Chris Johnson


My publications are also listed on Google Scholar, ORCID (0000-0003-2192-3616), Publons / ResearcherID (B-3163-2012), and on my University of Manchester Research Page.

  1. P. Gajjar, C. Johnson, J. Carr, K. Chrispeels, J.M.N.T. Gray and P. Withers (2021)
    Size segregation of irregular granular materials captured by time-resolved 3D imaging,
    Scientific Reports (accepted)
  2. Andrew Edwards, Sylvain Viroulet, Chris Johnson and Nico Gray (2021)
    Erosion-deposition dynamics and long distance propagation of granular avalanches,
    Journal of Fluid Mechanics 915 A9, doi:10.1017/jfm.2021.34
    [Show abstract] [PDFDownload PDF] The net erosion-deposition rate of an avalanche is fundamental to its dynamics and in determining its growth or decay. Small-scale experiments are performed by releasing a given volume of yellow sand onto a stationary erodible red sand layer on a rough inclined plane. Depending on the erodible layer depth and the slope angle, the avalanche is found to either decay, grow, propagate steadily or rapidly shed grains to produce secondary avalanches. The use of different coloured sand with identical properties shows that a particle exchange occurs, which eventually results in a flow that is comprised entirely of particles from the stationary layer rather than the initial release. It is notoriously difficult to model the erosion and deposition processes in granular flows, but it is shown that a two-dimensional depth-averaged avalanche model, with a hysteretic basal friction law, can reproduce all of the observed behaviours. The results illustrate how a continuous exchange of particles with the substrate layer is fundamentally important to the propagation of such avalanches. An investigation into long distance propagation behaviour reveals that avalanches can reach a steady state, the size and speed of which are independent of the initially released volume. In certain conditions avalanches can grow to steady states that are significantly more massive than the flows from which they are originally formed. This paper demonstrates the importance of correctly including erosion-deposition in operational forecast models of snow avalanches and other geophysical mass flows.
  3. Sylvain Viroulet, Chris Johnson and Nico Gray (2021)
    Modelling erosion and deposition in geophysical granular mass flows,
    Europhysics News 52/1, 29–32, doi:10.1051/epn/2021106
    (Magazine article)
    [Show abstract] [PDFDownload PDF] During hazardous geophysical mass flows, such as rock or snow avalanches, debris flows and volcanic pyroclastic flows, a continuous exchange of material can occur between the slide and the bed. The net balance between erosion and deposition of particles can drastically influence the behaviour of these flows. Recent advances in describing the non-monotonic effective basal friction and the internal granular rheology in depth averaged theories have enabled small scale laboratory experiments to be quantitatively reproduced and can also be implemented in large scale models to improve hazard mitigation.
  4. Thomas Barker, Matthas Rauter, Eoin Maguire, Chris Johnson and Nico Gray (2021)
    Coupling rheology and segregation in granular flows,
    Journal of Fluid Mechanics 909 A22, doi:10.1017/jfm.2020.973
    [Show abstract] [PDFDownload PDF] During the last fifteen years there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the 𝜇(𝐼)-rheology (where 𝜇 is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This paper details theoretical and numerical frameworks (based on OpenFOAM) which unify these currently disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible 𝜇(𝐼)-rheology, which is coupled to the gravity-driven segregation theory of Gray & Ancey (J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. Further inclusion of an excess air phase, which segregates away from the granular material, then allows the complex evolution of the free surface to be captured simultaneously. Three primary coupling mechanisms are identified: (i) advection of the particle concentrations by the bulk velocity, (ii) feedback of the particle-size and/or frictional properties on the bulk flow field and (iii) influence of the shear rate, pressure, gravity, particle size and particle-size ratio on the locally evolving segregation and diffusion rates. The numerical method is extensively tested in one-way coupled computations, before the fully coupled model is compared with the discrete element method simulations of Tripathi & Khakhar (Phys. Fluids, vol. 23, 2011, 113302) and used to compute the petal-like segregation pattern that spontaneously develops in a square rotating drum.
  5. Finn Box, Chris Johnson, Draga Pihler-Puzovic (2020)
    Hard Auxetic Metamaterials,
    Extreme Mechanics Letters (40) 100980, doi:10.1016/j.eml.2020.100980
    [Show abstract] [PDFDownload PDF] Auxetics are materials that contract laterally when compressed, rather than expand, in contrast to common experience. Here we show that common metals and plastics can be rendered auxetic through the introduction of a regular array of holes. Under compression, these hard holey materials bypass localized failure modes, such as shear banding, and instead deform via a global pattern transformation previously reported in elastomeric structures. Despite significant variations in internal structure, the pattern transformation responsible for auxetic behaviour in both metals and plastics is governed by the buckling of the slender struts that comprise the microarchitecture. Furthermore, in contrast to elastomeric structures, holey sheets made from hard materials exhibit significant negative post-buckling stiffness. This suggests that, beyond the geometrical nonlinearities associated with topological modifications, material nonlinearities which arise during plastic deformation offer further potential for altering the material properties of the constituent.
  6. 'Rapid crystallization of precious-metal-mineralized layers in mafic magmatic systems' on the cover of Nature Geoscience, May 2020 Luke N. Hepworth, J. Stephen Daly, Ralf Gertisser, Chris G. Johnson, C. Henry Emeleus & Brian O’Driscoll (2020)
    Rapid crystallization of precious-metal-mineralized layers in mafic magmatic systems,
    Nature Geoscience 13, 375–381, doi:10.1038/s41561-020-0568-3
    [Show abstract] [PDFDownload PDF][Article in The Times newspaper] The solidified remnants of mafic magmatic systems host the greatest concentrations of platinum-group metals in the Earth’s crust. Our understanding of precious-metal mineralization in these intrusive bodies is underpinned by a traditional view of magma chamber processes and crystal mush solidification. However, considerable uncertainty remains regarding the physical and temporal controls on concentrating these critical metals, despite their importance to modern society. We present high-precision 87Sr/86Sr analyses of plagioclase and clinopyroxene from within centimetre-thick precious-metal-enriched layers in the Palaeogene open-system Rum layered intrusion (northwest Scotland). Isotopic heterogeneity is present between plagioclase crystals, between clinopyroxene and plagioclase and within plagioclase crystals throughout the studied section. On the basis of these observations, we demonstrate that platinum-group element mineralization formed by repeated small-volume reactive melt percolation events. The preservation of strontium isotope heterogeneities at 10–100 µm length scales implies cooling of the melts that formed the precious-metal-rich layers occurred at rates greater than 1 °C per year, and cooling to diffusive closure within tens to hundreds of years. Our data highlight the importance of cyclic dissolution–recrystallization events within the crystal mush and raise the prospect that precious-metal-bearing mafic intrusions may form by repeated self-intrusion during cooling and solidification.
  7. C. G. Johnson (2020)
    Shocking Granular Flows,
    J. Fluid Mech. 890, doi:10.1017/jfm.2020.61 F1 (Invited Focus on Fluids review)
    [Show abstract] [PDFDownload PDF] When a lightning bolt darts across the sky, the thunderclap that reaches our ears a few seconds later is an example of a fluid dynamical shock: a wave across which flow properties such as pressure and density change almost discontinuously. In compressible fluids these shocks are associated with high-energy supersonic flows and so require specialist equipment to realise in steady state. But in granular media, shocks occur much more readily and at flow speeds easily obtainable in the laboratory. In the featured article, Khan et al. (J. Fluid Mech., vol. 884, 2020, R4) exploit this to explore a remarkable range of steady and oscillatory shocks and shock interactions, which demonstrate many of the unique rheological complexities of granular flow.
  8. Self-channelisation and levee formation in monodisperse granular flows on the cover of JFM 876 F. M. Rocha, C. G. Johnson and J. M. N. T. Gray (2019)
    Self-channelisation and levee formation in monodisperse granular flows,
    J. Fluid Mech. 876 591–641, doi:/10.1017/jfm.2019.518
    [Show abstract] [PDFDownload PDF] Dense granular flows can spontaneously self-channelise by forming a pair of parallel-sided static levees on either side of a central flowing channel. This process prevents lateral spreading and maintains the flow thickness, and hence mobility, enabling the grains to run out considerably further than a spreading flow on shallow slopes. Since levees commonly form in hazardous geophysical mass flows, such as snow avalanches, debris flows, lahars and pyroclastic flows, this has important implications for risk management in mountainous and volcanic regions. In this paper an avalanche model that incorporates frictional hysteresis, as well as depth-averaged viscous terms derived from the 𝜇(I)-rheology, is used to quantitatively model self-channelisation and levee formation. The viscous terms are crucial for determining a smoothly varying steady-state velocity profile across the flowing channel, which has the important property that it does not exert any shear stresses at the levee–channel interfaces. For a fixed mass flux, the resulting boundary value problem for the velocity profile also uniquely determines the width and height of the channel, and the predictions are in very good agreement with existing experimental data for both spherical and angular particles. It is also shown that in the absence of viscous (second-order gradient) terms, the problem degenerates, to produce plug flow in the channel with two frictionless contact discontinuities at the levee–channel margins. Such solutions are not observed in experiments. Moreover, the steady-state inviscid problem lacks a thickness or width selection mechanism and consequently there is no unique solution. The viscous theory is therefore a significant step forward. Fully time-dependent numerical simulations to the viscous model are able to quantitatively capture the process in which the flow self-channelises and show how the levees are initially emplaced behind the flow head. Both experiments and numerical simulations show that the height and width of the channel are not necessarily fixed by these initial values, but respond to changes in the supplied mass flux, allowing narrowing and widening of the channel long after the initial front has passed by. In addition, below a critical mass flux the steady-state solutions become unstable and time-dependent numerical simulations are able to capture the transition to periodic erosion–deposition waves observed in experiments.
  9. A. N. Edwards, A. S. Russell, C. G. Johnson and J. M. N. T. Gray (2019)
    Frictional hysteresis and particle deposition in granular free-surface flows,
    J. Fluid Mech. 875 1058–1095, doi:/10.1017/jfm.2019.517
    [Show abstract] [PDF Download PDF] Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness hstop(𝜁) on a rough slope inclined at an angle 𝜁 to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle 𝜁start , or the thickness is increased to hstart(𝜁) > hstop(𝜁). This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to hstop. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness hstop) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness h* it produces a deposit close to hstop for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to hstop for all initial thicknesses. This insensitivity to the initial conditions implies that is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the hstop and hstart curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth hstop.
  10. S. Viroulet, A. N. Edwards, C. G. Johnson, B. P. Kokelaar and J. M. N. T. Gray (2019)
    Shedding dynamics and mass exchange by dry granular waves flowing over erodible beds,
    Earth Planet. Sci. Lett. 523, doi:/10.1016/j.epsl.2019.07.003
    [Show abstract] [PDF Download PDF] A continuous exchange of particles between an erodible substrate and the granular flow above it occurs during almost all geophysical events involving granular material, such as snow avalanches, debris flows and pyroclastic flows. The balance between eroded and deposited material can drastically influence the runout distance and duration of the flow. In certain conditions, a perfect balance between erosion and deposition may occur, leading to the steady propagation of material, in which the flow maintains its shape and velocity throughout. It is shown experimentally how the erosion-deposition process in dense flows of sand (160-200 μm) on an erodible bed of the same material produces steadily propagating avalanches that deposit subtle levees at their lateral extent. Moreover, it is shown in this paper, by using two colours of the same sand, that although the avalanche is propagating at constant velocity and maintaining a constant shape, the grains that are initially released are deposited along the flow path and that the avalanche will eventually be composed entirely of particles that are eroded from the bed. Different steady travelling wave regimes are obtained depending on the slope angle, thickness of the erodible layer and the amount of material released. Outside of the range of parameters where steady travelling waves form, the avalanches loose mass and decay if the initial amount of material released is too small, or, if the initial release is too large, they re-adjust to a steadily propagating regime by shedding material and breaking into smaller avalanches at its rear side. Numerical simulations are performed using a shallow-water-like avalanche model together with a friction law that captures the erosion-deposition process in flowing to static regimes and a transport equation for the interface between layers of the two colours. The characteristic behaviours observed in the experiments are qualitatively reproduced. Specifically, the complex processes such as the exchange of particles leading to a change in colour of the avalanche and the formation of lateral levees are captured by the model. Finally a comparison is made with deposits in lunar craters, which are interpreted as closely analogous to the deposits formed in our laboratory experiments.
  11. A. S. Russell, C. G. Johnson, A. N. Edwards, S. Viroulet, F. M. Rocha and J. M. N. T. Gray (2019)
    Retrogressive failure of a static granular layer on an inclined plane,
    J. Fluid Mech. 869 313–340, doi:/10.1017/jfm.2019.215
    [Show abstract] [PDF Download PDF] When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwards et al. (J. Fluid. Mech., vol. 823, 2017, pp. 278–315, J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.
  12. P. Gajjar, Jakob. S. Jørgenson, Jose R. A. Godinho, Chris G. Johnson, A. Ramsey, P. J. Withers (2018)
    New software protocols for enabling laboratory based temporal CT,
    Review of Scientific Instruments 89 093702, doi:/10.1063/1.5044393
    [Show abstract] [PDF Download PDF] Temporal micro computed tomography (CT) allows the non-destructive quantification of processes that are evolving over time in 3D. Despite the increasing popularity of temporal CT the practical implementation and optimisation can be difficult. Here, we present new software protocols that enable temporal CT using commercial laboratory CT systems. The first protocol drastically reduces the need for periodic intervention when making time-lapse experiments, allowing a large number of tomograms to be collected automatically. The automated scanning at regular intervals needed for uninterrupted time-lapse CT is demonstrated by analysing the germination of a mung bean (vigna radiata), whilst the synchronisation with an in-situ rig required for interrupted time-lapse CT is highlighted using a shear cell to observe granular segregation. The second protocol uses golden-ratio angular sampling with an iterative reconstruction scheme and allows the number of projections in a reconstruction to be changed as sample evolution occurs. This overcomes the limitation of the need to know a priori what the best time window for each scan is. The protocol is evaluated by studying barite precipitation within a porous column, allowing a comparison of spatial and temporal resolution of reconstructions with different numbers of projections. Both of the protocols presented here have great potential for wider application, including, but not limited to, in-situ mechanical testing, following battery degradation and chemical reactions.
  13. K. van der Vaart, A. R. Thornton, C. G. Johnson, T. Weinhart, L. Jing, P. Gajjar, J. M. N. T. Gray, C. Ancey (2018)
    Breaking size-segregation waves and mobility feedback in dense granular avalanches,
    Granular Matter 20:46, doi:10.1007/s10035-018-0818-x
    [Show abstract] [PDF Download PDF] Through experiments and discrete particle method (DPM) simulations we present evidence for the existence of a recirculating structure, that exists near the front of dense granular avalanches, and is known as a breaking size-segregation (BSS) wave. This is achieved through the study of three-dimensional bidisperse granular flows in a moving-bed channel. Particle-size segregation gives rise to the formation of a large-particle-rich front and a small-particle-rich tail with a BSS wave positioned between the tail and front. We experimentally resolve the structure of the BSS wave using refractive-index matched scanning and find that it is qualitatively similar to the structure observed in DPM simulations. Our analysis demonstrates a relation between the concentration of small particles in the flow and the amount of basal slip, in which the structure of the BSS wave plays a key role. This leads to a feedback between the mean bulk flow velocity and the process of particle-size segregation. Ultimately, these findings shed new light on the recirculation of large and small grains near avalanche fronts and the effects of this behaviour on the mobility of the bulk flow.
  14. S. Viroulet, J. L. Baker, F. Rocha, C. G. Johnson, P. Kokelaar and J. M. N. T. Gray (2018)
    The kinematics of bidisperse granular roll waves,
    J. Fluid Mech. 848, 836–875, doi:10.1017/jfm.2018.348
    [Show abstract] [PDF Download PDF] Small perturbations to a steady uniform granular chute flow can grow as the material moves downslope and develop into a series of surface waves that travel faster than the bulk flow. This roll wave instability has important implications for the mitigation of hazards due to geophysical mass flows, such as snow avalanches, debris flows and landslides, because the resulting waves tend to merge and become much deeper and more destructive than the uniform flow from which they form. Natural flows are usually highly polydisperse and their dynamics is significantly complicated by the particle size segregation that occurs within them. This study investigates the kinematics of such flows theoretically and through small-scale experiments that use a mixture of large and small glass spheres. It is shown that large particles, which segregate to the surface of the flow, are always concentrated near the crests of roll waves. There are different mechanisms for this depending on the relative speed of the waves, compared to the speed of particles at the free surface, as well as on the particle concentration. If all particles at the surface travel more slowly than the waves, the large particles become concentrated as the shock-like wavefronts pass them. This is due to a concertina-like effect in the frame of the moving wave, in which large particles move slowly backwards through the crest, but travel quickly in the troughs between the crests. If, instead, some particles on the surface travel more quickly than the wave and some move slower, then, at low concentrations, large particles can move towards the wave crest from both the forward and rearward sides. This results in isolated regions of large particles that are trapped at the crest of each wave, separated by regions where the flow is thinner and free of large particles. There is also a third regime arising when all surface particles travel faster than the waves, which has large particles present everywhere but with a sharp increase in their concentration towards the wave fronts. In all cases, the significantly enhanced large particle concentration at wave crests means that such flows in nature can be especially destructive and thus particularly hazardous.
  15. T. Sauma-Perez, C. G. Johnson, Y. Li, T. Mullin (2018)
    An experimental study of the motion of a light sphere in a rotating viscous fluid,
    J. Fluid Mech. 847, 119–133, doi:10.1017/jfm.2018.330
    [Show abstract] [PDF Download PDF] We present the results of an experimental investigation of the motion of a light, solid sphere in a horizontal rotating cylinder filled with viscous fluid. At high rotation rates, the sphere sits near the axis of the cylinder. At lower rotation rates, a set of off-axis fixed points are observed for a range of sphere radii. The locations of these fixed points are in quantitative agreement with the predictions of a model based on available theory. The fixed points are observed to become unstable to periodic orbits below a critical Reynolds number Rec . The radius of the observed orbits increases with Reynolds number more slowly than a typical Hopf bifurcation, in this case, growing as 1/Rec2.
  16. C. G. Johnson, U. Jain, A. L. Hazel, D. Pihler-Puzovic and T. Mullin (2017)
    On the buckling of an elastic holey column,
    Proc. Royal Soc. A 473:20170477, doi:10.1098/rspa.2017.0477
    [Show abstract] [PDF Download PDF][Nature research highlight] We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from meters in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localised (hole- scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.
  17. Multiple solutions for granular flow over a smooth two-dimensional bump on the cover of JFM 815
    Segregation-induced finger formation in granular free-surface flows on the cover of JFM 809
    Asymmetric breaking size-segregation waves in dense granular free-surface flows on the cover of JFM 794
    S. Viroulet, J. L. Baker, A. N. Edwards, C. G. Johnson, C. Gjaltema, P. Clavel, J. M. N. T. Gray (2017)
    Multiple solutions for granular flow over a smooth two-dimensional bump,
    J. Fluid Mech. 815, 77–116, doi:10.1017/jfm.2017.41
    [Show abstract] [PDF Download PDF] Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.
  18. J. L. Baker, C. G. Johnson, J. M. N. T. Gray (2016)
    Segregation-induced finger formation in granular free-surface flows,
    J. Fluid Mech. 809, 168–212, doi:10.1017/jfm.2016.673
    [Show abstract] [PDF Download PDF] Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the μ(I)-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.
  19. P. Gajjar, K. van der Vaart, A. R. Thornton, C. G. Johnson, C. Ancey, J. M. N. T. Gray (2016)
    Asymmetric breaking size-segregation waves in dense granular free-surface flows,
    J. Fluid Mech. 794, 460–505, doi:10.1017/jfm.2016.170
    [Show abstract] [PDF Download PDF] Debris and pyroclastic flows often have bouldery flow fronts, which act as a natural dam resisting further advance. Counter intuitively, these resistive fronts can lead to enhanced run-out, because they can be shouldered aside to form static levees that self-channelise the flow. At the heart of this behaviour is the inherent process of size segregation, with different sized particles readily separating into distinct vertical layers through a combination of kinetic sieving and squeeze expulsion. The result is an upward coarsening of the size distribution with the largest grains collecting at the top of the flow, where the flow velocity is greatest, allowing them to be preferentially transported to the front. Here, the large grains may be overrun, resegregated towards the surface and recirculated before being shouldered aside into lateral levees. A key element of this recirculation mechanism is the formation of a breaking size-segregation wave, which allows large particles that have been overrun to rise up into the faster moving parts of the flow as small particles are sheared over the top. Observations from experiments and discrete particle simulations in a moving-bed flume indicate that, whilst most large particles recirculate quickly at the front, a few recirculate very slowly through regions of many small particles at the rear. This behaviour is modelled in this paper using asymmetric segregation flux functions. Exact non-diffuse solutions are derived for the steady wave structure using the method of characteristics with a cubic segregation flux. Three different structures emerge, dependent on the degree of asymmetry and the non-convexity of the segregation flux function. In particular, a novel ‘lens-tail’ solution is found for segregation fluxes that have a large amount of non-convexity, with an additional expansion fan and compression wave forming a ‘tail’ upstream of the ‘lens’ region. Analysis of exact solutions for the particle motion shows that the large particle motion through the ‘lens-tail’ is fundamentally different to the classical ‘lens’ solutions. A few large particles starting near the bottom of the breaking wave pass through the ‘tail’, where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate.
  20. M. Ungarish, C. G. Johnson and A. J. Hogg (2016)
    Sustained axisymmetric intrusions in a rotating system,
    Euro. J. Mech. B – Fluids 56, 110–119, doi:10.1016/j.euromechflu.2015.10.008
    [Show abstract] [PDF Download PDF] We analyse the effects of rotation on the propagation of an axisymmetric intrusion through a linearly stratified ambient fluid, arising from a sustained source at the level of neutral buoyancy. This scenario occurs during the horizontal spreading of a volcanic ash cloud, which occurs after the plume has risen to its neutral buoyancy level. A simple and well-accepted approximation for the flow at late times is that inertial effects are negligible. This leads to a lens-shaped intrusion governed by a balance between Coriolis accelerations and horizontal pressure gradients, with the radius scaling with time as rN∼t1/3. However, we show using a shallow-layer model that inertial forces cannot be neglected until significant times after the beginning of the influx. These inertial forces result in the flow forming two distinct domains, separated by a moving hydraulic jump: an outer ‘head’ region in which the radial velocity and thickness vary with time, and a thinner ‘tail’ region in which the flow is steady. Initially, the flow expands rapidly and this tail region occupies most of the flow. After about one half-revolution of the system, Coriolis accelerations halt the advance of the front, and the hydraulic jump separating the two regions propagates back towards the source of the intrusion. Only after approximately one and a half rotations of the system does inertia become insignificant and the Coriolis lens solution, with rN∼t1/3, become established. Importantly, this means that neither inertia nor Coriolis accelerations can be neglected when modelling intrusions from volcanic eruptions. We exploit the two-region flow structure to construct a new hybrid model, comprising just two ordinary differential equations for the intrusion radius and location of the hydraulic jump. This hybrid model is much simpler than the shallow-layer model, but nonetheless accurately predicts flow properties such as the intrusion radius at all stages of motion, without requiring fitted or adjustable parameters.
  21. S. Pouget, M. I. Bursik, C. G. Johnson, A. J. Hogg, J. C. Phillips, R. S. J. Sparks (2016)
    Interpretation of umbrella cloud growth and morphology: implications for flow regimes of short-lived and long-lived eruptions,
    Bulletin of Volcanology 78:1, doi:10.1007/s00445-015-0993-0
    [Show abstract] [PDF Download PDF] New numerical and analytical modeling shows that the growth of a volcanic umbrella cloud, expressed as the increase of radius with time, proceeds through regimes, dominated by different force balances. Four regimes are identified: Regime Ia is the long-time behavior of continuously-supplied intrusions in the buoyancy-inertial regime; regime IIa is the long-time behavior of continuously-supplied, turbulent drag-dominated intrusions; regime Ib is the long-time behavior of buoyancy-inertial intrusions of constant volume; and regime IIb that of turbulent drag-dominated intrusions of constant volume. Power-law exponents for spreading time in each regime are 3/4 (Ia), 5/9 (IIa), 1/3 (Ib), and 2/9 (IIb). Both numerical modeling and observations indicate that transition periods between the regimes can be long-lasting, and during these transitions, the spreading rate does not follow a simple power law. Predictions of the new model are consistent with satellite data from seven eruptions and, together with observations of umbrella cloud structure and morphological evolution, support the existence of multiple spreading regimes.
  22. C. G. Johnson, A. J. Hogg, H. E. Huppert, R. S. J. Sparks, J. C. Phillips, A. C. Slim and M. J. Woodhouse (2015)
    Modelling intrusions through quiescent and moving ambients,
    J. Fluid Mech. 771, 370–406, doi:10.1017/jfm.2015.180
    [Show abstract] [PDF Download PDF] Volcanic eruptions commonly produce buoyant ash-laden plumes that rise through the stratified atmosphere. On reaching their level of neutral buoyancy, these plumes cease rising and transition to horizontally spreading intrusions. Such intrusions occur widely in density-stratified fluid environments, and in this paper we develop a shallow-layer model that governs their motion. We couple this dynamical model to a model for particle transport and sedimentation, to predict both the time-dependent distribution of ash within volcanic intrusions and the flux of ash that falls towards the ground. In an otherwise quiescent atmosphere, the intrusions spread axisymmetrically. We find that the buoyancy-inertial scalings previously identified for continuously supplied axisymmetric intrusions are not realised by solutions of the governing equations. By calculating asymptotic solutions to our model we show that the flow is not self-similar, but is instead time-dependent only in a narrow region at the front of the intrusion. This non-self-similar behaviour results in the radius of the intrusion growing with time t as t3/4, rather than t2/3 as suggested previously. We also identify a transition to drag-dominated flow, which is described by a similarity solution with radial growth now proportional to t5/9. In the presence of an ambient wind, intrusions are not axisymmetric. Instead, they are predominantly advected downstream, while at the same time spreading laterally and thinning vertically due to persistent buoyancy forces. We show that close to the source, this lateral spreading is in a buoyancy-inertial regime, whereas far downwind, the horizontal buoyancy forces that drive the spreading are balanced by drag. Our results emphasise the important role of buoyancy-driven spreading, even at large distances from the source, in the formation of the flowing thin horizontally extensive layers of ash that form in the atmosphere as a result of volcanic eruptions.
  23. M. Ungarish, C. G. Johnson and A. J. Hogg (2015)
    A novel hybrid model for the motion of sustained axisymmetric gravity currents and intrusions,
    Euro. J. Mech. B – Fluids 49A, 108–120, doi:10.1016/j.euromechflu.2014.07.007
    [Show abstract] [PDF Download PDF] We consider the sustained propagation of axisymmetric intrusions and gravity currents through linearly stratified or unstratified ambient fluids. Such flow configurations are found in a number of atmospheric and oceanic flows, in particular the predominantly horizontal spreading of a volcanic ash cloud after it has ascended through the atmosphere. There is strong theoretical evidence that these flows consist of two domains: an outer annular ‘head’ at the front of the current in which the motion is unsteady; and an inner, much thinner ‘tail’, which is steady, but spatially varying. The transition between the regions is a moving hydraulic jump. While it is possible to investigate these motions by numerically integrating the governing shallow layer equations, here we develop a much simpler mathematical model, which reproduces the more complicated model accurately and addresses issues such as what determines the position of the front and the moving bore between the two regions; what is the partition of influxed volume between the tail and head; and what is the distribution of suspended particles in the flow if present at the source? In such settings a conventional integral model fails, as does scaling based on dimensional analysis and the anticipation of an underlying self-similar form; the predictions they yield for these flows are incorrect. Instead we present a new hybrid model, which combines exact results of the steady shallow-water equations in the tail with simplifying assumptions in the head. This model predicts the flow properties by the straightforward solution of three ordinary differential equations (for front and bore positions and the volume fraction of particles in the head), without using adjustable constants, and obtains the correct asymptotic behaviour for the radius of the current rN with respect to time t, namely rN~t4/5 for gravity currents and rN~t3/4 for intrusions. The predictions are obtained with negligible computational effort and accurately capture results from the more complete shallow water models. The model is also applied with success to gravity currents and intrusions that carry particles. For flows in which it is the presence of the particles alone that drives the motion, we identify length and time scales for the runout in terms of dimensional parameters that characterise the release, thus establishing the hybrid model as a useful tool also for modelling radial runout.
  24. C. G. Johnson and A. J. Hogg (2013)
    Entraining gravity currents,
    J. Fluid Mech. 731, 477–508, doi:10.1017/jfm.2013.329
    [Show abstract] [PDF Download PDF] Entrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow-water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as t0.447. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as t4/5. In both cases, entrainment is most significant close to the current front.
  25. M. J. Woodhouse, A. R. Thornton, C. G. Johnson, B. P. Kokelaar and J. M. N. T. Gray (2012)
    Segregation-induced fingering instabilities in granular free-surface flows,
    J. Fluid Mech. 709, 543–580, doi:10.1017/jfm.2012.348
    [Show abstract] [PDF Download PDF] [Movies] Particle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.
  26. C. G. Johnson, B. P. Kokelaar, R. M. Iverson, R. G. LaHusen, M. Logan and J. M. N. T. Gray (2012)
    Grain-size segregation and levee formation in geophysical mass flows,
    J. Geophys. Res. 117, F01032, doi:10.1029/2011JF002185
    [Show abstract] [PDF Download PDF] [Movies] Data from large-scale debris-flow experiments are combined with modeling of particle-size segregation to explain the formation of lateral levees enriched in coarse grains. The experimental flows consisted of 10m3 of water-saturated sand and gravel, which traveled ~80m down a steeply inclined flume before forming an elongated leveed deposit 10m long on a nearly-horizontal runout surface. We measured the surface velocity field and observed the sequence of deposition by seeding tracers onto the flow surface and tracking them in video footage. Levees formed by progressive downslope accretion approximately 3.5m behind the flow front, which advanced steadily at ~2m/s during most of the runout. Segregation was measured by placing ~600 coarse tracer pebbles on the bed, which when entrained into the flow, segregated upwards at 6–7.5cm/s. When excavated from the deposit these were distributed in a horseshoe-shaped pattern that became increasingly elevated closer to the deposit termination. Although there was clear evidence for inverse grading during the flow, transect sampling revealed that the resulting leveed deposit was strongly graded laterally, with only weak vertical grading. We construct an an empirical, three-dimensional velocity field resembling the experimental observations, and use this with a particle-size segregation model to predict the segregation and transport of material through the flow. We infer that coarse material segregates to the flow surface and is transported to the flow front by shear. Within the flow head, coarse material is overridden, then recirculates in spiral trajectories due to size-segregation, before being advected to the flow edges and deposited to form coarse-particle-enriched levees.
  27. C. G. Johnson and J. M. N. T. Gray (2011) Granular jets and hydraulic jumps on the cover of JFM 675
    Granular jets and hydraulic jumps on an inclined plane,
    J. Fluid Mech. 675, 87–116, doi:10.1017/jfm.2011.2
    [Show abstract] [PDF Download PDF] [Movies] A jet of granular material impinging on an inclined plane produces a diverse range of flows, from steady hydraulic jumps to periodic avalanches, self-channelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steady-state flow, in which the jet generates a closed teardrop-shaped hydraulic jump on the plane, enclosing a region of fast-moving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardrop-shaped, but exhibits a more complex ‘blunted’ shape with a steadily breaking wave. We explain these regimes by consideration of the supercritical or subcritical nature of the flow surrounding the shock. A model is developed in which the impact of the jet on the inclined plane is treated as an inviscid flow, which is then coupled to a depth-integrated model for the resulting thin granular avalanche on the inclined plane. Numerical simulations produce a flow regime diagram strikingly similar to that obtained in experiments, with the model correctly reproducing the regimes and their dependence on the jet velocity and slope angle. The size and shape of the steady experimental shocks and the location of sub- and supercritical flow regions are also both accurately predicted. We find that the physics underlying the rapid flow inside the shock is dominated by depth-averaged mass and momentum transport, with granular friction, pressure gradients and three-dimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a friction–gravity balance, and some flow features, such as a persistent indentation in the free surface, are not reproduced in the numerical solutions. On planes inclined at a shallow angle, the effect of stationary granular material becomes important in the flow evolution, and oscillatory and more general time-dependent flows are observed. The hysteretic transition between static and dynamic friction leads to two phenomena observed in the flows: unsteady avalanching behaviour, and the feedback from static grains on the flowing region, leading to levéed, self-channelised flows.
  28. C. G. Johnson and C. J. Davis (2006)
    The location of lightning affecting the ionospheric sporadic-E layer as evidence for multiple enhancement mechanisms,
    Geophys. Res. Lett. 33, L07811, doi:10.1029/2005GL025294
    [Show abstract] [PDF Download PDF] We present a study of the geographic location of lightning affecting the ionospheric sporadic-E (Es) layer over the ionospheric monitoring station at Chilton, UK. Data from the UK Met Office's Arrival Time Difference (ATD) lightning detection system were used to locate lightning strokes in the vicinity of the ionospheric monitoring station. A superposed epoch study of this data has previously revealed an enhancement in the Es layer caused by lightning within 200km of Chilton. In the current paper, we use the same data to investigate the location of the lightning strokes which have the largest effect on the Es layer above Chilton. We find that there are several locations where the effect of lightning on the ionosphere is most significant statistically, each producing different ionospheric responses. We interpret this as evidence that there is more than one mechanism combining to produce the previously observed enhancement in the ionosphere.
  29. C. J. Davis and C. G. Johnson (2005)
    Lightning-induced intensification of the ionospheric sporadic E layer,
    Nature 435, 799–801, doi:10.1038/nature03638
    [Show abstract] [PDF Download PDF] A connection between thunderstorms and the ionosphere has been hypothesized since the mid-1920s. Several mechanisms have been proposed to explain this connection and evidence from modelling as well as various types of measurements demonstrate that lightning can interact with the lower ionosphere. It has been proposed, on the basis of a few observed events, that the ionospheric 'sporadic E' layer—transient, localized patches of relatively high electron density in the mid-ionosphere E layer, which significantly affect radio-wave propagation—can be modulated by thunderstorms, but a more formal statistical analysis is still needed. Here we identify a statistically significant intensification and descent in altitude of the mid-latitude sporadic E layer directly above thunderstorms. Because no ionospheric response to low-pressure systems without lightning is detected, we conclude that this localized intensification of the sporadic E layer can be attributed to lightning. We suggest that the co-location of lightning and ionospheric enhancement can be explained by either vertically propagating gravity waves that transfer energy from the site of lightning into the ionosphere, or vertical electrical discharge, or by a combination of these two mechanisms.