  We study in a fully kinetic framework the generation of seed magnetic fields
through the Weibel instability driven in an initially unmagnetized plasma by a
large-scale shear force. We develop an analytical model that describes the
development of thermal pressure anisotropy via phase mixing, the ensuing
exponential growth of magnetic fields in the linear Weibel stage, and its
saturation when the seed magnetic fields become strong enough to instigate
gyromotion of particles and thereby inhibit their free-streaming. The predicted
scaling dependencies of the saturated seed fields on key parameters (e.g.,
ratio of system scale to electron skin depth, the forcing amplitude) are
confirmed by 3D and 2D particle-in-cell simulations using an electron-positron
plasma. This work demonstrates the spontaneous magnetization of a collisionless
plasma through large-scale motions as simple as a shear flow, and therefore has
important implications for magnetogenesis in dilute astrophysical systems.
