Density, with the planetary spirals indicated (left), and meridional velocity (right) for various altitudes in the disk. Phase predictions for the primary Lindblad arm (Eq. (13), thick line) and m = 3 buoyancy arm (Eq. (15), thin line) are overplotted. In all panels, the cylindrical radius, R = r sin θ, is plotted on the x-axis and the azimuthal angle, θ, on the y-axis. Due to the relatively rapid thermal relaxation in this simulation, buoyancy spirals are markedly weaker than Lindblad spirals in all cases.