# A New Degassing Model to Infer Magma Dynamics from Radioactive Disequilibria in Volcanic Plumes

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## Abstract

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^{210}Pb-

^{210}Bi-

^{210}Po radioactive disequilibria in gaseous emanations from the volcano. These new results [$({}^{210}\mathrm{Po}/{}^{210}\mathrm{Pb})=42$ and $({}^{210}\mathrm{Bi}/{}^{210}\mathrm{Pb})=7.5$] are in fair agreement with those previously reported. Previously published degassing models fail to explain satisfactorily measured activity ratios. We present here a new degassing model, which accounts for

^{222}Rn enrichment in volcanic gases and its subsequent decay into

^{210}Pb within gas bubbles en route to the surface. Theoretical short-lived radioactive disequilibria in volcanic gases predicted by this new model differ from those produced by the former models and better match the values we measured in the plume during the 2015 campaign. A Monte Carlo-like simulation based on variable parameters characterising the degassing process (magma residence time in the degassing reservoir, gas transfer time, Rn-Pb-Bi-Po volatilities, magma volatile content) suggests that short-lived disequilibria in volcanic gases may be of use to infer both magma dynamics and degassing kinetics beneath Mount Etna, and in general at basaltic volcanoes. However, this simulation emphasizes the need for accurately determined input parameters in order to produce unambiguous results, allowing sharp characterisation of degassing processes.

## 1. Introduction

_{2}flux measurements [1] and analysis of the sulfur content preserved in melt inclusions [2]. At subduction-zone-related volcanoes, the amount of degassing magma usually exceeds by one or several orders of magnitude the volume of lava actually erupting during the same period [3], which is notably the case of Mount Etna [4]. Moreover, the budget of magma entering the Mount Etna plumbing system, inferred from gravity changes and deformation data, matches the volume of degassing magma and exceeds by far the erupted lava volume [5]. It thus emphasizes the fact that degassing and actually erupted volumes of magma can be significantly unbalanced, a feature that is not observed at non-subduction-related volcanoes [3,6]. This feature also appears to depend on the eruptive style over a given temporal window [7]. At a volcano like Mount Etna (subduction-zone-related stratovolcano fed with volatile-rich-alkali basalts, e.g., [8]), it could be explained by a dynamic regime of magma redistribution beneath the volcano during which degassed magma is continuously removed by convection from the degassing reservoir and replaced by fresh undegassed magma [9,10,11].

^{238}U decay chain:

^{210}Pb,

^{210}Bi and

^{210}Po (see Figure 1). Because they all have short half-lives (22 years, 5 days and 138 days, respectively), these isotopes are suitable to study recent fractionations (younger than two years based on

^{210}Po half-life, the longest-lived

^{210}Pb decay product) associated with pre-eruptive and syn-eruptive magmatic processes. Furthermore, lead, bismuth and polonium are strongly fractionated upon degassing (polonium being more volatile than bismuth, which is in turn more volatile than lead), which gives birth to large radioactive disequilibria between

^{210}Pb-

^{210}Bi-

^{210}Po in the gas phase [14,16,17]. These properties have enabled radioactive disequilibrium measurements in volcanic plumes to be linked to degassing activity through the use of two models. Lambert et al. [14] first developed a static degassing model for which radionuclide exsolution takes place in a degassing cell containing a proportion $\mu $ of deep undegassed magma in radioactive equilibrium. They also considered the transfer time of radionuclides between the time of exsolution from the magma and the time of emission at the surface, as

^{210}Bi (five days half-life) is short-lived enough to significantly decay during gas transfer towards the surface. This approach has been extensively used to characterise gaseous emissions at Mount Etna [14,18,19]. More recently, Gauthier et al. [15] proposed a dynamic degassing model for which radioactive disequilibria in the gas phase also depend on the magma residence time in the degassing reservoir because of continuous regeneration of highly volatile

^{210}Po by decay of its less volatile parent (

^{210}Bi) within the reservoir. This latter model has been successfully applied to persistently degassing, open-conduit, basaltic volcanoes like Stromboli [15] or Ambrym [20]. At these two volcanoes, both the magma residence time in the degassing reservoir and the transfer time of the gas phase towards the surface were estimated and were shown to vary according to eruptive activity.

^{222}Rn can be significantly enriched in the gas phase when magma residence time in the degassing reservoir increases. Although it has been neglected so far, mostly because of its short half-life of 3.8 days, we show that

^{222}Rn plays a major role in controlling the magnitude of

^{210}Pb-

^{210}Bi-

^{210}Po disequilibria by producing, through its radioactive decay, a new generation of

^{210}Pb atoms within gas bubbles. By using a dataset of previously published values describing both trace element volatilities and volatile content in magmas from Etna, we present a Monte Carlo-like simulation that explains radioactive disequilibria measured in 2015 in Mount Etna gases well. Implications for the retrieval of quantitative information on degassing dynamics from radioactive disequilibria in a volcanic plume are presented at the end of the paper.

## 2. Short-Lived Radioactivity Measurements in Mount Etna Plume

#### 2.1. Field Description and Sampling Techniques

^{210}Pb-

^{210}Bi-

^{210}Po radioactive disequilibria in the volcanic plume. The field campaign took place during a brief, 5-day-long eruption at NSEC. After several days of tremor increase, eruptive activity started at NSEC on 12 May 2015 with loud explosions producing reddish to dark grey ash-rich plumes, followed by sustained strombolian activity at NSEC summit while a fissure opened on the eastern flank of the cone, emitting a lava flow that travelled towards and inside Valle del Bove. On 15 May, the intensity of the eruption gradually decreased until it reached an end on 16 May. The eruptive activity at NSEC during this short eruptive episode was too intense to grant safe access to the summit. We collected the diluted plume of Mount Etna downwind of the crater on the southern slope of the volcano, at remote sites near Torre del Filosofo (Figure 2). Samples collected at these locations were mostly from the main ash-rich gas plume emitted by the NSEC summit, although a minor contribution of the small gas plume released from the eruptive fissure cannot be ruled out.

#### 2.2. Analytical Techniques

^{210}Bi (5.01 days half-life) thoroughly decays away in about one month, filter samples were taken back to Laboratoire Magmas et Volcans in Clermont-Ferrand within a few days after collection. Untreated filters were analysed with a low-background noise alpha-beta counting unit (IN20 Canberra) following the procedure described in Gauthier et al. [21]. Repeated measurements were carried on over a month after collection in the field, each analytical cycle lasting 48 h (one cycle comprises eight 6-hour-long counting blocks that are ultimately averaged, the mean activity being considered as the “instantaneous” activity at the time ${t}_{i}$ + 24 h, where ${t}_{i}$ is the starting date and time of a given cycle). Both alpha and beta activities were simultaneously determined on filter samples. Alpha counts provide a direct measure of

^{210}Po activity while beta counts correspond to the detection of

^{210}Bi beta decay particles. Beta emissions of

^{210}Pb cannot be directly measured because of their energies that are too low, so that

^{210}Pb is measured via

^{210}Bi one month after sampling when both isotopes have reached radioactive equilibrium [21]. Radionuclide activities on filter samples were determined by subtracting the detector background (electronic noise and some unblocked cosmic rays) and by taking into account the detector efficiency and an attenuation factor for alpha particles [25]. Initial activities at the time of sampling were then retrieved by fitting radioactive decay trend during the one-month period of analysis by using classical radioactivity laws.

#### 2.3. Analytical Results

^{210}Pb and

^{210}Bi detection limit can be quantified at $0.8\phantom{\rule{0.277778em}{0ex}}\mathrm{m}\mathrm{Bq}/{\mathrm{m}}^{3}$ and that of

^{210}Po at $0.3\phantom{\rule{0.277778em}{0ex}}\mathrm{m}\mathrm{Bq}/{\mathrm{m}}^{3}$. Although activities are below detection limits for samples TDF1 and TDF3, and similar to the bêta detection limit for sample TDF2, all other samples appear significantly enriched in

^{210}Pb,

^{210}Bi and

^{210}Po compared to the atmospheric blank. As previously shown ([19], and references therein), this suggests that the plume of Mount Etna is considerably enriched in radionuclides over a standard atmosphere and that high-quality samples can be obtained at safe distances from the summit area. Figure 3 shows

^{210}Bi (Figure 3a) and

^{210}Po (Figure 3b) activities plotted against

^{210}Pb activities for all aerosol samples. Activities follow linear trends passing through the origin, which are interpreted as dilution trends of the volcanic gas (considerably enriched in

^{210}Pb,

^{210}Bi and

^{210}Po) into the atmosphere for which radionuclide activities are negligible. The two linear correlations in Figure 3 are well defined (${R}^{2}=0.99$ for

^{210}Bi vs.

^{210}Pb; ${R}^{2}=0.91$ for

^{210}Po vs.

^{210}Pb), which suggests that the dilution of volcanic gases in the atmosphere does not significantly affect their pristine isotopic signature within at least 1.5 km distance from the summit area. Furthermore, it was reported that radioactive disequilibria in gases released at summit craters significantly differ from those in gas emanations from eruptive vents along eruptive fissures [19]. Therefore, the well-defined linear correlations observed in Figure 3 suggest that the contribution of gases released at the eruptive fissure to the main plume is negligible or alternatively steady through time, which would be highly fortuitous. Samples collected in the diluted plume are thus taken to be representative of the chemistry of volcanic gases at the source. Radioactive disequilibria in the volcanic plume of the NSEC are retrieved by linear regression of the whole dataset at (

^{210}Bi/

^{210}Pb) $=7.5\pm 0.4$ and (

^{210}Po/

^{210}Pb) $=42\pm 6$. These values are in fair agreement with those previously reported for Mount Etna’s summit craters, in the range 10–30 for (

^{210}Bi/

^{210}Pb) and between 20 and up to 90 for (

^{210}Po/

^{210}Pb) ([19], and references therein).

## 3. Modelling of Radionuclide Degassing and ^{210}Pb-^{210}Bi-^{210}Po Radioactive Disequilibria in Volcanic Plumes

^{210}Pb-

^{210}Bi-

^{210}Po radioactive disequilibria in volcanic gases to degassing processes. Although radioactive disequilibria in volcanic gases from Mount Etna have been successfully explained by the model of Lambert et al. [14], Gauthier et al. [15] showed that Lambert’s model can be used only if the magmatic vapour is released from a rapidly overturned batch of deep magma in radioactive equilibrium prior to degassing. This is due to the fact that the model proposed by Lambert et al. [14] neglects, in the degassing magma, the radioactive ingrowth of

^{210}Bi and

^{210}Po (both moderately to highly volatile at magma temperature) from their parent

^{210}Pb, which is weakly volatile and mostly remains in the melt. If the magma residence time in the degassing reservoir is long enough for

^{210}Pb to decay, then both

^{210}Bi and

^{210}Po atoms are regenerated in the melt. The longer is the residence time, the more efficient is the regeneration. Because

^{210}Pb decay products have greater affinity for the gas phase than for the melt, they preferentially partition into the vapour phase according to their own volatility. When the magma residence time in the degassing reservoir increases, magmatic gases consequently become more and more enriched in the most volatile

^{210}Po and, to a lesser extent, in the moderately volatile

^{210}Bi over

^{210}Pb [15]. By neglecting radioactive ingrowth within the degassing magma, Lambert et al. [14] considered that the ratio of two radionuclides in the magmatic vapour at the time of exsolution could reach a maximum value equal to the ratio of their emanation coefficients $\u03f5$, the widely used parameter describing trace element volatility [14,26]. Gauthier et al. [15] showed that the $\u03f5$ ratio of two radionuclides corresponds instead to the minimum value for activity ratios in the magmatic vapour at the time of volatile exsolution. This minimum value is reached when magma residence time is negligible compared to

^{210}Po half-life i.e., less than about 10 days. For increasing values of the magma residence time, activity ratios in the gas phase increase up to a theoretical value, which is defined by the ratio of the liquid–gas partition coefficients D for the considered radionuclides [15].

^{210}Po for basaltic systems (including Mount Etna) is close to 100%, suggesting that most of polonium atoms are transferred to the magmatic vapour upon degassing [22,27,28,29,30]. The emanation coefficient of lead in basaltic systems appears to be higher in calk-alkaline systems than in other geodynamical settings [21]. At Mount Etna, like at other arc-related basaltic volcanoes, it has a value of approximately 1.0 ± 0.5% [14,17,31,32]. Therefore, the minimum value for (

^{210}Po/

^{210}Pb) activity ratios in Mount Etna gases should be around 100, which has never been measured, excepted at the beginning of the 1992 eruption [19]. In particular, the ratio of 42 ± 6 we find in the May 2015 plume is far below this minimum theoretical value and cannot be explained by using the dynamic degassing model with realistic parameters for metal volatility [15]. In order to reproduce our observations, the degassing model would indeed require either polonium emanation coefficients lower than 50%, which is in strong disagreement with analyses of freshly erupted lavas at Mount Etna [22], or else lead emanation coefficients up to 3% or even more, which has never been reported.

#### 3.1. ^{222}Rn Enrichments in Volcanic Gases: Towards a New Degassing Model for Short-Lived Radionuclides

^{222}Rn-rich gas phase may present significant

^{210}Pb excesses over

^{226}Ra [29,33,34,35,36]. Volcanic gases are also characterised, in some cases, by

^{210}Pb/Pb ratios significantly higher than those measured in lavas [23]. Although the origin of these

^{210}Pb enrichments in volcanic gases has not been fully understood, we tentatively assume that

^{210}Pb excesses could result from radioactive decay of

^{222}Rn atoms in the gas phase.

^{222}Rn was not taken into account. We present therefore a new degassing model that accounts for it. The conceptual framework of this new model matches that of Gauthier et al. [15]. Accordingly, we consider that volatile exsolution takes place in an open degassing reservoir, which has reached a dynamical and chemical steady-state. Dynamical steady-state implies that the degassing reservoir has a constant mass M (or volume V) through time. It means that any input flux ${\varphi}_{0}$ of deep undegassed magma (in radioactive equilibrium for

^{222}Rn and its daughters) has to be balanced by a flux of gas ${\varphi}_{G}$ and a flux of lava ${\varphi}_{L}$ leaving the reservoir. Let $\alpha $ be the fraction of volatiles initially dissolved in the deep magma and ultimately released, the fluxes ${\varphi}_{G}$ and ${\varphi}_{L}$ can be written:

_{k}(either

^{222}Rn,

^{210}Pb,

^{210}Bi,

^{210}Po, k depending on the position along the decay chain) in the degassing magma varies according to:

_{k}either in the undegassed magma (index 0), the degassed lava (index L) or the gas phase (index G), and where ${\lambda}_{k}$ stands for the radioactive decay constant of I

_{k}. From left to right, the right terms of Equation (2) correspond to the production of I

_{k}by I

_{k−1}decay, the loss of I

_{k}according to its own decay, the input of I

_{k}from the undegassed magma entering the reservoir, the output of I

_{k}by magma withdrawal and finally the output of I

_{k}by degassing. This equation can be expressed in terms of activity per unit of mass by multiplying each member by ${\lambda}_{k}/M$:

#### 3.2. Radioactive Disequilibria in Gases at the Time of Exsolution

#### 3.2.1. Specific Case of ^{222}Rn Exsolution

_{k}in the gas phase depends on the activity of its precursor in the decay chain, which means that a precursor for

^{210}Pb has to be taken into account so as to compute all other activities. All four radionuclides between

^{222}Rn and

^{210}Pb (

^{218}Po,

^{214}Pb,

^{214}Bi and

^{214}Po) have half-lives of a few minutes at most (Figure 1), which appears very short compared to the expected residence time of the magma in the degassing reservoir. They can thus be neglected in the model and we consider here, mathematically speaking, that

^{210}Pb is directly regenerated by

^{222}Rn decay. In order to quantify

^{222}Rn degassing efficiency Gauthier and Condomines [37] introduced a parameter f, ranging between 0 (no radon degassed) and 1 (total degassing of radon), and used this previous parameter to calculate

^{222}Rn activity in the degassing melt at steady-state:

^{222}Rn, replacing ${\left({}^{222}\mathrm{Rn}\right)}_{L}$ and ${\left({}^{226}\mathrm{Ra}\right)}_{L}$ by their expression in Equations (7) and (8), respectively, and ${\varphi}_{0}/M$ by $1/\tau $, we obtain:

^{222}Rn can be produced in the gas phase for long residence times. Providing that the escape time of gases is long enough (see Section 3.3),

^{222}Rn atoms could then decay and act as a significant additional source of

^{210}Pb in the gas phase.

#### 3.2.2. ^{222}Rn-^{210}Pb-^{210}Bi-^{210}Po Fractionation upon Exsolution

^{210}Pb,

^{210}Bi,

^{210}Po activities in the gas phase after exsolution can now be derived iteratively using Equation (5). Because short-lived

^{226}Ra daughters are thought to be in radioactive equilibrium in deep magmas prior to degassing [37], the application of Equation (5) to

^{210}Pb, taking into account the expression of ${\left({}^{222}\mathrm{Rn}\right)}_{L}$ given by Equations (7) and (8) leads to:

^{210}Bi, replacing ${\left({}^{210}\mathrm{Pb}\right)}_{L}$ by ${\left({}^{210}\mathrm{Pb}\right)}_{G}/{D}_{\mathrm{Pb}}$ and using the expression of ${\left({}^{210}\mathrm{Pb}\right)}_{G}$ given in Equation (10), leads to an expression of ${\left({}^{210}\mathrm{Bi}\right)}_{G}$. This expression is then used to determine ${\left({}^{210}\mathrm{Po}\right)}_{G}$, still using Equation (5). The obtained expressions are reproduced in Appendix A and, like Equations (9) and (10), they have the following form:

#### 3.3. Gas Phase Transfer towards the Surface: Radioactive Decay within Gas Bubbles

_{k}in the magma. Their expression are not reproduced here, but they can be explicited replacing M, ${M}^{-1}$, ${\left({}^{222}\mathrm{Rn}\right)}_{G}$, ${\left({}^{210}\mathrm{Pb}\right)}_{G}$, ${\left({}^{210}\mathrm{Bi}\right)}_{G}$ and ${\left({}^{210}\mathrm{Po}\right)}_{G}$ by their expressions provided before or in the appendix.

#### 3.4. Results and Discussion

^{210}Pb,

^{210}Bi and

^{210}Po in the gas phase according to our model are presented in Figure 4. Values of $\alpha $, f and volatility for Pb, Bi and Po (D or equivalently $\u03f5$) are fixed using reasonable estimates from the literature ($\alpha =5$ wt.%, $f=1$, ${\u03f5}_{\mathrm{Pb}}=1.5$%, ${\u03f5}_{\mathrm{Bi}}=36$%, ${\u03f5}_{\mathrm{Po}}=100$%). The choice of these values, and their impact on the model predictions, are discussed later (see Section 4). Theoretical values of disequilibria are plotted against the residence time in the degassing reservoir for several values of the transfer time. These new results are systematically compared to radioactive disequilibria produced by the model of Gauthier et al. [15].

^{222}Rn/

^{210}Pb) in the gas phase (not shown in Figure 4) upon exsolution ($\theta =0$) dramatically increases with the residence time and can reach values as high as 1000 for residence times up to 100 days. Therefore, huge

^{222}Rn enrichments can be generated because of the difference of gas-melt partitioning coefficients between

^{222}Rn and its non-volatile precursor

^{226}Ra. This prediction confirms the potential for important

^{210}Pb ingrowth in the gas phase during its transfer.

^{210}Bi/

^{210}Pb) and (

^{210}Po/

^{210}Pb) (Figure 4) in the gas phase at the time of exsolution ($\theta =0$) are identical between the two models, which was expected since

^{222}Rn has no time to decay within gas bubbles. The observed trends have been explained by Gauthier et al. [15]: ${({}^{210}\mathrm{Po}/{}^{210}\mathrm{Pb})}_{G}$ significantly increases with the residence time $\tau $, as a result of

^{210}Po regeneration in the liquid phase of the degassing reservoir; ${({}^{210}\mathrm{Bi}/{}^{210}\mathrm{Pb})}_{G}$ also increases with $\tau $ but at a slower rate. This is because the regeneration of

^{210}Bi by

^{210}Pb decay in the liquid phase is not as important, which is due to the smaller difference of volatility between Pb and Bi than between Pb and Po.

^{210}Bi/

^{210}Pb) activity ratio in the gas phase, as computed with the former model ([15], dashed lines in Figure 4a), decreases with increasing values of $\theta $ from 0 to 30 days owing to the short half-life of

^{210}Bi. In about one month (6 times

^{210}Bi half-life of 5.01 days),

^{210}Bi is back to equilibrium with its parent

^{210}Pb, leading to an activity ratio of 1. Our model suggests, however, that low values of ${({}^{210}\mathrm{Bi}/{}^{210}\mathrm{Pb})}_{G}$ can be produced for gas transfer time $\theta $ as short as a few days (solid lines, Figure 4a). This feature is explained by both the radioactive decay of

^{210}Bi according to its own half-life and the radioactive decay of

^{222}Rn within gas bubbles, which produces new

^{210}Pb atoms. Accordingly, ${\left({}^{210}\mathrm{Bi}\right)}_{G}^{\theta}$ decreases for increasing values of $\theta $ while ${\left({}^{210}\mathrm{Pb}\right)}_{G}^{\theta}$ increases, leading to a faster decrease in the ${({}^{210}\mathrm{Bi}/{}^{210}\mathrm{Pb})}_{G}$ activity ratio. However, for longer transfer times (e.g., $\theta =30$ days), the two models produce again similar values close to the equilibrium ratio of 1, as expected since both

^{222}Rn and

^{210}Bi have similar half-lives (3.82 days and 5.01 days, respectively). In other words, the most important difference between the two models happens for magma residence times $\tau $ higher than a few hundred days (significant regeneration of

^{222}Rn from

^{226}Ra in the degassing melt and subsequent radon enrichments in the gas phase) and gas transfer times $\theta $ shorter than a few days (significant

^{222}Rn-driven production of novel atoms of

^{210}Pb within gas bubbles with limited return of

^{210}Bi to radioactive equilibrium).

^{222}Rn decay is even more pronounced since the decrease in ${\left({}^{210}\mathrm{Po}\right)}_{G}^{\theta}$ according to the half-life of

^{210}Po (138.4 days) is rather limited for short transfer times $\theta $. The radioactive decay of

^{222}Rn thus strongly controls the magnitude of

^{210}Pb-

^{210}Po radioactive disequilibria. For instance, for values of $\tau $ higher than 150 days, it can be seen that our degassing model produces ${({}^{210}\mathrm{Po}/{}^{210}\mathrm{Pb})}_{G}$ activity ratios at $\theta =1$ h lower than those derived from Gauthier et al. [15] for $\theta =30$ days.

^{222}Rn into account in dynamic degassing models. They also suggest that activity ratios in the gas phase as low as those measured in the plume of Mount Etna could be explained and modelled within this novel theoretical framework.

## 4. Model Application

#### 4.1. Estimation of Input Paramaters

#### 4.1.1. Volatile Weight Fraction $\alpha $

_{2}O) to be flushed out of the magma [40]. The total volatile content of magmas is often estimated from volatile concentrations in melt inclusions trapped in crystals, assuming that these melt inclusions represent the deep undegassed magma. In magmatic systems, the main volatile species are, by decreasing order of importance, H

_{2}O, CO

_{2}, S-species (mostly SO

_{2}in basaltic systems), HCl and HF. Since these species have different solubilities in basalts, the depth-related pressure of inclusion entrapment has to be considered in order to derive a reliable total volatile content dissolved in the magma prior to degassing. At Mount Etna, olivine-hosted melt inclusions have been studied for long by different authors e.g., [41,42]. Their studies yield to close estimates of the volatile weight fraction $\alpha $ in the range 4–5 wt.%. Such high volatile content appears to be characteristic of alkali-rich basaltic magmas like those of Etna [42]. In other geodynamical settings and especially at non-arc-related volcanoes, the total amount of dissolved volatile usually is much lower [2]. Nevertheless, it must be pointed out that the model still applies to these volcanoes, provided that $\alpha $ is carefully quantified.

#### 4.1.2. Fraction of Degassed Radon f

^{222}Rn enrichments in the gas phase and, subsequently, the radioactive ingrowth of

^{210}Pb within gas bubbles. Very few data exist in the literature about radon degassing from basaltic magmas. Nevertheless, analyses of freshly erupted basalts and andesites [27,43] provide evidence for almost thorough radon degassing from erupting magmas. Further experimental studies [40,44] confirm that radon is entirely flushed out of mafic magmas upon degassing, provided that a major gas species can act as a carrier. Although f is most likely close to 1 at Etna, we use here a conservative estimate with f varying between 90% and 100%.

#### 4.1.3. Volatilities of Lead, Bismuth and Polonium (Emanation Coefficients $\u03f5$ and Gas-Melt Partitioning Coefficients D)

^{226}Ra/

^{210}Pb) in erupted basaltic lavas ([35], and references therein) including Etnean basalts [45] suggest a minimal loss of

^{210}Pb upon degassing and hence an emanation coefficient of lead of a few percents at most (1.5% according to [14]). Mather [26] computed a “volatility coefficient” (equivalent to a gas-melt partitioning coefficient) for lead using gas and lava data from Mount Etna, which is as low as 0.13. Using Equation (6) with $\alpha =5$ wt.%, it corresponds to an emanation coefficient of 0.7%. Therefore, we take ${\u03f5}_{\mathrm{Pb}}$ between 0.7 and 1.5% (corresponding to ${D}_{\mathrm{Pb}}$ values in the range 0.13–0.37).

#### 4.2. Inversion of the Model

#### 4.2.1. Methodology

- each parameter (see Table 2) is chosen randomly in its range of variation according to an uniform law.
- the residence time $\tau $ and the transfer time $\theta $ are also chosen randomly (between 0 and 5000 days for $\tau $, and between 0 and 15 days for $\theta $). The upper limit for $\tau $ is in agreement with the order of magnitude of Mount Etna magma residence time in shallow reservoirs: a few tens of years in Condomines et al. [45], one year in Armienti et al. [46]. The upper limit for $\theta $ is coherent with maximum estimates of the gas phase transfer time at Mount Etna [14].
- radioactive disequilibria in the gas phase are computed according to the model equations.
- if the computed values match the measured ones, then the set of parameters and the dynamic variables ($\tau $ and $\theta $) are stored in a database.
- these operations are repeated until a statistically relevant database (here 10,000 elements) is built. If enough combinations of parameters are simulated, the parameter space is sampled without any important gap.

#### 4.2.2. Discussion of Results and Implications for Magma Dynamics at Mount Etna

_{2}flux measured during the field campaign and the eruptive stage from 12 May to 16 May (5200 $\mathrm{t}$/$\mathrm{d}$). Scaled to the sulfur content of Etnean basalts (0.3 wt.% in [42]), considering complete SO

_{2}degassing and a magma density of 2700 $\mathrm{k}\mathrm{g}$/${\mathrm{m}}^{3}$, we find ${\varphi}_{0}=8.7\times {10}^{8}\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\mathrm{g}/\mathrm{d}=3.2\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}/\mathrm{d}$. Owing to the definition of the magma residence time, the volume of the degassing reservoir is merely given by $V={\varphi}_{0}\tau $. Our estimate of ${\varphi}_{0}$ thus yields a volume of 0.15–0.30 $\mathrm{k}{\mathrm{m}}^{3}$ for a degassing reservoir having a residence time of 500–1000 days (peak values in Figure 6). A residence time of 5000 days (highest value tested in our simulation; Figure 6) would correspond to a reservoir volume of 1.5 $\mathrm{k}{\mathrm{m}}^{3}$.

## 5. Conclusions

^{210}Pb-

^{210}Bi-

^{210}Po radioactive disequilibria in the plume of Mount Etna. Measured activity ratios in the gas phase are in good agreement with values previously reported in the literature for the volcano. However, they can not be explained by existing theoretical models accounting for radionuclide degassing.

^{222}Rn, ignored in previous models, in the exsolved magmatic gas phase is thought to play a major role on the

^{210}Pb-

^{210}Bi-

^{210}Po systematics by producing

^{210}Pb excesses in the gas phase during its transfer towards the surface. Here, this contribution has been modelled theoretically and it appears to produce radioactive disequilibria that can be twice as low as those predicted by the former degassing model of Gauthier et al. [15].

^{222}Rn enrichments in producing

^{210}Pb excesses in the gas phase. Precise quantification of magma dynamics (i.e., magma residence time $\tau $ and gas transfer time $\theta $) beneath active volcanoes through the use of our degassing model necessitates a sharp characterisation of the different input parameters, especially radionuclide emanation coefficients $\u03f5$. Nevertheless, using a range of published estimates for $\u03f5$ values, we found that measured activity ratios in the plume of Mount Etna are most likely explained by a magma residence time in the degassing reservoir of 500–1000 days and a transfer time no longer than seven days. These figures correspond to a volume of degassing magma of about 0.15 $\mathrm{k}{\mathrm{m}}^{3}$ with an exsolution depth of no more than 5 km bsl. This volume of magma and its location in the shallowest part of the volcanic edifice suggests that most of the degassing process takes place within the shallow feeding system of Etna whose dynamics controls eruptive activity at the summit craters.

^{222}Rn activity in diluted volcanic plumes in order to provide evidence of radon enrichments. It will have further implications, notably in better deciphering

^{210}Pb-

^{210}Bi-

^{210}Po desequilibria in volcanic gases, including at basaltic volcanoes from other geodynamical settings.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

^{210}Bi and

^{210}Po in the gas phase are determined by following the procedure described in Section 3.2.2. Corresponding expressions are reproduced hereafter:

## Appendix B

^{222}Rn,

^{210}Pb,

^{210}Bi and

^{210}Po, respectively.

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**Figure 1.**

^{238}U decay chain. Half-lives are indicated beneath the symbol of the element. Minor embranchments are not reproduced.

**Figure 2.**Map of Mount Etna summital zone in 2012 (aerial photography from Italian Geoportal). CC stands for Central Craters (Voragine and Bocca Nuova), NEC for northeast Crater and NSEC for new southeast Crater. Sampling sites (TDF1, TDF2-3, TDF4 and TDF5-6, where TDF stands for Torre del Filosofo) are pointed on the map.

**Figure 3.**Volumic activities in the plume. Bars represent one sigma errors and are derived from the uncertainty of radioactivity analysis. Full lines represent the linear regression between the points and the origin, and dotted lines represent two sigma standard error on the slope of the regression trend.

**Figure 4.**Radioactive disequilibria (

**a**) (

^{210}Bi/

^{210}Pb) and (

**b**) (

^{210}Po/

^{210}Pb) in the gas phase versus magma residence time in the degassing reservoir according to the new model (plain lines) and according the model of Gauthier et al. [15] (dashed lines). Curves for several values of transfer time $\theta $ (0, 1 h, 1, 5 and 30 days) are drawn. Fixed parameters of the model are $\alpha =5$ wt.%, $f=1$, ${\u03f5}_{\mathrm{Pb}}=1.5\%$, ${\u03f5}_{\mathrm{Bi}}=36\%$ et ${\u03f5}_{\mathrm{Po}}=100\%$.

**Figure 5.**Histograms of the model parameters ($\alpha $, f, ${\u03f5}_{\mathrm{Pb}}$, ${\u03f5}_{\mathrm{Bi}}$, ${\u03f5}_{\mathrm{Po}}$) for which the model can explain the measurements. The horizontal bar represents the uniform law used to generate random values. This line corresponds to the theoretical histogram that would be obtained for a parameter having no influence on the production of simulated results matching measured activity ratios. Its frequency is not relevant in itself and is merely equal to the reciprocal of the number of bars in the histogram.

**Figure 6.**Histograms of the dynamic parameters ($\tau $ and $\theta $) for which the model can explain the measurements. The horizontal bar represents the uniform law used to generate random values. See the caption of Figure 5 for further details.

**Figure 7.**Scatter plot of the dynamic parameters ($\tau $ versus $\theta $) for which the model can explain the measurements.

**Table 1.**

^{210}Pb-

^{210}Bi-

^{210}Po radioactivity in Mount Etna plume. Sample names correspond to sampling sites as described in Figure 2. Activities are reported in $\mathrm{mBq}$/${\mathrm{m}}^{3}$ with 1-$\sigma $ uncertainties. “bdl” stands for below detection limit. Sampling starting time is expressed in local time.

Sample | Date and time | Volume (m^{3}) | ^{210}Pb | ^{210}Bi | ^{210}Po |
---|---|---|---|---|---|

Atmospheric blank | 11/05/2015 12:00 | 3.9 | bdl | bdl | bdl |

TDF1 | 12/05/2015 12:07 | 3.9 | bdl | bdl | bdl |

TDF2 | 13/05/2015 11:04 | 6.6 | 0.8 ± 0.2 | 3.1 ± 1 | 14.6 ± 0.7 |

TDF3 | 13/05/2015 12:15 | 6.6 | bdl | bdl | bdl |

TDF4 | 14/05/2015 10:51 | 1.7 | 3.5 ± 1.0 | 29 ± 5 | 212 ± 3 |

TDF5A | 14/05/2015 11:38 | 3.9 | 4.0 ± 0.4 | 31 ± 3 | 162 ± 2 |

TDF5B | 14/05/2015 11:38 | 3.0 | 8.2 ± 0.4 | 62 ± 2 | 349± 2 |

TDF6 | 14/05/2015 12:31 | 6.6 | 5.9 ± 0.2 | 41 ± 2 | 217 ± 1 |

Parameter | Range | References |
---|---|---|

$\alpha $ (wt.%) | 4–5 | Métrich et al. [41], Spilliaert et al. [42] |

f | 0.9–1 | Gill et al. [27], Gauthier et al. [40], Sato and Sato [43], Sato et al. [44] |

${\u03f5}_{\mathrm{Pb}}$ (%) | 0.7–1.5 | Lambert et al. [14], Mather [26] |

${\u03f5}_{\mathrm{Bi}}$ (%) | 20–45 | Lambert et al. [14], Pennisi et al. [17] |

${\u03f5}_{\mathrm{Po}}$ (%) | 80–100 | Le Cloarec et al. [22], Gill et al. [27], Reagan et al. [28], Girard et al. [30] |

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**MDPI and ACS Style**

Terray, L.; Gauthier, P.-J.; Salerno, G.; Caltabiano, T.; La Spina, A.; Sellitto, P.; Briole, P. A New Degassing Model to Infer Magma Dynamics from Radioactive Disequilibria in Volcanic Plumes. *Geosciences* **2018**, *8*, 27.
https://doi.org/10.3390/geosciences8010027

**AMA Style**

Terray L, Gauthier P-J, Salerno G, Caltabiano T, La Spina A, Sellitto P, Briole P. A New Degassing Model to Infer Magma Dynamics from Radioactive Disequilibria in Volcanic Plumes. *Geosciences*. 2018; 8(1):27.
https://doi.org/10.3390/geosciences8010027

**Chicago/Turabian Style**

Terray, Luca, Pierre-J. Gauthier, Giuseppe Salerno, Tommaso Caltabiano, Alessandro La Spina, Pasquale Sellitto, and Pierre Briole. 2018. "A New Degassing Model to Infer Magma Dynamics from Radioactive Disequilibria in Volcanic Plumes" *Geosciences* 8, no. 1: 27.
https://doi.org/10.3390/geosciences8010027