Non-stationary, rotational, linear surface waves are considered where the underlying sheared current has constant vorticity. A time-dependent study is presented on the formation and persistence of a Kelvin cat-eye structure in the presence of bottom topography. The flow domain is two-dimensional, which allows for the use of a conformal mapping and working in a computational flat-bottom domain. In some cases an initial disturbance is prescribed, while in others the waves are generated from rest. Submarine particle dynamics numerically captures the horizontal critical layer, defined by closed orbits separating the fluid domain into two disjoint regions. In the wave’s moving frame, these recirculation regions are structured in the form of Kelvin cat-eyes. Owing to the interaction with topography, the usual travelling-wave formulation is abandoned and the critical layer is identified through a non-stationary set of equations. The respective time-dependent Kelvin cat-eye structure dynamically adjusts itself at the onset of wave–topography interaction, without losing its integrity. The formation of a Kelvin cat-eye structure is also studied in the case where the surface is initially undisturbed. Surface waves are generated from either the current–topography interaction or by a pressure distribution suddenly imposed along the free surface. Under the pressure forcing, an isolated cat-eye forms with a single recirculation region beneath the wave.