  We consider a 2-layer quasi-geostrophic ocean model where the upper layer is
forced by a steady Kolmogorov wind stress in a periodic channel domain, which
allows to mathematically study the nonlinear development of the resulting flow.
The model supports a steady parallel shear flow as a response to the wind
stress. As the maximal velocity of the shear flow (equivalently the maximal
amplitude of the wind forcing) exceeds a critical threshold, the zonal jet
destabilizes due to baroclinic instability and we numerically demonstrate that
a first transition occurs. We obtain reduced equations of the system using the
formalism of dynamic transition theory and establish two scenarios which
completely describe this first transition. The generic scenario is that two
modes become critical and a Hopf bifurcation occurs as a result. Under an
appropriate set of parameters describing midlatitude oceanic flows, we show
that this first transition is continuous: a supercritical Hopf bifurcation
occurs and a stable time periodic solution bifurcates. We also investigate the
case of double Hopf bifurcations which occur when four modes of the linear
stability problem simultaneously destabilize the zonal jet. In this case we
prove that, in the relevant parameter regime, the flow exhibits a continuous
transition accompanied by a bifurcated attractor homeomorphic to $S^3$. The
topological structure of this attractor is analyzed in detail and is shown to
depend on the system parameters. In particular, this attractor contains
  (stable or unstable) time-periodic solutions and a quasi-periodic solution.
