Primitive streak formation in the chick embryo involves significant coordinated cell movement lateral to the streak, in addition to the posterior-anterior movement of cells in the streak proper. Cells lateral to the streak are observed to undergo 'polonaise movements', i.e. two large counter-rotating vortices, reminiscent of eddies in a fluid. In this paper, we propose a mechanism for these movement patterns which relies on chemotactic signals emitted by a dipolar configuration of cells in the posterior region of the epiblast. The 'chemotactic dipole' consists of adjacent regions of cells emitting chemo-attractants and chemo-repellents. We motivate this idea using a mathematical analogy between chemotaxis and electrostatics, and test this idea using large-scale computer simulations. We implement active cell response to both neighboring mechanical interactions and chemotactic gradients using the Subcellular Element Model. Simulations show the emergence of large-scale vortices of cell movement. The length and time scales of vortex formation are in reasonable agreement with experimental data. We also provide quantitative estimates for the robustness of the chemotaxis dipole mechanism, which indicate that the mechanism has an error tolerance of about 10% to variation in chemotactic parameters, assuming that only 1% of the cell population is involved in emitting signals. This tolerance increases for larger populations of cells emitting signals.