Banded tremor at Miyakejima volcano, Japan: Implication for two-phase flow instability



[1] Volcanic tremor was recorded during a period of volcanic gas emission from Miyakejima volcano activity from 2001 to 2006. Similar to volcanic earthquakes and long-period events, the source of banded tremor lies at a shallow depth beneath the summit caldera. Tremor packets occurred every 20 to 45 min for 1 to 2 days. We modeled the source of this phenomenon with a two-phase hydrothermal instability flow model, consisting of subcool and superheated water regions, and found two modes, an excursion mode and a series of damping oscillations mode. The interval between banded tremor packets can be interpreted in terms of this two-phase flow oscillation with a heat supply in the order of 10–100 MW from the magma beneath the hydrothermal system.

1. Introduction

[2] The eruption of Miyakejima volcano, Japan, began on 26 June, 2000, with a subsurface dike intrusion [Ueda et al., 2005]. One of the most characteristic activities of this eruption was the formation of a caldera from July to August 2000 [e.g., Fujita et al., 2004; Geshi et al., 2002]. After these first specific and dynamic activities, tremendous volcanic gas emissions from the summit caldera have occurred since September 2000, and are still going on in 2006, although emissions are decreasing. Seismic activities from September 2000 to 2006 were less than those in the first dike intrusion stage, but we observed swarms of volcano-tectonic earthquakes and LP events. Among these events, we have intermittently recorded significant banded tremor since 2001. We believe that the banded tremor can reveal the state of the volcanic system. Probably, the shallow hydrothermal system beneath the summit plays an important role in controlling the emission of volcanic gases.

[3] Banded tremor has been observed at several volcanoes around the world at certain times [e.g., Mori et al., 1996]. At Pu'u O'o, Hawaii [Barker et al., 2003], banded tremor occurred in relation to the visible drain-back of the lava lake in the crater, as supported by shallow borehole tiltmeter records. The amplitudes of banded tremor, tilt change, and magma volume in the lava pond have a close positive correlation with each other. In Montserrat [Baptie and Thomson, 2003], banded tremor was closely related to the instability of the lava dome at the summit of Soufrière, and their amplitudes and periods were independent. In addition, Edmonds et al. [2000] reported banded tremor associated with changes of HCl and SO2 concentration. During the eruption of Mt. Spurr, Alaska, banded tremor was also observed and the amplitude of which was proportional to the intensity of the eruptive activity [Smithsonian National Museum of Natural History, 1992]. At Anathahan volcano, Mariana Islands, banded tremor had a strong correlation with phreatic eruptions and the ash-cloud elevation [Trusdell et al., 2001].

[4] Four types of models to explain the characteristics of the source mechanism of volcanic tremor have been proposed: (a) fluid-flow-induced oscillation [e.g., Julian, 1994; Balmforth et al., 2005]; (b) excitation and resonance of fluid-filled cracks [e.g., Chouet, 2003]; (c) bubble growth or collapse due to hydrothermal boiling of groundwater; (d) and oscillation of the volcanic plumbing system [Konstantinou and Schlindwein, 2003]. The source mechanism of banded tremor has been discussed in view of their characteristics, but no comprehensive models have been offered so far, especially to explain their periodical characteristics. Therefore we propose a model of a hydrothermal system to explain how its instability can trigger an oscillation.

2. Characteristics of Banded Tremor

[5] Miyakejima volcano is located about 120 km to the south of Tokyo (Figure 1). The NIED (National Research Institute for Earth Science and Disaster Prevention) Miyakejima volcanic observation network has five stations mainly equipped with 1-Hz velocity seismometers and tiltmeters installed at about 100 m beneath the ground surface, and broadband seismometers (Streckeisen STS-2) at 2 m depth. Figure 2 shows examples of banded tremor recorded at the station MKT, less than 2 km from the summit. Other stations except MYKA (Japan Meteorological Agency) cannot detect banded tremor on account of the low S/N ratio. Each plate shows a 1-day continuous seismograph of the 1-Hz velocity seismometer at MKT. Each trace corresponds to a 1-h record of the vertical component. Spindle-shaped packets occur intermittently, with almost similar onset, coda, and durations in each period of activity. This banded tremor is different from tectonic earthquake swarms, since most seismic events have clear P and S phases and do not last as long as banded tremor. Furthermore, the occurrence of seismic swarms are random in many cases, but the banded tremor occurs successively at almost regular intervals. This interval depends on their active period, as seen in Figure 2. Their amplitudes are about 10−4–10−5 m/s, almost constant throughout the observation period.

Figure 1.

Location of Miyakejima volcano, Japan. The observation network of the National Research Institute for Earth science and Disaster prevention consists of five 1-Hz velocity seismometers and tiltmeters installed at about 100 m beneath the surface. During the 2000 eruption, an large caldera with 1.6 km diameter and 400 m depth was formed at the summit. Seismicity around Miyakejima volcano is also shown. Volcano-tectonic earthquakes and long-period earthquakes with magnitude less than 3 occurred beneath the shallow part of the summit.

Figure 2.

Examples of waveform of banded tremor waveforms on (a) 15 August 2002, (b) 11 August 2003, and (c) 29 March 2004. Each plate includes a 1-day seismic record consisting of 24 traces of 1-h records of vertical components of a velocity seismometer installed at about 100 m depth at MKT, about 1.8 km from the summit. Spindle-shape packets, which we call banded tremor, are recorded intermittently at similar intervals. The duration and interval depends on the active periods.

[6] Banded tremor has been recorded since 2001 at Miyakejima volcano, as shown in Figure 3. Even though there are some periods with no records due to technical problems, the active periods of banded tremor are occasional and each period lasts about 1 to 2 days on average, ranging from a few hours to 10 days. Remarkable banded tremor was recorded in April and August 2002; March, April, August, and September 2003; March–April, May–June, July–August, and November–December 2004; and April and August 2005. It seems that the banded tremor has seasonal activity. There were some small summit eruptions during these years, as shown in the figure, but there is no clear relation between them. The characteristic frequencies of each packet of tremor are about 4 to 8 Hz (Figure 4), and their amplitudes have been almost constant throughout the period.

Figure 3.

Diagram of banded tremor active periods from 2002 to 2005. Each row corresponds to one day and has 24 cells horizontally. Black cells shows the hour in which we recorded banded tremor, and gray cells show a lack of records due to technical problems. Banded tremor showed active seasons in March and April and in August.

Figure 4.

Example of spectrograms of banded tremor. Each plate shows 1-h records of a velocity seismometer at MKT and its spectrogram. Each record includes two or three packets of banded tremor at intervals of about 20 min with characteristic frequencies of about 4 to 8 Hz, with some multiple peaks. The 0.5-Hz high energy is background ocean noise.

[7] We introduce some parameters to characterize banded tremor more quantitatively, especially the periodicity (Figure 5). The time marks T1 and T2 correspond to the start and end time of each packet, and the differences T2T1 and T1T1 are the duration of a packet and the interval between, respectively. Figure 6 shows the durations of tremor packets, and three points should be noted: (1) the durations of packets in each active period are similar: (2) two typical durations are about 7 min and 11 min on average; (3) the durations in spring season (March and April) are shorter than those in summer (May–August). Their average intervals (Figure 7) were about 40 and 32 min in 2002, 25 min in 2003 and 31 min in 2004. Figure 8 show the histograms of durations and intervals and this clearly shows that there are typical values for both characteristic times about 400 s and 1500 s, respectively. These distributions suggest the existence of some threshold to trigger the banded tremor.

Figure 5.

Definition of parameters defining banded tremor. Each packet of banded tremor has an onset time T1 and a stopping phase T2. The duration of each packet is T2T1 and the interval is calculated as T1T1.

Figure 6.

Duration of banded tremor packets observed in 2002, 2003, and 2004. There are some active periods, and the duration is similar in each period. The solid and dashed lines show the average and standard deviation values, respectively.

Figure 7.

Intervals of banded tremor packets observed in 2002, 2003, and 2004. The solid and dashed lines show the average and standard deviation values, respectively.

Figure 8.

The histograms of (a) duration and (b) intervals of banded tremor. It is noted that both have specific peaks about 400 s and 1500 s, respectively.

3. Source Mechanism of Banded Tremor

[8] The location of the source of the banded tremor is not clear, but it seems to lie at a shallow depth beneath the summit, since only the stations close to the summit detected the signal. In addition, the banded tremor was not observed throughout the entire period of volcanic activity but only intermittently between 2001 and 2006. During the most active period, i.e., in 2000, no banded tremor was observed. This fact may suggest that the source system may not be a simple resonator, because tremor should be triggered by nearby earthquakes. Therefore we infer that a hydrothermal system at less than 1 km depth is most plausible source mechanism of the observed banded tremor.

[9] A hydrothermal system with a steam-water mixture behaves like an oscillating source due to its instability, which is controlled by many factors, including system geometry, quantity of water, and heating rate. A shallow water reservoir is heated by the magma chamber beneath the conduit, appears to be capable of maintaining continuous gas emission for more than 5 years. This hydrothermal system may become unstable and trigger oscillations under some conditions. The characteristics of the relation between total flux and pressure loss of this two-phase system show N-shape characteristics, which cause instabilities due to negative resistance. This condition leads to many kinds of instabilities, such as runaway flow, pressure-drop oscillation [Fujita et al., 2004], density wave oscillation [Iwamura and Kaneshima, 2005], geysering, chugging, and slug oscillation [Fujii et al., 1999], depending on the gas volume fraction. Among these instabilities, we focus on the density wave oscillation and limit cycle of the system, since the observed banded tremor has two typical periods, i.e., the duration and the interval.

3.1. Formulation of Two-Phase Flow Instability

[10] Figure 9 shows a schematic view of a one-dimensional hydrothermal system. Water enters from the bottom (point 0) through the inlet to the inside (point 1). The heat is provided from the wall by thermal conduction at a total energy conduction rate of equation image. Just inside the inlet (region 1), the fluid velocity and the temperature are defined as u1 and T1, respectively. As water flows toward the outlet, its enthalpy increases, and boiling starts at a critical point (point 2). The volume of the mixture v begins to increase gradually from this point. This volume can be expressed as functions of enthalpy: i, time: t and position: z.

Figure 9.

(a) Schematic view of one-dimensional hydrothermal system beneath the volcano. The hydrothermal system is heated by thermal conduction from the magma chamber and/or hot rock. (b) Assuming a one-dimensional pipe flow, we labeled points as 0, the position of the inlet outside the valve, (1), the inlet inside, (2), the point where the vesiculation starts, and (3), the outlet. l, pipe length, equation image, total heat supply rate. The subcool region (from points 1 to 2) is the “heavy region”, and the supercritical region (from points 2 to 3) is the “light region”. (c) The relations between volume: v, enthalpy: i, time: t, and position: z are schematically depicted as on the right. The region between 2 and 3 is the transient area and the volume of the mixture gradually increases with respect to enthalpy, time and position.

[11] Here we define the region between the points 2 and 1 as the heavy region, and that between points 3 (outlet) and 2 as the light region. The heavy region is in the subcool state, that is, the temperature of water is under its saturation temperature, and the light region is in the supercritical state, that is, water is in the supersaturation. For these two regions, we will follow the formulation by Zuber [1966], developed for the evaluation of nuclear systems. For the heavy region, the equation of continuity is

equation image

the equation of energy is

equation image

the equation of motion is

equation image

and the equation of state is

equation image

[12] Here u, P, ρf, and i are velocity, pressure, density, and enthalpy, respectively, as functions of position z and time t. The parameters q, ξ, Ac, f, g, and D are heat flux density, heated perimeter, cross-sectional area of pipe, friction factor, gravitational constant, and diameter of the pipe, respectively. The notations of variables and parameters are summarized in Appendix A.

[13] For the light region, the equation of continuity is

equation image

the equation of energy is

equation image

the equation of motion is

equation image

and the equation of state is

equation image

[14] We assume that water is incompressible in the heavy region, i.e., ∂u/∂z = 0, so the velocity is a function of time only: u = u(t). Here we consider the perturbation from the steady state as

equation image

where equation image is the steady state velocity and s is the complex frequency of the oscillation.

[15] The initial and boundary conditions are

equation image

at z = 0, t = τ1,

equation image

at z = 0 and

equation image

at z = λ(t), where λ(t) is a “space-lag”, which is the length of the transition area from the subcool to the superheated region. Substituting the input velocity perturbation (9), integrating the equations using the boundary conditions and combining the heavy and light regions' equations, we get the characteristic equations of motion for a perturbation δu1.

equation image

where Ω is the reaction frequency (see in Appendix B). The detailed derivation procedure is shown in Appendix B and the definition of the factors of F1 to F7 are in Appendix C.

[16] The perturbation of gas region δug can be expressed as

equation image

then this equation can be rewritten by the perturbations in the heavy region δu1 as,

equation image

[17] And the space-lag perturbation is expressed by the inlet perturbation δu1 = equation imageest as

equation image

so equation (15) is rewritten only by the inlet perturbation δu1 by the time derivative of equation (15) as

equation image

[18] This second-order differential equation characterizes the oscillating and non-oscillating phenomena, depending on the geometry, heating and kinematic parameters of the system.

3.2. Characteristic Frequency

[19] From equation (17), we derive an eigen equation to obtain eigen frequencies of the hydrothermal system. In equation (9), we assume that the perturbation can be expressed as εest where s can be treated as a complex number whose real and imaginary parts show the oscillating frequency and attenuation factor, respectively [e.g., Fujita and Ida, 2003]. We therefore derive the eigen equation as

equation image

[20] Figure 10 shows an example of the absolute value of the left-hand side of the eigen equation (18) with real and imaginary frequency s along the horizontal and vertical axes, respectively. The zero-points, marked with the circle on the contour map, give the solutions of the eigen equation. It is noted that two kinds of solution exist, an excursion mode and a series of damping oscillations.

Figure 10.

Example of eigenvalues of two-phase flow instability obtained from the eigen equation (18). Horizontal and vertical axes show real and imaginary parts, respectively, of perturbation frequency s. It is noted that there are two kinds of mode, an excursion mode with positive Re[s] and a series of damping oscillations.

3.2.1. Low Frequency Phenomena: Excursion Mode

[21] When s is real and a positive small value, the characteristic equation is approximated as

equation image

[22] In this case, s is obtained by a simple analytic expression as

equation image

[23] We have assumed a positive s to derive equation (19), so the numerator in (20) must be negative. The positive value of s therefore expresses the excursion of the instability. This instability is called a Ledinegg instability [Fujii et al., 1999]. The criterion for this instability can be shown as

equation image

[24] The steady state pressure drop at each location is defined in Appendix D. This is also expressed by the total mass flow rate W as

equation image

where the total flux W is approximated as ρfAcequation image. The total pressure is

equation image


equation image
equation image
equation image

[25] Equation (23) gives two real solutions of W: that is, it has maximum and minimum values when

equation image

[26] In other words, only under this condition is there a region with dΣP/dW < 0, so the system can realize a limit cycle, that is, a self-exciting cyclic phenomena triggered by a non-linear system. The local maximum and minimum of the equation (23) are obtained therefore from

equation image

[27] The system has “N-type” characteristics in the relation between total mass flux W and pressure drop ΔP, suggesting that the system has both stable and unstable conditions depending on the flux W. In this case, the differential equation (15) can be rewritten as follows for the small perturbation of real positive s, using the roots derived in (19) [Fujita et al., 2004]:

equation image


equation image
equation image
equation image
equation image
equation image
equation image
equation image

[28] Solving this equation, we obtain the recursive feature of the instability, and see that the cycle of the instability is controlled by the model parameters. Figure 11 shows an example of the temporal change of the velocity perturbation δu1 for some sets of the parameters. It can be noted that both the duration and intervals of periodic disturbance are determined mainly by two parameters, ε and β. These parameters have close relations with Δi12, equation image and other parameters. The heat flux equation image can be estimated as,

equation image
Figure 11.

Example of an excursion mode oscillation obtained from the equation (29). The parameters in this equation are assumed as, R = 8.3 J/mol K, Cp = 2051 J/K kg, P = 1.013 MPa, q = 2.51 × 108 J/s, ξ = 6.28 m, Ac = 3.14 m2, F1 = 100, F2 = 0.01, F3 = 10, F4 = 10, F5 = 10, F6 = 10, F7 = 10, τ1 = 1, τ2 = 20, τ3 = 22. This is an non-linear oscillation determined by the N-type characteristics of the relation between flux and pressure drop relation. This mode enables a limit cycle with a long period.

[29] The following section discusses the dependency of physical parameters on the characteristics of the perturbation.

3.2.2. High-Frequency Phenomena: Damping Oscillations Mode

[30] When the perturbation frequency s is an imaginary number, the hydrothermal system instability becomes an oscillating phenomenon. As shown in Figure 10, we have a series of eigen frequencies whose real part is negative, suggesting damping oscillations. We can obtain the eigen values by the numerical calculation of the eigen equation (18), which introduces kinematic parameters of the hydrothermal system. Figure 12 is an example of these damping oscillation mode. The oscillating feature is closely related to the reaction frequency Ω, pipe length l, and the lengths of both heavy and light regions.

Figure 12.

Example of a damping oscillation obtained from the equation (17). The parameters in this equation are assumed as, R = 8.3 J/mol K, Cp = 2051 J/K kg, P = 1.013 MPa, q = 2.51e4 J/s, ξ = 10 m, f = 0.1, l = 10 m, D = 10 m, ΔV = 1/0.804–1/1000 m−3, Δi = 1506 KJ/kg, ki = 2, ke = 1, F1 = 10000, F2 = 0.01, F3 = 1, F4 = 1, F5 = 1, F6 = 1, F7 = 1, τ1 = 1, τ2 = 20, τ3 = 22 and equation image1) = 1.0 m/s. With the combination of eigen frequency and the reaction frequency Ω defined in (B26), the oscillation becomes a damping oscillation.

4. Discussion

[31] The banded tremor at Miyakejima has characteristic intervals, durations, and characteristic frequencies. The proposed two-phase flow instability model also has characteristic periods. The system has an excursive instability with a specific period when the perturbation in the inlet velocity δu1 is small and s is positive. When s is negative, the system has a series of damping oscillations.

[32] As demonstrated in the previous section, the characteristics of the instability depend on many parameters, i.e., the geometry of the system (Ac, cross-section of pipe, l, length of pipe, D, diameter of pipe, f, friction factor, ki, ke, inlet and outlet valve factors). Furthermore, we must pay attention to the coefficient (dv/di)P, which is equal to Δvfgifg for the sub-critical region and to R/PCp (Figures 11 and 12) for the supercritical region. In addition, the important factor Δi12 is the subcool enthalpy. The total heat flux equation image represents the rate of energy supply to the system.

[33] Our geophysical observations cannot constrain all of these parameters, unfortunately. Here we assume some parameters as listed in Table 1, based on observations of shallow parts of the volcano. The pressure differences in steady state (e.g., ΔP01, ΔP12, ΔP23) are assumed to approximate the ambient hydrostatic pressure, and the velocity in steady state is reasonably assumed to be about 1 m/s. Then we can focus on the effects of input heat rate equation image and inlet subcooling Δi12 on the characteristics of the unstable oscillation. To produce the episodic excursions of the instability, the system must satisfy the inequality (27). Figure 13 shows the conditions leading to an excursion mode as a function of subcool enthalpy Δi12 and total heat rate equation image. The excursion mode occurs only when equation image is larger than the critical value.

Figure 13.

Condition for the existence of an excursion mode defined by the relation between subcool enthalpy Δi12 and total heat flux equation image. This is the case with the parameters of f = 0.1, l = 10., D = 10, ki = 2, ke = 1. Other parameters are summarized in Table 1.

Table 1. Summary of Parameters of the Hydrothermal System to Estimate Heat Flux for Miyakejima Banded Tremor
ParameterAssumed Values
Gas constant: R8.31 J/mol · K
Specific heat for gas: Cp2051.0 J/K · kg
Pressure: P1.01325 × 105 Pa
Heated perimeter: ξ6.28 m
Liquid density: ρf1.0 × 103 kg/m3
Diameter: D1 m
Length: l10 m
Cross section: Ac3.14 m2
Coefficient of the inlet flow restriction: ki2
Coefficient of the outlet flow restriction: ke1
Specific heat capacity at constant Pressure: (dv/di)p4.1816 × 103 J/K · kg
Friction factor: f0.1
Mass of heavy region: F1100 kg/m2
Mass of light region: F20.01 kg/m2
Effect of inlet flow perturbation: F310 Pa · s/m
Effect of velocity perturbation: F410 Pa · s/m
Effect of density perturbation: F510 Pa · s/m
Effect of pressure drops at different positions: F610 Pa · s/m
Effect of the space-lag perturbation: F710 Pa · s/m
Change of specific volume in vaporization: Δv1.24278 m3/kg
Latent heat of vaporization: Δi1506 kJ/kg
Time at the inlet: τ11 s
Critical transit time: τ220 s
Time at the outlet: τ322 s
Steady state velocity in the heavy region: equation image11 m/s

[34] Since the differential equation (15) is non-linear, there is no unique solution to fit the observed banded tremor interval. However, we can give an example of a scheme to estimate the parameters of a hydrothermal system, especially the total heat flux rate equation image, if other parameters are assumed. The interval of the banded tremor can be interpreted as the recursive time of the excursion mode, which is primarily controlled by the parameters ε and β. Values for ε and β can be obtained by least squares comparison of the observed and theoretical intervals, and characteristic frequencies of the tremor packet. The parameters F′, F1, F2, ζ, α, α+ and equation image1 also depend on the hydrothermal system and they are unknown factors. We choose plausible values for a shallow hydrothermal system.

[35] In case of the banded tremor at Miyakejima volcano, we assume the parameters given in Table 1. From the observed interval between banded tremor packets of 20 to 45 min, equation image is estimated to be on the order of 10–100 MW. This estimate depends on the parameters in Table 1, and their ambiguity may distort the estimation. The heat flux equation image can be calculated by equation (37), in which the square of subcool enthalpy Δi12 is proportional to equation image. If the subcool enthalpy of the inlet water is half of the value given in Table 1, the heat flux estimation will be reduced to one fourth of our previous estimate, but will remain on the same order. Our heat flux estimate is less than 10% of the 1000 MW flux derived by thermal cameras observations [Matsushima and Nishi, 2001; Meteorological Research Institute, 2005]. This is because our analysis accounts for heat supply only to the source region producing banded tremor, rather than the entire hydrothermally active area.

[36] The short-period signal, i.e., the characteristic frequency of 4–8 Hz in each tremor packet, is interpreted to be due to the damping oscillation mode. This mode is physically equivalent to the density wave oscillation proposed by Iwamura and Kaneshima [2005].

[37] Our estimation of heat flux equation image is only from the interval and characteristic frequency of the banded tremor, so there are some uncertainties in our analysis. Incorporating information on tremor amplitude would provide more constraints on the evaluation of hydrothermal systems. Furthermore, it would be beneficial to include the effects of seismic wave propagation and attenuation from the source to the observatory. Deployment of a dense network of detectors around the source region would allow a more robust determination of many of the parameters of the hydrothermal system.

[38] Periodic phenomena generally suggest that the system is more or less stable; in other words, there should not be an abrupt change of pressurization, heat supply, or other kinematic conditions. Therefore banded tremor may indicate that there will not be a sudden large eruption, which would require significant pre-eruptive pressurization. On the other hand, continuous banded tremor with increasing amplitude seems to be closely related to the buildup to an eruption.

5. Conclusions

[39] We observed banded tremor at Miyakejima volcano and developed a hydrothermal instability model to infer some kinematic parameters for the source of this tremor. We have observed some important characteristics of the banded tremor: (1) Banded tremor occurs at intervals of about 20 to 45 min over 1 to 2 days; (2) The characteristic frequency of each tremor packet is about 4 to 8 Hz; (3) The source location is inferred to lie at a shallow depth beneath the summit; (4) Our formulation of a hydrothermal instability model suggests that there are two types of modes, an excursion mode and a damping oscillations mode; (5) From the model, we estimate the total heat flux rate which triggers banded tremor is on the order of 10–100 MW.

Appendix A:: Notations of Variables and Parameters


cross section of heated pipe, m2


friction factor


mass of heavy region (equal to Mf), kg/m2


mass of light region (equal to Mg), kg/m2


effect of inlet flow perturbation, Pa · s/m


effect of velocity perturbation, Pa · s/m


effect of density perturbation, Pa · s/m


effect of pressure drops at different positions, Pa · s/m


effect of the space-lag perturbation, Pa · s/m


mass flux density, kg/m2 · s


enthalpy, J/kg


latent heat of vaporization, J/kg


inlet subcooling, J/kg


coefficient of the inlet flow restriction


coefficient of the exit flow restriction


total length of the heated pipe, m


mass in the heavy region per unit area, kg/m2


mass in the light region per unit area, kg/m2


pressure, Pa


pressure rise of the external system, Pa


steady state pressure drop at the inlet, Pa

Δequation image12

steady state pressure drop due to friction in heavy region, Pa

Δequation imagebf

steady state pressure drop due to gravity in heavy region, Pa

Δequation imagea

steady state pressure drop due to acceleration in light region, Pa

Δequation imagebg

steady state pressure drop due to gravity in light region, Pa

Δequation image23

steady state pressure drop due to friction in light region, Pa

Δequation image34

steady state pressure drop at the exit, Pa


heat flux density, W/m2

equation image

total heat input rate, W


gas constant, J/mol · K


complex exponent of the inlet velocity perturbation, Hz


Period of the inlet velocity perturbation, s


time, s


velocity, m/s

equation image1

steady state velocity in the heavy region, m/s

equation imageg

steady state velocity in the light region, m/s

equation image3

steady state velocity at the exit, m/s


average velocity in the light region, m/s


log mean velocity in the light region, m/s


mean velocity in the light region, m/s


inlet velocity perturbation, m/s


velocity perturbation in the light region, m/s


specific volume of the heavy region, m3/kg


specific volume of the light region, m3/kg


change of specific volume in vaporization, m3/kg


total steady state mass flow rate, kg/s


length, m


heated perimeter, m


space lag, m


space lag perturbation, m


amplitude of inlet velocity perturbation, m/s


time lag from region 1 to 2, s

Δτ = τ3τ1

total transit time, s


critical transit time, s


characteristic reaction frequency, Hz


density in the heavy region, kg/m3


density in the light region, kg/m3


density at the exit, kg/m3


log mean density, kg/m3

Appendix B:: Derivation of the Characteristic Equation

[40] We combine equations (1) to (9) and boundary conditions (10)(12) for both the heavy and light regions to produce the characteristic equation (13). The detailed procedure is complicated but we summarize it as follows:

[101] For the heavy region, substituting equation (9) into (2), we have

equation image

[41] This equation has the general solution

equation image


equation image
equation image

[42] C1 and C2 are constants. Then equation (B1) is written as an ordinary differential equation, given by

equation image

[43] From the first and the second terms,

equation image

and from the first and the third terms,

equation image

[44] Substituting the boundary condition (10) into (B6), we get

equation image

[45] From the enthalpy, we obtain

equation image

and we substitute this equation to (B8) to give

equation image

[46] Here we define some parameters. The time taken by the flow to move from the inlet (region 1) to the boiling point (region 2) is defined as

equation image

[47] From equation (B10), the space-lag is defined as,

equation image

then, the steady state of the space lag is written as,

equation image

[48] Using the expression for τb in (B11), the space-lag is rewritten as,

equation image

[49] Substituting the boundary conditions (11) and (12) into the equation of motion (3), we derive the integral equation,

equation image

then we obtain the linearized equation adopting the terms of O(ε) as,

equation image

[50] For the pressure drop at the inlet, we introduce the valve factor at the inlet, ki, giving

equation image

then P1 is given as a function of δu1 by

equation image

[51] Here we define the steady state pressure drop for the gravity effect as,

equation image

for the friction effect as,

equation image

and the inlet orifice effect as

equation image

[52] Finally, we obtain the pressure drop for the heavy region i.e., between the region 2 and 0, by adding the equation (B16) and (B18) as,

equation image

[53] For the light region, the equation of state is given by (8), which is equivalent to v = v(i), such that,

equation image
equation image


equation image

[54] Introducing the boundary condition v = vf at i = i2, we find

equation image

[55] This equation can be written as

equation image

where Δvfg and Δifg are the volume change from fluid to gas and the latent heat, respectively. From the equation of continuity (5) and the equation of energy (6),

equation image

[56] We now define the reaction frequency as

equation image

[57] Then the velocity is written as

equation image

[58] The steady state velocity in the light region is

equation image


equation image

[59] The steady state velocity at the exit is

equation image

[60] To simplify notification, we define some velocities as,

equation image

for the average velocity in the light region,

equation image

for the log-mean velocity,

equation image

for the log-mean density and,

equation image

[61] From the equation of energy, we include the boundary conditions and integrate the differential equation to derive the following equations.

equation image
equation image
equation image

[62] Using these equations, we obtain the individual equations of motion, and substituting boundary conditions,

equation image

for the inertia term;

equation image

where Δequation imagea = ρlmulm(equation image3equation image1) for the convective term;

equation image

where Δequation imagebg = g(lequation image)〈ρg〉 for the gravitational term;

equation image

where Δequation image23 = f(lequation image)/2D · Gug〉 for the frictional term;

equation image

where Δequation image34 = keρ3equation image32 for the exit pressure drop. Therefore adding equations (B38), (B39), (B40), (B41) and (B42), we obtain the integrated equation of motion for light region as

equation image

[63] Finally, we add the equations of the heavy region (B19) and of the light region (B43) then we get

equation image

where ΔPex is the total sum of all pressure drops. This equation can be written as the characteristic equation (13) when we use the expressions defined in Appendix C and D.

Appendix C:: Definition of the Factors in the Characteristic Equation

[64] The factors F1F7 are defined as the steady state values including boundary and other conditions as follows,

equation image
equation image

[65] The factors F1 and F2 are the masses of the heavy and light region, respectively.

equation image

[66] Factor F3 shows the effect of inlet flow perturbation.

equation image

[67] Factor F4 is the velocity perturbation.

equation image
equation image

[68] These two factors F5 and F6 are the effects of density perturbation and the various pressure drops at different positions, respectively.

equation image

[69] Finally, factor F7 is the effect of the space-lag perturbation on the acceleration pressure drop in the light fluid region.

Appendix D:: The Steady State Pressure Drops

[70] The steady state pressure drops across each region are approximated by

equation image
equation image
equation image
equation image
equation image

where ki and kex are the valve factor at inlet and outlet, respectively.


[71] We are very grateful to Motoo Ukawa, Eiji Yamamoto, Hideki Ueda, Masae Kikuchi, Momoe Nakamura, and the Japan Meteorological Agency for observations at Miyakejima volcano. Stimulating discussions with Jonathan M. Lees, David Bridges, and Mario Ruiz helped to improve our manuscript. Finally, we thank Richard Arculus, Sarah Fagants, Takeshi Nishimura and one anonymous reviewer for providing us numerous stimulating comments.