Continuous convection from an isolated source of heat

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Abstract
Laws are derived for the variation of vertical velocity, w, and temperature, θ, with height, z, above a continuous circular source of heat of finite extent, on the assumptions that the lateral profiles are of similar shape at different distances above the source and that the resistance is a quadratic function of w. The lateral profiles are then taken to be Gaussian, though not all the properties deduced are dependent on this assumption. Complete solutions are obtained allowing for the thermal stratification of the environment; unless this stratification is neutral the common assumption that the vertical heat flux is constant with height is no longer true.It is shown that the width of the column will increase uniformly with height. In general w increases with z at low elevations and then decreases, but this second stage does not occur when the environmental lapse rate is large and the source area sufficiently extensive. Under superadiabatic conditions the second stage, when it occurs, is followed by a third stage in which w increases again without limit; under subadiabatic conditions w decreases rather abruptly to zero at a certain ‘ceiling’ height, for which a simple approximate formula is derived.Numerical values illustrate the application of the solutions to problems of atmospheric pollution, frost prevention, and dry thermals. The solution for neutral conditions is compared with Railston's laboratory measurements.
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