  To comprehensively understand saturation of two-dimensional ($2$D) magnetized
Kelvin-Helmholtz-instability-driven turbulence, energy transfer analysis is
extended from the traditional interaction between scales to include eigenmode
interactions, by using the nonlinear couplings of linear eigenmodes of the
ideal instability. While both kinetic and magnetic energies cascade to small
scales, a significant fraction of turbulent energy deposited by unstable modes
in the fluctuation spectrum is shown to be re-routed to the conjugate-stable
modes at the instability scale. They remove energy from the forward cascade at
its inception. The remaining cascading energy flux is shown to attenuate
exponentially at a small scale, dictated by the large-scale stable modes.
Guided by a widely used instability-saturation assumption, a general
quasilinear model of instability is tested by retaining all nonlinear
interactions except those that couple to the large-scale stable modes. These
complex interactions are analytically removed from the magnetohydrodynamic
equations using a novel technique. Observations are: an explosive large-scale
vortex separation instead of the well-known merger of $2$D, a dramatic
enhancement in turbulence level and spectral energy fluxes, and a reduced
small-scale dissipation length-scale. These show critical role of the stable
modes in instability saturation. Possible reduced-order turbulence models are
proposed for fusion and astrophysical plasmas, based on eigenmode-expanded
energy transfer analyses.
