Mark J. Woodhouse

Uncertainty analysis of a model of wind-blown volcanic plumes

Mathematical models of natural processes can be used as inversion tools to predict unobserved properties from measured quantities. Uncertainty in observations and model formulation impact on the efficacy of inverse modelling. We present a general methodology, history matching, that can be used to investigate the effect of observational and model uncertainty on inverse modelling studies. We demonstrate history matching on an integral model of volcanic plumes that is used to estimate source conditions from observations of the rise height of plumes during the eruptions of Eyjafjallajökull, Iceland, in 2010 and Grímsvötn, Iceland, in 2011. Sources of uncertainty are identified and quantified, and propagated through the integral plume model. A preliminary sensitivity analysis is performed to identify the uncertain model parameters that strongly influence model predictions. Model predictions are assessed against observations through an implausibility measure that rules out model inputs that are considered implausible given the quantified uncertainty. We demonstrate that the source mass flux at the volcano can be estimated from plume height observations, but the magmatic temperature, exit velocity and exsolved gas mass fraction cannot be accurately determined. Uncertainty in plume height observations and entrainment coefficients results in a large range of plausible values of the source mass flux. Our analysis shows that better constraints on entrainment coefficients for volcanic plumes and more precise observations of plume height are required to obtain tightly constrained estimates of the source mass flux.

Unsteady turbulent buoyant plumes

We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the conservation of mass, axial momentum and buoyancy. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly described by shape factors in the integral equations; the commonly-assumed top-hat profiles lead to shape factors equal to unity. The resultant model is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity, differs from unity. The solutions of the model when source conditions are maintained at constant values retain the form of the well-established steady plume solutions. We demonstrate that the inclusion of a momentum shape factor that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is identified, resulting in a range of its values where the amplitude of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution. We compute numerical solutions to illustrate the transient development following an abrupt change in the source conditions. The adjustment to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse are described by a similarity solution and, by constructing this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differ.

Charge structure in volcanic plumes: a comparison of plume properties predicted by an integral plume model to observations of volcanic lightning during the 2010 eruption of Eyjafjallajökull, Iceland.

Observations of volcanic lightning made using a lightning mapping array during the 2010 eruption of Eyjafjallajökull allow the trajectory and growth of the volcanic plume to be determined. The lightning observations are compared with predictions of an integral model of volcanic plumes that includes descriptions of the interaction with wind and the effects of moisture. We show that the trajectory predicted by the integral model closely matches the observational data and the model well describes the growth of the plume downwind of the vent. Analysis of the lightning signals reveals information on the dominant charge structure within the volcanic plume. During the Eyjafjallajökull eruption both monopole and dipole charge structures were observed in the plume. By using the integral plume model, we propose the varying charge structure is connected to the availability of condensed water and low temperatures at high altitudes in the plume, suggesting ice formation may have contributed to the generation of a dipole charge structure via thunderstorm-style ice-based charging mechanisms, though overall this charging mechanism is believed to have had only a weak influence on the production of lightning.

Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions

The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre & Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai & Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.

Rapid granular flows down inclined planar chutes. Part 1. Steady flows, multiple solutions and existence domains

The highly agitated flow of grains down an inclined chute is modelled using a kinetic theory for inelastic collisions. Solutions corresponding to steady, fully developed flows are obtained by solving numerically a nonlinear system of ordinary differential equations using a highly accurate pseudospectral method based on mapped Chebyshev polynomials. The solutions are characterized by introducing macroscopic, depth-integrated variables representing the mass flux of flowing material per unit width, its centre-of-mass and the mass supported within the flowing layer, and the influence of the controlling parameters on these solutions is investigated. It is shown that, in certain regions of parameter space, multiple steady solutions can be found for a specified mass flux of material. An asymptotic analysis of the governing equations, appropriate to highly agitated flows, is also developed and these results aid in the demarcation of domains in parameter space where steady solutions can be obtained.

A Framework for Probabilistic Multi-Hazard Assessment of Rain-Triggered Lahars Using Bayesian Belief Networks

Volcanic water-sediment flows, commonly known as lahars, can often pose a higher threat to population and infrastructure than primary volcanic hazardous processes such as tephra fallout and Pyroclastic Density Currents (PDCs). Lahars are volcaniclastic flows formed by water, volcanic debris and entrained sediments that can travel long distances from their source, causing severe damage by impact and burial. Lahars are frequently triggered by intense or prolonged rainfall occurring after explosive eruptions, and their occurrence depends on numerous factors including the spatio-temporal rainfall characteristics, the spatial distribution and hydraulic properties of the tephra deposit, and the pre- and post-eruption topography. Modelling such a complex system requires the quantification of aleatory variability in the lahar triggering and propagation. To fulfill this goal, we develop a novel framework for probabilistic hazard assessment of lahars within a multi-hazard environment, based on coupling a versatile probabilistic model for lahar triggering (a Bayesian Belief Network: Multihaz) with a dynamic physical model for lahar propagation (LaharFlow). Multihaz allows us to estimate the probability of lahars of different volumes occurring by merging varied information about regional rainfall, scientific knowledge on lahar triggering mechanisms and, crucially, probabilistic assessment of available pyroclastic material from tephra fallout and PDCs. LaharFlow propagates the aleatory variability modelled by Multihaz into hazard footprints of lahars. We apply our framework to Somma-Vesuvius (Italy) because: (1) the volcano is strongly lahar-prone based on its previous activity, (2) there are many possible source areas for lahars, and (3) there is high density of population nearby. Our results indicate that the size of the eruption preceding the lahar occurrence and the spatial distribution of tephra accumulation have a paramount role in the lahar initiation and potential impact. For instance, lahars with initiation volume ≥ 105 m3 along the volcano flanks are almost 60% probable to occur after large-sized eruptions (~VEI≥5) but 40% after medium-sized eruptions (~VEI4). Some simulated lahars can propagate for 15 km or reach combined flow depths of 2 m and speeds of 5-10 m/s, even over flat terrain. Probabilistic multi-hazard frameworks like the one presented here can be invaluable for volcanic hazard assessment worldwide.

Atmospheric processes affecting the separation of volcanic ash and SO 2 in volcanic eruptions

The separation of volcanic ash and sulfur dioxide (SO2) gas is sometimes observed during volcanic eruptions. The exact conditions under which separation occurs are not fully understood but the phenomenon is of importance because of the effects volcanic emissions have on aviation, on the environment, and on the earth’s radiation balance. The eruption of Grímsvötn, a subglacial volcano under the Vatnajökull glacier in Iceland during 21–28 May 2011 produced one of the most spectacular examples of ash and SO2 separation, which led to errors in the forecasting of ash in the atmosphere over northern Europe. Satellite data from several sources coupled with meteorological wind data and photographic evidence suggest that the eruption column was unable to sustain itself, resulting in a large deposition of ash, which left a low-level ash-rich atmospheric plume moving southwards and then eastwards towards the southern Scandinavian coast and a high-level predominantly SO2 plume travelling northwards and then spreading eastwards and westwards. Here we provide observational and modelling perspectives on the separation of ash and SO2 and present quantitative estimates of the masses of ash and SO2 that erupted, the directions of transport, and the likely impacts. We hypothesise that a partial column collapse or “sloughing” fed with ash from pyroclastic density currents (PDCs) occurred during the early stage of the eruption, leading to an ash-laden gravity intrusion that was swept southwards, separated from the main column. Our model suggests that water-mediated aggregation caused enhanced ash removal because of the plentiful supply of source water from melted glacial ice and from entrained atmospheric water. The analysis also suggests that ash and SO2 should be treated with separate source terms, leading to improvements in forecasting the movement of both types of emissions.