Sawtooth wave-like pressure changes in a syrup eruption experiment: implications for periodic and nonperiodic volcanic oscillations
Abstract
This study is based on the observation of sawtooth wave-like pressure changes (STW) observed during repetitive gas emissions in a syrup eruption experiment. Similar waveforms are observed at many active volcanoes as geodetic signals. By studying the physics of such experiments, we often find new ideas and insights that are applicable to natural volcanic phenomena. We consequently try identifying the features common to both our experimental system and natural volcanic systems. We infer that the oscillatory mechanism in our experiment is similar to flow-induced oscillation controlled by a coupling between elastic capacitance and variable flow resistance. We developed an elementary pipe–chamber system to quantitatively test this hypothesis. We observed three distinct oscillatory patterns: periodic STW, non-STW, and nonperiodic STW. A mathematical model is constructed to support the hypothesis and to enable comparison with existing models of volcanic systems. Models of flow-induced volcanic oscillations are mathematically similar to the model derived from our experimental system. Our results indicate that flow pattern transitions are essential for oscillatory behavior during gas emission. An important finding is that cycle periodicity and fluctuation are controlled by interactions between upward and downward flow within the pipe, especially during the termination of gas emission and the transition to the next cycle. Similar interactions occur in natural volcanoes: during the termination of an explosive eruption, fall-back or drain-back of ejected materials may modulate the explosivity and periodicity of the next eruption cycle. Our experimental system may provide a useful tool for understanding the oscillatory behavior of natural volcanic systems. This study may also give a proof that the original eruption experiment provides a useful tool for explaining the dynamics of oscillatory and eruptive behaviors of natural volcanic systems in educational (e.g., Open Day) demonstrations.
Keywords
Laboratory experiment Sawtooth waveform Dynamical system Flow-induced oscillation Lumped parameter model Educational outreach experimentIntroduction
Experiments using common materials to simulate geological processes are effective tools for Earth science education and outreach (e.g., Coffey 2008; Kurita et al. 2008; Deus et al. 2011). In addition, studying the physics of experimental systems frequently provides novel ideas and insights that can be applied to natural phenomena (e.g., Scott et al. 1986; Mader et al. 2004; Iga and Kimura 2007; Kumagai et al. 2008).
Inflation–deflation cycles of volcanic edifices are observed at many active systems and are interpreted as geodetic signals associated with magma and gas flow (e.g., Ohminato et al. 1998; Fujita et al. 2004; Iguchi et al. 2008; Anderson et al. 2010; Genco and Ripepe 2010; Fontaine et al. 2014). These geodetic oscillations display a variety of waveform shapes, commonly including asymmetric waveforms consisting of alternating gradual change and rapid recovery. Gradual inflation and abrupt deflation cycles are commonly associated with repetitive eruptions (e.g., Strombolian eruptions: Genco and Ripepe (2010); Vulcanian explosions: Iguchi et al. (2008); Takeo et al. (2013); and geysers: Nishimura et al. (2006)). During dome-building eruptions, similar cycles (Voight et al. 1998) and reverse cycles consisting of gradual deflation and rapid inflation (Anderson et al. 2010) have been reported. Reverse cycles have also been recorded during caldera collapse (Fujita et al. 2004; Fontaine et al. 2014) and magma drain-back (Fukuyama 1988).
Asymmetric volcanic oscillations are reproduced by flow-induced oscillation models that include coupling between elastic capacitance and variable flow resistance (Whitehead and Helfrich 1991; Ida 1996; Wylie et al. 1999; Maeda 2000; Barmin et al. 2002; Melnik and Sparks 2005; Kozono and Koyaguchi 2012; De’ Michieli Vitturi et al. 2013; Chen et al. 2018). Existing models commonly assume that elastic capacitance is related to chamber elasticity and/or magma compressibility. However, different mechanisms of variable flow resistance have been proposed, including viscosity dependence on temperature distribution in the conduit (Whitehead and Helfrich 1991); changes in conduit radius due to the deformation of surrounding rocks, which depends on overpressure (Ida 1996; Maeda 2000; Chen et al. 2018); velocity-dependent volatile and/or crystal contents, which control magma viscosity (Wylie et al. 1999; Barmin et al. 2002); and stick–slip motion with rate-dependent friction (Iverson et al. 2006). Flow-induced oscillation can also be generated by the rate-dependent effective density of magma in a conduit and the competition between bubble growth and escape (Woods and Koyaguchi 1994). The combined effects of these mechanisms have been investigated through advanced numerical methods (Melnik and Sparks 2005; Kozono and Koyaguchi 2012; De’ Michieli Vitturi et al. 2013).
Various mechanisms of flow-induced oscillations have been identified in laboratory experiments, and possible applications of such mechanisms to natural volcanic oscillations have been proposed. For example, Lane et al. (2008) reported cyclic pressure changes resulting from alternating periods of Poiseuille flow and plug flow in a vertical pipe and compared the oscillation to tilt cycles during dome-building eruptions. Self-induced pressure perturbations repeatedly occur during flow from a chamber through a pipe for fluids with complex rheologies with a negative correlation between shear stress and shear rate (Den Doelder et al. 1998). Similar oscillations might occur during flow of magma which has a complex rheology (Kurokawa 2016). A self-induced oscillation known as a “pressure drop oscillation” occurs in systems with a sufficiently compressible volume in a chamber connected upstream to a boiling pipe (Ozawa et al. 1979). Fujita et al. (2004) applied the mechanism to explain oscillatory behavior observed in the 2000 Miyakejima caldera collapse.
During the STW stage of our eruption experiment, large bubbles slowly ascended the pipe as a slug flow. Ascending slug flow is regarded as an analogue for Strombolian eruptions and has been studied in several experiments (Jaupart and Vergniolle 1989; James et al. 2004, 2006; Azzopardi et al. 2014; Del Bello et al. 2015; Pioli et al. 2017). In these experiments, pressure fluctuations in the conduit are due to bubble motion, which is controlled by viscous, inertia and gravity forces, by bubble expansion, and by the boundary conditions including the geometry of the system. The STW in our experiment were generated by slug flow in a pipe connected to a chamber and did not occur when the chamber was disconnected. Therefore, we assume that the STW were not generated by bubble motion alone but were similar to flow-induced oscillations controlled by coupling between elastic capacitance (the bottle) and variable flow resistance of the liquid syrup flowing up the pipe.
To quantitatively test this hypothesis, we conducted controlled experiments involving gas–liquid flow in a vertical pipe connected to a gas chamber. We observed three distinct oscillatory patterns: periodic STW, non-STW, and nonperiodic STW, all of which have been observed in natural volcanoes (e.g., Fukuyama (1988) for periodic STW, Voight et al. (1998) for non-STW, and Iguchi et al. (2008) for nonperiodic STW). We find that waveform shapes are related to flow patterns in the pipe. We then derived a mathematical model for the experimental system in which we treat the physical quantities as lumped parameters to represent the system by ordinary differential equations. This derivation allows the experimental system to be compared with models for natural volcanic systems. We then identify the mathematical similarities and differences among our system and existing models of volcanic oscillation. Although our experiment differs from scaled experiments that simulate slug flow in a volcanic conduit (e.g., Jaupart and Vergniolle 1989; James et al. 2004, 2006; Del Bello et al. 2015), our experimental observations, combined with comparison to its mathematical equivalent, provide new insights into volcanic systems.
Experimental method
Gas flux into the chamber (Q_{in}) was regulated by a microneedle valve (IBS FMNV2) that ensured a constant gas flux of between 10^{−7} and 10^{−6} m^{3}/s. We used sugar syrup diluted in distilled water as the liquid phase. The liquid behaved as a Newtonian fluid, and we measured its viscosity using a BOHLIN CVO Rheometer, and adjusted it to 1 Pa s at the experimental temperature of 25 °C. Injected gas flowed through the pipe and forced the liquid residing in the pipe upward, resulting in gas–liquid flow within the pipe.
A pressure sensor (KISTLER 701A with a 5011A charge amplifier) was installed to measure pressure changes in the chamber, and a microphone (Brüel & Kjær 4193+2669L with a Nexus 2690 signal conditioner) was mounted to the top of the pipe. Data were sampled at 10 kHz using a PC-based data acquisition system (DEWETRON DEWE-211).
A high-speed black-and-white camera (Photoron FASTCAM Mini) was synchronized with the data acquisition PC and was focused on the lowermost 200 mm of the pipe, where gas–liquid flow occurred. The field of view was illuminated by a flat backlight source. We used a pale-color syrup so as to improve the visibility of the liquid phase against the gas phase and the pipe. Dark areas were thus shaded according to the thickness of the liquid across the pipe. We used the resulting distribution of light intensity to observe flow patterns.
The controlled experimental parameters were V_{c}, Q_{in}, and the amount of syrup in the pipe, as defined by the initial syrup height H_{s}. These parameters were fixed during each experimental run. We conducted 143 runs in total, during which the parameters varied from 1.5 to 7.1 × 10^{−7} m^{3}/s for Q_{in}, 20 to 157 cm^{3} for V_{c}, and 60 to 150 mm for H_{s}. Data were recorded from the beginning of each run for at least 20 pressure cycles.
Experimental results
Relatively small Q_{in} and V_{c} yield non-STW waveforms (Figs. 3c, 4c, and 6b), displaying no abrupt drop in pressure. Moreover, these waveforms have no discernible periodic features, resulting in spectra that lack clear harmonic peaks. The peak frequencies of pressure changes can be used to determine the period of the experimental data (e.g., Fig. 3d). Slug flow is observed to ascend continuously within the pipe (Movie 2). The uppermost liquid slug undergoes a significant reduction in length, forming a membrane-like structure that then ruptures. Ruptures are not observed in the following liquid slugs; consequently, we observed no slug-to-annular flow transitions. Although there is a periodic rise and fall of the top surface of a gas/liquid column in a sawtooth wave-like manner which is similar to a pattern reported in Azzopardi et al. (2014), sawtooth wave-like pressure changes do not appear.
Nonperiodic STW are observed when Q_{in} is large (Figs. 3e, 4c, and 6c; Movie 3). These waveforms display STW features but contain no clear harmonic spectra (Fig. 3f). STW cycles are associated with slug-to-annular flow transitions (Fig. 6c), similar to the periodic STW; however, each slug-flow regime displays a distinct flow pattern.
Mathematical model
We now construct a mathematical model that includes the two essential factors of elastic capacitance and variable flow resistance, in a similar manner to the existing lumped parameter model for flow-induced instabilities in volcanic systems (Ida 1996; Wylie et al. 1999; Maeda 2000; Barmin et al. 2002; Nakanishi and Koyaguchi 2008).
Parameters and variables used in the mathematical model of our experiment
Parameter | Units | Description | Basic value | Range |
---|---|---|---|---|
Dimensional parameters for the experiment | ||||
P_{0} | Pa | Atmospheric pressure | 10^{5} | – |
V_{c} | m^{3} | Gas chamber volume | 100 × 10^{−6} | 20 – 157 × 10^{−6} |
Q_{in} | m^{3}/s | Gas volume flux into the chamber | 5 × 10^{−7} | 1.5 – 7.1 × 10^{−7} |
r | m | Radius of the gas flow | 1.85 × 10^{−3} | |
a | m | Pipe radius | 2.5 × 10^{−3} | |
T | K | Temperature | 293 | |
μ_{A} | Pa s | Viscosity of the air | 1.8 × 10^{−5} | |
μ | Pa s | Viscosity of the liquid | 1 | |
ρ | kg/m^{3} | Density of the liquid | 1400 | |
L | m | Length of the pipe | 0.59 | |
g | m/s^{2} | Acceleration due to gravity | 9.8 | |
H_{s} | m | Initial syrup height | 90 × 10^{−3} | 60 – 120 × 10^{−3} |
w | m/s | Average velocity of the downward film flow | 1.7 × 10^{−3} | |
W | m^{3}/s | wπa ^{2} | 5.2 × 10^{−8} | |
t_{c} | s | Time required for the slug reconstruction process | 1.5 | |
ξ | Pa s/m^{2} | Fitting parameter for Δp (see Appendix) | 1.11 × 10^{−6} | 0 – 1.11 × 10^{−6} |
Dimensional variables for the experiment | ||||
p | Pa | Excess pressure in the chamber | ||
t | s | Time | ||
Q | m^{3}/s | Volume flux in the pipe (=πa^{2}u) | ||
u | m/s | Velocity of liquid slug and gas in the pipe | ||
μ_{v} | Pa s | Average viscosity of the slug in the pipe (Eq. 3) | ||
ρ_{v} | Pa s | Average density of the slug in the pipe (Eq. 4) | ||
L_{∗} | m | Total slug length | ||
L_{c} | m | Critical liquid slug length | ||
Δp | Pa | Overpressure caused by surface tension (Eq. 5) | ||
μ_{E} | Pa | Effective viscosity (Eq. 7) | ||
H_{w} | m | Top of the film flow | ||
x | m | Head position of the slug flow | ||
Scaling units for the experiment | ||||
P_{u} | πa^{2}LP_{0}/(V_{c}A) | Pressure change at the gas chamber when gas with the volume of πa^{2}L/A is added in or effused from the gas chamber with V_{c} | 1.4 × 10^{4} | 8.9 × 10^{3} – 7.0 × 10^{4} |
Q_{u} | π^{2}a^{6}P_{0}/(8μ_{A}V_{c}A) | Characteristic flux of Poiseuille flow, πa^{4}P_{u}/8μ_{A}L | 2.0 × 10^{−2} | 1.3 × 10^{−2} – 1.0 × 10^{−1} |
T_{u} | 8μ_{A}V_{c}L/(πa^{4}P_{0}) | Characteristic time, (πa^{2} L/A)/Q_{u} | 6.9 × 10^{−4} | 1.4 × 10^{−4} – 1.1 × 10^{−3} |
Dimensionless variables for the experiment | ||||
P^{′} | p/P_{u} | Dimensionless excess pressure | ||
Q^{′} | Q/Q_{u} | Dimensionless outflux | ||
t^{′} | t/T_{u} | Dimensionless time | ||
\( {L}_{\ast}^{\prime } \) | L_{∗}/L | Dimensionless total slug length | ||
x^{′} | x/(Q_{u}T_{u}/(πa^{2})) | Dimensionless head position of the slug flow | ||
\( {H}_{\mathrm{w}}^{\prime } \) | H_{w}/(Q_{u}T_{u}/(πa^{2})) | Dimensionless top of the film flow | ||
\( {\mu}_{\mathrm{E}}^{\prime } \) | μ_{E}/μ_{A} | Dimensionless effective viscosity | ||
\( {\rho}_{\mathrm{v}}^{\prime } \) | ρ_{v}/ρ | Dimensionless average density | ||
\( {L}_{\mathrm{c}}^{\prime } \) | L_{c}/L | Dimensionless critical slug length | ||
Dimensionless parameters for the experiment | ||||
α | r^{2}/a^{2} | Area fraction of the gas slug in the pipe | 0.55 | |
Γ | ξa^{2}/(8μ) | Dimensionless parameter representing the effect of surface tension | 0.86 | 0 – 0.86 |
A | (1 − α)/α | Ratio of the area of the film flow to gas part at the cross section of gas slug | 0.83 | |
\( {Q}_{\mathrm{in}}^{\prime } \) | Q_{in}V_{c}8μ_{A}A/(π^{2}a^{6}P_{0}) | Dimensionless Q_{in}, (Q_{in}/Q_{u}) | 2.5 × 10^{−5} | 1.5 × 10^{−6} – 5.5 × 10^{−5} |
G | ρgV_{c}A/(πa^{2}P_{0}) | Ratio of hydrostatic pressure (ρgL) to P_{u} | 5.8 × 10^{−1} | 1.2 × 10^{−1} – 9.1 × 10^{−1} |
μ^{′} | μ/μ_{A} | Ratio of high-viscosity part (liquid) to low-viscosity part (gas) | 5.5 × 10^{4} | |
\( {t}_{\mathrm{c}}^{\prime } \) | t_{c}/T_{u} | Dimensionless time for slug reconstruction process | 2.1 × 10^{3} | 1.3 × 10^{3} – 1.0 × 10^{4} |
W^{′} | W/Q_{u} | Dimensionless W | 2.6 × 10^{−6} | 0.3 × 10^{−6} – 2.5 × 10^{−6} |
Basic equations
We consider a volume flux of gas Q_{in} into the chamber with volume V_{c}. Both the liquid slug and gas in the pipe move upward at velocity u. We define the volume flux in the pipe as Q = πa^{2}u and the area fraction of gas flow in the pipe as α = r^{2}/a^{2}, where a is the pipe radius and r is the radius of the gas flow (Fig. 7a–c). Then, the volume flux of gas in the pipe is defined as αQ.
Dimensionless equations
Parameters and variables used in the mathematical model for volcanic systems
Parameter | Units | Description | Basic value | Range |
---|---|---|---|---|
Dimensional parameters for volcanic system | ||||
γ | Pa | Effective volumetric modulus of the chamber | 10^{10} | 10^{10 – 11} |
V_{N} | m^{3} | Chamber volume | 2.7 × 10^{8} | 10^{6 – 11} |
Q_{in} | m^{3}/s | Influx to the chamber | 1 | 0.26–2.00 |
a_{N} | m | Radius of the conduit | 10 | 10–25 |
μ_{1} | Pa s | Viscosity of magma (before significant crystallization [NK model]) | 6.4 × 10^{5} | 10^{5 – 6} |
μ_{2} | Pa s | Viscosity after significant crystallization [NK model] | – | |
ρ_{N} | kg/m^{3} | Density of magma | 2500 | |
L_{N} | m | Length of the conduit | 10^{3} | 10^{3 – 4} |
t_{∗} | s | Viscosity of each fluid particle increases after the fluid particle ascends for a constant time t_{∗} [NK model] | 10^{6} | 10^{5 – 6} |
Dimensional variables for volcanic system | ||||
p_{ch} | Pa | Chamber pressure | ||
t | s | Time | ||
Q | m^{3}/s | Inflow to chamber | ||
μ_{vN} | Pa s | Average viscosity in the conduit | ||
L_{∗N} | m | Length of high-viscosity region [NK model] | ||
Scaling unit for volcanic system | ||||
P_{uN} | \( \pi {a}_{\mathrm{N}}^2{L}_{\mathrm{N}}\gamma /{V}_{\mathrm{N}} \) | Pressure change at the magma chamber when a magma with the volume of conduit (\( \pi {r}_{\mathrm{N}}^2{L}_{\mathrm{N}} \)) is added in or effused from the chamber with V_{N} | 1.2 × 10^{7} | 3.1 × 10^{4} – 2.0 × 10^{12} |
Q_{uN} | \( {\pi}^2{a}_{\mathrm{N}}^6\gamma /\left(8{\mu}_1{V}_{\mathrm{N}}\right) \) | Characteristic flux of Poiseuille flow, \( \pi {a}_{\mathrm{N}}^4{P}_{\mathrm{uN}}/8{\mu}_1{L}_{\mathrm{N}} \) | 7.1 × 10^{1} | 1.9 × 10^{−2} – 4.7 × 10^{7} |
T_{uN} | \( 8{\mu}_1{L}_{\mathrm{N}}{V}_{\mathrm{N}}/\left(\pi {a}_{\mathrm{N}}^4\gamma \right) \) | Characteristic time, \( \pi {a}_{\mathrm{N}}^2{L}_{\mathrm{N}}/{Q}_{\mathrm{uN}} \) | 4.4 × 10^{3} | 4.2 × 10^{−2} – 1.6 × 10^{8} |
Dimensionless variables for volcanic system | ||||
\( {P}_{\mathrm{N}}^{\prime } \) | p_{ch}/P_{uN} | Dimensionless chamber pressure | ||
\( {Q}_{\mathrm{N}}^{\prime } \) | Q/Q_{uN} | Dimensionless influx | ||
\( {t}_{\mathrm{N}}^{\prime } \) | t/T_{uN} | Dimensionless time | ||
\( {L}_{\ast \mathrm{N}}^{\prime } \) | L_{∗N}/L | Dimensionless length of High-viscosity region [N&K 2008] | ||
\( {\mu}_{\mathrm{vN}}^{\prime } \) | μ_{vN}/μ_{1} | Dimensionless viscosity | ||
Dimensionless parameters for volcanic | ||||
\( {Q}_{\mathrm{inN}}^{\prime } \) | \( {Q}_{\mathrm{inN}}{V}_{\mathrm{N}}8{\mu}_1/\left({\pi}^2{a}_{\mathrm{N}}^6\gamma \right) \) | Dimensionless influx | 1.4 × 10^{−2} | 2.1 × 10^{−8} – 5.2 × 10^{1} |
G_{N} | \( {\rho}_{\mathrm{N}}g{V}_{\mathrm{N}}/\left(\pi {a}_{\mathrm{N}}^2\gamma \right) \) | Ratio of magmastatic pressure, (ρ_{N}gL_{N})/P_{uN} | 2.1 × 10^{0} | 1.2 × 10^{−4} – 7.8 × 10^{2} |
\( {\mu}_{\mathrm{N}}^{\prime } \) | μ_{2}/μ_{1} | Ratio of the viscosity [NK model] | 10^{2} | 10^{1 − 3} |
t_{∗}^{′} | t_{∗}/T_{uN} | Dimensionless t_{∗} [NK model] | 6.1 × 10^{4} | 6.1 × 10^{−4} – 2.4 × 10^{7} |
Definitions and values for the dimensionless parameters are listed in Table 1.
Mathematical analysis of the system behavior
Calculation method
The model periodicity is defined by the duration from a completion of liquid slug formation to the next. For comparison with the experimental results, the model amplitude is evaluated using RMS. We removed the mean pressure change before calculating RMS.
Periodic STW and non-STW
Nonperiodic STW
Our model cannot explain nonperiodic STW (Figs. 3e and 6c) as each cycle is assumed to start with the same initial values. Figure 6c (e.g., t = 74 – 75 s) shows slug reconstruction within the middle of the pipe (hereafter termed “halfway reconstruction”), which is only observed during nonperiodic STW. We infer that large surface disturbances, resulting from violent liquid slug membrane rupturing under large Q_{in}, grow rapidly and induce halfway reconstruction.
Figure 9b–d shows the waveforms and periodicities with variable initial conditions among different cycles. STW cycles are stable provided that new liquid slugs are reproduced at the bottom of the pipe, irrespective of variations in their initial lengths and growth rates (Fig. 9b, c). However, variation in the slug reconstruction site has a significant effect on the amplitude and periodicity of the signal, generating pressure change waveforms similar to nonperiodic STW (Fig. 9d), as observed in the experiment (Fig. 6c). To determine the mechanism that controls the initial conditions, we will conduct additional experiments with higher-resolution images of flow during future work.
Discussion
Model comparisons
Model comparisons
Model for experiment | Model for volcanic system | Similarities and differences | |
---|---|---|---|
(a) Pressure change in the chamber and pressure drop in the pipe (conduit) | This study \( \left\{\begin{array}{c}\mathrm{d}{P}^{\prime }/\mathrm{d}{t}^{\prime }={Q}_{\mathrm{in}}^{\prime }-\alpha {Q}^{\prime}\\ {}\kern-0.55em {P}^{\prime }={\mu}_{\mathrm{E}}^{\prime }{Q}^{\prime }+{\rho}_{\mathrm{v}}^{\prime }G\end{array}\right. \) | E.g., Ida (1996), Wylie et al. (1999), Barmin et al. (2002), Nakanishi and Koyaguchi (2008) \( \left\{\begin{array}{c}\kern-0.80em \mathrm{d}{P}_{\mathrm{N}}^{\prime }/\mathrm{d}{t}_{\mathrm{N}}^{\prime }={Q}_{\mathrm{inN}}^{\prime }-{Q}_{\mathrm{N}}^{\prime}\\ {}{P}_{\mathrm{N}}={\mu}_{\mathrm{vN}}^{\prime }{Q}_{\mathrm{N}}^{\prime }+{\rho}_{\mathrm{vN}}^{\prime }{G}_{\mathrm{N}}\end{array}\right. \) | Waveform shapes are caused by coupling between elastic capacitance and variable-flow resistance. |
(b) Effective viscosity and density in the pipe (conduit) | \( {\mu}_{\mathrm{E}}^{\prime }=\left\{\begin{array}{c}1\ \left({L}_{\ast}^{\prime }=0\right)\ \\ {}1-{L}_{\ast}^{\prime }+{\mu}^{\prime }\ {L}_{\ast}^{\prime }+{\varGamma}_{\mathrm{L}}\ \left({L}_{\ast}^{\prime }>0\right)\end{array}\right. \) \( {\rho}_{\mathrm{v}}^{\prime }={L}_{\ast}^{\prime } \) Γ_{L}: Surface tension term given by \( {\varGamma \mu}^{\prime}\left({L}_{\mathrm{c}}^{\prime }-{L_{\ast}}^{\prime}\right) \) | \( {\mu}_{\mathrm{vN}}^{\prime }=\left\{\begin{array}{c}1\ \left({L}_{\ast \mathrm{N}}^{\prime }<0\right)\ \\ {}1-{L}_{\ast \mathrm{N}}^{\prime }+{\mu}_{\mathrm{N}}^{\prime }\ {L}_{\ast \mathrm{N}}^{\prime }\ \left({L}_{\ast \mathrm{N}}^{\prime }>0\right)\end{array}\right. \) \( {\rho}_{\mathrm{vN}}^{\prime }=1 \) Nakanishi and Koyaguchi (2008) | Effective viscosity is a function of L_{∗} This study: Liquid slug (L^{′}_{∗}) and gas slug (1 − L^{′}_{∗}) region Nakanishi and Koyaguchi (2008): High-viscosity (L^{′}_{∗N}) and low-viscosity (\( 1-{L}_{\ast N}^{\prime } \)) region |
(c) Effective viscosity (density) change | \( \frac{{\mathrm{d}\mu}_{\mathrm{E}}^{\prime }}{\mathrm{d}{t}^{\prime }}=\left\{\begin{array}{c}-{\mu}^{\prime}\left(1-\varGamma \right){W}^{\hbox{'}}\kern5.50em \left({x}^{\prime }<{H}_{\mathrm{w}}^{\prime}\right)\ \\ {}-{\mu}^{\prime}\left(1-\varGamma \right){Q}^{\prime }\ \left({x}^{\prime }={H}_{\mathrm{w}}^{\prime }\ \right)\end{array}\right. \) \( \frac{{\mathrm{d}\rho}_{\mathrm{v}}^{\prime }}{\mathrm{d}{t}^{\prime }}=\left\{\begin{array}{c}-{W}^{\hbox{'}}\kern1.75em \left({x}^{\prime }<{H}_{\mathrm{w}}^{\prime}\right)\ \\ {}-{Q}^{\prime}\kern2em \left({x}^{\prime }={H}_{\mathrm{w}}^{\prime }\ \right)\end{array}\right. \) Γ = 0 when \( {L}_{\ast}^{\prime }<{L}_{\mathrm{c}}^{\prime } \) | \( \frac{\mathrm{d}{\mu}_{\mathrm{vN}}^{\prime }}{\mathrm{d}{t}_{\mathrm{N}}^{\prime }}=-2\left({\mu}_{\mathrm{N}}^{\prime }-1\right)\left({Q}_{\mathrm{N}}^{\prime }-\frac{\mu_N^{\prime }-{\mu}_{\mathrm{vN}}^{\prime }}{\mu_{\mathrm{N}}^{\prime }-1}\frac{1}{t_{\ast}^{\prime }}\ \right) \) Nakanishi and Koyaguchi (2008) | For the experiment, there is no steady state when \( {Q}_{in}^{\prime }>0 \) under \( {L}_{\ast}^{\prime }>0 \). When \( {L}_{\ast}^{\prime }=0 \), P^{′} and Q^{′} are convergent to the stable steady state (\( {P}^{\prime }={Q}^{\prime }={Q}_{in}^{\prime }/\alpha \)). |
The dimensionless parameters \( {Q}_{\mathrm{in}}^{\prime } \), G, μ^{′}, and \( {t}_{\mathrm{c}}^{\prime } \) in our experimental model are equivalent to \( {Q}_{\mathrm{inN}}^{\prime } \), G_{N}, \( {\mu}_{\mathrm{N}}^{\prime } \), and \( {t}_{\ast}^{\prime } \) in models used to approximate natural volcanic systems, respectively (see Tables 1 and 2 for their definitions). Parameter \( {t}_{\mathrm{c}}^{\prime } \) is a dimensionless characteristic time for slug reconstruction, defined using t_{c} = L_{∗i}/(Aw) and \( {t}_{\mathrm{c}}^{\prime }={t}_{\mathrm{c}}/{T}_{\mathrm{u}} \). Parameter \( {Q}_{\mathrm{inN}}^{\prime } \) is the dimensionless influx to the magma chamber, G_{N} is the ratio of magmastatic pressure (ρ_{vN}gL_{N}) to characteristic pressure (P_{uN}), L_{N} is the length of the conduit, and \( {\mu}_{\mathrm{N}}^{\prime } \) is the ratio of high to low viscosity, as defined by Barmin et al. (2002) and used in the NK model. Parameters \( {Q}_{\mathrm{inN}}^{\prime } \) and G_{N} are common in existing models (e.g., Ida 1996; Wylie et al. 1999; Barmin et al. 2002; Nakanishi and Koyaguchi 2008; Chen et al. 2018).
The \( {\mu}^{\prime }{Q}_{\mathrm{in}}^{\prime } \) term has a physical meaning in our experiment, representing the ratio of pressure loss due to viscous resistance in the pipe to the pressure change due to elastic capacitance. It is equivalent to \( {\mu}_{\mathrm{N}}^{\prime }{Q}_{\mathrm{inN}}^{\prime }=8{\mu}_2{Q}_{\mathrm{inN}}{V}_{\mathrm{N}}/\left({\pi}^2{a}_{\mathrm{N}}^6\gamma \right) \) in the NK model, where μ_{2} is the viscosity after crystallization, Q_{inN} is the influx to the chamber, V_{N} is the chamber volume, a_{N} is the radius of the conduit, and γ is the effective volumetric modulus of the chamber (Table 2). Considering that the loss of pressure is mainly caused by the high-viscosity part, \( {Q}_{\mathrm{in}}^{\prime } \) and \( {Q}_{\mathrm{inN}}^{\prime } \) are multiplied by the corresponding viscosity ratio in comparison. Using the values listed in Table 2, we estimate the range in values for \( {\mu}_{\mathrm{N}}^{\prime }{Q}_{\mathrm{inN}}^{\prime } \) as 10^{−8} – 10^{4} which is consistent with the range in our experiment (\( {\mu}^{\prime }{Q}_{\mathrm{in}}^{\prime}\sim {10}^{-1}\hbox{--} {10}^0 \)).
The NK model assumes a contrast in viscosity before and after crystallization of \( {\mu}_{\mathrm{N}}^{\prime}\sim {10}^1\hbox{--} {10}^3 \). In our experiment, μ^{′} was much larger than \( {\mu}_{\mathrm{N}}^{\prime } \) and the contribution of the lower viscosity was negligible. This explains why the STW features in our system are sharper than those in the NK model.
In our model, we determined the relative importance of the effect of gravity and viscous flow resistance (using the higher viscosity) using \( G/\left({\mu}^{\prime }\ {Q}_{\mathrm{in}}^{\prime}\right)\sim {10}^{-1}-{10}^0 \). This value lies within the range of \( {G}_{\mathrm{N}}/\left({\mu}_{\mathrm{N}}^{\prime }\ {Q}_{\mathrm{inN}}^{\prime}\right)={G}_{\mathrm{N}}/\left({\mu}_{\mathrm{N}}^{\prime }\ {Q}_{\mathrm{inN}}^{\prime}\right)\sim {10}^{-1}-{10}^3 \) in the NK model, where ρ_{N} is magma density (Table 2). The values of these parameters suggest that gravity will affect the pressure change within the chamber if a change in fluid density occurs. When the viscosity is low, the gravity effect becomes dominant, namely, density change may play a role in controlling the oscillations (Kozono and Koyaguchi 2012).
In the NK model, the parameter \( {t}_{\ast}^{\prime }{Q}_{\mathrm{inN}}^{\prime }={t}_{\ast }{Q}_{\mathrm{inN}}/\left(\pi {a}_{\mathrm{N}}^2{L}_{\mathrm{N}}\right) \) represents the ratio of the timescale of viscosity change to the time during which fluid flows through the conduit, and determines whether pressure oscillations occur. We assume that t_{c} corresponds to the timescale (t_{∗}) because both represent the timescale of the change from low-viscosity flow (i.e., annular in our experiment) to high-viscosity flow (i.e., slug flow). In the NK model, \( {t}_{\ast}^{\prime }{Q}_{\mathrm{inN}}^{\prime } \) has an upper and lower limit for oscillations to occur. For the upper limit condition, \( {t}_{\ast}^{\prime }{Q}_{\mathrm{inN}}^{\prime }<1 \) must be satisfied, as an increase in viscosity occurs within the conduit. Correspondingly, our model yields \( {t}_{\mathrm{c}}^{\prime }{Q}_{\mathrm{in}}^{\prime}\sim {10}^{-2}\hbox{--} {10}^{-1} \). In our model, we disregard the shear stress on the inner surface during downward film flow to allow slug reconstruction. If we increased \( {t}_{\mathrm{c}}^{\prime }{Q}_{\mathrm{in}}^{\prime } \) with increasing Q_{in}, flow within the pipe would occur as a churn or co-current annular flow. However, as we focus on the range of Q_{in} where STW are observed, we did not study the physical processes occurring at such large Q_{in}. In the present study, we observe nonperiodic behavior at the upper range of the chosen Q_{in} values. Below the lower limit of \( {t}_{\ast}^{\prime }{Q}_{\mathrm{inN}}^{\prime } \) in the NK model, flow in the conduit achieves steady state. In contrast, our system does not achieve steady state, irrespective of the value of \( {t}_{\mathrm{c}}^{\prime }{Q}_{in}^{\prime } \), when Q_{in} > 0 and L_{∗} > 0 (Table 3(c)).
In summary, the lumped parameter model enables us to compare our experimental system with models of flow-induced pressure oscillations in volcanoes. Even if the flow structures appear different, the mechanisms responsible for oscillation may be described by the same mathematical model, using dimensionless parameters in ranges that span within the possible variations in natural volcanic systems.
Cyclic behavior due to flow pattern transitions
In terms of mathematical structures, our experimental system is similar to the models of flow-induced pressure oscillations during dome-building eruptions. These models however do not explicitly include flow pattern transitions, though some of them (Melnik and Sparks 2005; Kozono and Koyaguchi 2012) interpret rapid deflation phases with high-discharge rates as explosive phases. On the other hand, flow pattern transitions (i.e., slug-to-annular and annular-to-slug transitions) are essential for the cyclic behavior in our system. Therefore, the behavior of our experimental system is more similar to repetitive explosive eruptions than to the systems which do not include flow pattern transitions (Ida 1996; Wylie et al. 1999; Barmin et al. 2002; Nakanishi and Koyaguchi 2008; Chen et al. 2018).
In our experiment, disturbances in downward flow grow to halfway liquid slug reconstructions (Fig. 6c), which are shown to affect the periodicity by numerical calculation (Fig. 9b–d). Therefore, our experimental results indicate that “downward flow” within the conduit could interact with ascending flow and modulate the explosivity and periodicity of an eruption. When the termination states the same as the initial state of the cycle (Fig. 10b), the period remains constant; however, a strong explosion could disturb the termination state (Fig. 10b which would result in nonperiodic behavior.
Conclusions
Controlled experiments were conducted on gas and liquid flow in a pipe connected to a gas chamber. Under relatively large Q_{in} and V_{c}, periodic STW is observed, which consists of gradual pressure increase during slug flow and abrupt pressure drop at the slug-to-annular flow transition. Under small Q_{in} and V_{c}, only slug flow with non-STW features is observed. With a far larger Q_{in}, the periodicity of STW is lost and we observe large surface disturbances on downward film flow causing slug reconstruction at varying locations along the pipe. Experimental movies for these conditions are provided.
A mathematical model of the experimental system and its numerical solutions reveal the mechanisms of the experimental observations. The coupling between elastic capacitance and variable flow resistance is the main factor that generates the STW feature. Fluctuations in the location of the liquid slug reconstruction at each annular-to-slug flow transition have the marked effect in breaking periodicity.
Models of flow-induced volcanic oscillations are mathematically similar to our model, and our gas chamber is equivalent to a magma chamber in terms of an elastic capacitance. Regarding the flow viscosity, the length change of the liquid slug corresponds to the length change of the high-viscosity crystallized region in the dome-building eruption model of Nakanishi and Koyaguchi (2008). The ranges of the dimensionless model parameters for the experiments fall within those for the dome-building eruptions. Considering the essential roles of flow pattern transitions, the behavior of our experiment is more similar to repetitive explosive eruptions than to the nonexplosive dome-building models which do not include flow pattern transitions. Our study implies that “downward flow” in the conduit may interact with the ascending flow and modulate the explosivity and periodicity of the eruption. We can thus make use of this experimental system for understanding periodic and nonperiodic behavior during flow-induced oscillations in volcanic systems.
This study was inspired by the syrup eruption experiment conducted during an outreach activity. The results of this study also strengthen the educational outreach value of the simple experiment in Fig. 1. The experiment allows a clear understanding and active illustration of dynamical systems highlighting the importance of conduit–chamber geometry and flow pattern transitions for oscillatory behaviors (e.g., geodetic signals and eruption cycles) and possible effects of “downward flow” for disturbing eruption periodicity.
Notes
Acknowledgments
We would like to thank M. Ripepe, V. Vidal, K. Kurita, and O. Kuwano for useful comments that improved the experimental system and model. The original syrup eruption experiment was designed by S. Takeuchi and introduced to the authors by A. Namiki. The manuscript has been significantly improved by constructive comments by the two anonymous reviewers and the associate editor (L. Pioli) and thorough check by the executive editor (A. Harris).
Funding information
This work was supported by the Japan Society for the Promotion of Science KAKENHI Grant No. 15K13561 and JSPS Grant-in-Aid for JSPS Research Fellow No. 16J06324.
Supplementary material
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