MST Resistive Wall Tearing Mode Simulations

TL;DR

This paper investigates resistive wall tearing modes and resistive wall modes in the MST tokamak, showing they could cause disruptions but with longer thermal quench times than in other devices, aligning with experimental observations.

Contribution

It presents simulations demonstrating the stability characteristics of RWTMs and RWMs in MST, highlighting their potential role in disruptions and the transition between these modes.

Findings

MST is unstable to RWTMs and RWMs.

Predicted thermal quench times are longer than in other tokamaks.

Disruptions are likely caused by RWTMs and RWMs.

Abstract

The Madison Symmetric Torus (MST) is a toroidal device that, when operated as a tokamak, is resistant to disruptions. Unlike most tokamaks, the MST plasma is surrounded by a close fitting highly conducting wall, with a resistive wall penetration time two orders of magnitude longer than in JET or DIII-D, and three times longer than in ITER. The MST can operate with edge q_a < 2, unlike standard tokamaks. Simulations presented here indicate that the MST is unstable to resistive wall tearing modes (RWTMs) and resistive wall modes (RWMs). They could in principle cause disruptions, but the predicted thermal quench time is much longer than the experimental pulse time. If the MST thermal quench time were comparable to measurements in JET and DIII-D, theory and simulations predict that disruptions would have been observed in MST. This is consistent with the modeling herein, predicting that disruptions are caused by RWTMs and RWMs. In the low q_a regime of MST, the RWTM asymptotically satisfies the RWM dispersion relation. The transition from RWTM to RWM occurs smoothly at q_a = m/n, where m,n are poloidal and toroidal mode numbers.

Equations

$$S^{3/4}{S}_{wall}^{-5/4}({\hat{\gamma }}^{9/4}+g_s{\hat{\gamma }}^{5/4})={\Delta }_i\hat{\gamma }+g_s{\Delta }_n$$

$${\Delta }_i=-2\frac{n{q}_0-m+1-(r_0/r_w{)}^{2m}}{[n{q}_0-m+1-(r_0/r_s{)}^{2m}][1-(r_s/r_w{)}^{2m}]}.$$

$$\gamma {\tau }_{wall}=-2m\frac{n{q}_0-(m-1)}{n{q}_0-(m-1)-(r_0/r_w{)}^{2m}}.$$

Full text

1 Introduction

Tokamaks are subject to disruptions, events in which thermal and magnetic energy confinement is lost. It was not known what instability causes the thermal quench (TQ) in disruptions. Disruptions have been predicted to be a severe problem in large future devices such as ITER. Previous studies of JET [1], ITER [2], DIII-D [3] showed that disruptions can be caused by resistive wall tearing modes (RWTMs). These are tearing modes (TMs) whose resonant surface is inside the plasma, but is close to the wall. With a perfectly conducting wall, the TMs are stable, but with no wall, they are unstable. The TQ time is proportional to the RWTM growth time. With a highly conducting wall, the growth time is of order the resistive wall penetration time. In the Madison Symmetric Torus (MST) [4, 5] the growth time is much longer than the shot duration.

The main result of this paper is that the predicted thermal quench time in MST is much longer than in conventional tokamaks such as JET and DIII-D, and is longer than a prediction for ITER based on RWTMs. This is shown in Fig. 1. The TQ time in $ms$ is shown as a function of $S_{wall}$, the resistive wall penetration time normalized to the Alfvén time (defined below). For JET and DIII-D, the TQ time is based on experimental data and simulations. For ITER and MST, the TQ time is based on simulation. The MST case is for $q_{a}=2.6$, described in more detail below. In MST, a TQ does not occur during the experimental shot duration, which gives a lower limit to the possible TQ time.

Another result is that in the low edge safety factor $q_{a}\leq 2.6$ regime of MST, the RWTM growth time scales linearly in the resistive wall penetration time [3]. This is characteristic of large $S_{wall}$, in which the RWTM asymptotically satisfies the RWM dispersion relation. ITER could also be in this regime. The largest amplitude magnetic perturbation seen in simulations is the RWTM with rational surface radius $r_{s}$ at which $q(r_{s})=m/n$ is closest to $q_{a},$ although RWMs can have a comparable amplitude.

The MST experiment is a toroidal device which can be operated as a reversed field pinch (RFP) or as a tokamak. It is well known [6] that RFPs require a highly conducting wall, or feedback, in order to stabilize external kink modes. The MST device is shown in Fig. 2. It has a circular cross section with limiters, a single-turn TF winding, PF windings wrapped around an iron-core transformer, and a close-fitting conducting shell with wall penetration time $\tau_{wall}=800ms$.

The MST wall penetration time is more than three times longer than in ITER, with [7] $\tau_{wall}=250ms.$ MST has a pulse time of about $50ms,$ during which the wall is effectively an ideal conductor. It can operate with $q_{a}\leq 2.$ [5]. No disruptions have been seen to date when it is operated as a standard tokamak. Disruptions can occur under non standard conditions with very low density, in discharges dominated by runaway electrons (RE). It has internal MHD modes, including $(1,1)$ internal kinks, which produce sawteeth. According to the theory and simulations presented here, disruptions are suppressed by the highly conducting wall.

In the following, theory and simulations are presented which indicate that MST is unstable to RWTMs and RWMs, which could cause disruptions in conventional tokamaks [1, 2, 3]. These results suggest that disruptions are not observed in MST because the predicted thermal quench time $\tau_{TQ}$ is much longer than the experimental pulse duration. It is shown that RWTMs and RWMs in MST have the same mode growth time scaling linearly in the resistive wall magnetic penetration time $\tau_{wall}.$ It is also shown that there is a smooth transition from RWTMs to RWMs when the rational surface exits the plasma.

Simulations were done to obtain the scaling of the TQ time $\tau_{TQ}$ with wall resistivity. The simulations were performed with the nonlinear resistive MHD M3D code [8] with a resistive wall [9]. Simulations were initialized with MSTFit [10] equilibria having on axis $q_{0}=1,$ and several values of edge $q=q_{a}$ in the range $1.5\leq q_{a}\leq 2.6.$ The parameters were: Lundquist number $S=10^{5}$ (the experimental value), and parallel thermal conductivity $\chi_{\parallel}=10R^{2}/\tau_{A},$ (somewhat larger than the experimental value $4R^{2}/\tau_{A}.$) The experiment has $B_{T}=0.133T,$ and average density $n_{e}=4.5\times 10^{12}cm^{-3}.$ The major radius is $R=1.5m,$ and the minor radius is $0.52m.$ The fill gas was deuterium. The Alfvén time is $\tau_{A}=R/v_{A}=1.15\times 10^{-6}s.$ The resistive wall time is $\tau_{wall}=\mu_{0}\delta_{w}r_{w}/\eta_{w},$ where $\delta_{w}$ is the wall thickness, $r_{w}$ is the wall radius, and $\eta_{w}$ is the wall resistivity. With $\tau_{wall}=.8s,$ then $S_{wall}=\tau_{wall}/\tau_{A}=7\times 10^{5}.$ The value of $S=10^{5},$ which gives $T=69eV$ on axis. The parallel conductivity in the collisional regime is $\chi_{\parallel}=2.1v_{te}^{2}\tau_{e},$ or $\chi_{\parallel}=8\times 10^{11}cm^{2}/s.$ This can be expressed as $\chi_{\parallel}=4.1R^{2}/\tau_{A}.$ The simulations do not include rotation, which has a stabilizing effect [12, 13, 14] on RWTMs. The computational plasma extended to the wall, and the narrow limiters were not explicitly taken into account.

2 RWTM Theory

This section reviews some theory of RWTMs and RWMs. The RWTM dispersion relation [1, 11] is

[TABLE]

where $\hat{\gamma}=\gamma\tau_{wall},$ $S$ is the Lundquist number, $g_{s}=2m/[1-(r_{s}/r_{w})^{2m}],$ ideal wall stability parameter $\Delta_{i}=r_{s}\Delta_{ideal}^{\prime}/m,$ external stability parameter $\Delta_{x}=2r_{s}^{2m}/(r_{w}^{2m}-r_{s}^{2m})$, with rational surface radius $r_{s}$, in a cylindrical geometry large aspect ratio model. The no wall stability parameter is $\Delta_{n}=\Delta_{i}+\Delta_{x}.$ Resistive wall tearing modes have $\Delta_{i}\leq 0,$ and require finite $S_{wall}.$ The RWTM growth rate scalings vary as $\gamma\propto S^{-\alpha},$ with $4/9\leq\alpha\leq 1.$ In a JET example [1], $\alpha=4/9,$ while in a DIII-D example [3] $\alpha=2/3.$

In the following MST examples, $\gamma\sim\tau_{wall}^{-1}$, $\alpha=1.$ This is because of the smallness of the coefficient on the left side of (1). With MST parameters, $S^{3/4}S_{wall}^{-5/4}=2.8\times 10^{-4}.$ The left side of (1) is only significant when $\Delta_{i}$ is within a small range of $\Delta_{i}=0.$

In a step current model [11, 15] with a constant current density and $q=q_{0}$ contained within radius $r_{0}$, zero current density for $r>r_{0},$ $q=q_{0}(r/r_{0})^{2},$ Then $q_{w}=q_{s}(r_{w}/r_{s})^{2},$ where $q_{0}$ is the value on axis and $q_{s}=m/n.$ In the model $q_{a}=q_{w}$.

[TABLE]

It can be seen that as $r_{s}\rightarrow r_{w},$ that $\Delta_{x},-\Delta_{i}\approx 2/[1-(r_{s}/r_{w})^{2m}].$

The growth rate is, neglecting the left side of (1),

[TABLE]

Remarkably, this is also the growth rate of a RWM [11, 16] using the same model equilibrium. The crossover from RWTM to RWM occurs smoothly at $q_{a}=m/n.$ For $q_{a}<m/n,$ the mode becomes a RWM.

In Fig. 3(a) is plotted $\gamma\tau_{wall}$ in (3) for the cases $q_{0}=1.08,$ $(m,n)=(2,1),$ and $(3,2)$ as a function of $q_{w}.$ Also shown is $\Delta_{i}$ in (2). For $\Delta_{i}\geq 0,$ the mode is a tearing mode. For $(2,1),$ this occurs at $q_{w}\geq 3.7,$ in this model. There is a $(3,2)$ tearing mode for $q_{w}\geq 2.27.$ For the $(2,1)$ mode, $\gamma$ is in the regime of validity of (3) for $q_{a}\leq 1.98.$ The $(3,2)$ is in this regime for $q_{a}\leq 1.7.$ There is a pole in $\Delta_{i}$ when $r_{s}=r_{w}$ or $q_{w}=m/n.$ This is the transition from RWTM to RWM. There is no pole in $\gamma,$ so the transition from RWTM to RWM is smooth. There is also a pole in $\gamma$ at which the denominator of (3) vanishes. This is the same as the zero of $\Delta_{i},$ at which the RWTM becomes a tearing mode.

The scaling of $\gamma$ with $S_{wall}$ is shown in Fig. 3(b). The curves were obtained by solving (1) with parameters $(m,n)=(2,1)$, $q_{0}=1.08,$ $S=10^{5},$ and several $q_{w}$ values. For small $S_{wall},$ the RWTM satisfies a no wall tearing mode dispersion relation. For large $S_{wall},$ except for the case $\Delta_{i}=0,$ with $q_{w}=3.7,$ which has $\alpha=4/9,$ asymptotically $\gamma$ is given by (3), plotted as dashed curves. The range of $S_{wall}$ which deviates from (3) decreases as $q_{w}$ decreases.

The model used here is consistent with simulations. The RWTM and RWM have a growth time and thermal quench time proportional to $S_{wall}.$ The $(3,2)$ growth rate is larger than the $(2,1).$ The simulations show that the nonlinear behavior is dominated by the $(2,1)$ mode for $q_{a}\geq 2,$ and the $(3,2)$ for $q_{a}<2$.

The TQ is caused by the growth of RWTMs and RWMs. The linear growth rates indicate the $S_{wall}$ scaling of the TQ time, but are not sufficient to obtain $\tau_{TQ}$ quantitatively. Nonlinear simulations are needed, which are described in the following.

3 Case $q_{a}=2.6$

Nonlinear resistive MHD simulations are carried out with M3D [8] in which the resistive wall [9] time was varied. It is found that the MST is unstable to RWTMs and RWMs. The modes cause a TQ, on a timescale of order $200ms,$ much longer than the experimental pulse of $50ms.$

The simulations used $16$ poloidal planes, adequate to resolve low toroidal mode numbers. The simulations were dominated by $n=1,2$ modes.

The simulations were initialized with equilibrium reconstructions in which $q$ on axis $q_{0}=1,$ and $q_{a}=2.6,2.0,1.7$ and $1.5$ at the edge.

Fig. 4 shows contour plots from a simulation with $q_{0},q_{a}=1,2.6.$ It has $S_{wall}=3.3\times 10^{4}.$ The plots are at time $t=5300\tau_{A},$ near the end of the simulation. Fig. 4(a) shows $\psi$, which appears distorted by a $(3,2)$ tearing mode. The simulation contains $(3,2)$, and $(2,1)$ modes. Fig. 4(b) shows the perturbed $\psi,$ where $\tilde{\psi}=\psi-\psi_{avg},$ and $\psi_{avg}$ is the toroidal average. It predominantly has $(2,1)$ structure. Fig. 4(c) is the temperature $T$.

Fig. 5 shows the effect of wall penetration time $\tau_{wall}.$ Simulations were done with several values of $S_{wall}.$ Fig. 5(a) shows the time history of the volume integrated pressure $P$. The value of $S_{wall}$ is indicated by labels, with $\infty$ indicating an ideal wall. The TQ time is measured as the time difference $(t_{20}-t_{90})/0.7,$ where $t_{90}$ is the time at which the temperature is $90\%$ of its peak, and $t_{20}$ is the time when it has $20\%$ of its peak value.

The values of $\tau_{TQ}$ as a function of $S_{wall}$ are collected in Fig. 5(b). They are fit with $0.22S_{wall},$ and projected to the experimental value $S_{wall}=7\times 10^{5},$ $\tau_{TQ}\approx 1.5\times 10^{5}\tau_{wall}.$ This is $\tau_{TQ}\approx 180ms,$ the data point for MST plotted in Fig. 1.

The magnetic perturbations consist of primarily a $(2,1)$, with a smaller amplitude $(3,2)$ mode. Fig. 5(c) shows the magnetic energy in $n=1,2$ toroidal harmonics of the normal component of magnetic field at the wall, given by $MW_{n}=\oint|{\bf B}(n)\cdot\hat{\bf n}/B|^{2}dl/L,$ where ${\bf B}(n)=(2\pi)^{-1}\oint{\bf B}\exp(in\phi)d\phi,$ $L=\oint dl,$ $\hat{\bf n}$ is the unit normal to the wall, and $l$ is the length along the wall for fixed toroidal angle $\phi.$ The time dependent increase in the rate of the pressure drop in Fig. 5(a) is caused by the growth of the amplitude of the magnetic perturbations.

4 Cases $q_{a}=2,1.7,1.5$

It is possible to have $q_{a}\leq 2$ in MST [5]. The case $q_{a}=2$ is at the borderline between RWTM and RWM. It has a much slower TQ than the case $q_{a}=2.6.$

Fig. 6(a) shows the TQ time as a function of $S_{wall}$ for $q_{a}=2.$ Also plotted for comparison is $0.7S_{wall}.$ Projecting to the experimental $S_{wall}$ gives $\tau_{TQ}\approx 5\times 10^{5}\tau_{A}$ $\approx 0.57s.$ Fig. 6(b) shows $MW_{n}$ and $P$ for $S_{wall}=10^{4}.$ In this case the $(3,2)$ or $n=2$ mode is dominant. The $(3,2)$ RWTM has a larger amplitude than the $(2,1)$ RWM.

Simulations were also performed for $q_{a}=1.7$ and $q_{a}=1.5.$ The time history data is summarized in Fig. 7. Fig. 7(a) shows $\tau_{TQ}$ as a function of $S_{wall}$ for $q_{a}=1.7.$ The data is fit by $\tau_{TQ}\approx 0.16\tau_{wall},$ which is projected to $\tau_{TQ}\approx 1.15\times 10^{5}\tau_{A},$ $\approx 0.13s.$ Fig. 7(b) shows $\tau_{TQ}$ as a function of $S_{wall}$ for $q_{a}=1.5.$ The data is fit by $\tau_{TQ}\approx 0.13\tau_{wall},$ which is projected to $\tau_{TQ}\approx 0.9\times 10^{5}\tau_{A},$ $\approx 0.10s.$

The TQ time data is summarized in Fig. 8, showing $\tau_{TQ}$ as a function of $q_{a},$ at the experimental value of $S_{wall}.$ Except for $q_{a}=2,$ the values of $100ms<\tau_{TQ}<200ms.$ The $q_{a}=2$ case is much slower, $\tau_{TQ}=570ms.$ Also shown in Fig. 8 is $1/\gamma$ where $\gamma\tau_{wall}$ is taken from Fig. 3(a), using $\gamma(2,1)$ for $q_{a}\geq 2,$ and $\gamma(3,2)$ for $q_{a}<2.$ The slow TQ at $q_{w}=2$ is also seen in the model linear growth times. The agreement of simulations and theory is remarkable, considering the simplicity of the model used to obtain (3).

5 Conclusions

MST was originally operated as a reversed field pinch, which required a highly conducting, close fitting wall. When run as a tokamak, the MST is resistant to disruptions. This is consistent with simulations with $S_{wall}^{-1}=0,$ which is an ideal wall. When the wall is made resistive in the simulations, RWTMs become unstable and cause a thermal quench. The TQ time increases linearly with $\tau_{wall}$. This is characteristic of large $\tau_{wall}$, $S_{wall}\gg S^{3/5}.$ Asymptotically the RWTM satisfies the RWM dispersion relation, except when $\Delta_{i}\approx 0.$ MST has large $S_{wall}$, as does ITER when $S\ll 10^{9}$ in the edge.

Nonlinear simulations examined four cases, with edge $1.5\leq q_{a}\leq 2.6.$ In all cases, the TQ time $\tau_{TQ}>100ms,$ compared to the experimental pulse time $50ms.$ This is shown in Fig. 8.

The TQ time from the $q_{a}=2.6$ case was plotted in Fig. 1, since this case more nearly resembles a standard tokamak. The implication for other tokamaks is that a more conducting wall slows the RWTM and mitigates disruptions, especially in ITER.

Acknowledgement We thank Jay Anderson for help with the MSTFit code. This work was supported by U.S. DOE under grants DE-SC0020127, DE-SC0020245, and DE-SC0018266.