Nonlinear Magnetohydrodynamics in the Dag Confinement Configuration

H.R. Strauss
New York University, New York, New York
D. Weil
Hebrew University, Jerusalem, Israel

Abstract. The Dag magnetic fusion confinement configuration is a spheromak - like toroidal device. It consists of central vertical current channel, and an outer toroidal chamber with a toroidal current. It has a special magnetic topology. Whether this has consequences for plasma confinement is a motive for this study. A restricted class of computations, using the M3D code [W. Park et al., Phys. Plasmas 6, 1796 (1999)], indicate stability for $\beta < 15\%.$ For higher $\beta,$ the simulations exhibit turbulent magnetic behavior similar to spheromaks and reverse field pinches. A reverse field pinch - like variant should be capable of a less turbulent start up and higher $\beta$ than the spheromak - like Dag. The Dag [1] magnetic fusion confinement configuration is a proposed spheromak - like toroidal device. It consists of central vertical current channel, and an outer toroidal chamber with a toroidal current. It has a special magnetic topology [2]. Whether this has consequences for plasma confinement is a motive for this study. A restricted class of computations done so far, using the M3D [3,4] code, indicate ideal and resistive magnetohydrodynamic (MHD) stability for $\beta < 15\%.$ For higher $\beta$, the simulations exhibit turbulent magnetic behavior similar to spheromaks and reverse field pinches. A reverse field pinch (RFP) variant should be capable of a less turbulent start up and higher $\beta$ than the spheromak - like Dag.

The M3D (Multi-level 3D) project carries out simulation studies of plasmas using multiple levels of physics, geometry, and grid models. The present study is done with a resistive MHD model. M3D combines a two dimensional unstructured mesh with finite element discretization in poloidal planes [5], with a pseudo spectral representation in the toroidal direction. The code has both distributed memory and shared memory parallelization options. The code has had extensive benchmarking, especially for ideal and resistive physics models. The simulations assume a conducting wall boundary condition. An unstructured mesh, similar to the mesh used in the calculations, but with about $1/4$ the number of mesh points for clarity, is shown in Fig.. The part of the mesh adjacent to the boundary was made using the ellipt2d package, which incorporates the Triangle mesh generation code [6]. A difference from the original Dag is presence of a small central hole, which is required by the present version of M3D. A top view of the mesh, showing the small central hole, is shown in Fig..

The Dag configuration is obtained as a relaxed Taylor[7] state. Like the spheromak or RFP, it is a solution of the force free condition

$${\bf J}= \alpha {\bf B}$$ (1)

With an equilibrium magnetic field of the usual form

$${\bf B}= \nabla \psi \times \nabla \phi + I \nabla \phi$$ (2)

then $\psi$ satisfies

$${ \partial ^2\psi \over \partial R^2 } -{1\over R} { \partial... ... } + { \partial ^2\psi \over \partial Z^2 } = - \alpha^2 \psi$$ (3)

and

$$I = \alpha \psi$$ (4)

A special solution has the form [1]

$$\psi = \sum_{k=0} {a_k R \over (\alpha^2 - k^2)^{1/2}} J_1 [R(\alpha^2 - k^2)^{1/2}] \cos(kZ)$$ (5)

A Dag configuration has one elliptic and one hyperbolic axis, with toroidal magnetic field in opposite directions at the axes. This gives it a homotopic invariant of unity [1,2]. Homotopy is a topological property. Vector fields with the same homotopic invariant can be continuously deformed into one another, provided that the field is non vanishing and certain boundary conditions are satisfied. For the homotopy classes considered here, the normal component of the magnetic field must be nonzero on the boundary near the $z$ axis, and zero elsewhere, and the magnetic field on the boundary must never be antiparallel to its direction on the $z$ axis. The homotopic invariant is a constant of the motion, since it is a consequence of the continuity of the magnetic field, provided that the field is not null and satisfies the the boundary constraints. Unlike magnetic helicity, it is not affected by magnetic reconnection. It was conjectured that the additional constraint of nonzero homotopic invariant might prevent plasma relaxation to the lowest available energy state. In the case of a configuration with a single o point and x point, the homotopic invariant is $K = {1\over 2} (\sigma_o - \sigma_x)$ where $\sigma_{o,x}$ are the signs of the toroidal magnetic field at the $o$ and $x$ points respectively. Combinations of $k= 0$ and other $k$ terms in the sum have $K = 1$, as well as having closed flux surfaces. The following example has $\alpha = 25,$ $k = 23,$ and $a_{23} = -0.4 a_0.$

The force free equilibria have zero pressure. Pressure is added in the form

$$p = p_0 \psi^2$$ (6)

and the resulting state evolved in time. With viscous dissipation, a finite pressure equilibrium is found, similar to the zero pressure equilibrium.

The finite pressure initial states equilibrate on a rapid time scale. The pressure expands radially until force balance is reached. The current, and to a lesser extent, the toroidal magnetic field, also adjust. The pressure broadens, and there is some current at the x point, indicating magnetic reconnection and some loss of poloidal magnetic flux.

Equilibria of the kind have been found to be Mercier stable with an average $\beta > 20\%$[1]. The stability is a consequence of the relatively high magnetic shear. Stability to long wavelength kink and tearing modes is the main question addressed in the following. Infinite mode number ballooning modes are not examined, but moderate mode number ballooning modes would show up in the computations. Typically, resistive tearing modes and ideal kink modes have a lower beta threshold than ballooning modes.

The configuration is allowed to evolve, for hundreds of Alfvéntimes. The evolution occurs in three dimensions, so that instabilities can occur. A small three dimensional perturbation is added, from which unstable modes could grow. The resolution in the toroidal direction permits toroidal modes of the form $\exp{in\phi}$ with $n \le 8.$ This is adequate to test for long wavelength instabilities with $n q \sim 1,$ of the sort that occur in RFPs[8] and spheromaks[9]. The low pressure equilibrium shown here appears stable. In this case $\beta = 11\%,$ where $\beta = 2< p > / < B^2 >,$ the ratio of volume averaged plasma pressure to volume averaged magnetic pressure.

In the evolution, sources of toroidal current and pressure are included, to oppose dissipative diffusion. The ratio of resistive diffusion time to Alfvéntime, in all the computations, is $S = 10^3.$ The poloidal flux $\psi$ at $t = 328 t_A$ is shown in Fig., where $t_A = R / v_A$ is the Alfvéntime, $R$ is the major radius, and $v_A$ s the Alfvénspeed. The toroidal flux $I$ is shown in Fig.. The pressure $p$ is shown in Fig.. The toroidal current, in Fig., is nonzero on open field lines outside the separatrix.

The toroidal magnetic field $B_\phi$ and poloidal field $B_z$ as a function of $R$ in the midplane, are shown in Fig.. The toroidal magnetic field $B_\phi$ reverses inside the poloidal separatrix, at a larger radius $R$ than the poloidal field $B_z$ reversal. The $q(\psi)$ profile, Fig., shows that instabilities with toroidal mode number $n = 1$ cannot be resonant, because $q < 1$ everywhere. The $n = 1$ mode has been problematic for spheromak experiments[9]. There might be resonances at the separatrix with $n = -1,$ but these instabilities would have to be highly localized, and in fact are not seen in the simulations.

In the case of higher $\beta = 22\%,$ the configuration is unstable. The initial state is formed as before, increasing the pressure source and allowing the configuration to relax to equilibrium on a short time scale.

The pressure at $t = 123 t_A$ is shown in Fig.. The pressure is highly distorted by three dimensional instabilities. The pressure perturbations have a structure typical of resonant kink modes. They saturate at relatively low amplitude by flattening of current and pressure profiles, similar to the relaxation process in RFPs [8].

The instability has a predominant toroidal Fourier harmonic $n = 4,$ which is resonant in that it can satisfy $nq = 1$ for $q = .25.$

Further simulations indicate that the stability boundary of this configuration is about $\beta < 15\%.$

The central hole in the simulations suppresses instabilities of the central poloidal current column flowing vertically on open field lines into conductors (at $Z = \pm 0.925$ in the figures). These instabilities are important for a spheromak - like startup. A poloidal current is driven electrostatically between electrodes; the current becomes kink unstable and eventually relaxes into a more or less two dimensional steady state [9]. Future simulations with different central boundary conditions will be required to assess stability of the central current channel.

As an alternative for a less turbulent start up one might inductively drive a toroidal current, using a center hole, similar to an RFP. The vertical central current can be eliminated, and replaced by a simple induction coil. An advantage over a standard RFP is that there is no need for toroidal magnetic field coils.

It is a simple modification of the Dag initial state to set the poloidal and toroidal current equal to zero on open field lines. This state can then be evolved with nonzero pressure as before. Eliminating current on the open field lines prevents reversal of the toroidal magnetic field. It also eliminates the requirement of non vanishing magnetic field for the Dag homotopic topological invariant. This does not seem to have a substantial effect on the MHD stability limit, which is somewhat higher than the previous cases. A stable example is shown in the following.

The toroidal flux at $t = 350 t_A$ is shown in Fig., and is zero on the open field lines. Likewise, the toroidal current, shown in Fig., remains zero outside the poloidal separatrix. The poloidal flux and pressure at the same time are similar to Fig. and Fig. respectively. In this case $\beta = 16\%.$ The $\beta$ tends to be somewhat higher in this configuration, because it has somewhat lower total magnetic energy.

Because the toroidal magnetic field does not reverse sign, $q(\psi)$, shown in Fig., does not reverse sign. The spike in $q$ indicates the separatrix. The height of the spike would increase with higher numerical resolution.

In summary, three dimensional dissipative initial value simulations indicate the stability of low pressure Dag - like configurations. The main difference from the original Dag is the rigid boundary in the center of the computational boundary. For moderate $\beta < 15\%$ the Dag is stable. For higher $\beta$ it is unstable to internal kink like modes. These modes do not cause disruption, as in a tokamak, but rather produce a turbulent steady state, as in an RFP or spheromak.

It is possible to have zero current on open field lines, turning the Dag into a more RFP like configuration, which has a similar equilibrium and a somewhat higher $\beta$ stability limit.

Acknowledgment This work was supported in part by the U.S. Department of Energy.