Effects of debris entrainment and recycling on explosive volcanic eruption jets and columns

Abstract

A multiphase fluid dynamic model is used to explore the effects of entrainment of granular debris into sustained volcanic jets such as those which produce sub-Plinian to Plinian eruption columns. The debris may be sourced from processes such as avalanches from crater walls or from recycling of previously erupted material. The results indicate that debris is not immediately, homogeneously mixed into a jet but instead forms a dense sheath that is dragged upward around the jet margin. While very small volumes of debris relative to the eruptive discharge rate mix progressively into the jet with increasing altitude, the dense sheath can inhibit entrainment of air into the lower portions of the jet, which may explain signs of column instability such as increased stratification in fallout deposits where lithic content increases. As debris volume increases, the dense sheath can collapse from a range of elevations to feed pyroclastic currents. The presence of the sheath of entrained debris contradicts some assumptions such as the top-hat profile for density and velocity that is commonly used in 1-D models. Transitions from fallout-producing buoyant column to collapsing behavior can be related to debris entrainment without any changes in primary eruption parameters such as vent size, exit velocity, or gas content. Boiling-over behavior can also be caused by debris entrainment, including recycling of previously erupted material such as might occur in a crater with restricted outlet. When entrained debris is relatively fine-grained such that it can couple well with the erupting mixture, complex, highly transient overpressured jet processes can occur due to the pinching effect of debris flowing into the base of the jet. Increasingly coarse debris causes collimation of the jet within the sheath of entrained material. The results suggest that accounting for the effects of debris entrainment is likely important for theoretical assessment of many natural eruption sequences and for assessment of hazard scenarios for potential sub-Plinian to Plinian activity.

Appendix

Modeling approach

The governing equations are conservation of mass, momentum, and energy for a carrier gas phase and for one or two fields of dispersed particles. The equations are written in an Eulerian framework for both the gas and the particle fields, which are treated as overlapping continua with volume fractions within a control volume that sum to unity. Gas and particle fields interact with each other through momentum transfer (drag) and heat transfer. More details can be found in Benyahia et al. (2012), Sweeney and Valentine (2017), and Valentine and Sweeney (2018). The governing equations are as follows (nomenclature in Table

Table 3 Notation for Eqs. 1–6

3):

$$\frac{\partial }{\partial t}\left({\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}\right)+\frac{\partial }{\partial {x}_{\mathrm{j}}}\left({\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{U}_{\mathrm{gj}}\right)=0 .$$

(1)

$$\frac{\partial }{\partial t}\left({\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}\right)+\frac{\partial }{\partial {x}_{\mathrm{j}}}\left({\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}{U}_{\mathrm{mj}}\right)=0.$$

(2)

$$\frac{\partial }{\partial t}\left({\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{U}_{\mathrm{gi}}\right)+\frac{\partial }{\partial {x}_{\mathrm{j}}}\left({\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{U}_{\mathrm{gj}}{U}_{\mathrm{gi}}\right)=-{\epsilon }_{\mathrm{g}}\frac{\partial {P}_{\mathrm{g}}}{\partial {\mathrm{x}}_{\mathrm{i}}}+\frac{\partial {\tau }_{\mathrm{gij}}}{\partial {x}_{\mathrm{j}}}-\sum_{\mathrm{m}=1}^{M}{I}_{\mathrm{gmi}}+{{\epsilon }_{\mathrm{g}}\rho }_{\mathrm{g}}{g}_{\mathrm{i}}.$$

(3)

$$\frac{\partial }{\partial t}\left({\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}{U}_{\mathrm{mi}}\right)+\frac{\partial }{\partial {x}_{\mathrm{j}}}\left({\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}{U}_{\mathrm{mj}}{U}_{\mathrm{mi}}\right)=-{\epsilon }_{\mathrm{m}}\frac{\partial {P}_{\mathrm{g}}}{\partial {x}_{\mathrm{i}}}+\frac{\partial {\tau }_{\mathrm{mij}}}{\partial {x}_{\mathrm{j}}}+{I}_{\mathrm{gmi}}-\sum_{l=1}^{M}{I}_{\mathrm{mli}}+{\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}{g}_{\mathrm{i}}.$$

(4)

$${\epsilon }_{\mathrm{g}}{\rho }_{\mathrm{g}}{C}_{\mathrm{pg}}\left[\frac{\partial {T}_{\mathrm{g}}}{\partial t}+{U}_{\mathrm{gi}}\frac{\partial {T}_{\mathrm{g}}}{\partial {x}_{\mathrm{j}}}\right]=-\frac{\partial {q}_{\mathrm{gj}}}{\partial {x}_{\mathrm{j}}}+\sum_{m=1}^{M}{\gamma }_{\mathrm{gm}}\left({T}_{\mathrm{m}}-{T}_{\mathrm{g}}\right) +{\gamma }_{\mathrm{Rg}}\left({T}_{\mathrm{Rg}}^{4}-{T}_{\mathrm{g}}^{4}\right).$$

(5)

$${\epsilon }_{\mathrm{m}}{\rho }_{\mathrm{m}}{C}_{\mathrm{pm}}\left[\frac{\partial {T}_{\mathrm{m}}}{\partial t}+{U}_{\mathrm{mj}}\frac{\partial {T}_{\mathrm{m}}}{\partial {x}_{\mathrm{j}}}\right]=-\frac{\partial {q}_{\mathrm{mj}}}{\partial {x}_{\mathrm{j}}}-{\gamma }_{\mathrm{gm}}\left({T}_{\mathrm{m}}-{T}_{\mathrm{g}}\right) +{\gamma }_{\mathrm{Rm}}\left({T}_{\mathrm{Rm}}^{4}-{T}_{\mathrm{m}}^{4}\right).$$

(6)

Constitutive models that describe interphase heat and momentum transfer, and intraphase heat transfer and stress, are found in Syamlal et al. (1993), Syamlal and Pannala (2011), Benyahia et al. (2012), and Sweeney and Valentine (2015, 2017). Valentine and Sweeney (2018) include information related to model validation. In this and similar gas-particle multiphase approaches, stresses within the particle phase (Eq. 4) are modeled as a function of the so-called granular temperature (a.k.a. granular energy), which is a measure of the fluctuation of particle velocities. Here, I use an algebraic approximation for granular temperature rather than a full conservation equation (see Benyahia et al 2012; Breard et al. 2019; Valentine 2020). Additional volcanological applications of the MFIX code can be found in Dartevelle (2004), Dartevelle et al. (2004), Dufek and Bergantz (2007a, b), Dufek and Manga (2008), Dufek et al. 2009), and Breard et al. (2018, 2019).