Woodhouse, MJ, Hogg, AJ, Phillips, JC, & Sparks, RSJ (2013).

19 Pages·2017·1.55 MB·English

Woodhouse, M. J., Hogg, A. J., Phillips, J. C., & Sparks, R. S. J.

(2013). Interaction between volcanic plumes and wind during the 2010
Eyjafjallajokull eruption, Iceland. Journal of Geophysical Research:
Solid Earth, 118(1), 92-109. https://doi.org/10.1029/2012JB009592
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JOURNALOFGEOPHYSICALRESEARCH:SOLIDEARTH,VOL.118,92–109,doi:10.1029/2012JB009592,2013
Interaction between volcanic plumes and wind
during the 2010 Eyjafjallajökull eruption, Iceland
M. J. Woodhouse,1 A. J. Hogg,1 J. C. Phillips,2 and R. S. J. Sparks2
Received2July2012;revised1October2012;accepted12November2012;published31January2013.
[1] Estimatesofvolcanicsourcemassflux,currentlydeducedfromobservationsofplume
height,arecrucialforashdispersionmodelsforaviationandpopulationhazard.Thisstudy
addresses the role of the atmospheric wind in determining the height at which volcanic
plumesspreadintheatmosphereandtherelationshipbetweensourcemassfluxandplume
heightinawindfield.Wepresentapredictivemodelofvolcanicplumesthatdescribesthe
bendingoveroftheplumetrajectoryinacrosswindandshowthatmodelpredictionsarein
accord with a dataset of historic eruptions if the profile of atmospheric wind shear is
described. The wind restricts the rise height of volcanic plumes such that obtaining
equivalent rise heights for a plume in a windy environment would require an order of
magnitude increase in the source mass flux over a plume in a quiescent environment. Our
model calculations are used to calibrate a semi-empirical relationship between the plume
height and the source mass flux that explicitly includes the atmospheric wind speed. We
demonstratethatthemodelcanaccountforthevariationsinplumeheightobservedduring
the first explosive phase of the 2010 Eyjafjallajökull eruption using independently
measured wind speeds and show that changes in the observed plume height are better
explained by changing meteorology than abrupt changes in the source mass flux. This
study shows that unless the wind is properly accounted for, estimates of the source mass
flux during an explosive eruption are likely to be very significant underpredictions of the
volcanic source conditions.
Citation: Woodhouse,M.J.,A.J.Hogg,J.C.Phillips, andR.S.J.Sparks(2013),Interactionbetweenvolcanicplumesand
windduringthe2010Eyjafjallajo¨kulleruption,Iceland,J.Geophys.Res.SolidEarth,118,92–109,doi:10.1029/2012JB009592.
1. Introduction adopted a precautionary policy of ash avoidance, with no
concentration of ash in the atmosphere considered safe for
[2] A major hazard arising from explosive volcanic erup- aircraft. However, the disruption to transatlantic and
tions is the injection of volcanic ash into the atmosphere Europeanaviationduringthefirstweekofexplosiveactivity
and its subsequent dispersion and deposition. The largest at Eyjafjallajökull (14–18 April 2010) led to a relaxation of
eruptions can inject large volumes of ash at stratospheric
thispolicyinEurope,withtheUKCivilAviationAuthority
levels which have been responsible for global temperature
and Eurocontrol introducing ash concentration thresholds
changes and ash deposition over thousands of square kilo- forcommercialairtraffic[Bonadonnaetal.,2012].Ashcon-
meters with major infrastructural and societal impacts [Self, centrations below 2mgm(cid:1)3 are considered safe for flights
2006].
[ICAO, 2010; CAA, 2011; Langmann et al., 2012], while
[3] The weakly explosive phase of the 2010 eruption of flight operations at higher concentrations require a Safety
Eyjafjallajökull (magnitude volcanic explosivity index
Caseacceptedbynationalregulators[CAA,2011].Typically,
[NewhallandSelf,1982]of3)causedsignificantdisruption
SafetyCaseshavebeenacceptedforashconcentrationsupto
to aviation over European airspace, highlighting the severe 4mgm(cid:1)3 [CAA, 2011]. The introduction of ash concentra-
andextensive consequences of smaller eruptions to interna-
tionlevelsplacesincreaseddemandsonatmosphericashdis-
tional infrastructure and transport. Modern commercial jet
persion modeling for airspace management during volcanic
engines are susceptible to damage from low concentrations
crises[Bonadonnaetal.,2012].Crucialcomponentsoffore-
ofash,andairframescanbesubjecttoabrasionfromthesus-
castsofthemovementofashintheatmospherearethelevel
pended particulates. Prior to the 2010 Eyjafjallajökull erup- of neutral buoyancy of the volcanic plume in the stratified
tion, the International Civil Aviation Organization (ICAO) atmosphere(the‘plumeheight’)andthemassfluxofmaterial
released from the volcano. Accurately determining these
1SchoolofMathematics,UniversityofBristol,Bristol,UK. source conditions is an essential requirement for airspace
2SchoolofEarthSciences,UniversityofBristol,Bristol,UK. management during volcanic crises [Bonadonna et al.,
Correspondingauthor:M.J.Woodhouse,SchoolofMathematics, 2012].
UniversityofBristol,BristolBS81TW,UK.([email protected]) [4] The sourcemass fluxof avolcanicplume is currently
impossible to measure directly but is fundamentally related
©2012.AmericanGeophysicalUnion.AllRightsReserved.
2169-9313/13/2012JB009592 to the plume height as a result of the dynamics of buoyant
92
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
plumeriseintheatmosphere[Mortonetal.,1956].Thishas well as the vertical [Hewett et al., 1971]. The wind-driven
led to inversion methods to estimate the source mass flux plume model introduces an additional entrainment coeffi-
based on the approximate quarter-power relationship to the cient, denoted here by k , which parameterizes the entrain-
w
plume height in a density-stratified environment such as mentparalleltotheplumeasitbendsoverinthecrosswind.
the atmosphere [Morton et al., 1956; Wilson et al., 1978; Together, these models can be used to describe the rise of
Sparks, 1986; Sparks et al., 1997; Mastin et al., 2009]. volcanic eruption columns in a wind field [Bursik, 2001;
A small dataset of historic eruptions where the source DegruyterandBonadonna,2012].
duration, total erupted mass, and plume neutral buoyancy [8] Eyjafjallajökull is a stratovolcano on the south
height are known has been used to calibrate this relation- coast of Iceland, with a summit at 1666m above sea level
ship [Wilson et al., 1978; Sparks, 1986; Sparks et al., [Siebert and Simkin, 2002–2012]. The 2.5km-wide summit
1997;Mastinet al.,2009].Thisdataset(andthe calibrated caldera is covered by ice around 200m (and up to 400m)
plume height-mass flux relationship) is inevitably biased thick[Magnússonetal.,2012].Theexplosivephasesofthe
by the disproportionate number of large eruption events, Eyjafjallajökull eruption began on 14 April 2010 beneath
for which volcanic ash deposits are more easily assessed, the ice cover. Volcano-ice interactions rapidly melted
while there is less data available for the more frequent throughtheicecover,withdistinctcauldronsformingduring
yet smaller eruptions. Furthermore, plumes from smaller 14–16 April [Magnússon et al., 2012]. An ash-poor plume
eruptions are more strongly affected by atmospheric con- from Eyjafjallajökull was observed on the morning of
ditions, in particular atmospheric winds, during the ascent 14 April [Arason et al., 2011; Höskuldsson et al., 2011;
of material in the atmosphere. Wind-affected volcanic Magnússon et al., 2012], with a dark ash-rich plume rising
plumes are therefore under-represented in the historical from around 1830 UTC on 14 April and continuing until
eruption dataset, so application of calibrated inversion 18April. The volcano-ice interaction during the first explo-
methods to lower source mass flux plumes produced by sive phase (14–17 April) produced very fine-grained ash
smaller magnitude volcanic activity could be significantly [Dellino et al., 2012]. Between 18 April and 4 May, the
in error. We have re-analyzed the historic eruption dataset eruption intensity fell, but explosive activity resumed on
andfindthatvolcanicplumeheightdependssystematically 5May and continued with a varying intensity until 18 May
on atmospheric wind speed for a given source flux, and (the second explosive phase) [Gudmundsson et al., 2011;
have explored the underlying relationships using an integral Höskuldsson et al., 2011] producing fine-grained ash-rich
modeling approach accounting for the thermodynamic plumes. From18May,the eruption intensity declined,with
exchange of heat between volcanic ash, volcanic gas, and continuous activity ending on 22 May 2010. Some of the
entrained atmospheric air, and the entrainment of horizontal fine-grained ash, produced predominately during the first
momentum due to the atmospheric wind [Hewett et al., 1971; explosive phase and the early part of the second explosive
Bursik, 2001]. phase (5–7 May) [Stevenson et al., 2012], was carried over
[5] The key physical process controlling the ascent of a large distances by atmospheric winds, although most was
turbulentbuoyantplumeistheentrainmentofenvironmental deposited near to the volcano as aggregates [Bonadonna
fluid into the body of the plume by turbulent eddies on the et al., 2011; Stevenson et al., 2012].
plumemargins.Turbulencewithintheplumethenefficiently [9] In section 2, we derive an integral model to describe
mixes the entrained fluid, altering the density contrast volcanic plumes, composed of solid pyroclasts, magmatic
between the plume and the surrounding environment. In a gases, and entrained air, rising in a windy atmosphere. We
stratified environment, the plume density may eventually demonstrate that the predictions of the integral model for
match that of the environment, at which point the vertical thedependenceoftheplumeriseheightonthesourcemass
component of the buoyancy force on the plume vanishes. flux adequately describe observations from the historical
Thisisthelevelofneutralbuoyancy.Inertiacausestheplume record when wind shear is included in the integral model.
to rise above this level of neutral buoyancy, and the plume The integral model predictions are used to calibrate a new
density here exceeds the environment. The material in the semi-empirical relationship, akin to those of Sparks et al.
plume therefore falls back and begins to spread laterally [1997] and Mastin et al. [2009], that explicitly includes
aboutthelevelofneutralbuoyancy. the atmospheric wind speed. In order to assess the role
[6] Integral models of turbulent buoyant plumes [Morton of phase changes of water and the release of latent heat
et al., 1956] represent the entrainment process through a on the ascent of wind-blown volcanic plumes, we derive
simpleentrainmentvelocitywhich,inthemostbasicmodels, an integral model of moist volcanic plumes in a windy,
islinearlyproportionaltothecenterlinevelocityoftheplume moist atmosphere in section 3. We discuss the implications
withthecoefficientofproportionalityknownastheentrain- of our modeling in sections 4 and 5. In section 4, we com-
mentcoefficient,heredenotedbyk.Suchmodelshavebeen pare results of our integral plume models to a time series
s
utilizedwidelytoquantitativelydescribetheriseofindustrial of observed plumes rise heights during the first explosive
andenvironmentalplumes[Woods,2010].Anintegralmodel phase of the 2010 Eyjafjallajökull eruption. We demon-
ofvolcaniceruptioncolumnscanbeformulatedbyexplicitly strate that the inclusion of atmospheric wind in the integral
includingadescriptionofthethermodynamicsofheattrans- plume model allows observed variations in plume height
ferbetweensolidpyroclasts,magmaticgases,andentrained to be described, with significant implications for the esti-
air[Woods,1988]. mation of the source mass flux. We then comment on
[7] Plumeriseinacrosswindhasbeenmodeledbyinclud- the consequences of our results for ash dispersion model-
ing momentum conservation in the horizontal direction as ing and aviation, and on the estimation of the source mass
93
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
The adiabatic cooling term [Glaze and Baloga, 1996] is
typically much smaller than the cooling produced by the
entrainmentofambientatmosphericair,makingonlyasmall
contribution to the heat budget. Furthermore, it is not clear
how the presence of solid pyroclasts affects this adiabatic
cooling,particularly athighsolidsconcentrationneartothe
vent. While a significant proportion of the gas issuing from
volcanic vents is water vapor [Sparks et al., 1997], in this
section we assume there is no change of phase of the water
vapor, an assumption that is relaxed in section 3 where we
develop an extension of the dry wind-blown plume model
to describe the moisture content of the plume and the sur-
roundingenvironment.
[12] Models of the fallout of pyroclasts from the rising
Figure 1. A model of a volcanic plume in a crosswind. A
plumehavebeenproposedforplumesinquiescentenviron-
Cartesiancoordinatesystemisfixedwithxdenotingthedis-
ments [Ernst et al., 1996; Woods and Bursik, 1991; Sparks
tance downwind from the vent and z denoting the vertical
et al., 1997]. However, it is not currently known how the
distance from the vent. Equations describing the plume dy- interaction with the wind modifies the empirical settling
namics are derived in a plume-centered coordinate system,
models [Ernst et al., 1996; Bursik, 2001] that are used to
with s denoting the curvilinear distance (arc length) from
describe sedimentation of particles from plumes rising in
the vent along the plume axis, and θ(s) is the angle of the
quiescentenvironments.Plumemodelswhichincludeparti-
centerline with respect to the horizontal. A cross-section of
cle fallout in quiescent environments have shown that the
theplumenormaltothecenterlinehasareaAandcircumfer-
loss of mass associated with fallout has only a small effect
ence Ω. The wind speed is denoted by V(z), the centerline
ontheriseheightattainedbybuoyantplumesunless fallout
speed of the plume is U(s), and U denotes the entrainment
e occurs before pyroclasts have thermally equilibrated with
velocity at the plumemargins.
the gases in the plume [Woods and Bursik, 1991; Sparks
etal.,1997].Foreruptionsproducingpyroclastslargerthan
a few millimeters, there is a significant relaxation time to
flux for explosive volcanic eruptions, in section 5. Finally,
thermal equilibrium and pyroclasts may fall out before
in section 6, we present some concluding remarks.
thermal equilibrium is reached, reducing the supply of heat
(and therefore buoyancy) to the eruption column [Woods
and Bursik, 1991; Sparks et al., 1997]. Therefore, for
2. Integral Model of Dry Volcanic Eruption
coarse-grained eruption columns, particle fallout may play
Columns in a Crosswind
an important role in determining the plume rise height. In
[10] Anintegralmodelforasteadyvolcaniceruptioncol- contrast, since thermal equilibrium occurs rapidly for small
umninawindfieldcanbederivedbycombininganintegral grainsizes(within1kmoftheventforpyroclastsofdiameter
model of pure plumes in a horizontal wind [Hewett et al., uptoapproximately0.4cmejectedat100ms(cid:1)1)[Woodsand
1971] with an integral model of volcanic eruption columns Bursik, 1991; Sparks et al., 1997], the fallout of pyroclasts
in a quiescent atmosphere [Woods, 1988]. The volcanic haslittleeffectonfine-grainederuptioncolumns.Weexpect
plumemodelofWoods[1988]extendstheclassicalintegral thermal equilibration of the fine-grained pyroclasts and the
modelof turbulent buoyantplumes [Mortonetal.,1956]to gasestoalsooccurrapidlyinawind-blownplume,soexpect
include essential features of volcanic eruption columns. In the fallout of pyroclasts to have only a secondary effect on
particular, aspects ofthe multiphase character oftheplume, the rise height attained by the plume. We therefore neglect
which is a mixture of solid pyroclasts and gases, and the thefalloutofpyroclastsinourmodel.
thermodynamics of heat exchange between these phases [13] Theentrainmentofenvironmentalairintothebodyof
are included in the mathematical descriptionof the plume. theplumethroughtheactionofturbulenteddiesisparameter-
[11] The mathematical model presented here shares the ized empirically by an entrainment velocity that is directed
same entrainment formulation [Hewett et al., 1971] as that normaltothelocalplumeaxis(Figure1).Inawindyenviron-
applied by Bursik [2001] to volcanic plumes. However, ment,wheretheplumetrajectorydeviatesfromthevertical,
while Bursik [2001] adopts the quiescent plume model of theentrainmentvelocityhascontributionsfromthedifferen-
GlazeandBaloga[1996],ourmodelutilizestheformulation tialvelocitiestangentialandnormaltotheaxisoftheplume.
of Woods [1988] which additionally incorporates the influ- This can be modeled [Hewett et al., 1971] with an entrain-
ence of the solid pyroclasts on the bulk plume properties mentvelocitygivenby
(i.e.,theplumedensityandheatcapacity)andsoisapplica-
ble for large explosive eruptions where the solid content of U ¼kjU(cid:1)Vcosθjþk jVsinθj; (1)
e s w
the plume near the vent is high and the heat content of the
pyroclasts and transfer of heat from solids to entrained air whereUistheaxialcenterlinevelocityoftheplume,Visthe
has an important effect on the plume dynamics [Woods, horizontal velocity of the wind, θ is the local angle of the
1988; Sparks et al., 1997]. The model of Woods [1988] plumeaxistothehorizontal,k istheentrainmentcoefficient
s
neglects the contribution of the adiabatic cooling of the gas due to the motion of the plume relative to the environment,
phase in the energy conservation equation that appears in and k is the entrainment coefficient due to the alignment
w
the model of Glaze and Baloga [1996] for vapor plumes. of the wind field with the local normal to the plume axis.
94
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
In the absence of atmospheric wind, V=0, the entrainment pressure changes is then included. The total energy of the
velocity (1) reduces to U =kU, and therefore, k is the plume is the sum of the bulk enthalpy, kinetic energy and
e s s
entrainment coefficient for plumes rising in a quiescent potential energy, and the total energy changes due to the
environment [Morton et al., 1956; Woods, 1988]. When entrainment of atmospheric fluid. Conservation of energy
incorporated into an integral model of buoyant plumes in a is therefore given by
uniform crosswind, this form for the entrainment velocity Z (cid:2) (cid:3)
(1)isabletoreproduceplumetrajectoriesobservedinlabora- d r C TþU2þgz UdA
toryexperiments[Hewettetal.,1971]. dsI (cid:2)p 2 (cid:3) (6)
[14] A mathematical description of the variation of the ¼ r C T þUe2þgz U dΩ;
steady eruption column with distance from the volcanic a a a 2 e
source is formulated in a plume-centered coordinate system
within a Cartesian frame of reference (Figure 1). We let z where C and C are the specific heat capacities at constant
p a
denote the height of the plume, x denote the distance from pressure of the bulk plume and the atmospheric air,
the vent in the downwind direction and s denote the curvi- respectively.
linear distance from the vent along the centerline of the [18] If we assume top-hat profiles for r, U, and T (i.e.,
plume. Therefore, x and z are related to s through these quantities have constant values within the plume and
vanish outside the plume boundary) and that cross-sections
dx dz
¼cosθ; ¼sinθ: (2) of the plume normal to the axis are circular with radius
ds ds R(s), thentheintegralsin(3)–(6)canbeevaluatedtogive
(cid:4) (cid:5)
d
[15] Turbulencewithinthebodyoftheplumeensuresthat rUR2 ¼2r U R; (7)
thematerialremainswellmixed,andpropertiesoftheerup- ds a e
tioncolumncanbedescribedbytime-averagedbulkquanti- (cid:4) (cid:5)
d
ties,with the timeaveraging performed overatimeinterval rU2R2sinθ ¼ðr (cid:1)rÞgR2; (8)
ds a
greaterthantheeddy-turnovertime[Woods,1988].Thebulk
density of the plume, denoted by r(s), varies due to the d (cid:4) (cid:5)
entrainment, mixing, and expansion of atmospheric air, ds rU2R2cosθ ¼2raUeRV; (9)
which has density r . The bulk temperature of the column
a
isdenotedbyT(s),whiletheatmospherictemperatureisT . (cid:2) (cid:2) (cid:3)(cid:3)
a d U2
Equations describing the variation of r(s), U(s), and T(s) rUR2 C Tþ þgz
are derived by considering conservation of mass, momen- ds (cid:2) p 2 (cid:3) (10)
U2
tum, and energy in cross-sections normal to the plume axis ¼2r RU C T þ e þgz :
withareaAandboundaryΩ(Figure1).Neglectingthefallout a e a a 2
ofsolidpyroclastsfromthecolumn,themassofthecolumn
increases due to the entrainment of atmospheric air at the Otherprofiles,forexample,Gaussiandistributions,couldbe
boundaryoftheplume,somassconservationdemands adopted to describe the variation of density, velocity, and
Z I temperaturewithintheplume.However,adoptingsuchpro-
d rU dA¼ r U dΩ: (3) files has little effect on the predictions of plume models in
ds a e
quiescentenvironmentsifthevalueoftheentrainmentcoef-
ficient is appropriately adjusted [Kaye,2008].
[16] An equation for the conservation of vertical momen- [19] ThemassfluxpQ,axialmomentumfluxpM,andthe
tum can be written using Newton’s second law, with the enthalpy flux pE of the eruption column are defined as
changeinverticalmomentumbalancingthebuoyancyforce,
Z Z Q¼rUR2;M ¼rU2R2;E¼rUR2C T: (11)
p
d
rU2sinθdA¼ gðr (cid:1)rÞdA: (4)
ds a The system of equations(7)–(10) canbe combined to give
Here it is assumed that deviations of the vertical pressure dQ Q
¼2r U pffiffiffiffiffiffiffiffi; (12)
gradient from hydrostatic and stresses are negligible. The ds a e rM
horizontal momentum of the column changes only due to
the entrainment of fluid from the windy environment, so dM ¼gðr (cid:1)rÞ Q2 sinθþ2r pQffiffiffiffiffiffiffiffiU Vcosθ; (13)
conservation of horizontalmomentum canbe written ds a rM a rM e
Z I dθ Q2 Q
d rU2cosθdA¼ r U VdΩ: (5) ds ¼gðra(cid:1)rÞrM2cosθ(cid:1)2raMpffirffiffiMffiffiffiffiffiUeVsinθ; (14)
ds a e
(cid:2) (cid:3)
dE U2 dQ M2 dQ
[17] It is most convenient to formulate the total energy ds ¼ CaTaþ 2esffidffiffisffiffi þ2Q2 ds
of the eruption column at distance s in terms of the bulk (15)
r M
enthalpy of the plume material [Woods, 1988], as the work (cid:1) aQgsinθ(cid:1)2r U Vcosθ;
done in expanding gaseous phases due to temperature or r a r e
95
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
where linearly in the stratosphere. The atmospheric pressure in
the Standard Atmosphere is assumed to be hydrostatic
(cid:7) (cid:7) [Gill, 1982],
(cid:7) (cid:7)
Ue¼ks(cid:7)(cid:7)MQ(cid:1)Vcosθ(cid:7)(cid:7)þkwjVsinθj: (16) dPa ¼(cid:1) gPa : (22)
dz R T
a a
[20] Thebulkdensityoftheplumeisrelatedtothedensity Thedensityoftheatmosphereisfoundbyassumingthatthe
of the solids pyroclasts, r, and the density of the gaseous
s atmospheric gases behave as ideal gases,so
phase [Woods,1988] as
P
1¼1(cid:1)nþnRgT; (17) ra¼RaTaa: (23)
r r P
s a
wherenisthemassfractionofgas,Paisthepressureofthe [22] The mathematical model is completed by providing
atmosphere, and Rg is the bulk gas constant of the plume. closure relations for the entrainment coefficients. Typically,
Notethatin(17),itisassumedthatthepressureintheplume the entrainment coefficient for buoyant plumes in quiescent
isinstantlyequilibratedwiththeatmosphericpressure.Con- environmentsistakentobeaconstant,withk (cid:3)0.09.How-
s
servationofsolidpyroclasts,withnoparticlefallout,allows ever, there is some evidence from laboratory experiments
the gasmass fraction to be determined as thatk isnotconstant[Kaye,2008]butvariestowardsacon-
s
stantvalueastheplumeevolvestowardsaself-similarform
Q
n¼1(cid:1)ð1(cid:1)n Þ 0; (18) [Kaminski et al.,2005; Carazzo et al.,2006].The variation
0 Q in the entrainment coefficient is related to the profiles of
wherezerosubscriptsdenotequantitiesatthevent.Thebulk plume velocity, buoyancy, and turbulent shear stress within
gasconstantandbulkheatcapacityatconstantpressurecan the plume, and an empirical expression for the entrainment
thenbe determined [Woods,1988; Scase,2009] with coefficient has been determined for plumes in a quiescent
environment [Carazzo et al., 2006]. In a cross flow, it is
R ¼R þ(cid:4)R (cid:1)R (cid:5)n0ð1(cid:1)nÞ; (19) likelythattheseprofilesarealtered.However,therehasbeen
g a g0 a nð1(cid:1)n Þ noinvestigationofthedetailedinfluenceofthewindonthe
0
variation of the entrainment coefficient. We therefore adopt
C ¼C þ(cid:4)C (cid:1)C (cid:5) ð1(cid:1)nÞ ; (20) a simple model [Woods, 1988] to represent the variation of
p a p0 a ð1(cid:1)n0Þ theentrainmentcoefficientastheeruption columndevelops
from a momentum-driven jet near the vent to a buoyant
whereRaandCaarethegasconstantandtheheatcapacityat plume, with the eruption column separated into distinct
constantpressureoftheair,respectively.Weassumethatthe regions.Inthenear-sourceregion,thematerialissuingfrom
magmaticgasattheventiscomposedpredominatelyofwater the vent is more dense than the atmosphere due to the high
vapor,sotakethebulkgasconstantatthesourcetobethegas concentration of particulates and is driven upwards as a
constantofwatervapor,Rg0=Rv,andthebulkspecificheat dense jet. The entrainment coefficient in this gas-thrust
capacity to be given by Cp0=n0Cv+(1(cid:1)n0)Cs, where Cv region is a function ofpthffiffieffiffiffiffidffiffiffieffinsity contrast [Woods, 1988]
and Cs are the specific heat capacities at constant pressure andistakentobek ¼ r=r =16.Theentrainmentofatmo-
ofwatervaporandthesolidpyroclasts,respectively. s a
sphericairinthegas-thrustregionreducesthebulkdensityof
[21] If observations of the atmospheric temperature and the eruption column and may lead to the column becoming
pressureareknown,theycanbeutilizedintheplumemodel,
buoyant. In this buoyant region, we take the entrainment
with interpolation between data points used to approximate coefficient k =0.09. There have been fewer investigations
the atmospheric conditions at points of integration. Here s
of appropriate entrainment models for plumes in a cross-
weuselinearinterpolationasthisdoesnotintroducepossibly
wind. A study of the sensitivity of model predictions for
spurious local extrema in the atmospheric fields. In the the rise height of volcanic plumes in a wind field to the
absence of atmospheric observations, we adopt the U.S. values assigned to the entrainment coefficients [Barsotti
StandardAtmosphere [COESA,1976]todescribe theatmo-
et al., 2008] has shown that variation in the entrainment
spherictemperatureandpressurefields,withtheatmospheric coefficients, within the range 0.09≤k ≤0.15 and 0.6 ≤
temperaturegivenby k ≤ 1.0 suggested by experimental invsestigations, results
8 w
<T (cid:1)mz; forz<H ; insignificant changes in the calculated plume heights. Here
T ðzÞ¼ Ta0(cid:1)mH ; forH ≤z1≤H ; (21) we take a constant entrainment coefficient kw=0.9 deter-
a : a0 1 1 2
T (cid:1)mH þlðz(cid:1)H Þ; forz>H ; mined from a series of laboratory experiments [Hewett
a0 1 2 2
etal.,1971].
whereTa0isthetemperatureatsealevel,mandlarethelapse [23] Examplesofsolutionsoftheintegralmodelforvolca-
rates of temperature in the troposphere and stratosphere, nic plumes in a crosswind, with atmospheric conditions
respectively, H is the altitude at which the tropopause modeledwiththeU.S.StandardAtmosphereandparameters
1
begins, and H is the altitude at which the stratosphere giveninTable1,areshowninFigure2.Initialconditionsfor
2
begins. Note that the temperature in the Standard Atmo- the integration of the governing equations are given in
sphere decreases linearly in the troposphere and increases Table 2. The atmospheric wind profile is modeled with a
96
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
Table1. ParametersEmployedintheDryVolcanicPlumeModel Furthermore, the enhanced entrainment of environmental
fluidintoaplumerisinginawindfieldresultsinamorerapid
Parameter Symbol Value Unit
rate of decrease in the density contrast between the plume
Atmosphericpressureatsealevel Pa0 100 kPa and the atmosphere, and the rise height of the plume in a
EADntemtnrsaoiistnypmhoeefrnistcocltoiedmefpfipyecrrioeacntultariesntsaatbsseeancleevoelfwind kTras0 21092.03090 Kkgm(cid:1)3 ccroonsssiwdeinreddisroctoatnisoenquoefntthlyerwedinudcedfi.elNd.otTehheerientwegerahlavpelunmoet
s
Entrainmentcoefficientduetowind k 0.9 model can be extended to include changing wind direction
Gasconstantofatmosphere Rw 285 JK(cid:1)1kg(cid:1)1 by introducing a third coordinate axis, the azimuthal wind
GGarasvcitoantisotannatloacfcveolelcraantiiocngasatvent Rgag0 94.6821 mJKs(cid:1)(cid:1)12kg(cid:1)1 angle, and an additional equation for the conservation of
momentumalongthisthirdaxis.Anexaminationofsolutions
Heightofstratosphere H 20 km
2 to the integral model in wind fields with varying direction
Heightoftropopause H 11 km
Lapserateoftemperatureinstratosphere l1 2.0 Kkm(cid:1)1 (not shown here) suggests that rotation of the wind vector
Lapserateoftemperatureintroposphere m 6.5 Kkm(cid:1)1 has little effect on the rise height of volcanic eruption
SSppeecciifificchheeaattccaappaacciittyyooffactomluomspnhaetrevent CCa 9196824 JJKK(cid:1)(cid:1)11kkgg(cid:1)(cid:1)11 columns since the entrainment velocity is dependent on the
p0 wind speed but not on the wind direction and a changing
winddirectionusuallydoesnotaddsignificantlytothelength
constantwindshearuptothetropopause,withconstantwind ofthetrajectoryoftheascendingplume.
speedV above,
1
(cid:8) 2.1. Comparisonof Model Predictions to Observations
V z=H ; forz<H ;
VðzÞ¼ V1; 1 forz≥H1: (24) [25] We have re-analyzed the record of plume rise height
1 1 and mass flux of historic eruptions [Sparks et al., 1997;
Mastin et al., 2009] to investigate the effect of atmospheric
[24] The solutions demonstrate increasingly bent-over wind.Forsomeoftheeruptions inthedataset,typicalwind
plume trajectories as the wind speed V increases.
1 speeds at the time of the eruption (as recorded in the
SmithsonianInstitutionGlobalVolcanismProgramdatabase
[Siebert and Simkin, 2002–2012]) can be estimated from
ECMWF reanalysis meteorological data (ECMWF ERA-
Interim data have been obtained from the ECMWF Data
Server) (Figure 3). There is a degree of scatter in the data,
some of which could be attributed to varying atmospheric
conditions, for example, the variation in atmospheric lapse
ratesandaltitude ofatmospheric layerswithlatitude,which
are known to influence rise heights of volcanic plumes
[Woods,1995;Sparksetal.,1997].Inaddition,byadopting
the wind speed at a single altitude to characterize the atmo-
spheric wind conditions, we are unable to describe atmo-
sphericwindstructures,suchasjetstreams,whichmayhave
a significant influence on the ascent of the plume [Bursik,
2001; Bursik et al., 2009]. However, despite these limita-
tions, we find that the dataset records a systematic depen-
dence of volcanic plume height on atmospheric wind speed
foragivensourcemassflux(Figure3).Inparticular,athigh
windspeedsinexcessof30ms(cid:1)1,plumeheightstendtobe
limitedtoaltitudesbelow15km.
[26] Thepredictionsofourmodelforthevariationofplume
heightwithsourcemassfluxforincreasingatmosphericwind
speed are shown in Figure 3. Here the atmospheric wind is
modeled as a linear shear flow in the tropopause with con-
stant wind speed above (24), and the atmospheric tempera-
ture is described using the U.S. Standard Atmosphere (21)
[COESA, 1976]. A range of exit velocities and vent radii
Figure 2. Calculated centerline trajectories of volcanic
Table2. SourceConditionsforExampleProfilesofDryVolcanic
plumesinacrosswind.Thewindistakentoincreaselinearly
PlumesinaCrosswind(Figure2)
inthetropospheretoaspeedV atheightz=11kmandhas
1
constant speed above. We take V1=0 (with Ws ¼0 as Variable Symbol Value Unit
defined in equation 26), V =10ms(cid:1)1 ( W ¼0:09 ),
V1=20ms(cid:1)1 (Ws ¼0:17), V11=30ms(cid:1)1 (Wss ¼0:26), CExoiltuamnnglteemperature Tθ0 10200 K
and V1=40ms(cid:1)1(Ws ¼0:34). The temperature profile of Exitvelocity U00 100 ms(cid:1)1
the atmosphere is modeled using the U.S. Standard Atmo- Gasmassfraction n 0.03
0
sphere [COESA, 1976]. The complete set of model para- Ventaltitude z0 0 m
Ventradius R 100 m
metersisprovidedinTable2. 0
97
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
Figure3. Theriseheightofaneruptioncolumn,H,asafunctionofthemassfluxofmaterialfromthe
volcanic vent, Q. A data set of historical eruptions [Sparks et al., 1997; Mastin et al., 2009] where the
mass flux of the eruption, Q, and rise height of the plume, H, can be independently estimated is used to
calibrate a scaling law relationship between rise height and mass flux [Sparks et al., 1997; Mastin
et al., 2009] (as given on the figure, for H measured in km and Q measured in kgs(cid:1)1). A representative
wind speed at an altitude of 10km can be assigned, in some cases, using ECMWFReanalysis data. The
data show a tendency for plume rise heights from small eruptions (source mass flux Q<108 kgs(cid:1)1) to
be reduced in high winds (wind speed V >20ms(cid:1)1). Predictions of the integral model of dry volcanic
1
plumesinacrosswindthatincreaseslinearlywithaltitudeuptoaspeedV atthetropopauseatanaltitude
1
of11km(denotedbytheblackdashedline)arecomputedusingtheU.S.StandardAtmosphere[COESA,
1976] to describe the temperature profile in the atmosphere, for a range of exit velocities and vent radii
(the source conditions employed are given in Table 3).
are employed as given in Table 3 together with the other Mastinetal.,2009]areunderpredictedbyanorderofmagni-
model parameter values used. The model predictions repro- tude[seealsoBursik,2001].
duce the expected quarter-power scaling between the rise
heightandthesourcemassflux,particularlyforlargesource 2.2. Relating Mass Flux and Rise Height
mass flux. A deviation from the approximate quarter-power for Wind-blown Plumes
scalingisobservedforsmallersourcemassflux,whichispar- [28] The transition from strong plumes that are little
ticularly apparent for low wind speeds, when the plumes affectedbythewindfieldduringtheirascenttoweakplumes
reach the tropopause where there is a discontinuous change withtrajectoriesthatarestronglybentovercanbequantified
in the atmospheric lapse rate. If a constant wind speed is usingadimensionlessparameter
adopted in the volcanic plume model, the model over-
sporeudrcicetsmtahsesrfleudxucwtihoennicnomplpuamreedritsoethheeiogbhstefrovrataionspse(ctihfieesde Wp¼(cid:11)g(cid:9)CpT(cid:1)kCs1a=T2aV(cid:10)Q(cid:12)1=4N1=4; (25)
calculationsarenotshownhere).However,whenthevertical ra0 CaTa
profile of wind shear is accounted for, there is improved
agreement between the model predictions and the observa- whereVisarepresentativewindspeed,r isthedensityof
a0
tionaldataset(Figure3). theplumeatthesource,TandT arethetemperatureofthe
a
[27] Curve fits calibrated to observations of historical plume and environment, respectively, at the source, Cp and
eruptions[Sparksetal.,1997;Mastinetal.,2009](Figure3) C are the specific heat capacities at constant pressure of
a
donotexplicitlyaccountforcrosswindsontheriseofvolca- the plume and environment, respectively, g is the accelera-
nic plumes. Figure 3 demonstrates the strong influence of tion due to gravity, and N is the buoyancy frequency of the
atmospheric winds on the ascent of volcanic plumes. For atmosphere. The parameter W represents the ratio of the
p
small and moderately sized eruptions, a strong crosswind horizontal wind speed to the vertical buoyant rise speed,
canlimittheplumeriseheightsuchthatthesourcemassflux assumingthewindspeedisuniformwithaltitude.However,
estimatedusingthecalibratedcurvefits[Sparksetal.,1997; taking a uniform wind may not be representative of
98
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
Table 3. Source Conditions Employed in Model Predictions for [29] An estimate of the effect of the shear rate on the rise
RiseHeightofVolcanicPlumesinaCrosswind(Figure3) height of volcanic plumes can be obtained from a simple
integral model of pure plumes rising in a linear shear cross
Variable Symbol Value Unit flow, as described in Appendix A. In the pure plume
Columntemperature T0 1200 K model, the multiphase character of volcanic plumes and
EExxiittavneglolecity θU0 01–500 ms(cid:1)1 thethermodynamicsofthegasexpansionarenotconsidered.
0 Furthermore, the atmosphere is assumed to be uniformly
Gasmassfraction n 0.05
Ventaltitude z0 0 m stratified. Numerical solutions for pure plumes in a linear
0
Ventradius R 1–500 m shear flow can be readily calculated and the rise height of
0
pureplumesdetermined(Figure4).Fromthenumericalsolu-
tions(asdetailedinAppendixA),arationalfunctionapprox-
imation can be used to describe the effect of the parameter
atmospheric winds. The atmospheric wind can be usefully
approximatedasalinearshearflowintheloweratmosphere, Ws on the rise height. We find that the rise height above
takingVðzÞ¼g_zwhereg_istheshearrateandzistheheightin theventiswelldescribedby
theatmosphere.Inashearflow,dimensionalanalysisshows 1þ1:373W
the appropriate dimensionless parameter measuring the H (cid:3)H01þ4:266W þ0:3s527W2; (27)
strengthofthewindfieldis s s
g_ V where H0 is the rise height of a pure plume in a quiescent
Ws¼N ¼NH1 ; (26) environment. This approximation adequately reproduces
1 thenumericalsolutionofthepureplumemodelforW <5
s
whereV =V(H )isthewindspeedatareferencealtitudeH (Figure 4), so the approximation is appropriate for typical
1 1 1
(e.g.,atthetropopause)(seealsoAppendixA).Wenotethat atmospheric conditions.
thedimensionlessparameterW dependsonlyonproperties [30] An approximation of the rise height for volcanic
s
of the atmosphere and is independent of the plume source plumesinaquiescentatmospherethatremainwithinthetro-
conditions.TheparameterW canbeinterpretedastheratio posphere can be found from a fit to data obtained from the
s
of the time scale of vertical motions, given by 1/N, to the integral plume model in a Standard Atmosphere as
timescaleofhorizontalmotions,1=g_.Thus,forW ≫1,hor-
izontal motion of a parcel of fluid in the plume, isnduced by H0(cid:3)0:318Q0:253; (28)
thewind,occursonshortertimescalesthantheverticalrise
for rise height H measured in kilometer and source mass
oftheparcelintheplumeandsotheplumetrajectorybends flux Q measured0in kilogram per second, which is similar
over in the wind, while for Ws ≪1, the vertical motion to the expressions obtained from fits to observational data
occurs on a shorter time scale than the horizontal motion
and there is little deviation of the plume trajectory from
the vertical. A similar dimensionless parameter has been
identified by Degruyter and Bonadonna [2012], where
the column-averaged wind speed and buoyancy frequency
are adopted. Here a local wind speed and reference height
are taken in order to represent the vertical shear profile of
theatmosphericwind.Solutionsoftheintegralplumemodel
in a crosswind demonstrate the controlling influence of W
s
(Figure 2). For explosive eruptions of the magnitude of
Eyjafjallajökull 2010 and a wind speed of V =40ms(cid:1)1 at
1
H =10km,theparameterW ¼0:4,takinganatmospheric
1 s
buoyancy frequency N=0.01s(cid:1)1. In order to obtain weak
plumes, W >1, very strong wind shear or weak atmo-
s
spheric stratification is required. However, variations in the
vertical rise speed, wind speed, and temperature profile
cause local variations in the plume strength. In particular,
astheplumedeceleratesasitnearsthelevelofneutralbuoy-
ancy,the wind fieldwill inevitablycausea bending over of Figure4. Theheightofneutralbuoyancyforpureplumes
theplumetrajectoryasthemaximumaltitudeisapproached in a linear shear flow as a function of the wind strength pa-
(Figure2).Furthermore,itisnotappropriatetorepresentthe rameterW (bluesolidline).Theheightofneutralbuoyancy,
s
windprofileasalinearshearthroughouttheatmosphere,and H,isnormalizedbytheheightofneutralbuoyancyforapure
for larger eruptions, with plumes that ascend above the tro- plumeinaquiescentenvironment,H .Theambientenviron-
0
posphere, there may be interaction with jet streams where ment is uniformly stably stratified. A rational function ap-
the wind speed is locally high [Bursik, 2001; Bursik et al., proximation, equation (27), with three fitting parameters,
2009]. While any profile of the wind could be used, for well describes the numerically determined relationship for
smallandmoderatelysizederuptionsthatdonotrisesignif- W ≤5(reddashedline).ValuesofWf estimatedforEyjafjal-
s s
icantly above the troposphere and where the wind field can lajökull 14–17 April 2010 using radiosonde measure-
betakentoincreaselinearlywithaltitude,theparameterW ments of the meteorology at Keflavik International Airport
s
is appropriate to assess the strength of the wind. [Oolman,2012] are marked (black points).
99
WOODHOUSEETAL.:VOLCANICPLUMESANDWIND
[Sparksetal.,1997;Mastinetal.,2009],andthepower-law linear shear crosswind in the intermediate regime where
scaling is close to the one-quarter power expected from the plume rise speed and wind speed are comparable.
dimensional analysis. The pre-factor in (28) is determined
from solutions of the integral model using the parameters 3. Integral Model of Moist Volcanic Eruption
giveninTable1andthesourceconditionsgiveninTable3, Columns in a Crosswind
andhasadependenceonthesourceconditions,inparticular
the temperature contrast between the plume and the atmo- [33] Theadditionofwatervaporintotheeruptioncolumn,
sphere. The influence of the model parameters and source either from entrainment of moist atmospheric air during the
ascentoftheplumeorfromtheevaporationofsurfacewater
conditions can be assessed by determining the power-law
atthevent,canhaveasignificanteffectontheheightofrise
scaling (28) from model calculations in quiescent environ-
ofthe column[Woods,1993;KoyaguchiandWoods,1996;
ments or, alternatively, by using an approximate scaling
Sparks et al., 1997; Mastin, 2007]. Water vapor in the
law relationship for the rise height of volcanic plumes in a
column at low altitude is transported to higher altitudes
quiescentatmosphereasafunctionofthemodelparameters
where the column may become saturated with respect to
and source conditions [see, e.g., Wilson et al., 1978; Settle,
watervapor,andthewatervaporwillthencondensetoliquid
1978; Woods, 1988; Sparks et al., 1997; Degruyter and
water or ice, releasing latent heat to the column, increasing
Bonadonna,2012]givenby
thecolumntemperature,andpromotingtheriseoftheplume.
(cid:2) (cid:4) (cid:5)(cid:3) For phreatomagmatic eruptions, such as the first explosive
H (cid:3)0:p00ffiffi1ffiffi3 g Cp0T0(cid:1)CaTa0 1=4N(cid:1)3=4Q1=4; (29) phase of the 2010 Eyjafjallajökull eruption [Höskuldsson
0 ks ra0CaTa0 etal.,2011;Magnússonetal.,2012],therecouldbeasignif-
icantincorporationofmeltwaterintotheeruptioncolumnat
the source, decreasing the temperature of the plume at the
for H measured in kilometer.
0 source and increasing the gas content and moisture loading
[31] Assumingthattheshearrateoftheatmosphericwind
oftheeruptioncolumn[KoyaguchiandWoods,1996].
isconstantinthetroposphere,theshearratecanbewrittenas
g_ ¼V =H ; where H is the height of the tropopause and [34] The moisture content of an eruption column can be
1 1 1 included in an integral model of volcanic plumes [Morton,
V =V(H )isthewindspeedatthetropopause.Afunctional
1 1 1957; Woods, 1993; Koyaguchi and Woods, 1996; Glaze
approximationfortheheightofrise(abovethevent)ofvol-
canicplumesinaconstantshearwindfield,whichremainin et al., 1997; Mastin, 2007; Degruyter and Bonadonna,
2012] by accounting for phase changes of the water within
thetroposphere,withthewindspeedexplicitlyincludedcan
the column and the effect of phase changes on the energy
beconstructedbycombiningequation(27)with(28)togive
budget. Here we follow the formulation of Woods [1993]
[see also Sparks et al., 1997]. In contrast, Degruyter and
1þ1:373Wf Bonadonna [2012] adopt the formulation of Glaze et al.
H ¼0:318Q0:253 s ; (30)
1þ4:266Wf þ0:3527Wf 2 [1997], which additionally includes an adiabatic cooling of
s s thegaseous phasesappropriateforvapor plumes. However,
the equation for the conservation of heat flux presented by
with Wf ¼1:44V =ðNH Þ, where the dimensionless con- Degruyter and Bonadonna [2012] is obtained from the
s 1 1
stant here is chosen by fitting to numerical solutions of the Glaze et al. [1997] conservation of energy equation assum-
dry volcanic plume model with constant wind shear in a ing that the heat capacity of the atmosphere is independent
Standard Atmosphere. During the first explosive phase of ofthemoisturecontentoftheatmosphereandthebulkden-
the Eyjafjallajökull eruption, 14–17 April 2010, the wind sity of the plume is equal to the atmospheric density. Note
parameterisestimatedtotakevaluesintherange0<Wf < thatweneglectphasechangeofwatervaporandliquidwater
s
1:1 (Figure 4), where the wind speed at a height H =7km to ice. Although such phase transformations release latent
1
has been taken as representative of the wind conditions. heattothecolumn,thelatentheatoffreezingisaboutafac-
The approximation given in equation (30) well describes torof10smallerthanthelatentheatofvaporization[Sparks
theriseheightscalculatedusingtheintegralvolcanicplume et al., 1997]. Therefore, the effect of moisture on the erup-
model for eruption columns which remain within the tro- tion column dynamics can be assessed, to leading order, by
posphere, at altitudes below 11km (Figure 5). Above the neglecting the complicated phase change to ice.
tropopause, the wind field is modeled with a uniform wind [35] We assume that the gas released at the vent is com-
speed and the atmospheric stratification in the Standard posed entirely of water vapor released from magma in the
Atmosphere changes, and therefore, the simple approxima- conduitandwatervaporfromtheevaporationofgroundwa-
tion in equation (30) inevitably deviates from the model ter. Water vapor is entrained into the eruption column from
predictions. the moist atmosphere and is advected with the bulk flow.
[32] The semi-empirical relationship given by equation Therefore, conservation of water in the column can be
(30) is similar to the relationship between source mass flux written as
and plume height in a wind field proposed by Degruyter
and Bonadonna [2012]. However, whereas the relationship d ðQfÞ¼2r U Rf ; (31)
of Degruyter and Bonadonna [2012] is based on a linear ds a e a
combinationofasymptoticresultsforplumeriseinaquies-
cent atmosphere and for a plume which immediately bends wherefisthemassfractionofliquidwaterandwatervapor
over in a strong uniform wind field, the relationship (30) is inthecolumn,andf isthemassfractionofwatervaporinthe
a
obtained from a consideration of pure plumes rising in a atmosphere (i.e., the specific humidity of the atmosphere).
100