Scaling laws for extremely strong thermals

A rich variety of scaling laws for the evolution of convective buoyant thermals generated by an explosive release of energy has been established. Such events may correspond to a blast, spark, or volcano eruption. It was found that an occurrence of a particular scaling law depends on the interplay of many factors, viz., (1) amount of energy released in the environment, (2) time since the release, and (3) spatial scale of the release domain. The analytical treatment involves solutions of coupled equations for mass, momentum and buoyancy (heat) conservation in the Boussinesq and non-Boussinesq approximation. A model for the entrainment flow that accounts for a strong thermal flux has been proposed. For the limiting case of a weak thermal and the Boussinesq approximation (low density contrast between the buoyant thermal and the ambient environment) the celebrated Batchelor, Morton, and Turner scalings are recovered. Results have been favorably compared with limited data available in the literature.

Figure 1

A rising convective thermal after a nuclear explosion. Photo courtesy of the Los Alamos National Laboratory, U.S. Department of Energy [15].

Figure 2

Temporal dependency of the radius (a), (c) and elevation (b), (d) of a buoyant thermal in a log-log plot. The two top and two bottom graphs assumed an initial density contrast of $K_0\ll 1$ and $K_0\gg 1$, respectively. The lines on the plot are referred to also by numbers in the main text; see the legend.

Figure 3

Temporal dependency of power law exponents for elevation ($\gamma$) and radius ($\delta$) of a buoyant thermal in a log-log plot for weak and strong thermal: (a) $K_0\ll 1$, (b) $K_0\gg 1$ [see Eq. (1)].

Figure 4

Numerical simulation of the dependency of radius (a) and elevation (b) of the buoyant thermal on the total buoyancy. Two regions of the initial density contrast, $K_0\ll 1$ and $K_0\gg 1$, for short and large timescales are shown in a log-log plot.

Figure 5

Temporal evolution of the radius (a) and elevation (b) of a buoyant thermal in a log-log plot obtained by a simulation [35] and theoretical predictions, Eq. (1) and Table 1. The lines on the plot are referred to also by numbers in the main text; see the legend.

Figure 6

Transient behavior of the elevation of a buoyant thermal in a log-log plot originating from an atmospheric nuclear test [42] and theoretical predictions, Eq. (1) and Table 1. The lines on the plot are referred to also by numbers in the main text; see the legend.