Wave vortices constitute a large family of wave entities, closely related to phase singularities and orbital angular momentum (OAM). So far, two main classes of localized wave vortices have been explored: (i) transversely localized monochromatic vortex beams that carry well-defined longitudinal OAM and propagate and diffract along the longitudinal $z$ axis in space, and (ii) two-dimensional (2D)-localized spatiotemporal vortex pulses that carry the more elusive transverse (or tilted) OAM and propagate and diffract along both the $z$ axis and time. Here, we introduce another class of wave vortices which are localized in a 2D $(x,y)$ plane, do not propagate in space (apart from uniform radial deformations), and instead propagate and diffract solely along time. These vortices possess well-defined transverse OAM and can naturally appear in 2D wave systems, such as surface polaritons or water waves. We provide a general integral expression for time-diffracting 2D wave vortices, their underlying ray model, and examples of approximate and exact wave solutions. We also analyze the temporal Gouy phase closely related to the rotational evolution in such vortices. Finally, we show that time-diffracting 2D vortices can provide strong spatiotemporal concentration of energy and OAM at subwavelength and oscillation-period scales.

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