The nature of E1 low-energy strength (LES), often denoted as a "pygmy dipole resonance", is analyzed within the random-phase approximation (RPA) in 208 Pb using Skyrme forces in a fully self-consistent manner. A first overview is given by the strength functions for the dipole, compressional, and toroidal operators. More detailed insight is gained by averaged transition densities and currents where the latter provide a very illustrative flow pattern. The analysis reveals clear isoscalar toroidal flow in the low-energy bin 6.0-8.8 MeV of the LES and a mixed isoscalar/isovector toroidal/compression flow in the higher bin 8.8-10.5 MeV. Thus the modes covered by LES embrace both vortical and irrotational motion. The simple collective picture of the LES as a "pygmy" mode (oscillations of the neutron excess against the nuclear core) is not confirmed.

We investigate effects of pairing and of quadrupole deformation on two sorts of nuclear excitations, γ-vibrational K π = 2 + states and dipole resonances (isovector dipole, pygmy, compression, toroidal). The analysis is performed within the quasiparticle random-phase approximation (QRPA) based on the Skyrme energy functional using the Skyrme parametrization SLy6. Particular attention is paid to i) the role of the particle-particle (pp) channel in the residual interaction of QRPA, ii) comparison of volume pairing (VP) and surface pairing (SP), iii) peculiarities of deformation splitting in the various resonances. We find that the impact of the pp-channel on the considered excitations is negligible. This conclusion applies also to any other excitation except for the K π = 0 + states. Furthermore, the difference between VP and SP is found small (with exception of peak height in the toroidal mode). In the low-energy isovector dipole (pygmy) and isoscalar toroidal modes, the branch K π = 1 − is shown to dominate over K π = 0 − one in the range of excitation energy E < 8-10 MeV. The effect becomes impressive for the toroidal resonance whose low-energy part is concentrated in a high peak of almost pure K π = 1 − nature. This peculiarity may be used as a fingerprint of the toroidal mode in future experiments. The interplay between pygmy, toroidal and compression resonances is discussed, the interpretation of the observed isoscalar giant dipole resonance is partly revised.

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Two basic concepts of nuclear vorticity, hydrodynamical (HD) and Rawenthall-Wambach (RW), are critically inspected. As a test case, we consider the interplay of irrotational and vortical motion in isoscalar electric dipole E1(T = 0) modes in 208 Pb, namely the toroidal and compression modes. The modes are described in a self-consistent random-phase approximation (RPA) with the Skyrme force SLy6. They are examined in terms of strength functions, transition densities, current fields, and form factors. It is shown that the RW conception (suggesting the upper component of the nuclear current as the vorticity indicator) is not robust. The HD vorticity is not easily applicable either because the definition of a velocity field is too involved in nuclear systems. Instead, the vorticity is better characterized by the toroidal strength which closely corresponds to HD treatment and is approximately decoupled from the continuity equation.

The toroidal, compression and vortical dipole strength functions in semi-magic 124 Sn (and partly in doubly-magic 100,132 Sn) are analyzed within the random-phase-approximation method with the SkT6, SkI3, SLy6, SV-bas, and SkM * Skyrme forces. The isoscalar (T=0), isovector (T=1), and electromagnetic ('elm') channels are considered. Both convection jc and magnetization jm nuclear currents are taken into account. The calculations basically confirm the previous results obtained for 208 Pb with the force SLy6. In particular, it is shown that the vortical and toroidal strengths are dominated by jc in T=0 channel and by jm in T=1 and 'elm' channels. The compression strength is always determined by jc. Is also shown that the 'elm' strength (relevant for (e,e') reaction) is very similar to T=1 one. The toroidal mode resides in the region of the pygmy resonance. So, perhaps, this region embraces both irrotational (pygmy) and vortical (toroidal) flows.

The low-energy dipole excitations in ^{24}Mg are investigated within the Skyrme quasiparticle random phase approximation for axial nuclei. The calculations with the force SLy6 reveal a remarkable feature: the lowest I^{π}K=1^{-}1 excitation (E=7.92  MeV) in ^{24}Mg is a vortical toroidal state (TS) representing a specific vortex-antivortex realization of the well-known spherical Hill's vortex in a strongly deformed axial confinement. This is a striking example of an individual TS which can be much more easily discriminated in experiment than the toroidal dipole resonance embracing many states. The TS acquires the lowest energy due to the huge prolate axial deformation in ^{24}Mg. The result persists for different Skyrme parametrizations (SLy6, SVbas, SkM*). We analyze spectroscopic properties of the TS and its relation with the cluster structure of ^{24}Mg. Similar TSs could exist in other highly prolate light nuclei. They could serve as promising tests for various reactions to probe a vortical (toroidal) nuclear flow.

We review a recent progress in investigation of the isoscalar toroidal dipole resonance (TDR). A possible relation of the TDR and low-energy dipole strength (also called a pygmy resonance) is analyzed. It is shown that the dipole strength in the pygmy region can by understood as a local manifestation of the collective vortical toroidal motion at the nuclear surface. Application of the TDR as a measure of the nuclear dipole vorticity is discussed. Finally, an anomalous splitting of the TDR in deformed nuclei is scrutinized.