Explosive eruption cycles in a rotating Z-pinch

TL;DR

The paper investigates metastable MHD transport barriers in a rotating Z-pinch, drawing parallels with tokamak edge pedestals and ELMs. It develops a first-principles energy framework where available energy is minimized via flux-tube interchanges, reducing to a Linear Sum Assignment Problem in the low-$\beta$ limit and extended by a heuristic for finite $\beta$. The authors demonstrate cyclic pedestal formation, collapse via rapid filament eruptions, and pedestal rebuilding in high-resolution simulations, with energy and ejected-mass quantities predicted from combinatorial optimization. This work provides a principled link between microphysical interchange dynamics and macroscopic energy release, offering a framework applicable to predicting transport barriers and cyclical eruptive behavior in magnetized plasmas.

Abstract

A transonic shear flow directed along magnetic field lines can linearly stabilize a steep pressure gradient in a confined magnetohydrodynamic (MHD) plasma. In Z-pinch geometry, we show that, like the edge pedestal in tokamak devices, this transport barrier -- which we call the ``MHD pedestal'' -- is metastable, i.e., unstable to finite-amplitude displacements of flux tubes. We simulate the slow formation of an MHD pedestal in a heated and sheared Z-pinch, which collapses on reaching a critical height, expelling an order-unity fraction of the confined thermal energy. The MHD pedestal then rebuilds and the process repeats, in a manner analogous to the ELM cycle seen in fusion experiments. We show that the available energy of the metastable equilibrium, and the most energetically favorable amount of ejected plasma, can be calculated from first principles via combinatorial optimization of flux-tube interchanges.

Figures (6)

• Figure 1: Panel (a): Visualization of a rotating, hard-core Z-pinch. Fast-flowing cold plasma at large radii stabilizes a sharp gradient of thermal pressure, confining hot plasma to the red annulus. Panel (b): Numerical example of an equilibrium at marginal linear stability [$\mathcal{L}=0$ in Eq. \ref{['linear_stability_low_beta']}] with a $\tanh$ profile for $\ell(r)$ and $\chi(r)=\mathrm{const}$ (red and pink lines). The blue line shows the velocity of a flux tube erupting from $r<R_{\mathrm{ped}}$ and the black dashed line shows Eq. \ref{['singlefluxtubemotion']}.
• Figure 2: Left panel: Numerical simulation without toroidal flow damping [i.e., $\tau_{\mathrm{drag,\phi}}\to \infty$ in Eq. \ref{['utheta_damping']}]. The plasma is driven to an interchange-marginal state with $s/\chi\sim \mathrm{const}$. Right panel: Numerical simulation with $\tau_{\mathrm{drag,\phi}}\sim \tau_{\mathrm{heat}}$. The plasma is interchange-marginal both within and outside of a hot core, but with a sharp gradient of $s/\chi$ between these regions (the "MHD pedestal"). Annotations above each panel indicate the spatial localizations of the sources and sinks in Eqs. \ref{['modelthermal']} and \ref{['utheta_damping']}. A movie showing the transition from L-mode to H-mode is available at https://youtu.be/Ayh1-HcQd54.
• Figure 3: Eruption event in the H-mode simulation shown in the right panel of Fig. \ref{['fig:LH_comparison']}, at a later time. The first image in panel (a) shows the state of the plasma before the eruption; its $\beta=2p/B^2$ profile (averaged over $z$) is shown in red in panel (b). The next three images show, from left to right: a mid-eruption state; a post-eruption state [blue line in panel (b)]; and the post-eruption state after interchange-driven mixing in the core has homogenized the $s/\chi$ distribution there [thick gray line in panel (b)]. The times given are in units of $R_{\mathrm{heat}}/c_{s,0}(R_{\mathrm{heat}})$. The dashed, dash-dotted and dotted lines show the subsequent rebuild of the pedestal to marginal linear stability, before the next eruption in the cycle. A movie version of this figure is available at https://youtu.be/8U26HgsziPM.
• Figure 4: Panel (a): Interchange acceleration [Eq. \ref{['interchange_force']}] as a function of the displaced flux tube's radius $r_2$. Eruptions coincide with saddle-node bifurcations. Panel (b): Comparison of the available potential energy (APE) of the simulation state (blue line) with the poloidal kinetic energy (PKE; red line). The dashed red line shows the PKE of a simulation with $\gamma = 2$, for which the MHD pedestal is not metastable. Panel (c): Available heat load (AHL) compared with actual thermal drain in the simulation. Quantities in panels (b) and (c) are measured in units of $\Delta E_H$, defined to be the excess thermal energy in H-mode vs. L-mode.
• Figure 5: Illustration of the procedure used to obtain a 1D state for use in solution of the assignment problem, as explained in the text. Panel (a) shows the simulation state at a particular instant in time [the same one as in the final panel of Fig. \ref{['fig:collapse']}(a)]. Panel (b) shows the same state downsampled into $3000$ equal-flux macrocells over $N_{\mathrm{stripes}}=55$ radial stripes, as described in the text. Panel (c) shows the 1D representation of the simulation state (black line) and the corresponding ground state found by solving the LSAP (blue line).