Electron heat flux and propagating fronts in plasma thermal quench via ambipolar transport

When magnetic field lines suddenly intercept solid surfaces that provide a sink for plasma energy and sometimes particles, a magnetized fusion-grade plasma can undergo a thermal collapse. This can happen, for example, in the thermal quench (TQ) of fusion plasma during either naturally occurring or intentionally triggered, mitigated tokamak disruptions. The naturally occurring tokamak disruption can be triggered when large-scale MHD activities turn nested flux surfaces into globally stochastic field lines that connect fusion-grade core plasma directly to the divertor/first wall.^1–4 As a result, the substantial thermal energy of plasmas is released within a few milliseconds,^5–7 causing severe damage to plasma facing components⁸ (PFCs). Major disruptions are also intentionally triggered for disruption mitigation by injecting high-Z impurities, for example, in the form of deliberately injected solid pellets.^9–12 The injected high-Z impurities into a pre-disruption plasma are intended to have the hot plasma deposit its thermal energy onto the pellet, so pellet materials can be ablated and ionized to be assimilated within the flux surface. The ablated pellet materials, thus, initially form radiative cooling mass (RCM) that provides strongly localized radiative cooling for the thermal energy of the surrounding fusion-grade plasma. The fact that the background plasma undergoes a thermal quench by transporting energy into the RCM, which through radiation can spread the heat load over the entire first wall, is the logic behind this approach for thermal quench mitigation. In both situations, the plasma will attach to an energy sink, being a vapor-shielded wall or the ablated pellet, and lose its energy via the fast parallel transport to the energy sink. It must be emphasized that for fusion-grade plasmas, which are nearly collisionless provided that the plasma mean-free-path, $λ mfp ∼ 10 km$⁠, is much longer than the magnetic connection length L_B or the tokamak major radius, the TQ due to the presence of a cooling spot (energy sink) is in an exotic kinetic regime, in which the fast parallel transport along the field lines is expected to be the dominant mechanism. The normal expectation is that such nearly collisionless parallel transport of plasma thermal energy in the short magnetic connection length regime represents a worst-case scenario of a TQ in tokamaks, where the plasma thermal energy is released in the shortest possible time. Therefore, understanding the plasma TQ in such an exotic regime is critical for disruption mitigation.

However, it is worth noting that the inhibition of electron thermal conduction in the nearly collisionless plasma has been extensively studied in astrophysics by considering tangled magnetic fields^16,17 and plasma instabilities^18–20 in order to reach the convection-dominated scenario of the thermal conduction, which would naturally yield the cooling flow that aggregates masses onto the cooling spot in clusters of galaxies.^21–24 In a related vein for the fusion plasma, it is well known that the convective scaling of electron thermal conduction is obtained at the entrance of the steady-state sheath, in which the plasma is nearly collisionless, as a result of the ambipolar transport.^25–27 In this paper, we will show that the convective scaling with the parallel ion flow, in the electron thermal conduction itself or its spatial gradient, can be established throughout the bulk, quasineutral, and nearly collisionless plasma away from the wall due to ambipolar constraint. Such convective scaling comes about because the cooling of the surrounding plasma takes the form of propagating fronts that originate from the cooling spot and the fundamental ambipolar transport constraint between the slow fronts modifies the thermal conduction heat flux. As a result, a robust plasma cooling flow into the cooling spot will be developed due to the retained large electron temperature and hence pressure gradient.²⁸ Such weaker convective scaling of the electron thermal conduction, via itself or its gradient, is also critically important for a slower TQ of the fusion plasmas by modifying the core plasma cooling processes.²⁹ This should be contrasted with a straightforward application of the flux-limiting form for electron parallel thermal conduction, which would yield a much faster TQ on a timescale that is around a factor of $m i / m e$ faster. Had one unwisely deployed the Braginskii parallel thermal conductivity instead for such a nearly collisionless plasma, an even faster TQ would be obtained in numerical simulations if the plasma has collisional mean-free path longer than the system size.

Here, we follow the previous letter of Ref. 28 on the subject and present the details of numerical diagnostics, the analyses of the propagating fronts' characteristics, and the underlying physics, as well as the electron thermal conduction flux. First-principles fully kinetic simulations were performed with the VPIC³⁰ code to investigate the parallel transport physics in the TQ of a nearly collisionless plasma. A prototype one-dimensional slab model is considered with a normalized background magnetic field, where an initially uniform plasma with constant temperature $T 0 ∼ 10 keV$ and density $n 0 ∼ 10 19 m − 3$ is filled the whole domain. The plasma is signified as semi-infinite with the right simulation boundary simply reflecting the particles. We notice that such a boundary condition would not affect the plasma dynamics as long as the later-defined electron fronts have not arrived there yet. This indicates that we consider $t < L B / v t h , e$⁠. However, for longer timescale, the basic physics holds, which affects the TQ processes.²⁹ A cooling spot is modeled at the left boundary as a thermobath that mimics a radiative cooling spot, which conserves particles by re-injecting electron–ion pairs (equal to the ions across the boundary) with a radiatively clamped temperature $T w ≪ T 0$⁠. For comparison, an absorbing boundary, as a sink to both the particles and energy that absorbs all the particles hitting the left boundary, is also considered for simulations. We found that these two types of cooling spots show remarkable similarities in plasma cooling so the absorbing boundary is quite useful for understanding the underlying physics.

For both thermobath and absorbing boundaries, the TQ is found to be governed by the formations of propagating fronts (see Fig. 1). Particularly, for the former (latter), there are four (three) fronts: the first two have speeds scale with the electron thermal speed $v t h , e$ and, thus, are named electron fronts, while the other two (one) propagate at speeds that scale with the local ion sound speed c_s and thus named ion fronts. Based on the underlying physics and their roles in the TQ dynamics, these fronts can be named the precooling front (PF), precooling trailing front (PTF), recession front (RF), and cooling front (CF, for the thermobath boundary only), respectively, as illustrated in Fig. 1. The precooling and recession fronts describe the onset of $T ∥$ cooling of electrons and ions, respectively, which are independent of the cooling spot types. In contrast, the precooling trailing and cooling fronts will not only play a role in $T ∥$ cooling but also reflect the onset of $T ⊥$ cooling via the dilution with the cold recycled particles for the thermobath boundary. It should be noted that the propagating fronts for the rapid cooling of a nearly collisionless plasma, as described and reported here, are not the artifact of the boundary conditions deployed in the simulation. In a forthcoming paper that focuses on the impurity ion assimilation by a cooling plasma, the same front propagation physics are found in a plasma where a hot and dilute plasma cools against a cold and dense plasma that is initially in pressure balance between the two regions.

The rest of the paper is organized as follows: In Sec. II, we elucidate the underlying physics of electron fronts, while the ion front(s) physics are investigated in Sec. III. The electron thermal conduction flux within the recession layer (the region between the recession and cooling front for the thermobath boundary or between the recession front and sheath entrance for the absorbing boundary), which is essential for the formation of the plasma cooling flow, will be discussed in Sec. IV. We will conclude in Sec. V.

Here, $n e = ∫ f e d 3 v$ is the electron density, $V e ∥ = ∫ v ∥ f e d 3 v / n e$ is the parallel electron flow, $T e ∥ = ∫ m e v ̃ ∥ 2 f e d 3 v / n e$ is the parallel electron temperature, and $q e n = ∫ m e v ̃ ∥ 3 f e d 3 v$ is the parallel thermal conduction flux of the parallel degree of freedom,³¹ where $v ̃ ∥ ≡ ( v ∥ − V e ∥ )$⁠.

Interestingly, the faster PTF for the absorbing boundary nearly coincides with the $T e ⊥$ collapse front (cold recycled electron beam front) for the thermobath boundary as shown in Fig. 1. This is because, at the beginning of the thermal collapse, the number of recycled electrons is not high enough to cause an appreciable reduction of the reflecting potential compared to that for the absorbing boundary. Such nearly equalized reflecting potential $( Δ Φ ) RF$ will accelerate the recycled electrons, which dilutely cool $T e ⊥$⁠, to U_PTF for the absorbing boundary case. A fitting speed of $1.4 v t h , e$ for $T e ⊥$ precooling front is shown in Fig. 3.

The ion front(s) not only describes the plasma temperature cooling, but also controls the plasma density evolution and the associated cooling flow generation. This section is dedicated to study the underlying physics of the ion front(s), taking into account that the plasma thermal conduction is forced to be convective, in the form of either itself or its gradient, due to the ambipolar transport, which will be investigated in Sec. IV. As a result, the electron pressure gradient remains large to drive plasma flow toward the cooling spot, which is known in astrophysics as the cooling flow.^21–24 

The cooling front (CF) in the thermobath boundary case is where the hot ions meet the cold ions as illustrated in Fig. 6. In fact, it is a shock front, across which all the plasma state variables have jumps as shown in Figs. 1–5. Particularly, the ions undergo heating in the parallel direction when the plasma flow runs into the CF, where the substantial ion flow energy in the recession layer is converted into ion thermal energy as shown in Figs. 1, 4, and 5. Such conversion is via the mixing of the cooling flow ions with the cold recycled ions (see Fig. 6), the latter of which are accelerated from the boundary to the CF by the inverse pressure gradient due to the pressure pileup at the boundary. As a result, these cold recycled ions will offset the plasma flow generated by the surrounding ions as shown in Fig. 4. In sharp contrast to ions, the electrons will undergo cooling via dilution with high-density cold electrons so that $T e ∥ ∼ T w$ behind the CF as shown in Figs. 1 and 3. Moreover, the presence of the CF and the associated cooling zone behind the CF is of fundamental importance to $T i ⊥$ and $T e ⊥$ cooling via dilution and thus the CF represents a deep cooling of electrons.

In this section, we investigate the electron thermal conduction flux within the recession layer under the ambipolar transport constraint $V e ∥ ≈ V i ∥$⁠. Both model analyses and first-principles simulations using VPIC will be provided.

The first-principles kinetic simulations have confirmed all the above analytical results. Specifically, to check the convective transport scaling of q_en, i.e., $q e n ∝ m i − 1 / 2$⁠, we have conducted simulations by employing different ion-electron mass ratios, i.e., $m i / m e = 100 , 400 , 900$⁠, and 1600. Equation (19) indicates that both the ion flow and local recession speed ξ in the recession layer scales with $m i − 1 / 2$⁠. Therefore, to overlap the recession layer to the same location for different m_i, we should choose temporal moments with constant $ω p i t$ for different cases. Here, ω_pi is the ion plasma frequency.

In Fig. 7, we plot the profiles of q_en for the absorbing boundary case, which demonstrates that q_en itself indeed follows the convective transport scaling $q e n ∝ V i ∥$ within the recession layer, although the coefficient σ_e slightly increases with m_i. Such a variation of σ_e comes from the weakly positive dependence of v_c on m_i via $v c ∼ v t h , e 2 ln ( v t h , e / V i ∥ )$⁠.

For the case of a thermobath boundary, Fig. 8 shows that q_en is nearly independent of m_i and instead recovers a flux-limiting form for the heat conduction flux with $α e ∼ 0.2$ in the recession layer. However, its gradient within the recession layer still has the convective transport scaling, i.e., $⟨ d q e n / d x ⟩ ∝ m i − 1 / 2$⁠. As discussed in Eq. (36), the free-streaming form is recovered for q_en because of the dominating cold beam contribution of $− 2 n b v c T 0$⁠. This cold beam term, however, does not contribute to $⟨ d q e n / d x ⟩$ since the cold beam flux $n b v c$ in the collisionless recession layer is constant as shown in Eq. (41). To quantify the beam contribution to the electron heat flux as well as the particle flux, we have separated the cold recycled electrons from the original electrons. In Fig. 9, we show the contributions to the electron particle flux $n e V e ∥$ from both recycled $n b v c$ and the original trapped electrons $n m V m ∥ ≡ n e V e ∥ − n b v c$ for the $m i = 100 m e$ case. It shows that both the recycled and the original trapped electrons carry significant electron particle flux but nearly cancel each other to generate much smaller $n e V e ∥ ≪ n b v c$⁠. Notice that $n b v c$ has the electron scaling $n b v c ∝ v t h , e$ while $n e V e ∥$ has the ion scaling $n e V e ∥ ∝ v t h , i$ so that the difference between them is even stronger for larger m_i. Figure 9 also confirms that $n b v c$ is nearly constant within the recession layer as expected, so it does not contribute to $d q e n / d x$⁠. It is worth noting that the cold electron front is ahead of the PTF. Ahead of the PTF, $n b v c$ varies in space so that q_en itself and its gradient continuously follow the free-streaming scaling.

In conclusion, the thermal collapse of a nearly collisionless plasma interacting with a cooling spot is both theoretically and numerically investigated. For both types of cooling spots that signify, respectively, the radiative cooling masses (thermobath boundary) and a perfect particle and energy sink (absorbing boundary), we found that the thermal quench comes about in the form of propagating fronts that originate from the cooling spot. Particularly, for the thermobath boundary, two fast electron fronts have speeds that scale with the electron thermal speed $v t h , e$⁠, and two slow ion fronts propagates at local ion sound speed c_s. The former denotes the fast but moderate cooling of electrons, while the latter represents the slow but aggressive cooling of electrons and ions.

The underlying physics behind these propagating fronts have been investigated, in which the electron thermal conduction heat flux q_en is found to play an essential role. Specifically, the electron fronts are completely driven by $q e n ,$ which follows the free-streaming form (⁠ $q e n ∝ n e v t h , e T e ∥$⁠). Such large thermal conduction flux is reminiscent of a very limited amount of $T e ∥$ drop over a very large volume. In contrast, the ion fronts are formed as a result of the full transport physics, the crucial one of which is the ambipolar transport constraint. Due to such ambipolar transport constraint, q_en in the recession layer itself follows the convective energy transport scaling with the parallel plasma flow $V i ∥$ for the absorbing boundary. While for the thermobath boundary, the cold electron beam will restore the free-streaming limit of $q e n ∼ − 2 n b v c T 0 ∝ v t h , e$⁠, but its spatial gradient will follow the convective transport scaling since the beam particle flux (⁠ $n b v c$⁠) remains nearly constant within the recession layer. As a result, the electron temperature and, hence, the pressure retain large spatial gradient to drive the plasma flow toward the cooling spot.

For a thermobath boundary that provides an energy sink while re-supplying cold particles, the plasma cooling flow eventually terminates against the cooling spot via a plasma shock, which we named the cooling front since it signifies the deep cooling of electrons to the radiatively clamped temperature T_w. It is shown that such a shock front will convert the ion flow energy into the ion thermal energy via the mixing of hot and cold ions behind the front, which completely blocks the cold ions from migrating upstream. Therefore, the cooling front and the associated cooling zone behind the cooling front are of fundamental importance to $T i ⊥$ and $T e ⊥$ cooling via dilution. Unlike the recycled cold ions, part of the recycled electrons can penetrate through the cooling front to reach the electron fronts, causing further cooling of the electrons behind the electron fronts.

We thank the U.S. Department of Energy Office of Fusion Energy Sciences and Office of Advanced Scientific Computing Research for support under the Tokamak Disruption Simulation (TDS) Scientific Discovery through Advanced Computing (SciDAC) Project and the Base Theory Program, both at Los Alamos National Laboratory (LANL) (Contract No. 89233218CNA000001). This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated (Contract No. DE-AC02–05CH11231) and the LANL Institutional Computing Program, which is supported by the U.S. Department of Energy National Nuclear Security Administration (Contract No. 89233218CNA000001).

The authors have no conflicts to disclose.

Yanzeng Zhang: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Jun Li: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Xian-Zhu (X. Z.) Tang: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.