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The double plasma resonance (DPR) instability plays a basic role in thegeneration of solar radio zebras. In the plasma, consisting of the loss-conetype distribution of hot electrons and much denser and colder backgroundplasma, this instability generates the upper-hybrid waves, which are thentransformed into the electromagnetic waves and observed as radio zebras. In thepresent paper we numerically study the double plasma resonance instability fromthe point of view of the zebra interpretation. We use a 3-dimensionalelectromagnetic particle-in-cell (3-D PIC) relativistic model. First using themulti-mode model, we study details of the double plasma resonance instability.We show how the distribution function of hot electrons changes during thisinstability. Then we show that there is a very good agreement between resultsobtained by the multi-mode and specific-mode models, which is caused by adominance of the wave with the maximal growth rate. Therefore, for computationsin a broad range of model parameters, we use the specific-mode model. Wecompute the maximal growth rates of the double plasma resonance instability.The results are compared with the analytical ones. We find a very goodagreement between numerical and analytical growth rates. We also computesaturation energies of the upper-hybrid waves in a very broad range ofparameters. We find that the saturation energies of the upper-hybrid waves showmaxima and minima at almost the same values of$\omega_\mathrm{UH}/\omega_\mathrm{ce}$ as the growth rates. Furthermore, wefind that the saturation energy of the upper-hybrid waves is proportional tothe density of hot electrons. The maximum saturated energy can be up to onepercent of the kinetic energy of hot electrons. All these findings can be usedin the interpretation of solar radio zebras.

Double plasma resonance instability as a source of solar zebra emission