Linking the flow-induced tremor model to the seismological observation: application to the deep harmonic tremor at Hakone volcano, Japan

Abstract

Decades ago, Julian (J Volcanol Geotherm Res 101:19–26, 1994. https://doi.org/10.1029/93JB03129) proposed the lumped parameter model of non-linear excitation of an elastic channel vibration by fluid flow as a mechanism of volcanic harmonic tremor. Since then, his model and similar flow-induced oscillation models have been applied or considered to explain volcanic tremors and low-frequency earthquakes. Here we extended Julian’s model to allow quantitative comparison with observation data and applied it to deep harmonic tremor observed at Hakone volcano, Japan. We formulated the model in terms of the channel volume and linked the solution to the volumetric moment tensor. We also incorporated the turbulent flow effect to deal with both magma and super-critical fluid as the working fluid. Assuming the realistic material parameters at the tremor source depth ($\sim$ 30 km) beneath Hakone, we searched for the conditions in which tremor was generated at an observed frequency ($\sim$ 1 Hz). It is shown that both magma and super-critical fluids can generate realistic tremors with similar channel sizes of several-meter long and several-centimeter wide. We convolved the model solution with the Green’s function at each seismic station to compare the model with the data. The result showed that Julian’s model could produce synthetic tremor waveforms very close to the observed ones. Although the source waveform had only a single peak at each cycle, the convolved waveform exhibited an apparent secondary peak, like the observed waveforms. While the previous models generated such waveforms exhibiting alternative large and small peaks by a non-linear effect of period-doubling before the chaos, our model did not show such transitions, at least with the investigated parameters. Although most of the parameters and physical values of the solutions were in the realistic ranges, the only problem was the presumed low elasticity of the channel as small as $10^5$ Pa to generate oscillation at $\sim$ 1 Hz. We proposed that not the rock property alone but the channel structure consisting of rock and compressible fluids could generate the low effective elasticity. To fully validate our model, the mechanism of such small elasticity should be identified, which is our future work.

Graphical Abstract

Introduction

A volcanic tremor is a continuous seismic signal that persists for several minutes or days and is sometimes accompanied by volcanic eruptions or independently. Among the various characteristics of volcanic tremors, a harmonic tremor, with spectral peaks corresponding to a fundamental frequency and additional overtones, is often observed (e.g., Hellweg 2000; Ichihara et al. 2013; Kamo et al. 1977; Konstantinou and Schlindwein 2002; Lees et al. 2004). While the source location of these harmonic tremors was usually estimated at shallow parts of volcanic edifices, mostly just beneath a conduit (e.g., Matsumoto et al. 2013; Maryanto et al. 2008; Ripepe et al. 2009) and not deeper than 10 km (e.g., Konstantinou and Schlindwein 2002; Matsumoto et al. 2013), a few studies report volcanic tremors radiated from the root of a volcano (Aki and Koyanagi 1981; Ukawa and Ohtake 1987).

Although a physical understanding of the volcanic harmonic tremors has been elusive, several generation mechanisms have been proposed. Julian (1994) proposed a lumped-parameter tremor model in which a non-linear oscillation was excited by fluid flow through an elastic plane-like channel within a volcanic edifice. Following the idea, various flow-induced oscillation models have been proposed theoretically and experimentally (e.g., Lyons et al. 2013; Rust et al. 2008; Corona-Romero et al. 2012; Takeo 2020). Another model considered elastic waves along the fluid-filled crack walls (Chouet 1988; Ferrazzini and Aki 1987; Dunham and Ogden 2012; Lipovsky and Dunham 2015). Hellweg (2000) discussed several possible fluid-dynamical periodic waves like a saw-tooth wave, generating the harmonic spectral feature. Specifically at a shallow depth, a magma-wagging oscillation around a magma column (Jellinek and Bercovici 2011), the growth and collapse of bubbles due to groundwater boiling (Leet 1988), the two-phase flow instability (Iwamura and Kaneshima 2005; Fujita 2008), the resonance of bubble clouds (Chouet 1996; Konstantinou et al. 2019), gas accumulation beneath permeable media (Girona et al. 2019), and the self-oscillations of fluid filling a cavity (Konstantinou 2015) were also proposed models for the harmonic or semi-monochromatic volcanic tremor.

Among the above tremor models, the flow-induced oscillation proposed by Julian (1994) (hereafter denoted as J94 model) is the most frequently considered in interpreting observed tremors. Because it is a lumped-parameter model, it is easy to calculate the oscillation of the source. On the other hand, the model parameters and calculated source time functions are not directly comparable with the observations. Therefore, the applications of the J94 model or other similar models have been limited to explain the oscillation mechanism (e.g., Lyons et al. 2013; Ichimura et al. 2018; Yamada et al. 2021) or non-linear features of waveforms and spectra (e.g., Hellweg 2000; Julian 2000; Konstantinou 2002; Hagerty and Benites 2003; Natsume et al. 2018). Recently, Natsume et al. (2018) and Takeo (2020) applied the J94 model or its extension to explain the waveforms of the harmonic tremor observed during the 2011 Shinmoe-dake eruption at the Kirishima volcano complex, Kyushu, Japan. Also, Natsume et al. (2018) showed that the J94 model could reproduce waveforms of the low-frequency earthquakes (LFEs) that occurred during the same eruption and had the same frequency components as the harmonic tremor. The data at Shinmoe-dake were assumed to represent the source waveforms because they were recorded within 1 km from the source. On the other hand, their models did not include the seismic wave radiation and did not evaluate the absolute amplitudes of the observed ground motions. Takeo (2020) noted that the tremor model only explains the qualitative characteristics of the observed phase portrait but emphasized that none of the other tremor models can explain the observed phase portrait.

To allow quantitative applications of the J94 model, we extended the model, linked between the model waveforms and seismic moment tensor, and clarified the relationship between the model parameters and geophysical parameters. This study tests the model with the harmonic tremor observed at a depth of $\sim$ 30 km beneath the Hakone volcano, central Japan (Yukutake et al. 2022). Although a few studies previously reported harmonic or monochromatic tremors that originated from the deep part beneath volcanoes, depth ranges from the lower crust to upper mantle (Aki and Koyanagi 1981; Ukawa and Ohtake 1987), deep LFEs that occurs within this depth range have been widely investigated (Aso et al. 2013; Hasegawa and Yamamoto 1994; Nichol et al. 2011; Yoshida et al. 2020; Oikawa et al. 2021). Yoshida et al. (2020) considered the J94 model as one of the possible mechanisms of the deep LFEs. On the other hand, Oikawa et al. (2021) precluded this possibility, assuming that the J94 model consisting of fluid movement would make a single force or dipole. This study associates the J94 model with the volumetric moment tensor. Our extension of the J94 model will be useful in the investigation of the deep tremors and LFEs, which are crucial to elucidate the feeding mechanism of magmatic fluid at depth (Matoza 2020).

Theory

We formulate a model for the oscillation induced by the viscous incompressible fluid flow in a thin channel connecting two reservoirs (Fig. 1). The fluid has the viscosity, $\eta$, and density, $\rho$. The system is embedded in the elastic host rock, having the Lamé constants, $\lambda _r$ and $\mu _r$, and density, $\rho _r$. The effective pressures (the pressure subtracted by the hydrostatic pressure of the fluid) in the upstream and downstream ($p_1$ and $p_2$, respectively) are constant, and their difference, $p_1$–$p_2$ ($>0$), drives the flow. The fluid pressure in the channel changes with the flow speed, and the elastic channel deforms in response. The non-linear coupling between the flow and the channel generates the oscillation of the channel cross-section under certain conditions, which is derived by the linear stability analysis following J94. In the main text, we mainly explain our modification to the J94 model. The details of the derivation has been given in Supporting Information S1.

Fig. 1

The geometry of the tremor model, consisting of a thin channel connecting two large reservoirs (Julian 1994). The channel has an elliptical cross-section with half lengths of a and b ($\delta =a/b\ll 1$) and length L. The mean vertical flow speed is v, which is driven by the effective pressure difference between the upstream, $p_1$, and the downstream, $p_2$. The right-hand Cartesian coordinate (x, y, z) is taken with the x-axis in the flow direction and the y-axis parallel to the ellipse’s minor axis. The inset figure shows the position of the channel projected onto the z–y and z–x planes. The origin of the coordinate axes is set at the center of the channel

Equation of motion of the channel wall

Although J94 considered a two-dimensional flow between parallel plates, we approximate the channel cross section by an ellipse with half lengths of a and b ($\gamma \equiv a/b\ll 1$). Because of the thin geometry, the wall deformation is dominated by the change of the minor axis a(t) with time, t, keeping $b\sim b_0$, where the subscript 0 is used to indicate the reference value. The cross-sectional area is $\alpha (t)=\pi b_0 a(t)$. The length along the flow is a constant, L. We define the right-hand Cartesian coordinate (x, y, z), with the x-axis in the flow direction and y-axis in the ellipse’s minor axis direction. The coordinate origin is placed at the center of the channel (Fig. 1).

J94 formulated the equation of motion for the channel width h as

$$M^\prime \ddot{h}+A^\prime {\dot{h}}+k^\prime (h-h_0 )=F_p,$$

(1)

where the parameters of J94 are distinguished by the prime, and a dot and two dots indicate single and double differentiation with respect to time, respectively. The right-hand side, $F_p$, is the force on the two-dimensional channel wall exerted by the fluid pressure, which we discuss in “Resistive force for flow in the channel” section. On the left-hand side of Eq. (1), the first term represents the inertia. J94 approximated the effective mass, $M^\prime$, to be of order of $\rho _r L^2$. The second is the damping, and the coefficient $A^\prime$ was arbitrarily assumed. The third is the channel elasticity. To estimate $k^\prime$, J94 considered the opening of a dike, $\frac{w}{h}=\frac{\mu _r}{p(1-\nu )}$, where $\frac{w}{h}$ is the dike aspect ratio, and $\nu \equiv \frac{\lambda _r}{2(\lambda _r+\mu _r)}$ is the Poisson’s ratio of the rock, and p is the overpressure. From this relation, J94 defined $k^\prime =\mu _r \frac{L}{w}$ and assumed $\frac{L}{w}$ ranging from 0.01 to 0.1.

We modify Eq. (1) for the channel cross section as

$$C_m\rho _r\alpha _0\frac{\ddot{\alpha }}{\alpha _0}+A\frac{\dot{\alpha }}{\alpha _0}+K_ck_r\frac{\alpha -\alpha _0}{\alpha _0}=\frac{F_p}{L},$$

(2)

where the force due to the fluid overpressure on the right-hand side is

$$F_p=\int _{-L/2}^{L/2}p(x,t)\hbox {d}x.$$

(3)

In Eq. (2), the effective mass and elasticity parameters explicitly include the rock density, $\rho _r$, and the bulk modulus, $k_r\equiv \lambda _r+\frac{2}{3}\mu _r$ with dimensionless coefficients, $C_m$ and $K_c$.

We approximate

$$C_m^3=\frac{L^2}{48\pi ^2\alpha _0}=\frac{L^2}{48\pi ^3a_0b_0},$$

(4)

by the analogy of the effective mass that a spherical bubble in fluid feels on its expansion (Appendix 1). The representation of the elasticity in Eq. (2) refers to Mizuno et al. (2015). The coefficient $K_c$ represents the reduction of the effective elasticity due to the shape and depend on the channel shape and the Poisson’s ratio of the rock. Mizuno et al. (2016) provides a tool to calculate $K_c$, assuming $\lambda _r=\mu _r$ ($\nu =1/4$), for an ellipsoidal cavity for given aspect ratios $a_2/a_3$ and $a_1/a_3$, where $a_1$, $a_2$, $a_3$ are the axial lengths of the ellipsoid $(a_1<a_2<a_3)$. We may apply the tool to estimate $K_c$ by substituting $(a_1,a_2 )=(a,b)$ and assuming $a_3\gg b$. We refer to values from this calculation but use $K_c$ as a tuning parameter. It should be noted that $K_c k_r$ is not the effective bulk modulus of the host rock, but represents the elastic expansivity of the channel due to overpressure. For determining $K_c k_r$ for the thin channel, $\mu _r$ is more essential than $k_r$, as J94 formulated for $k^\prime$.

Resistive force for flow in the channel

the pressure within the channel, p, is uniform in the y–z plane and depends on x and t. The flow speed averaged in the y–z plane is denoted as v(x,t), which is subject to a resistive force, $F_v$, per unit cross section and unit length along the flow:

$$F_v(x,t)=\frac{4\pi \eta }{\alpha }\left( \gamma +\frac{1}{\gamma }\right) v(x,t)+\frac{C_d\rho |v(0,t)|}{4a}v(0,t),\quad \gamma +\frac{1}{\gamma }\simeq \frac{\pi a_0^2}{\alpha },\quad a\simeq \frac{\alpha }{\pi b_0}.$$

(5)

The first term of the right-hand side works in the laminar flow regime with small Reynolds number, $R_e=2a\rho v/\eta$. J94 used a corresponding term for a parallel-plane channel. We use the representation for an elliptic channel (Takeo, 2020). Considering $\gamma \ll 1$, we use the approximation of the second equation in (5). We introduce the second term to extend the model for turbulent flow conditions, where $C_d$ is the friction factor of turbulent flow. We approximate $C_d=0.01$ for simplicity (Wilson et al. 1980). For the tractability, the turbulent friction term is represented by the value at x = 0. The first term dominates the second when $R_e\ll 32/C_d=3200$, which is consistent with the stability conditions of laminar flows between parallel planes ($R_e\le 1000$) and in a circular pipe ($R_e\le 1800$) (Landau and Lifshitz 1987).

Equations of motion

We consider the mass and momentum conservation in the channel and assume Bernoulli’s theorem at the inlet and the exit of the channel, following J94. The detail is given in Supporting Information 1. Finally, we obtained the system equations, which are similar to those of J94.

The equation of motion of fluid is

$$\rho {\dot{v}}+\frac{4\pi ^2 b_0^2\eta }{\alpha ^2}v+\frac{\pi C_d\rho b_0}{4\alpha }|v|v=\frac{p_1-p_2}{L},$$

(6)

where v = v(0,t), and all v hereafter denotes the value at x = 0. Replacing the corresponding equation of J94 by Eq. (6), the fluid pressure force $F_p$ in Eqs. (2) and (3) is specified. Then, the equation of motion of the channel wall becomes

$$\left( C_m\rho _r+\frac{\rho L^2}{12\alpha }\right) \ddot{\alpha }+\left[ \frac{A}{\alpha _0}+\frac{L^2}{12\alpha }\left( \frac{4\pi \eta }{\alpha }\frac{\pi b_0^2}{\alpha } -\frac{\rho }{2}\frac{\dot{\alpha }}{\alpha } \right) \right] \dot{\alpha } +K_ck_r\frac{\alpha -\alpha _0}{\alpha _0}=\frac{p_1+p_2}{2}-\rho \frac{v^2}{2}.$$

(7)

Seismic moment tensor

The advantage of solving for the channel area is that we can relate the solution to the seismic moment tensor for the volume change, $\Delta V^C=L(\alpha -\alpha _0)$. The seismic moment tensor is directly related to the stress-free volume change, $\Delta V^T$, instead of $\Delta V^C$ (Aki and Richards 2002; Ichihara et al. 2016). Because the expansion is dominated in the y-direction in the current model, we may approximate $\Delta V^T=\Delta V^C$, and the seismic moment tensor becomes (Mizuno et al. 2015)

$$\left( \begin{array}{ccc} M_{xx} & 0 & 0\\ 0 & M_{yy} & 0\\ 0 & 0 & M_{zz} \end{array} \right) =3k_rL\Delta \alpha \left( \begin{array}{ccc} 1/5 & 0 & 0\\ 0 & 3/5 & 0\\ 0 & 0 & 1/5 \end{array} \right) ,$$

(8)

where $\Delta \alpha \equiv \alpha -\alpha _0$.

Corona-Romero et al. (2012) developed a flow-induced oscillation model assuming a fluid-filled pipe buried in elastic half-space and calculated the far-field seismic waveforms using the so-called cylindrical source with the moment tensor component ratio of 1:2:2. Mizuno et al. (2015) presented that the volumetric moment tensor of a thin prolate ellipsoid differs from the cylindrical source model without ends along the axis. They also showed that the identical moment tensor representation (8) holds regardless of the existence of ends in the plane normal to the opening axis.

Tremor condition analysis

J94 presented the linear stability analysis to determine the conditions under which the flow excites the continuous oscillation. We made the same analysis for the modified model.

Steady flow solution

Equations (5) and (6) are transformed for the steady state with $v=v_s>0$ and $\alpha =\alpha _s$ by setting all the time deviations to zero:

$$\frac{4{\pi ^2}b_0^2\eta }{\alpha _s^2}v_s+\frac{\pi C_d\rho {b_0}}{4\alpha _s}v_s^2= \frac{p_1-p_2}{L},$$

(9)

$${K_c}{k_r}\frac{\alpha _s-\alpha _0}{\alpha _0}= \frac{p_1+p_2}{2}-\rho \frac{v_s^2}{2}.$$

(10)

We solve Eqs. (9) and (10) numerically. Because both $v_s$, and $\alpha _s$ are positive, and all the parameters are also positive, a set of $(v_s, \alpha _s)$ is uniquely determined.

Linear stability analysis

Equations (6) and (7) are linearized considering small perturbation around the steady-state solution as $v=v_s+{\hat{v}}$ and $\alpha =\alpha _s+\hat{\alpha }$. The linearized equations are represented in the matrix form as

$$\frac{\hbox {d}}{\hbox {d}t}\left[ \begin{array}{c} \hat{\alpha }\\ \dot{\hat{\alpha }}\\ {\hat{v}} \end{array} \right] =\left[ \begin{array}{ccc} 0 &1 &0\\ -\frac{K_ck_r}{M_j+m_j} & -\frac{A+a_j}{M_j+m_j} & -\frac{\rho v_s\alpha _0}{M_j+m_j}\\ \frac{v_s}{\alpha _s}\left( \frac{2a_j}{m_j}+e\right) & 0 & -\left( \frac{a_j}{m_j}+2e\right) \end{array} \right] \left[ \begin{array}{c} \hat{\alpha }\\ \dot{\hat{\alpha }}\\ {\hat{v}} \end{array} \right] ,$$

(11)

where $(M_j, m_j, a_j)$ are defined to be comparable with (M,m,a) of J94 as

$$M_j\equiv C_m\rho _r\alpha _0, m_j\equiv \frac{\rho L^2\alpha _0}{12\alpha _s}, a_j\equiv \frac{L^2\alpha _0}{12\alpha _s}\frac{4\pi ^2b_0^2\eta }{\alpha _s^2}=m_j\Gamma ,$$

(12)

where

$$\Gamma \equiv \frac{4\pi ^2b_0^2\eta }{\alpha _s^2\rho }.$$

(13)

In addition, the parameter e is defined to incorporate turbulent friction as

$$e\equiv \frac{\pi C_d b_0 v_s}{4\alpha _s}.$$

(14)

It is noted that e dominates $\Gamma$ at large $R_e$ and vice versa.

We express the characteristic equation of the matrix in (11) as

$$\lambda ^3+C_2\lambda ^2+C_1\lambda +C_0=0,$$

(15)

where

$$C_0\equiv \frac{1}{M_j+m_j}\left[ \left( \Gamma +2e\right) K_ck_r+\frac{\rho v_s^2\alpha _0}{\alpha _s}\left( 2\Gamma +e\right) \right] ,$$

(16)

$$C_1\equiv \frac{1}{M_j+m_j}\left[ \left( \Gamma +2e\right) (A+m_j\Gamma )+{K_c}k_r\right] ,$$

(17)

$$C_2\equiv \frac{{A+m_j\Gamma }}{{M_j+m_j}}+\Gamma +2e.$$

(18)

These coefficients are all positive. Therefore, as J94 derived, the condition for tremor to occur is

$$R\equiv \frac{C_0}{C_1C_2}>1,$$

(19)

and the angular frequency, $\omega$, at the onset of instability is

$$\omega ^2=C_1.$$

(20)

Material Parameters

We assume that flow of magma or super-critical water drives the oscillation of channel. Since we model the oscillation at the lower crustal level, primary magma, namely basaltic magma, was assumed. The density ($\rho$) and viscosity ($\eta$) of basaltic magma at a depth of 30 km is assumed as 2800 kg/m$^3$ and 20 Pa s, respectively. The viscosity value is arbitrary because the temperature of the tremor source region has not been constrained. In the case of super-critical fluids, we assume an H$_2$O–CO$_2$ mixture with the CO$_2$ molar concentration of 0.6. The molar volume of the carbonated water under the conditions of 500 MPa and 500 $^\circ$C are 34 cm$^3$/mol (Bowers 1995). The corresponding density is $\rho \sim 1010$ kg/m$^3$. The viscosity is approximated as $2\times 10^{-4}$ Pa s by extrapolating the data for super-critical pure H$_2$O from 2 to 100 MPa at 500 $^\circ$C (JSME 1983) because it depends on the pressure only slightly up to 1 GPa (Audétat and Keppler 2004).

The elasticity is the problematic parameter when one wants to fit the observed tremor by the J94 model. We assumed that the channel wall is composed of the representative rocks at this depth. According to the tomographic result by Yukutake et al. (2021), the averaged P- and S-wave velocities at a depth of 21 km beneath the Hakone volcano are estimated at 6.7 km/s and 3.8 km/s, respectively. The P-wave velocity corresponds to the lower crust of Izu-arc composed of the gabbroic rock (Kodaira et al. 2007). According to Christensen (1996), we assumed the $\rho _r$ of gabbro as 2968 kg/m$^3$ that forms the host rock around the channel. Given the tomographic result by Yukutake et al. (2021), elastic constant $k_r$ of host rock is obtained as 76.1 GPa. The viscosity of channel wall can be negligible in the frequency range we considered.

Numerical methods

Model calculations

To select adequate sets of parameters, we searched for conditions in which tremor occurs, represented by $R>1$ in Eq. (19), and the oscillation frequency, f, is around 1 Hz, referring to the observation at Hakone. The main tuning parameters are the channel geometry $(L, \gamma )$ and its elasticity coefficient $K_c$, and the driving pressure $p_1$. The dependence on $b_0$ was minor. We kept the downstream effective pressure at the lithostatic value, that is $p_2=0$.

For each set of parameters, we numerically solved Eqs. (9) and (10) to determine the steady-state solution $(v_s, \alpha _s)$. The variable $\alpha _s$ can be eliminated from the two equations to give a quartic equation in $v_s$, which is solved by a Python function, numpy.roots. We employed the solution satisfying $v_s>0$ and $\alpha _s>0$ and used them to calculate R and $R_e$, defined here by

$$R_e\equiv \frac{2\alpha _s\rho v_s}{\pi b_0\eta },$$

(21)

where $\alpha _s/(\pi b_0)$ is the channel width in the steady state. On the other hand, f was determined from the numerical solutions of Eqs. (6) and (7). We obtained the solutions $(\alpha , \dot{\alpha }, v)$ as functions of the laps time, t, at 0.01-s intervals, with the initial condition of $(\alpha (0), \dot{\alpha }(0), v(0))=(\alpha _0, 0, 0)$. The calculation was performed for 500 s, which was long enough to achieve the limit cycle at each condition. We picked the times at which $\alpha$ passed $\alpha _s$ with $\dot{\alpha }>0$ by the linear interpolation for the last two cycles, sequentially as $t_i$ $(i=1,2,3)$. Then, we calculated the frequency by $f_i=1/(t_{i+1}-t_i)$ $(i=1,2)$. When $|f_2-f_1|<0.01$ s$^{-1}$, we regarded that the solution reached the limit cycle and employed $f_2$ as f.

We used a Python function, scipy.integrate.solve_ivp to calculate the model equations, (6) and (7). The relative and absolute tolerances ($r_{tol}$, $a_{tol}$) were set at $(10^{-6},10^{-8}$) for magma and $(10^{-8}, 10^{-10})$ for super-critical fluids. We compared the solutions with those obtained using $(r_{tol}, a_{tol})=(10^{-6}, 10^{-8})$ and $(10^{-8}, 10^{-10})$ for some cases and found no significant differences.

Comparison with the observation

To convert the source waveforms to the ground displacement at each station, we calculated the Green’s function using OpenSWPC code based on the finite difference method developed by Maeda et al. (2017). We used the one-dimensional velocity structure beneath Tanzawa Mountains near Hakone (Hiraga 1987) that is an initial model for the tomographic study (Yukutake et al. 2021), including the topography information at the surface layer. We consider two models of seismic attenuation structure; (1) the homogeneous attenuation model in which Qp and Qs are set as 600 and 300, and (2) the one-dimensional attenuation model based on the result by Kashiwagi et al. (2020). We set the grid nodes at the interval of 0.03 km in the horizontal and vertical directions and 0.001 s in the direction of the time axis. We configured the rectangular computational region with a length of 24 km in the horizontal direction and from − 4 to 35 km in the depth direction, centered at the epicenter of the tremor source. Green’s function was calculated for 20 s from the origin time. As a source model, we assumed an open crack, of which moment tensor is given in Eq. (8), oriented to the EW direction. The Green’s function was calculated for the moment value of $M_0\equiv M_{xx}+M_{yy}+M_{zz}=10^9$ Nm. This value of $M_0$ is not essential because we normalize the Green’s function by $M_0$ in taking the convolution with the source time function. We used the Kupper function with a duration time of 0.05 s as a source time function of the Green’s function. Using this duration time, we can consider it a delta function with a flat spectral response below 10 Hz, which is applicable to our target observed tremor with a fundamental peak of 1 Hz. We also convolved the response of the velocity seismometer to compare the observed waveforms.

Results

Parameter search

We investigate the dependence of R, f, and the steady-state channel volume, $V_0=\alpha _sL$, on the model parameters.

Figure 2 presents the result for magma as the work fluid. In Fig. 2a, we calculated Eqs. (9) and (10) to obtain $v_s$ and $\alpha _s$ for the given parameters in Table 1 and various L and $p_1$ as on the axes. The bottom panel shows $V_0$ with contours of which values (m$^3$) are presented on the individual curves. Then, R is calculated by Eqs. (12)–(19) and presented in the top panel with contours. Because of condition (19), oscillation occurs in the parameter range above the contour of R = 1. In the region of $R>1$, we solved Eqs. (6) and (7) and evaluated the frequency, f, as explained in the section, Model calculations. The frequency is shown in the middle panel by contours with values in Hz. Similarly, we investigated R, f, and $V_0$ as functions of $K_c$ and $p_1$ (Fig. 2b) and the channel width $\alpha _0/(\pi b_0)$ and $p_1$ (Fig. 2c). The parameters except those on the axes are fixed as listed in Table 1. The Reynolds number, $R_e$, is smaller than 1000 in the displayed parameter ranges of Fig. 2, indicating laminar flow in the channel.

Fig. 2

Oscillation condition diagrams for magma ($\rho =2800$ kg/m$^3$, $\eta =5$ Pa s). The parameter R determining the tremor condition, the oscillation frequency, f, and the channel volume, $V_0=L\alpha _0$, are shown from the top to the bottom as functions of the driving pressure, $p_1$, and either of the crack length, L (a), the channel elasticity coefficient, $K_c$ (b), and the minor half width of the channel cross section, $a_0$ (c). In each frame, all the parameters except those on the axes are fixed at the values in Table 1

Table 1 Parameters for magma flow

Also, Fig. 3 shows the corresponding results (only on the $p_1$–L space) for the hydrothermal fluid. The other parameters are fixed as listed in Table 2. In this case, the flow is turbulent ($R_e\sim 10^6$), for which the turbulent friction term in Eq. (5) works effectively.

Table 2 Parameters for hydrothermal flow

Fig. 3

Oscillation condition diagrams for super-critical fluid ($\rho =1010$ kg/m$^3$, $\eta =2\times 10^{-4}$ Pa s). The format follows Fig. 2a

Both Figs. 2 and 3 demonstrate that R increases as the driving pressure ($p_1$) increases, indicating that the oscillation is more effectively excited. It is interpreted as the result of the increasing flow speed. On the other hand, as L increases, it becomes more difficult for the oscillation to occur. The essential result here is that there are sets of parameters with which tremor was excited at a frequency around 1 Hz, as observed at Hakone volcano (Yukutake et al. 2022). We selected a representative set of parameters for each of magma and hydrothermal flows (Table 1), and calculated Eqs. (6) and (7) with the initial condition of $(\alpha , \dot{\alpha },v)=(\alpha _0,0,0)$ (Fig. 4). In either case, continuous oscillation at $f\sim 1$ Hz is generated. The waveform has a non-linear wave characteristics with a sharp peak and a round trough. The non-sinusoidal periodic waveforms have harmonic spectra.The violent oscillation presented in Fig. 4b includes $\alpha$ and $v_s$ almost zero in each cycle. The hydrothermal fluid more easily generates such violent oscillations than magma even with R close to unity. These waveforms are compared with the observed tremor at Hakone in the next section.

Fig. 4

Development of the channel oscillation by flow of magma (a) and super-critical fluid (b). The cross-sectional area, $\alpha$, and the flow velocity, v, are shown as functions of the lapse time. The inset in (b) shows a magnified waveform of $\alpha$ from 45 to 50 s. The parameter values are listed in Tables 1 and 2, and $\alpha _s=0.153$ and 0.059 m$^2$ for (a) and (b), respectively. Phase portraits $(\dot{\alpha }, \ddot{\alpha })$ are shown using the waveforms in the yellow-colored period (a: 20–25 s, b:45–50 s)

Application to the seismic data at Hakone

We assumed the location of source model at the northern part of Hakone volcano (Fig. 5a) at a depth of 30 km (Yukutake et al. 2022). The Green’s functions at some stations are calculated and the one at the OMZ station is presented in Fig. 5b as an example. We took the convolution of the calculated $\Delta \alpha =\alpha -\alpha _0$ (Fig. 4) multiplied by $3k_rL$ (see Eq. 8) and the Green’s function normalized by the assumed moment value, $M_0=10^9$ Nm, to obtain the model waveforms to be compared with the observation. We used $k_r=7.61$ GPa, ten times smaller than the host rock value, to make the calculated amplitude similar to the observation, which is discussed later.

Fig. 5

a Map of Hakone volcano. Triangles show the locations of seismic stations. Red star shows the optimal epicenter of volcanic tremor estimated by Yukutake et al. (2022). To calculate the Green’s function, we assumed a vertical open crack with EW strike (y-direction) at a depth of 30 km beneath the optimal epicenter. The inset indicates the target region for the Japanese Island. b The synthetic Green’s function at OMZ station

Figure 6 compares the calculated waveforms and the observations at three selected stations. Although the source waveform has only a single peak in each cycle (Fig. 4), the convoluted waveform has an apparent secondary peak. The previous models can generate similar waveforms exhibiting alternative large and small peaks by a non-linear effect of period-doubling (Julian 1994; Takeo 2020). However, the waveform in Fig. 6 is not due to the period-doubling but due to the amplification of the second mode due to the medium structure.

Fig. 6

Comparison of the calculated ground displacement waveforms and the observations at OMZ, KIN, and TNM stations (Fig. 5a). The left and central columns display the convolutions between the Green’s function at each station and $\alpha (t)$ in Fig. 4a for magma and Fig. 4b for super-critical fluid, respectively. The right column shows the observed ground velocity

The calculated ground displacement became much larger than the observation if we used the bulk modulus of the rock ($k_r=76.1$ GPa). Even if we used the Green’s function with a possible lower Q, the calculated amplitudes were reduced only by factors (Fig. 7). When R is only slightly above unity, namely the condition is close to the oscillation limit, the amplitude is smaller. However, the waveform of $\alpha$ becomes sinusoidal so that the convolved waveforms at the stations do not have overtones and are less similar to the observed waveforms than the example in Fig. 6. We discuss the assumption of the smaller $k_r$ in the next section.

Fig. 7

Comparison of the calculated waveforms assuming the one-dimensional attenuation model based on the result by Kashiwagi et al. (2020) and the observations at OMZ, KIN, and TNM stations. The meaning of each column is same as that in Fig. 6

Discussion

The small channel elasticity

The modified J94 model of this study was able to generate tremor around 1 Hz for either of magma or hydrothermal fluid with the range of parameter values presented in Figs. 2 and 3 and Tables 1 and 2. The material parameters of the rock and the fluids are realistic except the crack elasticity, $K_ck_r$ (Fig. 2b), Even though $K_c$ can be much smaller than unity for a thin crack, the values of $K_c$ in the order of $10^{-6}$–$10^{-7}$ are too small. For example, a thin ellipsoidal crack of $4\times 1 \times 0.02$ m$^3$ has $K_c\sim 0.017$ (Mizuno et al. 2016).

In the original model of J94, the corresponding parameter of $K_c$, which was the ratio of $k^\prime$ in Eq. (1) to $k_r$, was assumed in the order of 0.01. Figure 9 of J94 shows oscillation at $\sim$ 5 Hz, which is realized with a longer channel (L = 10 m) with a larger inertia, $M^\prime =3\times 10^5$ kg m$^{-1}$, where $M^\prime$ in Eq. (1) was approximated by $\rho _rL^2$ (Julian 1994). Equation (12) defines the corresponding inertia parameter of the elastic wall in Eq. (2) as $M_j=C_m\rho _r\alpha _0$, where $C_m\sim 1$ (Appendix 1 and Tables 1 and 2). The inertia coefficient scaled by $\rho _r$ and the channel cross-sectional area is consistent with Corona-Romero et al. (2012), who formulated the flow-induced tremor model in a circular pipe. Then, $M_j\ll \rho _rL^2$, which is, for magma, much smaller than the inertia coefficient of fluid in the channel, $m_j\sim \rho L^2/12$, defined in Eq. (12). Considering this condition and the observed amplitude of the tremor signals at Hakone, Appendix 2 mathematically shows that the current model cannot generate oscillation at $f\sim 1$ Hz with $K_c$ larger than $10^{-6}$.

The small $K_c$ indicates that the channel is more deformable than the expectation from the shape and the host rock property. When the host rock contains fluid in deformable configurations like partial melt network (Takei 2002) or fracture meshes (Sibson 1996), the effective elasticity can be reduced significantly (Mavko et al. 1998; Takei 2002). Although these mechanisms may explain the ten-times smaller $k_r$ we used to calculate the ground displacement amplitudes similar to the observations (Figs. 6, 7), they may not generate the rock elasticity as small as $\sim \,10^5$ Pa.

The sharp edges of the channel might be damaged or melted by the stress concentration, so the channel might have lost the elasticity of the rock. If the driving fluid is magma, it cannot penetrate the sharp edges due to its viscosity. Then, cavities filled with low-pressure volatiles exsolving from magma are generated at the tip and control the crack dynamics (Rubin 1993; Rubin and Gillard 1998). We assume cavities at both ends of the thin channel at $(y,z)=(0,\pm b_0)$ (Fig. 1). The cross-sectional area of each cavity on the yz-plane is $\alpha _c$, and the cavity length along the z-axis is $b_c$. The cavity is filled with super-critical fluids at pressure, $P_c$, of which bulk modulus is approximated by $P_c$. When the channel expands, $\alpha _c$ also increases by $\Delta \alpha _c$. Then, the cavity pressure decreases by $\Delta P_c$ such that

$$\Delta P_c = -P_c\frac{\Delta \alpha _c}{\alpha _c}.$$

(22)

This pressure change acts as a spring to pull the channel wall inward. Assuming that $P_c\sim 500$ MPa and representing $\Delta \alpha _c/\alpha _c=\epsilon (\alpha -\alpha _0)/\alpha _0$ and $b_c=\beta b_0$, where $\epsilon$ and $\beta$ are small constants, we may obtain the effective channel elasticity ($K_ck_r$) as $\epsilon \beta P_c$. The small elasticity is realized if $\epsilon \beta$ is small enough.

The above is one possible way to explain the small $K_c$ for magma. It may not work when the oscillation is driven by super-critical fluids. What we want to emphasize here is that the channel elasticity may not be determined by the material property of rock alone but by the structure consisting of the rock and fluids. To fully validate the current flow-induced oscillation model as the mechanism of the deep harmonic tremor at Hakone, we need to specify a mechanical and material model that explain the small $K_c$, which is our future work.

Implications for tremor observation at Hakone

We extended J94 model to deal with turbulent flow. The magma flow was laminar, while the hydrothermal fluid flow was turbulent. We found that both of basaltic magma and hydrothermal fluids can generate similar flow-induced oscillation with similar channel dimensions (Tables 1, 2 and Fig. 7), regardless of their contrasting viscosity. This means that we cannot identify the nature of the source fluid. Due to the different viscosity, the effective pressures required to drive the flow were different by an order. In the case of the hydrothermal flow, the effective pressure was in the same order as the buoyancy of the fluid in the mantle rock ($p_1\sim (\rho _r-\rho )gL$). In the case of magma, extra overpressure is sustained due to the viscosity. The overpressure in the order of 0.1 MPa (Table 1) is small compared to the lithostatic pressure at the tremor source depths ($\sim$ 500 MPa), so that it will not be observed.

Yukutake et al. (2022) show that the tremor signal at Hakone has a broad peak around 1.2 Hz during the initial part, while it represents the harmonic feature and the frequency gliding from 0.90 to 0.98 Hz at fundamental mode in the latter part. Then, the tremor terminated abruptly. These features were also reported for shallow harmonic tremors (Hotovec et al. 2013; Konstantinou et al. 2019; Takeo 2020). The flow-induced oscillation model like J94 and the current one may explain the transition between the broadband tremor and harmonic tremor (Konstantinou et al. 2019). Takeo (2020) reproduced the change of tremor waveform and amplitude by the change of the effective pressure. The harmonic tremor evolution at Hakone might also be explained by the change of conditions. In case of the magma flow, the shortening of L generates upward frequency gliding, while the decrease of $p_1$ can bring the condition from oscillation ($R>1$) to non-oscillation ($R<1$), keeping f at 1 Hz (Fig. 2a). On the other hand, in the hydrothermal system, the decrease of either L or $p_1$ may generate the upward frequency gliding (Fig. 3). Although the transition to chaos with the period doubling was the most interesting feature of J94 model, the current model did not exhibit such transitions, at least with the parameter ranges we tested. Thorough investigations of the behavior of the current model is beyond the scope of this study.

Yukutake et al. (2022) also reported that the deep LFEs within the depth ranges of 20–30 km above the tremor source region activated several hours prior to the occurrence time of volcanic tremor, suggesting that the fluid was supplied before the volcanic tremor. According to the results of theoretical modeling, the observed sequence might reflect the following procedure of fluid flow at the deep root of the volcano. At the onset of the activation for the deep LFEs, the supply of magmatic fluid started. At this timing, $p_1$ is not large enough to cause oscillation in the channel and the fluid flow triggered only the deep LFEs. At the start time of the volcanic tremor, $p_1$ increased, leading to oscillation of the channel. Moreover, during the latter part of the tremor, shortening of L or decrease in $p_1$ caused the frequency gliding in the harmonic tremor. Finally, the abrupt termination of tremor amplitude occurred due to decrease in $p_1$.

Conclusions

We extended the flow-induced oscillation model of Julian (1994) to incorporate the realistic material parameters beneath a volcano and to link the model waveforms to the seismometer data. Applying the model to the deep harmonic tremor observed at Hakone volcano, we obtained the following conclusions.

• (1) Both magma and super-critical fluids can generate tremors with realistic material and flow parameters and channel sizes (widths of centimeters and lengths of meters).

• (2) The model waveforms convolved with the Green’s function at each seismic station reproduced the observed waveform features.

• (3) Although the source waveform had only a single peak at each cycle, the convolved waveform exhibited an apparent secondary peak, which was similar to the observed waveforms. While the previous models generated similar waveforms exhibiting alternative large and small peaks by a non-linear effect of period-doubling before the chaos, our model did not show such transitions, at least with the investigated parameters.

• (4) Although most of the parameters and physical values of the solutions were in the realistic ranges, the effective elasticity of the channel as small as $10^5$ Pa was required to generate oscillation at $\sim$ 1 Hz. Also, we needed to assume a smaller bulk modulus around the channel to obtain the ground displacement amplitudes at the stations similar to the observation. The small channel elasticity and bulk modulus might be generated by the presence of compressible fluids in the system. To fully validate our model, the mechanism of such small elasticity should be identified, which is our future work.

Appendices

Appendix 1: Effective mass of the channel wall

We derive the effective mass coefficient, $C_m$, used in the equation of motion of the channel wall (Eq. 2), by the analogy of the equation of motion for bubble expansion in fluid (Plesset and Prosperetti 1977; Prosperetti 1982). First, we consider the spherical expansion of a source with a radius, S, and a volume, $V=4\pi S^3/3$. Approximating that the deformation around the source is incompressible,

$$r^2v_r=S^2{\dot{S}}=\frac{{\dot{V}}}{4\pi },$$

(23)

where $v_r$ is the radial velocity at a distance r from the center of the sphere. The equation of motion for the spherical bubble expansion is derived by integrating the radial component of Navier-Stokes equation. For simplicity, we neglect the non-linear advection term, so that

$$\rho _r\int ^\infty _{S}\frac{\partial v_r}{\partial t}\hbox {d}r=p_i-p_\infty +3\int ^\infty _S\frac{\tau _{rr}}{r}\hbox {d}r,$$

(24)

where $p_i$ and $p_\infty$ are the pressure in the bubble and in the medium at $r=\infty$, respectively, and $\tau _{rr}$ is the deviatoric stress. Substituting $v_r$ in Eq. (23), the inertia term of (24) is

$$\rho _r\int ^\infty _{S}\frac{\partial v_r}{\partial t}\hbox {d}r=\frac{\rho _r{\ddot{V}}}{4\pi S}.$$

(25)

We use Eq. (25) as the inertia term of the channel expansion, $C_m\rho _r\ddot{\alpha }$, by replacing ${\ddot{V}}$ by the volume acceleration of the channel, ${\ddot{V}}=L\ddot{\alpha }$, with an approximation of $(4\pi /3)S^3\sim \alpha _0L$. Namely, we use the following equations to obtain Eq. (4):

$$C_m\rho _r\ddot{\alpha }=\frac{\rho _rL\ddot{\alpha }}{4\pi S}, S^3\sim \frac{3\alpha _0L}{4\pi }.$$

(26)

Appendix 2: Possible range of $K_c$ that generates tremor around 1 Hz

We calculated the channel oscillations with $L\sim 4$ m (Tables 1 and 2) and obtained the amplitudes of $\Delta \alpha \sim 10^{-1}$ m$^2$. The corresponding seismic moment is $k_rL\Delta \alpha \sim 10^{10}$ Nm. The calculated ground displacement at the stations were too large, so that we used $k_r=7.61$ GPa instead of 76.1 GPa in Figs. 6 and 7. Yet, the calculated waveforms are about ten times larger than the observed seismic amplitude (Yukutake et al. 2022). This means that the expected value of $L\Delta \alpha$ is $\sim \,10^{-2}$. Assuming that $\Delta \alpha$, $\alpha _0$, and $\alpha _s$ are all in the same order,

$$L\Delta \alpha \sim L\alpha _s\sim L\alpha _0=L\pi a_0b_0\sim 10^{-2}.$$

(27)

We consider a thin channel with $a_0/b_0<10^{-2}$ with the length L in the same order as $b_0$. Then, we may estimate from (27) as $L\sim b_0 \ge 1$.

We use these estimations to rearrange the equations obtained by the stability analysis; Eqs. (11)–(19). Below we assume A = 0 for simplicity. Using (12) with (4) and above approximation, we obtain

$$M_j= \rho _r\left( \frac{L^2{\alpha _0^2}}{48\pi ^2}\right) ^{\frac{1}{3}}\sim \rho _r\left( \frac{10^{-4}}{48\pi ^2}\right) ^{\frac{1}{3}}\sim \frac{\rho _r}{4.73^{1/3}}\times 10^{-2}\sim 20,$$

(28)

$$m_j \sim \frac{{\rho L^2}}{12},$$

(29)

$$\frac{{M_j}}{{m_j}}\sim \frac{240}{\rho L^2}\ll 1.$$

(30)

Below we approximate $M_j+m_j\sim m_j$.

Equation (20) gives $C_1\sim (2\pi f_i)^2$, where $f_i$ is the frequency at the onset of instability. It is noted that, in the non-linear regime, the frequency f determined from the numerical solution is smaller than $f_i$ by a factor. For example, with the parameter set in Table 1 that generates the solution in Fig. 4a gives $C_1\sim 320$, which yields $f_i\sim 2.8$ Hz. Equation (17) is transformed to

$$K_ck_r\sim [(2\pi f_i)^2-\left( \Gamma +2e\right) \Gamma ]m_j\sim [(2\pi f_i)^2-\left( \Gamma +2e\right) \Gamma ]\frac{\rho L^2}{12},$$

(31)

Also, we rewrite the tremor condition (19) as

$$K_ck_r<\rho v_s^2\left( 2+\frac{e}{\Gamma }\right) -2(\Gamma +2e)(\Gamma +e)\frac{\rho L^2}{12},$$

(32)

From the definition (13) and (14),

$$\Gamma \sim \frac{4\eta }{a_0^2\rho },e\sim \frac{C_dv_s}{4a_0}.$$

(33)

For magma, $\Gamma$ dominates e. Then, Eqs. (31) and (32) become

$$K_ck_r\sim [(2\pi f_i)^2-\Gamma ^2]\frac{\rho L^2}{12}<2\rho v_s^2-2\Gamma ^2\frac{\rho L^2}{12}$$

(34)

We consider the realistic magma flow speed $v_s$ to be smaller than a few meters per second, namely, $v_s^2<10$. Then, with $\rho =2800$ kg/m$^3$, Eq. (34) requires $K_ck_r<2\rho v_s^2<5.6\times 10^4$ Pa, which yields $K_c<10^{-6}$. Equation (34) with (33) also requires

$$\frac{4\eta }{a_0^2\rho }< 2\pi f_i,$$

(35)

$$L^2< \frac{24v_s^2}{(2\pi f_i)^2+\Gamma ^2}<\frac{24v_s^2}{(2\pi f_i)^2}.$$

(36)

The relationships (35), (36), and (27) along with the magma properties (Table 1) require that $a_0$ should be larger than the order of 0.01, and $b_0$ and L should be smaller than several meters. It means that we cannot assume an extremely thin channel to allow $K_c$ to be small.

On the other hand, $\Gamma \ll e$ for hydrothermal fluid. Then, Eq. (31) becomes and (32) become

$$K_ck_r\sim [(2\pi f_i)^2-2e\Gamma ]\frac{\rho L^2}{12}<\rho v_s^2\frac{e}{\Gamma }-4e^2\frac{\rho L^2}{12}$$

(37)

The inequality relation of (37) is transformed with (33) to

$$\frac{\rho L^2}{12}<\frac{\rho v_s^2}{(4\pi f_i)^2+4e^2}\frac{e}{\Gamma }<\frac{\rho v_s^2}{4e\Gamma }\sim \frac{\rho ^2a_0^3v_s}{4\eta C_d}.$$

(38)

Using the relationship (38) and (37), one obtains

$$K_ck_r<(2\pi f_i)^2\frac{\rho ^2a_0^3v_s}{4\eta C_d}\sim 10^4 v_s,$$

(39)

using the parameter values in Table 2 and assuming $a_0<0.01$ m. The conditions that generate tremors comparable with the observations yield $v_s<10$ m/s. though $v_s$ of the hydrothermal fluid can be larger. Then, we estimate $K_c\le 10^{-6}$.