## 1. Introduction

[2] When a sound field impinges on a fluid, a bubble may grow due to a mass transport process called rectified diffusion. During the expansion phase of an oscillation, the solubility of volatiles decreases, and in the absence of any other diffusive process, volatiles exsolve into the bubble. During contraction, the opposite effect occurs.

[3] Bubbles take in more volatiles during expansion than they discharge during contraction [e.g., *Hsieh and Plesset*, 1961; *Eller and Flynn*, 1965]. The two most significant reasons for the asymmetry are: (1) the interface is larger during expansion than during contraction and (2) radial bubble expansion tangentially stretches the diffusion layer and sharpens the radial gradient of the volatile concentration in the diffusion layer, so that the volatile flux into the bubble is enhanced. This pumping of volatiles into the bubble is known as rectified diffusion.

[4] *Sturtevant et al.* [1996] and *Brodsky et al.* [1998] pointed out that rectified diffusion might explain volcanic eruptions triggered by distant earthquakes like those observed by *Linde and Sacks* [1998]. The earthquakes generate seismic waves and the dilatational strain of the waves transfers volatiles from magma into bubbles by rectified diffusion. It was argued that this volatile transfer increases the pressure of the system if bubble expansion is restricted in a confined liquid. *Brodsky et al.* [1998] recognized that the solubility of the volatiles must increase with increasing pressure and proposed that there existed a limit on the pressure increase attainable by rectified diffusion. They used the classic solution of *Hsieh and Plesset* [1961] to calculate the mass transfer from rectified diffusion and then solved for the limiting pressure of the system. They concluded that the maximum attainable pressure was determined by the degree of supersaturation of the system.

[5] However, the problem of the limiting pressure cannot be decoupled from the rectified diffusion solution in a confined system. Since the most important effect in generating a mass flux during a strain cycle is the change in solubility of volatiles, then changes in pressure (and hence solubility) must be fully coupled in computing the rate of volatile transfer via rectified diffusion. The *Hsieh and Plesset* [1961] solution assumed that the pressure, bubble radius, and volatile concentration all oscillate around the fixed values. The transient problem of mass transport by rectified diffusion has been solved for the bubble radius increasing with time under a constant mean pressure [*Skinner*, 1972; *Fyrillas and Szeri*, 1994]. However, to the best of our knowledge, no solution has yet been presented for the geophysically important transient problem in which the average pressure increases with time while the average bubble radius does not change. In this paper we derive such a solution to show that the increase in background pressure limits the rate of mass transfer which in turn limits the total pressure rise. The mathematical technique in this paper is based on *Fyrillas and Szeri* [1994] (hereinafter referred to as FS).

[6] We first use the advective-diffusion equation in spherical symmetry to describe the transport of gas with the appropriate time-dependent boundary conditions. Then we transform the equation into a Lagrangian reference frame and split the system into smooth and oscillating parts. We then recognize that the smooth part is a diffusion problem with a small perturbation of the diffusion layer due to the oscillatory bubble motion. We derive a numerically tractable solution that describes the long-term pressure rise as an expansion in terms of the amplitude of the imposed strain. We ultimately show that the pressure rise from rectified diffusion is limited by the imposed strain regardless of the degree of supersaturation of the system.