Tsunamis generated by pyroclastic flows: experimental insights into the effect of the bulk flow density

Abstract

For a better assessment of hazards related to tsunamis triggered by pyroclastic flows entering water, it is crucial to know and quantify the contribution of the physical parameters involved in the generation of waves. For this purpose, we investigate experimentally the effect of pyroclastic flow density on tsunami generation by considering variably fluidized granular flows denser or less dense than water, referred to as heavy and light granular flows, respectively, by varying the particle density. Qualitative observations show that differences in bulk flow density mainly affect the propagation of granular flows underwater. In contrast, the bulk flow density has little effect on the amplitude of the leading and largest wave. In fact, the wave amplitude is initially similar to the local water depth along the inclined plane, and then reaches a maximum value that depends mainly on the other flow parameters (i.e., velocity, thickness, volume of flows). Far from the shoreline, we provide evidence of the bulk flow density effect on the wave amplitude, while other characteristics of the leading wave remain broadly unaffected in the range of parameters considered. Finally, a main difference on the tsunami generation between light and heavy granular flows is related to the energy distribution between the leading largest wave and the wave train, which is attributed to different modes of interaction of the two flow types with the water. For tsunami hazard assessment, our study suggests that the contribution of the bulk flow density on tsunami generation has a second-order effect compared to other flow parameters.

Appendix. Other far-field features of the leading wave

We investigated also the effect of the bulk flow density on other characteristics of the leading wave, namely its period T, the propagation velocity c, and the asymmetry coefficient $A_s$ of the first positive crest (Fig. 8). As for the far-field amplitude $A_{ff}$ presented in the main text, the values of the period T and the asymmetry coefficient $A_s$ correspond here to measurements made at $x=2.4\,$ m (i.e., $x/H_o=9$) from the shoreline, while the error bars indicate the variation from $x=2$ to $2.8\,$ m (i.e., from $x/H_o=7.5$ to 10.5).

Fig. 8

(a) Scaled period $(g/H_o)^{1/2}T$ (inset: scaled velocity $c/(gH_o)^{1/2}$) and (b) asymmetry coefficient $A_s$ of the leading wave versus the degree of fluidization $\beta$. Both T and $A_s$ are estimated at $x=2.4\,$ m (i.e., $x/H_o=9$) from the shoreline, while error bars indicate the variation from $x=2$ to $2.8\,$ m (i.e., from $x/H_o=7.5$ to 10.5). (—) $(g/H_o)^{1/2}T= 6$ (inset: $(c/(gH_o)^{1/2}=1$); $A_s=0.17$ (Average values of data). The bottom inset of Fig. 8b shows $\eta /A_{ff}$ versus t/T (here, $t=0$ is the time at which $A_{ff}$ is measured) in the far-field after a light (blue, particles P143) or heavy (dark, particles G66) granular flow ($\beta =1.3$, $\theta =30^\circ$) enters water. Dashed lines (visible at small and large t/T) correspond to Eq. 5 with $\alpha ^-\sim 1.5$ (blue) and 1.2 (black), for $t/T<0$, and $\alpha ^+\sim 2.2$ (blue) and 1.5 (black), for $t/T>0$. Symbols: ($\square$) G66; ( ) G159; ( ) P143; ( ) P580, with $\theta = 15^\circ$ (opened) and $\theta =30^\circ$ (closed)

Figure 8a shows that there are no significant variations on the scaled period $(g/H_o)^{1/2}T=6$, and propagation velocity $c/(gH_o)^{1/2}=1$, of the leading wave far away from the shoreline, in the range of parameters considered. To quantify the wave asymmetry coefficient As, the method used by Bullard et al. (2019b) is considered, which consists of fitting the temporal evolution of the wave height profile $\eta (x,t)$ over the period T of the leading wave, in a stationary location (i.e., $x=0$ and $t=0$ defined as the position and the time at which the far-field amplitude $A_{ff}$ is measured), as

$$\begin{aligned} \eta (x,t)=A_{ff}\;\text {sech}^2\left[ \alpha ^{-/+}\sqrt{\frac{3A_{ff}}{4H_o^3}}\left( -ct\right) \right] , \end{aligned}$$

(5)

where the two fitting parameters $\alpha ^-$ and $\alpha ^+$ take a different value on either side of $t=0$, and the wave velocity is set to $c=(gH_o)^{1/2}$ according to our results (inset of Fig. 8a). Recall also that Eq. 5 is derived from the stationary solitary wave solution that describes a perfectly symmetrical wave with $\alpha ^-=\alpha ^+=1$ (Le Méhauté 1976). Then, the asymmetry coefficient is estimated as $As=(9/5\pi )\ln (\alpha ^+/\alpha ^-)$ using the empirical relation proposed by Bullard et al. (2019b). In any case, the main purpose of the asymmetry coefficient $A_s$ is to quantify the general shape of waves with three distinct types of asymmetry, namely asymmetry with steeper and flatter front faces for negative and positive values of As, respectively, and symmetrical shape with $As=0$. The inset of Fig. 8b reveals a slight asymmetry with the leading edge of the wave that is generally slightly steeper than the trailing edge, while fitting curves (dashed lines) are barely observed due to an excellent agreement with the data. Again, no significant variation in As is reported when varying the bulk flow density (blue vs. black symbols), as well as the degree of fluidization $\beta$, which can therefore be characterized by a constant value of $As=0.17$ (solid line, Fig. 8b).