Ion-acoustic waves can propagate in an unmagnetized plasma when the electron temperature T_e greatly exceeds the ion temperature T_p. In this limit, the electron density n_e can be approximated by n_e = n₀ exp[eΦ/(k_BT_e)], where n₀ is the mean electron density, and Φ is the electrostatic potential.

(a) Show that for ion-acoustic waves that propagate in the z direction, the nonlinear equations of continuity, the motion for the ion (proton) fluid, and Poisson’s equation for the potential take the form

Here n is the proton density, and u is the proton fluid velocity (which points in the z direction).

(b) Linearize Eqs. (21.35), and show that the dispersion relation for small-amplitude ion-acoustic modes is

where λ_D is the Debye length.

Transcribed Image Text:

(nu)e zę ne n+ = ue Ət + ne Ət дz 22Ф az2 = 0, = е аф дz m IP e - (n — nje&Ф/(kBTə). €0 === (21.35)