# Introduction

This is an excerpt of the modeling part of the paper, (Hong 2016). The abstract of the paper is included below for interested readers.

Strong ground motions induce large dynamic stress changes that may disturb the magma chamber of a volcano, thus accelerating the volcanic activity. An underground nuclear explosion test near an active volcano constitutes a direct treat to the volcano. This study examined the dynamic stress changes of the magma chamber of Baekdusan (Changbaishan) that can be induced by hypothetical North Korean nuclear explosions. Seismic waveforms for hypothetical underground nuclear explosions at North Korean test site were calculated by using an empirical Green’s function approach based on a source-spectral model of a nuclear explosion; such a technique is efficient for regions containing poorly constrained velocity structures. The peak ground motions around the volcano were estimated from empirical strong-motion attenuation curves. A hypothetical M7.0 North Korean underground nuclear explosion may produce peak ground accelerations of 0.1684 m/s2 in the horizontal direction and 0.0917 m/s2 in the vertical direction around the volcano, inducing peak dynamic stress change of 67 kPa on the volcano surface and  120 kPa in the spherical magma chamber. North Korean underground nuclear explosions with magnitudes of 5.0–7.6 may induce overpressure in the magma chamber of several tens to hundreds of kilopascals

# Verifying dynamic stress changes using a numerical models

## Model Setup

We compute dynamic stress changes induced in the magma chamber by a nuclear explosion using PyLith, “a finite-element code for dynamic and quasistatic simulations of crustal deformation, primarily earthquakes and volcanoes”(Aagaard 2013, Aagaard 2013a).

A magma chamber has been detected beneath Baekdu from low shear wave velocity in the depth range of 10-16 km (ZHANG 2002). Based on this observation, we assume that the magma chamber is a sphere of a 3 km radius and is contained in a 12 $$\times$$ 15 $$\times$$ 15 km rectangular box representing the crust (Fig. \ref{fig:planewave_domain}a) We approximate the waves propagating through the magma chamber as a pulse of plane wave. Our goal to estimate only the peak stress change justifies this simplification. The plane wave is generated by the time-dependent displacement uniformly applied on one boundary, which is given as $u(t) = u_{0}\frac{t}{\tau}\exp\left(-\frac{t}{\tau}\right),$ where $$u$$ is the boundary-normal component of displacement, $$t$$ is time and $$u_{0}$$ and $$\tau$$ are set to be 0.40 m and 1.5 sec. These values are chosen such that the peak ground velocity (PGV) becomes 0.004 m/s at the moment when the plane wave first reaches the magma chamber (0.8 sec in Fig. \ref{fig:planewave_domain}b). This value of PGV is acquired from the PGV attenuation equation constructed from the attenuation pattern around Baekdu. The opposite side of the box has an absorbing boundary condition while the sides parallel to the wave’s propagation direction are assumed to be reflecting boundaries (i.e., the boundary-normal displacement component is zero.) The stresses are initially zero everywhere in the domain and the gravitational body force is not included. These conditions are suitable to this study’s purpose of estimating only dynamic stress changes.

To represent various possible states of the magma chamber during an explosive volcanic eruption (e.g., Hong 2014), we assigned Vp/Vs ratios between 1.65 and 1.95. Vp, Vs, Vp/Vs as well as bulk and shear modulus (K and G) and Poisson’s ratio ($$\nu$$) are listed in Table \ref{tab:plane_wave_magma_chamber_parameters}. The uniform density of the magma chamber is fixed at 2500 kg/m$$^{3}$$.

\label{tab:plane_wave_magma_chamber_parameters}

 Model Vp/Vs Vp (km/s) Vs (km/s) K (GPa) G (GPa) $$\lambda$$ (GPa) E (GPa) $$\nu$$ 1 1.65 5.90 3.58 44.3 32.0 22.9 77.4 0.209 2 1.75 5.90 3.37 49.2 28.4 30.2 71.5 0.258 3 1.85 5.90 3.19 53.1 25.4 36.1 65.7 0.293 4 1.95 5.90 3.03 56.4 23.0 41.1 60.8 0.321

The surrounding crust has Vp and Vs equal to 6.17 km/s and 3.57 km/s. The crustal density is 2700 kg/m$$^{3}$$.

## Results

The process of wave propagation in the model with Vp/Vs equal to 1.85 is shown in Fig. \ref{fig:planewave_domain}b. The wave composed of compressional and dilatational pulses generates ground motion with velocity magnitude of about 0.007 m/s and induce transient stress changes in the magma chamber. The magnitude of $$\sigma_{xx}$$ reaches the maximum, 118 kPa, at 0.8 sec and stays at around 100-110 kPa until the wave exits the chamber.

The temporal variation of induced dynamic stress changes in the chamber can be illustrated by plotting the maximum of $$\sigma_{xx}$$ ($$\sigma_{xx,max}$$) and pressure ($$p_{max}$$) in the chamber over time. Such a plot for the magma chamber with Vp/Vs equal to 1.85 is show in in Fig. \ref{fig:planewave_stress_time}. The peak values of $$\sigma_{xx,max}$$ and $$p_{max}$$ are $$-$$118 kPa and $$-$$71 kPa, respectively. Peak values of $$\sigma_{xx,max}$$ and $$p_{max}$$ are plotted against Vp/Vs values assumed for the chamber. $$|p_{max}|$$ shows a positive correlation with Vp/Vs (Fig. \ref{fig:planewave_stress_VpVs}), increasing from 64 kPa at Vp/Vs = 1.65 to 75 kPa at Vp/Vs = 1.95. $$|\sigma_{xx,max}|$$, in contrast, stays at about 118 kPa, for all the Vp/Vs values because the maximum value is reached where the plane wave first enters the chamber and the traction continuity condition on the chamber boundary requires that the $$\sigma_{xx}$$ there be equal to that of the outside.