The geophysical detection of magma bodies and the estimation of the dimensions, physical properties, and the volume fraction of each phase is required to improve the forecasting of volcanic hazards and understanding of transcrustal magmatism. We develop an analytical model to calculate the speed of sound in a magma consisting of crystals and gas bubbles suspended in a viscous melt. We apply our model to calculate the speed of sound as a function of the temperature in three magmas with different chemical compositions, representative of the diversity that may be encountered in arc magmatism. The model employs the coupled phase theory that explicitly accounts for the exchange of momentum and heat between the phases. We show that the speed of sound varies non-linearly with the frequency of an acoustic perturbation between two theoretical bounds (Fig. 1). The dispersion of the sound in a magma results from the exchange of heat between the melt and dispersed phases that affects the magnitude of their thermal expansions. The lower bound of the sound speed occurs at low frequencies where all the constituents can be considered in thermal equilibrium, whereas the upper bound occurs at high frequencies for which the exchange of heat between the phases may be neglected. The presence of gas in a magma produces a sharp decrease of the velocity of compressional waves and generates conditions in which the dispersion of the sound is significant at the frequencies usually considered in geophysics (Fig. 2). Finally, we compare the estimates of our model with the one from published relationships (Fig. 3). Differences are largest at higher frequencies and are <10% for typical magma.
Figure 1: Dispersion curve of a magmatic suspension. The volume fraction in liquid, solids, and gas are ϕl = 0.6, ϕs = 0.35, and ϕg = 0.05, repsectively. The solid black curve indicates the results obtained with our model employing the coupled phase theory. The black, red, and blue dashed lines indicate the isothermal speed of sound, the isentropic speed of sound at thermal equilibrium, and isentropic speed of sound out of thermal equilibrium, respectively. The black and blue vertical doted lines indicate the critical frequencies above which the solid and gas bubbles are not in thermal equilibrium with the surrounding liquid, respectively.
Figure 2: Evolution of the phase assemblage, thermodynamic properties, and speed of sound of the magmas during the cooling simulations perfomed using the software MELTS. [A], [B], and [C] are the phases assemblage computed during the simulation of the cooling of the basalt (A), andesite (B), and dacite (C). [D] evolution of the bulk densities and bulk moduli of the magmas. [E] Evolution of the bulk coefficient of thermal expansion and bulk heat capacity at constant pressure as a function of temperature. [F] Evolution of the speed of sound in the magmas. The solid, dashed and dashed-doted curves indicate the thermal disequilibrium, thermal equilibrium, and isothermal bounds, respectively. The shaded area corresponds to the range of velocity that may be computed at different frequencies. ρ* is the bulk density of the magma, K* is the average bulk modulus of the magma, α* is the bulk coefficient of thermal expansion. CP* is the bulk specific heat capacity at constant pressure, c is the speed of sound.
Figure 3: [A] Comparison of estimations of sound speed in a suspension of gas bubbles in water for 10-3≤f≤103 Hz. [B] Comparison of the speed of sound computed with our model and with Eq. (41) in Commander & Prosperetti (1988) for 103≤f≤106. The properties of the suspension are the same as in (A). The sharp increases in sound speed predicted by Commander & Prosperetti (1988) occurs at the resonance frequency of the gas bubbles, which is not accounted in our model.
Related publication: Carrara, A., Lesage, P., Burgisser, A., Annen, C., Bergantz, G.W., 2020. The dispersive velocity of compressional waves in magmatic suspensions, submitted to Geophysical Journal international (In revision)
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