Plume height, volume, and classification of explosive volcanic eruptions based on the Weibull function

Abstract

The Weibull distribution between volume and square root of isopach area has been recently introduced for determining volume of tephra deposits, which is crucial to the assessment of the magnitude and hazards of explosive volcanoes. We show how the decay of the size of the largest lithics with the square root of isopleth area can also be well described using a Weibull function and how plume height correlates strongly with corresponding Weibull parameters. Variations of median grain size (Mdϕ) values with square root of area of the associated contours can be, similarly, well fitted with a Weibull function. Weibull parameters, derived for both the thinning of tephra deposits and the decrease of grain size (both maximum lithic diameter and Mdϕ), with a proxy for the distance from vent (e.g., square root of isoline areas) can be combined to classify the style of explosive volcanic eruptions. Accounting for the uncertainty in the derivation of eruptive parameters (e.g., plume height and volume of tephra deposits) is crucial to any classification of eruptive style and hazard assessment. Considering a typical uncertainty of 20 % for the determination of plume height, a new eruption classification scheme based on selected Weibull parameters is proposed. Ultraplinian, Plinian, Subplinian, and small–moderate explosive eruptions are defined on the ground of plume height and mass eruption rate. Overall, the Weibull fitting represents a versatile and reliable strategy for the estimation of both the volume of tephra deposits and the height of volcanic plumes and for the classification of eruptive style. Nonetheless, due to the typically large uncertainties (mainly due to availability of data, compilation of isopach and isopleth maps, and discrepancies from empirical best fits), plume height, volume, and magnitude of explosive eruptions cannot be considered as absolute values, regardless of the technique used. It is important that various empirical and analytical methods are applied in order to assess such an uncertainty.

Appendices

Appendix 1

According to the model of Bursik et al. (1992), based on the assumptions that i) the atmosphere in which the eruptive plume develops is continuously stratified; ii) the wind field at the spreading current level is constant; iii) the volume flux in the spreading current is constant with distance (i.e., there is no air entrainment within the spreading current); iv) particles are vertically well mixed by turbulence in the spreading current; v) the concentration of particles in the cross-wind direction of the spreading current has a Gaussian distribution (inherited from the eruption column); vi) particles sediment from the bottom of the spreading current where turbulence diminishes and the vertical velocity is negligible, the total mass of particles, M_i (kg), of a given size fraction (having a terminal v_i) carried by the spreading current beyond a certain distance x is:

$$ {M}_i\left(x;{v}_i\right)={M}_0\left({v}_i\right) \exp \left(-{\displaystyle {\int}_{x_{\mathrm{O}}}^x\frac{v_iw}{Q} dx}\right) $$

(A.1)

where M ₀ (v _i ) is the initial mass of particles injected into the current at H_b having terminal velocity v _i , w is the maximum cross-wind width of the current at the source, Q is the volumetric flow rate into the current at the neutral buoyancy level, and x₀ is the plume-corner position. For sake of simplicity, in the derivation, we will not consider effects of wind and assume that the distance x is directly proportional to the square root of the isopach areas. Therefore, considering the entire range of particle terminal velocities, the total mass carried by the spreading current beyond a certain distance x is:

$$ {M}_{Tot}(x)={\displaystyle {\int}_{v_{min}}^{v_{max}}{M}_i\left(x;{v}_i\right)d{v}_i={\displaystyle {\int}_{v_{min}}^{v_{max}}{M}_0\left({v}_i\right) \exp \left(-{\displaystyle {\int}_{x_{\mathrm{O}}}^x\frac{v_iw}{Q} dx}\right)d{v}_i}} $$

(A.2)

According to the first mean value theorem for integration there exist v _∗ ∈ (v _min ,v _max ) such that:

$$ {M}_{Tot}(x)= \exp \left(-{\displaystyle {\int}_{x_0}^x\frac{v_{\ast }w}{Q} dx}\right){\displaystyle {\int}_{v_{min}}^{v_{max}}{M}_0\left({v}_i\right)d{v}_i=}{M}_{0, tot} \exp \left(-{\displaystyle {\int}_{x_0}^x\frac{v_{\ast }w}{Q} dx}\right) $$

(A.3)

Physically v _* would represent an effective mean terminal velocity of the mixture particles up to the distance x. If we make the general assumption that v _* w/Q follows a power law with the distance such that:

$$ {v}_{\ast }w/Q=k{x}^m $$

(A.4)

for a generic positive m (generalizing the assumptions made by Bursik et al. 1992), we have:

$$ {M}_{Tot}(x)={M}_{0, tot} \exp \left[-\frac{k}{m+1}\left({x}^{m+1}-{x}_0^{m+1}\right)\right]\equiv {M}_{0,\mathrm{tot}} \exp \left[-C\left({x}^n-{x}_0^n\right)\right] $$

(A.5)

with $C=\frac{k}{m+1}\mathrm{and}\kern0.5em n=m+1$ Hence the mass accumulated on the ground till the distance x can be written as:

$$ {M}_G(x)={M}_{0, tot}\left\{1- \exp \left[-C\left({x}^n-{x}_0^n\right)\right]\right\}={\rho}_{dep}\;{V}_{0, tot}\left\{1- \exp \left[-C\left({x}^n-{x}_0^n\right)\right]\right\} $$

(A.6)

that at distances where x/x₀ ≫ 1 is formally equivalent to the Weibull distribution empirically proposed by Bonadonna and Costa (2012):

$$ {M}_G(x)={\rho}_{dep}\;V(x)={\rho}_{dep}\;{V}_{0, tot}\left\{1- \exp \left[-{\left(x/\lambda \right)}^n\right]\right\} $$

(A.7)

with C = (1/λ)ⁿ Note that the same derivation can be made for each given size fraction having a terminal v _i , that implies that mass distribution of each particle accumulated on the ground class follows a Weibull distribution.

Appendix 2

Table 4 Weibull parameters associated with the eruptions considered in our study that are not reported in Table 1 of main text. θ _th, θ _ML, and θ _Mdϕ are expressed in centimeter, while n _th, n _ML, and n _Mdϕ are dimensionless. References are in caption of Table 1