  In many plasma systems, introducing a small background shear flow is enough
to stabilize the system linearly. The nonlinear dynamics are much less
sensitive to sheared flows than the average linear growthrates, and very small
amplitude perturbations can lead to sustained turbulence. We explore the
general problem of characterizing how and when the transition from near-laminar
states to sustained turbulence occurs; a model of the interchange instability
being used as a concrete example. These questions are fundamentally nonlinear,
and the answers must go beyond the linear transient amplification of small
perturbations. Two methods that account for nonlinear interactions are
therefore explored here. The first method explored is edge tracking, which
identifies the boundary between the basins of attraction of the laminar and
turbulent states. Here, the edge is found to be structured around an exact,
localized, traveling wave solution; a solution that is qualitatively similar to
avalanche-like bursts seen in the turbulent regime. The second method is an
application of nonlinear, non-modal stability theory which allows us to
identify the smallest disturbances which can trigger turbulence (the minimal
seed for the problem) and hence to quantify how stable the laminar regime is.
The results obtained from these fully nonlinear methods provides confidence in
the derivation of a semi-analytic approximation for the minimal seed.
