Sediment waves and the gravitational stability of volcanic jets

Abstract

It is increasingly recognized that the gravitational stability of explosive eruption columns is governed by complex ash-pumice-gas (multiphase) interactions as well as the mechanics of turbulent entrainment in the lower momentum-driven (fountain) and upper buoyancy-driven (plume) regions of typical Plinian eruption columns (volcanic jets). We use analog experiments on relatively dense mono- and bi-disperse particle-freshwater and particle-saltwater jets injected into a linearly stratified saltwater layer to revisit, characterize and understand how transitions among Buoyant Plume (BP), Total Collapse (TC) and Partial Collapse (PC) multiphase jet regimes in a traditional source strength (−Ri₀) - particle concentration (ϕ₀) parameter space are modified by particle inertial effects expressed through a Stokes number (St) and particle buoyancy effects expressed through a Sedimentation number (Σ). We show that “coarse particles” (1.4 ≤St ≤ 6.0) enhance entrainment and modify significantly published conditions favoring BP and TC jets. Furthermore, the transition between BP and TC regimes occurs smoothly over a PC regime that extends a large $-\text {R}\text {i}_{0} \leftrightarrow \phi _{0}$ parameter space. Large volume annular sedimentation waves excited periodically at the fountain-plume transition height and the cloud level of neutral buoyancy (LNB) in PC and TC regimes lead to “phoenix clouds” spreading at multiple altitudes and build terraced deposits. Applied to volcanic jets, we develop a new set of conceptual models for jets in the BP, TC and PC regimes that make explicit links among source parameters, column heights, sedimentation wave properties, cloud structures and deposit architectures. These conceptual models make predictions for cloud structures and deposit characteristics that agree with observations made for well-studied historic and pre-historic eruptions and explain the origin of common but enigmatic features of proximal explosive eruption deposits, such as alternating air-fall and pyroclastic flow layering in subaerial deposits and terracing in submarine Catastrophic-Caldera Forming (CCF) eruption deposits. Additionally, our models provide guidance for real-time monitoring of eruption column stability for eruptions undergoing a typical BP→PC→TC regime evolution and predict pyroclastic flows to occur more frequently as columns transition from the PC→TC regime. Our experimental results combined with scaling considerations expressed through a set of new conceptual models provide exciting new pathways for future laboratory-, computer- and field-based studies of explosive eruptions.

Appendices

Appendix A: Mixture bulk density, particle drag factor and Reynolds number

The bulk density of a solid-fluid phase mixture is given by

$$ \rho = (1-\phi)\rho_{f} + \phi\rho_{p} $$

(18)

for a monodispere PSD with ϕ = ϕ_cp or ϕ_fp, particle density ρ_p = ρ_cp or ρ_fp, and for a bidisperse PSD

$$ \rho = (1-\phi_{cp}-\phi_{fp})\rho_{f} + \phi_{cp}\rho_{cp} + \phi_{fp}\rho_{fp} $$

(19)

Eqs. 18 and 19 can be defined at the source with the density of interstitial jet water ρ_f = ρ_w,0, ϕ = ϕ₀, ϕ_cp = ϕ_cp,0 and ϕ_fp = ϕ_fp,0 (Jessop et al. 2016).

The particle drag factor, f_p, which is used to calculate the particle response time in Eq. 13 (particle Stokes number), and Reynolds number, Re_p, are defined as

$$ \text{f}_{\text{p}} = 1+0.15\text{R}\text{e}_{\text{p}}^{0.687}+\frac{0.0175}{1+42,500 \text{R}\text{e}_{\text{p}}^{-1.16}} $$

(20)

$$ \text{R}\text{e}_{\text{p}} = \frac{U_{s}L}{\nu_{w}} $$

(21)

where U_s is the terminal settling velocity of the particle (Burgisser et al. 2005). f_p and Re_p can be defined at the jet source with L = d_p and ν_w is the kinematic viscosity of water.

Appendix B: 1D integral plume model: Effective entrainment parameter and jet conditions above source

One goal of this study is to investigate the effects of two-way and fully coupled particles and interstitial fluid buoyancy on the process of entrainment for the initial transient jet. Following the methods of Morton et al. (1956) and Bloomfield and Kerr (1998) and Carazzo and Jellinek (2012) , we numerically solve a 1D integral plume model with a Boussinesq approximation (density differences only accounted for in buoyancy terms) in which particles are conserved up to the maximum height of the jet, where $u(z)\rightarrow 0$, to determine an effective entrainment parameter value for the initial jet in an experiment. Using Eqs. 3a-3c

to set the initial conditions of the plume model, we solve for their change with height

$$ \begin{array}{@{}rcl@{}} \frac{dQ}{dz} &=& 2\alpha_er(z)u(z) \end{array} $$

(22a)

$$ \begin{array}{@{}rcl@{}} \frac{dM}{dz} &=& g^{\prime}(z)r(z)^2 \end{array} $$

(22b)

$$ \begin{array}{@{}rcl@{}} \frac{dB}{dz} &=& -N(z)^2r(z)^2u(z) \end{array} $$

(22c)

until $u(z)\rightarrow 0$. To account for the change in ϕ₀ with height, we assume conservation of particles where no particles exit the jet mixture during rise (Carazzo and Jellinek 2012)

$$ \ \phi(\text{z}) = \frac{{\phi_{0}}Q_{0}}{Q(z)} $$

(23)

This set of equations is closed with Eq. 1 and the model is initiated with source parameters determined by Eqs. 3a-3c and the source particle volume fraction ϕ₀. We note that this method is limited to modeling the initial jet when it enters the quiescent tank environment and, therefore, cannot model re-entrainment of collapsing material or account for the dynamics of the oscillating overshoot region.

Once an effective entrainment parameter is determined for initial jet rise in each experiment, we model the jet conditions at the observed LNB height of the jet fluid, where $g^{\prime }_{f}\rightarrow 0$, for jets in the BP and PC regimes. In particular, we estimate the momentum flux, M_LNB, buoyancy flux, B_LNB and particle volume fraction, ϕ_LNB, of the mixture. M_LNB and B_LNB are used in Eq. 12 to predict the oscillation frequency of the overshoot fountain, f_OS (see “1 1”), whereas ϕ_LNB is compared with the descent velocity of sediment waves. We note that during the steady-state period of an experiment, this simple modeling approach does not capture mass and momentum exchange between the jet and mixture descending from the overshoot fountain along the jet edge. Additionally, this modeling approach does not capture processes such as entrainment of collapsing mixture.

To further investigate the effect of coarse and fine particles, with 0.2 < St₀ < 6, on the process of entrainment, we use the Kaminski et al. (2005) and Carazzo et al. (2008) entrainment parameterization (variable entrainment model) to predict the maximum rise height of experimental jets

$$ \alpha_{e}(z) = |\text{Ri}(z)|\left( 1-\frac{1}{A}\right)+\frac{1}{2}\frac{d ln A}{dz}+\frac{1}{2}C $$

(24)

where |Ri(z)| is the local Richardson number. A = 1.1 and C = 0.135 and are constant for the model run. Maximum jet heights predicted with the variable entrainment model are compared to measured maximum jet heights during the initial transient phase of the experiment. These results are then used to determine if the variable entrainment model over- or underestimates entrainment for jets with particles.