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Context. A universally accepted definition of what a vortex is has not yet been reached. Therefore, we lack an unambiguous and rigorous method for the identification of vortices in fluid flows. Such a method would be necessary to conduct robust statistical studies on vortices in highly dynamical and turbulent systems, such as the solar atmosphere. Aims. We aim to develop an innovative and robust automated methodology for the identification of vortices based on local and global characteristics of the flow. Moreover, the use of a threshold that could potentially prevent the detection of weak vortices in the identification process should be avoided. Methods. We present a new method that combines the rigor of mathematical criteria with the global perspective of morphological techniques. The core of the method consists in the estimation of the center of rotation for every point of the flow that presents some degree of curvature in its neighborhood. For that, we employ the Rortex criterion and combine it with morphological considerations of the velocity field. We then identify coherent vortical structures by clusters of estimated centers of rotation. Results. We demonstrate that the Rortex is a more reliable criterion than are the swirling strength and the vorticity for the extraction of physical information from vortical flows, because it measures the rigid-body rotational part of the flow alone and is not biased by the presence of pure or intrinsic shears. We show that the method performs well on a simplistic test case composed of two Lamb-Oseen vortices. We combine the proposed method with a state of the art clustering algorithm to build an automated vortex identification algorithm. (Abridged)