A model of the formation of stable nonpropagating magnetic structures in the solar wind based on the nonlinear mirror instability

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Abstract
A simple model for the formation of stable nonpropagating structures in a magnetized collisionless plasma is presented. The model describes the evolution of an electron‐proton plasma from an initially spatially uniform, but unstable, configuration toward a final nonuniform and nonpropagating stable configuration. The model is based on the following hypothesis: (1) one‐dimensionality and spatial periodicity, (2) cold electrons, (3) bi‐Maxwellian protons as initial condition, (4) conservation of magnetic moment for all protons, (5) conservation of energy for magnetically non trapped protons, (6) spatial pressure balance, (7) evolved structure has a crenellated shape, (8) slow growth of the structure. Given these assumptions all the macroscopic properties of the plasma (density, pressure, and magnetic field) in the saturated state can be computed explicitly. The model shows that a spatially uniform and homogeneous plasma that is unstable against the linear mirror mode can form stable non propagating structures. Thus one can consider the model as a model for the nonlinear mirror instability where the magnetic trapping of protons in the low magnetic field region is the important saturation mechanism. A simple expression for the magnetic field saturation amplitude is found. The pressure balance, between high and low magnetic field regions, which is needed for the evolved structure to be a stable one, is obtained solely through betatron cooling of the trapped protons. Modification of the trapped protons energy due to the Fermi effect seems to be of secondary importance. The model predicts that the evolved structures are characterized by narrow and deep magnetic wells except in the case of very low magnetic pressure (ratio of thermal to magnetic pressure β ≳ 10) where the opposite situation becomes possible. This enforces the idea according to which the proton mirror instability is the driving mechanism for the formation of magnetic holes in high β(≳ 1) plasmas.