  A chemotaxis-diffusion-convection coupling system for describing a form of
buoyant convection in which the fluid develops convection cells and plume
patterns will be investigated numerically in this study. Based on the
two-dimensional convective chemotaxis-fluid model proposed in the literature,
we developed an upwind finite element method to investigate the pattern
formation and the hydrodynamical stability of the system. The numerical
simulations illustrate different predicted physical regimes in the system. In
the convective regime, the predicted plumes resemble B\'enard instabilities.
Our numerical results show how structured layers of bacteria are formed before
bacterium rich plumes fall in the fluid. The plumes have a well defined
spectrum of wavelengths and have an exponential growth rate, yet their position
can only be predicted in very simple examples. In the chemotactic and diffusive
regimes, the effects of chemotaxis are investigated. Our results indicate that
the chemotaxis can stabilize the overall system. A time scale analysis has been
performed to demonstrate that the critical taxis Rayleigh number for which
instabilities set in depends on the chemotaxis head and sensitivity. In
addition, the comparison of the differential systems of
chemotaxis-diffusion-convection, double diffusive convection, and
Rayleigh-B\'enard convection establishes a set of evidences that even if the
physical mechanisms are different at the same time the dimensionless systems
are strongly related to each other.
