Stationary response and singular limits in a linear zonal gyrokinetic model

We derive the stationary response of a linear periodic gyrokinetic model without assuming uniform kinetic relaxation. When the quadratic energy is coercive, the energy-orthogonal stationary projection is determined by the field operator D₀ = C − R∗ΠR. Cesàro and Abel means converge strongly, but their operator-norm errors remain one when nonzero frequencies accumulate at zero. Adding the same constant drift detuning to every species removes the stationary space and generates a slow unitary evolution on its former kernel, so the long-time and zero-detuning limits do not commute.

At vanishing polarization, a homogeneous kinetic-species model instead imposes charge and current constraints. We construct a self-adjoint evolution on the continuous-velocity constraint space and prove an O(√ε) error bound, in the energy coordinate, for one coercive regularization of constrained initial data; unrestricted data retain fast charge oscillations. In a specified 25-dimensional orbit discretization with trapped and passing particles, replacing the energy projection by ordinary orbit averaging changes the stationary field by 23.2%. The projection and singular-limit results are analytical; the percentage is stable under the reported grid refinements and is not a certified continuum value.