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\begin{document}
\title{Latitudinal changes in the morphology of submarine channels:
Reevaluating the evidence for the influence of the Coriolis force}
\author[1]{Zoltan Sylvester}%
\author[2]{carlos.pirmez}%
\affil[1]{University of Texas at Austin}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
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Published in "\textbf{Latitudinal Controls on Stratigraphic Models and
Sedimentary Concepts}", \textbf{SEPM Special Publication 108}, September
2017, DOI: 10.2110/sepmsp.108.02. Accompanying data and code available
at~\url{https://github.com/zsylvester/channel\_sinuosities}.
\section*{Abstract}
{\label{493732}}
Using a script that automatically calculates sinuosity and radius of
curvature for multiple bends on sinuous channel centerlines, we have
assembled a new data set that allows us to reevaluate the relationship
between latitude and submarine channel sinuosity. Sinuosity measurements
on hundreds of channel bends from nine modern systems suggest that there
is no statistically significant relationship between latitudinal
position and channel sinuosity. In addition, for the vast majority of
submarine channels on Earth, using flow velocities that are needed to
transport the coarse-grained sediment found in channel thalwegs,
estimates of the curvature-based Rossby number are significantly larger
than unity. In contrast, low flow velocities that characterize the upper
parts of turbidity currents in submarine channels located at high
latitudes can easily result in Rossby numbers of less than one; this is
the reason why levee deposits are often highly asymmetric in such
channels. However, even in channels with asymmetric levees, the
sinuosity of the thalweg is often obvious and must have developed as the
result of an instability driven by the centrifugal force. Analysis of a
simple centerline-evolution model shows that the increase in channel
curvature precedes the increase in sinuosity and that low sinuosities
are already associated with large curvatures. This suggests that the
Coriolis effect is unlikely to be responsible for the low sinuosities
observed in certain systems.
\section*{Keywords}
{\label{845749}}
submarine channels, Coriolis force, sinuosity, curvature
\section*{Introduction}
{\label{412424}}
Submarine channels are common---and often beautifully sinuous---
geomorphic features of the Earth's seafloor that serve as important
conduits of sediment transport from rivers and shallow water to the
continental slope and basin floor. In addition to their role in the
large- scale redistribution of clastic sediment, they often correspond
to locations of thick and relatively coarse-grained accumulations that
can host commercially significant hydrocarbon reservoirs. Ever since it
was recognized that these features exist~\citep{Henry_W_Menard_Jr_2__1955} and that
their planform patterns can be remarkably similar to the meandering
shapes familiar from rivers \citep{Damuth_1983,Clark_1992} the assumption has been
that the relevant physical processes are fundamentally the same across
the globe and, therefore, there is no need for facies and architecture
models of submarine channels that are specific for different latitudes.~
This line of thinking has been challenged by~\citet{Peakall_2011}, who
have looked at the relationship between submarine channel sinuosity and
latitude and suggested that channels closer to the poles had lower peak
sinuosities. They concluded that this is largely due to the Coriolis
force having a stronger influence at high latitudes. Experimental work
relying on a rotating flume tank showed that at low Rossby numbers (that
is, when the Coriolis force is larger than the centrifugal force)
turbidity currents do behave differently from the conventional
model~\citep{Cossu_2010,Cossu_2010a,Cossu_2012}. Building on these and similar experimental
results,~\citet{Cossu_2015} proposed that channel systems of the
Cretaceous Cerro Toro Formation, exposed in southern Chile and deposited
at high paleo latitude, display low sinuosity and an asymmetric
stratigraphic structure due to the Coriolis effect.
In a comment on the~\citet{Peakall_2011} study, we have presented evidence
that the apparent pole-ward decrease in submarine channel sinuosity is
unrelated to the Coriolis force \citep{Sylvester_2013}. In the present
article we expand on these ideas and present additional analysis (1) of
an improved and more consistent channel sinuosity measurement and (2) of
the magnitude of different forces as a function of channel size and flow
behavior. In addition, we briefly discuss the impact of the Coriolis
effect on overbank deposits, which is an important latitudinal effect in
these systems.
\section*{Methods: measuring sinuosity and radius of
curvature}
{\label{808832}}
To analyze the relationship between latitude and sinuosity, a relatively
precise and reproducible measure of sinuosity is needed. To estimate the
importance of the Coriolis force, we also need to calculate a
characteristic radius of curvature for each channel bend. Although it is
possible to collect these measurements one by one by analyzing each
channel bend separately, this manual approach would be fairly time
consuming, and the results would be difficult to check and replicate. To
avoid these issues, we have written a Python script that takes the ``x''
and ``y'' coordinates of a channel centerline as inputs and calculates
both sinuosity and radius of curvature for each bend. This methodology
also ensures that we are comparing sinuosity and radius of curvature
values that were derived in the same way for all channels. To perform
the analysis and generate the figures, we have used IPython (Jupyter)
Notebook, an interactive notebook-like, open- source computing platform
that is based on the Python programming language \citep*{Perez_2007}. The
processing steps that are performed are as follows:
1. Resample the channel centerline so that its defining points are
approximately equally spaced (50-m spacing). We used a parametric spline
representation of the curve to do this.
2. Smooth the centerline using the Savitzky-Golay
filter~\citep*{Savitzky_1964}. This filter is based on fitting successive
sets of data points with a polynomial, using linear least squares. For
our analysis, we used the scipy implementation of the algorithm and
adopted a window length of 21 centerline points that were convolved with
a third-order polynomial.
3. Calculate the curvature of the centerline and smooth it, using the
same filter as in step 2. The number of points over which the smoothing
is applied determines how many inflection points (points of zero
curvature) will be found. To obtain consistent results, this window
length has to scale with the channel, that is, the meander wavelength.
We have used values between 11 and 201 points (550 to 10,050 m),
depending on the scale of the channel, and a third-order polynomial.
4. Find inflection points and locations of maximum curvature. A strongly
smoothed curvature function will result in fewer zero crossings of the
function, fewer inflection points on the centerline, and a smaller
number of channel bends.
5. Calculate half wavelengths, arc lengths, and sinuosity for each
channel bend (1/4 segment between two consecutive inflection points).
Eliminate from further analysis straight channel segments with less than
1.01 sinuosity.
6. At each maximum curvature point, identify a centerline segment with a
length of one-tenth of the average arc length. Using least- squares
optimization, find a circle that matches these points best. One could
also simply take the average curvature of the centerline at these
locations to estimate the characteristic radius of curvature, but
least-squares optimization allows us to quickly plot the corresponding
circle and visually check the results
(Fig.~{\ref{309450}}).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/fig1/fig1}
\caption{{A) Example of a centerline segment from the Amazon Channel showing
channel bends and fitted circles that define the radius of curvature. B)
Raw and smoothed curvature calculated from the same centerline segment
as in A. Red and blue segments correspond to channel bends with opposite
signs of curvature.
{\label{309450}}%
}}
\end{center}
\end{figure}
The methodology used here builds on the method developed
by~\citet{nokey_2d8ed}, which was in turn derived from
\citet{nokey_e93a9}, to quantify curvature and wavelength of the Amazon
Channel. The minimum radius of curvature (which coincides with the
location of maximum curvature) is not a representative value for
calculating the centrifugal acceleration because it is sensitive to
measurement errors and would give an underestimate of the radius (or an
overestimate of curvature). On the other hand, using too many centerline
points in this estimation results in a significant mismatch between the
centerline and the fitted circles, plus an overestimation of the radius
of curvature values. The radius of curvature (R) values obtained here
are overall smaller than those calculated by \citet{nokey_2d8ed} as a
result of the fact that \citet{nokey_2d8ed} used the entire segment
between two inflection points in the calculation.
In contrast with \citet{Peakall_2011}, who have used only the peak
sinuosity for each channel system, we have argued that all available
measurements, not just the maximum values, should be used in such an
analysis \citep{Sylvester_2013}. This choice is justified by the fact that
the forces we are interested in act on all channel bends, and a
pole-ward decrease in sinuosity should be obvious not just in the
extreme values of the sample distributions but in other measures of the
upper range of the distributions as well.
The analysis shown in~\citet{Sylvester_2013} only includes new sinuosity
measurements from the Danube Channel; the data for the rest of the
systems come from~\citet*{nokey_f708d} and~\citet{nokey_2d8ed}. The
sinuosities were derived in three slightly different ways: for
example,~\citet{nokey_2d8ed} used full wavelengths in the calculation,
whereas \citet{Sylvester_2013} worked with half wavelengths. Although the
results should not be significantly different, for the present study we
have made an effort to compare only sinuosities derived using the same
scripted---and therefore fully reproducible--- methodology, described
above.
Using this methodology, we have digitized and analyzed channel
centerlines from nine systems (Amazon---\citealt{nokey_2d8ed};
Zaire---~\citealt{Babonneau_2010}; Danube---\citealt{Popescu_2001};
Monterey---~\citealt*{Fildani_2004},~\citealt{Paull:2011dt}; North-Atlantic
Mid- Ocean Channel {[}NAMOC{]}---\citealt{Klaucke:1997wi}; Knight
Inlet---~\citealt{Conway_2012}; Rhone---\citealt{TORRES_1997};
Nile---\citealt{Migeon_2010}; and Tanzania---\citealt{Bourget_2008}). The
sinuosity distributions are all strongly skewed, with lots of small
values (close to 1.0), and many fewer values that are larger than 2.
Both the channel centerline data and the scripts used for analysis and
plotting can be downloaded at the following data repository:
\url{https://github.com/zsylvester/channel\_sinuosities}.
\section*{\texorpdfstring{{Analysis and
interpretation}}{Analysis and interpretation}}
{\label{423656}}
\subsection*{\texorpdfstring{{Sinuosity--Latitude
Relationship}}{Sinuosity--Latitude Relationship}}
{\label{517069}}
For the sake of completeness, we reproduce here the results of our
previous analysis of the sinuosity--latitude relationship
(Fig.~{\ref{518222}}). Using sinuosity values for all
channel bends, as opposed to relying on peak sinuosities, and after
adding the Amazon and Danube channels to the~\citet*{nokey_f708d} data
set, we have shown that---at least for those channels under
consideration---both sinuosity and valley slope correlate with
latitude~\citep{Sylvester_2013}. Thus, the impact of the Coriolis force is
not the only possible explanation for the low sinuosities at high
latitudes; it has been suggested before that, just like in the case of
rivers, steeper valley gradients result in lower channel
sinuosities~\citep{Clark_1992,nokey_2d8ed}. Therefore, the pole-ward decrease in
sinuosity in this data set is likely a reflection of the fact that many
of the higher-latitude channels are steeper than the rest
(Fig.~{\ref{518222}}A). We have suggested that this
difference in gradient is primarily a reflection of the nature of the
sediment source for the turbidite system ~\citep{Sylvester_2013}: submarine
channels that are fed by large rivers with high sediment discharge are
located on extensive submarine fans or continental slopes with lower
gradients, and these are the settings in which high-sinuosity channels
tend to develop. Plotting only slope values for the bends with maximum
sinuosities,~\citet{Peakall_2011}, have essentially ruled out the
possibility that slope might play a significant role in the
sinuosity--latitude relationship that they have observed. Furthermore,
relying only on the extreme values of the sample distributions reduces
the robustness of the analysis, and our more inclusive approach does
show an overall increase of valley slope with latitude, at least for the
systems that were included in this initial, and still fairly limited,
data set (Fig.~{\ref{518222}}A).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/fig2/fig2}
\caption{{Plots of slope vs. latitude (A), sinuosity vs. latitude (B), and
sinuosity vs. slope (C). Same data as in \protect\citet{Sylvester_2013}; largely
based on data in
\href{https://www.authorea.com/users/157948/articles/200301-latitudinal-changes-in-the-morphology-of-submarine-channels-reevaluating-the-evidence-for-the-influence-of-the-coriolis-force\#nokey_f708d}{Clark
and Pickering (1996)}, with the exception of data points for the Amazon
and Danube systems. Blue dots correspond to channels that are directly
related to large rivers; red dots represent channels that are not
directly linked to large rivers.
{\label{518222}}%
}}
\end{center}
\end{figure}
Additional insight about the latitude--sinuosity relationship can be
gained if we look at the nine systems that we have analyzed for the
present study. Plotting sinuosity against latitude (Fig.
{\ref{978363}}) shows no clear trend, not even for the
peak sinuosities. High-sinuosity bends are present in the Nile, Danube,
and Knight Inlet channels, despite the fact that these channels are all
located at latitudes where the influence of the Coriolis force should be
stronger. Similarly, low-sinuosity systems like the NAMOC are not
restricted to high latitudes; the Tanzania Channel~\citep{Bourget_2008} is
located at latitudes similar to those of the highly sinuous Zaire
Channel.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/fig3/fig3}
\caption{{Plot of sinuosity vs. latitude, for all sinuosity values. Peak
sinuosities are highlighted with larger circles.
{\label{978363}}%
}}
\end{center}
\end{figure}
This analysis does not include some of the systems that were looked at
before; however, adding these systems would not change our conclusions
about the lack of correlation between latitude and sinuosity. We have
tested whether the combined data set of maximum sinuosities from the
systems in
~\href{https://www.authorea.com/users/157948/articles/200301-latitudinal-changes-in-the-morphology-of-submarine-channels-reevaluating-the-evidence-for-the-influence-of-the-coriolis-force\#nokey_f708d}{Clark
and Pickering (1996)} plus the ones added in ~\citet{Sylvester_2013} and in
this study (Zaire, Danube, Knight Inlet, Nile, Tanzania) shows a robust
latitude--sinuosity correlation. The resulting scatterplot suggests that
there is no correlation (Fig. {\ref{562478}}); the R2
value for the linear regression is 0.041. In other words, only about 4\%
of the variance in the peak sinuosities might be caused by a latitudinal
effect. More importantly, the large p- value (0.378) for the regression
suggests that any apparent correlation is actually not statistically
significant.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig4/fig4}
\caption{{Scatterplot of peak sinuosities as a function of latitude. Red data
points were added in this study (compared to Peakall et al. {[}2011{]}
and Sylvester et al. {[}2013{]}).
{\label{562478}}%
}}
\end{center}
\end{figure}
\subsection*{Estimation of the Impact of the Coriolis
Force}
{\label{605979}}
The Coriolis force is an apparent force that affects objects or fluids
moving within a rotating reference frame. In the case of planet Earth,
particles moving on the northern hemisphere are pushed to the right,
perpendicular to the direction of movement; the orientation of the force
vector is to the left in the southern hemisphere. The magnitude of the
Coriolis acceleration is given by
\begin{equation}a_c = 2\omega \sin(\phi)U \end{equation}
where~\(\omega\) is Earth's angular rotation speed,
\(\phi\) is latitude, and \(U\) is the velocity
of the particle. A particle that otherwise would move in a~straight line
is deflected and follows a circular trajectory, assuming that the effect
of other forces is negligible. In the case of a sinuous submarine
channel, the particle in a channel bend is moving along a curved
trajectory, characterized by a radius of curvature \(R\).
In a simplified view, we are interested in how the Coriolis force
compares with the centrifugal force. The magnitude of the centrifugal
acceleration is
\begin{equation}a_{cf} = \frac{U^2}{R}\end{equation}
The ratio between the centrifugal and Coriolis forces is the
dimensionless Rossby number:
\begin{equation}Ro_r = \frac{a_{cf}}{a_c} = \frac{U}{2\omega\sin(\phi)R} \end{equation}
In this formulation of the Rossby number, we use radius of curvature as
the length scale, as this is the parameter that determines the
centrifugal force. A large value of the Rossby number means that the
centrifugal force dominates; values of \(Ro_r\) below one
describe situations in which the Coriolis force is larger than the
centrifugal one. In theory, the Coriolis force would suppress the
development of highly sinuous channel bends if it counteracted the
centrifugal force, which is ultimately responsible for the instability
leading to sinuosity development in submarine channels.
From a quick inspection of Eq. 3, it is clear that the Rossby number
will tend to be smaller at higher latitudes, low flow velocities, and
low curvature channel bends. In addition, taking into account that the
angular rotation speed of the Earth is a relatively small number (7.29e-
05 radians/s) and that flow velocities of turbidity currents are
unlikely to exceed \textasciitilde{}20 m/s, the radius of curvature must
be on the order of \textasciitilde{}10 km in order to decrease the
Rossby number enough so that the Coriolis force really matters.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/fig5/fig5}
\caption{{Estimates of the Rossby number for seven submarine channels, as a
function of latitude, assuming a flow velocity of 2 m/s. Black lines are
lines of equal radius of curvature.
{\label{860856}}%
}}
\end{center}
\end{figure}
To get a better idea of the typical ranges for Rossby numbers in
submarine channels, we have estimated~\(Ro_r\) for a large
number of channel bends in seven channel systems: Amazon, Zaire,
Tanzania, Nile, Danube, Knight Inlet, and NAMOC
(Figs.~{\ref{860856}},~{\ref{444764}}).
To eliminate straight channel segments, we have only included channel
bends with sinuosities larger than 1.01. Assuming a flow velocity of 2
m/s, a value that is likely to be characteristic of large channelized
turbidity currents~\citep*{Pirmez_2003}, the results show that, with the
exception of NAMOC, the Rossby numbers in these systems are all larger
than 10 (Fig.~{\ref{860856}}). In the case of NAMOC,
the \(Ro_r\) values are less than 10 but larger than 1.
However, while a flow velocity of 2 m/s is a good estimate of turbidity
current speeds in the Amazon Channel ~\citep*{Pirmez_2003}, where
sand-sized grains dominate the channel thalweg, it is likely an
underestimate of the current velocities at the bottom of the NAMOC,
where gravel is not uncommon \citep{Klaucke:1997wi}. Assuming flow
velocities of 6.5 to 8 m/s for the lower part of the flow, as suggested
by \citet{Klaucke:1997wi}, the Rossby number increases about fourfold for
the NAMOC bends as well.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/fig6/fig6}
\caption{{Estimates of the Rossby number for seven submarine channels, as a
function of latitude, assuming a flow velocity of 0.1 m/s. Black lines
are lines of equal radius of curvature.
{\label{444764}}%
}}
\end{center}
\end{figure}
An obvious way to reduce the Rossby numbers is to consider much lower
flow velocities. At 0.1 m/s, most channel bends of the NAMOC and
Tanzania channels, and a significant proportion of the Danube data
points, fall below the Ror 1/4 1 threshold
(Fig.~{\ref{444764}}); for the Amazon and Zaire data
points, the centrifugal force is still about an order of magnitude
larger than the Coriolis force. Although flows with such low velocities
would not be able to transport most of the sediment that characterizes
the active channel thalwegs in virtually all systems, current speeds of
a few centimeters per second are probably common in the upper, more
dilute, much finer-grained parts of the flows~\citep{Andrieux_2013}. The
below-unity Rossby numbers for the Danube and NAMOC channels are
consistent with the observation that levees are strongly asymmetric in
both systems~\citep{Klaucke_1998,Popescu_2001}. The asymmetry of submarine levees at
higher latitudes has long been recognized in other systems as well
\citep{Komar_1969,Carter_1988}.
In summary, for the bottom-hugging parts of turbidity currents flowing
in submarine channels that cover a wide range of latitudes and channel
dimensions, the centrifugal force is at least an order of magnitude
larger than the Coriolis force. Therefore, it is unlikely that the
latter is responsible for the low sinuosity of some high-latitude
channels.
\section*{Discussion}
{\label{162398}}
If we revisit Eq. 3, the expression of the Rossby number, we can see
that the radius of curvature has a much larger impact on the value
of~\(Ro_r\) than latitude. Going from a latitude of 20\selectlanguage{ngerman}º to the
pole results in only a threefold decrease in the Rossby number, but
radius of curvature in submarine channels can cover several orders of
magnitude (from \textasciitilde{}100 m to tens of kilometers),
and~\(Ro_r\) is a hundred times smaller in the case of a
channel bend with~\(R\ =\ 10\ km\) compared to one
with~\(R\ =\ 100\ m\).
In other words, for channels at high latitudes, the overall size of the
channel is more important for determining the impact of the Coriolis
force than the precise latitudinal position. As is the case for rivers,
both radiuses of curvature and meander wavelengths of sinuous submarine
channels correlate with channel widths \citep*{Pirmez_2003}; the average
radius of curvature is a measure of the scale of the channel system and
of the typical flows that have carved and built the channel.
\citet{Peakall_2013} have suggested that ``as bend sinuosity decreases
with latitude, radius of curvature increases.'' However, low sinuosity
does not necessarily imply a large radius of curvature. To better
understand the relationship between sinuosity development and changes in
radius of curvature, we have used an implementation of the
\citet*{Howard_1984} curvature-based centerline model and briefly discuss
the results here.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/fig7/fig7}
\caption{{A) Sinuosity, half wavelength, mean radius of curvature, and mean
curvature through time in an implementation of the \protect\citet*{Howard_1984}
model. Inset shows how the centerline looks at four different stages of
the early evolution of the channel. B) Detail of the simulation output
that was used to derive the data in A.
{\label{693028}}%
}}
\end{center}
\end{figure}
As in all centerline evolution models, the first centerline is a
straight line, with some noise added. Initial curvature values of the
noisy centerline are large (and, therefore, the mean radius of curvature
is small), but they decrease as the centerline gets smoother and a
characteristic wavelength is being established. A peak value of R is
reached early on and then the radius of curvature declines rapidly
(Fig.~{\ref{693028}}). The increase in sinuosity lags
behind the increase in curvature, so that there is barely any departure
from a straight line during the early phase of high values of R (Fig.
{\ref{693028}}A). If the Coriolis effect was
suppressing the development of sinuosity, this phase of large radiuses
of curvature would represent the time period during which the Coriolis
force could and should limit the development of the instability that
leads to meandering. However, our numerical simulations suggest that by
the time the centerline has a visually noticeable sinuosity, curvature
values have increased significantly and the radius of curvature has
dropped to a value that remains characteristic for the system for the
rest of its evolution (Fig. 7A). A similar early phase of low curvature
values that rapidly transitions to larger curvatures characteristic of
the sinuous channel has been documented using other, more sophisticated
numerical models as well \citep{Camporeale_2005}.
The implication of this analysis is that channels with low but clearly
visible sinuosity are likely to have already established a
characteristic radius of curvature that is not going to significantly
decrease any further. More sinuous stretches of the NAMOC have an
overall sinuosity of \textasciitilde{}1.06, a value high enough to
suggest that this system is past its early phase, characterized by small
curvatures (Fig. {\ref{693028}}A). In other words, the
NAMOC has large values of R because it is a large system, not because of
its low sinuosity. The presence of an obvious sinuosity is evidence for
an inertial instability in the first place; as \citet{Klaucke:1997wi} have
stated in their study of the NAMOC, ``in general, the thalweg appears to
be located on the outside of meander bends, which demonstrates the
predominant effect of the centrifugal force on the lowest and fastest
parts of the flows.''
Thus, the low sinuosity of very large channels like the NAMOC and
Tanzania is unlikely to be caused by the Coriolis effect. Both of these
systems reflect important differences compared to the highly sinuous
channels that are more or less directly linked to their feeding
rivers/deltas: (1) they are about an order of magnitude larger than even
the largest ``typical'' submarine channel
(Fig.~{\ref{777085}}); and (2) they are fed by a
tributary channel system, as opposed to the typical avulsion-related
distributary channel pattern characteristic of channels on submarine
fans.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/fig8/fig8}
\caption{{Plotting a selection of submarine channels at the same scale shows how
much larger the NAMOC system is than any of the other systems.
{\label{777085}}%
}}
\end{center}
\end{figure}
Instead of comparing low-latitude channels fed by large rivers to
high-latitude, often steeper systems that are not directly linked to
rivers and their deltas, it is more insightful to compare systems that
have many characteristics in common but are situated at different
latitudes. The best candidate in this regard is the Danube Channel,
located at 43\selectlanguage{ngerman}º to 44º N, fed by the Danube River, and sharing many
similarities with large submarine channels close to the Equator. The
most interesting feature of the Danube submarine fan is that the
channels are highly sinuous, yet the levees display a strong asymmetry
that is clearly driven by the Coriolis effect \citep{Popescu_2001}. This
suggests that a turbidity current can be strongly influenced by the
centrifugal force in its lower, faster, coarser-grained part while the
upper, slower, and finer-grained layer is pushed preferentially to one
side by the Coriolis force. In addition to the velocity differences, the
geomorphology of many leveed submarine channels is likely to further
contribute to this effect: the lower part of the flow goes through
channel bends with large curvatures, while the upper part follows a much
straighter path, often only partly confined by the levees, with lower
sinuosity and large values of~\(R\)
(Fig.~{\ref{357103}}).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/fig9/fig9}
\caption{{Three-dimensional model of typical leveed submarine channel morphology.
The lower part of the flow is faster and following a more sinuous path
than the upper part. Modified from \protect\citet{Sylvester_2011}.
{\label{357103}}%
}}
\end{center}
\end{figure}
The levee asymmetry of the channels on the Danube Fan also influences
the large-scale stratigraphic architecture of the system. As a result of
the taller and wider right-side levees, avulsions preferentially occur
on the left side \citep{Popescu_2001}; this results in the asymmetric
overall structure of the Danube Fan, as most of the deposition---and
certainly most of the sand deposition---takes place on the northern side
of the initial channel levee system.
\citet{Peakall_2013} have suggested that there was a cutoff for
high-sinuosity channels at 50\selectlanguage{ngerman}º latitude; according to this view, this
cutoff would explain why some high-sinuosity channels like the Danube
and Knight Inlet still occur at latitudes less than or close to 50º. We
see no particular reason for a step change in the Coriolis effect at 50º
latitude. For the same velocity and radius of curvature, the Rossby
number decreases rapidly at low latitudes and then stays almost constant
at latitudes higher than \textasciitilde{}50º (Fig. 10).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig10/fig10}
\caption{{The variation of Rossby number as a function of latitude, for a flow
velocity of 2 m/s and three values of the radius of curvature.
{\label{294659}}%
}}
\end{center}
\end{figure}
One of the potential reasons why sinuous submarine channels are uncommon
on the modern seafloor at higher latitudes is the lack of large sources
of significant sediment input into the deep sea. The low gradient
systems with the highest-sinuosity channels (e.g., Amazon, Zaire, Indus,
Bengal, Danube) are all directly linked to a large fluvial sediment
source. Most of the larger rivers that drain Asia and North America into
the Arctic Ocean (e.g., Kolyma, Indigirka, Lena, Yenisei, Ob) have
relatively low sediment discharges \citep*{Milliman_2009}, and, under the
present-day conditions with high sea level, most of the limited amount
of sediment that reaches the sea never gets to the shelf edge. An
exception is the Mackenzie Delta, where thick accumulations of
deep-water clastic sediments are known from the subsurface (e.g.,
\citealp{nokey_d6435}). The analysis and arguments presented here suggest
that such settings, with their high sediment supply, low gradients, and
relatively long continental slopes, are favorable for the development of
highly sinuous submarine channels, regardless of their latitude.
\section*{Conclusions}
{\label{421900}}
A new look at a number of submarine channels, using an automated
workflow for calculating sinuosity and curvature, suggests that there is
no robust relationship between submarine channel sinuosity and latitude.
The apparent correlation between peak sinuosity and latitude for a
certain set of channels \citep{Peakall_2011,Wells_2013} becomes statistically
insignificant if more data points are added.
The Coriolis force is a weak force that becomes important only at large
scales: assuming a flow velocity of 2 m/s, the Coriolis force exceeds
the centrifugal force in the lower, channel-shaping parts of the flow
only in very large channel bends, those with a radius of curvature
larger than \textasciitilde{}10,000 m. The vast majority of submarine
channels on Earth do not reach these dimensions. Decreasing velocities
tilt the force balance in the favor of Coriolis, even in smaller
systems, and, as a result, the upper, more dilute and slower layers of
turbidity currents are more likely to be affected by the Coriolis
effect, even at relatively low latitudes.
Analysis of sinuosity development using a simple centerline evolution
model shows that the initial low-curvature phase---during which the
curvature-based Rossby number must be small---corresponds to extremely
low sinuosities, and by the time there is a visible undulation in the
centerline, the mean radius of curvature has dropped significantly
(Fig.~{\ref{693028}}). Channels with low but clearly
visible sinuosity are likely to have established a characteristic radius
of curvature early on, which is not going to significantly decrease any
further.
The Coriolis-driven asymmetry in levee height is well documented in
large systems located at higher latitudes. Unequal levee heights can
occur in systems with high overall sinuosities, suggesting that strong
Coriolis effects in the upper part of the flow can accompany a faster
lower part that is dominated by centrifugal forces. This phenomenon is
enhanced by differences in flow behavior: large curvatures associated
with the sinuous thalweg characterize the lower part, whereas
centrifugal accelerations are small in the upper part that tends not to
strictly follow the underlying sinuous pattern. Channels of the Danube
Fan are good examples of such systems: they are highly sinuous yet show
a strong levee asymmetry. The increased levee height on the right
channel banks of the Danube Channel results in preferential avulsion on
the low-levee side to the left; in the long term, this leads to a
characteristic large-scale channel pattern that might be possible to
recognize in ancient systems.
Even in large high-latitude systems, patterns of erosion and deposition
and the direction of channel migration alternate from one channel bend
to the other and are consistent with an instability-driven channel
evolution model. Although it is possible that the Coriolis force plays a
role in limiting bend growth in a few very large systems, in the
majority of submarine channels this force is unlikely to strongly affect
the higher-density, faster-moving lower parts of gravity flows, which
are driving the development of sinuosity.
\section*{Acknowledgements}
{\label{926020}}
We are grateful for discussions on submarine channel sinuosity with Zane
Jobe, Alessandro Cantelli, Nick Howes, Morgan Sullivan, Tao Sun, and
Jacob Covault. We are also grateful to Carmen Fraticelli for inviting us
to present this work at the SEPM (Society for Sedimentary Geology)
Research Conference and to Chevron Energy Technology Company for
permission to publish. Reviews by Kyle Straub and Michael Sweet have
greatly improved the paper.
\selectlanguage{english}
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