Water loss calculations:
In order to use water and grain size dependent remote sensing data (ESPAT) transects on DMDs to investigate the presence and variation of water in DMD source material, the water loss of clasts of different size and with different durations of time above blocking temperature must be calculated. In this manner, an eruption of a certain grain size distribution and initial water content can be modeled in order to calculate a numerical ESPAT transect that can be resolved with that of the DMD in order to find the initial water content that provides the best fit. The obstacle presented is that water loss is not only a function of clast size and total time spent above blocking temperature, but also previously unconstrained evaporation and cooling rates, as well as the initial concentration of water.
To constrain these evaporation and cooling rates and calculate water loss for clasts of different sizes experiencing different cooling histories, we expand the work of Saal et al. [2008]. They introduced an innovating new method in which the composition of pyroclastic source material could be investigated by modeling diffusion profiles of volatile species F, Cl, S, and H2O recorded in Delano’s green glasses. These profiles within a single bead are a function of the clast size, evaporative and cooling environments, total time spent above blocking temperature, and initial concentration of the given volatile species. Following the lowering of the detection threshold for H2O and CO2 by virtue of secondary ion mass spectrometry (SIMS), Saal et al. [2008] were able to detect for the first time measurable amounts of H2O and CO2, as well as Cl, F, and S, in a single glass bead (Delano’s green glass bead #5) which decreased from core to rim within the bead. Our expansion on the work of Saal et al. [2008] shows that the solution to the concentration profiles in the beads is non-unique and can be fitted satisfyingly for a broad range of total time above the blocking temperature because of trade-offs between cooling time, cooling and evaporation rates (see two examples of simulations for 10 and 300 seconds of evaporation in figures s1 -4 ). However, we find that the profiles can only be matched across a narrow range of evaporation and cooling rates (figures s3 -4 ) which provides constraints on these two parameters for the eruption that formed the deposits analyzed by Saal and coworkers. Both the cooling and evaporation rates are controlled to a first order by the environmental conditions sampled by the eruption. The DMDs investigated are presumed to be picritic in origin and emplaced through fire-fountaining eruptions, for which the environmental conditions are expected to share similarities with those of Delano’s glasses. From these constraints, we can calculate extreme water loss case scenarios using different combinations of evaporation rate and cooling rate determined from the fits to the data of Saal et al. [2008].
To calculate the diffusion profiles of the different chemical species, we solve for the diffusion of water in a sphere with radiation at the surface, described using Eq. (s1) [Crank , 1975]:
\(\frac{\partial C}{\partial t}=\ \frac{D(T)}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial C}{\partial r})\)(s1)
where C is the concentration of the given volatile species (F, Cl, S, or H2O), D is the temperature, and therefore time, dependent diffusivity, t is time spent above blocking temperature, and r is the distance along the radius of the clast. Using an implicit, centered-space, finite-difference approximation, Eq. (s1) can be written as:
\(\frac{C_{k}^{n+1}-\ C_{k}^{n}}{t}=\ \frac{2D}{r}\left[\frac{C_{k}^{n+1}-\ C_{k-1}^{n+1}}{r}\right]+D\left[\frac{C_{k+1}^{n+1}-\ 2C_{k}^{n+1}+C_{k-1}^{n+1}}{{(r)}^{2}}\right]\)(s2)
where k is used to index position within the clast along the radius and n is used to index time in cooling history.
The boundary and initial conditions for this problem are:
\(\frac{\partial C}{\partial r}|_{r=R}=\ \frac{-\beta}{D(T)}\left(C-\ C_{o}\right)\)(s3)
\(\frac{\partial C}{\partial r}|_{r=0}=\ 0\) (s4)
\(C\left(r,0\right)=C_{\text{ini}}\) , (s5)
with \(\beta\) being the species dependent evaporation rate, Co being the initial concentration of the volatile species at the surface of the clast, here assumed to be negligible, and Cini being the initial concentration of a given volatile species within the clast.
The inputs of forward diffusion calculations are the total time of diffusion, ttotal, the radius of the clast, R, the evaporation rate β, and the rate of cooling [Crank , 1975]. These calculations are made simultaneously for all four species and we use a Markov Chain Monte-Carlo approach to find the optimal set of evaporation and cooling rates that satisfy these profiles. The diffusion coefficients and activation energies used in the inversion are reported in Watson and Bender [1980] for Cl, Zhang and Stolper[1991] for H2O, and Dingwell and Scarfe[1984] for F. For S, we use the assumption made in Saal et al. [2008] that the sulfur partitions primarily as S2 at low f O2 [Baker and Rutherford , 1996] and that the activation energy should be similar to that of O2- reported in Wendlandt[1991]. Finally, the diffusion coefficient for S is taken to be that reported in Saal et al. [2008].