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Tarshish and Romps, A closure for the virtual origin of turbulent plumes, JAS, 2021

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A campfire, a wildfire, and a citywide conflagration all serve as localized sources of atmospheric heat. These heat sources, if sustained, lead to the formation of atmospheric plumes. Since these plumes carry soot from the fire, and because that soot can harm human health and even perturb Earth's climate, it is good to know where that plume's air ends up. For example, the consequences for human health are very different if the soot from a wildfire ends up high in the troposphere versus trapped in the boundary layer near an urban area. Or, in the event of a citywide firestorm triggered by a nuclear attack, soot that ends up in the troposphere where it can rain out quickly is a very different matter than a nuclear-winter-triggering injection of soot into the stratosphere.

In 1956, Morton, Taylor, and Turner introduced an analytic model for turbulent plumes that works well far above the surface (in an unstratified, unsheared fluid). That model has two parameters that must be determined empirically: the entrainment parameter and the virtual origin. If we picture an idealized plume as shaped like a cone, the entrainment parameter determines the angle of the cone and the virtual origin gives the location of the cone's vertex. Although the angle of the cone has been reported from numerous laboratory experiments and numerical simulations, there has been very little work done to measure the virtual origin.

In this paper, direct numerical simulations (DNS) are used to measure the virtual origin of a plume in an unstratified, unsheared fluid sustained by a circular source of buoyancy located at a no-slip lower boundary. The virtual origin is found to be below the surface a distance roughly equal to the source radius. This contrasts with an assumption commonly used in the literature that the virtual origin is below the surface a distance of nine times the radius. Whereas the usual assumption would predict that the Hiroshima firestorm plume remained trapped near the surface, the virtual origin reported here predicts that the plume reached well into the troposphere, consistent with archival photographs.

Instantaneous (top row) and time-averaged (bottom row) cross sections of buoyancy b and vertical velocity w for a three-dimensional turbulent plume with a Reynolds number of 1000 and sustained by a source of buoyancy F distributed over a circle of radius R.