Three photos from Kauai

The rest are here.

Colors in Alakai swamp

Green turtle at Tunnels Beach

Na Pali Coast

Hillslope diffusion

Modeling erosion and deposition of sediment using the diffusion equation is among the important subjects that are usually omitted from sedimentary geology textbooks. Part of the reason for this is that ‘conventional’ sedimentary geology tended to only pay lip service to earth surface processes and was more interested in describing the stratigraphic record than figuring how it relates to geomorphology. Nowadays, a good discussion of stratigraphy and sedimentology cannot ignore anymore what geomorphologists have learned about landscape evolution. (One textbook that clearly recognizes this is this one.)

But let's get back to the subject of this post. Hillslope evolution can be modeled with the diffusion equation, one of the most common differential equations in science, applied for example to describe how differences in temperature are eliminated through heat conduction. In the case of heat, the heat flux is proportional to the rate of spatial temperature change; on hillslopes, the sediment flux is proportional to the spatial rate of change in elevation. This last quantity of course is the slope itself. In other words,

q = -k*dh/dx,
or

q = -k*slope,
where q is the volumetric sediment flux per unit length, k is a constant called diffusivity, h is the elevation, and x is the horizontal coordinate.

We also know that sediment does not disappear into thin air: considering a small area of the hillslope, the amount of sediment entering and leaving this area will determine how large the change in elevation will be:


dh/dt = -dq/dx,
in other words, deposition or erosion at any location is determined by the change in sediment flux.

Combining this equation with the previous one, we arrive to the diffusion equation:

dh/dt = k*d²h/dx².
Note that the quantity on the right side is the second derivative (or curvature) of the slope profile. Large negative curvatures result in rapid erosion; places with large positive curvature have high rates of deposition. Through time, the bumps and troughs of the hillslope are smoothed out through erosion and deposition.

The simplest possible case is the diffusion of a fault scarp. The movie below illustrates how a 1 m high fault scarp gets smoothed out through time; the evolution of slope and curvature are also shown. The dashed line indicates the original topography, at time 0. [The plots were generated using Ramon Arrowsmith's Matlab code; right-click the flash animation for playback controls].



More complicated slope profiles can be modeled as well; here is an example with two fault scarps:



Note how both erosion and deposition get much slower as the gradients become more uniform.

The simplicity of the diffusion equation makes it an attractive tool in modeling landscape evolution. In addition to hillslopes and fault scarps, it has been successfully applied in modeling - for example - river terraces, deltaic clinoforms, cinder cones, fluvial systems, and foreland basin stratigraphy. However, it is important to know when and where the assumptions behind it become invalid. For example, steep slopes often have a non-linear relationship between sediment flux and slope as mass movements dramatically increase sediment flux above a critical slope value. Also, the models shown here would fail to reproduce the topography of a system where not all sediment is deposited at the toe of the steeper slope, but a significant part is carried away by a river. And that brings us closer to advection; a subject that I might take notes about at another time.

Further reading: 1) The book "Quantitative Modeling of Earth Surface Processes" by Jon Pelletier has a chapter with lots of details about the diffusion equation. 2) Analog and numerical modeling of hillslope diffusion - a nice lab exercise.

Upcoming conference on seismic geomorphology

Ever since geoscientists and engineers started using seismic waves to figure out what lies under our feet, seismic reflection technology kept improving and today most oil-rich sedimentary basins have a wide coverage of high-quality three-dimensional datasets. In addition to finding structures and locations in the subsurface that are likely to be filled with hydrocarbons, these huge data volumes can also be used to reconstruct landscapes that are long gone from the Earth's surface. After all, stratigraphy is what is left behind from an ever-changing topography, and it is a mistake to think that stratigraphy can be understood without knowing a few things about geomorphology and landscape evolution.

Seismic data confirms that the past, indeed, is not that different from the present: if you peek (that is, listen) into these volumes of rock, you see ancient meandering rivers and submarine channels, deltas, barrier islands, mouth bars, and estuaries. However, in addition to and beyond the excitement of seeing another beautiful example of a sinuous channel or other depositional and erosional features, a lot remains to be learned from the true and large-scale three-dimensionality of these datasets. For a geologist, there is no other data type that offers such a degree of three-dimensionality. Even the largest outcrops offer only random two-dimensional sections through a system; the temptation is strong to convince ourselves that we can extrapolate to get an idea about the third dimension, but more often than not we are probably wrong, at least in the details of our extrapolation. This is actually worse than the case of "The Blind Men and the Elephant"; it is more like the blind men and a random cut through the elephant (but I will stop this chain of analogies right there).

Long story short, I (re)started to blog about this subject because there is going to be an SEPM research conference in Houston, a conference that focuses on using three-dimensional seismic data to better understand how sediment moves or gets deposited on continental slopes. It should be an interesting collection of talks and papers.



The image above is from the conference website and announcement; it happens to come from a paper that I am going to present, on a shelf-edge delta and its related slope channels in the Gulf of Mexico (the higher-resolution version is coming soon...). Colors represent thickness (red is thick, blue means thin). There are two leveed channels taking sediment from the shelf-edge delta into the deep ocean.

Two gigapans from Cliffs of Moher, Ireland

I shot these gigapans recently, while we were visiting some deep-water rocks in County Clare, Ireland (see more detail on these rocks and a few photos from the trip). One afternoon we took some time off from the turbidites to do a bit of geo-tourism at the Cliffs of Moher, a series of spectacular escarpments along an 8 km long stretch of the western coast of Ireland. They are 702 feet (214 meters) high at the highest point and expose Late Carboniferous (Namurian) sandstones and shales that were mostly deposited as deltaic and fluvial sediments of the Tullig and Kilkee cyclothems.

This place is one of the most visited tourist attractions in Ireland, and for a good reason: the combination of the cliffs, the landscape, and abundant wildlife is, indeed, spectacular.

This is a view to the south (launch full screen viewer):


And this is a view to the north (from O'Brian's Tower; launch full-screen viewer):


Unfortunately, these stamp-sized windows do not do justice to the panoramas; it is a good idea to click on the "Launch full-screen viewer" links.

More reasons to conclude that coastal 'chevrons' are not related to mega-tsunamis

If there was any doubt left that coastal sand accumulations called 'chevrons' are *not* related to gigantic tsunamis (see previous thoughts on the subject here and here; Ole also has a recent blog post, and see a news release here), the May issue of Geology provides additional arguments to show that this is the case. Joanne Bourgeois of University of Washington and Robert Weiss of Texas A&M University, both experts in the sedimentology of tsunami deposits, present two lines of arguments. First they show that the orientation of the Madagascar chevrons is significantly different from what is predicted through modeling the tsunami. While the tsunami wave tends to hit the coast with an overall perpendicular orientation, due to wave refraction, the 'chevrons' are oriented at various angles to the coast, angles that are more consistent with predominant wind directions. Second, they look at the sediment transport conditions and suggest that even coarse sand must have been in suspension in flows deep enough to create the chevrons. However, dune-like bedforms cannot develop without sediment being transported as bedload; therefore, the bedforms must have a different origin than mega-tsunamis. The obvious alternative is parabolic dunes; these well-known bedforms show up when vegetation partially covers the dune's tails and slows down sediment transport. The authors don't hesitate to draw the conclusion that

The extraordinary claim of "chevron” genesis by mega-tsunamis cannot withstand simple but rigorous testing.

I am far from being a tsunami expert, but I find this subject fascinating. The issue of suspended load vs. bedload and stratified or laminated vs. graded bedding is equally important for deposition from tsunami waves and turbidity currents. It is worth spending a bit of time and blogspace to explore the kind of analysis of sediment transport conditions that this paper presents.

Although I see no reasons to disagree with the paper's conclusions (as it could be predicted from my previous posts on the subject), at first reading I didn't fully understand the line of reasoning about suspended load vs. bedload. So here goes my attempt to understand it.

The argument goes as follows. The Rouse number is the ratio between the settling velocity of a certain grain size and the shear velocity of the flow, multiplied by von Karman's constant (which is ~0.4): Ro = ws/k*u_shear. For a grain of a given size, if the Rouse number is larger than 2.5, the grain's settling velocity is much larger than the upward-directed component of the turbulence, and the grain tends to stay close to the bottom, in the bedload. [This is equivalent to saying that the settling velocity has to be larger or equal to the shear velocity, a condition also known as the suspension criterion]. If the Rouse number is less than 0.8, the flow is turbulent enough to keep the grain fully suspended. In between these values, there is a zone of transitional behavior. For the flows that might have deposited the chevrons, the Rouse number is always less than 2.5, regardless of how the other parameters like the Froude number, grain diameter, and roughness length are varied. Although the authors state that the flows must have been deeper than 8 m (because most chevrons are higher than 4 m, and the flow must be at least twice as high as the bedform), there seems to be no other constraint on tsunami behavior [note that I did not have access - yet - to the supplementary web material].

So the question is: doesn't this reasoning apply to other types of flows as well? For example, the Mississippi River is certainly deeper than 8 m in many places -- does this mean that it is able to suspend very coarse (2 mm diameter) sand? In other words, what is the difference between flow in a tsunami run-up and the Mississippi River? The answers might be obvious to many, but they are certainly not obvious to me.

One thing we can do is to create a different kind of plot: instead of plotting the Rouse number against flow depth, let's plot velocity vs. depth. I have a better feeling for what are reasonable velocities for different kind of flows than I do for Rouse numbers. The Rouse number would form the third dimension of the plot; one can visualize that as a contour map of Rouse numbers as a function of flow depth and velocity:


The Rouse numbers shown in this plot are valid for a grain diameter of 2 mm and roughness length of 1 m (using the same equations for settling velocity and shear velocity as in Bourgeois & Weiss 2009). Anything coarser than this cannot be called sand any more. So if this grain size doesn't stay in the bedload, there is no chance for finer sediment either. It is obvious from the plot that, for flows deeper than 8-10 m, very coarse sand will be part of the bedload unless flow velocity is larger than ~5 m/s. The Mississippi River at New Orleans has velocities on the scale 1.5 m/s, so 1-2 mm sand should definitely stay close to the bottom, and in fact it does.

We know however that tsunamis are not exactly tranquil flows like the big old Mississippi at New Orleans. The larger ones are fast and furious and Google Earth might need massive updates after they rearrange entire coastal landscapes. [Don't get me wrong, I am not trying to diminish the power and destructive force of the Mississippi.] In other words, the Froude number of a tsunami run-up is larger than the Froude number of the Mississippi River. The Mississippi is relatively slow and deep; the tsunami is fast and relatively shallow. The Froude number is the ratio between velocity and the square root of gravity multiplied by flow depth:


Fr = u/√(g*d)
This number for the Mississippi is much less than one (these flows are called subcritical flows). On the other hand, tsunamis are waves of very large wavelengths, and they behave even in the open ocean as shallow water waves (wavelength 20 times larger than water depth). For these kinds of waves, the velocity is solely a function of water depth:


u = √(g*d)
If we assume that the tsunami run-up has a comparable velocity to that of the tsunami wave in the nearshore zone, we find that the Froude number of the run-up must be around 1. This is obviously a very back-of-the-envelope argument, but the point is that these flows must have in general relatively large Froude numbers. If we plot the lines for Fr = 1 and Fr =1.5 on the depth-velocity diagram (see above), we can see how different likely tsunami behavior is from that of large rivers. It also becomes evident that even coarse sand would not be part of the bedload in these flows, especially not in flows deep enough to build the 'chevrons'. Which means that sandy tsunami deposits are likely to be largely unstructured or poorly structured sand sheets rather than several m thick accumulations of cross-bedded sand.

And that ends my Saturday exercise in Fluid Mechanics 101.

Reference
Bourgeois, J., & Weiss, R. (2009). "Chevrons" are not mega-tsunami deposits--A sedimentologic assessment Geology, 37 (5), 403-406 DOI: 10.1130/G25246A.1

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