A nonlinear model for coherent tripolar electric field structures in the Earth's auroral zone and solar wind

A theoretical explanation is given for tripolar electric field signals observed recently in the Earth's magnetosphere by the Cluster multi-spacecraft mission and in interplanetary space by the WIND mission, which feature an overall potential drop characteristic for weak double layers, with adjacent potential minima and maxima that trap both ions and electrons, respectively. A kinetic theory, based on Schamel's model for trapped particle distribution, is presented for electron and ion holes coupled with weak double layers, whose properties are studied analytically and numerically. In contrast to earlier models of double layers, which require external excitation in the form of asymptotically non-Maxwellian velocity distributions, tripoles only trap particles locally and the corresponding asymptotic particle distributions are Maxwellian. In the small amplitude limit, $e\phi\,{\ll}\, T_{\rm e}, T_{\rm i}$, such structures propagate with a speed that is in the ion thermal velocity range and they can only be excited if there is a relative drift between electrons and ions, whose speed is in the electron thermal velocity range. Conversely, non-propagating tripoles are found in the large amplitude regime and in the absence of particle streaming.