Relaxed MHD states of a multiple region plasma

Recently, Hole et al [1] proposed a model for a partially relaxed plasma-vacuum system. The purpose of the model, which abandons all but a small number of flux surfaces, is to provide a mathematically rigorous foundation for ideal MHD equilibria in 3D configurations. The model appeals to both chaotic field lines, that flatten the pressure gradient in chaotic regions, and Taylor relaxation, which force the plasma gradient to be zero in Taylor-relaxed regions. The model consists of a stepped pressure profile, where the steps correspond to ideal MHD barriers across which can be supported a pressure or field jump, or a jump in rotational transform. Our overarching objective is the development of an equilibrium solver for 3D plasmas built on a stepped pressure profile model. In the 3D case, we envisage that the barriers can be chosen to be non-resonant KAM surfaces that survive the onset of field line chaos intrinsic to 3D equilibria. In between the interfaces, the field is Beltrami, such that $\nabla\times\mathbf{B}=\mu\mathbf{B}$. The boundary condition across the interfaces is the continuity of total pressure $p+B^{2}/2\mu_{0}$. Such a model, which we term a MRXMHD (multiple relaxation regions MHD) model, raises a number of questions. How should the equilibrium be constrained? How much jump in pressure and/or rotational transform $\,\iota\!\!$- can each interface support? Are the interfaces stable to deformation? Can the class of stability shed information onto other quasi-relaxed phenomena?

Previous work has focused on the equilibrium constraints [2, 3], construction of a numerical algorithm for calculation of Beltrami fields between interfaces in 3D configurations [4], and a variational principle for the equilibrium and stability of the multiple interface configuration in cylindrical plasmas [1]. We have also explored the relationship between relaxed plasma equilibrium models discussed here, and entropy related plasma self-organisation principles [5]. The motivation of this paper is to understand the nature of MRXMHD modes identified from the variational principle. Our work complements a separate in-press publication [6] that unifies relaxed and ideal MHD principles for constructing global solutions comprising mixed relaxed and ideal regions.

Recently, Tassi et al [7], performed a tearing mode stability treatment on stepped $\mu$ force-free equilibria close to Taylor relaxed states. Their motivation was to develop a mechanism for the formation of cyclic Quasi-Single-Helicity (QSH) states observed in Reverse Field Pinches [8]. They consider a cylindrical plasma divided into two different Beltrami regions, and encased in a perfectly conducting shell, and compute the tearing mode stability parameter $\Delta^{\prime}$ at a resonant radius $r_{s}$ for a helical flux perturbation $\chi_{1}(r)=mB_{z1}(r)-\kappa rB_{\theta 1}(r)$. Here, $r$ is the radial coordinate, and $m$ and $\kappa$ the poloidal and axial wave number. Tassi et al find critical values of the jump in $\mu$, beyond which the RFP-like plasma is unstable. Based on these, they postulate the QSH state may be viewed as a small, cyclic departure from a Taylor-relaxed state.

In this work, we extend the tearing mode stability treatment of Tassi et al [7] to plasmas with finite pressure and a vacuum region, and compare stability conclusions of our variational treatment to that of a tearing mode stability analysis. Our paper is arranged as follows : Sec. 2 summarises the variational model of stepped pressure profile plasmas, presented in Hole et al [1], and introduces a tearing mode model. Section 3 treats MRXMHD plasmas in cylindrical geometry, yielding stability parameter expressions for both the variational and tearing mode treatments. In Sec. 4, we compute stability for an example configuration, draw comparisons between the stability conclusions based on variational and tearing mode treatments, and explore marginal stability limits in wave-number space as a function of pressure. Finally, Sec. 5 contains concluding remarks.

The system comprises $N$ Taylor-relaxed plasma regions, each separated by an ideal MHD barrier. The outermost plasma region is enclosed by a vacuum, and encased in a perfectly conducting wall. Figure 1(a) shows the geometry of the system, and introduces the nomenclature used to describe the region and interfaces. The regions $\mathcal{R}_{i}$ comprise the $N$ plasma regions $\mathcal{R}_{1}=\mathcal{P}_{1},...,\mathcal{R}_{N}=\mathcal{P}_{N}$ and the vacuum region $\mathcal{R}_{N+1}=\mathcal{V}$. Each plasma region $\mathcal{P}_{i}$ is bounded by the inner and outer ideal MHD interfaces $\mathcal{I}_{i-1}$, and $\mathcal{I}_{i}$ respectively, whilst the vacuum is encased by the perfectly conducting wall $\mathcal{W}$.

Solutions in an azimuthally-symmetric, axially-periodic cylinder (with axial periodicity length $L=2\pi R$) are available in Hole et al [1]. In the cylindrical co-ordinate system $(r,\theta,z)$ they are:

where $k_{i},d_{i}\in\Re$, and $J_{0},J_{1}$ and $Y_{0},Y_{1}$ are Bessel functions of the first kind of order 0, 1, and second kind of order 0, 1, respectively. The terms $B_{\theta}^{V}$ and $B_{z}^{V}$ are constants. The constant $d_{1}$ is zero in the plasma core $\mathcal{P}_{1}$, because the Bessel functions $Y_{0}(\mu_{1}r)$ and $Y_{1}(\mu_{1}r)$ have a simple pole at $r=0$. Radius is normalized to the plasma-vacuum boundary, located at $r=1$. The equilibrium is constrained by the $4N+1$ parameters:

where $r_{i}$ are the radial positions of the $N$ ideal MHD barriers, and $r_{w}$ is the radial position of the conducting wall. Equivalent representations, and the mapping between these solutions has been discussed in earlier work [1-3].

We have compared stability conclusions using variational and tearing mode treatments for an illustrative configuration. The example chosen is guided by earlier detailed working [1], where the core Lagrange multiplier was $\mu_{1}=2$. The first interface is placed at $r=0.5$, and the axial periodicity length chosen to be $20\pi$, such that the effective aspect ratio $r/R=1/10$ is small. The jump in safety factor between the internal interface and the plasma-vacuum boundary has been chosen to resemble Hole et al , subject to the different $R$ values used for the two treatments ($R=1/(2\pi)$ in Hole et al ). We have used $\mu_{2}=3.6$, which requires $d_{2}/k_{2}=0.77$ for the rotational transform profile to be continuous. A second motivation for this choice is the similarity to $q$ profiles of high confinement reverse field pinches, such as the Madison Symmetric Torus [12] and RFX-mod [13], although the change in $\mu$ is greatly exaggerated in this work. The plasma pressure is selected by the parametrization $\beta_{1}=p_{1}/(B_{V}^{2}/2\mu_{0}),\beta_{2}=p_{2}/(B_{V}^{2}/2\mu_{0})$. Except for the final scan over $\beta$, a pressure-less plasma is assumed (i.e. $\beta_{1}=\beta_{2}=0$). Figure 1(b) shows the $q$ profile for the chosen equilibrium, where $q=1/\mbox{$\,\iota\!\!$-}$.

Figure 2(a) is a dispersion curve for $m=1$ modes, showing $\lambda$ computed using the variational treatment, and $-r_{s}\Delta^{\prime}$ computed for modes resonant within the plasma. Marginal stability corresponds to $\lambda=0$ and $\Delta^{\prime}=0$: these overlap identically in Fig.2(a). Modes with $n=-16$ and $n=12$ corresponds to a perturbation near-resonant with the outer and inner interfaces ($q(r_{2})\approx-1/16$ and $q(r_{1})\approx 1/12$) respectively.

In our variational treatment, we have prescribed no relationship between $\xi$ and $\mathbf{b}$ in the relaxed regions. As such, excepting at the ideal interfaces, field line resonance in such plasmas is not explicitly resolved. Expressions can however be constructed which provide an estimate of the localization of the mode $r_{s,\mathrm{eff}}$, and a convenient choice is $r_{s,\mathrm{eff}}^{2}=\sum_{i=1}^{N}(r_{i}X_{i})^{2}$, where the eigenvectors are normalised such that $\mathbf{X}\cdot\mathbf{X}^{H}=1$, with $H$ the Hermitian.

Figure 2(b) shows a comparison of $r_{s,\mathrm{eff}}$ to $r_{s}$, in which modes unstable to variational and tearing modes have been identified. Agreement between $r_{s,\mathrm{eff}}$ and $r_{s}$ is qualitatively good in the interval over which the plasma in unstable, and excellent near the interfaces. The $n=-16$ and $n=9$ modes are near resonant with the outer and inner interface, respectively. As shown in Fig. 3(a), a stability scan with wall radius indicates that in the limit $r_{w}\rightarrow 1$, modes for $n<0$ are wall-stabilized. In the limit that the outer interface is made resonant with the $n=-16$ tearing mode (for example by changing $R$), $\Delta^{\prime}\rightarrow\infty$. This mode is the current driven external kink of ideal MHD. Conversely, the unstable range for $n\geq 12$ is only very weakly affected by the wall position. If the inner interface is mode resonant with the perturbation, $\Delta^{\prime}\rightarrow\infty$, and the mode is ideal unstable. This is the internal kink of ideal MHD.

Recently, Mills et al [6] demonstrated that one can unify ideal and relaxed variational treatments by extending the relationship between $\mathbf{b}$ and $\xi$ through the Newcomb gauge $\mathbf{a}=\mbox{\boldmath{$\xi$}}\times\mathbf{B}$. If this variational treatment is followed, rational surfaces do explicitly enter the expression for $\xi$ as derived from $b_{r}$, the variation in the radial part of the magnetic field, and so both relaxed and tearing modes become localised at the resonant surface $r_{s}$.

The findings of Fig. 2, obtained for a pressure-less plasma, agree with that of Furth et al [14], who showed that for cylindrical pressure-less plasma with no vacuum, $\Delta^{\prime}$ is linear with the second variation in the magnetic energy driving the tearing mode. We have also compared stability conclusions drawn from variational and tearing mode treatments as a function of $\beta$. We find that while both variational and tearing mode treatments have the same stability trends with $\beta$, the marginal stability limit of the variational treatment (for a given $m$ and $n$) is lower than that of tearing modes. A study is ongoing into the cause of this discrepancy, as well as formally relating $\delta^{2}W$ to $\delta^{\prime}$

Finally, Fig. 3(b) is a plot of the marginal stability boundary ($\lambda=0$) in $n-\beta$ space for $m=1$ eigenmodes of the variational treatment. The two pressure profile configurations that have been studied are $\beta_{1}=0$ and $\beta_{2}=0$. Trends in the marginal stability boundary can be understood by relating the radial location of the mode resonant surface to the analog of radial pressure gradient in the MRXMHD model: the sign and magnitude of nearby pressure jumps. For $n>0$ modes resonant near the first interface, an increasing core pressure increases the pressure drop across the first interface, and so destabilises the plasma. Conversely, increasing the edge pressure leads to a pressure jump across the first interface, and so stabilises the internal modes. For $\beta_{2}>0.3$ all $m=1$ internal modes ($n>0$) are completely stabilised. For the $n<0$ modes resonant near the edge, changes in the core pressure have little effect, while increasing the edge pressure destabilises the plasma.

We have computed the stability of multiple relaxation region MHD (MRXMHD) plasmas using both a variational and a tearing mode treatment, evaluated in a periodic cylindrical configuration. The marginal stability conclusions of the two treatments for a zero $\beta$ plasma, as well as the trends with $\beta$, appear to be identical, in agreement with earlier analytic working by Furth et al [14]. Some discrepancy exists between the marginal stability boundaries of variational and tearing mode treatments for nonzero $\beta$, with the stability limit of variational plasmas lower than that of tearing modes. A study is underway to resolve this discrepancy.

The overarching aim of this work is to elucidate the nature of perturbations available to MRXMHD equilibria, which in turn, are motivated by our quest for mathematically rigorous solutions of ideal MHD force balance in 3D geometry. Our working builds of Tassi et al to nonzero $\beta$ multiple region relaxed plasmas with vacuum, and complements Mills et al , who demonstrated that that one can unify ideal and relaxed variational treatments through the Newcomb gauge. Combined, these results suggest that MRXMHD configurations which are stable to ideal perturbations plus tearing modes are automatically in a stable equilibrium state.

In ongoing work we are developing a faster numerical algorithm for the construction of 3D MRXMHD plasmas, and exploring the maximum pressure jump an interface can support before it is destroyed by chaos.