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tokens: propagate over a distance that is an order of magnitude bigger than its diameter, as suggested by the weak nonlinear theory and numerical simulations (Stern Radko 1998 ), as well as by experiments in nonrotating water tanks with a rectangular shape (Voropayev et al. 1999 ). More recent water-tank experiments (Cariteau Flór 2006 ) have demonstrated that it still has a finite lifetime, because the secondary component of such strongly asymmetric vortex pair starts to wrap around the principal monopole, creating a strain that eventually gives rise to an elliptic instability due to the parametric resonance between the oscillation of inertial waves and the ambient strain field; for details, see Cariteau Flór ( 2006 ). It should be noted that the behavior of Chaplygin vortices in a fully 3D geometry is still an open question, because no reliable 3D simulations and experiments have been reported in the literature and we are unable to predict whether the dynamical 3D turbulence of the solar wind and of Earth’s magnetosheath is dominated by stable dipolar (Larichev Reznik 1976 ; Petviashvili Pokhotelov 1992 ; Kuvshinov et al. 1999 ) vortices, or by long-lived, mostly monopolar Chaplygin structures. Their monopolar components feature a jump of the plasma density and of the compressional magnetic field B z at the edge. In the presence of a compressional magnetic field, such jumps are associated with a current shaped as a thin hollow cylinder at r = r 0 , which probably reduces its stability in high- β turbulent plasmas. In a high- β plasma, we have found two distinct types of coherent vortices propagating in the perpendicular direction. The first is identified as a generalized shear Alfvén structure that possesses both the torsional and the compressional components of the magnetic field perturbation. It has a zero parallel electric field and, being homogeneous along its axis that is inclined toward the ambient magnetic field, it sweeps along the z -axis with a velocity u z that is in the Alfvén speed range; the transverse phase velocity is equal to , where θ is the (small) pitch angle between the structure and the background magnetic field. While in a sufficiently incompressible plasma , it has the structure of a moving Chaplygin vortex with a monopole superimposed on a dipole; in plasmas with β ∼ 1, its monopolar component is likely to be unstable and short-lived due to the emergence of a thin, (electron-scale) current layer and/or a sharp peak of charge density at its edge. The compressible magnetic field associated with such a vortex is restricted to the interior of the vortex core, while the transverse perturbation “leaks” from the core to larger distances. The second type of propagating structure obtainable analytically possesses a finite compressional magnetic field and parallel electric field, as well as perpendicular fluid velocity and density perturbation, but vanishing parallel ion fluid velocity and transverse perturbations of the magnetic