Chris Johnson
Publications
My publications are also listed on Google Scholar, ORCID (0000000321923616), Publons / ResearcherID (B31632012), and on my University of Manchester Research Page.

F. M. Rocha, C. G. Johnson and J. M. N. T. Gray (2019)
Selfchannelisation and levee formation in monodisperse granular flows,
J. Fluid Mech. 876, doi:/10.1017/jfm.2019.518
[Show abstract] [PDF] Dense granular flows can spontaneously selfchannelise by forming a pair of parallelsided 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 depthaveraged viscous terms derived from the š(I)rheology, is used to quantitatively model selfchannelisation and levee formation. The viscous terms are crucial for determining a smoothly varying steadystate velocity profile across the flowing channel, which has the important property that it does not exert any shear stresses at the leveeāchannel 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 (secondorder gradient) terms, the problem degenerates, to produce plug flow in the channel with two frictionless contact discontinuities at the leveeāchannel margins. Such solutions are not observed in experiments. Moreover, the steadystate 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 timedependent numerical simulations to the viscous model are able to quantitatively capture the process in which the flow selfchannelises 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 steadystate solutions become unstable and timedependent numerical simulations are able to capture the transition to periodic erosionādeposition waves observed in experiments.  A. N. Edwards, A. S. Russell, C. G. Johnson and J. M. N. T. Gray (2019)
Frictional hysteresis and particle deposition in granular freesurface flows,
J. Fluid Mech. 875, doi:/10.1017/jfm.2019.517
[Show abstract] [PDF ] Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steadyuniform granular flow is brought to rest it leaves a deposit of thickness h_{stop}(š) on a rough slope inclined at an angle š to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle š_{start} , or the thickness is increased to h_{start}(š) > h_{stop}(š). This simple phenomenology leads to a rich variety of flows with coexisting regions of solidlike and fluidlike granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of selfchannelized flows with static levees, retrogressive failures as well as erosionādeposition waves that travel through the material. This paper is motivated by the experimental observation that a travellingwave 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 h_{stop}. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278ā315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steadyuniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness h_{stop}) 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 steadyuniform flow of thickness h_{*} it produces a deposit close to h_{stop} for all inclination angles. Numerical simulations show that the twowave structure deposits layers that are approximately equal to h_{stop} 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 h_{stop} and h_{start} 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 steadyuniform velocity. Finally, some of the pathologies in earlier nonmonotonic 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 h_{stop}.  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 ] 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 erosiondeposition process in dense flows of sand (160200 Ī¼m) 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 readjust to a steadily propagating regime by shedding material and breaking into smaller avalanches at its rear side. Numerical simulations are performed using a shallowwaterlike avalanche model together with a friction law that captures the erosiondeposition 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.  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, doi:/10.1017/jfm.2019.215
[Show abstract] [PDF ] When a layer of static grains on a sufficiently steep slope is disturbed, an upslopepropagating 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 depthaveraged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the crossslope coordinate. The hysteresis is modelled with a nonmonotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and timedependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a onedimensional depthaveraged 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. 278ā315, J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly underpredicted, 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.  P. Gajjar, Jakob. S. JĆørgenson, 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(9), doi:/10.1063/1.5044393
[Show abstract] [PDF ] Temporal micro computed tomography (CT) allows the nondestructive 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 timelapse experiments, allowing a large number of tomograms to be collected automatically. The automated scanning at regular intervals needed for uninterrupted timelapse CT is demonstrated by analysing the germination of a mung bean (vigna radiata), whilst the synchronisation with an insitu rig required for interrupted timelapse CT is highlighted using a shear cell to observe granular segregation. The second protocol uses goldenratio 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, insitu mechanical testing, following battery degradation and chemical reactions.  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 sizesegregation waves and mobility feedback in dense granular avalanches,
Granular Matter 20:46, doi:10.1007/s100350180818x
[Show abstract] [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 sizesegregation (BSS) wave. This is achieved through the study of threedimensional bidisperse granular flows in a movingbed channel. Particlesize segregation gives rise to the formation of a largeparticlerich front and a smallparticlerich tail with a BSS wave positioned between the tail and front. We experimentally resolve the structure of the BSS wave using refractiveindex 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 particlesize 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.  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 ] 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 smallscale 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 shocklike wavefronts pass them. This is due to a concertinalike 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.  T. SaumaPerez, 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 ] 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 offaxis 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 Re_{c} . The radius of the observed orbits increases with Reynolds number more slowly than a typical Hopf bifurcation, in this case, growing as 1/Re_{c}^{2}.  C. G. Johnson, U. Jain, A. L. Hazel, D. PihlerPuzovic 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 ][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 Eulerās 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 finiteelement 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 (columnscale) 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.  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 twodimensional bump,
J. Fluid Mech. 815, 77–116, doi:10.1017/jfm.2017.41
[Show abstract] [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, smallscale experiments are performed to investigate the flow of a granular avalanche over a twodimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steadystate 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 depthaveraged terrainfollowing 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, timedependent 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 terrainfollowing theory is compared to a standard depthaveraged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steadyshock regime is captured significantly better by the terrainfollowing avalanche model, and that the standard theory is unable to predict the takeoff point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrainfollowing model, a coordinate mapping is used to transform the terrainfollowing equations from curvilinear to Cartesian coordinates. The terrainfollowing model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.  J. L. Baker, C. G. Johnson, J. M. N. T. Gray (2016)
Segregationinduced finger formation in granular freesurface flows,
J. Fluid Mech. 809, 168–212, doi:10.1017/jfm.2016.673
[Show abstract] [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 wellposed depthaveraged model for these segregationmobility 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 depthaveraged 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 higherorder terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Twodimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable largerich flow front, which subsequently breaks into a series of fingerlike structures, each bounded by coarsegrained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregationinduced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.  P. Gajjar, K. van der Vaart, A. R. Thornton, C. G. Johnson, C. Ancey, J. M. N. T. Gray (2016)
Asymmetric breaking sizesegregation waves in dense granular freesurface flows,
J. Fluid Mech. 794, 460–505, doi:10.1017/jfm.2016.170
[Show abstract] [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 runout, because they can be shouldered aside to form static levees that selfchannelise 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 sizesegregation 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 movingbed 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 nondiffuse 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 nonconvexity of the segregation flux function. In particular, a novel ālenstailā solution is found for segregation fluxes that have a large amount of nonconvexity, with an additional expansion fan and compression wave forming a ātailā upstream of the ālensā region. Analysis of exact solutions for the particle motion shows that the large particle motion through the ālenstailā is fundamentally different to the classical ālensā solutions. A few large particles starting near the bottom of the breaking wave pass through the ātailā, where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate.  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 ] 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 wellaccepted approximation for the flow at late times is that inertial effects are negligible. This leads to a lensshaped intrusion governed by a balance between Coriolis accelerations and horizontal pressure gradients, with the radius scaling with time as r_{N}ā¼t^{1/3}. However, we show using a shallowlayer 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 āheadā region in which the radial velocity and thickness vary with time, and a thinner ātailā region in which the flow is steady. Initially, the flow expands rapidly and this tail region occupies most of the flow. After about one halfrevolution 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 r_{N}ā¼t^{1/3}, become established. Importantly, this means that neither inertia nor Coriolis accelerations can be neglected when modelling intrusions from volcanic eruptions. We exploit the tworegion 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 shallowlayer model, but nonetheless accurately predicts flow properties such as the intrusion radius at all stages of motion, without requiring fitted or adjustable parameters.  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 shortlived and longlived eruptions,
Bulletin of Volcanology 78:1, doi:10.1007/s0044501509930
[Show abstract] [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 longtime behavior of continuouslysupplied intrusions in the buoyancyinertial regime; regime IIa is the longtime behavior of continuouslysupplied, turbulent dragdominated intrusions; regime Ib is the longtime behavior of buoyancyinertial intrusions of constant volume; and regime IIb that of turbulent dragdominated intrusions of constant volume. Powerlaw 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 longlasting, 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.  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 ] Volcanic eruptions commonly produce buoyant ashladen 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 densitystratified fluid environments, and in this paper we develop a shallowlayer model that governs their motion. We couple this dynamical model to a model for particle transport and sedimentation, to predict both the timedependent 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 buoyancyinertial 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 selfsimilar, but is instead timedependent only in a narrow region at the front of the intrusion. This nonselfsimilar behaviour results in the radius of the intrusion growing with time t as t^{3/4}, rather than t^{2/3} as suggested previously. We also identify a transition to dragdominated flow, which is described by a similarity solution with radial growth now proportional to t^{5/9}. 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 buoyancyinertial regime, whereas far downwind, the horizontal buoyancy forces that drive the spreading are balanced by drag. Our results emphasise the important role of buoyancydriven 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.  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 ] 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 āheadā at the front of the current in which the motion is unsteady; and an inner, much thinner ātailā, 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 selfsimilar form; the predictions they yield for these flows are incorrect. Instead we present a new hybrid model, which combines exact results of the steady shallowwater 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 r_{N} with respect to time t, namely r_{N}~t^{4/5} for gravity currents and r_{N}~t^{3/4} 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.  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 ] Entrainment of ambient fluid into a gravity current, while often negligible in laboratoryscale flows, may become increasingly significant in largescale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallowwater 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 longtime 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 nonentraining counterparts. The longtime solution of an inviscid entraining gravity current generated by a lockrelease 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 t^{0.447}. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as t^{4/5}. In both cases, entrainment is most significant close to the current front.  M. J. Woodhouse, A. R. Thornton, C. G. Johnson, B. P. Kokelaar and J. M. N. T. Gray (2012)
Segregationinduced fingering instabilities in granular freesurface flows,
J. Fluid Mech. 709, 543–580, doi:10.1017/jfm.2012.348
[Show abstract] [PDF ] [Movies] Particlesize 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 segregationmobility feedback effects occur the flow may form digitate lobate fingers or spontaneously selfchannelize to form lateral levees that enhance runout distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where runout distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depthaveraged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentrationdependent friction coefficient, which results in a system of nonstrictly 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 largerich lateral levees, as observed in smallscale 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 illposed. 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.  C. G. Johnson, B. P. Kokelaar, R. M. Iverson, R. G. LaHusen, M. Logan and J. M. N. T. Gray (2012)
Grainsize segregation and levee formation in geophysical mass flows,
J. Geophys. Res. 117, F01032, doi:10.1029/2011JF002185
[Show abstract] [PDF ] [Movies] Data from largescale debrisflow experiments are combined with modeling of particlesize segregation to explain the formation of lateral levees enriched in coarse grains. The experimental flows consisted of 10m^{3} of watersaturated sand and gravel, which traveled ~80m down a steeply inclined flume before forming an elongated leveed deposit 10m long on a nearlyhorizontal 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 horseshoeshaped 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, threedimensional velocity field resembling the experimental observations, and use this with a particlesize 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 sizesegregation, before being advected to the flow edges and deposited to form coarseparticleenriched levees.  C. G. Johnson and J. M. N. T. Gray (2011)
Granular jets and hydraulic jumps on an inclined plane,
J. Fluid Mech. 675, 87–116, doi:10.1017/jfm.2011.2
[Show abstract] [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, selfchannelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steadystate flow, in which the jet generates a closed teardropshaped hydraulic jump on the plane, enclosing a region of fastmoving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardropshaped, but exhibits a more complex ābluntedā 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 depthintegrated 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 depthaveraged mass and momentum transport, with granular friction, pressure gradients and threedimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a frictionāgravity 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 timedependent 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 levĆ©ed, selfchannelised flows.  C. G. Johnson and C. J. Davis (2006)
The location of lightning affecting the ionospheric sporadicE layer as evidence for multiple enhancement mechanisms,
Geophys. Res. Lett. 33, L07811, doi:10.1029/2005GL025294
[Show abstract] [PDF ] We present a study of the geographic location of lightning affecting the ionospheric sporadicE (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.  C. J. Davis and C. G. Johnson (2005)
Lightninginduced intensification of the ionospheric sporadic E layer,
Nature 435, 799–801, doi:10.1038/nature03638
[Show abstract] [PDF ] A connection between thunderstorms and the ionosphere has been hypothesized since the mid1920s. 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 midionosphere E layer, which significantly affect radiowave 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 midlatitude sporadic E layer directly above thunderstorms. Because no ionospheric response to lowpressure 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 colocation 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.