  A collection of microswimmers immersed in an incompressible fluid is
characterised by strong interactions due to the long-range nature of the
hydrodynamic fields generated by individual organisms. As a result, suspensions
of rear-actuated `pusher' swimmers such as bacteria exhibit a collective motion
state often referred to as `bacterial turbulence', characterised by large-scale
chaotic flows. The onset of collective motion in pusher suspensions is
classically understood within the framework of mean-field kinetic theories for
dipolar swimmers. In bulk 2-D and 3-D, the theory predicts that the instability
leading to bacterial turbulence is due to mutual swimmer reorientation and sets
in at the largest length scale available to the suspension. Here, we construct
a similar kinetic theory for the case of a dipolar microswimmer suspension
restricted to a two-dimensional plane embedded in a three-dimensional
incompressible fluid. This setting qualitatively mimics the effect of swimming
close to a two-dimensional interface. We show that the in-plane flow fields are
effectively compressible in spite of the incompressibility of the 3-D bulk
fluid, and that microswimmers on average act as sources (pushers) or sinks
(pullers). We analyse stability of the homogeneous and isotropic state, and
find two types of instability that are qualitatively different from the bulk,
three-dimensional case: First, we show that the analogue of the orientational
pusher instability leading to bacterial turbulence in bulk systems instead
occurs at the smallest length-scale available to the system. Second, an
instability associated with density variations arises in puller suspensions as
a generic consequence of the effective in-plane compressibility. We conclude
that confinement can have a crucial role in determining the collective
behaviour of microswimmer suspensions.
