Self-consistent Energetic Particle Acceleration by Contracting and Reconnecting Small-scale Flux Ropes: The Governing Equations

Previous application of our focused transport equation for energetic ion test particle acceleration by numerous active small-scale flux ropes to solar wind conditions near 1 au yielded the formation of hard power-law spectra with high particle pressure. We present an extended theory where the focused transport equation is coupled to a new MHD turbulence transport equation for coherent, quasi-2D magnetic island structures, based on nearly incompressible (N i) MHD turbulence theory. The latter equation includes new expressions for the magnetic island damping/growth rates that enable a self-consistent description of energy exchange between energetic particles and flux ropes during flux-rope acceleration for four flux-rope acceleration scenarios identified in focused transport theory. Revised, more detailed expressions for coherent acceleration in response to mean dynamic flux-rope properties and for stochastic acceleration due to fluctuations in dynamic flux-rope properties are presented. A comparison is made between the efficiencies of the different flux-rope acceleration scenarios for suprathermal protons in the solar wind near 1 au. Dynamic flux-rope-induced pitch-angle scattering and stochastic acceleration rates are compared with the corresponding rates generated by interaction with parallel-propagating Alfvén waves. The results stress the importance of parallel guiding center motion acceleration by the parallel reconnection electric field formed in merging flux ropes, combined curvature drift and generalized betatron acceleration in contracting/merging flux ropes in the compressible limit (flux-rope compression acceleration), and the fluctuating magnetic mirroring force in flux ropes for pitch-angle scattering.