#### Papers

- C. G. Johnson, U. Jain, A. L. Hazel, D. Pihler-Puzovic and T. Mullin (2017)

**On the Buckling of an Elastic Holey Column**,

*Proceedings of the Royal Society A*(Accepted),

[Show abstract] 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. - 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**, 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, 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. - 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**, 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 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. - 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 ] 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. - 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 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*r*. 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_{N}∼t^{1/3}*r*, 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._{N}∼t^{1/3} - 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 ] 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. - 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 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*t*^{3/4}, rather than*t*^{2/3}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*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 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. - 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***49**A, 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 ‘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*r*with respect to time_{N}*t*, namely*r*for gravity currents and_{N}~t^{4/5}*r*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._{N}~t^{3/4} - 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 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*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)

**Segregation-induced fingering instabilities in granular free-surface flows**,

*J. Fluid Mech.***709**, 543–580, doi:10.1017/jfm.2012.348

[Show abstract] [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. - 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 ] [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 10m^{3}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. - 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, 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. - 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 ] 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. - 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 ] 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.

#### Selected Conference presentations

- KITP Program on Fluid-Mediated Particle Transport in Geophysical Flows, Santa Barbara, November 2013:
*Grain segregation and levee formation in geophysical mass flows* - EGU General Assembly, Vienna, April 2013
*Mixing of effluent in tidal coastal waters* - APS Annual Fall DFD Meeting, San Diego, November 2012
*Entraining gravity currents* - AGU Fall Meeting, San Francisco, December 2011
*Grain-size segregation and levee formation in geophysical mass flows* - Worldwide Universities Network Earth Surface Sedimentary Flows Symposium, Bristol, April 2011
*Grain-size segregation and levee formation in geophysical mass flows* - Gordon Research Conference in Granular and Granular-Fluid Flow, Colby College, June 2010
*My poster was presented by Nico Gray in my absence.* - MPGF09: Joint IUTAM-ISIMM Symposium on Mathematical Modeling and Physical Instances of Granular Flows, Reggio Calabria, September 2009
*Granular jets and hydraulic jumps on an inclined plane* - GEOFLOWS09: Workshop on Mathematical and Computational Aspects of Modeling Hazardous Geophysical Mass Flows, Seattle, March 2009
- IMA Conference on Dense Granular Flows, Cambridge, January 2009
*The flow generated by oblique impingement of granular material on an inclined plane* - Gordon Research Conference in Granular and Granular-Fluid Flow, Colby College, June 2008
*Regularisation of the earth-pressure coefficient in time-independent Savage-Hutter flows*