Negative-energy modes in a magnetically confined plasma in the framework of Maxwell-drift kinetic theory

The general expression for the second-order perturbation energy of a Maxwell-drift kinetic system derived by Pfirsch and Morrison [Phys. Fluids B 3, 271 (1991)] is evaluated for the case of a magnetically confined plasma for which the equilibrium quantities depend on one Cartesian coordinate y. The conditions for the existence of negative-energy modes with vanishing initial field perturbations are also obtained. If the equilibrium guiding center distribution function ${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$ of any particle species \ensuremath{\nu} has locally the property ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)g0, where ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$ is the guiding center velocity parallel to the magnetic field, and if this holds in the minimum-energy reference frame, parallel and oblique negative-energy modes exist with no essential restriction on either the orientation or magnitude of the wave vector. This condition also holds for the equilibria of a homogeneous magnetized plasma and an inhomogeneous force-free plasma with sheared magnetic field. If ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)0, the oblique negative-energy modes possible in a magnetically confined plasma are nearly perpendicular. The condition for purely perpendicular negative-energy modes reads as (${\mathit{dP}}^{(0)}$/dy)(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}y)0, where ${\mathit{P}}^{(0)}$ is the plasma pressure.