Attracting structures in volcanic ash transport

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Atmospheric Environment 48 (2012) 230e239

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Atmospheric Environment

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Attracting structures in volcanic ash transport

Jifeng Peng*, Rorik PetersonDepartment of Mechanical Engineering, University of Alaska Fairbanks, PO Box 755905, Fairbanks AK 99775, USA

a r t i c l e i n f o

Article history:Received 26 January 2011Received in revised form18 May 2011Accepted 19 May 2011

Keywords:Volcanic ash transportAviationAttracting structures

* Corresponding author.E-mail address: [email protected] (J. Peng).

1352-2310/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.atmosenv.2011.05.053

a b s t r a c t

Volcanic ash clouds are a natural hazard that poses direct threats to aviation safety. Many volcanic ashtransport and dispersion (VATD) models have been developed to forecast trajectories of volcanic ashclouds and to plan safety measures in the events of eruptions. Predictions based on these models areheavily dependent on accuracy of wind fields and initial parameters of ash plumes. However, these dataof high accuracy are usually difficult to obtain, leading to possible inaccurate predictions of ash cloudstrajectories using VATD models. In this study, a new method is developed to predict volcanic ashtransport. In contrast to many existing VATD models that simulate the evolution of volcanic ash clouds,the new method focuses on the overall properties of the wind field in which volcanic particles aretransported and correlates particle motion to the attracting structures that dictate the transport. Asdemonstrated in the study of Eyjafjallajökull eruption in Iceland in April, 2010, these structures act asattractors in the atmosphere towards which volcanic ash particles are transported. These attractingstructures are associated with hazard zones with high concentrations of volcanic ash. The advantages ofthe method are that the attracting structures are independent of particle source parameters and are lessprone to inaccuracy in the wind field than particle trajectories. The new approach provides the hazardmaps of volcanic ash, and is able to help improve long-term predictions and to plan flight route diver-sions and ground evacuations.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Explosive volcano eruptions often release volcanic ash cloudsinto the atmosphere, which consist of tephra (submillimeter-sizedrock particles), water vapor and other gases such as carbon dioxide(CO2), sulfur dioxide (SO2), hydrogen sulfide (H2S), etc. Ash particlesfrom volcano eruptions are transported by wind to thousands ofkilometers away, or even over 10,000 km from their source forsome fine particles (Ram and Gayley, 1991). Volcanic ash plumescan reach over 20 km in altitude above sea level (Holasek et al.,1996), well into and above the cruising altitude of commercialaircrafts, thus they pose significant threats to aviation safety. Arecent analysis of aircraft encounters with volcanic ash cloudsindicates that there were 129 probable encounters between 1953and 2009 (Guffanti et al., 2010). Furthermore, 79 of these encoun-ters reported various degrees of damage to either airframes orengines, though no crash was caused. An encounter with volcanicash can be costly, such as the 2000 encounter with a cloud fromMiyakejima that resulted in $ 12 million of damage to a singleaircraft (Tupper et al., 2004). Near ground surface ash and its

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deposits can be dangerous as well. Exposure to SO2 and H2S cancause many health problems including eye irritation, bronchitis,and sore throat (Delmelle et al., 2002). When inhaled, ash particlesadversely affect the lungs, causing respiratory problems such asasthma, bronchitis, and silicosis (Horwell and Baxter, 2006).Although volcanic ash sometime has a beneficial long-term effecton agricultural soil, deposits have a short-term deleterious effect oncrops (Self, 2006).

The threat to aviation imposed by volcanic ash was recentlylearned by many after the Eyjafjallajökull eruption in Icelandduring April and May 2010. The explosive eruption that started on14 April 2010 generated plumes of an estimated 250 million m3 oftephra, reaching over altitude of 10 km above sea level (GlobalVolcanism Program, 2010). Transported by wind, ash clouds fromthe eruption reached thousands of kilometers away and resulted ina week-long closure of a vast airspace in Europe, during which over100,000 flights were canceled, causing an unprecedented disrup-tion in air travel and a loss of billions of dollars revenue for airlines(Wall and Flottau, 2010).

The decision of closing airspace was in part based on scientificpredictions on the volcanic ash transport. During 1990s, nineVolcanic Ash Advisory Centers (VAACs) were established aroundtheworld to issue specialized advisories to the aviation community.Over the past decades, many volcanic ash transport and dispersion

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Fig. 1. (a) The 2D FTLE field at 9 km altitude above sea level. (b) the attractingstructures (red), which are the level set of largest values of FTLE, superimposed withvelocity vector field at 9 km altitude.

J. Peng, R. Peterson / Atmospheric Environment 48 (2012) 230e239 231

(VATD) models have been developed for use by these VAACs andother response centers, with examples including MEDIA(Piedelievre et al., 1990), HYSPLIT (Draxler and Hess, 1998x), PUFF(Searcy et al., 1998), CANERM (D’Amours et al., 1998), and NAME(Jones et al., 2007). Coupled with Numerical Weather Prediction(NWP) data, these models are able to predict volcanic ash transportand dispersion in the atmosphere, providing scientific basis forauthorities to plan hazard responses.

Although these VATD models are remarkably useful, there arestill many limitations on wider applications of these models. Thebiggest challenge is the prediction accuracy, which depends mainlyon two factors. First, many VATD models use Lagrangian particletrajectories, whose accuracy is highly dependent on the quality ofNWP data, especially its spatial and temporal resolution. Comparedwith reference trajectories, the mean error for ash particle trajec-tories over 36 h travel time can be as high as 35% (Stunder, 1996).Using finer-grid NWP data would improve prediction, however,high-resolution NWP simulations generally require considerableamount of time. The delay in the availability of high quality NWPdata deters fast and accurate ash transport prediction.

Another factor that restrict wider applications on real-timemonitoring and response is the lack of accurate descriptions of theinitial eruptive plume. The initial parameters are critical to thesuccess of many VATD models, because even short-term predic-tions heavily depend on descriptions of initial plume, e.g. maximalheight, particle size and number density distribution, etc. However,it is usually difficult to collect these data, especially during the earlyhours after the eruption, when a fast response is desired. Eruptivevolumes normally have distinct flow characteristics at variousheights and therefore comprise distinct layers with different ashproperties (Kieffer, 1984). Moreover, values of these sourceparameters may change during eruption (Mastin et al., 2009).Coordinated and multidisciplinary efforts have been used toimprove the accuracy of source parameters, by establishing corre-lation between these parameters from well-documented pasteruption events (Mastin et al., 2009).

Therefore, the utilization of conventional VATD models islimited by the accuracy of both the physical properties of initialash plume and the wind field in which particles are transported.To overcome these limitations, a new approach is proposed in thisstudy to provide a fast, accurate prediction on volcanic ashtransport that is less dependent on the quality of NWP data andinitial ash plume parameters. In contrast to the previous methodsthat simulate evolution of ash clouds, the new method focuses onthe overall properties of the wind field in which volcanic particlesare transported and correlates particle motion to some under-lining structures that dictate the transport. To be more specific,the method uses a dynamical systems approach to identifyattracting structures in the wind field. These structures act asglobal attractors to which particles move towards. These struc-tures, as the underlining structure of wind transport process, areindependent to particle source parameters. To demonstrate theutilities and advantages of the method, it is applied to theEyjafjallajökull eruption in Iceland during April 2010. An existingVATD model, PUFF, is used to simulate the transport of a volcanicash cloud, as the comparison to attracting structures. The studydemonstrates that particles from the PUFF simulation areattracted to the attracting structures. Due to turbulent dispersion,volcanic ash particles are scattered near the attracting structures.These structures coincide with high particle concentration,therefore indicating hazard regions. Comparisons of analysesbased on two NWP data sets with different resolutions demon-strated that the attracting structures are less prone to inaccuracyin the wind field than particle trajectories, especially in long-termprediction.

2. Methods

2.1. Weather forecast data

Two NWP wind data sets generated for the Eyjafjallajökulleruption are used in this study. These two data sets are defined atthe same atmospheric space, but are from different weather fore-cast models and are also different in both spatial and temporalresolutions.

The first data set is the Global Forecast System (GFS) simulationobtained from the National Weather Service on a 0.5-degree grid.This original data was then interpreted onto a polar stereographicgrid centered at (55�N 0�E) with longitudinal and latitudinalresolution of 100 km. The domain had a 41 (longitudinal) � 44(latitudinal) grid. Vertical nodes are available at 26 altitudes from100 to 34000 m. The temporal resolution is 6 h.

The other data set is generated by a custom simulation of theWeather Research and Forecasting (WRF) model performed at theUniversity of Alaska Fairbanks Arctic Region Supercomputing Center

J. Peng, R. Peterson / Atmospheric Environment 48 (2012) 230e239232

specifically for this analysis. The simulation is initialized using theglobal GFS simulation at 0000 Coordinated Universal Time (UTC)April 14 2010, and runs for 138 h with an output at every 3 h. Thedomain is a polar stereographic grid (199� 199) with a resolution of20 km centered on (55�N 0�E). There are 25 vertical levels spanning100e20000 m in elevation. This data set has higher spatial andtemporal resolutions than the GFS data. All the analysis in this studywas based on this high-resolution WRF data except for the compar-ison with the low resolution GFS data in the Discussion section.

2.2. Finite time Lyapunov exponents and attracting structures

Instead of modeling volcanic ash transport based on the evolu-tion of the plume from an eruption, a dynamical systems approachis used to identify global coherent structures in the wind field. Weuse the finite time Lyapunov exponents (FTLE) to locate attractors ina velocity field (Haller and Yuan, 2000; Haller, 2001; Shadden et al.,2005). FTLE describe the maximal rate of extension of a lineelement advected in the flow. In other words, FTLE measure themaximal separation/converging rate of nearby tracers in the flow.

Assuming that a tracer particle in the wind field follows thewind velocity, i.e., vðt; t0; x0Þ ¼ wðt; xÞ. Given its velocity v(t; t0, x0)at a series of time instants, the Lagrangian trajectory of a tracerparticle x(t; t0, x0) is the solution of

dxðt; t0; x0Þ=dt ¼ vðt; t0; x0Þ (1)

with the initial condition xðt0; t0; x0Þ ¼ x0. Notice that location andvelocity of the particle is Lagrangian marked by (t0, x0) whereas thevelocity of wind field is Eulerain.

To avoid confusion between particle trajectories and spatialcoordinates, the notation f is used for particle trajectories and x forspatial coordinates. By following particle trajectories over a dura-tion of time T after initial time t0, a flow map fT

t0 ðxÞ is obtained thatmaps particles from their position x at initial time t0 to their posi-tion at time t ¼ (t0 þ T), as

fTt0ðxÞ ¼ xðt0 þ T; t0; x0Þ ¼ f0

t0ðxÞ þZt0þT

t0

uðs; xðs; t0; x0ÞÞds (2)

From the flow map fTt0 ðxÞ, we compute the FTLE as

sTt0ðx0Þ ¼ 1jT jlnlmaxðt; x0; t0Þ (3)

where lmax(t, x0, t0) is the square root of the largest eigenvalue ofthe right CauchyeGreen deformation tensor

D ¼"dfT

t0ðxÞdx

#*$

"dfT

t0ðxÞdx

#(4)

In Eq. D ¼ dfTt0 ðxÞ=dx is the deformation gradient tensor. The

denotation ‘*’means the transpose of a tensor and ‘,’ represents thetensor product.

The physical meaning of FTLE sTt0 ðx0Þ can be explained byfollowing particle trajectories over a duration of time T after initialtime t0. Consider the trajectories for a slightly perturbed particle aty ¼ xþ dxð0Þ at time t0. After a time interval T, this perturbationbecomes

dfTt0ðxÞdx

��ðxi;jðtÞ;yi;jðtÞÞ ¼

26664xiþ1;jðt0 þ TÞ � xi�1;jðt0 þ TÞ

xiþ1;jðt0Þ � xi�1;jðt0Þxi;jþ1ðt0 þ TÞ �

yi;jþ1ðt0Þ �yiþ1;jðt0 þ TÞ � yi�1;jðt0 þ TÞ

xiþ1;jðt0Þ � xi�1;jðt0Þyi;jþ1ðt0 þ TÞ �

yi;jþ1ðt0Þ �

dxðTÞ ¼ fTt0ðyÞ � fT

t0ðxÞ ¼ dfTt0ðxÞdx

dxð0Þ þ O��dxð0Þ��2�

¼ Ddxð0Þ þ O��dxð0Þ��2� (5)

By dropping high order terms of dxð0Þ, the magnitude of theperturbation is given by

kdxðTÞk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidxð0Þ$D$dxð0Þ

qThe magnitude of the perturbation is maximal when dxð0Þ is

aligned with the eigenvector associated with the maximumeigenvalue of D. That is, if lmaxðDÞ is the square root of themaximum eigenvalue of D, then

kdxðTÞk ¼ lmaxðDÞkdxð0Þk (6)

Therefore the FTLE represents the maximum linear growth rateof a small perturbation,

sTt0ðxÞ ¼ 1jTjlnlmaxðDÞ ¼ 1

jT jlnkdxðTÞkkdxð0Þk (7)

The analysis above canbeapplied to two-dimensional (2D)aswellas three-dimensional (3D) velocity fields. FTLE fields are scalar fieldsand visualized as contour plots. The ridges on FTLE contour plots,which have local maximal values, indicate the structures that havethe maximal separation and converging rate. Intuitively, for a 2DFTLE, a ridge line is a curve normal towhich the topography is a localmaximum. Similarly, for a 3D FTLE, a ridge is a surface normal towhich the topography is a localmaximum.The ridges, i.e., level sets ofmaximal separation and converging rate, represent either repellingstructures or attracting structures. When fluid particle trajectoriesare integrated forward in time (i.e. T> 0), ridges of the forward FTLEare repelling structures. These structures are said to be repellingbecause particles on either side of the structures are stronglyrepelled. Conversely, backward-time integration of fluid particletrajectories (T < 0) calculate backward FTLE whose ridges revealattracting structures, alongwhich fluid particles on either side of thestructures are attracted to them. Only the attracting structures areused in this study because they are of interests to this study.

2.3. Computation of FTLE and identification of attracting structures

The input, the wind velocity field from NWP data, consists ofa time sequence of velocity data defined on a mesh of discretepoints. To calculate FTLE sTt0 ðxÞ, a region of interest is first definedon which FTLE will be calculated. A Cartesian mesh is constructedover the region and used as the initial grid. For every point on theinitial grid, its trajectories from x(t0) to x(t0 þ T) is calculated. A 4thorder RungeeKutta algorithm is used to calculate the integration. Acubic interpolation scheme is used to compute the velocity atarbitrary positions (Lekien and Marsden, 2005).

After trajectories are calculated, the (right) Cauchy-Greendeformation tensor dfT

t0 ðxÞ=dx is evaluated for every node on theinitial mesh. For a 2D velocity field, the deformation tensor ata given node x ¼ (xi,j, yi,j) is given by

xi;j�1ðt0 þ TÞyi;j�1ðt0Þyi;j�1ðt0 þ TÞyi;j�1ðt0Þ

37775 (8)


The largest eigenvalue of D is calculated to determine FTLE sTt0 ðxÞat every node. This process can be repeated to calculate FTLE fordifferent t0 at a series of time frames. Interested readers can refer toLeiken et al. (2007) for 3D FTLE calculation.

The extraction of ridges from FTLE fields can be accomplished bya variety of ad hoc methods including thresholding or gradientsearches of the FTLE field to identify local maxima. Shadden et al.(2005) derives more rigorous criteria; however, for practicalpurposes, identification of attracting structures fromwell-resolvedFTLE fields is relatively insensitive to the implemented method ofextraction. In this study, the ridges are extracted visually from FTLEcontour plots.

2.4. Comparison with PUFF

The analysis described above determines the attracting struc-tures of a wind field. To demonstrate the utilities of these attractingstructures in ash transport, we compare these structures withprediction from an existing VATD model: PUFF (Searcy et al., 1998).The PUFF model has been used in studying many volcano eruptionevents (Dean et al., 2004; Webley et al., 2010) and is briefly intro-duced as follow.

Fig. 2. Temporal evolution of the attracting structures and the motion of an ash plume at theeruption of 12 h starting at 1800 UTC on April 14th at Eyjafjallajökull. Turbulent dispersionvelocity w(t). The 4 different frames are at (a) 12, (b) 24, (c) 36 and (d) 48 h after the start of tto the attracting structures, the evolution of ash particles follows that of the attracting stru

PUFF is an individual particle based Lagrangian model. Aneruption plume is represented by a large number of ash particlesand the evolution of an ash cloud can be simulated by trajectories ofall the particles. After convection simulation, particle concentrationis calculated by counting tracer particles in unit volume ofatmosphere.

The motion of ash particles are due to a combination of severalfactors including wind advection, turbulent dispersion and gravi-tational deposition. The velocity of a tracer particle v(t) is expressedas

vðtÞ ¼ wðtÞ þ zðtÞ þ sðtÞ (9)

where w(t), z(t), and s(t) represent wind velocity from NWP data,turbulent dispersion velocity at length scales smaller than winddata grid, and particle deposition velocity.

The dominant component of particle velocity is the windadvection velocity w(t), which is available from NWP data.However, due to the large size of grid used in NWP models, nor-mally of O(1) w O(100) in km, there exists wind turbulence atlength scales smaller than wind data grid. The motion of ashparticles due to this sub-grid scale turbulence is estimated bya random walk process.

altitude of 9 km. Particles are shown as gray dots. The ash plume is from a continuousand gravitational fallout are neglected and volcanic ash is only transported by wind

he eruption. Particles are attracted to the attracting structures. After the particles movectures on which they are located.


zðtÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nK=d

q(10)

in which r is a unit vector with random direction, n is the dimen-sionality of space, K is the effective diffusivity, and d is the time stepused in integration. In addition, due to gravity, ash particles fall toground over time. And the deposition process is estimated by theStokes’ law

sðtÞ ¼ 29

�rp � ra

�m

r2g; (11)

where rp and ra are ash particle and air densities, m is the airviscosity, r is particle radius, and g is the gravitational constant. Ashparticle density varies for different composition where as airdensity and viscosity change with altitude. For simplicity, it iscommon to assume a constant for the term (rp�ra) g/m ¼ 1.08 � 109 m�1 s�1 (Searcy et al., 1998).

The particle grain size strongly influence the setting velocity.Larger particles with radius greater than 100 mm generally fall outto the ground within hours from over 10 km initial height (Durantand Rose, 2009). Usually they do not experience regional or globaltransport and their threat to aviation is limited. On the other hand,smaller particles, especially the so-called Class III particles(Koyaguchi and Ohno, 2001), which are fine ash less than tens of

Fig. 3. Temporal evolution of the attracting structures and the motion of an ash plume with tin Fig. 2, but has its motion subject to turbulent dispersion. Figures (a) to (d) represents the sscatter around the attracting structures. But attraction from the attracting structures keeps

microns, can stay airborne for days and pose greatest hazard toaviation (Mastin et al., 2009). In this study, we only focus on thistype of particles. For these small particles, the magnitudes for threevelocity components compare as wðtÞ >> zðtÞ >> sðtÞ.

3. Results

To better present the attracting structures and to demonstratetheir roles in the volcanic ash transport, the analysis is first appliedto a 2D wind field from the WRF data at the altitude of 9 km. Themotion of volcanic ash was assumed to be limited at this altitude. Inother words, all vertical motion for volcanic ash particles wereneglected. A snapshot of the FTLE field of this 2D wind field isplotted in Fig. 1a. Multiple attracting structures are extracted fromthe FTLE contour and are plotted in Fig. 1b together with thevelocity vector field of that time instant. It is clear that the onecannot determine the structures directly from the Eulerian velocityfield. In other words, the pattern of instantaneous velocity fielddoes not correlate directly with the geometry of attractingstructures.

The attracting structures represent attractors in the flow fluid.Because the flow field is time-dependent, the attracting structuresalso evolve with time. To illustrate that volcanic ash particles areattracted to these structures over time, we simulate an eruption

urbulent dispersion. Particles are shown as gray dots. The ash plume is the same as thatnapshots at the same time instants as in Fig. 2. Turbulent dispersion causes particles toparticles limited to a band along the attracting structures.


starting at 1800 UTC on April 14, 2010 at the location of Eyjafjal-lajökull. An ash plume from a continuous eruption of 12 h isreleased. Turbulent dispersion and gravitational fallout areneglected in the volcanic ash transport and particles are onlytransported by wind velocity w(t). Fig. 2 plots the motion of theparticles at four difference instants: 12, 24, 36 and 48 h after thestart of the eruption. Attracting structures at these time instants arealso plotted in Fig. 2. From the evolution of attracting structures andash particles, it is clearly seen that the particles move towards oneof the attracting structures. At 12 h after eruption, the ash plumeerupted earlier moves towards the structure (Fig. 2a). At 24 h,a portion of the ash plume already arrives at the structure. Only thelater erupted portion does not coincides with the structure, but is invery close proximity (Fig. 2b). After all the particles reach theattracting structure, they stay on and evolve exactly with theattracting structure (Fig. 2c and d). Therefore the structures here actas attractors in the flow field. Neglecting turbulent dispersion andgravitational fallout, ash particles are attracted to the attractors ofthe flow field in which they are transported. After a certain timeinterval during which particles move from the source to reach theattracting structures, their motion strictly follows the evolution ofattracting structures on which they are located.

Besides advected by wind, volcanic ash particles are alsotransported by turbulent dispersion, which is accounted for in the

Fig. 4. Distribution and concentration of ash particles for the same eruption but different va(c) K ¼ 50 km2 h�1, and (d) K ¼ 75 km2 h�1. Particles are shown as dots with their colors instart of the eruption. The concentration is normalized by the highest concentration in Figturbulent dispersion effective diffusivity. The location of high particle concentration coincid

PUFF model. The same ash plume as that shown in Fig. 2 is simu-lated by PUFF and ash particles are transported by wind withvelocity w(t) as well as by turbulent dispersion with velocity z(t).The motions of ash particles are shown in Fig. 3 at the same timeinstants as in Fig. 2, together with the attracting structures. Due tothe turbulent dispersion, ash particles are scattered over a bandregion along the attracting structures. Also, the ash plume stillfollows the motion of the attracting structures, with the distribu-tion of particles centered at the attracting structures. Comparedwith results without turbulent dispersion (Fig. 2), it can beconsidered that dispersion causes particles that would have stayedon the structures to deviate from them. However, because they arecontinuously subject to the attraction from the attracting struc-tures, they move back towards the attracting structures, either theoriginal structure fromwhich theymove away or a different nearbystructure. Therefore, their motion is limited to a certain distanceaway from these structures.

To show that volcanic ash particles are scattered aroundattracting structures in a band region whose width depends on thestrength of turbulent dispersion, the same eruption event wassimulated for four different turbulent dispersion effective diffu-sivity. Shown in Fig. 4 are particle location and concentration 48 hafter the start of the eruption for these four different values ofdispersion diffusivity. The concentration is normalized by the

lues of turbulent dispersion effective diffusivity. (a) K ¼ 10 km2 h�1, (b) K ¼ 25 km2 h�1,dicating the concentration at the location. Snapshots are at the frame of 48 h after the. 4a. Particles are scattered into a larger region with smaller concentration for largeres with the attracting structures.


highest concentration in Fig. 4a.With larger diffusivity, particles arescattered into a larger regionwith smaller concentration. However,for all values of turbulent diffusivity, particles distributions arecentered on the attracting structures, with highest particleconcentration coinciding with the attracting structures. Becausethe attracting structures are independent of turbulent dispersion,they can be used as the indicator of location of highest particleconcentration. The value of turbulent diffusivity only changes thedistribution of particles around the attracting structures.

As described in the PUFF model, another major factor in ashtransport is gravitational fallout. The 3D FTLE analysis is used toextract the 3D attracting structures based on the 3D wind velocityw(t). Instead of being 2D curves on a plane, the attracting structuresare surfaces in 3D space. The 3D attracting structures at the samefour time instants are shown in Fig. 5. A 3D ash plume from thesame eruption as above and released at an initial altitude of 11 kmis simulated and the particle motion shown in Fig. 5. Only smallClass III particles (Koyaguchi and Ohno, 2001) are considered in theanalysis and larger particles that fall to ground in a short time areneglected. The grain size distribution in the simulation is assumedto have a mean of 10 mm and a standard deviation of 2 mm. Eventhough ash particles gradually falls in altitude, they are stillattracted to the attracting structures. The reason is that windtransport is the attracting mechanism for particles tomove towardsthese structures. In essence, there is competition between windtransport, which move particles towards the attacking structures,

Fig. 5. 3D evolution of attracting structures and particles (dots). Only major structures arParticles are still attracted to the attracting structures though they gradually fall in altitude

and gravitational settling, which move particles away from them.So if there were no gravitation, particles would strictly movetowards the 3D attracting structures and stay on them. On the otherhand, gravitational settling can be interpreted as a disturbance tothe wind transportation and it makes particles move away (verti-cally) from the 3D attracting structure. So considering thata particle falls out from the 3D attracting structure at an altitudedue to gravitation, the wind will keep carrying it back towards theattracting structures at a new altitude.

4. Discussion

Most of the existing VATD models aim to predict the exactevolution of volcanic ash clouds in the atmosphere which stronglydepends on the accuracy of input parameters such as properties ofinitial ash plume, NWP wind data, turbulent dispersion, etc. Themethod proposed in this study, in contrast, does not directly predictexact trajectories of ash clouds in the atmosphere. Rather, it look forproperties of the most dominant driving force of ash transport, thewind field. By considering the wind field as a dynamic system, thenew method determines the attracting structures in generaltransport processes in the wind field, which also dictates volcanicash transport in particular. By comparing with simulations ofvolcanic ash motion using an existing VATD model PUFF, the studyshows that volcanic ash clouds are attracted to these structures.Demonstrated by the study on Eyjafjallajökull eruptions in April

e shown. Particles are subject to wind velocity, turbulent dispersion and gravitation..


2010, volcanic ash clouds move towards the attracting structuresafter eruption. After clouds reach these attracting structures, theyfollow the evolution these structures.

The attracting structures based on FTLE are more coherent thanarbitrary structures and are robust to even large errors of thevelocity field as long as these errors are local in time (Haller, 2002).The errors in calculating attracting structures are much less thanthose of particle trajectories, thus enable the analysis to generatereliable results from even approximate, low resolution NWP data.All the results shown above are based on the high-resolution WRFdata. To demonstrate attracting structure are more robust to errorsin velocity data than particle trajectories, we repeat the analysisusing the low resolution GFS data set. Fig. 6 shows the attractingstructures as well as the motion of the volcanic ash clouds from thesame eruption in Fig. 3, using the GFS data. Compared with Fig. 3,the same initial plume apparently have very different trajectoriespredicted by PUFF, especially in long-term. This highlights thestrong dependence of ash clouds trajectories on the accuracy ofNWP data and the difficulty to predict the accurate trajectories ofash clouds. However, attracting structures from these two data setsare much more similar. Therefore, the new method is less prone toerrors in velocity data and can achieve higher accuracy from a lowresolution NWP data compared with Lagrangian trajectories. Notethat in the two PUFF simulations using different data sets, the ash

Fig. 6. Temporal evolution of the attracting structures and the motion of an ash plume withare shown as gray dots. Figures (a) to (d) represents the snapshots at the same time instantespecially in the later time instants shown in (c) and (d). However, the attracting structures aattracting structures compared with Fig. 3.

plume moves towards to the different portion of the sameattracting structure. So the attracting structures represent regionsacting as attractors in thewind field which canwithstand the errorsin ash particle Lagrangian trajectories.

The attracting structures are solely based on the wind field,which is the most dominant factor in the transport of volcanic ashclouds. It is well known that other factors also contribute to thetransport, with turbulent dispersion and gravitational fallout beingthe major ones. These two factors are not included when deter-mining the attracting structures. Even though their presence in ashtransport alters the motion of particles from transport solely bywind field, particles are still attracted to the attracting structures.Turbulent dispersion and gravitational fallout can be considered asdisturbance to the ideal case of particles falls exactly on theattracting structures. For the turbulent dispersion, the disturbanceis to scatter volcanic ash particles around the attracting structures,with the largest concentration located on the attracting structures.As for the gravity, the disturbance is to cause particles deviate awayfrom intersections of the 3D attracting structures at the altitudethey are passing. As particles fall, they are attracted to the struc-tures at the new altitude.

Besides turbulent dispersion and gravitational fallout, there areadditional factors affecting volcanic ash particle transport. Forexample, even if they are perfectly spherical and have a uniform

turbulent dispersion as in Fig. 3, but based on the low resolution GFS data set. Particless as in Fig. 3. The trajectories of volcanic ash particles are different from that in Fig. 3,re more similar to those in Fig. 3. Particles are attracted to different portions of the same


density, due to their inertia, finite-sized particles do not strictlyfollows the flow velocity in a flow field (Maxey and Riley, 1983).Their trajectories are generally different from the simple integra-tion of flow velocity over time, especially when they pass throughregions with hyperbolic stagnation points (Babiano et al., 2000).The effect of particle inertia on FTLE and the coherent structureshas been studied (Sapsis and Haller, 2009; Peng and Dabiri, 2009).In volcanic ash transport, because the sizes of particles havea much smaller length scale than their trajectories, their inertiaeffect can be neglected. Due to the same reason, the irregular sizeof tephra should not playing a significant role in their transport,either.

The computational cost of FTLE and the attracting structures issufficiently low to guarantee fast analysis and response. Usinga single processor personal computer, the 2D analysis took severalminutes and the 3D analysis less than an hour for analyses in thisstudy. The computation can speed up using high performancecomputers. Because NWP models usually need longer computa-tional time, apparently the availability of high-resolution NWP datais the limit in time consumption of the analysis, not the FTLEcomputation. In addition, to prevent large errors in particletrajectories, existing VATDs constantly use updated NWP data torecalculate. However, because the attracting structures are lessprone to errors in NWP data than particle trajectories, the newmethod does not require as frequent recalculations, with thepotential to overcome the limitation on the availability of frequentupdate of high-resolution NWP data. All the analysis in this study isbased on singe NWP predictions on April 14, 2010. Using a singleNWP prediction as the input, results of attracting up to 48 h areshown to be more consistent than trajectories of ash plumes. Thefurther attracting structures can be calculated using updated NWPdata when it is available.

One subject of future studies is to compare the attractingstructures with satellite images of ash clouds from volcano erup-tions, especially ones with clear images of old plumes days aftereruptions. Another focus will be to implement the method in real-time monitoring and to assist aviation authorities to make plans onflight diversions and airspace closure. Oneway to apply themethodin practice is to use the attracting structures in wind transport asthe skeleton of volcanic ash hazard zones, where high concentra-tion of ash clouds are located. Then the size of the zones can bedetermined by the turbulent dispersion strength. Even though ashclouds may not cover all attracting structures in the predictions,they should be still avoided by aircrafts because, if there is any errorin predictions of clouds trajectories, actual clouds would beattracted to other parts of the attracting structures than thesimulation predicts. So using attracting structure provides a largermargin of error in locating hazard regions.

After a sufficient time is allowed for particles to reach theattracting structures, ash particles are located on or very near theattracting structures (due to turbulent dispersion), and theirevolution follows that of the attracting structures for an extendedtime. In the long-term, the attracting structures are more reliablethan particle trajectories, which make them more valuable forextended predictions of ash clouds with high accuracy. Therefore itis a very valuable complement to existing VATD models, which aremore accurate for near-term prediction, and provides a robust wayfor long-term prediction of volcanic ash transport. The ability oflong-term prediction with high accuracy can be particularly usefulto identify safe airspaces for flights after a volcano eruption haselapsed. Because areas far way from the attracting structures can beconsidered less dangerous or even clear, the new method canpotentially reduce the size of hazard zones predicted by conven-tional VATD models and let authorities to sooner resume aviationactivities.

Acknowledgement

The authors thank Dr. Don Morton and Kayla Harrison forgenerating the WRF model simulations. This study is supported byan award from the Alaska Space Grant Program to J.P.

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Attracting structures in volcanic ash transport