Many swimming micro-organisms are bottom-heavy and so, although their swimming directions are fairly random, they naturally tend to swim upwards (negative gravitaxis or geotaxis). Such creatures tend to aggregate at the upper boundaries of the fluid and, in sufficiently shallow layers and at low concentrations, may form a horizontally uniform, top- heavy equilibrium distribution in which the flux of cells due to upswimming is balanced by diffusion down cell concentration gradients, due to a degree of randomness in their swimming behaviour. If the micro-organisms have a higher density than the ambient fluid, then aggregations of the cells at the upper surface can initiate an overturning instability, reminiscent of thermal or Rayleigh–Bénard convection, and thus produce spatial concentration patterns. The bulk motion of the fluid exerts viscous (or frictional) torques on the micro-organisms and the resulting balance with the gravitational torque is called gyrotaxis (Kessler, 1984b, 1985a,b). The micro-organisms’ geometry and mass distribution imply that a component of their swimming velocity is towards regions of downwelling fluid and away from upwelling fluid. In this way, the micro- organisms increase the average density of downwelling regions of fluid and cause them to sink faster. This second ‘gyrotactic’ instability mechanism, together with the overturning instability, drives complex patterns in suspensions of the micro-organisms which are termed bioconvection. When viewed from above, the patterns are characterized by highly concentrated aggregations of cells in both one- and two- dimensional structures. Childress et al. (1975) modelled bioconvection for upswimming cells in the absence of gyrotaxis and used the model to predict the wavelengths of the initial instabilities. Their analysis predicted large pattern wavelengths limited only by the size of the experimental apparatus. At any point in space, a population of micro- organisms has a random distribution of possible swimming directions, characterized by an average swimming direction and a direction- and flow-dependent diffusivity tensor (Pedley and Kessler, 1990). A deterministic model for gyrotactic bioconvection using a constant diffusivity was first analysed in layers of infinite and finite depth by Pedley et al. (1988) and Hill et al. (1989), respectively, and more realistic wavelengths were predicted. This model was further extended in a completely self-consistent fashion by Pedley and Kessler (1990) and Bees (1996), who modelled bioconvection using a probability distribution function for the cell swimming direction in a stochastic formulation of gyrotaxis. In all these works, it was found that the gyrotactic instability mechanism depends on the absolute cell concentration, unlike the overturning instability which depends on the gradient of the cell concentration. The purpose of the present investigation is to attempt to quantify observations of pattern formation by swimming micro-organisms in a rational and reproducible manner in order to compare them with the predictions made from mathematical models of bioconvection (see Pedley and Kessler, 1990; Bees, 1996; M. A. Bees and N. A. Hill, in preparation). Observations of pattern formation have been recorded previously by such authors as Wager (1911), Loeffer and Mefferd (1952), Wille and Ehret (1968), Levandowsky et al. (1975) and Kessler (1984b), but the results have tended to be of a qualitative nature. The present study reports one of the first, controlled experiments aimed at quantitatively cataloguing aspects of the bioconvection patterns. Methods will be described that we have developed for measuring the attributes of these patterns in suspensions of a particular micro-organism, the alga Chlamydomonas nivalis. Fourier analysis is used to extract the dominant unstable wavenumber from the patterns as a function 1515 The Journal of Experimental Biology 200, 1515–1526 (1997) Printed in Great Britain © The Company of Biologists Limited 1997 JEB0824 Bioconvection occurs as the result of the collective behaviour of many micro-organisms swimming in a fluid and is realised as patterns similar to those of thermal convection which occur when a layer of water is heated from below. A methodology is developed to record the bioconvection patterns that are formed by aqueous cultures of the single-celled alga Chlamydomonas nivalis. The analysis that is used to quantify the patterns as a function of cell concentration, suspension depth and time is described and experimental results are presented. Key words: Chlamydomonas nivalis, bioconvection patterns, wavelengths, Fourier transforms, swimming micro-organisms. Summary Introduction WAVELENGTHS OF BIOCONVECTION PATTERNS M. A. BEES* AND N. A. HILL Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK Accepted 17 March 1997 *e-mail: amt5mab@amsta.leeds.ac.uk.