On the generation of Alfvén waves by solar energetic particles

A simple analytical theory of Alfven waves amplified by streaming solar energetic particles (SEPs) is studied. It is pointed out that a finite time-integrated net flux of energetic protons has to pass each point in space before we can expect Alfven waves to be significantly modified by the streaming instability. The time-integrated net proton flux needed for the time-integrated wave growth rate (or wave growth, for short) to exceed unity is evaluated. Assuming that protons stream much faster than the waves, we evaluate the wave growth as a function of position and wavenumber for a specified proton injection energy spectrum, $\mathrm{d}N/\mathrm{d}E$. The wave growth is found to be proportional to $vp\, \mathrm{d}N/\mathrm{d}E$, where v and p are the particle speed and momentum, and to the local Alfven speed $V_{\mathrm{A}}$. Thus, maximum wave growth is achieved at the location of maximum $V_{\mathrm{A}}$ (at a few solar radii), and the minimum value of $\mathrm{d}N/\mathrm{d}E$ required for the wave growth to exceed unity there is a few times 10$^{32}/vp$ protons per unit solid angle (in coordinate space) at the solar surface. If $\mathrm{d}N/\mathrm{d}E$ is below this value, test-particle theory is a valid description of particle transport and acceleration. The value is not exceeded (above 1 MeV energies) in small gradual SEP events having peak 1-MeV proton intensities below ∼ 10 protons (cm$^{2}\, $sr s MeV) -1 at 1 AU. The spatial and momentum dependence of the wave growth can also be used to estimate the maximum emission strength of a moving proton source in the interplanetary medium. For a strong source moving through the solar wind at constant super-Alfvenic speed, the number of escaping particles per unit time and flux-tube cross section is approximately constant in time, predicting a plateau-type time–intensity profile observed ahead of the source. The model reproduces observations of streaming-limited intensities at energies around 10 MeV and explains the double peaked injection profiles observed in large SEP events.