# Combining Spherical-Cap and Taylor Bubble Fluid Dynamics with Plume Measurements to Characterize Basaltic Degassing

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## Abstract

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## 1. Introduction

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^{4}Pa·s [1], which enable the free flow of gas bubbles within the melt, in contrast to the behavior of more viscous silicic systems [2]. In basaltic magmas spherical bubbles are generated following exsolution of gas from the melt [3]. These bubbles grow via diffusion, decompression-based expansion, or coalesce to form non-spherical bubbles, e.g., of spherical-cap morphology [4,5,6], which can transition into Taylor bubbles (also called gas slugs), which nearly span the conduit width, and are of a length greater than, or equal to, the conduit diameter (see Figure 1 for further details on the morphological characteristics of spherical-cap and Taylor bubbles) [4,7,8]. These distinct bubble morphologies give rise to a variety of potential classes of surface degassing activity, specifically, passive degassing of spherical bubbles [2]; puffing, from bursting of non-spherical bubbles or non-over-pressurized Taylor bubbles [9,10,11]; and explosions from over-pressurized Taylor bubbles [12,13,14]. The latter scenario is associated with strombolian volcanism, as manifested on the eponymous Stromboli volcano, e.g., [9,15], where the activity has been well characterized through measurements of the erupted gas masses, e.g., [11,16,17,18], and studies into the explosive dynamics, e.g., [19,20,21].

_{2}fluxes, which enables the capture of rapid degassing phenomena in unprecedented detail [34,35,36]. This work is also one of only a few in recent years, focused on defining transitions between basaltic degassing modes, building on pioneering work performed in this area a number of decades ago, e.g., [37], that of Palma et al. [38] who identified the relationship between bubble bursting strength and the duration of the styles of basaltic volcanic activity relevant to this study, and the more recent work of Gaudin et al. [26], who categorized explosions based on bubble sizes and eruptive properties.

## 2. Modelling Transitions between Spherical, Non-Spherical and Taylor Bubble Flow Regimes

#### 2.1. Interactions between Taylor Bubbles

#### 2.2. Interactions between Spherical-Cap Bubbles

^{−3}, a condition satisfied in volcanic scenarios, e.g., [14]. The spherical-cap bubble rise velocity (${u}_{CB}$) can then be calculated using the following relationship, rearranged from Joseph [47], Equation (2.1):

_{b}/2) [48]. For spherical-cap interaction length (${c}_{min}$), there is no available prior modelling literature to refer to here, hence, we take this to be four times greater than ${c}_{wake}$, as this is the approximate scaling between the wake and wake interaction lengths in the Taylor bubble case, although, as spherical-cap bubble volume decreases, the influence of the interaction length will also decrease.

#### 2.3. Bubble Interactions

_{min}, can be defined as a function of bubble volume (i.e., length) below which it would be highly improbable for an independent trailing bubble to burst at the surface. In this case, any following bubble would be travelling within the leading bubble’s wake, hence, would be liable for imminent coalescence; t

_{min}was, therefore, taken to be equal to the rise time of the trailing bubble (i.e., the ascent velocity of the spherical-cap or Taylor bubble) through a column of liquid of thickness equal to that of the leading wake length (${l}_{wake}$ or ${c}_{wake}$) plus the height of fluid (${l}_{film}$), arising from complete drainage of the film surrounding the leading bubble, following each burst. The latter was constrained, in the case of Taylor bubbles, from the film volume, as a function of the slug volume, i.e., height, with knowledge of the film thickness, as per Equation (5). For spherical-cap bubbles, this was constrained from the bubble volume and conduit volume around the bubble (i.e., applying the formula for the volume of a cylinder, for given conduit and/or bubble radii):

## 3. Model Application to Target Volcanoes and Comparison with Field Data

^{−3}, viscosity of 2000 Pa·s, and conduit radius of 1.5 m, e.g., [49]. In general, the field data clearly fall above the repose gap area, affirming the model suggestion from Pering et al. [24] that independently-bursting, large-volume, low repose time events would be improbable, due to inter-bubble coalescence in the conduit. The majority of bursts (62%) fall within the rapid bursting Taylor and spherical-cap bubble area, with 15% contained in the single Taylor bubble explosive area, 12% in the single Taylor bubble puffing area, and most of the remainder in the single puffing area, indicative of activity spanning the strombolian explosive-puffing spectrum. In the case of this rapid bursting scenario, the model points towards bubble interaction playing a key role in the fluid dynamics, as previously suggested by Pering et al. [24].

^{−3}[50,51], 300 Pa·s [52], and 1.5 m, respectively [53,54]. Figure 3b also includes a range of field data points, based on literature-derived main vent burst volumes [11,17,18]. These data generally fall within the single explosive Taylor bubble region, in line with the classical strombolian activity associated with this target. However, the very smallest bursts from the Tamburello et al. [11] dataset fall within the single Taylor bubble puffing region, capturing the spectrum of activity exhibited at the volcano. For the specifically-described ‘puffing’ events from Tamburello et al. [11], all the data points are located away from the rapid Taylor bubble bursting area. Note that in all the above cases, minimum repose times from the literature have been assigned, e.g., 50 s from Ripepe et al. [10] for puffing. Hence, even for these rather extreme prescriptions of inter-burst temporal resolution, the model points towards clearly independent bubble flow behavior.

^{−3}and 1000 Pa·s, with a conduit radius of 1.5 m [22] within the model. In particular, we converted the Kremers et al. [33] data to the volume (see Table 1) using the formula for the volume of a cylinder (the infrasound derived length data in Kremers et al. [33] already account for pressure and viscous effects), and plot against repose time in Figure 3c. In this case all data fall within the single explosive Taylor bubble region, in line with the existent strombolian activity, and indicating independent bubble flow, well outside the repose gap region.

## 4. Discussion and Limitations

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Notation and Greek Letters

$R{e}_{b}$ | Bubble Reynolds number |

${\rho}_{m}$ | Magma density (kg·m^{−3}) |

${u}_{sb}$ | Ascent velocity of a spherical bubble (m·s^{−1}) |

$l$ | Bubble length (m) |

$\mu $ | Magma viscosity (Pa·s) |

${\rho}_{g}$ | Gas density (kg·m^{−3}) |

$g$ | Gravitational acceleration (m·s^{−2}) |

${P}_{slim}^{*}$ | Dimensionless burst vigour |

${A}^{\prime}$ | Ratio of bubble radius to pipe radius |

${P}_{surf}$ | Atmospheric pressure at the surface (Pa) |

${r}_{TB}$ | Taylor bubble radius (m) |

${r}_{c}$ | Conduit radius (m) |

${\lambda}^{\prime}$ | Falling film thickness (m) |

${N}_{f}$ | Dimensionless inverse viscosity |

$Fr$ | Froude number |

${u}_{TB}$ | Taylor bubble base ascent velocity (m·s^{−1}) |

${l}_{wake}$ | Taylor bubble wake length (m) |

${l}_{min}$ | Taylor bubble interaction length (m) |

${d}_{e}$ | Equivalent diameter (m) |

${V}_{b}$ | Bubble volume (m^{3}) |

$Re$ | Reynolds number |

${C}_{d}$ | Bubble drag coefficient |

${u}_{CB}$ | Spherical cap bubble base ascent velocity (m·s^{−1}) |

${d}_{b}$ | Bubble diameter (m) |

${V}_{w}$ | Volume of spherical cap bubble wake (m^{3}) |

${c}_{wake}$ | Spherical cap bubble wake length (m) |

${l}_{CB}$ | Cap bubble length (m) |

${c}_{min}$ | Spherical cap bubble interaction length (m) |

${t}_{min}$ | Minimum repose time (s) |

${t}_{transition}$ | Transition time (s) |

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**Figure 1.**An illustration of (

**a**) spherical-cap and (

**b**) Taylor bubble morphologies including the bubble features relevant to the model described here. Any bubble falling within the wake length of the bubble ahead of it is considered liable for imminent coalescence. Any bubble beyond the interaction length would be considered to flow independently (e.g., in the single regions of the model outputs shown in Figure 2 and Figure 3). Bubbles within the interaction length are affected by the leading bubble (e.g., falling within the rapid regions of Figure 2 and Figure 3).

**Figure 2.**Left—an illustrative example of the zonation between activity classes associated with the model, plotted on arbitrary inter-bubble repose time vs. bubble volume axes. Within area (

**a**) are spherical-cap bubbles, which produce passive activity or light puffing, (

**b**,

**c**) are Taylor bubble flow scenarios resulting in non-explosive and explosively-bursting scenarios, respectively. Area (

**d**) represents cases where bubbles rise in sufficient proximity to one another to affect one another’s fluid dynamics, while (

**e**) corresponds to a region in which independent bubble bursting is unlikely due to coalescence between neighboring bubbles. Right—a further illustrative example with only defining lines between sections of the model. Important equations and bubble features are highlighted. Please see the text for full details on these.

**Figure 3.**Outputs from the model run with input conduit and fluid dynamic parameters appropriate to: (

**a**) Mt. Etna; (

**b**) Stromboli; and (

**c**) and Yasur volcanoes. Overlain on the plots are data points derived from field measurements on these targets, for Etna from Pering et al. [24], and Yasur from Kremers et al. [33]. In the Stromboli case, a number of literature sources are referred to, as detailed above. In each case a single repose interval is applied, which is a minimum for the observed activity. Figure 3b also just shows the maximum, mean, and minimum burst volumes from each of the Stromboli papers to simplify the graphic.

**Table 1.**Slug length and repose time data associated with rapid strombolian activity in the Yasur volcano (Kremers et al. [33] Table 2, note, only bursts with defined repose times are taken and this is then taken as the time between bursts) in addition to calculated explosive gas volumes.

Length (m) | Volume (m^{3}) | Repose (s) |
---|---|---|

118.5 | 285 | 82 |

174.9 | 354 | 29 |

138.5 | 308 | 160 |

68.6 | 207 | 59 |

109.7 | 271 | 73 |

149.5 | 322 | 149 |

142.1 | 313 | 185 |

102.6 | 262 | 380 |

83.7 | 235 | 82 |

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**MDPI and ACS Style**

Pering, T.D.; McGonigle, A.J.S. Combining Spherical-Cap and Taylor Bubble Fluid Dynamics with Plume Measurements to Characterize Basaltic Degassing. *Geosciences* **2018**, *8*, 42.
https://doi.org/10.3390/geosciences8020042

**AMA Style**

Pering TD, McGonigle AJS. Combining Spherical-Cap and Taylor Bubble Fluid Dynamics with Plume Measurements to Characterize Basaltic Degassing. *Geosciences*. 2018; 8(2):42.
https://doi.org/10.3390/geosciences8020042

**Chicago/Turabian Style**

Pering, Tom D., and Andrew J. S. McGonigle. 2018. "Combining Spherical-Cap and Taylor Bubble Fluid Dynamics with Plume Measurements to Characterize Basaltic Degassing" *Geosciences* 8, no. 2: 42.
https://doi.org/10.3390/geosciences8020042