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A nonlinear approach to transition in subcritical plasmas with sheared flow

In plasma systems driven by bouyancy instabilities, somewhat analogous to Rayleigh-Bernard convection, introducing a small background shear flow is often 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. This is distinct from the related problem of transition from turbulence, where the collapse back to a laminar state is studied as the stability of the system is increased. The transition to turbulence is fundamentally nonlinear, and investigations 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 provided confidence in the derivation of a semi-analytic approximation for the minimal seed.