Dissipation efficiency of internal shocks in GRB

Previous simulations of internal shocks in GRB ejecta (e.g. [2-5]) indicated that a small fraction epsilon similar to 0.1 of the ejecta kinetic energy is dissipated in this way. The low e can lead to an energy crisis of the model: prompt GRB should then be strongly dominated by the early afterglow from the external shock, in conflict with observations. More detailed analysis shows that epsilon depends on the initial amplitude of the velocity fluctuations in the ejecta and at high amplitudes epsilon can approach 100% [1]. These results are briefly summarized here and then a simple theory of outflows with Gaussian velocity fluctuations is developed. It is shown that the history of internal dissipation is governed by the spectrum of the fluctuations.