  In this article we will present pure two-dimensional analytic solutions for
the coupled non-compressible Newtoniain Navier-Stokes
  --- with Boussinesq approximation --- and the heat conduction equation. The
system was investigated from E.N. Lorenz half a century ago with Fourier series
and pioneered the way to the paradigm of chaos. We present a novel analysis of
the same system where the key idea is the two-dimensional generalization of the
well-known self-similar Ansatz of Barenblatt which will be interpreted in a
geometrical way. The results, the pressure, temperature and velocity fields are
all analytic and can be expressed with the help of the error functions. The
temperature field has a strongly damped oscillating behavior which is an
interesting feature.
