Toward an Integrated View of Ionospheric Plasma Instabilities: 3. Explicit Growth Rate and Oscillation Frequency for Arbitrary Altitude

Abstract

General analytic expressions are derived for the growth rate γ and oscillation frequency in the ion frame ωr′ of unstable plasma waves generated by ionospheric plasma instabilities including the Farley‐Buneman instability (FBI) and the gradient‐drift instability (GDI). The explicit expressions are developed for arbitrary altitude and scales in the local approximation. Limits of applicability are carefully considered focusing on the dependence on the electron density gradients G=∇n/n and wavelengths λ. It is shown that the key parameter that controls the applicability is the growth rate γ normalized to the ion collision frequency νi, with the developed expressions being valid for slow growths γ/νi<0.1. It is also shown that the commonly used assumption about the equivalency of the wave phase velocity Vph and the plasma drift velocity Vd fails in the F region at gradients as weak as G=10−5m−1. The developed analytic expressions for arbitrary altitude/scale offer a straightforward way of reconciling various altitude‐ and scale‐specific cases (e.g., FBI/GDI modes in the E region), with the often‐neglected ion inertia shown to play a critical role in the reconciliation. The new ion inertia effect is found to be represented by the quantity ()νi2+ωr′2−1 in the growth rate expression. The effect is found to reduce the standard FBI factor and amplify the GDI factor, and, due to the inverse relationship with the ion inertia, the effect becomes progressively stronger at larger altitudes and/or wavelengths.

Toward an Integrated View of Ionospheric Plasma Instabilities: 4. Behavior in the Transitional Valley Region

“…Traditional FBI treatment, however, does not include an additional inertial effect associated with the 1− I −1 term in equation . In the F region, this factor is indeed strongly negative (i.e., stabilizing) at long wavelengths, but can also be positive (destabilizing) at shorter wavelengths (Makarevich, , their Figure 4). The above analysis shows that this destabilization occurs for supercritical convection.…”

“…In this section, a general expression for the growth rate is analyzed focusing on behavior with altitude and plasma drift speed V E = E / B . A version of the general expression for the growth rate that describes fundamental ionospheric plasma instabilities including FBI and GDI for arbitrary altitude has been derived by Makarevich (, equation (32)) as …”

“…One should note that in the above notation convention, both Ω e and r e are negative. The above expression for the growth rate is valid within the limits of the linear fluid theory assuming local approach G ≪ k , slow growth approximation ( γ ≪ ν i ), isotropic ion and electron temperatures, and nearly magnetic‐field‐perpendicular propagation (Makarevich, ). The fluid theory fails for wavelengths comparable with the ion mean free path or the ion Larmor radius, whichever is larger (e.g., Fejer et al, ).…”

“…In this section, we analyze the instability behavior in the transitional valley region focusing on regimes where GDI contribution becomes important. We start by noting that the growth rate in the long‐wavelength limit λ → ∞ simplifies to GV E f since and and hence (Makarevich, ). It is useful therefore to consider a normalized growth rate since it approaches unity at λ → ∞ for all combinations of parameters.…”

“…The wavelength dependence also appears implicitly, through the inertial factor . The oscillation frequency from Makarevich (), equation , for our case of the field‐perpendicular propagation becomes and the inertial factor can be rewritten as where a new notation is introduced …”