Dissipative trapped particle modes in tandem mirrors

The theory for the dissipative trapped particle modes is developed for a tandem mirror taking into account equilibrium rotational effects, electron temperature gradients, and an axial ambipolar potential, for a broad range of collisional parameters. The electrons are treated as a Maxwellian plasma that occupies the central cell and anchor cell regions. It is assumed that the eigenfunction is piecewise constant with abrupt transitions between the anchor and central cell regions. It is found that when ω/νp≳1, with ω the mode frequency and νp the Pastukhov loss rate, that the energy conservation structure of the collision operator produces important changes to previously developed theories. A solution to the problem is achieved by using the solution for the lifetime of an electron in an ambipolar trap, taking into account the global energy conservation. The energy conservation structure also allows a self-consistent description of dissipative instabilities when thermal gradient and electric fields are present. At very high collision frequency, a new dispersion relation is obtained, which exhibits an axially rotational shear drive coupled to radial temperature gradients producing instability. Numerical studies are presented for some parameters, with the deviation from previous theory highlighted.