Fig. 5

Definition of the transitional level between gas thrust and buoyant regimes and pulse velocity derived model (PVDM) used to estimate the mass flux Q of an individual pulse: a The onset of transition t _I is marked by radial expansion, increased occurrence of vortices, and starting overturns of the pulse margins. The location of the maximum pulse width d _trans(t _I) determines the lower boundary of the transitional level. The upper boundary is defined by the tip of a pulse at t _II, when due to air entrainment, d _trans(t _II) is at maximum. The transitional uplift velocity v _trans is measured between t _I and t _II. The PVDM approach approximates the volumetric flux as that of an “ash package” through a cylindrical cross section with d _trans(t _I) being its diameter and v _trans being its flow velocity, persisting for a duration τ. This characteristic time parameter is determined by measuring the temporal distance between t _I and t _III, i.e., the moment when the pulse diameter decreased to d _trans(t _I)/e^0.5. b By multiplying the volumetric flow rate with the density of the pulse (taken to be equal to that of the surrounding air at the transitional level), it is possible to calculate the peak mass flux Q(t _I). Two cases are calculated, constraining the unknown flow rate evolution (center). While for a minimum estimate, an exponential temporal decrease of the flux is assumed (left), a maximum flow rate assumes a step function (right) which would correspond to a cylinder shaped ash package, as illustrated in a