Bioconvection in a suspension of phototactic algae

Abstract

In this paper we develop a new generic model for phototaxis in a suspension of microscopic swimming algae. Phototaxis is a directed swimming response dependent upon the light conditions sensed by the microorganisms. Positive phototaxis consists of motions directed toward the source of illumination and negative phototaxis is motion directed away from it. The model also incorporates the effect of shading whereby microorganisms nearer the light source absorb and scatter the light before it reaches those further away. This model of phototaxis and shading is then used to analyse the linear stability of a suspension of phototactic algae, uniformly illuminated from above, that swim in a fluid which is slightly less dense then they are. In the basic state there are no fluid motions and the up and down swimming caused by positive and negative phototaxis is balanced by diffusion, with the result that a horizontal, concentrated layer of algae forms, the vertical position of which depends on the light intensity. This leads to a bulk density stratification with a gravitationally stable layer of fluid overlying an unstable layer. When the resulting density gradient becomes large enough, a Rayleigh–Taylor type instability initiates fluid motions in the lower, unstable region that subsequently penetrate the upper, locally stable region. The behaviour of the suspension is characterized by four parameters: a layer depth parameter d, a Rayleigh number R, a Schmidt number Sc, and a sublayer position parameter C that specifies the vertical position of the sublayer in the fluid. Linear stability analysis of the basic state indicates that initial pattern wavelengths and the critical value of R at which the suspension becomes unstable are affected by the position of the sublayer, which is determined by the intensity of the light source, and by the type of boundary at the top of the fluid. Oscillatory modes of disturbance are also predicted for certain parameter ranges, driven by cells swimming vigorously upwards towards darker, concentrated downwelling regions and thus creating lower, positively buoyant regions which oppose the fluid motion in the convection patterns.