A simple integral model of buoyancy-generating plumes and its application to volcanic eruption columns

This paper discusses the importance of the increase in buoyancy flux due to thermal expansion of entrained air to clarify the fundamental features of volcanic eruption columns. A one-dimensional steady state model, which is simplified as much as possible, is developed to elucidate the essential differences in behavior between a typical incompressible turbulent plume and an eruption column in which the buoyancy flux significantly increases with height. An analytical solution of a simple form is specifically derived for an idealized axisymmetric turbulent plume of a linearly increasing buoyancy flux in a uniform environment. The solution predicts that the upward velocity of the plume is constant along the height, in contrast to the upward velocity of a common incompressible plume, the velocity of which decreases inversely proportional to the third root of the height. More realistic plumes in both uniform and density-stratified environments are also investigated by modifying the one-dimensional model. The model yields several scaling parameters, some of which are used to estimate the terminal height of an eruption column. Numerical investigations using the parameters indicate that the thermal energy in an eruption column is exhausted for heating and expanding entrained air before reaching the terminal height. Numerical investigations also imply that the buoyancy flux in an eruption column may be less than half of that predicted by the conventional theory of an incompressible plume. The discussion based on the present model also sheds new light on the physical background of the superbuoyant behavior of an eruption column.