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A large-scale ordered magnetic field plays a very important role in theformation and acceleration of a gamma-ray burst (GRB) jet. During the GRBprompt phase, some dissipation processes may happen and disturb the magneticfield, making the field become random to some extent. Therefore, a mixedmagnetic field consisting of an ordered component and a random component isplausible for the early afterglow phase. Here we investigate the polarizationevolution and light curve of an afterglow under such a 3-dimensional mixedmagnetic field. Three kinds of ordered component (i.e., aligned, toroidal andradial) are discussed. We find that the three cases are distinguishable throughjet polarization evolution. The polarization angle for a 3D mixed magneticfield with an aligned ordered component can evolve gradually but only changesabruptly by $90^\circ$ in the toroidal and radial cases. Furthermore, duringthe reverse shock crossing time, the polarization degree (PD) can be non-zerofor the toroidal case but roughly zero for the radial case. Since an alignedcomponent in a jet corresponds to a magnetar central engine and a toroidalcomponent corresponds to a black hole, GRB central engines are distinguishablethrough polarization observations even if the magnetic field is mixed in a jet.In addition, our polarization calculation can be applied to GRB 170817Aassociated with GW170817. Using the recently-observed PD upper limit $12\%$ ofGRB 170817A at $t=244$ days, the magnetic field strength ratio of the orderedto random components in this afterglow is constrained to be $\lesssim0.9$.

Polarization with a 3-dimensional Mixed Magnetic Field and Its Application to GRB 170817A