One- and two-dimensional simulations of electron beam instability: Generation of electrostatic and electromagnetic 2f_p waves

We have performed computer simulations of the self-consistent nonlinear evolution of electrostatic and electromagnetic 2f_p waves excited by electron beams with electromagnetic particle code. In both one- and two-dimensional periodic systems an electrostatic 2f_p wave is generated at twice the wave number of forward propagating Langmuir waves by wave-wave coupling. This wave grows with the forward propagating Langmuir wave in the nonlinear stage of the simulations. The electrostatic 2f_p wave in the simulations is saturated at about -20 ∼ -30 dB of that of the Langmuir waves. It is larger than the value expected from observations in the terrestrial electron foreshock. The electromagnetic 2f_p wave is only excited in two-dimensional systems. The magnitude of the electromagnetic 2f_p wave is correlated with the backward propagating Langmuir wave, not with the electrostatic 2f_p wave. This result suggests that the electromagnetic 2f_p wave is excited by the wave-wave coupling of forward and backward propagating Langmuir waves. The typical power density estimated from a reasonable amplitude of Langmuir wave is of the same order or much weaker than the value typically observed around the electron foreshock.