New Route to Shallow Electron Phase-Space Holes via a ``Velocity-Notch'' Instability$^*$

Properties of weak bipolar fields observed in space are found to be consistent with a theory for shallow electron phase space holes.$^1$ Here, we show that shallow phase space holes can develop \textit{dynamically} as a result of trapping during the saturation of a new electron ``velocity-notch'' instability. This instability occurs when there is a ``notch'' of width $\Delta v$ and density deficit $\Delta n$ in a unimodal electron velocity distribution with density $n_{e0}$ and thermal speed $v_{e0}$, provided $\Delta v/v_{e0}$ is sufficiently smaller than $\sqrt{\Delta n/n_{e0}}$. In the narrow-notch limit, the growth rate is the \textit{plasma frequency of the missing notch electrons}. The nonlinear saturation of this instability is studied using Vlasov simulations initiated with two different classes of electron distributions: Spatially uniform electron distributions with a shallow velocity notch result in holes whose form depends on the degree to which the instability threshold is exceeded. Distributions initialized with a \textit{spatially local} temperature enhancement develop a notch in velocity due to time-of-flight effects. This notch becomes progressively narrower until the instability threshold is crossed. The bipolar fields in the simulations are compared with those corresponding to the weak potential solutions $\phi=\phi_{\mathrm{max}}\mathrm{sech}^4(x/\alpha)$ from theory.$^1$ \\ $^*$ Work supported by NSF, NASA, and DOE \\ $^1$ M.~V.~Goldman, \textit{et al}., this meeting.