We have developed an aggregation scheme for use with the Lagrangian atmospheric transport and dispersion model NAME (Numerical Atmospheric Dispersion modelling Environment), which is used by the London Volcanic Ash Advisory Centre (VAAC) to provide advice and guidance on the location of volcanic ash clouds to the aviation industry. The aggregation scheme uses the fixed pivot technique to solve the Smoluchowski coagulation equations to simulate aggregation processes in an eruption column. This represents the first attempt at modelling explicitly the change in the grain size distribution (GSD) of the ash due to aggregation in a model which is used for operational response. To understand the sensitivity of the output aggregated GSD to the model parameters, we conducted a simple parametric study and scaling analysis. We find that the modelled aggregated GSD is sensitive to the density distribution and grain size distribution assigned to the non-aggregated particles at the source. Our ability to accurately forecast the long-range transport of volcanic ash clouds is, therefore, still limited by real-time information on the physical characteristics of the ash. We assess the impact of using the aggregated GSD on model simulations of the 2010 Eyjafjallajökull ash cloud and consider the implications for operational forecasting. Using the time-evolving aggregated GSD at the top of the eruption column to initialize dispersion model simulations had little impact on the modelled extent and mass loadings in the distal ash cloud. Our aggregation scheme does not account for the density of the aggregates; however, if we assume that the aggregates have the same density of single grains of equivalent size, the modelled area of the Eyjafjallajökull ash cloud with high concentrations of ash, significant for aviation, is reduced by

In volcanic plumes ash can aggregate, bound by hydro-bonds and electrostatic forces. Aggregates typically have diameters

The theoretical description of aggregation is still far from fully understood, mostly due to the complexity of particle–particle interactions within a highly turbulent fluid. There have been several attempts to provide an empirical description of the aggregated grain size distribution (GSD) by assigning a specific cluster settling velocity to fine ash

There have been only a few attempts to model the process of aggregation explicitly.

Here we introduce an aggregation scheme coupled to a one-dimensional steady-state buoyant plume model, which uses a discrete solution of the Smoluchowski coagulation equations based on the fixed pivot technique

We use a one-dimensional steady-state integral plume model, where mass, momentum, and total energy are derived for a control volume

Ice is produced whenever

The Smoluchowski coagulation equations are solved using the fixed pivot technique, which transforms a continuous domain of masses (while conserving mass) into a discrete space of sections, each identified by the central mass of the bin, i.e. the pivot. The growth of the aggregates is described by the sticking efficiency between the particles and their collision frequencies. The approach is computationally efficient but can be affected by numerical diffusion if the number of bins is too coarse compared to the population under analysis. The coupling of the fixed pivot technique with the one-dimensional buoyant plume model is applied at the level of the mass flux conservation equations. The mass flux is modified such that the mass fractions of the dry gas (

We assume a discretized GSD composed of

We assume that ash can stick together due to the presence of a layer of liquid water on the ash, following

The influence of the ambient conditions, such as the relative humidity, on liquid bonding of ash aggregates still remains poorly constrained. Moreover, when trying to derive environmental conditions from one-dimensional plume models, it should be remembered that this description of a three-dimensional turbulent flow simply represents an average of the flow conditions and lacks details on local pockets of liquid water due to clustering of the gas mixture

List of Latin symbols. Quantities with a superscript of 0 indicate values at the source.

List of Greek symbols.

To consider the influence of uncertainty on the source and internal model parameters on the simulated aggregated GSD, we have conducted a simple sensitivity study whereby the input parameters are varied one at a time. As such, we assess the difference between the simulated output using the set of default parameters (the control case) from a perturbed case. This approach assumes model variables are independent when considering the effects of each on model predictions.

For our case study, we consider the 2010 eruption of the Eyjafjallajökull volcano, Iceland (location 63.63

The control values used for the source and internal model parameters in the aggregation scheme are given in Table

Mixing ratios of water vapour (

Modelled aggregated GSDs corresponding to the times and phase conditions shown in Fig.

Model variables used in the aggregation scheme to represent the eruption conditions. The control values listed for each parameter are based on the defaults used in the existing literature. The range of parameter values considered in the sensitivity study are also given.

Properties of the simulated aggregated GSD from the model sensitivity runs. The output is for 19:00 UTC on 4 May 2010. Using control values (Table

The mode and

The aggregated GSD shows little sensitivity to the model values assigned to define the plume conditions within the ranges investigated, namely the entrainment parameters (

Sensitivity of the output aggregated GSD to the sticking efficiency parameters,

To gain insight into the dependence of the aggregation kernel (

First, we consider how the behaviour of the collision rate (

The variation in the collision rate (

We now turn to the sensitivity of the sticking efficiency (

Figure

Figure

The collision kernel inherits its sensitivity to

The variation in sticking efficiency (

The variation in the collision kernel (

We now investigate the impact of representing aggregation on dispersion model simulations of the distal ash cloud from the eruption of the Eyjafjallajökull volcano in 2010. We consider the period between 4 and 8 May 2010, as we have measurements of the GSD and density of the non-aggregated grains for this time

Figure

However, it is expected that porous aggregates, specifically cored clusters which consist of a large core particle (

The GSD of the Eyjafjallajökull (2010) non-aggregated tephra (dark grey bars; from

Input parameters for the NAME runs.

Modelled 1 h averaged total column mass loadings of the Eyjafjallajökull ash cloud at 00:00 UTC on 5 May 2010, using

Modelled 1 h averaged total column mass loadings of the Eyjafjallajökull ash cloud at 00:00 UTC on 5 May 2010 when 25 %, 50 %, and 75 % of the mass is on aggregates with densities of 1000 and 500 kg m

Relative areas of the Eyjafjallajökull ash cloud with concentrations

We have integrated an aggregation scheme into the atmospheric dispersion model NAME. The scheme is coupled to a one-dimensional buoyant plume model and uses the fixed pivot technique to solve the Smoluchowski coagulation equations to simulate aggregation processes in an eruption column. The time-evolving aggregated GSD at the top of the plume is provided to NAME as part of the source conditions. This represents the first attempt at modelling explicitly the change in the GSD of the ash due to aggregation in a model which is used for operational response, as opposed to assuming a single aggregate class

Previous sensitivity studies of dispersion model simulations of volcanic ash clouds have highlighted the importance of constraining the GSD of ash for operational forecasts, as this parameter strongly influences its residence time in the atmosphere

Dispersion model simulations are influenced by the interplay between the size and density distributions assigned to the particles. Aggregates can have higher fall velocities than the smaller single grains of which they are composed and, therefore, act to reduce the extent and concentration of ash in the atmosphere

Our dispersion model set-up in this study reflects the choices used by the London VAAC; as such we examine the transport and dispersion of ash with diameters

It should be remembered that operational forecasts are also sensitive to other eruption source parameters needed to initialize dispersion model simulations.

The grain size distribution of the non-aggregated Eyjafjallajökull tephra, determined using ground sampling and satellite retrievals, indicated that

To be considerate of the computational costs for operational systems, we have limited aggregation processes to the eruption column only. However, it is likely that, while ash concentrations remain high, aggregation will continue in the dispersing ash cloud. As we do not represent electric fields in our scheme, we are also unable to explicitly simulate aggregation through electrostatic attraction

Volcanic plumes are highly turbulent flows characterized by a wide range of interacting length and timescales. The length scale of the largest eddies (the integral scale) is the plume radius

We consider that particle sticking can occur due to viscous dissipation in the surface liquid layer on the ash

Using scaling analysis (Sect.

For the eruption considered in this study, liquid water is only present in top

Our one-dimensional treatment of the plume does not fully represent the three-dimensional turbulent flow and may be missing local pockets of liquid water. Initial experimental studies also suggest that aggregation can occur at relatively low humidity

When liquid water and ice did form, mass mixing ratios suggest that our modelled plumes are liquid water/ice rich; the maximum mass mixing ratio of liquid water (at the top of the plume) was

Fine ash could also be preferentially removed from both the plume and dispersing ash cloud due to other size-selective processes currently not described in NAME, such as gravitational instabilities, which represent a dominant process for this eruption

Finally, our one-dimensional treatment of the Smoluchowski coagulation equations does not allow us to represent the change in density of the simulated aggregates or track explicitly the mass fraction of aggregates versus single grains within a given bin size. Our scheme could be significantly improved by using a multi-dimensional description which represents the fluctuation in the density of the growing aggregates and retains information on the mass fraction of aggregated particles. To implement this change effectively would also require a better understanding of the structure (porosity) of aggregates.

We have integrated an aggregation scheme into the atmospheric dispersion model NAME. The scheme uses a buoyant plume model to simulate the eruption column dynamics, and the Smoluchowski coagulation equations are solved with a sectional technique which allows us to simulate the aggregated GSD in discrete bins. The modelled aggregated GSD at the top of the eruption column is then used to represent the time-varying source conditions in the dispersion model simulations. Our scheme is based on the assumption that particle sticking is due to the viscous dissipation of surface liquid layers on the ash, and scale analysis indicates that our output-aggregated GSD is strongly controlled by under-constrained parameters which attempt to represent these liquid layers. The modelled aggregated GSD is also sensitive to the physical characteristics assigned to the particles in the scheme, namely the initial GSD and density distribution. Our ability to accurately forecast the long-range transport of volcanic ash clouds is, therefore, still limited by real-time information on the physical characteristics of the ash. We found that using the time-evolving aggregated GSD in dispersion model simulations of the Eyjafjallajökull (2010) eruption had very little impact on the modelled extent of the distal ash cloud with mass loadings significant for aviation. However, our scheme neither represents all the possible mechanisms by which ash may aggregate (i.e. electrostatic forces), nor does it distinguish the density of the aggregated grains. Our results indicate the need for more field and laboratory experiments to further constrain the binding mechanisms and composition of aggregates, their size distribution, and density.

In order to gain more insight into the dependence of the collision kernel

In the special case that

In the case that

For

We now turn to the sticking efficiency. On making use of Eqs. (34) and (35), we can write

Consider first the special case

In the case

In the case

Consider first the variation in

Turning now to the variation of

It should be noted that

Data used in this paper may be requested from the corresponding author and can be downloaded from

The supplement related to this article is available online at:

FB performed the sensitivity study, the aggregation NAME modelling, and wrote the first draft. ER designed the aggregation scheme as part of his doctoral project, as supervised by CB. FB, ER, and BD integrated the scheme into NAME. BD performed the scaling analysis. CW supervised the project. CB oversaw the analysis and instigated the project. All authors contributed to the project design and the finalization of the paper.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Frances Beckett would like to thank Matthew Hort, for his enthusiastic support and for reviewing a draft of this paper. The authors would like to thank Larry Mastin and an anonymous reviewer, for their thoughtful and thorough reviews, which helped to greatly improve this paper.

This research has been supported by the Horizon 2020 research and innovation programme (grant no. 731070; EUROVOLC).

This paper was edited by Qiang Zhang and reviewed by Larry Mastin and two anonymous referees.