An adiabatic mode solution is derived for azimuthally advected gravity waves. Horizontal variations in the medium are assumed to be sufficiently gradual so that the coupling of energy between modes can be neglected. The wind speed is assumed to be small relative to the wave speed. The mode coefficients satisfy the same horizontal wave equation for both gravity and acoustic waves, which satisfy three-dimensional wave equations that are fundamentally different. The horizontal wave equation can be solved efficiently with the parabolic equation method. The adiabatic mode solution is used to model the propagation of gravity waves in the atmosphere of Jupiter.