Statistical properties of small-amplitude Langmuir waves in the Earth's electron foreshock

We present the results of a statistical study of Langmuir waves observed by the Cluster spacecraft in the Earth's electron foreshock. To classify the probability density distributions of the logarithms of the wave energies, a Pearson technique is used. We show that experimental distributions can be better approximated, in a statistical sense, by Beta distribution or Pearson Type IV distribution rather than by a normal distribution predicted by the stochastic growth theory. This conclusion agrees with the results of numerical simulations that were previously performed with the use of a model for Langmuir wave propagation in unstable plasma with random inhomogeneities. The main reason for deviations of empirical distributions from a normal distribution is probably related to the effective number of regions, where the waves grow, which is not sufficiently large for the central limit theorem to be applicable under typical conditions in the Earth's electron foreshock. For two of seven events such deviations may be partially attributed to the effects of thermal Langmuir waves.