Excitation of compressional waves and the formation of shocklets in the Earth's foreshock

Large‐amplitude waves with typical frequencies of 0.01‐0.05 Hz are often observed in the foreshocks of earth and other planets. Large‐amplitude waves in the earth's foreshock are sometimes (but not always) observed in a highly time‐developed form, either as steepened pulses or as discrete oscillatory wave packets of finite length. This implies that nonlinearities are strong enough to modify their waveforms before the solar wind carries them out the foreshock. The instabilities and steepening of upstream waves in the earth's foreshock caused by backstreaming ions are discussed in the first part of the paper. For typical foreshock “diffuse” ion distributions, right and left‐hand polarized (RHP and LHP) waves propagating parallel to the local magnetic field are preferentially excited. Such noncompressional waves neither steepen nor grow fast enough to account for the amplitude polarizations and waveforms observed in the diffuse ion foreshock. Oblique waves develop a density compression and their magnetic field polarization is elliptical. Although these characteristics match the observations of the steepened waves in the diffuse ion zone, the growth rates of those waves oblique enough to steepen are too small to account for the observed amplitudes. On the other hand, the parallel propagating waves excited by the “reflected” ion distribution at the leading edge of the foreshock do grow fast enough, but do not steepen. We suggest that parallel propagating waves grow to finite amplitude in the “reflected” and “intermediate” ion zones of the earth's foreshock and refract as they are carried by the solar wind into the “diffuse” ion region, so that they become increasingly oblique and compressional. The more compressional they become, the more rapidly they steepen. Some steepen to the point where finite ion inertia dispersion wave creates a nonlinear wave train—a shocklet. Because wave refraction is less important in the very large foreshocks of interplanetary shocks, it is less likely that oblique, compressive, steepened waves will be generated in them, in agreement with observation. In the second part of this paper, we will simulate the time evolution of oblique low‐frequency compressive waves using a one‐dimensional hybrid code in which main ions are treated as superparticles, diffuse ions as a double‐adiabatic fluid, and electrons as an isothermal fluid. Unlike conventional hybrid codes, parallel and perpendicular pressures of fluids are treated independently. This is necessary in order to well describe the compressional properties of obliquely propagating right‐hand polarized waves in a high β plasma. We initialize the simulations with an oblique sinusoidal low‐frequency wave of finite amplitude. Nonlinear steepening, formation of discrete dispersive wave packets, and subsequent ion cyclotron damping of the wave packets are found to occur.