The bioconvection equations, based on the continuum model of Pedley et al [J. Fluid Mech.195, 223 (1988)], consist of the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved efficiently using a semi-implicit second-order accurate conservative finite-difference method. The structure and stability of a three-dimensional plume in deep rectangular boxes with stress-free sidewalls are investigated. Comparisons are made with the two-dimensional and axisymmetric bioconvection. In deep chambers, the three-dimensional plume that forms initially along the central axis of the chamber typically breaks down via a meandering instability.

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