# The Staircase Structure of the Southern  Brazilian Continental Shelf

Authors: M. S. Baptista; L. A. Conti
Journal: Mathematical Problems in Engineering
 (2009)

---

## Abstract

We show some evidences that the Southeastern Brazilian Continental Shelf (SBCS) has a devil's staircase structure, with a sequence of scarps and terraces with widths that obey fractal formation
rules. Since the formation of these features is linked with the sea-level variations, we say that the
sea level changes in an organized pulsating way. Although the proposed approach was applied in
a particular region of the Earth, it is suitable to be applied in an integrated way to other shelves
around the world, since the analyses favor the revelation of the global sea-level variations.


---

## Body

## 1. Introduction

During the late  quaternary period, after the last
glacial maximum (LGM), from 18000 years ago till present, a global warming was
responsible for the melting of the glaciers leading to a fast increase in the
sea level. In approximately 13000 years, the sea level rised up to about 120
meters, reaching the actual level. However, the sea level did not go up in a
continuous fashion, but rather, it has evolved in a pulsatile way, leaving
behind a signature of what actually happened, the continental shelf, that is,
the seafloor.Continental shelves are located at the boundary with
the land so that they are shaped by both marine and terrestrial processes.
Sea-level oscillations incessantly transform terrestrial areas in marine
environments and vice versa, thus increasing the landscape complexity [1]. The presence of regions
with abnormal slope as well as the presence of terraces on a continental shelf
are indicators of sea-level positions after the last glacial maximum (LGM),
when large ice sheets covered high latitudes of Europe and North America, and
sea levels stood about 120–130 m lower than today [2]. Geomorphic processes
responsible for the formation of these terraces and discontinuities on the
bottom of the sea topography are linked to the coastal dynamics during eustatic
processes associated with both erosional or depositional forcing (wave-cut and
wave-built terraces, resp., [3]).The irregular distribution of such terraces and
shoreface sediments is mainly controlled by the relationship between shelf
paleo-physiography and changes on the sea level and sediment supply which
reflect both global and local processes. Several works have dealt with mapping
and modeling the distribution of shelf terraces in order to understand the
environmental consequences of climate change and sea-level variations after the
LGM [4–6].In this period of time the sea-level transgression was
punctuated by at least six relatively short flooding events that collectively
accounted for more than 90 m of the 120 m rise. Most (but not all) of the
floodings appear to correspond with paleoclimatic events recorded in Greenland
and Antarctic ice cores, indicative of the close coupling between rapid climate
change, glacial melt, and corresponding sea-level rise [7].In this work, we analyze data from the Southeastern
Brazilian Continental Shelf (SBCS) located in a typical sandy passive margin
with the predominance of palimpsests sediments. The mean length is
approximately 250 km, and the shelf break is located at 150 m depth. It is a
portion of a greater geomorphologic region of the southeastern Brazilian coast
called São Paulo Bight, an arc-shaped part of the southeastern Brazilian
margin. The geology and topography of the immersed area are very peculiar,
represented by the Mesozoic/Cenozoic tectonic processes that generated the
mountainous landscapes known as “Serra do Mar.” These landscapes (with mean
altitudes of 800 m) have a complex pattern that characterizes the coastal
morphology and leads to several scarps intercalated with small coastal plains
and pocket beaches.This particular characteristic determines the
development of several small-size fluvial basins and absence of major rivers
conditioning low sediment input, what tends to preserve topographic signatures
of the sea-level variations.For the purpose of the present study, we select three
parallel profiles acquired from echo-sounding surveys, since for all the
considered profiles, the same similar series of sequences of terraces were
found. These profiles [8–10] are transversal to the coastline and the isobaths
trend, and they extend from a 20 m to a 120 m depth.The importance of understanding the formation of these
ridges is that it can tell us about the coastal morphodynamic conditions, inner
shelf processes, and the characteristics of periods of the sea-level regimes
standstills (paleoshores). In particular, the widths of the terraces are
related to the time the sea level “stabilized.” All this information is vital
for the better understanding of the late quaternary climate changes dynamic.We find relations between the widths of the terraces
that follow a self-affine pattern description. These relations are given by a
mathematical model, which describes an order of appearance for the terraces.
Our results suggest that this geomorphological structure for the terraces can
be described by a devil's staircase [11]; a staircase with infinitely many steps in between
two steps. This property gives the name “devil” to the staircase,
once an idealized being would take an infinite time to go from one step to
another. So, the seafloor morphology is self-affine (fractal structure) as
reported in [12, 13], but according to our findings, it has a special kind
of self-affine structure, the devil's staircase structure.A devil's staircase as well as other self-affine
structure are the response of an oscillatory system when excited by some
external force. The presence of a step means that while varying some internal
parameter, the system preserves some averaged regular behavior; a consequence
of the stable frequency-locking regime between a natural frequency of the
system and the frequency of the excitation. This staircase as well as other
self-affine structures are characterized by the presence of steps whose widths
are directly related to the rational ratio between the natural frequency of the
system and the frequency of the excitation.In a similar fashion, we associate the widths of the
terraces with rational numbers that represent two hypothetical frequencies of
oscillation which are assumed to exist in the system that creates the structure
of the SBCS, here regarded as the sea-level dynamics (SLDs), also known as the
sea-level variations. Then, once these rational numbers are found, we show that
the relative distances between triples of terraces (associated with some
hypothetical frequencies) follow similar scalings found in the relative
distance between triples of plateaus (associated with these same frequencies)
observed in the devil's staircase.The seafloor true structure, apart  from the dynamics
that originated it, is also   a very relevant issue, specially for practical
applications. For example, one can measure the seafloor with one resolution and
then reconstruct the rest based on some modeling [14]. As we show in this work
(Section 5), a devil's staircase structure fits remarkably well the
experimental data.Our paper is organized as follows. In Section2, we
describe the data to be analyzed. In Section 3, we describe which kind of
dynamical systems can create a devil's staircase and how one can detect its
presence in experimental data based on only a few observations. In Section 4,
we show the evidences that led us to characterize the SBCS as a devil's
staircase, and in Section 5 we show how to construct seafloor profiles based on
the devil's staircase geometry. Finally, in Section 6, we present our
conclusions, discussing also possible scenarios for the future of the sea-level
dynamics under the perspective of our findings.


## 2. Data

The data consists of the tree profiles given in
Figures1(a)-1(b). The profile considered for our analyses is shown in Figure 1(b), where we show the continental shelf of the State of São Paulo, in a
transversal cut in the direction: inner shelf (coast) → shelfbreak
(open sea). The horizontal axis represents the distance to the coast and the
vertical axis, the sea level (depth), d. We are interested in the terraces widths and their
respective depths.Profiles (depth
versus the distance to the  coast) 	 of the
Southeastern Brazilian Continental Shelf. The arrows indicate the terraces
considered in our analyses. The profile shown with a thick black line is the
profile chosen for our derivations, reproduced also in (b). The other two
profiles had their original position of the two axes shifted by a constant
value such that one can also identify the terraces observed in the chosen
profile in these other two. Note that a translation of the profiles by a
constant value has no effect on any of the scalings observed. The reason of
this mismatch between the profiles is due to the local geometry of the coast at
the time the sea reached that level.
(a)(b)The profiles shown in Figure1 were the result of a
smoothing (filtering) process from the original data collected by Sonar
(the profiles used in this work were obtained by the use
of polinomial spilines in a two dimensional grip of points
10 km appart. This also imposes limits in the indentification
of terraces that have small widths). The smoothing
process is needed to eliminate from the measured data the influence of the
oscillations of the ship where the sonar is located and local oscillations on
the sea floor probably  due to the stream flows.Smaller topographic terraces could be smoothed or
masked due to several processes such as coastal dynamic erosional during
sea-level rising, Holocene sediment cover, and erosional processes associated
with modern hidrodynamic pattern (geostrophic currents). For that reason we only
consider the largest ones, as the ones shown in Figure2 (located at d=−30.01m with the width
of l=6.06km). As one can
see, the edges of the terraces are not so sharp as one would expect from a
staircase plateau. Again, this is due to the action of the sea waves and stream
flows throughout the time. To reconstruct what we believe to be the original
terrace, we consider that its depth is given by the depth of the middle point,
and its width is given by the minimal distance between two parallel lines
placed along the scarps of the terrace edges. Using this procedure, we
construct Table 1 with the largest and more relevant terraces found.Table 1
Terrace widths and depths. While the depths
present no representative deviation, the deviation in the widths becomes larger
for deeper terraces. The deviation in the widths is estimated by calculating
the widths assuming many possible configurations between the placement of the
two parallel lines used to calculate the widths.
ndn (m)ln (km)1−30.016.06± 0.052−41.861.59± 0.053−54.012.93± 0.054−61.141.73± 0.055−66.692.21± 0.056−74.330.80± 0.17−79.750.80± 0.18−85.300.80± 0.1Figure 2
Reconstruction of the
terraces. The width of the terrace is given by the minimal distance between the
two parallel dashed lines.We identify a
certain terrace introducing a lower indexn in l and d, according to their chronological order of
appearance. More recent appearance (closer to the
coast, less deep) smaller is the index n. We consider the more recent data to
have a zero distance from the coast, but in fact, this data is positioned at
about 15 km away from the shore, where the bottom of the sea is not affected by
the turbulent zone caused by   the breakout of the waves. The profile of Figure 1(b) was the one chosen among the other tree profiles because from it we could
more clearly identify the largest number of relevant terraces by Sonar (the profiles used in this work were obtained by the use
of polinomial spilines in a two dimensional grip of points
10 km appart. This also imposes limits in the indentification
of terraces that have small widths).


## 3. The Devil's Staircase

Frequency locking is a resonant response occurring in
systems of coupled oscillators or oscillators coupled to external forces. The
first relevant report about this phenomenon was given by Christian Huygens in
the 17th  century. He
observed that two clocks back to back in the wall, set initially with slightly
different frequencies, would have their oscillations coupled by the energy
transfer throughout the wall, and then they would eventually have their
frequencies synchronized. Usually, we expect that a harmonicPw1 of one
oscillatory system locks with a harmonic Qw2 of the other
oscillatory system, leading to a locked system working in the rational ratio P/Q [15].To understand what are the dynamics responsible for
the onset of a frequency-locked oscillation, that is, the reasons for which a
system either locks or unlocks, we present the simplest model one can come up
with to describe a more general oscillator. This model is described by an angleθ, which is changed (after one period) to the angle f(θ). So, f(θ)=θ+Ω.  Then, this model (a
rotation) is perturbed by an external force g, resulting in the
following model:(3.1)f(θ)=θ+Ω−g(θ,K)(mod1),where(3.2)g(θ,K)=K2πsin2πθ.Despite this map simplicity, the same cannot be said
about its complexity [16]. Arnol'd (see [17]) studied this map in detail aiming to understand how
an oscillatory system would undergo into periodic stable state when perturbed
by an external perturbation.ForK=0, (3.1) represents a pure rotation, which is
topologically equivalent to a twice continuously differentiable orienting
preserving mapping of the circle onto itself (Theorem of Denjoy; see [18]). Therefore, the simple (3.1)
can be considered as a model for many types of oscillatory systems. In fact,
(3.1) represents a more complicated system, a three-dimensional torus with
frequencies w1 and w2, when viewed by a Poincaré map. Thus, Ω in (3.1)
represents the ratio w1/w2. When w1/w2=p/q (with p≤q) is rational,
this map has a period p motion, and its
trajectory, that is, the value of θ, assumes the same value after q iterations. For K=0, the so-called winding number W is exactly
equal to Ω, that is, W = p/q.ForK≠0 (nonlinear
case), W is defined by (3.3)W(Ω,K)=limn→∞h(θ0,K)+h(θ1,K)+⋯+h(θn−1,K)n,where(3.4)h(θ,K)=Ω+g(θ,K).ForK<1, (3.1) is monotonic and invertible. For K=1, it develops a cubic inflection point at θ=0. The map is still invertible but the inverse has a
singularity. For K>1, the map is
noninvertible.Arnol'd wanted to understand how periodic oscillations
would appear as one increasesK from zero to
positive values. He observed that a quasiperiodic oscillation, for an
irrational Ω and K=0, would turn into a periodic oscillation as one varies K, from zero to positive values. He demonstrated that a
periodic oscillation has probability zero of being found for K=0 (rational
numbers set is countable while the irrational numbers set is uncountable) and
positive probability of being found for K>0. He also observed that fixing a positive value K, the winding number W in (3.3) is a
continuous but not differentiable function of Ω, as one can see in Figure 3, forming a stair-like
structure.Figure 3
A complete devil's
staircase, obtained from (3.1), for K = 1.IfW(Ω,K) is rational, it
can be represented by the ratio between two integer numbers as W=P/Q. At this point, the frequency Ω and the
frequency of the function g are locked,
producing the phenomenon of frequency locking, when W(K,Ω) does not change
its value within an interval ΔΩ (a plateau) of
values of Ω. In fact, the smaller the denominator Q is, the larger
the interval ΔΩ is.As one changesΩ, plateaus for W rational appear
following a natural order described by the Farey mediant. Given two plateaus
that represent P1/Q1 and P3/Q3 winding numbers,
with plateau widths of ΔΩ1 and ΔΩ3, respectively, there exists another plateau
positioned at a winding number W within the
interval [P1/Q1,P3/Q3] given
by(3.5)P2Q2=P1+P3Q1+Q3.The Farey mediant gives the rational with the smallest integer denominator
that is within the interval [P1/Q1,P3/Q3]. Therefore, the ΔΩ(P2/Q2) plateau is
smaller than ΔΩ(P1/Q1) and ΔΩ(P3/Q3), but is bigger than any other possible plateau.
Organizing the rationals according to the Farey mediant creates a hierarchical
level of rationals, which are called Farey Tree. The plateaus ΔΩ1 and ΔΩ3 are regarded as
the parents, and ΔΩ2 as the daughter
plateau.The interesting case of (3.1) for our purpose is exactly
when K = 1. For that
case, one can find periodic orbits with any possible rational winding number,
as one varies Ω. What means
that the probability of finding quasiperiodic oscillation in  (3.1)  is
zero by a random choice of the  Ω value. Also,
for K>1, due to the overlap of resonances (periodic
oscillation), chaos is possible.The devil's staircase can be fully characterized by
the relations between the plateaus widths and the relations between the gaps
between two of them. While the plateau widths are linked to the probability
that one has to find periodic oscillations, the
gaps widths between plateaus are linked to the probability
that one has to find quasiperiodic oscillations, in
(3.1).There are many scaling laws relating the plateau
widths [15, 19, 20]. There are local scalings,
which relate the widths of plateaus that appear close to a specific winding
number, for example, the famous golden mean WG=(5−1)/2. However, we will focus our attention on the global
scalings, which can be experimentally observed, and only for the case where K = 1. For this
case, we are interested in two scalings; the one that relates the plateau
widths with the respective winding numbers in the form 1/Q (the largest
plateaus), and the one that describes the structure of the complementary set to
the plateaus ΔΩ, that is, the structure of the gaps between plateaus.
The structure of the plateaus is a Cantor set as well as the structure of the
complementary set. Therefore, a characterization of these sets can be done in
terms of the fractal dimension D0 [21] of the complementary set.The first scaling is [15](3.6)ΔΩ(1Q)∝1Qγ(γ>3),The second scaling relates the widths of the complementary set as one
goes to smaller and smaller scales. These widths are related to a power-scaling
law whose coefficient D0 is the fractal
dimension of the complementary set. For K=1, we have that the fractal dimension of the
complementary set is D0≅0.87. This is a universal scaling. Since the complementary
set of the plateaus represents the irrational rotations, the smaller is its
fractal dimension, the smaller is the probability of finding quasiperiodic
oscillation.For experimental data, the determination ofD0 is difficult to
obtain because the dimension measures a microscopic quantity of the plateaus
widths, and in experimental data one can only observe the largest plateaus.
Fortunately, an approximation D' for D0 can be obtained
from the largest plateaus by using the idea proposed in [22],(3.7)(S'S)D'+(S''S)D'=1.where S', S, and S'' are represented
in Figure 4.Figure 4
Representation of the
gapsS'', S', and S, used to estimate the fractal dimension D0 of the
complementary set, using (3.7).In case one hasK≅1 (K<1), we do not have a complete devil's staircase. In
other words, winding numbers with denominators larger than a given Q˜ are cut off
from the Farey Tree. Using this information we can estimate the value of K through the
largest denominator observed [15],(3.8)Q˜≥11−K.Finally, we would like to stress that while in (3.1) the
plateaus of the devil's staircase are positioned at winding numbers defined by
(3.3), in nature, devil's staircases have plateaus positioned at some accessible
measurement.In the driven Rayleigh-Bénard experiment [23, 24], convection rolls appear in
a small brick-shaped cell filled with mercury, for a critical temperature
difference between the upper and lower plates. As one perturbs the cell by a
constant external magnetic field parallel to the axes of the rolls and by the
introduction of an AC electrical current sheet pulsating with a frequency fe and amplitude B, a devil's staircase is found in the variable fi, the main frequency of the power spectra of the fluid
velocity. As one varies the external frequency fe, stable oscillations take place at a frequency ratio fi/fe, for a given value of fe. In analogy to the devil's staircase of (3.1), fe should be
thought of as playing the same role of Ω in (3.1), and the
ratio fi/fe as playing the
same role of the winding number W.A devil's staircase can also be observed (see
[25]) in the amount of
information H (topological
entropy) that an unstable chaotic set has in terms of an interval of size ϵ, used to create the set. To generate the unstable
chaotic set, we eliminate all possible trajectories of a stable chaotic set
that visits this interval ϵ. In analogy with the devil's staircase of the circle
map, ϵ should be
related to ΔΩ, while H to W.The first proof of a complete devil's staircase in a
physical model was given in [26], in the one-dimensional ising model with convex
long-range anti-ferromagnetic interactions.In [27], it was found that a model for the El Niño, a
phenomenon that is the result of a tropical ocean-atmosphere interaction when
coupled nonlinearly with the Earth's annual cycle, could undergo a transition
to chaos through a series of frequency-locked steps. The overlapping of these
resonances, which are the steps of the devil's staircase, leads to the chaotic
behavior.


## 4. A Devil's Staircase in the Southern Brazilian Continental Shelf

Understanding the sea floor is a way for understanding
how the sea level changed. Models associating local features in continental
shelves and glacio global sea-level variations have been developed for the
formation of wave-cut and wave-built terraces at various sea levels. In
[28] a
process-response model for the development of barrier beaches during sea level
rise is described, and in [29] a model was developed for the formation of rock
platforms during changing sea level [30]. These models, though, consider the formation of
local features, as a particular terrace, and do not intend to make a large
picture of a continental shelf.On the other hand, fractal geometry approach has been
applied to describe large-scale topographic features associated with geomorphic
processes [31–34]. These models are
considered to be of potential methodological interest in sea floor since “it
is heuristically clear that seafloor or landscape topography is best described
by fractal geometry, because of the different and increasingly detailed
features appearing at increasing resolution” [35].The purpose of this work is to show how to find a
fractal geometry that fits reasonably good the SBCS.The main premise that guides the application of the
devil's staircase model to the shelf is that the rules found in quantities
related to the widths and depths of the terraces obey the same rules found in a
complete devil's staircase for the frequency-locked intervalsΔΩ and their
rational winding number, W. Thus, we assume that the terrace widths ln play the same
role of the frequency-locked intervals ΔΩ, and the terrace depths play the same role of the
rational winding number W.In order to interpret the shelf as a devil's
staircase, we have to show that the terraces appear in positions which respect
the Farey mediant, the rule that describes the winding number “positions”
of the many plateaus. For that, we verify whether the position of the terraces
atdn can be
associated with hypothetical frequency ratios, denoted by wn=pn/qn, which respects the Farey mediant. In doing so, we
want the metric of three adjacent terraces to respect the Farey mediant. In
addition, we also assume that the larger terraces are the parents of the Farey
Tree, while the smaller terraces between two larger ones are the daughters.
Thus, for each triple of terrace, we want(4.1)dn+2+dndn+1=wn+2+wnwn+1.One could have considered other ways to relate the depths and the
frequency ratios. The one chosen in (4.1) is used in order to account for the
fact that while dn is negative, wn is not.From (4.1) it becomes clear that for the further
analysis the depth of a particular terrace does not play a very important role
as the ratio between the depths of triple of terraces that contains this
particular terrace. These ratios may eliminate the influence of the local
morphology and the influence of the local sea-level dynamics into the formation
of the shelf. Therefore, the proposed quantity might be suitable for an
integrated analysis of the different shelves all over the world, specially the
ones affected by local geomorphological characteristics.From the Farey mediant, we have a way to obtain the
frequency ratios associated to each terrace,(4.2)wn+1=pn+pn+2qn+qn+2.Therefore, combining (4.1) and (4.2), we obtain(4.3)pn+pn+2qn+qn+2=pn+2/qn+2+pn/qnE,which results in(4.4)(pn+pn+2)(E−1)qnqn+2pn=pnqn+22+qn2pn+2,where E is defined
by(4.5)dn+2+dndn+1=E.We do not expect to have (4.4) satisfied. We only
require that the difference between the left- and right-hand  sides of this
equation, regarded as δϵ, is the lowest possible, among all possible values
for pm and qm (with m=n,n+2), for a given E, with the restriction that the considered largest
terraces are related to largest plateaus of (3.1), and thus pm+2 = pm and qm+2 = qm+1, and δϵ≪1. Doing so, we find the rationals associated with the
terraces, which are shown in Table 2. The minimal value of δϵ, denoted by min[δϵ], is min[δϵ(d1,d2,d3)] = 0.032002, with p1/q1 = 1/8 for the terrace
1, and p3/q3 = 1/9, for the terrace 3. We also find that min[δϵ(d6,d7,d8)] = 0.002344, with p6/q6 = 1/17, for the terrace 6, and p8/q8 = 1/18, for the terrace 8. These minimal values can be seen
in Figures 5(a) and 5(b), where we show the values of δϵ(d1,d2,d3), in  Figure 5(a),  and the values of δϵ(d6,d7,d8), in  Figure 5(b),  for different values of p and q. Using bigger values for p has the only
effect to  increase the value of δϵ.Table 2
Integers associated with then considered
terraces, with n=1,…,8.
npnqn11822173194——5——611772358118(a) Values ofδϵ(d1,d2,d3) and (b) δϵ(d6,d7,d8), for different values of p and q. δϵ is the
difference between the left- and the right-hand sides of (4.4).
(a)(b)We have not identified rationals that can be
associated with the terraces4 and 5, which means that for p and q within p=[1,50] and q=[1,400], we find that δϵ>1. We have assumed that they could be either a daughter
or a parent.From now on, when convenient, we will drop the indexn and represent
each terrace by the associated frequency ratio. So, the terrace 1, for n = 1, is
represented as the terrace with w=1/8.Table2 can be represented in the form of the Farey
Tree as shown in Figure 6. The branch of rationals in the Farey Tree in the
form 1/q belongs to the
most stable branch, which means that the observed terraces should have the
largest widths. We believe that the other less important branches of the
complete devil's staircase present in the data were smoothed out by the action
of the waves and the flow streams throughout the time, and at the present time
cannot be observed.Figure 6
Farey Tree representing
the frequency ratios associated with the major terraces.Notice that as the time goes by, the frequency ratios
are increasing their absolute value, which means that if this tendency is
preserved in the future, we should expect to see larger terraces.In the following, we will try to recover in the
experimental profile, the universal scaling laws of (3.6) and (3.7). Regarding (3.6),
we find that l scales as 1/q−3.60, as shown in Figure 7, which is the expected global
universal scaling for a complete devil's staircase. Regarding (3.7), and
calculating S', S'', and S using the
triple of terraces with widths l(w=1/8), l(w=2/17), and l(w=1/9), as represented in Figure 4, we find D' = 0.89. Using
the triple of terraces (n = 3, n = 4, n = 5), we find
that D' = 0.87. Both
results are very close from the universal fractal dimension D0≅0.87, found for a complete devil's staircase.Figure 7
Scaling between the1/q-terrace widths
and the value of q.


## 5. Fitting the SBCS

Motivated by our previous results, we fit the observed
shelf as a complete devil's staircase, using (3.1). Notice that the only
requirement for (3.1) to generate a complete devil's staircase is that the
function g has a cubic
inflection point at the critical parameter K=1. Whether (3.1) is indeed an optimal modeling for the shelf
is beyond the scope of the present study. We only chose this map because it is
a well-known system, and it captures most of the relevant characteristic a
dynamical system needs to fulfill in order to create a devil's staircase.We model the SBCS as a complete devil's staircase, but
we rescale the winding numberW into the
observed terrace depth. So, we transform the complete devil's staircase of
Figure 8 as good as possible into the profile of Figure 1(b), by rescaling the
vertical axis of the staircase in Figure 8.Magnifications of the
small box of Figure3, showing the plateaus of the devil's staircase of (3.1)
that appear for the same frequency ratios associated with the triple of
terraces w=(1/8,2/17,1/9),
in (a), and w=(1/17,2/35,1/18),
in (b).
(a)(b)We do that by first obtaining the functionF (see Figure 9)
whose application into the terrace depth d(w) gives the
frequency ratio wn=pn/qn associated with
the terrace.Figure 9
The functionF, which is a linear relation between the frequency
ratios associated with the terraces and their depths for the triple of terraces w=(1/8,2/17,1/9)
and w=(1/17,2/35,1/18).For the triple of terracesw=(1/8,2/17,1/9),
we obtain(5.1)F(d[km])=0.14219+0.00057853d[km],and for the triple of terraces w=(1/17,2/35,1/18),
we obtain(5.2)F(d[km])=0.080941+0.00029786d[km].Therefore, we assume that, locally, the frequency ratios are linearly
related to the depth of the terraces.Then, we rescale the vertical axis of the staircases
in Figures8(a)-8(b) and calculate an equivalent depth, d, for the winding number W by
using(5.3)d=F−1(W).We also allow tiny adjustments in the axes for a best
fitting. The result is shown in Figure10(a) for the triple of terraces w=(1/8,2/17,1/9)
and in Figure 10(b) for the triple of terraces w=(1/17,2/35,1/18).
We see that locally, for a short time interval, we can have a good agreement of
the terrace widths and positions, with the rescaled devil's staircase. However,
globally, the fitting in Figure 10(a) does not do well, as it is to be expected since
the function F is only locally
well defined, and it changes depending on the depths of the terraces. Notice
however that this short time interval is not so short since the time interval
correspondent to a triple of terraces is of the order of a few hundred years.(a) Rescaling of Figure8(a) in black,
showing that the devil's staircase fits well the terraces with w=(1/8,2/17,1/9)
of the profile of Figure 1(b), in gray. (b) Rescaling of Figure 8(b) in black,
showing that the devil's staircase fits well the terraces with w=(1/17,2/35,1/18)
of the profile of Figure 1(b), in gray.
(a)(b)The assumption made thatK=1 is also
supported from (3.8). Using this equation, we can obtain an estimation of the
maximum value of K from a terrace
with a frequency ratio that has the largest denominator. In our case, we
observed w=2/35. Using Q˜ = 35 in (3.8), we
obtain K≤0.97.In Figure10(a), we see a 1/7 plateau positioned in
the zero sea level. That is the current level. Thus, the model predicts that
nowadays we should have a large terrace, which might imply in an average
stabilization of the sea level for a large period of time. However, this
prediction might not correspond to reality if the sea dynamics responsible for
the creation of the observed continental shelf suffered structurally
modifications.


## 6. Conclusions

We have shown some experimental evidences that the
Southern Brazilian Continental Shelf (SBCS) has a structure similar to the
devil's staircase. That means that the terraces found in the bottom of the sea
are not randomly distributed but they occur following a dynamical rule. This
finding lead us to model the SBCS as a complete devil's staircase, in which,
between two real terraces, we suppose an infinite number of virtual (smaller)
ones. We do not find these later ones, either because they have been washed out
by the stream flow or simply due to the fact that the time period in which the
sea-level dynamics (SLDs) stayed locked was not sufficient to create a terrace.
By our hypothesis, the SLDs creates a terrace if it is a dynamics in which two
relevant frequencies are locked in a rational ratio.This special phase-locked dynamics possesses a
critical characteristic; large changes in some parameter responsible for a relevant
natural frequency of the SLDs might not destroy the phase-locked regime, which
might imply that the averaged sea level would remain still. On the other hand,
small changes in the parameter associated with an external forcing of the SLDs
could be catastrophic, inducing a chaotic SLDs, what would mean a turbulent
averaged sea-level rising/regression.In order to interpret the shelf as a devil's
staircase, we have shown that the terraces appear in an organized way according
to the Farey mediant, the rule that describes the way plateaus appear in the
devil's staircase. That allows us to “name” each terrace depth,dn, by a rational number, wn, regarded as the hypothetical frequency ratio.
Arguably, these ratios represent the ratio between real frequencies that are
present in the SLDs. It is not the scope of the present work to verify this
hypothesis; however, one way to check if the hypothetical frequency ratios are
more than just a mathematical artifact would be to check if the SLDs has,
nowadays, two relevant frequencies in a ratio 1/7, as predicted.The newly proposed approach to characterize the SBCS
relies mainly on the ratios between terraces widths and between terraces
depths. While single terrace widths and depths are strongly influenced by local
properties of the costal morphology and the local sea-level variations, the
ratios between terrace widths and depths should be a strong indication of the
global sea-level variations. Therefore, the newly proposed approach has a
general character, and it seems to be appropriated as a tool of analysis to
other continental shelves around the world.Reminding that the local morphology of the studied
area, the “Serra do Mar,” does not have a strong impact in the
formation of the shelf and assuming that the local SLDs is not
directly involved in the formation of the large
terraces considered in our analyses, thus, our results should reflect mainly
the action of the global SLDs.In fact, analytical models [36–38] that study the relation
between the sea-level variation and the climate change during the last three
decades have shown that there are preferred oceanic locations at which the
global sea-level variation can be well described by the local variation.If the characteristics observed locally in the São
Paulo Bight indeed reflect the effect of the global SLDs, then the global SLDs
might be a critical system. Hopefully, the environmental changes caused by the
modern men have not yet made any significant change in a relevant parameter of
this global system.



---

*Source: 624861-2009-04-26.xml*