I conducted numerical experiments to evaluate the condition for the second nucleation of bubbles when a sudden change in the decompression rate occurs in magmas with preexisting bubbles. I took into account homogeneous nucleation and diffusive bubble growth. Results from numerical experiments with variable number densities and sizes of preexisting bubbles were able to provide the limiting condition for the decompression rate, above which the second nucleation effectively takes place. The limiting condition as a function of number density and size of preexisting bubbles can be divided into two regimes that are controlled by the mass ratio of dissolved volatiles to exsolved volatiles as preexisting bubbles at the initial state. In the case with a realistically high mass ratio, namely, with a relatively small preexisting bubble volume, the detectable second nucleation requires decompression rates that are higher by a factor of about 10 than those that are hypothetically required to form the preexisting bubbles. By consulting a simple scaling argument, we derive the expression for the limiting decompression rate as a function of the number density and size of preexisting bubbles. The graphical display of the expression allows us to evaluate whether the second nucleation will effectively take place for practical cases with a given number density and size of preexisting bubbles. In natural pumice and scoria, the preexisting bubbles may be observed as larger pheno-bubbles that provide buoyancy at the storage regions, which is needed to trigger the eruption. Such pheno-bubbles are isolated in smaller matrix-bubbles that are formed by the rapid decompression that takes place during the ascent of magma to the surface.
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