Satellite observations at midlatitudes (≈20,000 km) near the earth's dayside polar cusp boundary layer indicate that the upward electron beams have a narrow latitudinal width up to 0.1¿. In the cusp boundary layer where the electron population consists of a finite-size electron beam in a background of uniform cold and hot electrons, the electron acoustic mode is unstable inside the electron beam but damped outside the electron beam. Simulations of the electron acoustic instability for a finite-size beam system are carried out with a particle-in-cell code to investigate the heating phenomena associated with the instability and the width of the heating region. The simulations show that the finite-size electron beam radiates electrostatic electron acoustic waves. The decay length of the electron acoustic waves outside the beam in the simulation agrees with the spatial decay length derived from the linear dispersion equation.

The ambient cold electrons in a diffusion region surrounding the beam are heated to a higher temperature by absorbing the radiated electron acoustic waves, with heating occurring mainly in the parallel direction. In the heat diffusion region, the temperature of the cold electrons decreases with the distance from the beam with a temperature gradient length smaller than the decay length of the wave energy. This feature of electron heating is modeled by using a heat diffusion equation derived from the second-order theory. In solving the heat diffusion equation, the wave energy inside the beam is assumed to be saturated by trapping the beam and the cold electrons. The dominant wave number is determined by assuming that the wave frequency outside the beam is equal to the frequency corresponding to the maximum growth rate inside the beam. These results are discussed with respect to the DE 1 plasma and wave observations in the polar cusp region. Using the typical parameters in the cusp boundary layer, the spatial decay length of the electron acoustic waves is estimated to be in the range of 1-10 km and the heat diffusion region is about 5 km. The results suggest that the electron acoustic waves generated by cusp upward electron beams are probably confined in a small region surrounding the beams. ¿ American Geophysical Union