High-resolution three-dimensional numerical experiments show that initially balanced (void of waves) geophysical flows, static and inertially stable, generate spiral patterns of small-amplitude inertia–gravity waves (IGWs). The spiral wave patterns are due to the spontaneous generation of IGW packets emitted from fluid volumes (the IGW sources) experiencing large local changes of potential vorticity. The IGW packets spread away from the vortical flow and cause spiral wave patterns of the same sense of spiralling, cyclonic or anticyclonic, as the moving IGW sources. The spiral patterns are noticeable in the vertical velocity in deep layers, away from the large-amplitude balanced vertical velocity. The generation of the spiral wave patterns is illustrated through several examples, namely, the single ellipsoidal vortex (cyclone and anticyclone), the merging of two spherical vortices, the dipole, and the anticyclonic shear instability.