Radiation Spectrum of the Photospheric Emission for a Turbulent Relativistic Jet

The prompt $γ$-rays of gamma-ray bursts (GRBs) may originate from the photosphere of a relativistic jet. However, only a few GRBs have been observed with evident blackbody-like emission, for example, GRB~090902B. It has been demonstrated that internal dissipation processes, such as magnetic reconnection, can occur within the relativistic jet and thereby drive violent turbulence in the dissipation region. In this paper, we study the photospheric emission of a jet with turbulence below its photosphere. Here, the turbulence is modeled phenomenological under the assumption that the four-velocity of its eddies follows a Gaussian distribution in the jet's co-moving frame. It is found that the turbulence scatters photons to high energies and thus intensifies the emission in the high-energy regime. The corresponding distortion of the radiation field can be preserved if and only if the turbulence occurs in a region with incomplete photon-electron coupling. Consequently, the observed radiation spectrum can be reshaped into a Band-like spectrum.