The combined effect of winds from a cluster of stars in the central 0.8 pc of the Galaxy is modeled as uniform power and mass input over the central region. The flow becomes supersonic outside the central region, and the expected decrease in pressure is in approximate accord with observations. The pressure variations on a larger scale suggest that the Galactic center wind passes through a shock front at a radius of a few pc, leading to a shocked wind bubble on a scale of 100 pc. The tangential magnetic field can come to dominate the pressure in the shocked wind flow even if the energy density of the magnetic field in the initial wind is only 0.1 percent of the wind kinetic energy density. The magnetic region produced in this way may be related to some of the apparently magnetized structures observed in the central region of the Galaxy.