We analyze the sensitivity of a mathematical model of volcanic ash dispersion in the atmosphere to the representation of key physical processes. These include the parameterization of subgrid-scale atmospheric processes and source parameters such as the height of the eruption column, the mass emission rate, the size of the particulates, and the amount of ash that falls out close to the source. By comparing the results of the mathematical model with satellite and airborne observations of the ash cloud that erupted from Eyjafjallajökull volcano in May 2010, we are able to gain some insight into the processes and parameters that govern the long-range dispersion of ash in the atmosphere. The structure of the ash cloud, particularly its width and depth, appears to be sensitive to the source profile (i.e., whether ash is released over a deep vertical column or not) and to the level of subgrid diffusion. Of central importance to the quantitative estimates of ash concentration in the distal ash cloud is the fallout of ash close to the source. By comparing the mass of the ash and the column loadings in the modeled and observed distal ash cloud, we estimate the fraction of fine ash that survives into the distal ash cloud albeit with considerable uncertainty. The processes that contribute to this uncertainty are discussed.
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 The unusually long duration of the Eyjafjallajökull eruption in April and May 2010 afforded many opportunities to measure the resulting ash cloud that spread over the North Atlantic and Europe. A point-by-point comparison of observations with numerical simulations of the ash cloud [Webster et al., 2012; Folch et al., 2012; Heinold et al., 2012; Matthias et al., 2012] has shown that dispersion models can produce substantial scatter about the observations. Many of these errors arise from a problem common to dispersion modeling of any airborne contaminant and not just ash, namely, that small errors in the wind field can lead to large errors in the concentration of the contaminant. However, the concentration is also strongly affected by the source properties such as the mass emission rate, the height at which particulates are released into the atmosphere, and the size of the particulates. In the case of volcanic ash, the amount of fallout that occurs close to the source is also of great importance. Furthermore, the structure of the distal ash cloud and hence the concentration of ash are also affected by the parameterization of subgrid-scale processes. (Note that by distal ash cloud we mean the ash cloud sufficiently far downwind of the source such that the fallout of large grains and aggregates has already occurred and that processes associated with the eruption column and initial spread of the ash cloud—such as its buoyancy—are no longer important. Conversely we define near-source to describe that part of the ash cloud that is affected by these processes.) The purpose of this study is to investigate in more detail one episode during the eruption of Eyjafjallajökull in May 2010: We focus on 14 May when particularly large concentrations were observed. By assessing the sensitivity of the model to a range of factors including the height of the eruption column, the source profile (the variation of source strength with height), and “turbulent” or subgrid diffusion (representing the dispersion by the unresolved eddies), we can gain some insight into the relative importance of these factors in modeling the long-range dispersion of ash in the atmosphere.
 The model we use here is the UK Met Office's operational dispersion model, NAME (Numerical Atmospheric-dispersion Modelling Environment). We make a qualitative and quantitative comparison of the model results with satellite observations and aircraft measurements. In particular, we compare the column loadings (vertically integrated concentrations) and areal masses obtained from the model with the observed values. In the simulations presented here we use an estimate of the total mass emitted from the volcano and release all of this material as particles with a diameter of 100 μm or less. Because larger particles (both larger ash grains and composite particles formed through near-source aggregation) will fall out rapidly, we expect to overpredict the column loadings and areal masses. However, the comparison between the model and observed values can be used to estimate the fallout of large particles near the volcano, although with significant levels of uncertainty. This provides an estimate of the fraction of ash that survives into the far field, which is called the distal fine ash fraction [Dacre et al., 2011]. Note that this is not the same as the fraction of fine ash that erupts from the volcano, since much of this will form larger aggregates and be deposited, often close to the volcano. The choice of 100 μm is a little arbitrary, but we will show below that particles of this size or larger fall out very rapidly, within 100 km of the source, whereas smaller particles, say, 30 μm, travel much farther. Independent estimates of the magnitude of near-source fallout are difficult to establish, but we will show that this method for estimating the distal fine ash fraction gives results similar to those of earlier studies [Rose et al., 2000]. For the modeling approach described here, the degree of near-source fallout is one of the largest sources of uncertainty in the quantitative modeling of the long-range dispersion of ash. Moreover, an order of magnitude estimate of the near-source fallout, however uncertain, could be used as a constraint for detailed models of the eruption column and its initial spread.
2. The NAME Model
 NAME is a Lagrangian stochastic model in which the trajectories of many model particles are followed through a realistic wind field which is usually provided by a Numerical Weather Prediction (NWP) model. As well as responding to the resolved flow field, the trajectories of the model particles contain a stochastic element which represents turbulent dispersion due to the unresolved eddies. Each particle carries a certain mass of the dispersing substance. When the substance consists of particulate material (as for volcanic ash), each model particle represents a number of real particles (either individual grains or aggregates) of a certain diameter. The diameter remains fixed for the duration of the simulation (we thus ignore any agglomeration or breakup of aggregates during dispersion) and is drawn from a specified distribution. Sedimentation is treated by giving each model particle a sedimentation velocity appropriate for the size of the real particles it represents. Particles are assumed to be spherical in shape with a Reynolds number dependent drag coefficient [Maryon et al., 1999] and a slip flow “Cunningham” correction factor [Pruppacher and Klett, 1997, p. 416] which becomes important for small particles. Dry deposition is treated using the resistance method to estimate a deposition velocity which is combined with the sedimentation velocity as described by Webster and Thomson . Wet deposition is treated using washout coefficients [Maryon et al., 1999]. For more details on NAME and its applications, see Jones et al. .
 In the simulations to be presented below, the flow field is provided by the UK Met Office's NWP model, the Unified Model (UM). The global configuration of the UM is used with a horizontal grid spacing of about 25 km and 70 unequally spaced vertical levels extending into the mesosphere with a typical resolution of 300–400 m in the mid-troposphere. Flow field data were provided at 3 hourly intervals consisting of alternate analyses and short-period forecasts.
 Each model particle is treated as a solid particle with a density of 2300 kg m−3. This may be an underestimate for some individual grains or an overestimate when the model particle represents an aggregate. However, the density and the particle size distribution together can be regarded as referring to the effective density and size needed to give the right sedimentation velocity. The height of the eruption column is estimated from radar measurements made by the Icelandic Meteorological Office [Arason et al., 2011]. Unless otherwise stated, the model particles are released uniformly in the vertical between the volcano summit and the top of the eruption column. The emitted mass is divided uniformly among the emitted model particles. There is no explicit modeling of the plume rise process itself with the model treating the ash as a passive, but sedimenting and depositing, tracer. We therefore regard the NAME source as the effective source. The particle release rate is 200,000 h−1. Tests showed that this produced results which were not unduly compromised by statistical noise. The simulation covers the period from midnight on 8 May until midnight on 14 May 2010. (Although we are primarily interested in 14 May, a 6 day simulation ensures that any impact of ash emitted well before 14 May is not excluded.) The source strength is calculated according to the empirical relationship between the height of the eruption column above the volcano summit, zmax, and the mass emission rate, Qm, given by Mastin et al.  (assuming a density of 2500 kg m−3):
where zmax is measured in km and Qm in kg s−1. Note that from Mastin et al. [2009, Figure 1], there is approximately a 50% chance that, for a given zmax, Qm can differ from (1) by a factor of 3–4 or more. Equation (1) is similar to the relationship given by Sparks et al. [1997, section 5.2], and the exponent, 4.15, is close to the value expected from dimensional analysis and integral plume models for vertically rising plumes [Morton et al., 1956; Sparks et al., 1997, section 2.6]. Equation (1) takes no account of the state of the atmosphere at the time of the eruption (such as the effects of atmospheric stability, wind speed and direction, and ambient moisture), nor does it account for the effects of moisture at the source (such as the melting of the glacier above Eyjafjallajökull). Several studies [e.g., Woods, 1993; Bursik, 2001; Mastin, 2007; Tupper et al., 2009] have demonstrated that the state of the atmosphere can have a considerable effect on the height attained by the volcanic plume, and this is one of the causes of the errors in (1). Since the eruption was relatively weak, the ambient wind is likely to give rise to potentially significant differences in the actual mass emission rate compared with the value given by (1). The evolution of the height of the eruption column and mass emission rate over the course of the simulation is shown in Table 1. The emission heights for the simulations were chosen subjectively to be at the upper end of the scatter seen in the 5 min radar values given by Arason et al. , while not reflecting the most extreme values. The error in the eruption-column height estimated from the radar can be more than a kilometer. During periods of little or no data the value prior to the period was retained.
Table 1. Evolution of the Height of the Eruption Column Above Mean Sea Level, H, and the Mass Emission Rate, Qm, With Timea
Qm (kg s−1)
Volcano summit height is taken to be 1666 m.
3.72 × 104
2.99 × 105
2.09 × 104
 All the emitted mass is distributed uniformly among the model particles with each model particle representing the mass of many real particles. For a given model particle, the real particles are all assumed to have the same diameter which is drawn from the probability distribution shown in Figure 1. Within each bin, log d is chosen from a uniform distribution over the appropriate range where d is the particle diameter. In reality, a large fraction of material is likely to be concentrated in particles with much larger diameters (due to the intrinsic grain size, the aggregation of grains into larger particles, or the accretion of ice or water when ash particles act as cloud condensation nuclei), but these particles fall out close to the source and so do not play a role in ash that travels far from the source. Of course, it would be possible to include these large particles in the model (though with no attempt to model the dynamics of the eruption column), but this would increase the computational expense (since more model particles would be needed to produce reasonable statistics) and is still subject to a large degree of uncertainty regarding the size distribution. A more straightforward approach is to reduce the emission rate by a given factor to account for the near-source fallout. In this paper, though, we do not reduce the emission rate. Instead we are interested in estimating the near-source fallout: By comparing the column loadings and areal masses calculated from the model data with the observed values, we can estimate the fraction of ash which is lost close to the source.
 The ash concentration is calculated by averaging over grid boxes that are approximately 40 km in each horizontal direction and 100 m in the vertical direction. Unless otherwise stated, all statistics are averaged over the hour preceding the stated time of the data. All times are UTC, and all the model data are calculated with NAME III version 6.0.
 Airborne measurements of the ash cloud were made by the Facility for Airborne Atmospheric Measurements (FAAM), a joint UK Met Office–UK universities consortium. Those measurements by the FAAM research aircraft that are considered here took place between approximately 10:30 and 15:30 on 14 May. We present satellite observations at 12:00 and 15:00 in section 3.1 and aircraft measurements in section 3.2, both of which will be compared with the model in section 4.
3.1. Satellite Observations
Figure 2 shows “dust red-green-blue” (RGB) images of the ash cloud taken from the Spinning Enhanced Visible and Infrared Imager (SEVIRI) on the Meteosat Second Generation (MSG) satellite. The images contain information on the equivalent black body temperatures of radiances measured at three infrared wavelengths (for more details, see Francis et al. ). RGB images were originally developed at the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) and have been extensively used as a qualitative volcanic ash monitoring tool [Francis et al., 2012]. The images show the ash cloud to the east of Iceland and extending as far as Britain and also depict the ash cloud splitting into two main branches, one extending northward back toward eastern Iceland (from 63.4°N northward) and the other extending southeastward as far as northern England. Analysis of ash that was emitted at earlier times reveals that the northern branch consists predominantly of ash released on 12 May that was formed as the ash cloud folded back on itself, in part because of its interaction with the occluded front shown in the surface pressure chart in Figure 3. Note also how Figure 2a shows the ash cloud splitting into two aligned narrow bands over the Atlantic between Iceland and Scotland around (12°W, 61°N). Also shown in Figure 2 is a small region immediately to the west of Eyjafjallajökull (bounded by 30°W and 23°W and 62°N and 68°N) in which ash was detected. For reference, Eyjafjallajökull is located at (19.6°W, 63.6°N).
 The height of the ash cloud and its associated column loading are estimated from the satellite data by applying a radiative transfer model iteratively (varying the height and column loading of the ash cloud and the effective radius of the particle size distribution) to a thin layer of ash until the modeled radiances agree well with the observed radiances [Francis et al., 2012]. The detection threshold for ash is generally of order 0.5–1.0 g m−2 and tends to vary with height: As the altitude of the ash increases, the detection threshold decreases. This can be understood as follows: As the temperature difference between the ash and the surface increases, as is the case for ash at high altitudes, detection becomes easier. This method of estimating a specific ash height is easiest to interpret when the optical thickness is large; in this case, as the ash cloud becomes more dense, the estimated height approaches that of the top of the ash cloud. Conversely, the estimated height of a deep, optically thin ash cloud effectively becomes more of a weighted average of the ash concentration profile, and by necessity occurs at some level below the real ash top. The andesite refractive indices of Pollack et al.  were used to calculate the column loadings and heights shown in Figures 4 and 5, respectively. Note that the largest column loadings over Scotland and the neighboring Atlantic are approximately 6 g m−2 and that the estimated height of the ash cloud is generally around 6–6.5 km. The heights and associated column loadings quoted here were derived while allowing both these parameters (along with the effective radius) to vary simultaneously within the retrieval scheme [Francis et al., 2012]. If the height is fixed within the retrieval scheme to be 6 km (of order the airborne lidar observations in Figure 7), then the column loadings remain consistent with those in Figure 4. If the height is fixed at a higher value, then the column loadings decrease below those in Figure 4. The presence of low-level cloud is likely to distort the satellite retrievals and lead to an overestimate of the column loading. The emissivity of land shows more variability and uncertainty compared with the emissivity of the sea and thus contributes to the uncertainty in the values for the column loading and ash cloud height given here. More detailed analysis of the satellite retrievals including a discussion of the assumptions and uncertainties behind the figures quoted here is given by Francis et al. .
 The fraction of the total mass in the ash cloud at 12:00 that is transported into the two branches of the distal ash cloud is 20% for the southern branch (0.109 Tg in the region bounded by 13°W and 2°W, 52°N and 63.4°N) and 63% for the northern branch (0.340 Tg in the region bounded by 23°W and 8°W, 63.4°N and 72°N, excluding the southwest corner with the volcano); only 17% is transported to the west of the volcano (0.092 Tg in the region defined above). We define the typical area of the ash cloud to be the ratio of the areal mass to the maximum column loading in that area. For the southern branch we find that the typical area is approximately 18,200 km2.
3.2. Aircraft Observations
Figure 6 shows the column loading along the flight path followed by the FAAM research aircraft estimated from measurements of the ash cloud made with a downward pointing lidar [Marenco et al., 2011]. Although the aircraft measurements necessarily survey a limited region of the ash cloud, a comparison with the satellite imagery indicates that the flight path is close to the centerline of the ash cloud (where the concentration is largest). The maximum column loading measured by the lidar was approximately 1.4 g m−2 over southern Scotland (at approximately 13:45), which is significantly less than that derived from the satellite data. However, the column loading near this location at about the same time was also estimated from vertical profiles of the ash cloud measured by the Cloud and Aerosol Spectrometer (CAS) on board the aircraft. The maximum column loading calculated in this way was estimated to be 6 g m−2, though it should be noted that this particular flight encountered relatively high levels of ice which may have mixed with or coated the ash and may have disrupted the CAS instrument more than the lidar, potentially leading to an overestimation of the column loadings [Johnson et al., 2012]. In general, estimates of the column loadings from the CAS instrument and the lidar agreed better on other flights than is the case here, and the error was then estimated to be a factor of 2 [Marenco et al., 2011; Johnson et al., 2012].
Figure 7 shows the vertical structure of the ash cloud as measured by the lidar along the flight path corresponding to that in Figure 6 [Marenco et al., 2011]. Notice that the ash cloud tends to form patchy layers that are typically located between 6 km and 7 km. The evolution of the typical layer depth, defined to be the ratio of the column loading to the maximum concentration over the layer, along the flight path is shown in Figure 8 and indicates that it tends to fluctuate between approximately 500 m and 1000 m. The vertical profiles measured by the CAS instrument over the Atlantic (7°W, 60°N) and southwest Scotland (3°W, 55°N) in Figure 9 show a similar depth over which the ash tends to be concentrated. The height measured by the aircraft is generally in good agreement with the satellite retrievals. More comprehensive analyses of the in situ and lidar data are given by Johnson et al.  and Marenco et al. , respectively.
4. Comparison of NAME With Observations
4.1. Comparison With Satellite Data
Figure 10 shows horizontal contour plots of the column loading calculated from NAME assuming no significant near-source fallout. Qualitatively, the model ash cloud exhibits many features of the observed ash cloud including the splitting of the ash cloud into two branches over the Atlantic to the northwest of Scotland. The northern branch is heading back toward Iceland while the southern branch reaches as far south as southern Scotland, although not as far south as the satellite observations. Figure 10b shows some evidence of the main ash cloud splitting into two aligned narrow bands around (12°W, 61°N) in a manner consistent with the satellite imagery (see Figure 2; we will return to this feature in section 5.3). However, further comparison with the satellite imagery shows that the modeled ash cloud lies too far to the west. Furthermore, NAME predicts that the highest column loadings in the distal ash cloud occur around (8°W, 60°N), in the vicinity of where the ash cloud diverges, and not over southern Scotland as indicated by the observations (see Figures 4 and 6). The possible reasons for this will be investigated in section 5. NAME predicts that approximately 25% of the total mass reaches the southern branch (as defined in section 3.1), which agrees well with the satellite retrievals, but less than 3% of the mass reaches the northern branch. In contrast with the satellite retrievals, approximately 58% lies in the region immediately to the west of the volcano. The reasons for this discrepancy are not clear but will to some extent be addressed in section 5.
 The error in the position of the NAME ash cloud means that a quantitative point-by-point comparison of the data could be misleading. Since the northern branch of the ash cloud is not modeled very well by NAME whereas the fraction of mass that is transported into the southern branch is close to the observed fraction, we estimate the distal fine ash fraction by comparing the observed and modeled masses in this part of the distal ash cloud. In NAME this gives 2.87 Tg at 12:00 and 2.39 Tg at 15:00. Since we assume no significant near-source fallout in the NAME simulations used here, these figures indicate that NAME, as expected, overpredicts the total mass by a factor of approximately 22–26 (the results at 12:00 giving the upper bound and those at 15:00 giving the lower bound). The ratio of the observed mass to the modeled mass suggests that approximately 95.5–96.2% of the ash falls out close to the source due to processes not included in the model such as emission of material with large grain sizes or aggregation of small grains into large particles. We expect such processes to become negligible beyond a few hundred kilometers of the source for a weak eruption of this type [Sparks et al., 1997, chapter 16; Durant et al., 2010]. We estimate that approximately 3.78–4.54% of the ash survives into the distal ash cloud, the distal fine ash fraction. There are significant uncertainties in this figure due to uncertainties in the mass emitted, uncertainties in the observations, and uncertainties in the modeling. The last of these uncertainties will be addressed to some extent by sensitivity tests in section 5. However, this estimate of the distal fine ash fraction is similar to that calculated during the initial phase of this eruption in April 2010 [Devenish et al., 2012; Dacre et al., 2011] and for other eruptions [Rose et al., 2000]. Because the fraction of ash that survives into the far field is much less than 100%, it indicates that near-source fallout is important and provides some indication of its magnitude despite the significant uncertainties involved. Indeed, because the estimated distal fine ash fraction is very small, it suggests that errors in the mass emission rate, while very large, do not dominate the estimate to the extent that it would become unphysical.
Figure 11 shows the ash cloud height calculated from the NAME data. The ash cloud height in a given column is taken to be the height of the maximum concentration in that column; results are only displayed for those columns in which the maximum concentration is greater than 1 mg m−3. In general, NAME gives a similar value for the ash cloud height over southern Scotland and the neighboring Atlantic as the satellite imagery (see Figure 5) and the aircraft measurements (see Figures 7 and 9), though, of course, there are some discrepancies in a point-by-point comparison.
 The NAME value of the maximum column loading in the southern branch at 12:00 is 52.6 g m−2, which, as expected, is much larger than the equivalent value calculated from the satellite data since we are assuming no significant near-source fallout. The typical area of the southern branch of the NAME ash cloud is approximately 54,600 km2, which indicates that the NAME ash cloud is more widely spread than the ash cloud observed by the satellite.
4.2. Comparison With Aircraft Measurements
 Since the aircraft measurements necessarily represent a limited sample of the ash cloud, it is not possible to estimate the total ash in the distal ash cloud from these measurements. Thus, we use a different approach to estimate the distal fine ash fraction from the aircraft measurements. Since a point-by-point comparison of the data could be misleading, we calculate the distal ash fraction from ratios of the observed to modeled local maximum values of the column loading. The local maximum is defined as the maximum value over the southern branch defined as in section 3.1 (i.e., between 13°W and 2°W, 52°N and 63.4°N). In NAME the local maximum column loading at 14:00 (the nearest available time) is 39.7 g m−2 assuming no significant near-source fallout. Comparing the maximum value recorded by the onboard lidar, we find that NAME is approximately 28 times the observed value, which gives a near-source fallout of 96.5% and a distal fine ash fraction of 3.53%. This degree of near-source fallout is similar to that in section 4.1. However, when NAME is compared with the maximum value recorded by the CAS instrument, the near-source fallout is considerably smaller at 85%. Note that had we used the local maximum column loading as estimated from the satellite data, which is similar to that measured by the CAS instrument, we would of course have found similar values of the near-source fallout and distal fine ash fraction. The possible reasons for the discrepancy between the lidar and CAS data have been discussed in section 3.2; it should be emphasized that these estimates are based on limited data samples and are subject to considerable uncertainty.
Figure 12 shows the vertical structure of the modeled ash cloud along the centerline of the ash cloud, which is defined to be the position of the maximum column loading along each line of longitude. Comparing Figures 12 and 7 it is readily seen that the NAME ash cloud is less patchy than the observations. However, it is possible to discern some similarities: The top of the modeled ash cloud is no higher than 8 km, which is consistent with the observations; the downward protruding tongue of ash that can be seen in Figure 12 between 10°W and 8°W is similar (allowing for some error in position) to that of around midday in Figure 7 (between (6.1°W, 60°N) and (8.6°W, 60°N)). Figure 13 shows the typical layer depth, calculated as in Figure 8, along the centerline of the modeled ash cloud. It is readily seen that the modeled ash cloud is much deeper than the observed ash cloud. In section 5 we consider some potential reasons for the differences between the modeled and observed structure of the ash cloud.
Figure 14 shows vertical profiles of the model ash concentration in the vicinity of the CAS measurements in Figure 9 and at similar times. The NAME profiles share many qualitative similarities with the observed profiles including the approximate maximum height of the ash cloud, though, as in Figure 12, the depth of the modeled ash cloud is much greater than the observed ash cloud. Comparing Figures 9a and 14a we see that the modeled ash concentrations are much larger than the observed ash concentrations (as expected due to lack of near-source fallout in these NAME simulations), whereas comparing Figures 9b and 14b we find the opposite. This is consistent with a modeled ash cloud that does not extend as far south as the observations and highlights the stringent nature of a point-by-point comparison of the model with the observations.
5. Sensitivity Studies
 We conduct a number of simulations to investigate the sensitivity of NAME to changes in (1) height of the eruption column; (2) vertical profile of the eruption column; (3) subgrid diffusion; and (4) particle size. For simplicity we illustrate the effect of these changes at 12:00; similar differences occur at other times.
5.1. Source Height
 Radar observations of the eruption column were made in discrete bands (corresponding to different elevation angles of the instrument) from which the height is estimated by interpolating between these bands [Arason et al., 2011]. Thus, there is a margin of error that ranges over the height of the particular band in question. In this section we consider the sensitivity of the dispersion statistics to changes in the eruption column height. We conducted several simulations with a variety of revised heights, and increasing the eruption column height on 12 and 13 May improves the agreement of the horizontal extent of the model ash cloud with the observations. The change in the eruption column height shown in Table 2 gives the best agreement while still lying within the observed radar band. Stohl et al.  noted that dense low cloud obscured the eruption column on 12 May, resulting in considerable uncertainty in estimates of its height and hence significant differences between numerical simulations and observations of the ash cloud downwind of the source. By increasing the height of the eruption column on 12 May from its original estimate of approximately 5 km [Arason et al., 2011] to approximately 7 km, they were able to achieve better agreement of the modeled ash cloud downwind of the source (over Great Britain) with the satellite observations.
Table 2. Evolution of the Revised Eruption Column Height With Time
Qm (kg s−1)
3.72 × 104
1.47 × 105
2.99 × 105
2.09 × 104
Figure 15 shows the column loadings calculated from the simulation using the eruption column heights and corresponding mass emission rates in Table 2. The local maximum column loading has increased relative to that in Figure 10a, as expected, and the southward penetration of the ash cloud now extends into northern England and the northern extent of the ash cloud is greater than that in Figure 10a for comparable values of the column loading relative to its (local) maximum value. The position of the local maximum column loading remains in the vicinity of (8°W, 60°N) unlike the corresponding satellite (Figure 4) and aircraft (Figure 6) measurements, both of which show a local maximum that is located over southern Scotland. A simulation with a constant eruption column height of 5 km (not shown) depicts an ash cloud with a structure similar to that in Figure 10a, that is, the splitting of the ash cloud into northern and southern branches (though of significantly less horizontal extent compared with Figure 10a) and with a distinct local maximum over the Atlantic to the northwest of Scotland (located slightly farther north than in Figure 10a). This suggests that the formation of this local maximum is not due to “pulsing” of the volcano (though such fluctuations may influence both its exact location and its value) but is primarily a property of the meteorology, most likely variations in the wind at the source and associated with the occluded front shown in Figure 3.
 While changes to the eruption column height appear to have only a small effect on the position of the local maximum, they do affect its value; the resulting estimate of the distal fine ash fraction calculated from the aircraft data is given in Table 3. The mass of ash in the southern branch (i.e., the integral of the column loading over the same area as in section 4.1) is approximately 74% larger than for the ash cloud in Figure 10a. This leads to a lower distal fine ash fraction (see Table 3) compared with section 4.1. It is worth noting that the increase in the masses (74%) is considerably larger than the increase in the local maximum column loadings (19%), which is consistent with an increased column height leading to a distal ash cloud that spreads farther.
Table 3. Estimates of the Distal Fine Ash Fraction for Several Different Simulations at 12:00 Based on Ratios of the Observed to Modeled Mass (Satellite Retrieval) and the Observed to Modeled Local Maximum Column Loadings (Aircraft Lidar and Aircraft CAS)a
Satellite Retrieval (Mass)
Aircraft Lidar (1.4 g m−2)
Aircraft CAS (6 g m−2)
The maximum observed column loadings recorded by the aircraft lidar and CAS instruments are shown at the top of the “aircraft lidar” and “aircraft CAS” columns, respectively. The local maximum column loading derived from the satellite data has the same value as that measured by the CAS instrument. The near-source fallout is simply (100 − x)%, where x is the distal fine ash fraction.
 In the absence of a suitably strong crosswind, a volcanic plume typically forms a radially spreading umbrella cloud near the top of the eruption column [see, e.g., Sparks et al., 1997, section 11.2]. Detrainment of material into the surrounding atmosphere occurs predominantly in this region. Observations of umbrella clouds suggest that its depth, Δz, is approximately zmax/4 with the height of the top and bottom of the umbrella cloud approximately coincident with respectively the maximum rise height of the eruption column and the level at which the eruption column first becomes neutrally buoyant. Since the lateral spreading of the umbrella cloud is most pronounced at the midpoint between the top and bottom of the umbrella cloud, where the density of the cloud equals that of the environment, the effective source profile of material can be modeled by a Gaussian distribution centered on this midpoint. The standard deviation of the Gaussian distribution is chosen to be equal to that of a uniform distribution over Δz, i.e., . We call this case the “vertically rising case.” As a special case of this case we also consider a point source located at the top of the eruption column. Such a source may be justified as an approximation to reality, since volcanic intrusions downstream of the source tend to form thin layers due to gravity currents.
 Relatively weak volcanic eruptions can be significantly distorted by the ambient wind. If the wind is sufficiently strong the plume becomes bent-over (so that its axis is horizontal). For a bent-over plume, the radius of the plume, b, at its maximum rise is given by βzc, where β is the entrainment constant associated with bent-over plumes, which typically has a value in the range 0.4–0.6 [see, e.g., Devenish et al. , and zc is the maximum height of the plume axis above the volcano summit (i.e., zmax = zc + b). We choose β = 0.5, which gives zc = (2/3)zmax, and hence the depth of the bent-over plume is Δz = 2b = (2/3)zmax. The spread of material in a bent-over plume about its axis can be approximated by a Gaussian distribution on a vertically orientated disc (although we only model the vertical profile and keep the horizontal source size as zero as for the other cases). We choose a standard deviation of b/2(= Δz/4 = zmax/6) so that it is equal to the standard deviation of a uniform distribution on a disc of radius b. We call this case the “bent-over case.” The vertically rising and bent-over cases represent the two limiting cases of the effective source profile. The Eyjafjallajökull profile most likely lies somewhere in between, although the true profile may be more complex due to fluctuations in emissions associated with “pulsing” of the eruption column.
Figure 16 shows the effect of changes to the vertical source profile. In all three cases the width of the ash cloud (more precisely the width of the horizontal projection of the ash cloud as produced by the column loading) located over western Scotland and the neighboring Atlantic is less than that from the uniform source (see Figure 10a). Comparing the three cases, the point source results in the least distinct northern branch of the ash cloud (aligned with the occluded front), suggesting that the ash that is transported along this branch emanates from the lower part of the eruption column. Conversely, the point source produces the greatest southward penetration of the ash cloud over Scotland, suggesting that the ash that arrives here is predominantly released toward the top of the eruption column. It should be emphasized, though, that the differences between the three cases are small.
 The estimated distal fine ash fractions for each case are given in Table 3. It can be readily seen that the distal fine ash fraction is smallest for a point source, followed by the vertically rising case and largest for the bent-over case. These estimates of the distal fine ash fraction for the three different source types considered here are all smaller than the distal fine ash fraction that was calculated for a uniform source (see section 4). These trends hold whether the distal fine ash fraction was computed from the local maximum column loadings (section 4.2) or the mass in the southern branch (section 4.1): Indeed the fractional change between the uniform case and the point source is almost the same irrespective of whether the distal fine ash fraction is based on the mass or the local maximum column loading. Both the mass of the southern branch and its local maximum column loading increase as the effective source becomes more concentrated toward the top of the eruption column (i.e., they are lowest for the uniform release in section 4 and highest for the point source). The same is true of the mass of the whole ash cloud and its overall maximum column loading. However, the fraction of modeled ash that is in the southern branch is approximately the same (25%) regardless of the source type. The same is true of the fraction of ash that is immediately to the west of Eyjafjallajökull: About 60% of the mass of the ash cloud is in the vicinity of the volcano regardless of source type. This suggests that the differences in the mass of the whole ash cloud (and hence that of the southern branch) arise from a larger degree of deposition occurring when material is released close to the ground. Thus, the uniform source exhibits the lowest mass (in either the whole ash cloud or the southern branch) and the point source the highest mass. Simulations with no deposition or sedimentation confirm this. The differences in the estimates of the distal fine ash fractions here and in section 4 are not larger than the error in (1) and so not the only major source of uncertainty.
 We remark that the vertical structure of the ash cloud along its centerline does not differ much between the three source types considered in this section (not shown). However, there are significant differences compared with Figure 12 in that the downward protruding tongue of ash visible in Figure 12 is less pronounced in the bent-over case and is almost nonexistent in the vertically rising and point-source cases. This suggests that it is due to ash released at low levels.
5.3. Subgrid Diffusion
 As stated in section 2, NAME includes a parameterization of subgrid diffusion to account for turbulent mixing due to unresolved eddies. In addition to parameterizing the effect of three-dimensional (3-D) turbulence, NAME also includes a parameterization for the dispersive effects of lower frequency “meandering” motions which are larger than the 3-D turbulence but which are not resolved by the input wind field from the UM. The meander applies only in the horizontal and has a much larger effect than the horizontal component of the 3-D turbulence. Both the horizontal meander and the subgrid diffusion due to the 3-D turbulence are assumed to have constant values in the free troposphere. The root mean square (RMS) velocity associated with vertical component of the 3-D turbulence (“vertical diffusion”), σw, is 0.1 m s−1 [e.g., Schumann et al., 1995; Muschinski et al., 2001], and the time scale associated with this process, τw, is 100 s. This gives an eddy diffusivity, Kw = σw2τw, of 1 m2 s−1 [e.g., Kurosaki et al., 1996; Satheesan and Krishna Murthy, 2002]. The RMS velocity and time scale associated with horizontal meander are respectively σu = 0.8 m s−1 and τu = 4 hours, which give an eddy diffusivity, Ku, of approximately 9200 m2 s−1 (H. N. Webster, private communication, 2011). Thus, in 24 hours the standard deviation of the distance traveled by a model particle due to horizontal meander, , is approximately 40 km whereas that due to vertical diffusion is approximately 415 m. These distances are small compared with the distances the particles travel due to the resolved motions yet not so small as to be insignificant. In particular, these subgrid processes potentially allow particles to experience possibly very different resolved wind speeds thus increasing the spread of the ash cloud.
 It is possible to produce an ash cloud that penetrates farther south over Britain (compared with the ash cloud in Figure 10a) by increasing the level of horizontal meander by an order of magnitude. However, this comes at the expense of a greatly increased horizontal extent of the ash cloud for which there is no observational evidence. Nor is there any other evidence for significantly higher levels of horizontal meander in the free troposphere. Indeed, when horizontal meander is neglected, Figure 17 shows that NAME captures more of the horizontal structure of the ash cloud than in Figure 10a. In particular, the width of the ash cloud is reduced and so agrees better with the satellite data in Figures 2 and 4. In addition, the split in the ash cloud between approximately 10°W and 15°W into two narrow aligned bands resembles that in Figure 2. The effect of reducing the vertical diffusion to negligible levels has little qualitative or quantitative effect on the dispersion. The combined effects of negligible vertical diffusion and no horizontal meander do not differ significantly from the results shown in Figure 17. Note that the local maximum column loading in Figure 17 is higher than in Figure 10: Intuitively one may expect that less diffusion would result in higher column loadings as the results indeed suggest. Conversely, the mass of the southern branch differs from that in section 4 by less than 1%; this is consistent with reduced subgrid diffusion leading to less lateral spreading of the ash cloud. The estimated distal fine ash fractions for this case are given in Table 3: As expected when calculated from the mass as in section 4.1, the value is almost the same for those simulations with subgrid diffusion, whereas the distal fine ash fraction that is based on the local maximum column loading shows significant differences.
Figure 18 shows the vertical cross section of the ash cloud along its centerline for the simulation with no subgrid diffusion. Compared with Figure 12, we see that the absence of subgrid diffusion leads to an ash cloud that is less deep (in the vertical) than is the case when subgrid diffusion is included and closer to the lidar observations in Figure 7. This is confirmed by a typical layer depth (shown in Figure 13) which is less than that seen in the model results in section 4. However, it is still often deeper than the observed ash cloud. It is possible that the limited vertical resolution of the wind field from the driving NWP model (the UM) is responsible for the greater-than-observed depth of the model ash cloud.
5.4. Particle Size
 We consider the effect of the particle size on the dispersion statistics. In Figure 19 we show the effect of using a fixed particle diameter for all the model particles, that is, we conducted four simulations each equivalent to the simulation in section 4 except with d = 1 μm, d = 10 μm, d = 20 μm, or d = 30 μm. Comparing Figures 19a–19d we see that the simulations with d = 1 μm and d = 10 μm are almost indistinguishable from Figure 10a, whereas Figures 19c and 19d are significantly curtailed in horizontal extent (especially Figure 19d for which d = 30 μm). A further simulation (not shown) with d = 100 μm showed that the ash cloud traveled less than 100 km from the source. Taken together, this suggests that the ash cloud that was observed over northern Britain is dominated by particles with diameters less than 30 μm. This is consistent with the observed particle size distributions in the vicinity of (8°W, 60°N) which show a peak in the range 3–10 μm [Johnson et al., 2012]. The local maximum column loadings in the simulations with d = 1 μm, d = 10 μm, and d = 20 μm do not differ significantly from that in Figure 10a (for d = 1 μm, there is a small increase in the spatial extent of the ash cloud, whereas for d ≥ 20 μm there is a decrease) whereas that in Figure 19d is significantly less, suggesting that there is a substantial amount of fallout of particles with d = 30 μm. However, when comparing any two simulations in Figure 19, the differences in the mass are larger than the differences in the column loadings. For example, the mass in the simulation with d = 10 μm is 89% that of the simulation with d = 1 μm, indicating that even for d = 10 μm there is some deposition. For comparision, the mass of the southern branch in the default simulation in section 4 is 67% of the simulation with d = 1 μm.
 For d = 1 μm and d = 10 μm, the vertical structure of the ash cloud along its centerline (not shown) is similar to that in Figure 12 (though with higher concentrations) except that the downward protruding tongue of ash is less pronounced, particularly for d = 1 μm. This is not the case for d = 20 μm: There is almost no ash above 6 km in this case, and the concentration of ash in the downward protruding tongue is considerably higher than in Figure 12. For d = 30 μm the ash is entirely concentrated close to the surface and at very low values.
5.5. Combined Sensitivities
 In this section we combine some of the sensitivities discussed above. We start by considering three cases: the three source types discussed in section 5.2 combined with the revised eruption column heights (given in Table 2 and discussed in section 5.1). Figure 20 shows the effect of these changes: All three cases exhibit ash clouds whose northern and southern branches extend farther than the ash cloud shown in Figure 16 (analogous to the difference between Figures 10a and 15). A comparison of the three plots in Figure 20 with each other shows that material is transported farther south by a point source, followed by the vertically rising case and the least amount in the bent-over case (as in Figure 16). Conversely, more material is transported along the northern branch in the bent-over case, followed by the vertically rising case and the least amount by a point source (again as in Figure 16). This suggests, as in section 5.2, that ash in the southern branch originates primarily high up in the eruption column and ash in the northern branch comes primarily from lower down in the eruption column. However, it is important to note that the ash in the northern branch has a larger travel time than the ash in the southern branch regardless of source type.
 We now modify the three cases discussed above by changing the amount of subgrid diffusion as in section 5.3 (i.e., negligible vertical diffusion and no horizontal meander). Figure 21 shows the effect of these changes along with a uniform source with both revised eruption column heights (see Table 2) and negligible subgrid diffusion (i.e., combining the changes of sections 5.1 and 5.3). In keeping with the differences between Figures 10a and 17, the ash clouds in Figures 21a–21d are all narrower than the respective simulations with subgrid diffusion (i.e., Figures 15 and 20) and closer to the satellite images in Figure 4. However, Figures 21a–21c do not show any evidence of the ash cloud splitting into two aligned narrow bands around (12°W, 61°N) as in Figures 17, 19a, 19b, and 21d. Figures 21b–21d all show the maximum column loadings occurring in the vicinity of (8°W, 60°N) whereas that in Figure 21a lies at the southern extent of the ash cloud. Moreover, in Figure 21a there is also a significant secondary peak around (6°W, 57°N). The spatial variation in the column loading in Figure 21a is closer to that seen in the observations (see especially Figure 6) albeit with differences in the exact location of the ash cloud. The estimated distal fine ash fraction for each case is given in Table 3. These results highlight their sensitivity to the amount of subgrid diffusion in the free troposphere and possibly its spatial variation.
 The results of this study suggest that NAME captures many qualitative and quantitative aspects of the ash cloud generated by the eruption of Eyjafjallajökull on 14 May 2010. In common with all models of the distal ash cloud which do not treat the near-source processes in detail and which use an effective source, NAME is sensitive to uncertainties in the effective source properties, namely, emission rate, near-source fallout, vertical profile of emissions, and particle size distribution. In addition, the cumulative effect of small errors in the modeled meteorology along the trajectory of the ash cloud cause timing and positional errors common to all dispersion problems regardless of the source type. Indeed, the case considered here clearly showed errors in the position of the ash cloud relative to the observations, in particular in the position of the local maximum column loading in the vicinity of (8°W, 60°N), which remained largely unchanged by any of the sensitivity tests in section 5 except for the combination of a point source, revised eruption column height and no subgrid diffusion (see Figure 21a). The results of this study show that material that is released high in the eruption column tends to move southward, whereas material that is released low in the eruption column tends to move northward. A similar picture was noted during the initial phase of the eruption in April 2010 [Devenish et al., 2012; Dacre et al., 2011] when material released high or low in the eruption column moved in different directions over northern Europe. However, it is important to note that what has been presented here is not a steady state and that the shape of the ash cloud is the result of a complex interaction with the wind field over many days. As discussed in section 4.1, NAME predicts that a similar mass fraction of ash reaches the southern branch of the distal ash cloud to that in the satellite retrievals. However, this is not true of the mass fractions that reach the northern branch and the ash cloud immediately to the west of the volcano. The reasons for this are not clear, and the structure of the ash cloud remained largely unchanged by any of the sensitivity tests described in section 5. The most likely sources of error are in the resolved meteorology, the height of the eruption column, and its interaction with the wind.
 We estimated the fraction of fine ash that survives into the distal ash cloud basing our estimates on either the mass in the southern branch of the distal ash cloud, which we compared with satellite retrievals, or the local maximum column loading, which we compared with two independent aircraft measurements. The former method and the latter method, when based on the aircraft lidar measurements, produce results of similar order of magnitude which are also consistent with an earlier phase of this eruption and from other eruptions [Devenish et al., 2012; Dacre et al., 2011; Rose et al., 2000]. Those based on the aircraft CAS instrument produced values which were larger by approximately a factor of 3; interestingly, if the distal fine ash fraction is calculated using the local maximum column loading from the satellite retrievals, similar values are obtained. In addition to this uncertainty in the observations, when all the various sensitivity analyses in section 5 are taken into account, it is not surprising that there is some variation in the estimated distal fine ash fraction: The ratio of the largest to smallest value is smaller when using the areal mass of the ash compared with that using the using the local maximum column loading. Perhaps this is not surprising since one might expect the local maximum column loading to be more sensitive than its integrated value: As the model ash cloud becomes narrower, so the maximum column loading increases, as shown in sections 5.3 and 5.5. In this regard, we remark that the magnitude of prescribed subgrid diffusion can have a significant impact on the width (and depth) of the ash cloud. A similar result was noted during the initial phase of the eruption [Devenish et al., 2012], although there the effect of vertical diffusion was considerably more pronounced than in this study. These results indicate a need for better measurements of (subgrid) diffusion coefficients in the free troposphere and better understanding of what governs their spatiotemporal variation.
 The results of section 5 show that the areal mass in the distal ash cloud is sensitive to the vertical profile of the effective source as well as the height of the eruption column (which is used to determine the mass emission rate). As shown in section 5.2, a larger fraction of the total ash cloud is lost through deposition when particles are released uniformly over the depth of the eruption column compared with a point source at the top of the eruption column; particles that are released closer to the ground will clearly reach the ground sooner than similarly sized particles released higher in the eruption column. Another factor in the variability of the estimated distal fine ash fraction is the fact that the ash cloud splits, with part of the ash cloud transported northward over Iceland and another part transported southward over northern Britain. As shown in section 5, the fraction of material transported in each direction can be sensitively dependent on the source height and its vertical profile as well as the modeling of the structure of the wind field. Compared with the satellite imagery, the model does not predict a northern branch of the ash cloud that reaches as far to the north. Furthermore, the model predicts too much mass in the near-source region on the west side of Iceland and not enough mass in the northern branch on the east side of Iceland. Clearly, the area over which the mass is calculated will affect the value of the distal fine ash fraction: e.g., comparisons with the satellite retrievals can be made between the southern branches, the northern branches, or combined. Since the mass fraction of the ash in the southern branch is similar to that in the satellite retrievals, we chose to base our estimates of the distal fine ash fraction on a comparison of the southern branch. However, it is arguable that the mass of the distal ash cloud over a wider region (e.g., the southern and northern branches together) should be used: Estimates then lead to distal fine ash fractions that are approximately a factor of 3–4 larger.
 The wide range of estimates of the distal fine ash fraction suggests that further ways of constraining the near-source fallout are required. As discussed in section 2, the state of the atmosphere at the time of the eruption can have a significant effect on the height of the eruption column, and hence the inferred mass emission rate, and the wind clearly plays an important role in determining the vertical profile of the eruption column. The volcano itself, though, can exhibit rapid “pulsing,” as was the case for Eyjafjallajökull, which could mask some of the effects of the atmosphere on the source profile and mean that the effective source may sometimes be better represented by a uniform profile.
 The distance traveled by the ash cloud is clearly sensitive to the size of the ash particles (as shown in section 5.4) with the best qualitative agreement between the observed ash cloud and the modeled ash cloud occurring for those simulations which contain sufficient numbers of particles with diameters less than 20 μm. While there is considerable uncertainty in the size distribution, what is important for long-range modeling of volcanic ash is that the density and the particle size distribution together give the correct sedimentation velocity. To some extent, this justifies our assumption of neglecting particles with diameter larger than 100 μm, though it should be pointed out that truncating the size distribution at 20 μm or 30 μm rather than 100 μm would lead to different values of the areal mass of the ash cloud and hence the estimate of the distal fine ash fraction.
 Several studies have shown the importance of aggregation in removing fine ash prematurely from the eruption column and proximal ash cloud [Sparks et al., 1997; Durant and Rose, 2009; Durant et al., 2010]. The amount of ash that is transported to large distances is determined both by the total amount of fine ash generated by explosive fragmentation and by the factors which determine the efficiency of aggregation. Fragmentation of magma is typically greater in silica-rich magmas, most likely due to the much smaller bubble sizes in the magma fragments, and in hydrovolcanic eruptions where extra fragmentation is thought to take place due to thermal shock as hot magma fragments are cooled by mixing with water [Sparks et al., 1997, section 8.5.1]. As a generality, more powerful and intense eruptions occur in more silica rich magmas [MacDonald, 1972, chapter 10; Walker, 1973], but there is great variability. The eruption of Eyjafjallajökull was of trachyandesite magma whose silica content is intermediate between rhyolite and basalt. Its behavior is likely to be more typical of silica-rich rhyolitic magmas than basalt magmas and so would be expected to generate abundant fine ash [Rose and Durant, 2009], and this indeed seems to be the case. Strong interactions with the glacier were confined to the first few days of the eruption in mid-April, and so hydrovolcanic fragmentation was not a factor in the period under consideration here (mid-May). The difference between mid-April and mid-May in the role of glacier melting may have affected aggregation rates. If abundant water was involved in the mid-April eruptions, then aggregation could well have been more efficient than in the magmatic phase (post 18 April 2010).
 The sensitivity tests outlined in section 5 have provided some insight into the physical processes that govern long-range ash dispersion in the atmosphere. By adjusting various parameters within some range of uncertainty, we have demonstrated that the model can produce results that are closer to the observations. For example, with an increased eruption column height (as in Table 2) the ash cloud spreads farther over Scotland in accordance with the observations. When a more realistic source profile is used, subgrid diffusion is reduced, or the particle size is reduced, a narrower ash cloud is produced that agrees better with the observations. However, no single change to the model captures the observations better than any other change, and this highlights the potential importance of temporal variations in the source parameters and spatiotemporal variations in the meteorology especially in the magnitude of subgrid diffusion. This wide range of plausible scenarios inevitably leads to a wide range of near-source fallout estimates using the method we have presented above. The near-source fallout is of central importance to the quantitative modeling of long-range volcanic ash dispersion in the atmosphere. By comparing model results with observations, we have provided a method for estimating this fraction, but, as we have also demonstrated, it is subject to a large degree of uncertainty arising from many factors.
 Airborne data were obtained using the BAe-146-301 Atmospheric Research Aircraft (ARA) flown by Directflight Ltd. and managed by the Facility for Airborne Atmospheric Measurements (FAAM), which is a joint entity of the Natural Environment Research Council (NERC) and the Met Office.