Long wavelength limit of the current convective instability

A linear theory is presented of the current convective instability in the long wavelength limit, i.e., kL ≪ 1 where k is the wave number and L is the scale length of the density inhomogeneity. A relatively simple dispersion equation is derived that describes the modes in this limit. Analytical solutions are presented in both the collisional (νin ≫ ω) and inertial (νin ≪ ω) limits where ω is the wave frequency and νin is the ion‐neutral collision frequency. It is shown that the growth rate scales as k in the collisional limit and as k2/3 in the inertial limit. The analytical solutions are compared to exact numerical solutions, and very good agreement is found. Applications to the auroral ionosphere are discussed.