High‐Frequency Hall Current Instability

In a partially ionized gas, if electrons and ions have a relative drift V d perpendicular to a static B field, Farley and Buneman have shown that an electrostatic wave with propagation vector K perpendicular to B and wavelength λ ≥ 0.375 meter will be unstable if K · V d / K exceeds the ion thermal speed. We have extended Farley's calculation for K perpendicular to B by the following procedure. First, we examine modes of higher frequencies and shorter wavelengths. Second, we retain the term K 2 λ DE 2 (λ DE is the electron Debye length), since this term cannot be neglected for short wavelength modes. This makes the dispersion relation density dependent. Third, we obtain plots of growth rate versus frequency for various values of K · V d / K . With the use of the Culler‐Fried on‐line computer system, the dispersion relation has been solved numerically by contour integration. There are four results of the numerical calculation. First, there is an electron density threshold for excitation of higher frequency modes. Second, for a fixed electron density of 3.5 × 10 5 /cm 3 , the fastest growing mode shifts to higher frequency as K · V d / K increases. For K · V d / K equal to three times the ion thermal speed, 3 a l , the fastest growing mode oscillates at ω r ≃ 0.7(Ω E Ω l ) ½ , where Ω E and Ω l are electron and ion cyclotron frequencies. The frequency ω r is far above those calculated by Farley. Third, the same behavior cited above is also exhibited for fixed K · V d / K but increasing electron density. Fourth, modes with oscillating frequency ω r ≃ (Ω E Ω l ) ½ can be excited for K · V d / K ≥ 2.3 a l and current density greater than 1.0 × 10 −5 amp/m 2 . These are reasonable values for the equatorial electrojet and the auroral electrojet in the ionosphere during disturbed times.