
conceptualized
as
a
map
of the
system's
time-evolution
on
an
n-dimensional
plot.
This
map
is a
portrait
of behavior
in
phase
space.
In
a
chaotic
system
pseudorandom
patterns
and
trajectories
are
controlled
by
one
or
more
attrac-
tors-regions within the
system
phase
space
to
which
all
trajectories
are attracted.
Attractors
exist
in
all
nonlinear
dynamical systems;
the attractors
in
a chaotic
system are
called
strange
or
chaotic
attractors.
There
are three general
methods
for
identifying
and
describing
deterministic chaos.
(1)
Numerical
modeling
simulates the
time-evolution of
a
system
or
the
different
system
states
associated
with
small
variations
in
initial
conditions.
These
numerical
simulations
result
in
phase portraits,
probably
the
clearest
way
to assess chaotic
behavior.
Unfortu-
nately,
this
approach
does
not
address
the
issue of
identifying
chaotic
behavior
from
field
evidence.
(2) Algorithms
such
as
those
of
Grassberger
and
Procaccia
(1983)
or
Wolf
et al.
(1985)
can
be
used
to
analyze
a single
realization
of
a
system
(i.e.,
a
time
series for
any
system variable)
to
detect
chaos.
This
is
clearly
applicable
to
field
situations
where
long
time
series
are
available,
although
questions have
recently
been
raised
as
to whether
the
nature
of
the
record
influences
or
determines
the
chaos
analysis
(Provost
et
al.
1991).
Stoop
and
Parisi (1991)
are
developing
methods
to
determine
chaotic
behavior
from
time
series that
require
fewer
data
points,
but
Islam
et
al.
(1991)
suggest
that
analysis
of time
series
for
geophysical
systems
where
the
dynamic
equations
are
unknown
may
yield
spurious
results.
The
third
general
approach,
employed
here,
ex-
amines
whether
the
increasing
divergence
over
time and
sensitive
dependence upon
initial
con-
ditions-characteristic
of
chaos-are
present. The
disadvantages
of
this
approach
are
that
it provides
little
or
no
information
on
the
nature
or
dimension
of
the
strange
attractors
in
a system
and
no
map
of
the
behavior
in
phase
space.
Linear
stability
analy-
ses
around
equilibria
cannot
provide reliable
infor-
mation
on
subsequent
evolution
of
a
perturbed
chaotic
system. Such
analyses can,
however,
detect
chaos.
The
advantage is
that
the
approach
is highly
robust
and
can
be
applied
to partially
specified
sys-
tems.
The
latter
often
typify
the kind
of
field
un-
derstanding
available
to geomorphologists
(see
Slingerland
1981).
Lyapunov
Exponents and
Divergence.
The
theory
of
Lyapunov exponents
is
a generalization
of
lin-
earized
stability
theory (Wiggins
1990,
p.
607).
Lya-
punov
exponents
can
thus be
used
to extend
the
stability
analysis of
partially
specified
geomorphic
systems
(Slingerland 1981;
Phillips
1987,
1990)
to
chaotic
analysis. Lyapunov
exponents measure the
evolution
of
neighboring
phase
trajectories
and
thus
measure
sensitivity
to
initial
conditions and
increasing
divergence
over
time.
An
n-component
dynamical
system
will
have
n
Lyapunov
expo-
nents.
If
a
system
is
allowed
to
evolve
from
two
slightly
different
initial
states,
x
and
x
+
A,
their
divergence
over
time
t
may
be
characterized
ap-
proximately
as:
A(t)
-
AoeLt
(1)
366
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GEOLOGICAL
NOTES
where
L
is
the Lyapunov
exponent.
L
gives
the
av-
erage
rate
of
divergence,
taken
over
many initial
conditions. If
L
is zero,
then A(t)
= Ao.
If
L
is nega-
tive,
slightly
different
initial
conditions
converge
and
chaos
is
not
present.
If
L
is
positive
nearby
trajectories
diverge;
the
evolution
is
sensitive
to
initial
conditions
and
chaotic.
Lyapunov
exponents
are
not disputed
as
reliable
indicators
of chaos.
As
chaos is
not
formally
de-
fined,
however,
some
disagreement
exists as
to
whether
any
positive
L
or
whether an average
posi-
tive
L
is
necessary
and sufficient
for chaos to
exist.
Wolf
et
al.
(1985)
hold
that
any
system
with
one
or
more
L
>0
is
chaotic.
Berryman
and
Millstein
(1989) consider
this a
necessary, but
not
sufficient,
condition,
arguing
that
the
average
L
must
be
posi-
tive
for
a
system
to
be
chaotic.
Note
that
Berryman
and
Millstein
(1989)
are
concerned with
the prob-
lem
of
whether
ecological
systems
in
general
ex-
hibit
chaos,
as
opposed
to
exhibiting
chaos only
under
certain
circumstances.
The
issue
is
circumvented by
referring
to
poten-
tially
and
generally
chaotic
systems.
Many
nonlin-
ear systems behave
regularly
under certain
circum-
stances
and chaotically
under
others
(May
1976).
In
a numerical
model,
a
generally
chaotic
system
would exhibit
chaos
at
most
parameter values,
while a
potentially
chaotic
system
would
do
so
only
at
particular
values
or
when
some threshold
is exceeded.
A
generally
chaotic system that typi-
cally
exhibits chaotic
behavior
has a
positive
aver-
age
Lyapunov
exponent-chaotic
behavior,
while
not
always
present,
is
the
norm
in
such
systems.
In
a potentially
chaotic
system,
deterministic
chaos is
not
necessarily the norm
but occurs
under
certain
circumstances, and
no
evolutionary trend
toward chaos
is implied.
Any
L
>0
shows
a
system
is
potentially
chaotic.
Asymptotic
Stability
and
Chaos.
The
presence of
positive
Lyapunov
exponents can be
detected
by a
qualitative
asymptotic
stability
analysis.
It
can be
argued,
in
fact,
that
the
theory
of
Lyapunov expo-
nents, an
established
part
of
nonlinear
dynamical
systems analysis, is
a special
case
of
linearized
stability
around
an
equilibrium (Wiggins
1990,
p.
607).
Asymptotic stability
is
defined
in
terms of
system
response
to
mathematically
small
pertur-
bations or
disturbances
to
a
system
near an
equilib-
rium
state
(and is
thus
sometimes
referred to
as
local
stability).
In geomorphic
terms, small
pertur-
bations
can
be
defined
as
those
which
do not
oblit-
erate
feedback
mechanisms
within
the
system
or
introduce
new
mechanisms.
The
"neighborhood"
can
be
defined
as
the
spatial
or
temporal
domain
over
which
external
controls
(such
as
climate and
lithology)
are
constant or
negligibly
variable.
For
example,
consider
stream
channel
response
to
im-
posed
flows.
Changes
in imposed
discharge
accom-
modated
by the
bankfull
channel
via
changes
in
velocity,
energy
grade
slope,
hydraulic
radius, or
flow
resistance
may be
viewed
as
small
perturba-
tions.
Overbank
flood
flows
are
not
small
perturba-
tions
in
this
sense,
as
the spreading
of
water onto
the
floodplain
changes
the feedback relationships
in the
system.
Loewenherz
(1991)
provides
an
addi-
tional
example of what
constitutes
a
small
pertur-
bation
near
an equilibrium
in the
context
of fluvial
erosion
and
channel
initiation
problems.
A
nonlinear
dynamical
geomorphic
system
can
be
represented by a
set
of
nonlinear
partial differen-
tial equations. These
may
be
transformed
to
a set
of
n
ordinary differential
equations
dxi/dt
=
f,(x,
x2, ..
x), i
=
1,
2,..
n
(2)
with
n
variables
x.
The
growth rate
of
initially
small
perturbations of
the
system
(8x)
is
governed
by
a
set
of
linear
differential equations
dbx,/dt
=
Ai8x, i =
1,
2,...
n.
(3)
Ai
are
elements
of
the
Jacobian
matrix
of f = (fl,
f2,
.. , )
defined
by
S
fi(x,
x2,
.
.
.
x
)
A" ax.
(4)
where Xo
is the initial
equilibrium state.
By
analyz-
ing
small
perturbations
in
this
manner we
are
in
essence
linearizing
the
nonlinear system
around
an
equilibrium
by
a
Taylor series
as
described
by
Puc-
cia
and Levins
(1985,
p.
244-248).
The
stability
properties
of
the linearized
system
in
this
sense
are
identical
to
those
of
the
nonlinear
"parent"
system.
The eigenvalues
(X)
of
the
Jacobian
with
ele-
ments A,
are the Lyapunov
exponents
of
the
sys-
tem.
The
Jacobian is
exactly
equivalent
to
an
inter-
action
matrix
A
whose
elements
a1
signify
the
positive, negative,
or
zero
influences
of
the
ith
component
on
the
jth
component
of
the
system
as
reflected
in
the dynamic
equations
of
the
system
(Puccia
and
Levins
1985,
p.
246-248).
The
sum
of
the
diagonal
elements
of
the
matrix
are
equal
to
the
sum of
the
real
parts
of
the
eigenvalues and
thus
the
Lyapunov
exponents:
la,
=
2(re)X, =
EL,.
(5)
Joumrnal
of
Geology
367
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GEOLOGICAL
NOTES
Thus,
one
crucial
piece
of information
on
cha-
otic behavior
is
given
indirectly
by
the
diagonals
of the
system
interaction
matrix,
which represent
self-enhancing
or
self-damping
feedbacks
of
system
components. A
necessary
condition for
general
chaos is
a
positive
average
Lyapunov
exponent
(Berryman
and
Millstein
1989).
Therefore,
a,
>0,
at
least
one
i;
and
|ay,
>0|
>
|
a <0|.
(6)
These conditions are
simply that
there must be
at
least
one
positive
self-effect
and
that
the
strengths
or magnitudes
of
positive
self-effects must
be
greater than
those
of
any
self-limiting
links.
The
presence
of
a
self-effect
term
(a
#
0)
occurs
when
the first
partial
derivative
of a
variable
with
itself,
evaluated at equilibrium,
is nonzero:
a(dxi/dt)
ax
=
a.
(7)
i
x*
Typically
it
is
the
understanding
of
the
process-
response
relationships
of
a
geomorphic
system
that
provides
the
signs
of
the
a1.
Where
such
under-
standing is
insufficient
to
reveal
self-enhancing
loops, some
empirical
test
for
self-effects that
will
allow
an examination
of eq. (7)
is needed.
Even
if
a
system
is not
generally
chaotic
it
may
be
potentially
chaotic if
chaos
is
possible
under
some
circumstances,
for
example,
if
a
system
is
asymptotically
unstable.
This
will
occur
if
at
least
one
L is
positive.
The
interaction
matrix
A is
of special
interest
in
this
regard
because
any
positive L
can
be
de-
duced
from
the
characteristic
polynomial. Asymp-
totic
stability can
be
determined
from
qualitative
information
(i.e,
whether
a,
=
0;
ai1
<0;
or
ai,
>0)
alone.
The
eigenvalues
of
the
system
are
the
roots
of
the
characteristic
polynomial
of
A
Oon
+
ilkn-1
+
2X
n-2
(8)
+
...
+
an-ik +
=
0
where a
are
coefficients.
The
signs
of
the real
parts
of
the
eigenvalues
can be determined
from
the
Routh-Hurwitz
criteria (Cesari
1971).
Puccia
and
Levins
(1985)
show
that the
coefficients
ak
of
the
characteristic equation
are
equivalent
to
the feed-
back
Fk
at
level
k
of
the
system,
where
feedback
(Fk)
represents
the
mutual
influences
of
system
components
on
each
other
at
level
k.
Feedback
at
k =
2,
for
example,
represents
bivariate
relation-
ships
of
the
form
aa
j
$
i,
or
ammakk,
m
k
and
m,k
$
i,
j.
Feedback
is
computed by
(9)
Z(m,k)
is the
product of
m
disjunct
loops
with
k
system
components. Disjunct
loops
are
sequences
of
one
or
more
a,
that
have
no
common
compo-
nent
i
or
j.
At
level
zero,
feedback
is
-
1.
The
char-
acteristic
equation
is
then
Fok
+ F1ik-1
+
F2
-2
(10)
+...
+
Fn-
i
+
F,=0.
Necessary
and
sufficient
conditions
for
all
real
parts
of
the
eigenvalues
to
be
negative,
according
to
the
Routh-Hurwitz
criteria,
are
that
F,
<0
for
all
i
and that
successive
Hurwitz
determinants
are
positive,
though
only
alternate
determinants have
to
be
tested
(Puccia
and
Levins
1985,
p.
167-170).
The
second
condition
can be
expressed algebrai-
cally,
and
for
n
=
3
or
n
=
4
(for
example),
is
F,
F,
+
F3
>0.
(11a)
For
n =
5
or n
=
6
(as
is
the
case
in
the
example
to
follow)
the
second
condition
is
FF4
+
FFs
- FF,F, -
F3,
>0
(lib)
If
the Routh-Hurwitz
criteria
are
met,
all
eigen-
values
have
negative
real
parts,
all
Lyapunov
expo-
nents
are negative, and
the
system
is
asymptoti-
cally
stable. If
the
criteria
are
not
met, the
system
is
unstable
and
potentially chaotic.
Summary. The
discussion
above
suggests
a
rec-
ipe for
determining
whether
a
geomorphic
system
is
non-chaotic,
potentially
chaotic,
or
generally
chaotic.
The
steps
are
summarized
as
follows:
(1)
Identify
the
critical
components
of the
geomor-
phic system. (2)
Identify
the
relationships
of com-
ponents
with
each
other;
i.e.,
whether
an increase
or
decrease
in
one component
produces a
corre-
sponding,
opposite,
or
negligible
change
in
the oth-
ers.
This
may
be
determined
from
an
equation
sys-
tem or
from
knowledge
or assumptions
about
how
the
system
works. Separate
analyses
are
necessary
if
any a, might
have
a
different
sign
under
different
circumstances
or
if
the
sign
may
change
as
the
sys-
tem
evolves.
(3)
Determine
the
asymptotic stabil-
ity
of
the
system
using
the
Routh-Hurwitz
criteria.
If
the
system
is asymptotically
stable,
it
is
non-
chaotic.
(4)
If
the
system
is
asymptotically unsta-
ble,
but
the sum
of
the self-effect
terms
is
less
than
or equal to
zero
(a.
<0),
it
is potentially
chaotic.
368
Fk
=
'C(-l)m+lZ(mlk)~
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GEOLOGICAL
NOTES
(5)
Determine
whether any
a
>0
and whether
the
self-enhancing links
are
stronger
(operate
at
a
faster rate
or
a
greater
intensity)
than
self-damping
(ai
<0)
links. If
so,
the
system
is
generally
chaotic.
(6)
If
the
outcome
is
the
same
for
all
feasible
signs
for
the
a, in
the
system,
the
result is
globally
valid.
Otherwise,
the
system
is locally
stable,
potentially
chaotic,
or generally
chaotic
for
particular
system
configurations.
Use
of
this
procedure
to
assess
the
likelihood
of
chaotic
behavior
of
a
geomorphic
sys-
tem
is
illustrated
in
the
next
section.
Wetland
Response
to
Sea
Level Rise
Salt
marshes
and
other
coastal
wetlands
are
typi-
cally
geologically
ephemeral
features. Existing at
a
dynamic
land-water
interface,
these
wetlands
may
grow or
disappear
in
response
to
a variety
of
geo-
morphic,
hydrologic,
climatic,
and
anthropic
stim-
uli. The
response
of
coastal wetlands
to
current
and
predicted
near-future sea
level
rise
is
a
ques-
tion
of
some
urgency, not in
the
least
because
of
the
critical
ecological,
hydrological,
water
quality,
recreational,
aesthetic,
and
economic
roles
such
wetlands play (Titus 1988).
In a
geoscience
context
wetland
deterioration
and
accretion
is
a
sensitive
indicator
of
sea
level
change,
which may
in
turn
be
a
response
to
climate
signals
(Kearney
et
al.
1991).
A
number
of
recent
review
articles
provide
a
synthesis
of
the
literature
on coastal
wetland
re-
sponse
to
sea
level
change
(Reed
1990;
Stevenson
et al.
1988;
Titus
1988;
Orson
et
al.
1986).
The
focus
here
is
on mid-latitude
salt
marshes.
Implications
of
Chaos.
Chaotic
behavior
in
coastal
wetland geomorphic
systems
would
have
important
implications
in
understanding
the
rela-
tionships between
wetland
erosion
and
accretion
and changes
in
sea
level.
First, because tide
gage
records
rarely
go back
more
than
100
yr
and
there
are
few
data points
for
the past
1000 yr in
long-
term
sea
level
curves,
geomorphic
changes
in
coastal
wetlands
linked
to
sea
level
change
may
provide
a
critical source
of
information
on
late
Ho-
locene
sea level
(and possibly
climate)
trends
(Kear-
ney et
al.
1991). If
marsh
evolution
in
response to
sea
level
change
is
chaotic,
then
irregularities
in
marsh
growth or
deterioration
over
time
may
be
due to
inherent
system dynamics
rather
than
to
sea
level
trends,
and
trends
in
marsh
accretion
or
submergence
could be
quite complicated
and
ir-
regular.
Second,
the
presence
or
absence of
chaos
is also
important
for
interpreting
spatially
complex
pat-
terns
of
marsh
response.
Because
of
sensitive
de-
pendence
upon
initial
conditions, temporal
chaos
implies
spatial
chaos-slight
spatial
variation
in
initial
conditions
at
time
zero
would
be
exagger-
ated
over
time
in
a chaotic
system.
Kearney
et
al.
(1988),
for
example,
have
mapped
a
complex
spatial
pattern
of
marsh
surface
deterioration
in
response
to
local
coastal
submergence in
the
Nanticoke
es-
tuary
of
Chesapeake Bay.
Estuarine
shoreline
ero-
sion
has
been
linked
to
coastal
submergence
along
Delaware
Bay (Phillips
1986b),
and
an
extremely
complex
alongshore
pattern of
erosion
rates
has
been
documented
there (Phillips
1986a, 1989).
If
the
system is
chaotic,
then
irregular spatial
pat-
terns
of
marsh
response may
be attributable
to
deterministic
nonlinear
processes. There is
no
es-
sential
need
to
appeal to local
variations
in
envi-
ronmental
controls, although
the latter
could
con-
tribute
to
observed irregular
patterns.
Salt
Marsh
Response
Model.
Current
knowledge
of
the
important
processes
and relationships
in-
volved
in
marsh
response
to
sea
level rise
allows
only
partial
specification
of
the relationships
in-
volved.
For
example,
it
is
known
that
a
longer
hy-
droperiod
(inundation
regime) associated
with
ris-
ing
water
levels
stimulates
sediment deposition on
the
surface
and
inhibits
vegetation
production
(Reed
1990), but
no
dynamic
equation
or
generally
applicable
numbers
can
be used
to
describe these
relationships
for the
general case.
The important components
of
the
system
are
the
marsh
surface
elevation (relative
to local
mean
sea level),
the
hydroperiod,
inorganic
depositional
processes,
vegetative
growth
(organic
deposition),
net
vertical accretion,
and the
balance
between
net
vertical
accretion
and sea level
rise.
Tidal
re-
gime
and
climatological
factors influence
the hy-
droperiod
but
can be viewed
as
external
influ-
ences.
Relative
marsh
surface
elevation has
a
negative
effect
on
hydroperiod
(i.e.,
lower
elevations
lead
to
longer
hydroperiods,
and
vice-versa),
while
the
external
tidal
and
climate
(storm) factors
positively
influence
hydroperiod.
Hydroperiod
in
turn
posi-
tively
influences depositional
processes
and
nega-
tively
influences vegetative
growth.
Inorganic
de-
position
and
vegetation
production
have mutually
positive
influences
in
coastal
marshes
due
to
the
sediment-trapping
characteristics
of
vegetation
and
the substrate
and nutrients
provided
by
deposi-
tion.
Both
depositional
processes
and
organic
pro-
duction
have
positive, direct effects
on
net
vertical
accretion.
In
the
absence
of sea
level
rise,
vertical
accretion directly
influences
marsh
surface
ele-
vation.
Otherwise,
the
relationship
between
ac-
cretion
and
elevation depends on
the
balance
be-
tween
the
rates
of
vertical
accretion
and
sea
level
rise.
The
relationships
listed
above
and
their bio-
Journal
of
Geology 369
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GEOLOGICAL NOTES
Figure
1.
Interactions
between
sea
level rise,
hydroperiod,
and
organic
and inorganic depositional processes
controlling response
of coastal
marshes
to
sea
level
rise.
Redrawn
from
Reed
(1990,
figure
2).
physical
justifications
are
well
known
to
coastal
scientists
and
are
reviewed by
Reed
(1990),
Orson
et
al.
(1986)
and Stevenson et
al.
(1988).
They
are
depicted
in
figure 1
in
a
diagram
redrawn
from
Reed
(1990). By
considering
tidal
range
and
meteo-
rological
forcing
to
be
external
to
the
system, by
assuming
that
sea
level
is
rising,
and
by
depicting
the
relationship between
accretion
and sea
level
rise
rates
as
a
ratio,
Reed's
diagram
is
translated
into
figure
2.
This
in
turn
can be
translated
into
an
interaction
matrix
(table
1).
Stability
and
Chaos Analysis. The
interaction
matrix faithfully
represents
the geomorphologist's
knowledge
of
the
system,
which
includes
the
rela-
tionships
among
system components
but
not
spe-
cific
parameter values
or
transfer
functions.
Table
1
and figure 2
contain
one feature
not shown
in
figure
1:
a
self-effect
loop
for
vegetation.
A
system
with
no
a,
#
0 cannot
be
asymptotically
stable;
a
system
with
no a,
>0
cannot
be
generally
chaotic;
Figure
2.
Interaction
diagram for
response
of
coastal
marshes
to
rising
sea
level.
Adapted from
figure
1
by
assuming
sea
level
rise,
excluding
tidal
and
climate
forcings,
and by
adding
vegetation
self-effects
(see
text).
therefore, zeros on
the diagonal
predetermine an
outcome
of
asymptotic
instability
and
potential
(but
not
general)
chaotic
behavior.
The
literature
reviews
cited
above and
the
author's
personal
expe-
rience in
coastal
wetland
erosion-deposition
stud-
ies
(Phillips
1986a,
1986b,
1987b,
1989)
suggest
that
none
of
the components
except
vegetation
have
any
self-effects.
All
feedback
responses
must
involve
other
components.
Vegetation,
however,
through
the
well-known
effects
of
density
depen-
dence
(May
1973),
may have
self-effects
indepen-
dent
of
the other
system
components.
At
low
den-
sities
the effect
may
be
positive,
as
vegetation
provides
a
seed
source
and
creates
a
favorable
envi-
ronment
for
more vegetation,
thus
creating
a situa-
tion
where
the
partial
derivative of
vegetative
growth
with
respect
to
itself
is
positive.
At high
densities
the
self-effect
may
be
negative,
due
to
competition.
Recalling the
first
stability
criterion
is
that
all
370
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GEOLOGICAL
NOTES
Table
1.
Interaction
Matrix for
System
Shown
in
Fig-
ure
2
RMSE
HP VEG
DEP
NVA
VA/SL
RMSE
0
-a12
0 0
0
0
HP
0
0 -a23
a24
0
0
VEG
0
0
±a33
a34 a35
0
DEP
0
0
a43
0
a45
0
NVA
0 0
0
0
0
a56
VA/SL
a61
0
0
0 0 0
Note.
RMSE
=
relative
marsh
surface
elevation;
HP
=
hy-
droperiod;
VEG
=
vegetative
production;
DEP
=
depositional
processes;
NVA =
net
vertical
accretion;
VA/SL
=
ratio
of
vertical
accretion
rate
to
rate
of
sea
level rise.
Fi
<0,
the
feedbacks
for
the
system
in
table 1 and
figure 2
are
F1
=
a33
orFi =
-a33 (12a)
F2
= a34a43 >0
(12b)
F3
=
0
(12c)
F4
=
0
(12d)
Fs
=
-
a12a24
a45
a56
a61
<0
(12e)
F6 = (-a12a24a43a35a56a61)
(12f)
-
[(±a33)(-
a12a24a45a56
a61)].
Thus,
F6 will
be
negative
if
a33
<0
or if
(-a12a24a43a35a56a61)
> a33F5.
The
first
Routh-Hurwitz
criterion
is
clearly
vio-
lated,
and
the
system
is asymptotically
unstable.
It
is
potentially
chaotic if
a33
<0
and
generally
cha-
otic if
a33
>0.
Any
attractor
is
an
asymptotically
stable
state
of
the
system.
In
a chaotic
system
phase space
trajectories
are
attracted
to
and orbit
around
the attractors,
though
they
do
not
converge
there.
By
identifying
plausible
modifications to
the
system
(adding,
deleting, or
changing
the
sign
of
one
or
more
a,)
possible
stable,
nonchaotic
states
may
be
identified.
In
the
system of
table
1
and
figure
2
a
plausible
stable state
can be
envisioned
by
changing
the
sign
of
a43
to
negative
and
by
adding
a
link,
-
a64.
The
a43
entry represents the
effects
of
depositional
pro-
cesses
on
vegetation.
While
this
is
generally posi-
tive,
it
could be
negative
in
situations
of
rapid
deposition
where
vegetation
is
smothered by
ac-
creting
material.
This
has
been
observed
in
fringe
marshes
of
Delaware
Bay
(Phillips
1987).
It is
like-
wise
at
least conceivable
that
a
slowing
of
rapid
deposition
could
enhance
vegetation (re)establish-
ment.
The
matrix
element
-
a64
represents
a
nega-
tive
influence
of
the accretion/sea
level
balance on
depositional
processes,
independently
of
effects
via
marsh
surface
elevation
and
hydroperiod.
This
ef-
fect
could
occur
where
accretion
or
erosion/deteri-
oration (negative
accretion) directly
constrains
de-
positional
processes
via sediment
supply.
In
the
case
of
rapid
accretion
and
an
increasing ratio,
this
effect
would
depress
the
deposition
rate
by "lock-
ing
up" available
sediment.
In
the
case
of
erosion
or
deterioration, deposition
rates
may
be
enhanced
by
the
newly available
sediment.
This sort
of
supply-limited
marsh deposition
is
at
least plausi-
ble
in
a
rising
sea
level
environment,
where
the
locus of
deposition
of
fluvial
sediment
is pushed
upstream,
and
local
estuarine
sediments
become
a
major source for the
marshes.
Once again, this has
been
observed
in
the
Delaware
Bay
system
(Weil
1977; Phillips
1986a,
1986b).
The hypothetical
conditions
described
above
would
be most
likely
in
situations
of
actively
erod-
ing
or
accreting
marsh.
Thus
when
the
system
is
behaving
chaotically
the
phase
space trajectories
will
be attracted
in general toward
states
of pro-
nounced
erosion/deterioration
or accretion/expan-
sion,
although
the
specific
rates
of
erosion or
accre-
tion
will
be
quite
variable
and
the
patterns
quite
complex.
The
analysis
shows
that
where
vegetation
is
self-enhancing
(a33
>0), the
marsh
system
is
gener-
ally
chaotic. Where
vegetation
is
self-limiting,
the
system
is
potentially chaotic.
The
state
of
accre-
tionary
balance,
where
deposition rates
balance
the
rate
of
sea
level
rise,
is
unstable and in
the
chaotic
regime,
the
complex
dynamics
will
tend
toward
accretion-
or
erosion-dominated
states.
The
primary
purpose
here
has
been
to
illustrate
the qualitative
analysis
of chaos
rather
than
to
ad-
dress
the
wetland
response
to
sea
level
rise.
It
is
noted
in
passing,
however, that
the
analysis
sug-
gests that
complexities
and irregularities
in
spatial
and
temporal
patterns
of
marsh growth
or decline
could
occur
even
where
sea
level
rise
is
constant
and
in
the
absence
of
stochastic
forcings.
The
re-
sults
also point
to
the
importance
of
linking
vege-
tation and sedimentation dynamics.
Applicability
of
the
Case
Study.
The
example
above was
provided
to
illustrate
the
application
of
qualitative
chaos
analysis
to
a
specific
geomorphic
problem. The
problem
was
chosen on
the basis
of
the
author's
experience
and
the
presence
of
a
par-
tially
specified model
where, at
least
for
the
gen-
eral
case, the important
components
and their rela-
Journal
of
Geology
371
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GEOLOGICAL
NOTES
tionships
are
not
controversial.
No
model
was
devised
specifically
for
this
paper
to
eliminate
any
temptation
to
choose
or
construct
a
system
with
unusual
or
convenient
properties.
Nevertheless,
it
would
not
be unreasonable
to ask
whether
other
systems
are
as
amenable
to
the
method
described
as
the
example
above.
In
the
marsh/sea
level
example the
first
Routh-
Hurwitz criterion
was
violated,
making
evaluation
of
the
second
(eq.
1
lb)
unnecessary.
Because F3 and
F4
are
zero
the
second
criterion
would only
have
been
met
where
F1
is
positive. If
all
F1,
were
nonzero
and
if
it
were
unclear simply
from their signs
whether
eq.
(1
lb)
was
satisfied,
one
would
simply
determine
under
what
conditions
the
criterion
would
be
met.
Since each
F, represents
a particular
set
of
feedback
loops, these
conditions
have
a
geo-
morphic
interpretation
in
terms
of
which
would
have to be
stronger
for
stability.
Discussion
It
has
been
argued
elsewhere
that
geomorphic
sys-
tems,
as
a
rule, tend
to
be
asymptotically unstable.
Scheidegger
(1983) suggested
a
general
"instability
principle
of geomorphic
equilibrium"
best for-
mally
expressed as
qualitative
asymptotic
instabil-
ity. Phillips
(1992a,
1992b)
has argued,
on
the
basis
of
general
mass
flux
system models,
that
earth
surface
systems are
inherently
unstable
in
many
cases.
Given the link
between
instability
and
chaos,
it
seems
reasonable
to
suggest
that
many geomorphic systems are
at
least
potentially
chaotic.
The method
described here
can
be used
to
test
for
chaos
in quite
general,
partially specified geo-
morphic systems.
The question of
whether
an
as-
ymptotically
unstable
system
is
generally
or
poten-
tially
chaotic
can
be
addressed
by
looking
for
self-enhancing
positive
feedback.
Any
enterprising
geomorphologist
can
envision
positive feedbacks
in
geomorphic
systems.
Postulating
positive
self-
effects,
however,
may involve
unrealistic system
formulations
where
important
system
components
are
omitted. For
example,
in
some
cases
soil
ero-
sion
could
be
viewed
as
self-enhancing-as
erosion
proceeds
or
as
erosion
rates increase,
conditions
conducive
to
continued
or
increasing
erosion
are
often
created.
However, these
conditions-such
as
reduced
soil
moisture
storage
capacity, exposure
of
low-permeability
subsurface
horizons,
or a
de-
crease
in
vegetation
cover
or
vigor-invariably
in-
volve other
important
system
components.
A
dy-
namic
model of
soil
erosion
by water
that did
not
account,
at
least in
some
general
way,
for
vegeta-
tion
cover and
runoff
response of soils
would
be
inadequate
for
most purposes.
A
model
that
did
include
vegetation
and
runoff-response compo-
nents
would
reflect
the positive
feedback
as
multi-
component
loops
rather
than as positive
self-effect
loops.
Where
ai
>0
exist
and
are
stronger than
any
ai
<0,
an
asymptotically
unstable
system will
be
generally
chaotic. Note
that
the
most
famous ex-
ample
of
a
chaotic
dynamical
system,
the
Lorenz
equations
for
atmospheric
convection
(see
Lorenz
1963
or
any
recent
nonlinear
dynamical
systems
text),
contains
self-enhancing
feedback.
Because
the
presence of
positive
self-effects
is
a
key
indica-
tor
of
chaotic
dynamics, future
studies
should
fo-
cus
on the
extent
to
which
such
effects
occur
in
earth
surface
processes.
Conclusions
Geomorphic
systems
are
often partially
specified
in
that the
qualitative
nature
of
the
typically
non-
linear
relationships
among
system
components
are
known,
but
the
exact form of
the
governing
equa-
tions
or
the
quantitative
magnitudes of the
interac-
tions
are
not.
It
has
previously
been
shown that
the
stability
of
such partially
specified
systems
can
be
assessed
using the Routh-Hurwitz
criteria.
This
is
extended to
show
that
qualitative
analysis
of par-
tially
specified
systems
can
determine
whether
de-
terministic
chaos exists,
and whether
this
chaotic
behavior
occurs
under
a
wide
range
of
circum-
stances
or
only
under specific
and
limited
condi-
tions.
Application
of qualitative
chaos analysis is
illustrated
by
examining coastal marsh
response
to
sea
level
rise. This
system
is
unstable
and
poten-
tially
chaotic if
vegetation
is
self-limiting,
and
gen-
erally
chaotic
if
vegetation is
self-enhancing.
The
presence
of
positive self-effects
is
shown
to be a
key
diagnostic
of
the
likelihood
of
chaos
in
geo-
morphic
systems.
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