We construct a lattice Boltzmann model of a single-layer, "reduced gravity" ocean in a square basin, with shallow water or planetary geostrophic dynamics, and boundary conditions of no slip or no stress. When the volume of the moving upper layer is sufficiently small, the motionless lower layer outcrops over a broad area of the northern wind gyre, and the pattern of separated and isolated western boundary currents agrees with the theory of Veronis (1973). Because planetary geostrophic dynamics omit inertia, lattice Boltzmann solutions of the planetary geostrophic equations do not require a lattice with the high degree of symmetry needed to correctly represent the Reynolds stress. This property gives planetary geostrophic dynamics a significant computational advantage over the primitive equations, especially in three dimensions.