(TH/P3-4) Equilibria and Stability in Partially Relaxed Plasma-Vacuum Systems

Abstract.  Since the early working of Grad (Phys. Fluids 10(1), 1967), the existence and stability of 3D magnetic configurations in ideal MHD has bedeviled magnetic containment theory. The 3D problem is related to a lack of integrability of the Hamiltonian poloidal flux function. In non-integrable systems, this Hamiltonian can be written as the sum of an integrable part plus a non-integrable perturbation. If the non-integrable perturbation is vanishingly small (for instance, a tokamak), flux surfaces exist everywhere. As the perturbation is increased, flux surfaces with rational rotational transform are destroyed. The last surviving flux surfaces are KAM surfaces with strongly irrational rotational transform. In regions between these KAM surfaces, the magnetic field ergodically spans the volume, and so the pressure gradient is zero. In this work, we develop a stepped-pressure profile model, in which the pressure across the plasma is piece-wise constant, and the field obeys the Beltrami equations in regions between ideal MHD barriers, at which pressure jumps occur. As a first step en-route to fully 3D ideal MHD solutions, we solve for a multiple-interface cylindrical plasma-vacuum system, where analytic solutions are available. In turn, these permit a detailed exploration of the equilibrium constraints, magnetic configuration and stability. For a given number of interfaces, we explore optimization of the field configuration (via variation of constraints) to achieve peak beta. Also, we compare the convergence of the field structure of a smooth pressure profile equilibrium to a stepped pressure profile as the number of interfaces increases. The existence of advanced tokamak-like magnetic configurations (weak reverse shear in core) also prompts an investigation into an energy based study of the existence of Internal Transport Barriers (ITB’s), particularly around the q=2 and q=3 surfaces, where ITB's are often observed. Finally, in separate working, an algorithm to construct stepped pressure profile equilibria in arbitrary 3D geometry is developed, and an example in 3D computed.

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