The use of a numerical weather prediction model to simulate 1

near-field volcanic plumes 2

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R. R. Burton1, M. J. Wooodhouse2 , S. D. Mobbs1, A. Gadian1 4

1National Centre for Atmospheric Science, University of Leeds, Leeds, UK 5

2 School of Mathematics, University of Bristol, Bristol, UK 6

1 corresponding author, email: ralph@env.leeds.ac.uk 7

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Key Words 9

volcanic plumes; numerical weather prediction; WRF; ash dispersion; plume modelling 10

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Abstract 19

In this paper a state-of the art numerical weather prediction (NWP) model is used to simulate 20

the near-field plume of a Plinian-type volcanic eruption. The model is run at very high 21

resolution (of the order of 100 m) and includes a sophisticated representation of physical 22

processes, including turbulence. Results are shown for plume characteristics in an atmosphere 23

at rest and in an atmosphere with background wind. These results agree well with those from 24

theoretical models. A more complex implementation is given, to show how the model may 25

also be used to examine the dispersion of heavy volcanic gases. The modifications to the 26

standard version of the NWP model that are needed to model volcanic plumes are minimal, 27

and the model can be run on a desktop computer easily. This suggests that the modified NWP 28

model can be used in the forecasting of future volcanic events, in addition to providing a 29

virtual laboratory for the testing of hypotheses regarding plume behaviour. 30

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1 Introduction. 32

Forecasting the dispersion of volcanic ash and gases produced during volcanic eruptions is 33

crucial for the management of hazards during volcanic eruptions. Accurate forecasts rely on 34

the imposition of boundary conditions that model the volcanic source. Of particular 35

importance is the rate at which ash and gases are erupted from the volcano (referred to as the 36

source mass flux or the mass eruption rate) and the height at which this material is injected 37

into the atmosphere. Together, these two values are often referred to as the source term, and 38

they are related by the dynamics of the multiphase mixture proximal to the vent. The 39

complicated dynamics in this regime has led to simplified descriptions to estimate the source 40

term. For example, dispersion models such as the UK Met Office NAME model [Jones et al. 41

2007] often use calibrated algebraic relationships between the source mass flux and the 42

plume height [Sparks et al. 1997; Mastin et al. 2009] to estimate the source term for 43

explosive eruptions that produce buoyant eruption columns. These relationships are based 44

upon a series of historical eruptions, and relate the mass flux to the observed plume height, 45

and thus allow a mass flux to be approximated, given an observation of plume height (the 46

latter being routine to estimate in situ, the former extremely difficult). We subsequently refer 47

to these relationships as the Sparks-Mastin curves. Importantly, the Sparks-Mastin curves do 48

not discriminate between regimes of differing background wind speed or atmospheric 49

stratification (or rather, they contain this information only implicitly, and variations are 50

aggregated into the calibrated curve fit). Recent work (Woodhouse et al. 2013) has sought to 51

enhance the functionality of these curves by segregating the underlying historical data 52

according to the appropriate background wind speed (determined from meteorological 53

reanalyses). This type of approach, and other semi-theoretical approaches (e.g. Degruyter & 54

Bonadonna, 2012), cannot, by their very nature, allow sophisticated representations of 55

atmospheric processes and features such as three-dimensional turbulence, microphysics, 56

radiation and temperature inversions (and the interactions between these), but must include 57

their effects via simplified parametrisations, or by inferred relationships between wind speed 58

(for example) and plume height. 59

An alternative approach has been to develop simplified mathematical descriptions of 60

turbulent plumes (Morton, Taylor & Turner, 1956) and volcanic eruption columns (Woods, 61

1988; Sparks et al., 1997). These models average over the turbulent time scale and integrate 62

over a cross section of the plume, resulting in a system of ordinary differential equations (in 63

the steady case) that can be solved efficiently using standard computational methods. These 64

models allow the influence of atmospheric conditions (including non-uniform stratification 65

and wind) on the plume dynamics to be explored (e.g. Woods, 1988; Sparks et al. 1997; 66

Woodhouse et al. 2013), and their rapid implementation allows inverse calculations to be 67

performed to estimate the source conditions from plume height observations (Woodhouse et 68

al., 2015). However, in deriving the simplified description, a parameterization of some 69

physical processes, notably turbulent mixing, is required and the model predictions are 70

strongly dependent on the calibration of these parameterisations (Woodhouse et al., 2015). 71

Furthermore, the coupling of the plume dynamics to the local atmospheric structure is not 72

modelled. 73

The use of numerical weather prediction (NWP) models in volcanic ash dispersal modelling 74

is usually implemented at large spatial scales, commonly by driving an advection-diffusion 75

model such as NAME with meteorological fields. In this study we use a NWP model at very 76

high resolution to simulate the near-field plume behaviour; the plume is simulated in the body 77

of the NWP code. 78

The near-field of a volcanic plume is of significance since this is the region where pollutant is 79

particularly susceptible to atmospheric turbulence, thermal stratification, etc. which influence 80

the height and layering of the plume, and the plume dynamics can influence the atmospheric 81

structure. The use of a NWP model allows sophisticated meteorological and dynamical 82

processes to be included in the development of the modelling framework. This paper is 83

concerned with near-field plume dynamics, but the use of a NWP model additionally allows 84

the dispersal of pollutant over extended temporal and spatial scales to be investigated, with 85

the potential of incorporation into routine operational meteorological forecasts. Furthermore, 86

NWP models are subject to continuous revision and improvement to their parametrisations 87

and representations of physical effects, and so should represent the state-of-the-art in both 88

numerical methods and physical understanding of atmospheric processes. Thus the ability to 89

incorporate a near-field plume package into a NWP model would be of great benefit. 90

In Section 2, the NWP model to be used is introduced; Section 3 discusses some results from 91

the model, with further, more complex examples of its use given in Section 4. The work is 92

summarised in Section 5. 93

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2 The use of the Weather Research and Forecasting model 95

2.1 Model Description 96

The Weather Research and Forecasting model (WRF; Skamarock et al. 2008) model has been 97

used widely in meteorological modelling, both operationally and as a method of analysing 98

individual test cases. It is a state-of-the-art numerical weather prediction program, developed 99

at NCAR and other US agencies and institutes, and it is relatively easy to learn to use and to 100

implement. WRF is non-hydrostatic, compressible, and fully three-dimensional. It has a full 101

range of physical parametrisations, including representations of the boundary-layer, clouds, 102

microphysics, radiation and turbulence. In addition, a sophisticated land-surface model that 103

includes vegetative and evapotranspiration effects is included. The model is capable of being 104

operated in a mode akin to large-eddy-simulations, with the boundary layer parametrisation 105

switched off; the largest eddies are thus those explicitly represented in the model, with sub-106

grid turbulence being parametrised (see below for further details). WRF incorporates the very 107

latest developments in its physical parametrisations and numerical methods and has a major 108

new release typically every six months. The model can be implemented at a variety of spatial 109

scales, from hundreds of metres to tens of kilometres (Fernando et al. 2012). The 110

computational implementation is fully parallel and so the scale of run is limited only by the 111

resources available. The model can be run on relatively low numbers of processors; the 112

results in this paper were obtained on an inexpensive multi-processor machine. Of course, 113

the more processors used, the quicker the simulation will complete: the results presented here 114

took of the order of several hours using 24 processors. However, the ability to use hundreds 115

of processors means that the same results can be obtained very quickly if desired, making this 116

approach viable for disaster mitigation and operational use. The model can also be run in a 117

nested mode, whereby domains are nested within "parent" domains of lower resolution, 118

providing locally high spatial resolution in regions where small scales are strongly 119

influencing the dynamics without committing computational resources where they are not 120

required. Thus, there is considerable benefit in using such a model to simulate the behaviour 121

of volcanic plumes.1 122

2.2 initialisation strategies 123

WRF is capable of being initialised from host-model analyses, e.g. the National Centre for 124

Environmental Prediction's Global Forecast System (NCEP GFS) model data; this data then 125

forms the boundary conditions for the model, the boundaries updated at every model time-126

step (based upon an interpolation of the T-hourly analysis/forecast data). Alternatively it can 127

(when run in “idealised” mode) be initialised via a single vertical profile of wind, temperature 128

and moisture, corresponding to a case of interest. In this case the boundary conditions can be 129

open, periodic or symmetric; open boundaries are implemented in the following cases. At the 130

1 A version of WRF exists which includes chemistry - WRF-CHEM (REF). The authors have

chosen not to use this model since WRF-CHEM is very computationally expensive to use at

the high resolutions used in this paper. WRF-CHEM does include a volcano model, however

the source-term / plume-rise height relationship is determined from the Sparks-Mastin curves.

Since we wish to determine an alternative to the use of such curves, the use of WRF-CHEM

was avoided. Additionally, the authors plan (in future work) to add further momentum

equations to the WRF equation set, and thus turn WRF into a full multiphase model.

upper boundary, a sponge layer is used to gradually diminish signals that are propagating out 131

of the computational domain while minimizing spurious reflections. While these boundary 132

conditions are designed to allow information to propagate out of the computational domain, it 133

is possible that numerical artefacts due to imprecise boundary conditions can affect the 134

calculated solution. Therefore, in the tests that follow, the positions of the lateral and upper 135

boundaries are chosen so that any boundary-condition artefacts are far from the source 136

region. The lower boundary models the topography of the ground; in Sections 3, the surface 137

is taken to be flat, while in Section 4 we introduce orography. 138

To model the volcanic source, a large thermal perturbation is imposed at the surface, at the 139

location of the vent. The vent is taken to be circular with a radius of 200 m (the model regular 140

grid spacing means the vent is never exactly circular, however) at ground level and 141

horizontally centred in the domain. A passive tracer is released at a constant rate and is co-142

located with the thermal perturbation. The actual thermal perturbation determines the plume 143

rise height in this case (as will be discussed later). These thermal perturbations produce large 144

upwardly directed vertical velocities, on the order of 50m/s. These locally large flow speeds 145

result in small time steps since WRF is an Eulerian model and thus the time-step is severely 146

restricted by the Courant-Friedrichs-Lewy (CFL) condition (see, e. g., Innes and Dorling 147

2013). In the following results, the maximum acceptable time step was found to be 0.25s. We 148

note that acoustic modes are dealt with separately by the model dynamics and are typically 149

0.025s. In this framework the severe updraughts representing the volcanic eruption are 150

notionally equivalent to those of a very severe thunderstorm (albeit one which has a 151

continuous forcing mechanism). It is possible (and quite straight-forward) to implement an 152

additional “jet” of fluid from the source to model the momentum-driven ejection of material 153

at the volcanic vent. However, here we do not model the near-vent jet, i.e. all updraughts 154

arise from buoyancy. 155

The model domain is configured to be 30km x 30km with a resolution of 100m in the 156

horizontal. The vertical resolution is variable: there are 141 vertical levels (7 levels in the 157

lowest 1000m) with the model top at 30 hPa. This configuration prevents the gravity waves 158

generated in the model, and their subsequent reflections at model boundaries, from interfering 159

with the region of interest. The surface thermal perturbation is here taken to be constant in 160

time, although it is trivial to introduce time variation in the source to model unsteady 161

eruptions, such as Strombolian explosions (i.e. successive, discrete eruptions from a vent). 162

The model results presented in Section 3 are effectively one-way coupled: atmospheric 163

motion affects the tracer, but not vice-versa. In a real eruption this is not strictly the case, and 164

the ash particles can themselves generate turbulence and affect the motion of surrounding air 165

(REF). To include two-way coupling with ash – in other words, to allow the model to be run 166

in full multiphase mode – is an ongoing subject of study. However, a two-way coupling is 167

possible if the ejected matter is considered as a dense gas (such as sulphur dioxide), and in 168

Section 4 will present results illustrating the dynamics in this regime. 169

In this study, the microphysics, radiation and land-surface schemes are switched off, to 170

enable comparison with theoretical models that do not include the effects of these physical 171

processes. Such physical schemes can be run concurrently with the plume simulations, but 172

our aim is to compare the results with standard theoretical approaches, and no such simple 173

theoretical approaches currently exist to deal with the effect of (for example) short-wave 174

radiation upon plume height. 175

All simulations are conducted in large-eddy simulation mode, where the boundary-layer 176

parametrisation is switched off and the larger eddies (of the order of 100m) are explicitly 177

resolved. Smaller eddies are resolved via a turbulence-kinetic-energy-based subgrid 178

turbulence (TKE) scheme, acting on full (not perturbation) fields; second-order horizontal 179

diffusion acts on model surfaces. For a full description of the WRF TKE scheme, see 180

Skamarock et al. (2008). 181

Unless otherwise stated, the thermal stratification is taken to be that of the dry standard U.S. 182

atmosphere (U.S. Government Printing Office, 1976) at rest. In Section 4 the effect of 183

altering the background winds is investigated. In the majority of idealised test cases presented 184

here, the surface is taken to be flat; the exception being a demonstration of the method to 185

simulate the spread of dense gas that is erupted from a fissure in Iceland, presented in Section 186

4. 187

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3. Results: validation via selected numerical tests 189

3.1 Plume characteristics 190

Dimensional considerations (Morton et al. 1956, Sparks 1986) show that the rise height H of 191

the plume depends upon the heat flux Q density at the vent and should scale as 192

𝐻~ (𝑄𝑔𝑅2

𝜌0𝑐𝑝𝑇0)

1/4

𝑁−3/4 in a quiescent uniformly stably stratified ambient with buoyancy 193

frequency N, where R is the vent radius, 𝜌0 and 𝑇0 are the air density and temperature at 194

ground level, respectively, and 𝑐𝑝 is the specific heat capacity of dry air. As can be seen in 195

Fig. 1, the modelled results suggest a dependency of the form H~ Q0.29 , although there is 196

some scatter in the plot. In Fig. 1 the height is plotted as a function of the vent thermal 197

forcing Q, which can be related to the modelled vent temperature perturbation ΔT as shown 198

in Table 1. The plume height in Fig. 1 is determined either by examining the return of the 199

density profile to its background state, or by the examining the location of a suitably judged 200

iso-surface of concentration. We find no appreciable difference between these two methods. 201

The thermal perturbations used here are much lower than those that would be observed at an 202

explosive volcanic eruption. However, the model here is required to lift air (rather than an 203

air-particle mixture) and so lower values of heat flux are required to reach the typical heights 204

to which volcanic plumes ascend. In this and successive plots, the plume is examined at 205

t=1200s after the onset of the eruption, at which time the plume has reached its final height 206

and is spreading radially (in the absence of a background wind). A visualisation of a typical 207

plume extent is shown in Fig. 2(a) where the characteristic shape of a volcanic plume is seen, 208

with an approximately conical shape at lower elevations that is statistically steady on a time 209

scale that is longer than the eddy turnover time, and an evolving ‘umbrella cloud’ above in 210

which the air is overshooting the level of neutral buoyancy before collapsing back and 211

spreading approximately radially. The overshoot region reaches approximately 14 km above 212

ground level (AGL) and the level of neutral buoyancy is at approximately 10 km AGL 213

(corresponding to a surface temperature perturbation of 285K). The spreading rate of the 214

ejection column in the conical region at lower elevations (Fig. 2b) agrees with the theoretical 215

value (e.g. Turner 1973) of approximately 0.1 that is found if a constant entrainment rate is 216

assumed; this gradient of ash concentration with height reflects the underlying entrainment of 217

unpolluted air and will be considered in more detail below. 218

Run

number

Vent heat flux

density Q

Associated

temperature

difference at

lowest model level

1 1.8 x 106 Wm-2 200 K

2 3.2 x 106 Wm-2 285 K

3 4.4 x 106 Wm-2 380 K

4 6.4 x 106 Wm-2 505 K

5 8.8 x 106 Wm-2 615 K

6 1.1 x 107 Wm-2 700 K

Table 1. Selected values of temperature surface heat flux and perturbation temperature for Figure 1. 219

The maximum vertical velocities within the eruption column at low elevations above the vent 220

are shown in Fig. 3. We note that the velocity increases with distance from the source for a 221

fixed source heat flux, as expected for a ‘lazy plume’ (i.e. where the momentum flux at the 222

source is lower than the momentum flux that would be observed for a plume from a point 223

source of buoyancy at the point where the plume radius equals the source radius). The present 224

results also agree with results from LES studies of intense fires (Devenish et al. 2009) where 225

the vertical velocity increases up to some level of neutral buoyancy and then decreases above 226

this level, Fig. 4(a). This is also coupled with an increase in turbulence above this level (see 227

Fig. 4 (b) , using the root mean square velocity as a proxy for turbulence). Indeed, subject to 228

appropriate linear transformations of the axes, Fig. 4 (a) bears a strong similarity to Fig. 3 (a) 229

of Devenish et al. (2009). 230

Furthermore, Fig. 3 shows that vertical velocity, W, increases with heat flux as 𝑊~𝑄1/3. 231

Batchelor (1954) found (from dimensional analysis) that for a maintained convective point 232

source the plume vertical velocity w scales as w ~ F1/3 where F is the heat flux density 233

rewritten as 234

𝐹 = 𝑄𝑔

𝐶𝑝𝜌0𝑇0 235

and 𝐶𝑝, 𝜌0 and 𝑇0 are the specific heat and background density and temperature, respectively. 236

3.2 Entrainment of unpolluted air 237

As an example of the use of the model to test various hypotheses regarding more idealised 238

types of plume modelling, the above technique can be used to estimate entrainment 239

coefficients within the ejection column. The theoretical treatment of entrainment can be 240

found in Turner (1973). In the case of no ambient wind, it is commonly assumed that air is 241

entrained from the atmosphere with a velocity U that is proportional to the vertical velocity of 242

the plume, W (where W is plume vertical velocity along the vertical axis of a Gaussian 243

plume) so that 𝑈 = 𝛼𝑊, where 𝛼 is the entrainment coefficient. In integral models 𝛼 is often 244

taken to be constant, and this assumption has been shown to provide robust predictions of 245

plume dynamics across a wind range of scales (Turner 1973) from oil fires to volcanic 246

eruptions. However, recent work has shown that the development of the turbulent structures 247

of the plume evolves away from the vent leads to a non-constant entrainment coefficient 248

(Kaminski, Tait & Carazzo, 2005; Ezzamel, Salizzoni & Hunt, 2015). 249

For plumes rising from a point source of buoyancy in an unstratified ambient, dimensional 250

analysis demands that the plume must be conical, i.e. 𝑟 = 𝛽𝑧 where r is a characteristic radial 251

length scale of the plume, z is the distance from the source and 𝛽 is a dimensionless constant. 252

The equations for the conservation of mass, momentum and buoyancy, integrated over a 253

cross-section of the plume, with the application of the entrainment assumption, lead to 𝛽 =254

6𝛼/5 so that the entrainment coefficient can be determined from the spreading rate of the 255

plume (Morton, Taylor and Turner, 1956; Turner, 1973). While we model plumes in a 256

stratified atmosphere, measurements of the spreading rate can still be used to estimate the 257

local entrainment rate, provided the spreading rate is examined on a vertical length scale that 258

is much shorter than the scale height of the atmosphere (so that variations in the density of 259

the ambient are negligible). Furthermore, while we adopt an areal source (rather than an 260

idealized point source) that introduces an additional length scale into the problem, 261

experiments and analysis of integral models suggest that the plume rapidly adjusts to the 262

point-source solution. We note, however, that the spreading rate is not a suitable approach to 263

quantifying entrainment in the fountaining region near to and above the neutral buoyancy 264

level. 265

To determine the width of the modelled plumes, the best Gaussian fit to the vertical velocity 266

at a specified height is first determined. That this is feasible is demonstrated in Fig. 5 where 267

cross-sections of the vertical velocity in the plume are shown, and a Gaussian structure is 268

evident at all heights. Gaussian profiles of the vertical velocity in plumes are found in 269

laboratory experiments (e.g. Papanicolaou & List, 1988). From the Gaussian curves fitted to 270

the data, the characteristic radial length scale r is taken to be the point where the velocity has 271

fallen to e-1 of its maximum (as in Turner, 1973, and Papanicolaou & List, 1988). Fig. 6 272

shows that, despite some scatter (as would be expected, given this is a “snapshot” of a 273

simulated turbulent response), the spreading rate 𝛽 (defined here as the best linear fit between 274

r and z) varies between 0.11 and 0.10 giving a range for the entrainment coefficient 0.083 ≤275

𝛼 ≤ 0.091, agreeing well with values reported by Turner (1973) and also those deduced from 276

laboratory studies, e.g. (List, 1982). It is worth noting that the lowest value of Q used here 277

lends to a more robust linear relationship between r and z; all higher values of Q show 278

evidence of eddies at the plume edge (i.e., more scatter in the plots), thus distorting the shape 279

of the ejection column. Additionally there is no apparent relation between α and the time 280

elapsed since eruption, suggesting that if the entrainment assumption is valid then there is no 281

evidence to suggest it is time-dependent. 282

To further examine the entrainment assumption, we analyse the relationship between the 283

velocity of entrained air, measured directly from the numerical results, and the core 284

updraught velocity. We therefore consider the parameter U/WMAX where U = U (r, z, t) is the 285

velocity of entrained air at a distance r from the plume axis, z is the altitude above the vent, t 286

is the time after the onset of the eruption, and WMAX = WMAX(z, t) is the maximum updraught 287

at height z in the plume. U/WMAX was calculated for various heights and times with a surface 288

forcing of Q = 1.1 x 107 Wm-2, as shown in Fig.7. In all plots there is clearly a structural 289

similarity with U/WMAX decreasing away from the plume edge (with distance here normalised 290

by plume half-width ρ0); variability is of course present, and is associated with the “snapshot” 291

nature of these plots, as described above. However, if the mean (across all times and all 292

heights) of U/WMAX is taken the results approximate reasonably well the expected Gaussian-293

type dependence U/WMAX = α [1-exp( - X2)]/X2 where X = r/ρ0. This is shown in Fig. 8 where 294

additional and analogous calculations have been performed for Q = 4.4 x 106 Wm-2. There are 295

differences, however: the present values of U/WMAX do not decrease as rapidly from the vent 296

when r/ρ0 is greater than approximately 1.75. 297

This suggests that there is indeed an underlying functional relationship between U and WMAX - 298

the appropriate function being apparently independent of time, height and surface forcing in 299

the cases considered. Importantly, this suggests that the entrainment assumption is valid in 300

these cases. Furthermore, the analysis demonstrates that the current model implementation 301

may be used to examine models of turbulent entrainment. 302

3.3 Bent over plumes 303

The results presented above have adopted an atmosphere that is at rest. Recently, there has 304

been a renewed interest in plumes rising in a background wind field, following several 305

volcanic eruptions (notably the 2010 eruption of Eyjafjallajökull, Iceland) where the plume 306

was significantly influenced by atmospheric winds. Integral models of volcanic plumes that 307

included a background wind field (e.g. Bursik, 2001; Degruyter & Bonadonna, 2012; 308

Woodhouse et al., 2013) have demonstrated that the rise height of plumes in a wind field is 309

diminished in comparison to an equivalent source in a quiescent atmosphere. The neglect of 310

the wind can then lead to very significant under-predictions of the source mass flux 311

(Woodhouse et al., 2013). 312

Explosive volcanic eruptions are often characterized as producing “strong plumes” or “weak 313

plumes” on the basis of the effect of the background wind on the plume rise (e.g. Sparks et 314

al., 1997). This is change in regime is quantified using the dimensionless ratio of the wind 315

speed to the plume velocity, here denoted by 𝒲 and given by 𝒲 = 𝑉 𝑊⁄ = 𝑉 (𝐹0𝑁)1/4⁄ 316

where 𝐹0 =𝑄𝑔𝑅2

𝜌0𝑐𝑝𝑇0 is the buoyancy flux from the source. For strong plumes the background 317

wind field has little influence on the plume (𝒲 ≪ 1), whereas weak plumes (𝒲 ≫ 1) are 318

bent-over by the ambient wind. 319

Figs 9(a)-(e) show that when the background wind speed shear (here taken to be linear with 320

height) increases the plume height is reduced as the plume bends over (cf. Bursik, 2001; 321

Woodhouse et al., 2013). In Figure 10, the plume height is normalised by the height in the 322

absence of no shear (H0), and the shear is normalised by Brunt- Väisälä frequency, N. This 323

figure also shows several theoretical estimations of the normalised plume height based upon 324

models of varying complexity. As can be seen, the WRF results fall within the envelope of 325

theoretical results for normalised shear values less than approximately 0.75. For values above 326

this, the present results indicate the plume height to be diminished more than would be 327

estimated by the theoretical models. If one of the theoretical models had to be chosen as 328

closest to the present results (based upon root-mean square differences), it would be the 329

Devenish et al. 2010 three-parameter model. 330

In addition, for the weakest plumes in our simulations the plume bifurcates, and consists of 331

buoyant vortex pairs. The bifurcation has been attributed to the development of lateral 332

pressure gradients moving along the margins of the plume (Ernst et al. 1994; Sparks et al., 333

1997), which should be accompanied by a linear increase in vortex separation with distance. 334

This type of linear separation can be seen in Fig. 11(a) where an isosurface of concentration 335

at c=10-4 is used to visualize the plume margin and the distance from plume axis to plume 336

margin is approximately proportional to downwind distance. We note that in this simulation 337

there are no “clear” regions in between the bifurcating branches of the plume – merely 338

substantially lowered concentrations. In Fig. 11(b), the vortical motions can be inferred from 339

the vertical velocities. In addition to the central vortex pair, there are further, smaller eddies 340

at the edge of the bifurcated plume. At the ground surface, at a cross-plume distance of 0 km, 341

there is upward motion, flanked by downward motion at approximately 1.5 km AGL. This 342

suggests a separate vortex pair in addition to the vortex pair associated with the stretching 343

and tilting of vorticity near the plume source. Thus, these results show that the motion of the 344

air in the vicinity of the plume is extremely complex. 345

3.4 Entrainment in Bent-over plumes 346

The reduction in the height of rise for plumes rising in a wind field indicates that the ambient 347

wind enhances the mixing of atmospheric air into the plume. The density deficit that drives 348

the vertical motion is therefore reduced more rapidly in wind-blown plumes. In integral 349

models of turbulent buoyant plumes in a cross wind, the enhanced mixing is described by 350

modifying the entrainment assumption of Morton, Taylor & Turner (1956). Several modified 351

entrainment closures have been suggested (e.g. Hewett, 1971; Ooms & Mahieu, 1981), and 352

there has been some testing of the predictions of these models against experiments (e.g. 353

REFs). However, the calibration of these parameterizations of turbulent mixing has been 354

limited to settings that are accessible in the laboratory, which may differ from conditions 355

found in the natural environment. For example, it is difficult to produce non-uniform velocity 356

profiles in a moving density stratified ambient to represent a windy atmosphere into which a 357

volcano eruptions, and therefore the wind entrainment parameterizations have commonly 358

been calibrated in experiments adopting a uniform ambient flow. Numerical experiments 359

provide an alternative to laboratory experiments. 360

361

362

4. More complex implementations 363

Example: Volcanic degassing 364

To demonstrate that WRF may be applied to a real-life volcanic eruption – and thus to take 365

advantage of the model’s capability of numerical weather prediction - a further modification 366

to the model has been made, allowing dense gases to be tracked. This is achieved using the 367

method of Burton et al. (2017) where an extra term is added to the WRF microphysical 368

scheme to mimic, via the change in buoyancy, the effect of a dense gas, without interfering 369

with the existing microphysics in any way. This method was shown to reproduce both 370

theoretical models, and NWP test cases, of gravity current flow. 371

Burton et al. (2017) showed how the modification could be used to simulate the spread of 372

CO2 from a limnic eruption at Lake Nyos. Here we show how the method could be used to 373

simulate SO2 degassing from a volcanic fissure using the Bárðarbunga eruption in Iceland, 374

2014-2015 as a case study. The model is initialised with NCEP GFS analyses (supplying 375

background winds, thermal stratification and lateral boundary conditions) for 18Z on the 2nd 376

November 2014, at which time the fissure was approximately 20 km long and 2 km wide 377

(Fig. 10a) and was still emitting SO2 . The model horizontal resolution used is 100m (with 378

underlying topography gathered from a 30-arc-second database) and a fissure surface heat 379

flux of Q = 104 Wm-2 was imposed along the Bárðarbunga fissure (see Fig. 12). JUSTIFY 380

CHOICE OF Q HERE. Since the concentrations of the gases exsolved from the magma are 381

not known precisely, we perform two simulations, using SO2 concentrations of 6 g kg-1 and 382

200 g kg-1 of gas. It should be emphasised that now the motion of the volcanic fluid (dense 383

gas plus ambient air) is affecting the motion of the ambient air, as well as being affected by it. 384

This is unlike chemical transport models (e.g. the chemistry version of WRF, WRF-CHEM) 385

and thus the simulation is incorporating complex feedbacks between the volcanic gas and the 386

ambient air. 387

As expected, the lighter gas (SO2 concentration of 6 g kg-1) reaches a higher altitude in the 388

atmosphere (Figs. 13, right-hand panels), with the cloud top approximately 8.5 km above 389

ground level, while the denser gas (SO2 concentration of 200 g kg-1) reaches only 390

approximately 4km above ground level (Figs 13, left-hand panels). The horizontal spread of 391

dense gas at the surface and lower levels is consequently greater, and is seen to collect in the 392

valleys (Fig. 13(f)). Accompanying the spread of gas is a modification to the atmospheric 393

flow (as noted above, the dense gas affects the flow). This can be seen in Fig. 14 where the 394

footprint of the gas is clearly associated with (quite significant) changes to the flow. These 395

changes may be important and are certainly not included in conventional NWP simulations 396

(MORE?). These results are used only as illustration of the WRF implementation, since the 397

fissure heat flux and release rates are unknown. However, the results demonstrate that the 398

model may be modified to include the effects of (and interaction between) orography, dense 399

volcanic gases and convection, and may be useful for predicting the spread of volcanic gases 400

in the event of a degassing episode, if the heat flux and source strength are known (or can be 401

estimated). 402

403

5. Summary. 404

A method has been presented which allows near-field volcanic plume behaviour to be 405

modelled via a standard (albeit state-of-the-art) numerical weather prediction model. Results 406

have been presented which demonstrate that the model reproduces very well the theoretical 407

predictions (in terms of the various plume characteristics) for idealised cases (using a passive 408

tracer and no orography) for both the atmosphere at rest and for cases with background wind. 409

The entrainment of unpolluted air into the modelled plume is in accord with the entrainment 410

assumption across a range of times, heights, and surface forcings. 411

In the idealized setting the wind profile is taken to be constant with height, for simplicity and 412

in order to identify features that have been observed in laboratory experiments and from 413

simplified model descriptions to test our new implementation of the WRF model. In 414

applications to case studies of volcanic plumes, a non-uniform velocity profile for the wind 415

should be applied. This is straightforward in WRF and allows an examination of the effects of 416

shear in the wind velocity on the dynamics of the plume in a stratified atmosphere; a ambient 417

setting that is extremely challenging to reproduce in the laboratory. The simulation of plumes 418

using WRF can then be exploited as a surrogate for laboratory experimentation in the 419

development of mathematical models of turbulent buoyant plumes in atmospheric conditions 420

that are difficult to reproduce in the laboratory. Also, this framework provides an excellent 421

means of examining the effect of more complex processes (such as microphysics) upon 422

plume characteristics. 423

An example of a more complex implementation has been presented which demonstrates the 424

wider applicability and usefulness of this modelling framework. A particularly intriguing 425

aspect is the two-way coupling of the plume and atmospheric dynamics; here the use of a 426

NWP model is essential. 427

Acknowledgements 428

The authors would like to thank the UK Natural Environment Research Council (NERC) for 429

funding this work via the VANAHEIM consortium. Additionally, we would like to thank 430

Wei Wang of NCAR, USA for help in configuring the WRF model to accommodate tracer 431

releases. 432

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Contini (2011) REF for Fig. 10 439

Costa, A. et al. (2016). Results of the eruptive column model inter-comparison study. Journal 440

of Volcanology and Geothermal Research, X, Y. 441

Devenish 2010 REF for Fig. 10 (i.e. the WRF results against theory). 442

Devenish, BJ and Edwards, JM (2009). Large-eddy simulation of the plume generated by the 443

fire at Buncefield oil depot in December 2005. Proc. R. Soc. A., 465, 397-419 444

Innes, PM and Dorling, S (2013). Operational Weather Forecasting. Wiley-Blackwell, 445

London. 446

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Barriero, B (1999). Santorini Volcano, Geological Society Memoir No. 19 448

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Ezzamel, A, Salizzoni, P, Hunt, GR (2015) Dynamical variability of axisymmetric buoyant 451

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Pollution, Modeling, and Measurements. CRC Press, p352 454

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Application XVII, edited by: Borrego, C. and Norman, A.-L., (Proceedings of the 27 459

NATO/CCMS International Technical Meeting on Air Pollution Modelling and its 460

Application), Springer, 580-589 461

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buoyancy. J. Fluid Mech. 526, 361–376 464

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dependence on rotation, turbulent mixing, and cross-flow strength, Jour. Gephys. Res. 3405 466

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Mastin, LG, Guffianti, M, Servranckx, R, Webley, PW, Barsottie, S, Dean, K, Denlinger, R, 468

Durant, A, Ewert, JW, Gardner, CA, Holliday, AC, Neri, A, Rose, WI, Schneider, D, Siebert, 469

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multidisciplinary effort to assign realistic source parameters to model of volcanic ash-cloud 471

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: Special Issue on Volcanic Ash Clouds, edited by: Mastin, L. and Webley, P., 186, 10-21 473

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maintained and instantaneous sources. Proc. Roy. Soc., A., 234, 1196 475

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forcast-system-gfs (accessed 27th February 2017) 477

Ooms (1971) REF for Fig. 10 478

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source model for a stack plume. Applied Scientific Research, 36(5-6), 339—356 480

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and volcanic plume height: insights from three-dimensional numerical simulations. Journal of 489

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(accessed online 26/02/13) 494

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volcanic plumes and wind during the 2010 Eyjafjallajkul eruption, Iceland. Journal of 496

Geophysical Research: Solid Earth, 118 497

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507

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510

511

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513

List of Figures 514

Figure 1. Dependence of plume height log H upon associated surface heat flux Q. The line of 515

best fit connecting the simulated heights is marked with a dotted line and the inset text shows 516

the gradient and correlation coefficient for this best-fit line. 517

Figure 2. Structure of plume at t = 1200s. (a) An isosurface of constant arbitrary 518

concentration; (b) ejection column showing concentration (colour shaded; arbitrary units). In 519

(b) a dashed line represents a distance/height gradient of 0.11. 520

Figure 3. Maximum vertical velocities within the plume at several heights above ground 521

level for increasing surface heat flux. The best linear fit is also shown for each group of 522

simulations. 523

Figure 4. (a) The mean centre-line vertical velocity WMEAN as a function of height ; (b) the 524

profile of difference between root-mean square vertical velocity WRMS and WAVE as a function 525

of height, indicating regions of enhanced turbulence. Both plots use time averaging. 526

Figure 5. Vertical velocity in plume ejection column for vent heat flux Q = 1.8 x 106 Wm-2, 527

30 minutes after eruption. Individual panels represent various heights above ground (denoted 528

in each panel.) 529

Figure 6. Radial spread against height after (a) 15 minutes, (b) 30 minutes, (c) 45 minutes 530

and (d) 60 minutes after eruption. Coloured dots represent vent heat fluxes of Q = 1.8 x 106 531

Wm-2 (●), Q = 6.4 x 106 Wm-2 (●), Q = 8.8 x 106 Wm-2 (●), Q = 1.1 x 107 Wm-2 (●). The 532

best-line fit (see text for properties) for all the data is shown by the solid green line. 533

Figure 7. Velocity of entrained air U(x, z, t) normalised by maximum plume velocity 534

WMAX(z, t) (where z is altitude and t is time) against normalised distance from vent (ρ0 (z,t) is 535

Gaussian plume width). Plots show results for 15 minutes (black lines), 30 minutes (red 536

lines), 45 minutes (green lines) and 60 minutes (blue lines) after eruption. Surface forcing is 537

Q = 1.1 x 107 Wm-2. The altitude at which each plot is valid is inset in each plot. 538

Figure 8. The mean of all the curves shown in Fig. 6 (solid black line). The equivalent mean 539

for a surface forcing of Q = 4.4x106 Wm-2 is also shown (dotted black line). The red, green 540

and black thin solid lines represent the expected theoretical variation for three values of 541

entrainment coefficient (see text for details). 542

Figure 9. Dependence of plume height for background wind speeds (constant with height) of 543

(a) 10m/s, (b) 15 m/s,(c) 20m/s, (d) 30m/s and (e) 40 m/s. Results are for t=1200s after 544

eruption. 545

Figure 10. Normalised plume height versus normalised shear for WRF results (solid black 546

line) and various theoretical predictions (details of the latter can be found from the 547

References). Error bars on the WRF results show ±1 standard deviation. H0 is the height in 548

the absence of wind shear; N (equal to 0.01 s-1 here) is the Brunt-Väisälä frequency. 549

Figure 11. (a) Isosurface of concentration c=10-4 at t=1200s for background wind speed of 550

5m/s looking down on the plume. The straight black dotted line shows the near-linear spread 551

of the plume edge, the distance L being a near-linear function of downwind distance; the red 552

dotted line corresponds to the cross section shown in Fig. 11 (b). In the latter plot (looking 553

toward the vent), vertical velocities (red, negative; green, positive; solid black, zero) are 554

shown for the cross-section defined in Fig. 11 (a), together with the isosurface of 555

concentration c=10-4. 556

Figure 12. Topography of Iceland (coastline in solid blue line) with altitude contour intervals 557

every 100 m; the solid black shaded area represent the Bárðarbunga fissure. 558

Figure 13. Visualisation of the spread of SO2 from Bárðarbunga fissure, November 2014. 559

The 1 g / kg SO2 mixing ratio is shown in red, for gas release rates of 200 g / kg (left hand 560

panels) and 6 g / kg (right hand panels), for (a) 20, (b) 40, (c) 60, (d) 80, (e) 100 and (f) 120 561

minutes after initial release (eruption). The viewpoint is from an elevated position looking 562

due north, showing orography (colour-shaded.) 563

Figure 14. (a) Left-hand panel: wind barbs in the absence of any gas release for the 564

November 2014 case study. Middle panel: difference in 10 m wind speed between 6 g /kg 565

release and no-release case. Right-hand panel: difference in wind speed between 200 g / kg 566

release and no-release case. In all three panels wind speed is shown colour-shaded, orography 567

is shown with solid contours (250 m intervals) and the solid blue line represents lowest-level 568

concentration of SO2 equal to 1 g / kg. (a) 10 minutes after eruption, (b) 60 minutes after 569

eruption, (c) 120 minutes after eruption. In all plots the horizontal extent is 345 km (west-570

east) by 265 km (south-north). 571

572

573

574

Figure 1. Dependence of plume height log H upon associated surface heat flux Q. The line of 575

best fit connecting the simulated heights is marked with a dotted line and the inset text shows 576

the gradient and correlation coefficient for this best-fit line. 577

578

(a) (b)

579

Figure 2. Structure of plume at t = 1200s. (a) An isosurface of constant arbitrary 580

concentration; (b) ejection column showing concentration (colour shaded; arbitrary units). In 581

(b) a dashed line represents a distance/height gradient of 0.11. 582

583

584

585

Figure 3. Maximum vertical velocities within the plume at several heights above ground 586

level for increasing surface heat flux. The best linear fit is also shown for each group of 587

simulations. 588

589

590

591

592

593

594

Figure 4. (a) The mean centre-line vertical velocity WMEAN as a function of height ; (b) the 595

profile of difference between root-mean square vertical velocity WRMS and WAVE as a function 596

of height, indicating regions of enhanced turbulence. Both plots use time averaging. 597

598

599

600

Figure 5. Vertical velocity in plume ejection column for vent heat flux density Q = 1.8 x 106 601

Wm-2, 30 minutes after eruption. Individual panels represent various heights above ground 602

(denoted in each panel.) 603

604

605

606

607

608

Figure 6. Radial spread against height after (a) 15 minutes, (b) 30 minutes, (c) 45 minutes 609

and (d) 60 minutes after eruption. Coloured dots represent vent heat fluxes of Q = 1.8 x 106 610

Wm-2 (●), Q = 6.4 x 106 Wm-2 (●), Q = 8.8 x 106 Wm-2 (●), Q = 1.1 x 107 Wm-2 (●). The 611

best-line fit (see text for properties) for all the data is shown by the solid green line. 612

613

614

Figure 7. Velocity of entrained air U(x, z, t) normalised by maximum plume velocity 615

WMAX(z, t) (where z is altitude and t is time) against normalised distance from vent (ρ0 (z,t) is 616

Gaussian plume width). Plots show results for 15 minutes (black lines), 30 minutes (red 617

lines), 45 minutes (green lines) and 60 minutes (blue lines) after eruption. Surface forcing is 618

Q = 1.1 x 107 Wm-2. The altitude at which each plot is valid is inset in each plot. 619

620

Figure 8. The mean of all the curves shown in Fig. 6 (solid black line). The equivalent mean 621

for a surface forcing of Q = 4.4x106 Wm-2 is also shown (dotted black line). The red, green 622

and black thin solid lines represent the expected theoretical variation for three values of 623

entrainment coefficient (see text for details). 624

625

626

627

628

629

630

631

(a)

(b)

(c)

(d)

(e)

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633

Figure 9. Dependence of plume height for background wind speeds (constant with height) of 634

(a) 10m/s, (b) 15 m/s,(c) 20m/s, (d) 30m/s and (e) 40 m/s. Results are for t=1200s after 635

eruption. 636

637

638

639

640

641

642

Figure 10. Normalised plume height versus normalised shear for WRF results (solid black 643

line) and various theoretical predictions (details of the latter can be found in the References). 644

Error bars on the WRF results show ±1 standard deviation. H0 is the height in the absence of 645

wind shear; N (equal to 0.01 s-1 here) is the Brunt-Väisälä frequency. 646

647

648

649

650

651

(a)

(b)

652

Figure 11. (a) Isosurface of concentration c=10-4 at t=1200s for background wind speed of 653

5m/s looking down on the plume. The straight black dotted line shows the near-linear spread 654

of the plume edge, the distance L being a near-linear function of downwind distance; the red 655

dotted line corresponds to the cross section shown in Fig. 11 (b). In the latter plot (looking 656

toward the vent), vertical velocities (red, negative; green, positive; solid black, zero) are 657

shown for the cross-section defined in Fig. 11 (a), together with the isosurface of 658

concentration c=10-4. 659

660

661

662

663

664

665

666

667

668

669

670

(a)

671

Figure 12. Topography of Iceland (coastline in solid blue line) with altitude contour 672

intervals every 100 m; the solid black shaded area represent the Bárðarbunga fissure. 673

674

675

676

677

678

(a)

(b)

(c)

(d)

(e)

(f)

679

Figure 13. Visualisation of the spread of SO2 from Bárðarbunga fissure, November 2014. 680

The 1 g / kg SO2 mixing ratio is shown in red, for gas release rates of 200 g / kg (left hand 681

panels) and 6 g / kg (right hand panels), for (a) 20, (b) 40, (c) 60, (d) 80, (e) 100 and (f) 120 682

minutes after initial release (eruption). The viewpoint is from an elevated position looking 683

due north, showing orography (colour-shaded.) 684

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689

690

691

692

693

694

695

696

697

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700

(a)

(b)

(c)

701

Figure 14. (a) Left-hand panel: wind barbs in the absence of any gas release for the 702

November 2014 case study. Middle panel: difference in 10 m wind speed between 6 g /kg 703

release and no-release case. Right-hand panel: difference in wind speed between 200 g / kg 704

release and no-release case. In all three panels wind speed is shown colour-shaded, orography 705

is shown with solid contours (250 m intervals) and the solid blue line represents lowest-level 706

concentration of SO2 equal to 1 g / kg. (a) 10 minutes after eruption, (b) 60 minutes after 707

eruption, (c) 120 minutes after eruption. In all plots the horizontal extent is 345 km (west-708

east) by 265 km (south-north). 709