Resistive wall tearing mode disruptions in DIII-D and ITER tokamaks

Disruptions are a serious problem in tokamaks, in which thermal and magnetic energy confinement is lost. This is thought to be a severe problem in large tokamaks such as ITER. It was not known what instability causes the thermal quench (TQ) in disruptions and how to avoid it. A recent work identified the thermal quench in JET locked mode disruptions with a resistive wall tearing mode (RWTM).¹ The RWTM was also predicted in ITER disruptions.² This paper uses data from the DIII-D experiment, theory, and simulations to demonstrate that resistive wall tearing modes cause the thermal quench in a typical locked mode shot.

There can be many sequences of events leading to a disruption,³ which generally culminate in a locked mode. In locked mode shots, the plasma is at first toroidally rotating and relatively quiescent. The rotation slows, and the plasma becomes unstable to tearing modes (TMs). Locked modes are the main precursor of JET disruptions, but they are not the instability causing the thermal quench. Rather, the locked mode indicates an “unhealthy” plasma, which may disrupt.⁴ It was conjectured that at a critical amplitude, tearing modes would overlap and cause a disruption,⁵ although in many cases, such as in Fig. 1, the mode amplitude does not increase before the TQ occurs. Prior to and during the locked mode, the plasma can develop low current and temperature outside the q = 2 surface. This can be caused by tearing mode island overlap⁶ and impurity radiation.⁷ After the TQ, a current quench (CQ) occurs, as in Fig. 1, caused by high plasma resistivity.

In the following, we discuss experimental data of a locked mode disruption in DIII-D, linear theory of resistive wall tearing modes, linear and nonlinear simulations, and the onset conditions of the RWTM. The experimental data show that the TQ occurs in about half a resistive wall time. A linear theory presented in Sec. II shows the connection of tearing modes and resistive wall tearing modes. The scaling with resistive wall time is found. Linear simulations show that the mode is stable for an ideal wall and unstable with a resistive wall. Nonlinear simulations in Sec. III show that the mode grows to large amplitude, sufficient to cause a complete thermal quench. The thermal quench time is proportional to the reciprocal of the mode growth rate. The mechanism of the thermal quench is shown to be parallel thermal transport. The peak amplitude of the magnetic perturbations agrees with experimental measurement. The mode onset occurs when the radius of the q = 2 resonant surface is sufficiently close to the plasma edge as shown in Sec. IV, consistent with a database of DIII-D disruptions.⁹ 

A summary and the implications for ITER are given in Sec. V. It is not known whether ITER will rotate and lock, but it can have RWTMs.² The long timescale of the TQ implies that the mitigation requirements of ITER disruptions¹⁰ could be greatly relaxed.

Figure 1 shows data from DIII-D shot 154 576.⁸ A locked mode persists until the thermal quench. The TQ occurs in the time range $27.5–30 ms,$ marked with vertical lines. The upper frame shows the temperature T_e on a core flux surface. Also shown is the toroidal current I_p, which spikes after the TQ and then quenches on a slower timescale. The lower frame shows magnetic probe signals, dominated by the n = 1 toroidal harmonic. The harmonics of the magnetic field are mapped to their respective rational surface.⁸ The TQ time is $τTQ≈2.5 ms≈0.5τwall$⁠, where the resistive wall penetration time is $τwall=5 ms$⁠. The TQ time is proportional to the experimental growth time of the n = 1 magnetic perturbations. Before the TQ occurs, there is a locked mode, which consists of low amplitude precursors, identified as TMs.⁸ 

This is similar to JET shot 81 540¹ with TQ time $τTQ=0.3τwall=1.5 ms$ and $τwall=5 ms$⁠. The growth time of the mode is proportional to the TQ time, indicating the mode growth causes the TQ. These results suggest that a resistive wall mode (RWM) or resistive wall tearing mode causes the TQ.

The RWTM dispersion relation generalizes the standard tearing mode by taking into account diffusion of the magnetic perturbation across the resistive wall of a mode $ψ∝ exp (γt+imθ+inϕ)$⁠. This can be expressed for a thin wall as $γψ=(ηw/μ0δw)(ψ′vac−ψ′)$⁠, where ψ is the magnetic potential at the wall, $ψ′$ is its radial derivative on the plasma side of the wall, γ is the growth rate, $ηw,δw$ are the wall resistivity and thickness, respectively, and $ψ′vac=−(m/rw)ψ,$ assuming $ψ=ψvac$ at the wall, and $ψvac∝r−m,$ where r_w is the wall radius. This can be expressed as

giving a logarithmic derivative boundary condition for ψ at a resistive wall, instead of ψ = 0 at an ideal wall. The resistive wall penetration time^1,11–15 is $τwall=μ0rwδw/ηw$⁠. The RWTM dispersion relation is derived like that of a tearing mode, using boundary condition (1).

The dispersion relation^1,11 is

where $γ̂=γτA,$S is the Lundquist number, $τA=R/vA$ is the Alfvén time, R is the major radius, v_A is the Alfvén speed, $Sw=Swall(1−xs2m)/(2m),$$Swall=τwall/τA,$ ideal wall stability parameter $Δi=rsΔideal′/m,$ external stability parameter $Δx=2xs2m/(1−xs2m)$⁠, and $xs=(rs/rw)$ with rational surface radius r_s.

Ideal wall tearing modes are obtained from Eq. (2) in the limit $Sw→∞,$ while modes with a highly resistive wall are obtained in the limit $Sw→0,$

Resistive wall tearing modes have $Δi≤0,$ and require finite $Sw$⁠. The RWTM growth rate scalings can be approximated from Eq. (2). If $Δi=0,$ then assuming $γ̂Sw≫1$⁠,¹ 

If $Δi<0$ and $Δx+Δi>0$⁠, then neglecting the left side of Eq. (2) gives a kind of RWM with a rational surface in the plasma

If $Δi+Δx<0$⁠, there are no unstable solutions of Eq. (2). Intermediate asymptotic scalings of $γ̂$ are possible depending on the ratio $Δi/Δx,$ as in Fig. 2(a).

The dispersion relation can include a generalized Ohm's law in the tearing layer, including diamagnetic drifts. These effects are expected to be small, particularly in the limit (5) which does not contain the resistivity. Toroidal rotation is known to stabilize these modes.^12–14 The required rotation frequency is comparable to the TM linear growth rate. After mode locking, the residual rotation is not enough to stabilize the mode.

We now turn to numerical solutions of the equilibrium reconstruction of DIII-D shot 154576. Linear stability was studied using the M3D-C1¹⁶ code with a resistive wall.¹⁵ The reconstruction had on axis safety factor $q0>1$ to prevent the (1, 1) mode from dominating the simulations. It represents the equilibrium at time 3312 ms (⁠$27.2 ms$ in Fig. 1) just before the TQ. The equilibrium is axisymmetric and does not include magnetic islands. Figure 2(a) shows the growth rate as a function of $Swall$⁠. The curve labeled $−1/5$ has $Δx=1,$ $Δi=−1/5$⁠, which asymptotes to $γ∝Swall−2/3$⁠. The S_wall = 0 limit is a tearing mode with a highly resistive wall. The numerical solution is intermediate between scalings (4) and (5).

Figure 2(b) shows the perturbed magnetic flux $ψ,$ showing a (2, 1) structure. When the wall is ideally conducting, the mode is stable. This shows that the mode is not an ideal wall TM. It must necessarily have $Δi≤0.$

The linear simulations establish that the equilibrium reconstruction is unstable to a RWTM and stable to an ideal wall TM. Nonlinear simulations show that the mode grows to large amplitude, sufficient to cause a thermal quench. The simulations were performed with M3D¹⁷ with a thin resistive wall.¹⁸ The simulations used the same parameters as Refs. 1 and 2, in particular, $S=106$⁠, parallel thermal conductivity $χ∥=10R2/τA,$ perpendicular thermal conduction, and viscosity $χ⊥=μ=10−4a2/τA,$ where a is the minor radius. The simulations used 16 poloidal planes, which are adequate to resolve low toroidal mode numbers. The simulations and the experimental data were dominated by n = 1 modes. The simulations were initialized with small arbitrary perturbations, including all toroidal mode numbers.

Figure 3 shows a simulation with M3D¹⁷ with a resistive wall¹⁸ of the same equilibrium reconstruction of DIII-D 154 576. The simulation had $Swall=104$⁠. Experimentally, $Swall=1.2×104$⁠. The initial magnetic flux ψ is shown in Fig. 3(a), and the perturbed ψ is in Fig. 3(b), at a time late in the simulation, when the TQ is almost complete. The nonlinear perturbed ψ is predominantly (2, 1), similar to the linear structure of Fig. 2(b). The pressure, shown at the same time in Fig. 3(c), has a large perturbation that causes the TQ. The pressure is shown when the total volume integral of the pressure P is about 20% of its initial value.

Figure 4 shows several M3D simulations with different values of S_wall. Figure 4(a) shows time histories of P and $bn,$ where b_n is the perturbed normal $δB/B$ at the wall. The curves are labeled by the value of $Swall$⁠. All mode numbers n > 0 are included in $bn$⁠. Figure 4(b) shows the TQ time τ_TQ measured from the time histories of P. Also shown is $τ∥,$ the parallel transport time, given by^1,2

using the maximum value of b_n for each S_wall in Fig. 4(a). The fits are to $3.4/γs,$ where the growth rate γ_s is measured from the time histories of b_n in Fig. 4(a) and to $Swall2/3$⁠. This agrees with Fig. 2(a). The relation $τ∥∝Swall2/3$ gives the scaling $bn∝Swall−1/3,$ which also agrees with Fig. 5. The vertical line is the experimental value of $Swall$⁠. At the vertical line $τTQ/τA≈Swall/2,$ as in the experimental data. The mode growth occurs on the same time scale as the TQ, as in the experiment. The small drop in P in Fig. 4(a) at $t≈3000τA$ is due to internal modes. This resembles the minor precursor disruptions observed in JET¹ and DIII-D,⁸ which can be seen in Fig. 1.

The reason the mode grows to large amplitude may be the external stability parameter $Δx$⁠. TMs have internal stability parameter, which depends on the current profile. Growth of an island flattens the current gradient and stabilizes the TM at a moderate amplitude. The external stability parameter Δ_x depends only on $xs,$ independent of island size. It is not saturated by local flattening of the current profile.

The experimental value of $δB/B$ is in agreement with the simulations. At its maximum value, the (2, 1) magnetic perturbation in Fig. 1 is $δB≈480G,$ or $δB/B=2.8×10−2,$ taking $B=1.7T$⁠. The value of δB, measured by probes, was estimated⁸ at the (2, 1) rational surface r_s, as noted in the discussion of Fig. 1. To obtain its value b_w at the wall, $δB/B$ must be multiplied by $xs3=0.3,$ where $xs=.67,$ yielding $bw=8.4×10−3$⁠. To compare with the simulation, Fig. 5 shows b_l, the peak transverse perturbed magnetic field at the wall as a function of $Swall,$ and peak b_n, also shown in Fig. 4(a), in units of $10−3$⁠. The b_n signal would be measured by saddle coils in the experiment, while b_l would be measured by probes, like $bw$⁠. The peak value of b_l can be fit using Eq. (1), with $(bn,bl)∝(mψ/rw,ψ′)$⁠, noting that $γτwallbn=const$⁠. The fit is $bn≈0.05Swall−1/3,$ and $bl=bn+6.7×10−3$⁠. The maximum value $bl=8.8×10−3$ at the experimental value $Swall=1.2×104$ is in agreement with $bw.$

Having shown that the equilibrium reconstruction is unstable to a RWTM that grows large enough to produce a TQ, we consider the onset condition for the RWTM, $Δi=0$⁠. Both Δ_x and Δ_i depend on $xs$⁠. In a step current model^11,19 with a constant current density contained within radius r₀ with $rs/r0=(2/q0)1/2$ and $(m,n)=(2,1)$

For $Δi>0$⁠, this requires a positive numerator and a negative denominator. If this is not satisfied, the RWTM is unstable

For example, if $q0=1.05,$ then $Δi=0$ for $xs=0.65$⁠. For larger $xs>0.65$⁠, the RWTM is unstable. Other current profiles are similar,¹⁹ in that larger x_s causes $Δi≤0$⁠. There is additional experimental support for this. Whether a disruption occurs or not depends on the normalized q = 2 radius $ρq2$ in a database of DIII-D locked modes.⁹ Here, $ρq2≈rs/a,$ and $rw/a≈1.2$⁠. The disruption onset boundary in the database is $ρq2>0.75$ or $xs>0.625$⁠. In the simulations, $ρq2=0.8,$ so that $xs=0.67$⁠. The onset condition is that the q = 2 surface is close enough to the plasma edge. The database also shows that $ρq2$ increases in time from the beginning to the end of the locked mode, as the disruption is approached. This connects the onset condition to profile evolution.

To summarize, theory and simulations were presented of resistive wall tearing modes in an equilibrium reconstruction of DIII-D shot 154 576. The linear tearing mode dispersion relation with a resistive wall showed the parameter dependence of the modes, especially on $Swall$⁠. Linear simulations found that the equilibrium was stable with an ideally conducting wall and unstable with a resistive wall. For the particular example studied here, the growth rate scales as $γ∝Swall−2/3$⁠. The RWTMs grow to large amplitude nonlinearly. The thermal quench time is proportional to the RWTM growth time. The amplitude of the simulated peak magnetic perturbations is in agreement with the experimental data. The onset condition for disruptions is that the q = 2 rational surface is close enough to the plasma edge, consistent with a DIII-D disruption database.

These results are very favorable for ITER disruptions. The ITER resistive wall time, $250 ms,$ is 50 times longer than in JET and DIII-D. The TQ time, instead of being $1.5–2.5 ms$ in JET and DIII-D, respectively, could be $75–125 ms,$ assuming the TQ is produced by a RWTM with $τTQ≈τwall/2$ scaling. If the TQ is caused by a RWTM with $Swall−4/9$⁠, and the edge temperature is $500 eV,$ then² $τTQ=70 ms$⁠. The highly conducting ITER wall strongly mitigates RWTMs. It might relax the requirements of the disruption mitigation system, disruption prediction, and mitigation of runaway electrons.

This work was supported by U.S. DOE under Nos. DE-SC0020127 and DE-FC02-04ER54698 and by Subcontract No. S015879 with Princeton Plasma Physics Laboratory. The help of R. Sweeney with the DIII-D data and for discussions is acknowledged.

The authors have no conflicts to disclose.

Henry Strauss: Formal analysis (lead); Writing – original draft (lead). Brendan C Lyons: Software (equal); Writing – review & editing (equal). Matthias Knolker: Software (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.