Stability of the ion-acoustic surface waves in a Lorentzian plasma

We investigated the stability of ion-acoustic surface waves propagating on a boundary of semi-infinite Lorentzian (kappa) plasma. The real and imaginary parts of the wave frequency $\omega =\omega _r +i\gamma$ are obtained as functions of the normalized wave number $k_x \lambda _e$ where $k_{x}$ is the x-component of the wave number and $\lambda _e$ is the electron Debye length. The phase speed of the wave is found be decreased as the spectral index of the Lorentzian distribution function is decreased. In the long wavelength limit, the scaled phase velocity ${\left( {{\omega _r} \mathord{\left/ {\vphantom {{\omega _r } {\omega _{pi} }}} \right. \kern-\nulldelimiterspace} {\omega _{pi} }} \right)} \mathord{\left/ {\vphantom {{\left( {{\omega _r } \mathord{\left/ {\vphantom {{\omega _r } {\omega _{pi} }}} \right. \kern-\nulldelimiterspace} {\omega _{pi} }} \right)} {k_x \lambda _e }}} \right. \kern-\nulldelimiterspace} {k_x \lambda _e }$ becomes $\sqrt {\mu _K } $ where $\omega _{pi}$ is the ion plasma frequency and $\sqrt {\mu _\kappa } $ is a constant. The wave displays the resonance at ${\omega _r } \mathord{\left/ {\vphantom {{\omega _r } {\omega _{pi} }}} \right. \kern-\nulldelimiterspace} {\omega _{pi} }=1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-\nulldelimiterspace} {\sqrt 2 }$ as expected as the case of Maxwellian plasma. The imaginary part of the wave frequency appears to be negative which exhibits the linear wave dissipation in a collisionless plasma called Landau damping. The maximum damping rate is obtained as $\lambda _{Max} =0.14M_\kappa \sqrt {m \mathord{\left/ {\vphantom {m M}} \right. \kern-\nulldelimiterspace} M} $ at $k_x \lambda _e ={0.44} \mathord{\left/ {\vphantom {{0.44} {\sqrt {\mu _\kappa } }}} \right. \kern-\nulldelimiterspace} {\sqrt {\mu _\kappa } }$ where $M_\kappa$ is a kappa-dependent function, and $m$/$M$ is the electrion-ion mass ratio. The damping of the wave disappears as $k_x \lambda _e \to \infty $.