The prolonged temporal evolution of a magnetohydrodynamic (MHD) vortex influenced by a steady magnetic field along its axis is investigated both numerically and experimentally within a cubic domain. We directly validate the theory proposed by Davidson [J. Fluid Mech. 299, 153 (1995)] through numerical analysis: the angular momentum parallel with a magnetic field of a single vortex is conserved, whereas the perpendicular angular momentum decayed exponentially during a free-decay evolution. Moreover, as observed by Sreenivasan and Alboussière [J. Fluid Mech. 464, 287 (2002)], the initial linear phase, characterized by the dominant influence of the Lorentz force over the inertial force ($N_t\gg 1$, where $N_t$ represents the true interaction parameter), and the subsequent nonlinear phase, marked by an equilibrium between the Lorentz force and the inertial force ($N_t\sim 1$), are successfully corroborated through numerical and experimental means, particularly at a substantial initial interaction parameter ($N_0>1$, where $N_0$ denotes the initial interaction parameter). The transition time from linear phase to nonlinear phase varies with the square of $N_0$. The relative magnitude of the Lorentz force and the inertial force plays a pivotal role during the free-decay evolution, and we propose the governing equations for such a flow. Nevertheless, numerical simulations and experiments indicate that these two phases and the ensuing transition, as depicted by the velocity decay curve, are primarily limited to the vicinity of the vortex's periphery, exhibiting a certain degree of locality. By considering the scaling of the global kinetic energy $[\sim {(t/\tau )}^{-0.8}$, where $t$ denotes the physical evolution time normalized by the Joule time $\tau ]$, the Joule dissipation $[\sim {(t/\tau )}^{-2}]$, and the parallel length $[\sim {(t/\tau )}^{3/5}]$, it becomes evident that the global behavior of a vortex bears greater resemblance to the nonlinear phase rather than undergoing a direct transformation from one phase to another, even when subjected to a substantial initial interaction parameter.

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• Magnetohydrodynamics
• Vortex breakdown
• Vortex flows