In a three-dimensional incompressible inviscid fluid, the most representative traveling waves are vortex rings. Their two-dimensional analogues are counter-rotating vortex dipoles. In general, a certain portion of fluid containing these waves is carried along at the same speed; we refer to such a region as the vortex atmosphere. In this talk, we compare the properties of vortex atmospheres associated with several 3D rings and 2D dipoles. In particular, we establish the existence and stability of a Sadovskii vortex patch, a special 2D dipole in which the vorticity is evenly distributed within its atmosphere. Here, the Sadovskii vortex patch arises as a maximizer of the kinetic energy under a small-impulse condition together with an upper bound on the circulation. Its stability is obtained via Lions’ concentration–compactness argument. This talk is based on joint works with Ken Abe, Kyudong Choi, In-Jee Jeong, and Kwan Woo.