Einstein, tea leaves, meandering rivers, and beer

ResearchBlogging.org If you make your tea the old-fashioned way, ending up with a few tea leaves at the bottom of the teacup, and you start stirring the tea, you would expect the leaves to move outward, due to the push of the centrifugal force. Instead the leaves follow a spiral trajectory toward the center the cup. The physical processes that result in this ‘tea leaf paradox’ are essentially the same as the ones responsible for building point bars in meandering rivers. It turns out that the first scientist to make this connection and analogy was none other than Albert Einstein.

In a paper published in 1926 (English translation here), Einstein first explains how the velocity of the fluid tea flow is smaller at the bottom of the cup than higher up, due to friction at the wall. [The velocity has to decrease to zero at the wall, a constraint called ‘no-slip condition’ in fluid mechanics.] To Einstein it is obvious that “the result of this will be a circular movement of the liquid” in the vertical plane, with the liquid moving toward the center at the bottom of the cup and outward at the surface (see the figure below). For us, it is probably useful to think things out in a bit more detail.

Einstein’s illustration of secondary flow in a teacup

A smaller velocity at the bottom means a reduced centrifugal force as well. But overall, the tea is being pushed toward the sidewalls of the cup, and this results in the water surface being higher at the sidewalls than at the center. The pressure gradient that is created this way is constant throughout the whole water tea column, and overall it balances the centrifugal force (unless you stir so hard that the tea spills over the lips). This means that the centrifugal force wins at the top, creating a velocity component that points outward, but loses at the bottom, creating a so-called secondary flow that is pointing inward. The overall movement of the liquid has a helical pattern; in fact, those components of the velocity that act in a direction perpendicular to the main rotational direction are usually an order of magnitude smaller than the primary flow.

Einstein goes on to suggest that the “same sort of thing happens with a curving stream”. He also points out that, even if the river is straight, the strength of the Coriolis force resulting from the rotation of the Earth will be different at the bottom and at the surface, and this induces a helical flow pattern similar to that observed in meandering rivers. [This force and its effects on sedimentation and erosion are much smaller than the ‘normal’ helical flow in rivers.] In addition, the largest velocities will develop toward the outer bank of the river, where “erosion is necessarily stronger” than on the inner bank.

Secondary flow in a river, the result of reduced centrifugal forces at the bottom

I find the tea-leaf analogy an excellent way to explain the development of river meanders and point bars; just like tea leaves gather in the middle of the cup, sand grains are most likely to be left behind on the inner bank of a river bend. Yet Einstein’s paper is usually not mentioned in papers discussing river meandering — an interesting omission since a reference to Einstein always lends more weight and importance to one’s paper (or blog post).

A recent and very interesting exception is a paper published in Sedimentology. It is titled “Fluvial and submarine morphodynamics of laminar and near-laminar flows: a synthesis” and points out how laminar flows can generate a wide range of depositional forms and structures, like channels, ripples, dunes, antidunes, alternate bars, multiple-row bars, meandering and braiding, forms that are often considered unequivocal signs of turbulent flow. [This issue of Sedimentology is open access, so do click on the link and check out the paper!]. As they start discussing meandering rivers and point bars, Lajeunesse et al. suggest that Einstein’s teacup is extremely different dynamically from the Mississippi River, yet it can teach us about how point bars form:

A flow in a teacup with a Reynolds number of the order of 102 cannot possibly satisfy Reynolds similarity with the flow in the bend of, for example, the Mississippi River, for which the Reynolds number is of the order of 107. Can teacups then be used to infer river morphodynamics? 

The answer is affirmative. When dynamical similarity is rigorously satisfied, the physics of the two flows are identical. However, even when dynamical similarity is not satisfied, it is possible for a pair of flows to be simply two different manifestations of the same phenomenon, both of which are described by a shared physical framework. Any given analogy must not be overplayed because the lack of complete dynamic similarity implies that different features of a phenomenon may be manifested with different relative strengths. This shared framework nevertheless allows laminar-flow morphodynamics to shed useful light on turbulent-flow analogues.

Apart from helping understand river meandering, the tea leaf paradox has inspired a gadget that separates red blood cells from blood plasma; and helps getting rid of trub (sediment remaining after fermentation) from beer.

That explains the ‘beer’ part of the title. And it is time to have one.

References

Einstein, A. (1926). Die Ursache der Meanderbildung der Flusslaufe und des sogenannten Baerschen Gesetzes Die Naturwissenschaften, 14 (11), 223-224 DOI: 10.1007/BF01510300

Lajeunesse, E., Malverti, L., Lancien, P., Armstrong, L., Metivier, F., Coleman, S., Smith, C., Davies, T., Cantelli, A., & Parker, G. (2010). Fluvial and submarine morphodynamics of laminar and near-laminar flows: a synthesis Sedimentology, 57 (1), 1-26 DOI: 10.1111/j.1365-3091.2009.01109.x

The complexity of sinuous channel deposits in three dimensions

ResearchBlogging.org The beauty of the shapes and patterns created by meandering rivers has long attracted the attention of many geomorphologists, civil engineers, and sedimentologists. Unless they are fairly steep or have highly stable and unerodible banks, rivers do not like to follow a straight course and tend to develop a sinuous plan-view pattern. The description and mathematical modeling of these curves is a fascinating subject, but that is not what I want to talk about here and now. It is hard enough to understand the plan-view evolution of rivers, especially if one is interested in the long-term results – when cutoffs become important -, but things get really complicated when it comes to the three-dimensional structure of the deposits that meandering rivers leave behind. The same can be said about sinuous channels on the seafloor, created and maintained by dirty mixtures of water and sediment (called turbidity currents). An ever-increasing number of seafloor and seismic images show that highly sinuous submarine channels are almost as common as their subaerial counterparts, but much remains to be learned about the geometries of their deposits that accumulate through geological time.

Using simple modeling of how channel surfaces migrate through time, two recent papers attempt to illustrate the three-dimensional structure of sinuous fluvial and submarine channel deposits. In the Journal of Sedimentary Research, Willis and Tang (2010) show how slightly different patterns of fluvial meander migration result in different deposit geometries and different distribution of grain size, porosity and permeability. [These properties are important because they determine how fluids flow – or don’t flow – through the pores of the sediment.] River meanders can either grow in a direction perpendicular to the overall downslope orientation, or they can keep the same width and migrate downstream through translation. In the latter case – which is often characteristic of rivers incising into older sediments -, deposits forming on the downstream, concave bank of point bars will be preferentially preserved. These deposits tend to be finer grained than the typical convex-bank point bar sediments. In addition to building a range of models and analyzing their geometries, Willis and Tang also ran simulations of how would oil be displaced by water in them. One of their findings is that sinuous rivers that keep adding sediment in the same area over time (in other words, rivers that aggrade) tend to form better connected sand bodies than rivers which keep snaking around roughly in the same horizontal plane, without aggradation.

Map of deposits forming as river meanders grow (from Willis and Tang,  2010).
Cross sections through the deposits of two meander bends (locations shown in figure above). Colors represent permeability, red being highly permeable and blue impermeable sediment. From Willis and Tang, 2010.

Check out the paper itself for more images like these, plus discussions of concave-bank deposition, cutoff formation, and filling of abandoned channels.

The second paper (Sylvester, Pirmez, and Cantelli, 2010; and yes, one of the authors is also the author of this blog post, so don’t expect any constructive criticism here) focuses on submarine channels and their overbank deposits, but the starting point and the modeling techniques are similar: take a bunch of sinuous channel centerlines and generate surfaces around them that reflect the topography of the system at every time step. However, we know much less about submarine channels than fluvial ones, because it is much more difficult to collect data at and from the bottom of the ocean than it is from the river in your backyard. The result is that some of the simplifications in our model are controversial; to many sedimentary geologists, submarine channels and their deposits are fundamentally different from rivers and point bars, and there is not much use in even comparing the two. Part of the problem is that not all submarine channels are made equal, and, when looking at an outcrop, it is not easy – or outright impossible – to tell what kind of geomorphology produced the  stratigraphy. In fact, the number of exposures that represent highly sinuous submarine channels, as observed on the seafloor and numerous seismic images, is probably fairly limited. One thing is quite clear, however: many submarine channels show plan-view migration patterns that are very similar to those of rivers, and this large-scale structure imposes some significant constraints on the geometry of the deposits as well.

That being said, nobody denies that there are plenty of significant differences between real and submarine ‘rivers’ [note quotation marks]. A very important one is the amount of overbank – or levee – deposition: turbidity currents often overflow their channel banks as thick muddy clouds and form much thicker deposits than the overbank sediment layers typical of rivers. When these high rates of levee deposition combine with the strong three-dimensionality of channel migration, complex geometries result that are quite tricky to understand just by looking at a single cross section.

Cross section and chronostratigraphic diagram through a submarine channel system with inner and outer levees (from Sylvester et al., 2010).

One of the consequences of the channel migration is the formation of erosional surfaces that develop through a relatively long time and do not correspond to a geomorphologic surface at all (see the red erosional zones in the Wheeler diagram above). This difference between stratigraphic and geomorphologic surfaces is essential, yet often downplayed or even ignored in stratigraphy. In terms of geomorphology, the combination of channel movement in both horizontal and vertical directions and the extensive levee deposition results in a wide valley with scalloped margins and numerous terraces inside:

Three-dimensional view of an incising channel-levee system (from Sylvester et al., 2010).

This second paper is part of a nice collection focusing on submarine sedimentary systems that is going to be published as a special issue of Marine and Petroleum Geology, a collection that originated from a great conference held in 2009 in Torres del Paine National Park, Southern Chile.

PS. As I am typing this, I see that Brian over at Clastic Detritus is also thinking about submarine channels and subaerial rivers… Those channels formed by saline density currents on the slope of the Black Sea are fascinating.

Willis, B., & Tang, H. (2010). Three-Dimensional Connectivity of Point-Bar Deposits Journal of Sedimentary Research, 80 (5), 440-454 DOI: 10.2110/jsr.2010.046

Sylvester, Z., Pirmez, C., & Cantelli, A. (2010). A model of submarine channel-levee evolution based on channel trajectories: Implications for stratigraphic architecture Marine and Petroleum Geology DOI: 10.1016/j.marpetgeo.2010.05.012

Deep-sea landscapes from the ice age

The upcoming edition of Accretionary Wedge is going to focus on geo-images. I was always fascinated by the beauty of landscapes and landforms, natural patterns and textures, as many of the posts on this blog can testify; that is one of the reasons why I became a geologist.

However, this time I want to show a different kind of geo-image. These are not usual photographs; they are pictures of landscapes that existed thousands or millions of years ago. The ‘photographer’ uses acoustic waves instead of light. Once the data is recorded, a whole lot of processing and editing is required to get a reasonable result. Most often it is not trivial to make sure that the final image indeed comes close to capturing one geological moment in time, and part of it is not hundreds of thousands or millions of years older than the rest. It is a bit like stacking vertically pictures that come from time-lapse photography, but parts of the older images are erased later and get replaced with pixels that belong to more recent shots.

I am talking about maps that come from three-dimensional seismic surveys, especially their shallower sections located near the seafloor. Using this kind of data, it is possible to reconstruct ancient landscapes through careful mapping. The result is never going to be perfect, or even comparable to present-day satellite imagery, on one hand due to the limited lateral and vertical resolution, and on the other hand due to the removal of significant parts of the stratigraphic record through erosion.

Still, it is amazing that it is possible to reconstruct for example how the Gulf of Mexico looked like during a glacial period. The images below come form the continental slope of the Gulf, and are buried a few hundred feet below the seafloor. This morphology most likely formed during a glacial period when rivers were crossing the exposed shelf and delivering sediment directly onto the upper slope.

source: Virtual Seismic Atlas

Two submarine channels are visible, both of them directly linked to a delta that was deposited at the shelf edge. Colors correspond to thickness: red is thick, blue is thin. The next image shows the surface underlying the channels; in this case, the topographic surface is draped with seismic amplitude:

source: Virtual Seismic Atlas

There are more images from this ancient landscape available at the Virtual Seismic Atlas, a great resource for geo-imagery in general (see this post at Clastic Detritus for more detail). It is best to view these ‘photographs’ at larger resolution (which is pretty big in this case!) — you can do that if you go to the VSA website.

Evapor-art from the Permian Castile Formation, west Texas

The Late Permian Castile Formation is a ~500 m thick accumulation of evaporites in west Texas and south-eastern New Mexico. Its most striking feature is the vast number of alternating thin layers of lighter- and darker-colored deposits, layers that seem to be continuous across most of the Delaware Basin. The white laminae are mostly gypsum and anhydrite; the darker layers consist of calcite and organic matter.


Most experts agree that these laminations reflect seasonal changes; that is, a pair of white and dark layers corresponds to one year. The thicker gypsum layer was deposited during the dry season; the thinner calcite layer with the organic material formed during the humid season when algae were more abundant and only carbonates could precipitate from the lower-salinity water [for more details, see this paper].


The image above gives an idea how laterally persistent these laminations can be; the two photographs come from cores that are 24 km (~15 miles) apart (source: Kirkland, D.W., 2003, An explanation for the varves of the Castile evaporites (Upper Permian), Texas and New Mexico, USA, Sedimentology 50, p. 899-920).

These evaporites are often affected by small-scale faulting and folding; the resulting patterns are quite variable and aesthetically pleasing (well, at least according to me). I shot these photos on a recent geological trip to Guadalupe Mountains National Park, in a roadcut near highway 180; more pictures here.

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 animation 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].

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.

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

ResearchBlogging.org 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

Links to this post:
Scientia Pro Publica #4

Normal grading

In sedimentology, the word ‘grading’ has nothing to do with exams and assignments. Instead, it refers to a regularly decreasing or increasing grain size within one sedimentary layer. Because it is much more common than the other alternative, upward decreasing grain size is called ‘normal grading’. Grains that consistently increase in size toward the top of the bed are responsible for ‘inverse grading’. Upward fining and coarsening are related terms that are often used to describe grain-size trends in not one, but multiple beds.

Normal grading in a turbidite from the Talara Basin, Peru

The simplest way to generate normal grading is to put some poorly sorted sand and water in a container, shake it up, and then let it settle. The larger grains will settle faster than the smaller ones (as Stokes’ law tells us) and most of the large grains will end up at the bottom of the deposit. [Note that some fine grains will be at the bottom as well – the ones that were already close to the bottom at the beginning of sedimentation.] This kind of static suspension settling is not how most sediment is deposited on a river bed or a beach; even if a grain is part of the suspended load, it usually goes through a phase of bedload transport, that is, a phase of jumping and rolling and bouncing on the bed, before it comes to rest. The resulting deposit usually has lots of thin layers, laminations, and there are no obvious and gradual upward changes in grain size.

What is needed is a sediment-rich flow that suddenly slows down or spreads out and looses its power to carry most of its sediment load. Grains are getting to the bottom so fast that there is not much time for the flow to keep them rolling and bouncing around; instead they quickly get buried by the other grains that are ready to take a geological break. While this is still quite different from static suspension settling (because the flow did not come to a full stop), it can be thought of as a modified version of static settling: all is needed is a horizontal velocity component, in addition to the vertical one. Of course, the segregation of the coarser grains to the bottom of the flow may have started much earlier. Typically, they never made it to the top in the first place.

Conglomerate bed in the Cretaceous Cerro Toro Formation, Torres del Paine National Park, Southern Chile. There is some inverse grading at the base of this bed, before the size of the clasts starts decreasing

Such large, sediment-laden flows are not very common, certainly not on a human timescale. When they do occur, they tend to show up in the news, especially if human artifacts, or humans themselves, become part of the normally graded deposits. Deposits of snow avalanches, volcanic ash-laden pyroclastic flows, subaerial debris flows, tsunamis, submarine turbidity currents can all show normal grading. The images shown here all come from deposits of large submarine gravity flows. Some of them (like the one below) have a muddy matrix, but the grading is still obvious (the two large clasts at the top of the bed have lower densities).

Normally graded conglomerate layer with a muddy matrix, Cerro Toro Formation, Chile

In recent years, some questions have been raised about the common presence of normal grading, especially in turbidites. The fact is that normal grading is often seen in rocks of all ages, and, in a simple view, it is a reflection of larger grains getting quickly to the bottom.

Normal grading is normal, after all.

Description does not suffice for an explanation

On February 3, 1967, J. R. L. Allen gave the fifth “Geologists’ Association Special Lecture”, entitled “Some Recent Advances in the Physics of Sedimentation”. This is from the introduction:

“Two stages can generally be recognised in the historical growth of a reasonably advanced scientific discipline. There is an early, descriptive stage in which with little guide from theory, an attempt is made to collect, define and analyse phenomena. In the later, explanatory stage we see that efforts are concentrated on the production of generalisations and on the explanation of the reduced phenomena in terms of general laws. Of course, there is never a single point in time at which there is change over the entire scope of a discipline from the descriptive to the explanatory stage. The change is, rather, uneven, taking place earlier in some branches than in others, and more gradually in one branch than in another.

Sedimentology stands today in a period of transition. Its subject matter is sedimentary deposits, and its goal the origin and meaning of these in the context of planetary studies in general. But it is apparent, except to adherents of geological phenomenalism, that sedimentary deposits cannot be explained in terms of themselves. Already we are in possession of major generalisations about these deposits, and our chief task for some years should be to explore and ratify them in terms of general laws in order that our understanding of the sedimentary record can be made sharper. In those parts of the field where major generalisations have already been established, the provision of further descriptive data is of little value, except in so far as light is shed on the problems of particular deposits. These are validly a part of the subject, leading to a refinement of certain planetary laws. But the other and no less important laws in terms of which we should seek to frame our understanding are those of general chemistry, physics and biology. In order to achieve this framework in the case of detrital sediments, it will be necessary to set aside for a while the problems of particular deposits. This will, of course, be unacceptable to those who claim that geology, or sedimentology, is only to do with rocks as conceived in a historico-geographical manner. But they will be proved wrong, provided we keep our major goals in mind, for it is a mistake to suppose that a description will suffice for an explanation. Most of our explanations will probably turn out to be no better than qualitative, so complex are most sedimentary systems, but we should nevertheless attempt them and try to frame them as exactly as possible.”

Forty years after publication of the paper, this seems as timely as ever.

Reference:
Allen, J. R. L., 1969. Some Recent Advances in the Physics of Sedimentation. Proceedings of the Geologists’ Association 80:1-42.

The science of espresso, with a dash of geology

Almost every morning, I start the day with an experiment on flow in porous media. First, I generate some fine-grained sediment with a well-defined average grain size and proper sorting; then I use that sediment to fill a little basin of sort and try to mimic compaction. Finally, I use a machine to put water under pressure and force it to flow through this miniature sedimentary basin. Then I sit down to drink the fluid which is not simple water anymore, due to its interaction with the grains; and its taste and consistency tell me whether I got the grain size and the porosity right.

That’s a geologist’s view of making espresso. Unless you have a fully automated and ultra-expensive espresso machine, creating a high-quality caffeine concoction is not trivial, because the water must have the right temperature and has to spend the right amount of time in contact with the coffee grains that have the right size. The right temperature is 85–95 °C (185–203 °F), and, at least with our simple machine, the trick is to start the brewing at the right time. Better espresso machines do not use steam to generate pressure because that makes the water too hot; instead, they have a motor-driven pump that generates the ~9 bars of pressure. The correct grain size is easily achieved with a burr grinder (as opposed to a simple blender); a good espresso grind is a fine grind, because the water spends relatively little time in contact with the grains.

The duration of this contact is the most difficult bit to get right. To get a good shot with lots of crema, it cannot be less or more than 20 to 30 seconds. Not just grain size, but grain sorting as well play a role. If the coffee grinder produces a poorly sorted ‘sediment’ (and that’s what a blender does), the coffee will not be porous and permeable enough. Another factor is how well the sediment is compacted; that is, how much pressure do you apply to the coffee during tamping. This affects permeability again. Finally, it matters how much coffee you put in the coffee holder; the thicker the layer that the water has to go through, the longer the trip becomes for the same amount of water.

After using the machine hundreds of times, I still manage from time to time to produce something undrinkable. The art and science of espresso making started to make more sense once I started to think of it in terms of Darcy’s Law.

Henry Darcy was a French engineer who initially had made a name for himself by designing an enclosed and gravity-driven water-supply system for the town of Dijon. Later he had time and opportunity to do experiments of his own interest. In 1855 he measured the discharge of water under variable hydraulic heads through sand columns of different heights, and found that the discharge was directly proportional with the hydraulic head and inversely related to the height of the sand column:

Q = AK(H1-H2)/L,
where A is the cross sectional area of the sand column, L is the height of the sand column, K is the hydraulic conductivity (which is constant for the same granular material and same fluid), and H1-H2 is the hydraulic head. This is Darcy’s drawing of his experimental setup:

The hydraulic conductivity depends on both the properties of the fluids and of the granular material; these properties are the viscosity and density of the fluid, and the permeability of the sediment:

K = kρg/μ,

where K = hydraulic conductivity, k = permeability, ρ = fluid density, and μ is the dynamic viscosity.

In coffee speak, the hydraulic head is given by the pressure generated by the machine, and is fixed; one cannot change the density and viscosity of water either. The most important variable is coffee permeability, which is influenced by size, sorting, and packing (compaction) of the coffee grains. Also, it helps if you get the value of L right, that is, you shouldn’t try to save coffee.

Darcy’s Law was established with some simple experiments, and it has since then been generalized and derived from the Navier-Stokes equations, but it has a huge range of applicability, from ground-water hydrology to soil physics and petroleum engineering.

Add to that list everyday espresso making.

ps. Fantastic resource on Darcy’s work and his law here.