A three-dimensional numerical simulation of spreading umbrella clouds

During explosive volcanic eruptions, an eruption column buoyantly rises as a turbulent plume, and an umbrella cloud spreads laterally as a gravity current at the neutral buoyancy level. The source conditions of explosive eruptions such as mass discharge rates of magma at vents have been estimated from the field observations (e.g., satellite images) on the height of the eruption columns and the spreading rate of umbrella clouds on the basis of the one‐dimensional (1‐D) plume model and the gravity current model. However, these simplified models contain empirical constants (entrainment coefficient of turbulent plume, k, and Froude number of gravity current, λ) that should be justified. We developed a time‐dependent three‐dimensional (3‐D) numerical model of eruption clouds to independently determine the values of these constants. The 3‐D model is designed to calculate quantitative features of turbulent mixing between an eruption cloud and the ambient air without any a priori empirical constants by applying a sufficiently fine grid size with a third‐order accuracy scheme. It has reproduced the fundamental features of eruption clouds including eruption columns, pyroclastic flows, coignimbrite ash clouds, and umbrella clouds. The altitudes of the spreading umbrella clouds in the 3‐D simulations are consistent with those of the neutral buoyancy level of eruption columns estimated by the 1‐D plume model with the entrainment coefficient k ∼ 0.1. The relationship between the volumetric flow rate and the spreading rate of the umbrella cloud in the 3‐D simulations is explained by the axisymmetrical gravity current model with λ ∼ 0.2 for eruption clouds in the tropical atmosphere and λ ∼ 0.1 for those in the midlatitude atmosphere. On the other hand, the total column height in the 3‐D simulations strongly oscillates even for a constant mass discharge rate, and its time average tends to be substantially greater than the column height estimated by the 1‐D plume model with k ∼ 0.1, particularly for large‐scale eruption clouds in the midlatitude atmosphere. We recommend a method to estimate the mass discharge rate from the observation of the column height or the spreading rate of umbrella cloud. The method and the accuracy of the 3‐D simulations are tested on the basis of the field data (e.g., the total height of the eruption column, the altitude and the spreading rate of the umbrella cloud, and the mass discharge rate) of the Pinatubo 1991 eruption.