We study a quasi-two-dimensional electrostatic drift kinetic system as a
model for near-marginal ion temperature gradient (ITG) driven turbulence. A
proof is given of the nonlinear stability of this system under conditions of
linear stability. This proof is achieved using a transformation that
diagonalizes the linear dynamics and also commutes with nonlinear E×B
advection. For the case when linear instability is present, a corollary is
found that forbids nonlinear energy transfer between appropriately defined sets
of stable and unstable modes. It is speculated that this may explain the
preservation of linear eigenmodes in nonlinear gyrokinetic simulations. Based
on this property, a dimensionally reduced (∞×∞→1)
system is derived that may be useful for understanding dynamics around the
critical gradient of Dimits
