Coupled hydromagnetic wave excitation and ion acceleration upstream of the Earth's bow shock

A self‐consistent theory is presented for the excitation of hydromagnetic waves and the acceleration of ‘diffuse’ ions upstream of the earth's bow shock in the quasi‐equilibrium that results when the solar wind velocity and the interplanetary magnetic field are nearly parallel. For the waves the quasi‐equilibrium results from a balance between excitation by the ions, which stream relative to the solar wind plasma, and convective loss to the magnetosheath. For the diffuse ions the quasi‐equilibrium results from a balance between injection at the shock front, confinement to the foreshock by pitch angle scattering on the waves, acceleration by compression at the shock front, loss to the magnetosheath, loss due to escape upstream of the foreshock, and loss via diffusion perpendicular to the average magnetic field onto field lines that do not connect to the shock front. Diffusion equations describing the ion transport and wave kinetic equations describing the hydromagnetic wave transport are solved self‐consistently to yield analytical expressions for the differential wave intensity spectrum as a function of frequency and distance from the bow shock z and for the ion omnidirectional distribution functions and anisotropies as functions of energy and z . In quantitative agreement with observations, the theory predicts (1) exponential ion spectra at the bow shock in energy per charge, (2) a decrease in intensity and hardening of the ion spectra with increasing z , (3) a 30‐keV proton anisotropy parallel to z increasing from −0.28 at the bow shock to +0.51 as z → ∞ (4) a linearly polarized wave intensity spectrum with a minimum at ∼6 × 10 −3 Hz and a maximum at ∼2–3 × 10 −2 Hz, (5) a decrease in the wave intensity spectrum with increasing z , (6) a total energy density in protons with energies >15 keV about eight times that in the hydromagnetic waves.