This complete introduction to the use of modern ray-tracing techniques in plasma physics describes the powerful mathematical methods generally applicable to vector wave equations in nonuniform media, and clearly demonstrates the application of these methods to simplify and solve important problems in plasma wave theory. Key analytical concepts are carefully introduced as needed, encouraging the development of a visual intuition for the underlying methodology, with more advanced mathematical concepts succinctly explained in the appendices, and supporting MATLAB code available online. Covering variational principles, covariant formulations, caustics, tunneling, mode conversion, weak dissipation, wave emission from coherent sources, incoherent wave fields, and collective wave absorption and emission, all within an accessible framework using standard plasma physics notation, this is an invaluable resource for graduate students and researchers in plasma physics.

Numerical studies of magnetohydrodynamic (MHD) instabilities with feedback control in reversed field pinches (RFPs) are presented. Specifically, investigations are performed of the stability of m=1 modes in RFPs with control based on sensing the normal and tangential magnetic fields at the resistive wall and applying two-parameter feedback proportional to these fields. The control scheme is based on that of [J. M. Finn, Phys. Plasmas 13, 082504 (2006)], which is here modified to use a more realistic plasma model. The plasma model now uses full resistive MHD rather than reduced MHD, and it uses three realistic classes of equilibrium parallel current density profiles appropriate to RFPs. Results with these modifications are in qualitative agreement with [J. M. Finn, Phys. Plasmas 13, 082504 (2006)]: the feedback can stabilize tearing modes (with resistive or ideal-wall) and resistive wall ideal modes. The limit for stabilization is again found to be near the threshold for ideal modes with an ideal-wall. In addition to confirming these predictions, the nature of the instabilities limiting the range of feedback stabilization near the ideal-wall ideal-plasma threshold are studied, and the effects of viscosity, resistive wall time, and plasma resistivity are reported.

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Abstract. In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, ∆ = ∆(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of ∆(t) and the orbital motion. In the second category the time step is a function of phase space variables ∆ = ∆(q, p). In this case, the system of equations to be solved is analyzed by introducing a new time variable τ with dt = ∆(q, p)dτ. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant ∆. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.

In three-dimensional magnetic configurations for a plasma in which no closed field line or magnetic null exists, no magnetic reconnection can occur, by the strictest definition of reconnection. A finitely long pinch with line-tied boundary conditions, in which all the magnetic field lines start at one end of the system and proceed to the opposite end, is an example of such a system. Nevertheless, for a long system of this type, the physical behavior in resistive magnetohydrodynamics (MHD) essentially involves reconnection. This has been explained in terms comparing the geometric and tearing widths [1,2]. The concept of a quasi-separatrix layer [3,4] was developed for such systems. In this paper we study a model for a line-tied system in which the corresponding periodic system has an unstable tearing mode. We analyze this system in terms of two magnetic field line diagnostics, the squashing factor [3][4][5] and the electrostatic potential difference used in kinematic reconnection studies [6,7]. We discuss the physical and geometric significance of these two diagnostics and compare them in the context of discerning tearing-like behavior in line-tied modes.

Numerical results are presented on control of magnetohydrodynamic (MHD) modes in reversed field pinches (RFPs) for a geometry with two resistive walls. We use measurements of the normal component of the magnetic field and introduce the use of both tangential components. In Richardson et al (2010 Phys. Plasmas 17 112511), RFP control studies were performed sensing the radial (normal) component of the magnetic field and a single tangential component just inside the wall, showing that it is possible to stabilize the MHD modes in an RFP for current up to the ideal plasma-ideal wall limit in that configuration. Here, we extend our modeling by including two resistive walls, in a configuration relevant to experiments such as RFX-mod, and measuring all three magnetic field components, i.e. including a second tangential component, as an exploratory effort. We present our study incrementally, starting with a single resistive wall, and conclude that with the first tangential sensor located inside the wall, the plasma can be stabilized up to the ideal plasma-ideal wall limit, as in Richardson et al. With the first tangential sensor outside the wall, stabilization is possible only up to the ideal wall-resistive plasma (tearing) limit. We then show that for experimentally relevant parameters the thin-wall approximation is indeed valid for the MHD modes of interest but invalid for the high-frequency magnetosonic mode (Richardson et al) driven by the (first) tangential component feedback. In fact, when a thick-wall formulation with realistic parameters is considered, the high-frequency magnetosonic mode is found to be destabilized only for a very high gain parameter, and we conclude that this mode can be ignored for an experimentally relevant analysis. Consequently, the plasma can be stabilized in a much larger region of feedback gain parameter space than found in Richardson et al . In the presence of two walls, with the first tangential component measured just outside the inner wall and with RFX-mod relevant time constants, we show that feedback control can stabilize the plasma at currents much larger than the ideal wall-resistive plasma limit. The current limit is still less, however, than the ideal plasma-ideal wall limit. Use of the second tangential component appears in all cases to lead to significantly different but not necessarily improved feedback stabilization. These results may lead to better understanding and improved stability properties in current-day RFP experiments through robust access to quasi-single-helicity states.

The ability to morph electrostatic plasma turbulence into electromagnetic has promising applications, including the possibility of actively influencing the near-Earth plasma state, aka the space weather. This dual (electrostatic/electromagnetic) nature is a fundamental property of plasma turbulence, which has not been well explored but could explain many phenomena including the formation of a resonant cavity that can amplify the turbulence energy. The upcoming Space Measurement of A Rocket-Released Turbulence (SMART) mission is designed to understand the evolution of plasma turbulence and the nonlocal consequences of its dual nature. This includes the flow of energy into all possible wavelengths, as well as the transport of energy over a large geographical volume. The resulting energy redistribution in both waves and particles in an extended geographical volume creates a unique electromagnetic environment, which is important for space weather.

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