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6.2.4: Mud Cracks - Geosciences

6.2.4: Mud Cracks - Geosciences


Figure (PageIndex{1}): Modern Mud Cracks

Figure (PageIndex{2}): Modern Mud Cracks in a Playa

Figure (PageIndex{3}): A Closer View of Modern Mud Cracks in a Playa

Figure (PageIndex{4}): Ancient Mud Cracks

Figure (PageIndex{5}): Ancient Mud Cracks in Both Vertical and Plan Views

Figure (PageIndex{6}): Ancient Mud Cracks and Wave Ripple Crests

Figure (PageIndex{7}): Ancient Mud Chips in Sandstone

Figure (PageIndex{8}):Ancient Mud Chips in Sandstone

Figure (PageIndex{9}):Syneresis cracks due to dewatering of mud under water


Return to Sedimentary Structures


Chapter 6 Tide-Dominated Estuaries and Tidal Rivers

Physical and biological processes in nearly all estuaries are influenced by tides. The degree of influence is governed by estuarine morphology, tide range, water and sediment discharge, winds, and shelf processes. Tide dominated estuaries are those in which tidal currents play the dominant role in the fate of river-borne sediments, resulting in appreciable upstream transport of bedload sediment and, in extreme cases, little or no density-driven circulation. Tidal rivers, which have many of the same morphologic and sedimentologic features, are estuaries that occur in the lower reaches of large rivers where the penetration of tide extends farther than, and is decoupled from, the upstream penetration of salt. Here, subaqueous deltaic sedimentation is common.

Most tide dominated estuaries and tidal rivers have a funnel shape, bidirectional sediment transport, mutually-evasive transport pathways, a tide- or density-induced turbidity maximum, and extensive regions of fine-grained sediment deposition, often in the form of fluid mud. Bottom sediments range from mud to gravel. As tides move upstream through smaller cross-sectional areas, the tidal currents become progressively more asymmetric in both speed and direction. In many cases, this leads to net landward transport of the bedload sediments. Characteristic bedforms include tidal sand ridges, large sand waves, and megaripples characteristic sedimentary structures include cross bedding, tidal bedding, reactivation surfaces, and flaser, wavy, and lenticular bedding. Tidal flats, mangrove swamp, or marsh grass usually form the margins of the estuaries.

Examples of tide-dominated estuaries and tidal rivers can be found in a wide variety of settings: for example, in the Rio de la Plata and Amazon tidal rivers, where respective tide ranges are less than 1 m and 4–8 m, much of the sedimentation occurs in the form of subaqueous deltas that have built over and around transgressive sands in the Gironde, where tide range is range 4 m, a highly transitory turbidity maximum characterizes the estuary yet 60% of the suspended sediment leaves the estuary and accumulates on the shelf in the Severn, where tide range is 8 m, sedimentation rates are so low that exposed bedrock covers extensive sections of the estuary and, in the Cobequid Bay-Salmon River, where tide range can exceed 12 m, extensive progradation is occurring from sands derived seaward of the estuary.


X-ray Powder Diffraction (XRD) Instrumentation - How Does It Work?

The geometry of an X-ray diffractometer is such that the sample rotates in the path of the collimated X-ray beam at an angle θ while the X-ray detector is mounted on an arm to collect the diffracted X-rays and rotates at an angle of 2 θ . The instrument used to maintain the angle and rotate the sample is termed a goniometer. For typical powder patterns, data is collected at 2 θ from

5 ° to 70 ° , angles that are preset in the X-ray scan.


Pressure Controlled Permeability in a Conduit Filled with Fractured Hydrothermal Breccia Reconstructed from Ballistics from Whakaari (White Island), New Zealand

4 × 10 −15 m 2 (Figure 6). Although there is a general trend of increasing permeability with increasing porosity, as observed in previous studies on the permeability of volcanic rocks [57,58], there is also substantial scatter within and between lithologies (Figure 6). For example, the permeability of samples with a porosity of

4 × 10 −15 m 2 (Figure 6a). The relatively unaltered lava ballistics generally have lower porosity and permeability than the altered lava, altered ash tuff, and sulfur flow (Figure 6a). Compared to rocks collected from the surface (data from [13,59] shown in grey on Figure 6a), the ballistic samples (yellow symbols on Figure 6a) generally have lower porosity and a narrower range of permeability. The samples with experimentally created tensile fractures are 4–5 orders of magnitude more permeable than the unfractured rocks at both confining pressures of 1 and 3 MPa (Figure 6a,b). Irrespective of the initial permeability, the permeabilities of the fractured samples are very similar at low confining pressures (

10 −12 m 2 at 1 and 3 MPa Figure 6a,b).

1 order of magnitude in the unaltered lava and fractured unaltered lava, a decrease of 1-2 orders of magnitude in the unfractured altered lava, and a decrease in permeability of 2–4 orders of magnitude in the fractured altered lava (Figure 7). Importantly, our data show that the permeability reduction as confining pressure is greater in the fractured altered samples than in the fractured unaltered samples (Figure 7).

3.3. Results Summary


Desiccation Cracks and their Patterns: Formation and Modelling in Science and Nature

Sujata Tarafdar joined the faculty of Jadavpur University, Kolkata, in 1990 and at present is a Professor of Physics there. Her research interest covers: fractals and disordered systems, pattern formation in nature, viscous fingering and crack networks, soft condensed matter, porous media and polymer electrolytes. She is coordinator of the Condensed Matter Physics Research Centre, Jadavpur University and general secretary of the Indian Society of Nonlinear Analysts. She is on the editorial board of Frontiers in Physics (Nature publishing group) as a review editor.

Akio Nakahara is the Associate Professor of Physics at Nihon University, Japan. Having obtained his PhD degree at Kyushu University, he spent four years as a PostDoc at Chuo University, and then joined Nihon University. He has worked on the Physics of Pattern Formation, and recently he found a method to control morphology of crack patterns using memory effect of plastic fluid.

Tapati Dutta is an Associate Professor in the Physics Department of St. Xavier's College, Kolkata, India. Having completed her Master's degree from Calcutta University and her PhD from Jadavpur University, she joined the Physics Department of St. Xavier's College in 1990 and has been continuing since then. She formerly was Head of the Department and is at present Dean of Research of St. Xavier's College. She is a member of the academic committee of the Condensed Mater Physics Research Centre of Jadavpur University. Dr. Dutta is also a member of the Indian Society of Non-linear Analysts. Her research interests are condensed matter physics and nonlinear dynamics.

So Kitsunezaki is an Associate professor of the department of physics at Nara women's university, Japan. After receiving his PhD at Kyoto University in 1997, he has taught in the university until now. He specialises in nonlinear dynamics and pattern formation. He studied the pattern formation of mud cracks and columnar joints of starch, theoretically and experimentally, and his current interests include the fracture mechanics of paste and wet granular materials.

Lucas Goehring leads a research group at the Max Planck Institute for Dynamics and Self-Organization, and teaches at the nearby Georg-August Universität Göttingen. He obtained his PhD at the University of Toronto, for his work on columnar jointing, and was recently a Fellow of Wolfson College, at the University of Cambridge. There he worked on the drying and cracking of colloidal films. His current research interests involve pattern formation, multi-phase flow, soft-matter physics, and geophysics.


Pattern formation in the geosciences

Pattern formation is a natural property of nonlinear and non-equilibrium dynamical systems. Geophysical examples of such systems span practically all observable length scales, from rhythmic banding of chemical species within a single mineral crystal, to the morphology of cusps and spits along hundreds of kilometres of coastlines. This article briefly introduces the general principles of pattern formation and argues how they can be applied to open problems in the Earth sciences. Particular examples are then discussed, which summarize the contents of the rest of this Theme Issue.

1. Pattern formation

Patterns exist throughout nature and are attractive to the trained observing eye of the scientist. Many of the questions that are presented in this issue on pattern formation were already under the attention of the earliest members of the Royal Society. The origin of springs ‘running down by the Valleys or Guttes between the ridges of the Hills, and coming to unite, form little Rivulets or Brooks’ was discussed by Edmund Halley [1], while the prismatic forms of columnar joints were first brought into the records of the Society by travellers' reports forwarded by Sir Richard Bulkeley [2]. The quantitative investigation of such patterns, however, is now a field of active research. For example, Petroff et al. [3] here demonstrate that springs eroding into a slope will generically split, or bifurcate, at an angle of 2π/5, while I present a crack-ordering mechanism that links columnar joints with polygonal terrain and mud-cracks [4]. There are many other situations where regular patterns are generated, from the largest scales of geomorphology, such as the curving subduction arcs of the Earth's crust [5] or the meanders of river networks, to ripples on the beach [6] and periodic chemical precipitation patterns [7]. The details of such patterns can provide quantitative information about the systems in which they form and the mechanisms underlying them.

The science of pattern formation, a physically motivated study of spatio-temporal patterns, and an attempt to explain how, why and when they arise, was formalized during the second half of the twentieth century [8]. Feedback can do much more than amplify or control a signal, and as understanding of nonlinear dynamical systems grew, unexpected universal features of these systems were found. The complex response of even simple systems, such as a double pendulum, or Lorenz's first three-component atmospheric model [9], where arbitrarily close initial conditions rapidly diverge, is now known as chaos. In other situations, complex interactions can give rise to simple regular patterns. This is the case we focus on here, although often both chaos and patterns may be found in the same system (see [7] on geochemical precipitation patterns). In both cases, the systems are usually dynamical, nonlinear and out of equilibrium: the patterns are dissipative, entropy-producing states, driven by an energy flux between some external energy source and heat sink.

Patterns typically arise as the result of competition between two opposing forces. Rayleigh–Bénard convection is the result of a competition between buoyancy, which acts to lift warmer, lighter fluids, and viscous dissipation, which tends to damp out any motion. The ratios of these forces define the dimensionless groups whereby a dynamical system can be parametrized. In most cases, the relevant number of such groups is small, because when groups are too dissimilar in magnitude, the weaker terms are usually (but not always) negligible in effect. Thus, for example, Wells & Cossu [10] describe how a balance between centrifugal and Coriolis forces is captured by a dimensionless Rossby number and explain how the dominant driving force, and the consequent morphology, of submarine channelling differs at high and low Rossby numbers.

Finally, the conditions under which patterns change form have been shown to be particularly revealing. The theory of such bifurcations, or transitions between states, was originally developed by Poincaré [11], and modern physical use grew out of specific applications of these ideas to phase changes in condensed matter physics [8]. Bifurcations can frequently be classified into one of a small number of instability types, based on considerations of dimensionality and symmetry. These dictate what the leading nonlinear contributions can be, and what generic type of response is expected around a critical point. For example, in many cases an instability in a system with reflection (e.g. left–right or up–down) symmetry leads to its classification as a pitchfork bifurcation. Near critical points of bifurcations, the behaviours of diverse systems can reduce to those of a few universal responses, which are independent of the detailed physics of these systems (i.e. all members of the same universality class share the same behaviour close to a supercritical bifurcation). For example, the transition from a straight to wavy crack [12], and the onset of oscillations during certain chemical reactions [8,7] are Hopf bifurcations and near their transitions these very different problems can be mapped onto each other. Just as critical points are the key to understanding the behaviour of classical thermodynamic phase diagrams, they are generally a very powerful tool for establishing and testing the physics of any pattern-forming system.

2. Why are geophysical patterns interesting?

The Earth is not in thermodynamic equilibrium. Heat flows from the molten core to the crust, through a convecting mantle. Winds blow, rain falls and mountains erode, as the Sun's energy is processed and re-radiated to the cooler background of space. Life happens, and in doing so, changes the world. The means that shaped the geography that we see today are dynamic, complex, non-equilibrium and nonlinear. These are the natural conditions in which to expect self-organization, and the patterns that can be found tell us about the physics of these systems. Further, pattern-forming mechanisms can be very robust they are not only seen in controlled laboratory situations, but also survive (and can even thrive on) the noise of real-world geomorphic environments.

One strength of the modern approach to patterns is its universality. The same instability can appear in a great variety of situations. Turing's seminal paper on ‘The chemical basis of morphogenesis’, for example, outlined the conditions necessary for generating a linear instability in a system with two reacting, diffusing, chemical morphogens [13]. In general, it applies to any two (or by a straightforward generalization, more) interacting fields u and v, where

The universality of pattern-forming systems can also be a challenge. The patterned ground of permafrost soils has evident structure. Detailed numerical models can reproduce the shapes of this terrain well. However, as discussed by Hallet [17], such models can generate a similar pattern as the result of either frost heave or ground water convection. The frost-heave model is now preferred, but only as the result of cycles of prediction and validation. In another example, certain labyrinthine patterns in grasses look superficially like the reaction–diffusion-based vegetation patterns of larger plants, but may result from the entirely different mechanism of porous-media convection [18]. Similarity of form alone, no matter how beautiful, is insufficient for proof—a useful model of pattern formation is necessarily quantitative, and describes additional features such as wavelength selection, scaling, rates or the location of bifurcation points between patterns in parameter space. In turn, these quantitative details guide further testing of models against field measurements and aid in the design of meaningful analogue experiments. Once an understanding is gained and tested, one can turn to interpretation, and here patterns can be powerful diagnostics of conditions that no longer exist or which (e.g. other planets, long time scales) are difficult to access directly.

3. Contents of this issue

This issue is dedicated to the application of the tools of pattern formation to geophysical problems. Richard Feynman recalled of his student days that he acquired a ‘reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me’ [19]. Although pattern formation in physics and applied mathematics is a mature field, and the phenomena included here are mostly very well known, it is only relatively recently that this box of tools, including linear stability analysis, bifurcation theory, symmetries and symmetry breaking, and dynamical systems theory, has been brought to bear on geophysical problems. Progress has been impressive, as will hopefully be demonstrated. However, despite drawing on similar methods, the applications here have also tended to be developed piecemeal and represent a growing community with considerable opportunity to learn from each other. One additional aim of this Theme Issue is to draw these topics together.

We begin with a review of one of the most well-known and commonly seen geophysical patterns, that of sand dunes. When fluid flows over a granular bed, the grains can be picked up and transported with the flow. If the bed has an uneven surface, grains can be preferentially eroded from regions of higher wind stress and deposited elsewhere. Andreotti & Claudin [6] review how this mechanism leads to the linear instability of dunes of well-defined wavelength. They show how such dunes scale under different fluids, such as liquid water or the atmospheres of Earth, Mars and Venus, and how these small-amplitude dunes grow and coarsen until they are limited by the thickness of the flowing fluid layer. Finally, they discuss the current challenges in modelling subaqueous ripples, which may lie between turbulent and laminar flow conditions, and summarize extensions of these ideas to other situations, for example alternating river bars.

Thomas et al. [20] consider the same hydrodynamic forcing that gives rise to dunes but apply it to the viscoelastic substrate of microbial mats. In their paper on kinneyia [20], they consider the formation of a class of wrinkled fossil biofilms that are common in the Precambrian fossil record but absent from more modern times. By coupling linear stability theory to analogue experiments and field observations, they show how kinneyia may form through a generic Kelvin–Helmholtz-type instability of a sheared interface. This raises interesting questions of whether these fossils can be used to interpret the prevailing conditions under which they formed and as to why they are no longer observed.

Just as sands and biofilms are moved by flow, near shore sediments can be transported by wave action. This leads to an instability of coastlines to periodic perturbations and the growth of small bumps into cuspate spits and bays, for example. Murray & Ashton [21] review recent work on self-organized coastline patterns. They show how the along-shore sediment flux can be simplified into the form of a nonlinear diffusion equation, which becomes unstable at locations, or conditions, where the effective diffusivity changes sign. They then show how the finite-amplitude forms of this instability have been explored numerically, in a range of conditions ranging from open coastlines with symmetric or highly asymmetric wave forcing, to enclosed lakes, where coastline features on either side of a body of water can interact in surprising ways.

The near shore sediment flux in tidal regions can also be controlled by more local conditions, including vegetation. Da Lio et al. [16] describe how competition between specialized species in such wetlands can lead to the stabilization of a series of salt marsh platforms of different heights, each dominated by one species of vegetation. They explore a dynamical model that couples biomass of different species with sediment production and transport, and show how stable, attractive solutions that describe a series of platforms would naturally develop. As sea-level rise is a timely concern for tidal marshes, they also study the effects of the relative rate of rise on the stability of the structure of salt marshes and demonstrate that biogeomorphic feedback may make such wetland patterns more resilient than previously thought.

In other situations, the feedback between life and its environment can also give rise to patterns, and vegetation patterns in landscapes have become well studied in recent years. In arid environments, these typically take the form of spots, stripes or labyrinths of alternating vegetation and bare soil (e.g. [22]). Zelnik et al. [14] survey a broad range of the models that have developed to describe the interaction of vegetative spots with water, given a limiting precipitation rate. In particular, they study the process of desertification, which in these models results from multiple stable states, and hysteresis, in the vegetative cover. By investigating multiple models, they attempt to answer the question of whether desertification should proceed by gradual growth of desert patches, or by the rapid non-local shift between patterns. In terms of a perhaps more familiar phase change like freezing, these are equivalent to heterogeneous and homogeneous nucleation, respectively.

Although pattern formation in vegetation is common, observational tests of the theoretical predictions regarding such landscapes are few. Penny et al. [15] present a field study of patterns in the drylands of Texas, where alternating stripes of vegetative cover and bare soil cover several hillsides. Using Fourier methods, they analyse the directions of stripes and compare them to local slopes. The stripes generally run perpendicular to the slopes, but they show local excursions from this behaviour that depend on other heterogeneities of the environment. For example, the authors show that soil depth or type may be important in the feedbacks of vegetative stripes, features that have not previously been considered important.

Having now considered pattern formation in deserts, dry lands, wetlands and shorelines, we turn to surface patterns in icy regions. Hallet [17] presents a field study of sorted circles in permafrost. This pattern is thought to arise from a convection-like overturning of the soil over many years, an instability driven by freeze–thaw cycles. Interestingly, in contrast to vegetation patterns, it can also manifest as stripes on hillsides, but which lie perpendicular to the topographic contours. Hallet has studied the dynamics of this pattern on the flat plains of Spitzbergen for over 20 years, and he presents here a long-time-scale series of measurements of the evolution of their shape, the displacement and rotation of tracers in their surface and interior, along with local temperature records. These compare well to numerical models of instabilities of sorted circles and help to quantify our understanding of the near-surface transport of soils in polar environments.

Polar terrain is patterned by two different mechanisms, the freeze–thaw mechanism just considered and thermal contraction cycles. The ice-cemented soil that frequently underlies the active layer of permafrost is hard and tough, but can crack under the stresses caused by seasonal temperature variations. Here, I consider the ordering of these fracture patterns, and other hexagonally ordered crack structures, through analogue experiments on drying clay [4]. I show how a simple model of cracking, where cracks open and heal repeatedly, guided by their previous positions but changing their order of appearance, can explain quantitative details of such networks mechanistically.

Similar networks can also be found in surprising places, for example in fluid mixing. Fu et al. [23] present a numerical study of the dissolution of CO2 that is expected as a result of carbon capture and storage in underground aquifers. The convective mixing initially selects a most unstable wavelength, but the pattern rapidly develops into a well-defined cellular network of columnar fingers of CO2-rich fluid. The coarsening of this network occurs by discrete cellular rules that are shown to be similar to how foams and crack patterns coarsen, and gives rise to a non-equilibrium steady state with universal scaling.

Petroff et al. [3] also consider the dynamics of a network, when analysing the branching of springs and streams. By applying a complex potential method to the seepage of groundwater around a spring, they map this dynamics to the more general problem of growth in a potential field. They find an instability of the stream tip (such as the source or spring of the stream), and predict that it should give rise to a stream branching angle of 2π/5. This result is compared to other tip-splitting systems, such as solidification of a melt or vascular networks, and the branch angle of several thousand streams is measured and found to agree precisely with their model.

We close this issue with two further review pieces. The sinuous meander of rivers is another of the more well-known instabilities seen in nature. However, sinuosity is not only present in fluvial cases, but is seen in submarine channelling as well. Wells & Cossu [10] review interesting recent observations which show that high-latitude channels are significantly straighter than low-latitude channels. They connect this with the symmetry-breaking mechanism of the Coriolis force, and show how a dimensionless Rossby number, which describes the relative importance of this force, becomes significant at high latitudes.

Finally, L'Heureux [7] reviews a class of geochemical patterns that are related to the case of periodically precipitating Liesegang rings. He shows that the same generic reaction–diffusion instability can lead to the rhythmic banding of precipitates in volcanic, sedimentary and metamorphic rocks, and presents a particularly detailed novel account of this process in sapropels. These processes can also occur during the precipitation of single crystals and a brief review of this oscillatory banding is also given.

4. Concluding remarks

We have attempted to gather a representative collection of topics in this Theme Issue on pattern formation in the geosciences. It is not exhaustive, but gives an overview of what is possible when the techniques of pattern formation and nonlinear dynamics are applied to geophysical problems. Some topics have been well studied over several decades, such as sand dunes and sinuous rivers. The results that are presented here show how understanding of these ideas, once developed, can be fluidly applied in other contexts, such as sand bars [6] or submarine channels [10]. The inclusion of kinneyia [20] shows how new applications of linear stability theory can give insight into processes that happened long ago in our planet's past. Similarly, the papers on permafrost [4,17] and vegetation patterns [15] are relevant to the study of remote locations, such as the Earth's polar and desert regions, or Mars, where satellite imaging is available but direct studies can be challenging.

The systems discussed here are dynamic, and their steady states can be expected to evolve as conditions change. Prediction, and in some cases even control, of geophysical patterns is an area where future developments can be expected to be particularly influential. Pattern-forming systems can be susceptible to catastrophic regime changes [22], such as the desertification process studied by Zelnik et al. [14], when a bifurcation point is reached. Alternatively, the approaches of Murray & Ashton [21] to coastline migration, or Da Lio et al. [16] to wetland development use nonlinear feedback to make explicit testable predictions, which will be important in attempts to minimize disruption of coastal regions by changing climates. Finally, large-scale experiments on carbon capture and storage will require understanding and engineering patterns of reactive porous media flow over many cubic kilometres and inspired modelling such as that of Fu et al. [23].

It is our hope that this collection serves to demonstrate the strengths of the current research of this highly multidisciplinary field and to encourage its potential for future development.


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