Instability of coherent whistlers propagating along field lines in the magnetosphere*

We report on analytic and simulation studies of nonlinear instability triggered by a whistler propagating along a geomagnetic field line. For simplicity of interpretation, the electron distribution is taken to be the highly unstable ring distribution f(\textbf{v})=$\delta $(v$_{\vert \vert }-$v$_{\vert \vert 0})\delta $(v$_{\bot }-$v$_{\bot 0})$. The variation (quadratic near the equator) of the geomagnetic field B(z) along a field line is important, even though $\lambda \quad \sim $ 1 km while the field gradient scale $\sim $1000 km. The instability is triggered by an initial wave pulse of finite duration $\tau _{p}$; the value of $\tau _{p}$ also plays an important role. Instability occurs initially at the resonant points where $\omega -$kv$_{\vert \vert }-\Omega $=0, but is carried backwards in the pulse by the stream of resonant electrons. The fresh flow of unperturbed electrons into the pulse plays an important role, and in the non-uniform B(z), phase-trapped electrons can continue to drive the nonlinear stage of the instability, which is characterized by both growth and strong spatio-temporal variations of the wave frequency.