Abstract We study the formation of quasi-two-dimensional (thin pancake) vortex structures in three-dimensional flows and of quasi-one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at large Reynolds numbers, when their evolution is described in the leading order by the Euler equations for an ideal incompressible fluid. We show numerically and analytically that the compression of these structures and, as a consequence, the increase in their amplitudes are due to the compressibility of the frozen-in-fluid fields: the field of continuously distributed vortex lines in the three-dimensional case and the field of vorticity rotor lines (divorticity) for two-dimensional flows. We find that the growth of vorticity and divorticity can be considered to be a process of overturning the corresponding fields. At high intensities, this process demonstrates a Kolmogorov-type scaling relating the maximum amplitude to the corresponding thicknesses-to-width ratio of the structures. The possible role of these coherent structures in the formation of the Kolmogorov turbulent spectrum, as well as in the Kraichnan spectrum corresponding to a constant flux of enstrophy in the case of two-dimensional turbulence, is analyzed.