Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 https://doi.org/10.1186/s40623-021-01472-7

FULL PAPER Open Access Excitation of airwaves by bubble bursting in suspensions : regime transitions and implications for basaltic volcanic eruptions Kana Hashimoto and Ikuro Sumita*

Abstract Basaltic magma becomes more viscous, solid-like (elastic), and non-Newtonian (shear-thinning, non-zero yield stress) as its crystal content increases. However, the rheological efects on bubble bursting and airwave excitation are poorly understood. Here we conduct laboratory experiments to investigate these efects by injecting a bubble of volume V into a refractive index-matched suspension consisting of non-Brownian particles (volumetric fraction φ ) and a Newtonian liquid. We show that depending on φ and V, airwaves with diverse waveforms are excited, covering a frequency band of f = O(10 − 104) Hz. In a suspension of φ ≤ 0.3 or in a suspension of φ = 0.4 with a V smaller than critical, the bubble bursts after it forms a hemispherical cap at the surface and excites a high-frequency (HF) wave (f ∼ 1 − 2 × 104 Hz) with an irregular waveform, which likely originates from flm vibration. However, in a suspension of φ = 0.4 and with a V larger than critical, the bubble bursts as soon as it protrudes above the surface, and its aper- ture opens slowly, exciting Helmholtz resonance with f = O(103) Hz. Superimposed on the waveform are an HF wave component excited upon bursting and a low-frequency (f = O(10) Hz) air fow vented from the defating bubble, which becomes dominant at a large V. We interpret this transition as a result of the bubble flm of a solid-like φ = 0.4 suspension, being stretched faster than the critical strain rate such that it bursts by brittle failure. When the Helmholtz resonance is excited by a bursting bubble in a suspension whose surface level is further below the conduit rim, an air column (length L) resonance is triggered. For L larger than critical, the air column resonance continues longer than the Helmholtz resonance because the decay rate of the former becomes less than that of the latter. The experiments suggest that a bubble bursting at basaltic volcanoes commonly excites HF wave by flm vibration. The Helmholtz resonance is likely to be excited under a limited condition, but if detected, it may be used to track the change of magma rheology or bubble V, where the V can be estimated from its frequency and decay rate. Keywords: Magmatic suspension, Viscous-brittle transition, Bubble bursting, Broad frequency band, High-frequency wave, Helmholtz resonance, Air column resonance, Air fow

Introduction contains crystals (e.g., Gurioli et al. (2014)), which not In Strombolian and Hawaiian eruptions, bubbles burst only increase the viscosity but also give rise to viscoelas- upon surfacing and excite airwaves whose main energy ticity and non-Newtonian rheology (shear-thinning, non- is contained in the infrasonic band (Johnson and Rip- zero yield stress) (Mader et al. 2013; Namiki and Tanaka epe 2011; Fee and Matoza 2013). Magma commonly 2017). Such complex rheology is considered to afect the style of bubble bursting and airwave excitation, but their details are unknown. *Correspondence: [email protected] In order to better understand how such complex Earth and Environmental Science Course, Division of Natural System, rheology controls bubble bursting and airwave excita- Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920‑1192, Japan tion, laboratory experiments have been conducted by

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injecting an air bubble or continuous flow of air into Physical properties of suspensions a non-Newtonian gel and analyzing the recorded air- For the particles, we use spherical silicone powder (Shin- wave (Divoux et al. 2008; Vidal et al. 2009; Lyons et al. Etsu Chemical, KMP-590) with a diameter of dp ≃ 2 µ m 3 2013). These studies have shown that the characteris- and a density of ρp = 1300 kg/m . We prepare a silicone tic frequencies of the airwaves depend on the bubble oil mixture that has a refractive index close to that of volume and revealed temporal switching of the style the silicone powder, such that the suspension becomes ◦ of outgassing and airwave excitation. The magma sur- clearest at 23 C. Te oil has a density of ρl = 984.2 3 face at which the bubble bursts is usually below the kg/m , a Newtonian viscosity of ηl = 0.1 Pas, and a sur- conduit rim. Previous experiments have also modeled face tension coefcient of σ ≃ 0.021 N/m (Shin-Etsu such situation and have shown that air column reso- Chemical). Te particle - liquid density diference is 3 nance can be excited in the conduit above the bubble �ρpl = ρp − ρl = 315.8 kg/m . Further details on the (James et al. 2004; Kobayashi et al. 2010). Experiments particles, oils, the efects of gravity, thermal agitation, have also shown that resonance can be excited in the viscous and inertial stresses on the particle-scale motions cavity below the ruptured film (Vidal et al. 2006, 2010). under the strain rate of bubble ascent and rheometry, are These results indicate that airwaves excited by differ- given in the Additional fle 7: Material. ent mechanisms may be superimposed and that we In comparison, the ρp of the crystals consisting the need to distinguish them. Although previous experi- magmatic suspension (e.g., olivine, pyroxene, plagioclase ments using gels have been conducted by varying (Gurioli et al. 2014)) is in the range of ρp = 2630 − 3700 3 their concentrations which change their rheology, it kg/m , and the density of basaltic magma is 3 remained unclear how this corresponded to the crystal ρ = 2600 − 2650 kg/m (Philpotts and Ague 2009). Tus O 3 3 content in the magma. A more direct way to model the the �ρpl = (10 − 10 ) kg/m overlaps the �ρpl of our effects of crystals is to use suspensions, which however suspension. A fnite �ρpl is a cause for the non-Brown- are usually opaque. In this study we match the refrac- ian nature and the yield stress (Fall et al. 2009) of these tive indices of particles and the liquid (Wiederseiner suspensions. et al. 2011), which allows us to visualize the subsurface We thoroughly mix the particles and the oil to form bubble. suspensions with volumetric packing fractions of φ = Tis paper is organized as follows. First we describe 0.10, 0.30, 0.40, and 0.51. We hereafter use φ rounded the rheology of the suspensions we use for our experi- to 1 digit to denote a suspension with a specifed φ . Te ments. Next we show how the styles of bubble burst- bulk densities are ρ = 1016.1, 1082.5, 1110.3, and 1143.7 3 ing and airwave excitation transition as a function of kg/m for suspensions of φ = 0.1, 0.3, 0.4, and 0.5, respec- the particle volumetric fraction φ of the suspension and tively. Te σ of suspensions of φ = 0.3 and 0.4 measured bubble volume V. We analyze the high-speed images using a pendant drop technique are ∼ 20% smaller than and waveforms of the airwaves to infer the excitation that of particle-free oil. mechanisms. Ten we lower the suspension level below the tank rim to study how the air column resonance is simultaneously excited, and obtain the critical air col- Rheology of suspensions umn length above which it determines the duration of We measured the rheology of suspensions using a the airwave. Finally, we provide implications for basal- rheometer (Anton-Paar, MCR 301) in cone-plate geom- tic volcanic eruptions. etry (Additional fle 7: Material). Figure 1a shows the fow

(See fgure on next page.) Fig. 1 Rheology of suspensions (packing fraction φ ). a Flow curves and their fts to Eq. (1). Fitting parameters are as follows ; φ = 0 : τy = 0 n n n (Pa), n = 0.998 , K = 0.10 ( Pa s ) , φ = 0.1 : τy = 0.07 (Pa), n = 0.96, K = 0.14 ( Pa s ) , φ = 0.3 : τy = 1.2 (Pa), n = 0.91, K = 0.50 ( Pa s ) , n n φ = 0.4 : τy = 4.7 (Pa), n = 0.87, K = 2.0 ( Pa s ) , and φ = 0.5 : τy = 27 (Pa), n = 0.75, K = 19.2 ( Pa s ) . b The φ dependence of relative viscosity ′ ηr = η/ηl . A red curve indicates ηr calculated from Eq. (2) with φc = 0.60 (indicated by a blue broken line) . c Stress sweeps of storage (G : large markers) and loss (G ′′ : small markers) moduli measured under small to large amplitude oscillatory shear at f = 1 Hz. Arrows indicate the linear and non-linear regions of the φ = 0.4 suspension. G′ ≃ 0 of φ = 0, 0.1 suspensions are not plotted. × indicates the G′ , G′′ crossover which defnes the −3 ′ ′′ yield stress τy (corresponding to a strain γ ∼ 4 × 10 ). d Frequency sweeps of G (large markers) and G (small markers), measured under a small −4 amplitude (γ = 10 ) oscillatory shear. The thick (thin) broken line indicates the instrumental inertia efect (G inertia ) of the spindle used to measure the φ = 0 ( φ ≥ 0.1 ) suspension (Additional fle 7: Eq. (3)). e Creep curves of φ = 0.4 suspension sheared under the τ indicated in the legend. τ was applied during 0 ≤ t ≤ 5 s and released at t = 5 s (response time < 0.01 s). The γ is normalized by the respective maximum values γmax at t ≃ 5 s, −3 which increases from γmax = 1.4 × 10 (τ = 0.2 Pa) to γmax = 6.17 (τ = 5 Pa). f φ dependence of the yield stress τy (with errors) obtained from the fow curves (a) and stress sweeps (c). The bubble buoyancy pressures p (Eq. (5)) are plotted for comparison Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 3 of 25

Fig. 1 (See legend on previous page.)

curves (stress τ vs. strain rate γ˙ ). We ft the plots to a Her- where τy is the yield stress, K is the consistency, and n is schel-Bulkley model (Mader et al. 2013) the fow index (see legend for ftted values). n decreases n ≃ 1.00 φ = 0 n = 0.75 = + ˙ n from (Newtonian) for oil ( ) to τ τy Kγ , (1) Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 4 of 25

(shear-thinning) for a suspension of φ = 0.5 . Tese val- Amplifier Logger ues of n are similar to those of basaltic magma (Tran et al. (2015), Additional fle 7: Material). Viscosity η is Signal Microphone 2 calculated from η = τ/γ˙ (see Additional fle 7: Fig. S3 for Microphone 1 5 cm η vs. γ˙ plot). Figure 1b shows relative viscosity ηr = η/ηl High speed camera 3 cm 2.5 cm (η l : liquid viscosity) vs. φ . We ft the measurements at γ˙ = 100 1/s using the Maron-Pierce correlation (Guaz- L L1 zelli and Pouliquen 2018), ∆L −2 Bubble φ V Plane light source ηr = 1 − , φ (2) 25 cm c  where φc = 0.60 (maximum packing fraction) is the ft- ting parameter. Te ft using the Einstein-Roscoe equa- φ = 0.66 tion results in c , a value larger than the random Suspension ≃ close packing of spheres φc 0.64. Packing fraction: Φ Figure 1c shows the stress sweeps of storage and ′ ′′ Valve Pipe: ID = 2.5 mm loss moduli (G , G ) measured under a small to large Valve amplitude oscillatory shear. Te φ = 0, 0.1 suspensions ′′ ′ ′ are liquid-like (G ≫ G ) and we do not plot G . Te ′ ′′ Syringe φ ≥ 0.3 suspensions are solid-like (G > G ) at small τ G′ G′′ τ Fig. 2 Schematic diagram of the experimental setup. An acrylic tank and the , do not depend on (the linear region), W = a = G′′ > G′ τ is square in the cross section with an inner width 2 3 cm and but fuidize ( ) at large (the non-linear region). an outer width of 5 cm We classify the φ ≥ 0.3 ( φ ≤ 0.1 ) suspensions as solid- like (liquid-like) using the results at small τ , though we emphasize that solid-like suspensions become liquid- which the elastic and viscous strains become compara- like at large τ (Sumita and Manga 2008). For solid-like ble, becomes shorter. Tis rheology is similar to that of a suspensions, we defne the yield stress τy as the τ at ′ ′′ magmatic suspension which is also shear-thinning (e.g., which the G , G crossover (Bonn et al. 2017). ′ ′′ Ishibashi (2009); Gurioli et al. (2014)) and Maxwellian Figure 1d shows the frequency sweeps of G , G meas- −4 (viscous creep at t > tr ; e.g., Cordonnier et al. (2012)). ured under a small amplitude (γ = 10 ) oscillatory Figure 1f shows τy vs. φ obtained from the fow curves shear, in the linear region for the φ ≥ 0.3 suspensions. ′′ (Fig. 1a) and stress sweeps (Fig. 1c). Te τy obtained from For φ = 0 , 0.1, since G ∝ f , the rheology is liquid-like the 2 methods agree within a factor of 2 - 5. (viscous) (Larson 1999) (see Additional fle 7: Mate- rial 3 and 4.3 for other efects). On the other hand, for ′ ′′ ′ Experimental method φ = 0.3 − 0.5 , since G > G and G becomes nearly Figure 2 shows the experimental setup. We inject a bub- frequency independent, indicating a solid-like (elastic) ble from the bottom of a square tank and record images rheology. of the bubble ascent and bursting near the surface using Te rheology shown in Fig. 1a–d were obtained under a high-speed camera (IDT, M3) at 3000 fps (for runs steady or oscillatory states. Now we show a transient using suspensions of φ = 0, 0.3 ) and 2000 fps (for runs response. Figure 1e shows the creep curves of a solid-like using suspension of φ = 0.4 and η = 3.2, 10.8 Pas oil). φ = 0.4 suspension when τ is applied and released. Since Te excited airwave is recorded using 2 vertically aligned the strain γ increases with τ , we normalized the γ by their microphones (Primo, MX5307 ; sensitivities 5.7, 5.4 mV/ respective maximum values γmax . At small τ , an initial Pa) with a fat response in the frequency band of 10 - elastic response (a rescaled close-up is shown in Addi- 20000 Hz (calibrated from 0.5 Hz). Te voltage outputs tional fle 7: Fig. S4) is followed by a viscous fow, and an of the microphones are amplifed by ×74.645 (Primo, elastic recovery after the stress release. However at large MX5002) and recorded at 1 MHz using a data logger τ , the viscous strain becomes dominant. Tis result indi- (Graphtec, GL980) synchronized to high-speed imagery. cates that the rheology can be most simply approximated We mainly analyze the large-amplitude data recorded as a shear-thinning Maxwell fuid consisting of a viscous by the lower microphone 1 closer to the bubble. Unless component η(γ)˙ and an infnite frequency elastic com- noted otherwise, waveforms shown are those recorded G∞ τ γ˙ η(γ)˙ ponent . As (hence ) increases, the decreases, by Mic 1. Te signal recorded by the upper microphone and the relaxation time tr = η(γ˙ )/G∞ , the time scale in Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 5 of 25

2 is used to study the time lag and decay of the airwave. (a) Aperture (b) (c) R Tese microphones are located in the near or far feld of a l’ the wavefeld, depending on the frequency and excitation h h P mechanism (Additional fle 7: Material). We checked the b R b Rb V response of our microphones to a pressure pulse gener- ρ l ρ P ated by a hand fan. We confrmed that in addition to the b sound wave, the microphone detects the pressure fuctu- O ation arising from the air fow with a velocity vaf = (1) m/s and f ∼ 20 Hz and that the waveforms recorded Fig. 3 Schematic diagrams showing several mechanisms of airwave excitation. a Film vibration of a bubble with a hemispherical cap and by the 2 adjacent microphones correlate well. Te room 23◦ a hemispheroid below the surface. b Film vibration of a hemispherical temperature is maintained close to C so that the sus- cap bubble. c Helmholtz resonance at the neck (efective length l′ ) of pension remains as clear as possible. an aperture (radius Ra) In the frst series of experiments, we use a nearly full tank and vary the φ of the suspensions ( φ = 0, 0.3, 0.4 ) 3 and bubble volume V (V = 0.1 − 12 cm ). We also con- shape of a static bubble protruding above the surface duct experiments using high viscosity (η = 3.2 , 10.8 Pas) and the mechanism of bubble flm drainage (Nguyen silicone oils to compare with the results using suspen- Eo ≫ 1 L et al. 2013). For , relevant to our experiments sions. Te level 1 before the bubble injection is speci- O 1 2 ( Eo = (10 − 10 ) ), a static bubble forms a hemi- fed as follows. When the bubble is injected, the fuid spherical cap, and the flm drainage is gravity driven level rises by L (see Fig. 2), and the air column length (Additional fle 7: Material). Similarly, the Eo of a d = 1 L between the tank rim and the fuid surface decreases 4 m bubble in basaltic magma is Eo ∼ 7 × 10 ≫ 1 , where as L = L1 − L . Accordingly we specifed L1 such that 3 we used ρ = 2650 kg/m and σ ∼ 0.35 N/m (Murase and L becomes L ∼ 0 − 1 cm after the bubble is injected. In McBirney 1973). the second series of experiments, we study L depend- 3 Bubble buoyancy pressure p is evaluated from ence by injecting a V = 5 cm bubble into a suspension of φ = 0.4 . We lower the level in 7 steps in the range of L1 = �ρgV 2 p = = �ρgd . 1 - 10 cm. L is similarly calculated by correcting for L. 2 be (5) πdbe/4 3

Bubble size, overpressure and buoyancy In Fig. 1f, we indicate the range of p and compare with τ τ Te bubble size is characterized by an equivalent bubble the yield stress y . We fnd that p is larger than y by at diameter dbe (or radius Rbe ) defned as least a factor of 6 - 10. Tis is consistent with the obser- vation that the bubbles rise without being trapped. 6V 1/3 d = 2R = . be be π (3) Mechanisms of airwave excitation and decay  Here we summarize the mechanisms of airwave excita- Te dbe values of the bubbles we inject are tion and decay which are relevant to our experiments. = − dbe 5.8 28.4 mm, which when normalized to the First we consider flm vibration. Figure 3a shows flm / = − tank width W, become dbe W 0.19 0.95 . For vibration excited by a bursting bubble that partially pro- dbe/W > 0.6 (corresponding to a large bubble of V > 3 3 trudes above the surface. Te following derivation is cm ), the bubble becomes vertically elongated to form a essentially the same as those of previous works (Vergn- slug (Clift et al. 1978). We note that dbe is more than 3 iolle and Brandeis 1994; Kobayashi et al. 2010; Lyons orders of magnitude larger than the diameter of the par- et al. 2019), and is modifed from the vibration of a fully ≃ µ ticles ( dp 2 m) suspended in the oil. Tis indicates immersed bubble (Leighton 1994). We consider a bub- that the suspension can be considered a homogeneous ble consisting of a hemispherical cap (radius Rb ) above medium for the rising bubble. Te overpressure P of the surface and a hemispheroid (length l) below the sur- �P ∼ 2σ/R = 1 − 8 these bubbles is estimated as be Pa. face. Ten the total height of the bubble Rb + l becomes d 1/3 2/3 A dimensionless form of be is the Eötvös (or Bond) Rb + l = (3V /2π) (1 + l/Rb) . We assume that a number hemispherical flm with a thickness of h and a density of ρ �ρgd2 vibrates while the fuid below the surface is rigid. From be π 2ρ (ωδ )2 Eo = Bo = , (4) the balance between the kinetic energy Rb h R and σ 2 + 2 potential energy 3πγhPb(Rb/(Rb l))δR ( δR : amplitude ω γ which compares buoyancy (�ρ : suspension - air den- of oscillation; : angular frequency; h : ratio of specifc sity diference) to capillary stress. Eo governs the Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 6 of 25

′ ′ heats; Pb : pressure inside the bubble), the flm vibration 8Ra/3π ≤ l ≤ 16Ra/3π (Spiel 1992), we use l = 4Ra/π ∝ 1/2 frequency fV becomes as a representative value. It follows that fH Ra and fH increases as Ra grows. A frequency gliding resulting from 1 3γhPb the aperture growth has been observed for bubbles burst- fV = (6) 2π ρh(Rb + l) ing in water (Spiel 1992; Deane 2013; Poujol et al. 2021). We also consider the decay of pressure amplitude P during Helmholtz resonance. When the air in the neck of 31/3 γ P 1/2 l −1/3 = h b 1 + V −1/6. the aperture vibrates, P decays with time as (2π)5/6 ρh R (7)  b  = − P P0 exp ( βt), (12) Here we neglected an aperture and assumed that the where β is the decay rate. Here we consider 2 mecha- whole flm above the surface with a uniform thickness of nisms of decay (Kinsler et al. 2000). First is the decay h vibrates. When h is not uniform or if a localized sec- from radiation ( βr ), expressed as tor of the flm vibrates, then h should be regarded as an efective value. h can be estimated from the aperture (πR f )2 β = a H . growth velocity which we show later. For our experi- r ′ (13) 5 cl ments, γh = 1.4 (air) and Pb = 1.01325 × 10 Pa. Figure 3b shows a situation where a hemispherical cap Second is the decay from viscous and thermal dissipation bubble excites flm vibration. fV can be derived from Eqs. ( βvt ) at the neck wall, expressed as (6),(7) using l = 0 as 1 πηafH γh − 1 βvt = 1 + √ , (14) 1 3γhPb Ra ρa Pr fV = (8)   2π ρhRb where ηa is viscosity, ρa is density, and Pr = ηa/ρaκa is the Prandtl number where κa is the thermal difusivity. For 1/3 1/2 −5 3 γhPb − η = 1.827 × 10 ρ = 1.193 = V 1/6. our experiments (air), a Pas, a (2π)5/6 ρh (9) kg/m3 Pr = 0.71 β β R  , and . Both r and vt depend on a . Substituting Eq. (11) into Eqs.(13) and (14), and using ′ 2 −1 −3/4 −1/4 Second, we consider an airwave excited by instant l = 4Ra/π , we obtain βr ∝ RaV , βvt ∝ Ra V . removal of the bubble flm (Vergniolle and Brandeis Tus, for a given V, as Ra grows, βr becomes the domi- 1994). For an ideal spherical balloon, an N-shaped wave nant decay mechanism. f is excited (Deihl and Carlson 1968), whose frequency N Fourth, when the bubble becomes vertically elongated is expressed as and the bubble flm bursts rapidly, resonance may be c −1/3 excited within the bubble cavity. Vidal et al. (2010) stud- fN = ∝ V , 2Rb (10) ied the situation in which a soap flm enclosing a cavity (length L) bursts. Tey compared the bubble rupture time where c is the sound velocity. For our experiments, we ( tburst ) to the airwave propagation time ( tprop = 2L/c ) = ◦ use c 345.3 m/s at 23 C. For a hemispherical cap, Rb and showed that the pressure amplitude of the cavity res- = ( / π)1/3 becomes Rb 3V 2 . onance becomes large when tburst < tprop. Tird, we consider Helmholtz resonance where the Fifth, the overpressurized air in the bubble eventually air within the neck of the aperture of a bubble vibrates vents out, and the bubble defates. Kobayashi et al. (2010) (Fig. 3c), and has been used to model volcanic infrasound observed pressure fuctuation arising from such venting (Vergniolle and Caplan-Aucherbach 2004; Montalto and called it an air fow. et al. 2010). Te Helmholtz resonance frequency fH is expressed as (Kinsler et al. 2000), A regime diagram of bubble bursting and airwave cR 1 = a excitation fH ′ , (11) 2 πl V Figure 4a shows the 3 main styles of bubble bursting and ′ airwave excitation revealed by our experiments. Fig- where Ra is the aperture radius and l is the efec- ure 4b is a regime diagram mapping the diferent styles tive neck length. Te Ra dependence in Eq. (11) arises in the parameter space of volumetric packing fraction from the restoring pressure of a compressible gas scal- 2 ′ φ of the suspensions and bubble volume V. Varying φ ing as ∝ Ra . l is determined by the end correction of ′ results in changing the rheology (Fig. 1). Here we injected the aperture and scales as l ∝ Ra . From the estimate Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 7 of 25

bursts thereafter, exciting a high-frequency (HF) wave (a) (regime I). On the other hand, for runs using a suspen- ? sion of φ = 0.4 and bubbles larger than critical (V ≥ 5 3 cm ), the bubble bursts while it is rising when it partially protrudes above the surface and excites Helmholtz reso- nance (mostly with HF wave component) and/or air fow (Regimes III and IV). Regime II consists of a mixture of these 2 styles. In what follows, we describe the details of each of the styles shown in Fig. 4a. (b) I :Non-bursting II :Transitional I :Wall attached III :Helmholtz resonance I :HF wave IV :Air flow Diversity of bubble bursting and airwave excitation 101 ) A hemispherical cap bubble bursts and excites 3 high‑frequency wave

(cm Figure 5 shows a case in which a hemispherical cap bub- V ble bursts and excites a high-frequency (HF) wave (case 0 10 3 in regime I). Here the bubble arrived at the surface, formed a cap, and after ∼ 25 s elapsed (i.e., the drain- age time, see Additional fle 7: Fig. S8), it burst, and the flm disappeared within ∼ 1 ms. Since Eo (Eq. (4)) of the Bubble volume 10-1 injected bubble is Eo = 71 ≫ 1 , we infer that the bubble flm thinned by gravitational drainage (Additional fle 7: 00.3 0.4 Material). Packing fraction In our experiments, an HF wave is defned as a wave Fig. 4 a Schematic diagram showing the 3 main styles of bubble with a peak frequency fp > 9200 Hz having an irregular bursting and airwave excitation. Left : a hemispherical cap bubble waveform. A spectrogram indicates that the HF wave sits at the surface and bursts, exciting a high-frequency (HF) wave. A ∼ 14000 “?” indicates that the modes are unknown. Middle : a bubble bursts shown here has primary and secondary peaks at ∼ 23000 while it is rising and excites Helmholtz resonance (mostly with HF Hz and Hz, respectively, and these peaks are wave component), and later vents an air fow. Right : a bubble bursts also evident from the power spectrum (Fig. 10). Fig- while it is rising and vents an air fow. b A regime diagram in the ure 5a, b shows that the timing of the aperture opening parameter space of suspension φ and bubble volume V. A broken is within the time frame of 0.07 < t < 0.4 ms, where the curve indicates the main regime boundary (see text for details) uncertainty arises from the camera exposure time (0.33 ms). Since the travel time of the sound wave from the bubble to the microphone is < 0.07 ms, we infer that a large pressure amplitude (∼ 0.09 Pa) at t ∼ 0.58 ms was 3 bubbles with 21 diferent V (= 0.1 − 12 cm ) into suspen- excited after the bubble burst. sions with 3 diferent φ (= 0, 0.3, 0.4 ). We conducted 5 We further compared the timings of the airwave onset runs per each combination of V and φ , and then classi- and bursting of 87 runs, which we classifed as HF waves fed the result into 8 cases, which were then grouped into (case 3). We fnd that for all runs but one (98.9 %), the the following 4 regimes (I-IV) ; I : non-bursting, wall- onset of the airwave was coincident or after the time attached, high-frequency (HF) wave, II : transitional, III frame that showed bursting. Tis result indicates that : Helmholtz resonance, and IV : air fow. Te details of the HF wave is excited after bursting. It is also consist- these 8 cases and the method used to classify them into ent with the result of Kobayashi et al. (2010), who showed the 4 regimes are described in the Additional fle 7: Mate- that airwave is excited after bursting when a V = 8 , 32 3 rial. In what follows, non-bursting and wall-attached cm bubble ascends and bursts in a syrup solution with cases are not analyzed further. a viscosity of η ≥ 0.17 Pas, where V and η are comparable A key feature of Fig. 4b is that the style of bubble to those in our experiments. bursting can be broadly classifed into 2 types, whose boundary is indicated by a broken curve above which A rising bubble bursts and excites Helmholtz resonance Helmholtz resonance is excited. For all runs using sus- Figure 6 shows a case in which a bubble excites Helm- pensions of φ = 0, 0.3, and for runs using a suspension of holtz resonance (case 5 in regime II). In contrast to the 3 φ = 0.4 with bubbles smaller than critical (V ≤ 0.4 cm ), case shown in Fig. 5, the bubble bursts while it is rising the bubble forms a hemispherical cap after surfacing and and when it partially protrudes above the surface, i.e., Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 8 of 25

Fig. 5 An example of HF wave (case 3). V = 0.6 cm3 hemispherical cap bubble bursts in a suspension of φ = 0.3 . a Time-lapse images (see Additional fle 1: Movie 1). Blue × : before bursting, : after bursting, red × : flm disappearance. Red arrows indicate the aperture. b Waveform (detrended, highpass fltered at 800 Hz). Markers correspond to the timing shown in (a) and are plotted at the end of the exposure time of each image. c Spectrogram of (b) using a 0.512 ms sliding window and a 0.5 ms overlap

before it forms a hemispherical cap (see also Additional delay is close to the estimated 0.145 ms (= 5 cm /c) con- fle 2: Movie 2). Here a dimple forms at the top of the frming that this signal is a sound wave. Te waveform bubble before it bursts. After bursting the aperture grows of Mic 1 shows that a low-frequency (period ∼ 20 ms) and the bubble defates. Te waveform shows a damped signal follows after the Helmholtz resonance. Tis signal oscillation with a peak frequency of fp ≃ 4360 Hz, which has an amplitude which exceeds the noise (≤ 0.02 Pa) and decays in about 4 ms. Tis waveform, characterized by is not observed by Mic 2 further away from the bubble. a lower frequency that continues for a longer time, is Since these results are reproducible (Additional fle 7: clearly distinguishable from that of the HF wave (Fig. 5b). Fig. S7), we interpret this signal as an air fow vented Te onset of the airwave coincides with the timing of the by the defating bubble (Kobayashi et al. 2010). We cal- aperture opening within the camera frame rate (0.5 ms). culate the lower limit air fow velocity vaf , assuming that Te spectrogram (Fig. 6c) shows that the peak frequency the air fow was vented immediately after the Helmholtz glides toward a higher frequency. It also shows that an HF resonance decayed. Using the time lag and the distance wave component of fp ∼ 21270 Hz is excited. We calcu- between the bubble and Mic 1, we obtain vaf ∼ 1.6 m/s. late fH (Eq. (11)) using the aperture radius Ra measured Te overpressure P needed to generate this air fow can ∼ 2 ∼ from the images and plot on the spectrogram. Te meas- then be estimated as �P ρavaf /2 1.5 Pa, which is of ured fp agrees well with the calculated fH , from which the same order of magnitude as �P ∼ 2σ/Rb = 6.5 Pa. we interpret that frequency gliding is a consequence of We note that an air fow must always exist, since over- aperture growth (see below Eq. (11) for an explanation). pressurized air in the bubble would always fow out after We highpass and bandpass flter the waveform to sepa- the bubble bursts. However, the waveform we identifed rate the signals arising from the HF wave component and as an air fow is detected only in limited cases. We also Helmholtz resonance, respectively, which we show in note that the amplitude of the waveform will be afected Fig. 6d. Te onset and duration of these 2 waves are close by the location of aperture opening, which difers among to each other. the runs. Compared to Fig. 5, the bubble aperture grows Figure 6e shows the waveforms of both Mic 1 (lower) slowly and it takes about ∼ 36.5 ms after the aperture and Mic 2 (upper) covering a longer time span (70 ms). opens for the bubble flm to disappear, which is indicated Te waveform of the Helmholtz resonance recorded by by × in Fig. 6e. Tis experiment demonstrates that bub- Mic 2 is delayed relative to that recorded by Mic 1 by ble bursting can simultaneously excite a Helmholtz reso- O 3 O 4 0.147 ms, and its amplitude is smaller. Te measured time nance ( (10 ) Hz), an HF wave component ( (10 ) Hz), Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 9 of 25

Fig. 6 An example of Helmholtz resonance (small bubble, case 5). V = 0.6 cm3 bubble bursts while it is rising in a suspension of φ = 0.4 . a Time-lapse images (see Additional fle 2: Movie 2). × and indicate before and after aperture opening, respectively. b Waveform (detrended, highpass fltered at > 800 Hz). c Spectrogram of (b) using a 1.024 ms sliding window and a 1 ms overlap. indicates fH calculated from Eq. (11) using measured Ra . d High- (> 10000 Hz, left axis) and bandpass (800 - 10000 Hz, right axis) fltered result of (b) shown for a shorter time span. e Waveform shown for a longer time span (running-averaged raw data) indicating Helmholtz resonance followed by a low-frequency air fow. A red × indicates the time of flm disappearance Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 10 of 25

Fig. 7 An example of Helmholtz resonance (large bubble, case 5). V = 7 cm3 bubble bursts while it is rising in a suspension of φ = 0.4 . a Time-lapse images (see Additional fle 3: Movie 3). b Waveform (detrended, highpass fltered at 800 Hz). c Spectrogram of (b) using a 2.048 ms sliding window and 2.030 ms overlap. indicates the fH calculated from Eq. (11) using measured Ra

O and an air fow ( (10) Hz), covering a broad frequency bursts, both the Helmholtz resonance (∼ 3000 Hz) and 4 band. an HF wave component in the range of ∼ 1 − 3 × 10 Hz Figure 7 shows another example of Helmholtz reso- are excited and these are superimposed on the waveform. nance (case 5 in regime III). Here the bubble has a Te measured fp agrees well with the calculated fH plot- 3 larger volume (V = 7 cm ) than that shown in Fig. 6 and ted on the spectrogram. Te higher fp than that shown becomes vertically elongated to form a slug. Te bub- in Fig. 7 despite a similar V can be explained as a result ble bursts while it is rising and the waveform indicates of a larger Ra . When the thin flm bursts, the aperture damped oscillation. Te calculated fH plotted on the opens quickly at a velocity comparable to that in the case spectrogram is initially smaller than the measured peak where a hemispherical cap bubble bursts and excites an frequency fp but the agreement improves as the aperture HF wave. grows, confrming that Helmholtz resonance is excited. Here we consider the possibility of cavity resonance Tere are at least 2 explanations for fH being initially within the bubble. Te cavity resonance frequencies of O 3 smaller than fp . Te frst is that the aperture of this run is the the runs shown in Figs.7 and 8 are also (10 ) Hz an ellipse, and the measured Ra corresponds to its minor (Additional fle 7: Material). However Helmholtz reso- axis. Te second is that the expression of estimated neck nance better explains the measured fp , its dependence ′ length l has a range (see below Eq. (11) for explanations), on Ra , and the glide toward a higher frequency. In our and may have been overestimated. Compared to Fig. 6c, experiments, bubble rupture was slow and the cavity was the peak frequency ( fp ∼ 1670 Hz) is smaller and con- short such that tburst > tprop , from which we infer that tinues longer (∼ 10 ms). Te spectrogram also shows the excitation of cavity resonance was inefcient (Vidal simultaneous excitation of an HF wave component with et al. 2010). a frequency of fp ∼ 6500 Hz, which is lower than that shown in Fig. 6c. A rising bubble bursts and vents an air fow Figure 8 is a further example of Helmholtz resonance Figure 9 shows an example of an air fow dominant case excited by a rising bubble (case 4b in regime II). Here a (case 6a in regime IV). As in Figs.6, 7, 8, the bubble bursts thin flm forms at the apex of the bubble, and when it when it partially protrudes above the surface. However Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 11 of 25

Fig. 8 An example of Helmholtz resonance excited after a thin flm bursts (case 4b). V = 6 cm3 bubble bursts while it is rising in a suspension of φ = 0.4 . a Time-lapse images (see Additional fle 4: Movie 4). Red arrows indicate the aperture. b Waveform (detrended, highpass fltered at 800 Hz). c Spectrogram of (b) using a 1.024 ms sliding window and a 1.000 ms overlap. indicates the fH calculated using Eq. (11) and measured Ra

diferent from Fig. 7, the amplitude of the waveform is fp decreasing with V. Tere is also a trend of the fp of dominated by an air fow, which continues for ∼ 100 ms. HF wave component decreasing with V. Te V depend- In the spectrogram (Fig. 9c), we identify a weak signal ence of fp obtained using all available data and com- (indicated by an arrow) at f ∼ 1400 Hz superimposed on parison with scaling laws will be described later. Finally, O the main air fow signal at (10) Hz. Te signal is excited when a rising bubble excites an air fow ( φ = 0.4 , V = 12 3 immediately after the bubble bursts, and its frequency cm case; Fig. 9), the power of the Helmholtz resonance increases with time. Te calculated Helmholtz resonance becomes much smaller but can still be identifed on the frequency fH plotted on the spectrogram (see legend for spectrum. the values of fH ) bound the observed frequency from which we conclude that Helmholtz resonance was also Aperture growth and estimates of flm thickness excited. High-speed images indicate that the aperture growth velocity va = dRa(t)/dt is slower when Helmholtz reso- Comparison of power spectra nance is dominantly excited (Figs.6, 7) compared to other Figure 10 compares the power spectra of 5 wave- cases (Figs.5, 8, 9). To quantify the diference in va , we forms, which we showed in Figs.5, 6, 7, 8, 9. Here we measured the aperture radius Ra(t) as a function of time, highlight the characteristics of the spectra and their V which we show in Fig. 11a. Te fgure indeed shows that dependences. when the aperture opens slowly, Helmholtz resonance First, when a hemispherical cap bubble excites an is dominantly excited, whereas when the aperture opens 3 HF wave ( φ = 0.3 , V = 0.6 cm case, Fig. 5), the power rapidly, the HF wave or air fow becomes dominant. We = spectrum consists of 2 main peaks at fp > 10000 Hz. also measured the bubble surface radius Rb at t 0.5 ms, Next when a rising bubble excites Helmholtz resonance which we use to normalize Ra and the result is shown in 3 ( φ = 0.4 , V = 0.6, 6, 7 cm , Figs.6, 7, 8), fp values are Fig. 11b. Te fgure shows that the aperture grows rapidly O 3 ∼ (10 ) Hz, and the spectra show that there is a trend of to become Ra/Rb 1 when the HF wave is dominant. Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 12 of 25

Fig. 9 An example of an air fow (case 6a). V = 12 cm3 bubble bursts while it is rising in a suspension of φ = 0.4 . a Time-lapse images (see Additional fle 5: Movie 5). b Waveform (raw data) of the airwave recorded by 2 microphones. × and correspond to the times indicated in (a). The pink ⊲ and ⊳ indicate the times when the aperture radius Ra was measured. c Spectrogram (time window 16.384 ms, overlap 16.3 ms) of Mic 1 (lower) data shown in (b). ⊲ and ⊳ indicate fH = 870 and 2161 Hz, calculated from Eq. (11) using Ra measured at the respective times shown in (b). The pink arrow indicates the Helmholtz resonance excited after bursting

Following the method of Kobayashi et al. (2010), we σ R (t) = R exp t . use 2 models of aperture growth velocity to estimate a 0 ηh (17)  the flm thickness h using the Ra(t) measurements shown in Fig. 11a. First is the inviscid model (Taylor Tese 2 models are applicable during the initial stage of 1959; Culick 1960), in which the aperture grows under aperture growth when we may approximate that the flm the balance of capillary and inertial stresses at a con- near the aperture is horizontal and h is constant. stant velocity vi and is expressed as To estimate h, we ft our Ra(t) measurements using the 2 models (see Additional fle 7: Fig. S9 for ftted exam- 2σ ples), using the time span t which maximizes the cor- vi = . (15) ρh relation coefcient r. We did not ft Ra(t) for cases 4a and 4b because we could not time-resolve the initial Second is the viscous model (Debrégeas et al. 1998), in stage of rapid aperture growth. η in Eq. (17) is estimated which the aperture grows under the balance of capillary as follows. Te strain rate γ˙ of the aperture growth is O 2 4 and viscous stresses at a velocity vv which accelerates as γ˙ ∼ va/Ra ∼ (10 − 10 ) 1/s (Debrégeas et al. 1998), O Ra(t) grows, and is expressed as where we used the measured va ∼ (0.1 − 10) m/s. Our σ viscosity η measurements (Fig. 1b) indicate that at the v = R (t), γ˙ η v ηh a (16) high limit, approaches the value estimated from Eq. (2), from which we estimate η = 0.9 Pas for a suspension φ = 0.4 such that Ra(t) grows as of . h thus calculated from the 2 models are sum- marized in Table 1. Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 13 of 25

3 (a) = 0.3 , V = 0.6 cm (case 3, HF-wave) 3 HF-wave 5 Helmholtz resonance = 0.4, V = 0.6 cm3 (case 5, Helmholtz) 4a HF-wave 6a Airflow 4b Helmholtz resonance 6b Airflow = 0.4, V = 6 cm3 (case 4b, Helmholtz) 12 = 0.4, V = 7 cm3 (case 5, Helmholtz) = 0.4, V = 12 cm3 (case 6a, Air flow) 10 (mm) 45 a

R 8

40 6

35 4

30 2 Aperture radius / Hz )

2 25 0 (b) 0246810 Pa 1.2

20 b 1 R

15 0.8 10 0.6 5 Power (dB re 20 0.4 / Bubble radius 0 a 0.2 R

-5 0 0246810 -10 Time (ms) 3 4 4 R t 10 10 3 10 Fig. 11 Aperture radius vs. time. a a( ) vs. time. Thick lines indicate R (t) Frequency (Hz) the a of the runs shown in Figs.5, 6, 7, 8, 9 (the thickest pink line indicates the Ra(t) for Fig. 6). Thin lines indicate runs using Fig. 10 Power spectra of the waveforms (highpass fltered at > 800 suspensions of φ = 0.4 , where we used all runs whose timings of Hz) shown in Figs.5, 6, 7, 8, 9. Primary peak frequencies of the HF aperture opening could be determined accurately. Ranges of V ( cm3 ) wave (♦ ), Helmholtz resonance ( ), and secondary peaks of HF are as follows ; case 3 : 0.3 - 3; case 4a : 2 - 4; case 4b : 3 - 6; case 5 : 0.6 wave component excited together with Helmholtz resonance (♦ ) - 7; case 6a : 10 - 12; case 6b : 10 - 12. b Normalized ( R /R ) version of V 3 a b are indicated. Marker sizes indicate V. For the φ = 0.4, = 12 cm (a) (see text for details) case, data were not detrended and bandstop flters were applied at multiples of 2000 Hz to reduce camera noise. For other cases, data were detrended

h h = O(1 − 10) µ Table 1 Film thickness h (standard deviation δ ) calculated by Our results indicate that m for both ftting Ra(t) ( φ = 0.4 suspension) to inviscid and viscous models. models and that the inviscid model gives a better ft HF : high-frequency wave; H : Helmholtz resonance; AF : air fow; t (larger r and a longer ). Tis result is consistent with N: number of runs used; r : correlation coefcient (standard those of Kobayashi et al. (2010), who obtained the h of deviation δr ); t : Time span used for the ft. In terms of N, the bubbles bursting in a syrup solution with a comparable inviscid model gives a higher r for 69 % of the runs η ≃ 1.9 viscosity of Pas. Te values in Table 1 indicate Case N Inviscid h(δh)(µm) Viscous h(δh)(µm) that the h estimated for cases 5 and 6a (Helmholtz reso- nance) are thicker than those estimated for cases 3 (HF 3 (HF) 35 5 (4) 26 (21) wave) and 6b (air fow) as a result of smaller va . We note 5 (H) 16 69 (93) 44 (36) that the estimated h is comparable to or larger than the 6a (AF+H) 4 65 (60) 36 (25) particle diameter ( dp ≃ 2 µ m) suspended in the liquid. 6b (AF) 3 4 (4) 22 (12) From the viscous model, we obtained h/dp ∼ 11 − 22 . Total, r ( δr) 58 0.984 (0.002) 0.969 (0.003) Tis model uses suspension viscosity η , which assumes Total, t (ms) 58 2.4 ± 2.4 1.2 ± 1.0 O a continuum. However h/dp = (10) suggests that this Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 14 of 25

fp f ( h = 2 m) frequency range above the primary and above 5000 = 0 (HF wave) V f ( h = 20 m) = 0.3 (HF wave) V Hz. f = 0.4 (HF wave) N fp f ( R = 1 mm) Te values thus determined are plotted as a func- = 0.4 (Helmholtz resonance) H a f ( R = 10 mm) × = 0.4 (HF wave component) H a tion of V in Fig. 12. Each (red, green, blue) indicates

4 the fp of the HF wave, scattered over a range of 9200 - 4 10 26250 Hz, and they are poorly correlated with V (corre- lation coefcient r =−0.103 ). and △ (black) indicate (Hz) p the fp of the Helmholtz resonance and HF wave com- f 4 10 ponent, respectively, both of which decrease with V (negative correlation of r =−0.864 and −0.878 , respec- tively). A possible reason for the diferent r for the HF wave and HF wave component will be explained in the

Peak frequency next section. 103 We calculated the frequencies excited by the flm 800 0.11 10 15 vibration fV , N-wave fN , and Helmholtz resonance fH , Bubble volume V (cm3) which we show in Fig. 12. We assumed h in the range = − µ Fig. 12 Bubble volume V dependence of the peak frequency fp (with of h 2 20 m to calculate fV , such that fV bound errors) of the airwaves. An HF wave is excited by a hemispherical most of the measured fp of the HF wave. Tese h values cap bubble. Helmholtz resonance and the HF wave component are are comparable to the h ≃ 5 , ≃ 26 µ m (case 3, HF wave) excited together by a bubble which bursts when it partially protrudes above the surface. f , f , and f are calculated from Eqs.(9), (10), and estimated from aperture growth velocity (Table 1). Sim- V N H R (11) respectively, using h and R values given in the legend (see text ilarly, we assumed an aperture radius a in the range of a = − for details). A hemispherical cap bubble was assumed to calculate fV , Ra 1 10 mm, to calculate fH , such that fH bound f l R f N . For a partially protruding bubble with / b = 1 (Eq. (7)), V becomes the measured fp . Tese Ra are consistent with the meas- ∼ = smaller by a factor of 0.8 . ρ of φ 0.4 suspension was used to ured Ra (Fig. 11a). calculate f V Te fgure shows that the measured fp of the HF wave and Helmholtz resonance can be explained by fV and fH , respectively, using reasonable values of h and Ra . assumption may have been only marginally valid. Here Te scatter of fp for the same V can be attributed to the we assumed that the 2 models are applicable. Teir variation of h or Ra . We note that the power spectra of applicability need to be further checked using direct the HF wave and HF wave component (Fig. 10) have measurement of h and a sufciently time-resolved multiple peaks with comparable power, which may also high-speed imagery. be the reason for the scatter. We add that fN can also become a candidate to explain the HF wave. Bubble volume dependence of peak frequency When Helmholtz resonance is excited, we may use and decay the decay rate β to infer the damping mechanism. We determined the primary peak frequency fp of the Figure 13a shows an example of such analyses. We ft HF wave and Helmholtz resonance as follows. First we the envelope of the waveform to an exponential decay applied a highpass flter (> 800 Hz) to remove the low- function (Eq. (12)) and obtain β . We calculated β for frequency air fow and bandstop flters (2000, 4000 Hz, all runs that excited Helmholtz resonance (cases 4b, or 3000, 6000 Hz) to reduce camera noise arising from 5, 6a) and Fig. 13b shows the obtained β plotted vs. frame rate. We did not use 2 runs from case 3 (HF wave) peak frequency fp . Here fp is obtained using the same because the amplitude was small (< 0.02 Pa) near the method we used in Fig. 12. Te fgure shows that there noise level, and 4 runs (of which 3 had an amplitude is a trend of β increasing with fp . Tere are 2 outliers at = < 0.02 Pa) from case 6b (air fow dominant) because they fp 1200 and 1500 Hz, which belong to case 4b. Tese did not excite an HF wave or Helmholtz resonance. We runs have complex waveforms at the beginning, which then calculated the power spectrum of each waveform causes the fp to become smaller than that of the Helm- and determined the fp . When the primary fp corre- holtz resonance. Excluding these 2 outliers, we ft the ∝ m sponded to the Helmholtz resonance, we also determined data to a power law relation β fp and obtain an expo- = ± = the secondary fp , which corresponds to the HF wave nent m 1.66 0.01 ( r 0.948 ). Tis m is closer to the = component (e.g., Figs.6, 7, 8). In order to determine the m 2 predicted by radiation damping (Eq. (13)) than = secondary fp , we smoothed the power spectrum by a 7 to the m 0.5 for visco-thermal damping (Eq. (14)). = point running average, and searched for the peak in the We calculated βr , βvt using Ra 2 and 7 mm which we Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 15 of 25

Origin of high‑frequency wave (a) 0.3 We do not fully understand the origin of the HF wave excited by a hemispherical cap bubble or the HF wave 0.2 component excited by a rising bubble together with Helmholtz resonance. Airwaves that appear to be the 0.1 same as an HF wave have been reported for a bubble

(Pa ) bursting in water (Spiel 1992; Deane 2013) and in gel 0 (Divoux et al. 2008). Tese works report that such a wave is excited when the subsurface bubble volume is small or when a longer time has elapsed after surfac- Pressure P -0.1 ing, which is similar to our result in which the HF wave is excited by a hemispherical cap bubble formed after -0.2 Data Envelope bubble surfacing. Deane (2013) also reported that such Exponential fit wave is excited together with Helmholtz resonance, -0.3 0 0.5 1 1.5 22.5 33.5 which is the same as our results for cases 4b and 5 Time (ms) (Figs.6, 7, 8). Our tentative interpretation is that it originates (b) 104 from the flm vibration fV , which scales with the flm −1/2 −1/6 thickness h and volume V as fV ∝ h V (Eq. (7)). In detail, we showed that the frequency of the HF wave and HF wave component correlate difer- 3 10 ently with V (Fig. 12). Here we explain that this may (1/s ) have arisen from the efective h depending diferently on V. Images of the runs in which the HF wave com- ponent was excited show that when the bubble is small 102 V ≤ 4 cm3 Decay rate ( ), in all runs, the bubble bursts after a thin r ( Ra = 2 mm) ( R = 7 mm) flm patch forms near the bubble apex (e.g., Fig. 6). On r a V ≥ 5 cm3 vt ( Ra = 2 mm) the other hand, when the bubble is large ( ), ( R = 7 mm) vt a in the 52 % of the runs, the bubble bursts before a thin 101 flm patch forms (e.g., Fig. 7), indicating that its flm is 103 104 Peak frequency f (Hz) still thick. Tis observation suggests that the h at burst p increases with the V, which explains the negative corre- Fig. 13 Decay of Helmholtz resonance. a Waveform of the Helmholtz lation. On the other hand, a thin patch cannot be iden- resonance excited by a V = 0.6 cm3 bubble bursting in a suspension of φ = 0.4 . Red and blue lines indicate an envelope and its tifed when the hemispherical cap bubble bursts (e.g., exponential ft, respectively from which we obtain β = 1.434 ± 0.006 Fig. 5). We thus infer from a poor correlation with V, 1/ms. The waveform was detrended, and a highpass (> 800 Hz), that the h depend little on V. Direct measurement of h bandstop (2000, 4000 Hz) flters were applied. b β vs. peak frequency is needed however, to verify this. f p (both with errors) for the case in which Helmholtz resonance was Te flm vibration model is nevertheless simplifed. excited by a bubble bursting in a suspension of φ = 0.4 . The gray line indicates a power law ft excluding 2 outliers (see text). Thick red and We note that this model does not include an aperture, thin blue lines indicate the βr and βvt calculated from radiation (Eq. whereas in our experiments, the HF wave is excited after (13)) and visco-thermal (Eq. (14)) damping models, respectively, using the aperture opens. When the aperture becomes too R the a values given in the legend large, the bubble can no longer be compressed to excite vibration. In addition, power spectra (Fig. 10) show that HF waves commonly consist of more than one frequency R indicate in Fig. 13b. Tese a values were chosen so that component, suggesting that vibration of higher modes β β r bounds the measured , excluding the outliers, and may also be excited. We also note that the N-wave (Eq. are consistent with the measurements (Fig. 11a). We (10)) may also be excited, which explains the frequency of Ra > 3.5 O 4 fnd that for mm, radiation damping becomes (10 ) Hz as shown in Fig. 12. However, our waveform is β >β the main mechanism ( r vt ). Visco-thermal damp- not an ideal N-shape, and the aperture growth velocity va β >β ing is important ( vt r ) only during the initial stage is in the range of va = 0.3 − 9 m/s, which is much slower R < 2 V > 2.7 of the aperture opening ( a mm) of a large ( than the sound velocity c. Tis indicates that bursting is cm3 f < 2350 ) bubble (corresponding to p Hz). not instantaneous as assumed for the N-wave, and for Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 16 of 25

this case, Eq. (10) becomes an upper bound (Kulkarny fle 7: Fig. S3). Together with the creep curves (Fig. 1e), 1978). this indicates that shortly after the stress is applied, its rheology can be approximated as a shear-thinning Max- well fuid, from which we consider that Eq. (18) is appli- Condition for the excitation of Helmholtz cable. Such rate-dependent De(γ)˙ (and tr ) has been used resonance for shear-thinning silicic magmatic suspension (Cordon- Our experiments indicate that Helmholtz resonance is nier et al. 2012) and polymer (Arigo and McKinley 1998). ≤ ≤ 3 excited only when a bubble of 0.5 V 9 cm rises and De has been used as a criterion for the viscous - brit- = bursts in a suspension of φ 0.4 (Fig. 4). Here we con- tle transition of magma and its analogues (see Wads- sider the origin of this condition. worth et al. (2017) for a review). Previous works have −2 shown Dec ∼ 10 for silicate melts and syrup, and HF wave ‑ Helmholtz resonance transition that Dec criterion is applicable to a wide range cover- O −4 1 First we consider the main transition indicated by a bro- ing at least tr = (10 − 10 ) s. Importantly, Cor- ken curve in Fig. 4b above which Helmholtz resonance donnier et al. (2012) showed that Dec of magmatic is excited. Moitra et al. (2018) showed that when a dense suspension ( φ = 0.15 − 0.65 ) decreases with φ such that φ = O −3 suspension ( 0.55 ) is elongated faster than the critical Dec = (10 ) . Dingwell (1996) proposed a regime dia- ˙ strain rate γ , it fails brittly, whereas no fracturing occurs gram mapping the fow - fragmentation transition in the in a particle-free oil. Our experiments similarly show that td − tr parameter space. Our regime diagram (Fig. 4b) ≥ 3 φ = when V 0.5 cm bubble rises in a 0.4 suspension, resembles his, which motivated us to evaluate Dec at the the bubble bursts while it is rising when its flm is still transition. thick by being stretched faster. A close similarity indi- Our method used to calculate De(γ)˙ closely follows cates that the bubble flm failed brittly above the critical that of Cordonnier et al. (2012) who measured the τ ˙ γ. (Eq. (18)) needed to deform a magmatic suspension We show further evidence which indicate that the brit- under a controlled γ˙ , and used G∞ measured separately tle failure does not occur in a particle-free oil. Previous under a small strain γ and high frequency f. Similarly works (Debrégeas et al. 1998; Nguyen et al. 2013) have we calculate τ = η(γ)˙ γ˙ for bubble flm stretching and = ′ shown that a bubble rising in high viscosity (η 10 - 1000 use G measured under small amplitude oscillatory Pas) Newtonian silicone oil forms a hemispherical cap at shear for G∞ . High-speed images of the experiments in the surface and bursts thereafter. Tis indicates that the which Helmholtz resonance is excited show that before bubble flm does not fail brittly, even when the viscosity of bursting, the bubble forms a hemisphere (height Rh , ∼ 2 the oil is high. To confrm, we conducted the same experi- surface area S 2πRh ) above the surface, and its flm = ments using high viscosity (η 3.2, 10.8 Pas) silicone oils, is stretched as it rises. Te areal strain rate γ˙ of the flm which are liquid-like (Additional fle 7: Fig. S5) and New- immediately before bursting can be expressed as tonian, and the results are shown in Additional fle 7: Fig. 1 dS 2 η = 3.2 ˙ ∼ ≃ S10. Here a viscosity of Pas is the same as that for γ U, (19) a φ = 0.4 suspension, which we estimate next. Te experi- S dt Rh ments indeed showed that the bubble bursts after it forms where U is the ascent velocity U ∼ dRh/dt . From the a hemispherical cap and excites an HF wave. 3 images of rising V = 0.5 cm bubble, we obtain Rh ≃ 5.7 Here we non-dimensionalize γ˙ and defne the Deborah mm, U ≃ 0.011 m/s, and substituting these values number De(γ)˙ for a shear-thinning Maxwell fuid, into Eq. (19), we obtain γ˙ ≃ 3.7 1/s; from Additional t η(γ)˙ γ˙ τ η = 3.2 G∞ ˙ = r = = fle 7: Fig. S3, we estimate Pas. For , we use De(γ) . (18) G′ = 8210 φ = 0.4 γ = 10−4 td G∞ G∞ Pa ( ) measured under (in the linear region) and f = 100 Hz (Fig. 1d) which best ˙ = ˙ De(γ) compares the relaxation time tr η(γ )/G∞ , to the approximates the high frequency. Substituting these val- = /γ˙ γ˙ −4 −3 deformation time td 1 , where is the strain rate of ues we obtain tr ∼ 4 × 10 s and Dec = 1.4 × 10 . η(γ)˙ O −3 deformation. Here is the shear-thinning viscosity Te Dec = (10 ) is the same as Dec obtained for and G∞ is the infnite-frequency shear modulus, repre- φ = 0.4 silicic magmatic suspension (Cordonnier et al. senting the viscous and elastic component, respectively 2012). Estimates of the De(γ)˙ for other V are given in the ≪ and τ is the shear stress. For De Dec , the fuid deforms Additional fle 7: Table S3, which shows that the De(γ)˙ ≫ viscously ; conversely for De Dec , it deforms elastically increases with V. Tis is consistent with the estimate 1/3 and fails brittly. Our solid-like suspension is elastic under De(γ)˙ ∼ p/G∞ ∝ V derived from τ ∼ p (p : bubble small τ (Fig.1d), but fuidizes under the large τ (Fig. 1c) buoyancy). which is also apparent as shear-thinning η(γ)˙ (Additional Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 17 of 25

Next, we consider the transition between φ = 0.3 Helmholtz resonance ‑ Air fow transition 3 and φ = 0.4 suspension. φ = 0.3 suspension is also Next we consider the upper limit Vc = 10 cm , above solid-like and non-Newtonian albeit weaker com- which Helmholtz resonance is no longer excited and is pared to those of φ = 0.4 suspension. We interpret this dominated by air fow (case 6b). Here we emphasize that transition as a consequence of Dec decreasing with φ . the air fow will always exist, but the pressure signal aris- 1/3 ′ From De(γ)˙ ∝ V /G∞ and since G increases from ing from air fow becomes dominant only for these large φ = 0.3 to φ = 0.4 (Fig. 1d), it follows that for a given V, bubbles. For these large bubbles, the period of Helmholtz De(γ)˙ decreases with φ . Tus we inferred that the Dec resonance becomes longer (∼ 1 ms), and the aperture dropped with φ as shown by Cordonnier et al. (2012). grows faster (Fig. 11) than in the case of smaller bub- De(γ)˙ defned in Eq. (18) assumes a simple viscoelas- bles. Tus Ra is no longer small relative to the bubble size, ticity, but we consider it sufcient to explain the vis- which is assumed for Helmholtz resonance and this may cous - brittle transition we observe. Yield stress τy need have suppressed its excitation. not be included in the criterion because it is small com- pared to the buoyancy pressure p and capillary stress Simultaneous excitation of air column resonance ∼ which we show below. As we showed in Fig. 1f, τy 4 Now we study the situation where a bubble bursts in a = Pa of the φ 0.4 suspension is more than 1 order of suspension whose level is further below the rim, such ≃ = 3 magnitude smaller than the p 71 Pa (V c 0.5 cm ) that air column resonance is simultaneously excited in that drives bubble ascent and flm stretching. τy is also 2 addition to Helmholtz resonance. - 3 orders of magnitude smaller than the capillary stress O 2 3 2σ/h ∼ (10 − 10 ) Pa (Debrégeas et al. 1998) that Frequency and decay rate of air column resonance drives flm retraction during aperture growth. Te air column resonance frequency fA in an open- Nevertheless we emphasize that the universality of the closed pipe (length L) can be expressed by extending the De φ value of c , its dependence on , the liquid rheology, formula in Kinsler et al. (2000) as and the deformation style, are poorly understood. We (2n + 1)c also do not well understand the condition under which f = , A + (20) De is sufcient to explain the viscous - brittle transi- 4(L δL) tion noting that other criteria have been proposed (see where n = 0 indicates the fundamental mode, n = 1 Gonnermann and Manga (2013) for a review). We also indicates the third-order mode and so on, c is the sound note that the tr of silicic magmatic suspension measured velocity, and δL is the open-end correction. For our by Cordonnier et al. (2012) is 3 - 6 orders of magnitude square tank (half width a = 1.5 cm) surrounded by a larger than the tr of our suspensions. Tus more suspen- square fange (half width 2.5 cm), δL = 0.802a (Dalmont sion rheology measurements which determine the fail- et al. 2001). ure condition and its parameter dependence, covering a Te decay rate βa of an air column resonance ( n = 0 ) broad range of tr are needed. due to radiation damping is expressed as a function of L We estimate that a similar transition will occur if the as same experiments which we did were conducted using 2 2 other fuids (e.g., gel, syrup) which fail brittly above a πfA π a c γ˙ βa = = , critical , provided that De is close to Dec and the sur- Q(L) 16 L(L + δL)2 (21) face tension is small (a large Eo : Eq. (4)). Brittle failure is not required to excite Helmholtz resonance when Eo is where we used the expression for quality factor Q(L) small, because the static bubble protrudes less above the given in Johnson et al. (2018a). Both fA and βa have been surface. Tis is the situation for a small bubble (V ≤ 0.14 used to constrain L at volcanoes and to monitor their 3 cm ) bursting in water (Spiel 1992; Deane 2013; Poujol changes over time (Johnson et al. 2018b). et al. 2021). Te Eo of bubbles in water used in previous When both Helmholtz and air column resonances are works are in the range of Eo < 6 , which is smaller than simultaneously excited, their waveforms having difer- O 1 2 the Eo = (10 − 10 ) of bubbles used in our experi- ent frequencies and decay rates, will be superimposed. ments. Since the maximum aperture radius Ra is less Te condition for the air column resonance to continue than the bubble surface radius Rb , for the same V, the longer than the Helmholtz resonance can be expressed as maximum Ra becomes smaller for a bubble in water. Tis βa <βr + βvt, seems to be geometrically favorable for the excitation of (22) Helmholtz resonance. Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 18 of 25

Fig. 14 Waveforms and spectrograms (re 20 µPa2 ) of airwaves excited at varying levels. V = 5 cm3 bubble bursts in a suspension of φ = 0.4 . Air column lengths L (cm) are annotated (see Additional fle 6: Movie 6 and the Graphical Abstract for the images of (e)). Waveforms are detrended and highpass fltered at 500 Hz. Running windows and overlaps are (a-d) 4.096 ms and 4.000 ms, (e-f) 8.192 ms and 8.100 ms, respectively. Blue and indicate the fA ( n = 0, 1 , respectively) calculated from Eq. (20). Pink ⊲ and ⊳ indicate the fH calculated from Eq. (11) using aperture radii Ra measured at the times indicated on the time axis. In (f), fH is not calculated because the timing of the aperture opening could not be determined from the images

where the right-hand side indicates the decay of Helm- Air column length dependence of waveform, spectrum holtz resonance resulting from radiation ( βr , Eq. (13)) and decay rate and visco-thermal ( βvt , Eq. (14)) dampings. Tis inequal- Figure 14 shows how the waveforms of the excited air- ity can be used to estimate the critical air column length waves and their spectrograms change as L increases from Lc above which the air column resonance continues L = 0.73 cm to L = 9.5 cm, as the suspension level is longer than the Helmholtz resonance. lowered (Fig. 2). Here a large bubble (slug) with a fxed Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 19 of 25

50 (a) 10000 L = 0.73 cm f L = 1.8 cm A 40 L = 3.6 cm f H ( Ra = 1.1 mm)

L = 5.5 cm f H ( Ra = 2.5 mm) /Hz)

2 L = 7.5 cm 30 L = 9.5 cm Pa (Hz) p f

20 2000 10 Peak frequency Power (dB re 20 0 1000

-10 600 0.10.5 1510 15 500 103 104 3 104 Air column length L (cm) Frequency (Hz) (b) Fig. 15 Power spectra of the waveforms shown in Fig. 14. Bandstop 105 flters were further applied at 2000 and 4000 Hz. , , and , a

indicate the peaks which we identifed as air column resonances r+ vt ( Ra=1.7 mm) n = 0 ( , 1) and Helmholtz resonance, respectively. and plotted on r+ vt ( Ra=2.5 mm) 4 the upper frequency axis indicate the fA calculated from Eq. (20) using 10 n = 0, 1 and their sizes correspond to L (1/s )

103 3 V (= 5 cm ) rises in a suspension of φ = 0.4 and excites

Helmholtz resonance (case 5 in regime III). Decay rate Figure 14a shows the result for L = 0.73 cm. For this 102 case, the waveform consists of a signal arising from Helmholtz resonance, and the spectrogram shows gliding toward a higher frequency as a result of aperture growth. 101 > 10000 0.10.5 1510 15 Te HF wave component ( Hz) is also excited Air column length L (cm) upon bursting, the same as in Fig. 6c. From the images, Fig. 16 Peak frequency fp , decay rate β (both with errors) vs. air Ra 3 we measured the aperture radius at the beginning column length L. a fp vs. L of airwaves excited by a V = 5 cm bubble and end of the resonance, and their times are marked bursting in a suspension of φ = 0.4 . A green curve indicates fA (Eq. f R on the time axis of the waveform. Using these Ra , we (20)). Pink lines indicate H (Eq. (11)) calculated using a values given calculated the fH values (Eq. (11)), which are plotted on in the legend. b β vs. L; otherwise the same as (a). A green curve indicates the decay rate β (Eq. (21)). Pink lines indicate β + β (Eqs. the frequency axis of the spectrogram. Te calculated a r vt (13), (14)) calculated using Ra values given in the legend. Ra is chosen fH explains the measured peak frequency and gliding. to calculate the minimum and maximum values of βr + βvt in the We also calculated the air column resonance frequen- range of Ra = 1.1 − 2.5 mm used in (a) cies fA (Eq. (20)), which are indicated on the frequency axis of the spectrogram. However, the signal indicat- ing air column resonance could not be identifed on the For L ≥ 7.5 cm, the air column resonance continues for a spectrogram. long time, ∼ 3 − 50 ms. Figure 14c-f shows that as L increases, the waveform Figure 15 compares the corresponding power spec- becomes complex, and another frequency component tra. Te fgure shows that since V is fxed, the peak fre- appears on the spectrogram. Unlike Helmholtz reso- quencies fp of Helmholtz resonance ( ) are also fxed nance, the frequency of this component does not change in the 1000 - 2000 Hz band. On the other hand, fp cor- with time and, importantly, decreases with L. Since the responding to the air column resonance decrease with L frequencies of this component agree fairly well with the and agree fairly well with the calculated fA . Te spectra f calculated A , we interpret that they correspond to the air show that for L ≤ 3.6 cm, the primary fp corresponds n = 0 n = 1 column resonance. We fnd that and modes to fH ( ), whereas for L ≥ 5.5 cm, it corresponds to fA ≥ ≥ are excited at L 1.8 cm and L 7.5 cm, respectively. ( ), indicating that the power of air column resonance Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 20 of 25

2 4 3 Table 2 Dimensionless numbers for a rising bubble : the Eötvös Eo = �ρgdbe/σ (bubble size, Eq. (4)), the Morton Mo = gη /ρσ (fuid property), and the Reynolds Re = ρUdbe/η (bubble velocity) numbers (see Tables 3, 4 for notations). For magma, 4 η = 230 − 1.7 × 10 Pas (Additional fle 7: Material), dbe = 1 − 10 m are assumed and the Re is calculated from the Eo, Mo using an empirical function (Clift et al. 1978) Case Eo Mo Re fow type

Expt. ( φ = 0.4) 20 − 520 3 × 103 − 1 × 108 8 × 10−4 − 5 viscous Expt. (all) 20 − 520 0.1 − 1 × 108 8 × 10−4 − 1 × 102 viscous - transitional Basaltic magma 7 × 104 − 7 × 106 2 × 108 − 7 × 1015 0.02 − 800 viscous - inertial

becomes larger. Te power of the air column resonance PSD of the Helmholtz resonance was larger than that of increases with L, whereas that of the Helmholtz reso- the air column resonance. nance does not show L dependence. To summarize, our experiments showed that when Figure 16 shows how the peak frequency fp and decay Helmholtz resonance is excited at a lowered suspension rate β depend on L. Here we used all available data for level, air column resonance is excited simultaneously. 3 a V = 5 cm bubble bursting in a suspension of φ = 0.4 When L exceeds critical, the air column resonance deter- and excited Helmholtz resonance (case 5). We deter- mines the decay rate, corresponding to the condition mined fp and β , the same as in Fig. 13(b). We also cal- βa <βr + βvt. culated fA , βa for air column resonance and fH , βr + βvt for Helmholtz resonance, which are shown in the fg- ure. Based on our measurements, we defne 2 regimes Implications for basaltic volcanic eruptions separated at L = 5 cm: a Helmholtz resonance domi- First we explain the two efects which were not mod- nant regime ( L < 5 cm) and an air column resonance eled in our experiments. One is the inertial efect. Te Re dominant regime ( L > 5 cm), which are indicated as (Table 2) indicates that the experiments model a bubble diferent background colors in the fgure. rising viscously in basaltic magma. Another is the bub- In the Helmholtz resonance dominant regime, fp and ble overpressure P which can arise by ascending from β do not depend on L. fp = 1300 − 1917 Hz can be depth, in addition to that which originates from surface explained by fH with Ra = 1.1 − 2.5 mm, as shown in tension. It is difcult to accurately estimate P , but a Fig. 16a. To check that this range of Ra is reasonable, static model (James et al. 2009; Del Bello et al. 2012) can 3 we measured the Ra of 10 runs (case 5, V = 5 cm ) and be used to obtain the limiting criterion for P to arise. found that for all of these runs, Ra grows and exceeds A bubble rising in a wide conduit expands in accord to 2.5 mm within 0.5 - 9.5 ms after bursting (Fig. 11a). the decrease in the static pressure, and is stable (Leighton Tis time scale is comparable to the mean decay time 1994). However when a slug rises in a narrow conduit, 1/β ∼ 3.5 ms obtained from L < 5 cm data, indicating the fuid above the slug partly drains as a falling flm. As that the above values of Ra are reasonable. Measured a result, a slug longer than critical becomes unstable and β = 293 ± 59 (1/s) agrees fairly well with βr + βvt cal- overpressurized. A maximum slug length llim (at 1 atm) culated from the aperture radius of Ra = 1.7, 2.5 mm as which can rise to the surface without P can be defned. indicated in Fig. 16b (see legend for the choice of these For basaltic magma in a conduit, we estimate llim ∼ 10 m Ra values). (Additional fle 7: Material). A static model gives a lower In the air column resonance dominant regime, fp and limit estimate of P. β decrease with L, and the measured fp and β agree fairly Next we explain the applicability and similarity of the well with the calculated fA and βa . Figure 16b shows that experiments to volcanoes. It follows that our experi- O βr + βvt intersects with βa at L ∼ 5 cm. Tis corresponds ments approximate a dbe ∼ (0.1 − 1) m bubble ris- to the condition given in Eq. (22) and thus explains criti- ing in a basaltic magma, such that Eo ≫ 1 (Table 2), 2 5 cal L well. We notice, however, that in Fig. 16a, there is Re ≤ 10 (laminar fow), and P ≪ Pa ∼ 10 Pa. Te −3 an outlier at L = 9.5 cm, fp = 2050 Hz (PSD = 37.78 dB/ Dec = 1.4 × 10 (Eq. (18)) which we obtained, is com- Hz), which is close to fH rather than to fA . Te power parable to the Dec obtained in the previous works. Tus spectrum of this data has a secondary peak at 750 Hz it may be used to estimate the stress τ ∼ DecG∞ needed (PSD = 35.94 dB/Hz), which is close to the estimated for a magmatic bubble flm to fail brittly, or the modu- fA = 830 Hz. Tus, for this case, although L > 5 cm, the lus G∞ ∼ τ/Dec of the magma, when the G∞ or the τ is Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 21 of 25

known. We remark that the experiments and volcano enhanced by the P . Tis difers from the experiments share a geometrical similarity. In the experiments, the dbe where the the flm is stretched solely by the p. scaled to the conduit width W was dbe/W = 0.19 − 0.95 , Figure 17a, b is an example of an eruptive event which includes a slug ( dbe/W > 0.6 ), a geometry com- observed at Aso (Ishii et al. 2019). Te spectrum (black mon at volcanoes. Te experiments in which a slug bursts line in Fig. 17c) shows a primary peak at fp = 0.5 Hz, at diferent levels L below the conduit rim were con- which has been interpreted as fA with L ∼ 120 m (Yokoo ducted for L/W = 0.24 − 3.2 , which covers a wide range et al. 2019). It appears similar to Fig. 15 ( L = 5.5 cm ; including the case for Aso ( L/W ∼ 2.4 ) (Yokoo et al. L/W = 1.8 ) whose primary peak is also fA . We analyzed 2019). a total of 10 events (30 s each) covering a total time span Now we describe the implications. Our regime diagram of 44 minutes and stacked the power spectra, shown as (Fig.4b) showed that the HF wave is excited in suspen- a red line in Fig. 17c. Te stacked spectrum shows that sions of all φ which we used and for the widest range of the 0.5 and 2.2 Hz peaks are well-defned. However, the bubble volume V, where we emphasize that the HF wave peaks within the 3.5 - 16.5 Hz band vary among events. component is excited together with the Helmholtz reso- Figure 17c also shows the background spectrum during nance (e.g., Figs.6d, 8b). Tis suggests that the HF wave, the low activity period between the events. In addition to excited at burst, may be also common at volcanoes. On the incessant fA ∼ 0.5 Hz peak (Yokoo et al. 2019), the the other hand, the Helmholtz resonance is excited only ∼ 2.2 Hz peak also exists. Comparing the spectra, we when a bubble bursts while it is rising, which occurs in a infer that the 3.5 - 16.5 Hz band (red arrow) originates solid - like ( φ = 0.4 ) suspension and for a V above criti- from the eruptive event. Te ∼ 2.2 Hz peak deviates from cal. Tis suggests that if the Helmholtz resonance could the predicted third-order fA ∼ 1.5 Hz of the air column be detected at volcanoes, it may be used to track the resonance. Unlike the experiments, the Helmholtz reso- change of magma rheology or bubble V. Here we may nance in a bubble is unsuited as a mechanism. We specu- estimate the aperture radius Ra , by combining the meas- late that it may be a resonance in a separate cavity. ured frequency fp and decay rate β (Fig. 13b), and then Since the 3.5 - 16.5 Hz band appears similar to the HF estimate the V, an analyses similar to that by Vergniolle wave excited at burst, we estimate the bubble radius Rb and Caplan-Aucherbach (2004); Montalto et al. (2010). which excites this band, assuming fV and fN . From Eq. It is to be noted that there has been an observation at a (7), we fnd that a Rb ∼ 3 m bubble excites fV ∼ 10 Hz, supraglacial pond, suggestive of Helmholtz resonance in where we assumed a flm thickness of h ∼ 30 mm from a bubble (Podolskiy 2020). We showed that an air fow the scoriae size (Namiki et al. 2018) (see Additional fle 7: vented from a defating bubble can be detected (Fig. 6e, Materials for other parameters). Tis bubble is spherical Additional fle 7: Fig. S7) and that we may estimate its because it is smaller than the conduit radius W /2 ∼ 25 lower limit velocity vaf from the time lag of the air fow. m (Yokoo et al. 2019). Te observed frequency range may Tis analyses may complement the ballistic method used be explained by the variation of Rb and h (Fig. 12). From to estimate the vaf (Tsunematsu et al. 2019). We showed Eq. (10), we fnd that a Rb ∼ 20 m bubble excites fN ∼ 16 that when the fuid level is low, an air column resonance Hz, where we assumed c = 650 m/s for a hot gas with ash including its higher modes (frequency fA ) is triggered by (Ishii et al. 2019). Tus both fV , fN can become candi- bubble bursting, which itself excites resonant frequen- dates to explain this band. cies that difers from fA (Fig. 15). Tis suggests that the air column resonance at volcanoes (Johnson et al. 2018b) Conclusions may be common. Experiments on bubble bursting and airwave excitation Here we analyze the infrasound observed at Aso and revealed 2 strikingly diferent regimes. When a bub- compare with the experiments, bearing in mind that the ble rises in a suspension of φ ≤ 0.3 or a small bubble experiments approximate a bubble with a small over- ≤ 3 φ = 5 (V 0.4 cm ) rises in a suspension of 0.4 , it forms P ≪ Pa ∼ 10 P pressure ( Pa). An estimate of can be a hemispherical cap after surfacing and bursts thereaf- obtained from energy conservation (Wilson 1980). Using ter, exciting an HF wave, which is likely to be excited by the maximum gas fow velocity at Aso (Tsunematsu et al. ≥ 3 flm vibration. In contrast when a large bubble (V 5 P ∼ 7 × 10 3 2019), we estimate Pa (Additional fle 7: cm ) rises in a suspension of φ = 0.4 , it bursts when it Materials), an upper limit using this method. Although partially protrudes above the surface, and excites Helm- �P < Pa , this P is not negligible compared to the buoy- 4 holtz resonance and/or vents a detectable air fow. An p ∼ O(10 ) 3 ancy pressure Pa of a meter-sized bubble. intermediate-sized bubble ( 0.5 ≤ V ≤ 4 cm ) rising in Tus for a bubble rising at Aso, the flm stretching will be Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 22 of 25

Fig. 17 Infrasound observed at the Aso volcano, Japan, on April 21, 2015 (JST) (Ishii et al. 2019). a An example of a waveform (detrended). b Spectrogram of (a) (time window 2.56 s, overlap 2.50 s). c Power spectrum (smoothed) of (a) (event#6), stacked spectrum of 10 events during 19:07:10 - 19:51:00 (JST), background between the events (19:26:00 - 19:26:30). Arrows indicate the peaks at 0.5, 2.2 Hz, and the 3.5 - 16.5 Hz band a suspension of φ = 0.4 , bursts with a mixture of above of a silicic magmatic suspension. When a bubble excites 2 styles. We interpreted that the regime transition corre- Helmholtz resonance in the conduit with a lowered sus- sponds to the viscous - brittle transition, and estimated pension level, an air column resonance is excited simul- −3 the critical Deborah number as Dec = 1.4 × 10 , which taneously. Te critical air column length L above which is comparable to the Dec at the viscous - brittle transition the air column resonance continues longer than the Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 23 of 25

Table 3 Notation Table 4 Subscripts a tank or conduit half width (m) A air column resonance c sound velocity (m/s) af air fow d diameter (m) a aperture, air column, air f frequency (Hz) b bubble G modulus (Pa) be equivalent bubble g gravitational acceleration (m/s 2) c critical n K consistency (Pa s ) d deformation h flm thickness (m) H Helmholtz resonance L air column or cavity length (m) i inviscid δL open-end correction (m) l liquid l submerged bubble length (m) N N-wave l′ efective neck length (m) p particle, peak f m r relative, radiation, relaxation m power law exponent ( β ∝ p ) V flm vibration n fow index v viscid O the order of vt visco-thermal P pressure (Pa) y yield P overpressure (Pa) p bubble buoyancy pressure (Pa) R radius (m) r correlation coefcient Additional fle 1: Movie 1. An example of a bursting bubble dominantly exciting an HF wave (case 3) shown in Fig. 5. A hemispherical cap bubble t time (s) of a volume V = 0.6 cm3 bursts in a suspension of φ = 0.3 (air U ascent velocity (m/s) column length L ∼ 1 mm) and the aperture grows rapidly. Replayed at V bubble volume ­(m3) 1/3000 speed. v velocity (m/s) Additional fle 2: Movie 2. An example of a bursting bubble dominantly W tank or conduit width (m) exciting Helmholtz resonance (case 5) shown in Fig. 6. A bubble of a vol- = 3 = β decay rate (1/s) ume V 0.6 cm bursts while it is rising in a suspension of φ 0.4 ( L < 1 mm). The aperture grows slowly compared to Movie 1. Replayed γ˙ strain rate (1/s) at 1/200 speed (1/1000 during slowdown motion). γ ratio of specifc heats h Additional fle 3: Movie 3. An example of a bursting bubble dominantly η viscosity (Pa s) exciting Helmholtz resonance (case 5) shown in Fig. 7. A bubble of a 3 ρ density (kg/m3) volume V = 7 cm bursts while it is rising in a suspension of φ = 0.4 �ρ density diference (kg/m3) ( L ∼ 7 mm). The aperture grows slowly compared to Movie 1. Replayed at 1/500 (1/100) speed at t < 15 ms (t > 15 ms). σ surface tension coefcient (N/m) φ volumetric packing fraction Additional fle 4: Movie 4. An example of a bursting bubble exciting Helmholtz resonance after a thin flm bursts (case 4b) shown in Fig. 8. A τ shear stress (Pa) bubble of a volume V = 6 cm3 bursts while it is rising in a suspension of φ = 0.4 ( L ∼ 6 mm). A thin flm forms at the top of the bubble, which bursts and the aperture grows rapidly compared to the cases shown in Movies 2 - 3. Replayed at 1/2000 speed. Helmholtz resonance is explained by the decay rate of the Additional fle 5: Movie 5. An example of a bursting bubble dominantly former becoming less than that of the latter. Our work exciting an air fow (case 6a) shown in Fig. 9. A bubble of a volume V = 12 cm3 bursts while it is rising in a suspension of φ = 0.4 provides an experimental evidence that a bursting bubble ( L ∼ 13 mm). The aperture grows rapidly compared to Movies 2 - 3. A excites diverse airwaves, including the triggering of an air pink arrow on the spectrogram indicates Helmholtz resonance. Replayed column resonance. Te airwaves cover a frequency range at 1/100 speed. of 3 orders of magnitude, suggesting that the same broad- Additional fle 6: Movie 6. A bubble of a volume V = 5 cm3 bursts = band excitation occurs at basaltic volcanoes. Te HF wave while it is rising in a suspension of φ 0.4 with a lowered suspen- sion level (air column length L = 7.5 cm), shown in Fig. 14e and in is commonly excited, whereas the Helmholtz resonance is the Graphical Abstract. The air column resonance (n = 0 , n = 1 ), excited under a limited condition. Tis implies that if the the Helmholtz resonance, and an HF wave are simultaneously excited. Helmholtz resonance could be detected, it may be used to Replayed at 1/400 speed. track the change of magma rheology or bubble V. Additional fle 7. Additional materials, tables and fgures.

Supplementary Information The online version contains supplementary material available at https://​doi.​ org/​10.​1186/​s40623-​021-​01472-7. Hashimoto and Sumita Earth, Planets and Space (2021) 73:143 Page 24 of 25

Acknowledgements Gurioli L, Coló L, Bollasina AJ, Harris AJL, Whittington A, Ripepe M (2014) Dynam- We thank Roberto Sulpizio and anonymous reviewers for their helpful com- ics of strombolian explosions : inferences from feld and laboratory studies ments which improved the manuscript, A. Namiki for discussions, A. Yokoo of erupted bombs from stromboli volcano. J Geophys Res 119:319–345 and K. Ishii for providing a preprint and infrasound data, and K. Tsunematsu for Ishibashi H (2009) Non-Newtonian behavior of plagioclase-bearing basaltic a video of eruption. magma: subliquidus viscosity measurement of 1707 basalt of Fuji volcano, Japan. J Volcanol Geoth Res 181:78–88 Authors’ contributions Ishii K, Yokoo A, Kagiyama T, Ohkura T, Yoshikawa S, Inoue H (2019) Gas fow KH conducted the experiments and data analyses. IS contributed to devising the dynamics in the conduit of Strombolian explosions inferred from seismo- experimental method and modeling the results. Both authors contributed to acoustic observations at Aso volcano, Japan. Earth Planets Space 71:13. interpreting the data, preparing and revising the manuscript. Both authors read https://doi.​ org/​ 10.​ 1186/​ s40623-​ 019-​ 0992-z​ and approved the fnal manuscript. James MR, Lane SJ, Chouet B, Gilbert J (2004) Pressure changes associated with ascent and bursting of gas slugs in liquid-flled vertical and inclined Funding conduits. J Volcanol Geoth Res 129:61–82 This work was supported by JSPS KAKENHI (JP15K13591, JP17H05417, James MR, Lane SJ, Wilson L, Corder SB (2009) Degassing at low magma-viscos- JP18K03800) and Sasakawa Scientifc Research Grant (2018-6020). ity volcanoes: Quantifying the transition between passive bubble-burst and Strombolian eruption. J Volcanol Geoth Res 180:81–88 Availability of data and materials Johnson JB, Ripepe M (2011) Volcano infrasound: a review. 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