Thermal quench induced by a composite pellet-produced plasmoid

Injecting shattered pellets is the critical concept of the envisaged ITER disruption mitigation system (DMS). Rapid deposition of large amounts of material should presumably result in controlled cooling of the entire plasma. A considerable transfer of thermal energy from the electrons of the background plasma to the ions accompanies a localized material injection due to the ambipolar expansion along the magnetic field line of the cold and dense plasmoid produced by the ablated pellet. Radiation initially plays the dominant role in the energy balance of a composite plasmoid containing high-Z impurities. A competition between the ambipolar expansion and the radiative losses defines the Thermal Quench scenario, including the amount of pre-quench thermal energy radiated on a short collisional timescale—possibly detrimental for the plasma-facing components. The present work quantifies plasmoid energy balance for disruption mitigation parameters. For pure hydrogen injection, up to 90% of the pre-pellet electron thermal energy may go to the newly injected ions. We also demonstrate that a moderate high-Z impurity content within the plasmoid can reduce highly localized radiation at the beginning of the expansion. The thermal energy will then dissipate on the much longer ion collisional timescale, which would be attractive for ITER DMS.


Introduction
Large pellets injected into a tokamak plasma are used to mitigate detrimental disruptions.Excessive localized thermal loads and runaway electrons are high-priority issues for a disruption mitigation system (DMS).The use of pellets involves multiple physics aspects: material penetration, material shielding from the ambient plasma by a neutral-gas cloud and an electrostatic potential, ablation process, plasmoid dynamics, including its drift across the magnetic field, excitation of instabilities, etc [1].
In the present work, we address a part of the pellet assimilation problem: the 1D expansion of a pellet-produced plasmoid along the field line.This expansion occurs on the ambient plasma collisional timescale and is crucial for the thermal quench scenario.The thermal energy of the ambient plasma is either radiated or absorbed by the injected material during this phase.A reliable understanding of the assimilation phase is, therefore, essential for the ITER DMS.
Pellet injection is often considered within a fluid model (for example [2]) or using a 'solid body' approximation [3].It is, however, possible that the mean-free-path of the ambient particles exceeds the size of the pellet-produced plasmoid.
Because of that, the ambient plasma and the plasmoid cannot be treated as a fluid.They generally need a kinetic description [4,5].A significant effort in code development is also underway to treat such a multi-component plasma [6,7].
In this work we rely on the model developed in references [8][9][10][11], extending it to cover the role of high-Z impurities in disruption mitigation pellets.The amount of material in the pellet-produced plasmoid will be a free parameter in our study.We discuss the volumetric collisional heating of a transparent plasmoid and radiation losses in the presence of high-Z species.
The characteristics of the injected pellet and the ambient plasma determine the amount of material deposited onto a field line during the ablation process.In an initially cold plasmoid, the ambient ions experience much stronger collisional drag than the ambient electrons, as discussed in more detail in [10].Such a plasmoid is not transparent to the ambient ions until heated.During that initial 'non-transparent' phase, the plasmoid experiences pressure from the retarded ions impinging on it from the ambient plasma.However, as the plasmoid heats up, its internal pressure quickly becomes higher than the ambient pressure, and the plasmoid thus expands essentially freely.The internal pressure decreases during the expansion and eventually becomes comparable to the ambient plasma pressure.But the hot plasmoid is now transparent to the ambient particles and is, therefore, unaffected by external pressure.In what follows, we assume the plasmoid to be transparent throughout most of the expansion.This assumption limits the applicability of our model to the plasmoids with lineintegrated densities, N, below the inverse Coulomb collisions cross section, N < 1 σ = 3 4 √ π (4π ϵ0) 2 (kBT) 2 e 4 ln Λ , which signifies the 'number of collisions' required to completely stop an incoming particle.
We will also assume that the pellet instantaneously deposits material on the field line.A case of slow pellets, resulting in a slowly fueled plasmoid, was considered in [9].It was found that the net energy balance is altered only slightly, compared to the case of instantaneous deposition.Note that both these cases assume that the solid non-transparent pellet itself is not present on the considered field line.This condition is satisfied since the diameter of the ablation cloud usually exceeds that of the solid pellet by a large factor.

Self-similar expansion of a heated plasmoid
The self-similar expansion of a single-component heated plasmoid into rarefied plasma was studied in [8][9][10]12].In these studies, the ambipolar electric field is the only force accelerating the cold pellet ions, and the energy balance equation determines the electron temperature: the deposited heat equals the sum of the electron thermal energy and the ion kinetic energy.Aleynikov et al [8] provides the following self-similar solution for the case of a constant per-particle heating rate 2τ : where N is the field-line-integrated plasmoid density, u is the ion velocity, τ = 1 3N ´∞ −∞ Q dx is twice the per-particle heating rate, Q is the local volumetric heating rate, and M denotes the plasmoid ion mass.A notable feature of this solution is that the energy deposited into the plasmoid is split equally between the electron thermal energy and the ion kinetic energy.As for any self-similar solution, this solution is insensitive to initial conditions when the plasmoid length is much greater than the initial length.
The assumption of constant heating rate limits the applicability of this solution to early stages of plasmoid assimilation, at which the plasmoid is much colder than the ambient plasma, so that the heating rate Q is insensitive to the plasmoid temperature and given by where τ T is the 'temperature equilibration time' (see equation (20.6) in [13]) and τ e is the electron collision time, equation (2.5e) in [14].The other limitation is that the plasmoid is much denser than the ambient plasma.Under these limitations, the expansion dynamics do not depend on the plasmoid's initial size and density.These features enable a rather universal characterization of the plasmoid assimilation in a plasma of a given density and temperature.For instance, in an ITER-relevant scenario with the ambient plasma temperature of T a = 7 keV and density of n a = 10 20 m −3 the electron collisional time is τ e ≈ 150 µs.Within that timescale, plasmoid reaches the ambient temperature and the deuterium plasmoid 'length', L e || ≈ √ , reaches 110 m, irrespective of other initial conditions.If the plasmoid line integrated density, N, is much higher than n a L e || ≈ 10 22 m −2 , then the plasmoid density is still much higher than the ambient plasma density, and the subsequent homogenization will continue at equal temperatures of the dense plasmoid and the ambient plasma.If, however, the plasmoid line integrated density is lower than n a L e || , then the density homogenizes first and temperature equilibration follows later.
In the following Sections, we will use the solutions (1)-(3) in considering impurity physics.We start with a discussion of the impurity profile within the plasmoid.

Impurity profile in a composite plasmoid
In the cold-ion limit, the two forces acting on the impurity admixture ions in a deuterium plasmoid are the friction force from the bulk ion species and the ambipolar electric field.Both of them accelerate ions along the magnetic field.If the ratio of the ion charge, Z, to its mass, M, is the same for all ion species, then all the ions move together.That is not necessarily the Snapshot of the hydrogen (dashed) and neon (solid) profiles at 50 µs.Hydrogen profile is according to equation (1), neon is according to equation (6).
case because the ionization level for a high-Z impurity changes as the plasmoid temperature grows.The Z/M ratio is usually smaller for heavier impurities than for the H or D main species during the early stage of the expansion.The acceleration of impurities is then predominantly due to the friction force.
The friction force acting on an impurity ion with energy E α (velocity v α ) and mass m α in a plasma with temperature T β (and the bulk ion mass m β ) is given by equation (18.1) in [13]: where µ(x 2 ) = erf(x) − erf ′ (x)x, with the collisional time α n β ln λ , and x α/β = m β mα Eα T β .The velocity, v α , and energy, E α , are calculated relative to the speed of the bulk ions, which are assumed to be Maxwellian.
The ambipolar electric field in the process of self-similar expansion is eE = T ∂ ln n ∂x = 3 4 Mx t 2 .The resulting equation of motion for impurity ions with mass M i is with ∆v = ( 3 2 x t − ẋ) and T i denoting plasmoid ion temperature (we keep it small but finite , i.e. 10 eV, in our calculations for numerical stability reason).
Figure 1 shows a normalized profile for neon impurity in comparison with the self-similar profile of the bulk hydrogen ions after 50 µs of expansion.In this figure, the line-integrated density of the plasmoid is N = 10 22 m −2 , the ambient plasma density is n a = 10 20 m −3 , and the ambient temperature is T a = 7 keV.Neon is assumed to be initially at rest with the same density profile as hydrogen.We note that the neon profile is very close to the bulk hydrogen profile, except at the wings of the plasmoid, where the hydrogen density is too low to provide strong coupling.The same is true for heavier deuterium plasmoids.
In the high-velocity limit, the friction force, equation (5), is inversely proportional to the particle energy, which implies that the 'lag' between the profiles should increase with the difference between the velocities.The resulting impurity profile can thus steepen up to front breaking.Because of such steepening, the hydrodynamic treatment of the impurity profile can become problematic.
Despite the subtleties at the wings of the plasmoid, we find that the impurity density profile follows that of the bulk species closely in most cases of interest.We, therefore, treat the two profiles as identical throughout this paper, which simplifies the analysis significantly.

Ionization balance in a dense plasmoid
Although the high collisionality of a cold and dense plasmoid ensures a relatively quick ionization process, the transition through multiple ionization stages from a neutral to a highly charged ion may take significant time.In order to assess the effect of a non-equilibrium charge-state distribution during high-Z plasmoid expansion, we solve a set of timedependent ionization-recombination rate equations assuming temperature and density dependencies given by the selfsimilar equations equations ( 1) and ( 3).We ignore radiation losses effects at this stage, since such effects will slow down the temperature growth allowing for more time for the ionization to reach an equilibrium.
The system of equations describing the evolution of the ionization-level population density, y i , is ) where I {c,a} i (T) and R {c,a} i (T) are temperature-dependant ionization and recombination coefficients as given in ADAS database [15], with the superscript indices c and a referring to ionization and recombination due to collisions with the cold plasmoid electrons (i.e.I c i (T c )) and the hot ambient (i.e.I a i (T a )) electrons.
Figure 2 shows the resulting evolution of the average charge state < Z > and the corresponding volumetric radiation power As in the above examples, we assume a trace amount of neon in a deuterium plasmoid with N = 10 23 m −2 in a plasma with 7 keV temperature and density n a = 10 20 m −3 .Both n Ne and n evolve according to equation (1).The population densities of the levels y(t) refer to the center of the plasmoid.The red curves represent a collisional-radiative (CR) equilibrium charge-state distribution for comparison.Despite a quick temperature increase (7 keV by 100 µs) the mean charge state of the time-dependent solution follows closely the equilibrium charge state.The agreement between the two improves quickly for plasmoids with larger line-integrated density.
The total radiated energy ´Prad dt is 20% higher in the time-dependent solution than in the CR equilibrium.This discrepancy increases to 70% for a plasmoid with line-integrated NNe in an expending plasmoid.Non-equilibrium charge distribution (blue curve) is computed according to equation (7).
density N = 10 22 m −2 .However, as discussed in the next section, a cold and dense plasmoid should reabsorb the line radiation, which makes the CR model inapplicable in the first place.
In what follows, we take advantage of the fast charge state equilibration expected in realistic dense plasmoids and rely on the time-independent equilibrium charge state distribution Z(T) calculated using the CR model [15].

Radiation losses
In a plasma with high-Z impurities, radiation losses are predominantly due to line radiation of partially ionized atoms and continuum recombination radiation.The optical thickness of a plasmoid is different for different types of radiation.And while typical plasmoids are transparent for the continuum radiation, the mean free path of a line radiated photon can be significantly shorter than the plasmoid width (see, for example, [16]).Reabsorption of line radiation then reduces the line radiation power considerably.The plasmoid transitions to volumetric line radiation as it heats up, expands, and becomes optically thin.
In general, plasma opacity calls for a description of radiation transport.A rough assessment is possible via the socalled line escape factor approximation, which requires knowledge of each transition probability and the statistical weights of the levels.Yet, the atomic data is not always available for all atoms of interest.Instead, in this work, we use an upper estimate for the radiated power, knowing that the spectral radiance of radiation cannot exceed the black-body level given by Planck's formula [17,18].
Figure 3 demonstrates the modified spectral intensity (per unit surface area) of an r = 10 cm slab of neon plasma with n i = 10 22 m −3 at 5 eV.The spectral intensity of the strong lines is limited to the black body radiation (blue curve).
We use line-emissivity coefficients, ε l , from the CR model [15], which assumes transparency to line radiation.The photon emissivities should change as the density of the photons increases, but we ignore this circumstance in our model.
At extremely high densities, pressure broadening, such as the quadratic Stark effect, is the dominant mechanism for line broadening [18].As the plasmoid expands and its density decreases, Doppler broadening becomes most important, and we assume that it dominates throughout the entire expansion.Doppler broadening is due to the finite temperature of the emitting ions.As discussed in [10], collisional coupling between plasmoid electrons and ions is very strong early in the heating process so that their temperatures are equal.The electrons and ions decouple when the temperature reaches a few tens of eV.The ions then remain at low but finite temperature, T i , but the electrons are heated further.An estimate for the decoupling temperature is given by equation (2.15) in [10].It is typically in the range of 10 to 40 eV.In our model, we assume T i = min(T e , T decouple ), where T decouple is given by equation (2.15) in [10].
We thus assume lines to have the Doppler broadening profile capped at the black-body spectral radiance level as shown in the bottom panel of figure 3.In other worlds, the lines which lie entirely under the black-body spectrum, such as the 23 nm line in figure 3, remain unmodified in our model, whereas the lines which exceed the black body spectrum are 'cut down' to the level of the black body spectrum.The resulting radiance, R(λ), is thus the minimum of the black-body spectral intensity B(λ) and the sum of the line intensities, The summation here is over all atomic lines with their emissivity coefficients ε l and corresponding ion density n k l i .n e denotes the electron density and P l (λ) describes a Gaussian line broadening profile with a width σ l = λ l √ kTi mc 2 of a λ l line, The total radiation is an integral over this capped spectrum: and the effective cooling coefficient becomes Continuum radiation is ignored in equation (11).This radiation is important only at higher temperature, when the plasmoid is already transparent to line-radiation and the lines are well below the black-body intensity.
Our model implements integral (11) with line emissivity coefficients ε l obtained from the OPEN-ADAS database [15], and we still assume ionization balance to be according to CR equilibrium.
Note that both the black-body expression or completetransparency limit for radiation would lead to erroneous (excessive) radiation losses, as shown in figure 4. The curve P CR lines = rn i n e L lines (T) represents the surface radiation from a transparent plasmoid, with L lines (T) being the CR line cooling coefficient, whereas P Opaque lines represents equation (11).The figure also shows the continuum radiation for comparison.We observe that the plasmoid becomes transparent to line radiation after being heated to over 100 eV.As discussed next, the substantial reduction of radiation at low temperatures affects the early stages of plasmoid expansion in a significant way.

Electron kinetic effects in pellet plasmoids
The ambipolar electric field along the plasmoid creates a potential well for the electrons.This potential well confines low-energy electrons, whereas hot ambient electrons pass through the well and lose some energy via Coulomb collisions with the colder ones.The energy exchange between the hot and cold electrons requires a kinetic treatment with a selfconsistent calculation of the ambipolar potential.Because the potential evolves slowly during plasmoid expansion, an adiabatic invariant is the most appropriate variable for describing electron kinetics.
Arnold et al [4,5] provide a corresponding kinetic description of the electron energy exchange in an evolving ambipolar potential.
The energy of the ambipolar electric field is much smaller than the electron thermal energy in a quasineutral plasma.Therefore, the ambipolar electric field does not affect the overall energy balance.It, however, reduces the rate of energy exchange between the hot passing electrons and the colder electrons trapped in the plasmoid.
Arnold et al [5] provides an expression for the ambipolar potential assuming that the electron distribution is isotropic and is composed of two joined cut-off Maxwellians: We note that this potential can exceed the one given by the Boltzmann relation when T ≪ T a .The potential well does not trap most ambient electrons, which instead pass through, moving faster within the well and decreasing their density inside.The net effect is that the deeply-trapped population of cold electrons experiences 3/4 of the collisional heating it would in a homogeneous plasma ( [4], equation (7.7)).The 3/4 factor reflects the fact that the selfconsistent potential is approximately parabolic.We take that into account by multiplying the 'usual' heating coefficient for the plasmoid (4) by 3/4: where n e refers to the plasmoid electron density.

Parallel dynamics
Assuming that all ion species are cold and move together and considering ionization equilibrium with the opacity model for the radiation, the 1D dynamics of a heated plasmoid is governed by the following system of hydrodynamic equations.
where n H and n i are the main spices and the impurity admixture densities (assumed to be much greater than the background plasma density), M H and M i are the ions masses, u denotes their velocity, T is the plasmoid electron temperature (assumed uniform), Z(T) is the mean impurity ion charge, and L op is the effective cooling coefficient equation (12).Equations ( 15)-( 17) admit the following self-similar ansatz [8]: This ansatz reduces the hydrodynamic system to a set of three ordinary differential equations: The per-particle heating rate q ≡ Q ne , where Q is given by equation ( 14) (see also (20.3) in [13]), where τ T is the temperature equilibration time given by equation (20.6) in [13].
When the plasmoid is on a finite-length field line, the ambient temperature will decrease gradually as the ambient plasma loses energy to the plasmoid.Our model describes this temperature evolution by an additional energy balance equation where A denotes the length after which the field line closes on itself.This model implies that the ambient electrons remain Maxwellian when they cool down.The cold ion model precludes accounting for the collisional ion effects, such as viscosity.We have treated these effects rigorously in [10] within a set of Braginskii equations.We have found that the role of viscosity is negligible for the considered parameters for the majority of the plasmoid during the expansion time.The role of viscosity is observable in the outer regions of the plasmoid, where viscosity contributes to ion heating and deceleration.We ignore these effects in the new model.
In the subsequent sections we solve equations ( 20)-( 23) numerically for parameters relevant to ITER plasmas.

Deuterium plasmoid
We first consider pure deuterium plasmoids.In this case, the role of radiation is negligible, and the deposited energy is divided between the electron thermal energy and the ion kinetic energy.The Shattered Pellet Injection (SPI) system in ITER injects large pellets that break up into many small fragments before reaching the plasma.In this work, we sidestep the problem of pellet ablation in the SPI plume and adopt only crude geometric parameters, such as plume radius and plasma size.We then run our calculations for various amounts of ablated material and check that the chosen parameters are within the validity range of our assumptions.Most importantly, we verify that plasmoids are transparent to the ambient plasma.
The SPI system launches a plume of shattered material with an angular spread of 20 • .The characteristic perpendicular plume size is then r ≈ 0.5 m at mid-radius.The surface area of a mid-radius flux surface in ITER is approximately 500 m 2 , and the corresponding length of the field line of interest is 1000 m.We stop the calculations if the plasmoid spreads over the entire flux surface, i.e. if it reaches 1000 m in length.
Figure 5 shows the evolution of plasmoid length and other plasmoid and ambient electron parameters integrated over the flux surface.All energies are normalized to the pre-quench ambient electron thermal energy E 0 T a .The plasmoid electron thermal energy is E T = 3 2 ´ne TdS/E 0 T a , the ambient electron thermal energy E Ta = 3  2 ´na T a dS/E 0 T a and the ion kinetic energy dS/E 0 T a .The initial values of T a = 7 keV, n a = 10 20 m −3 and N = 2 • 10 23 m −2 correspond to tripling of the pre-pellet density on the flux surface.This calculation stops when the peak plasmoid density drops to 10 21 m −3 , by which time W i = 0.66, implying that most of the pre-pellet electron thermal energy has been transferred to the ions.The corresponding electron temperature is T = T a = 860 eV.Note that, for uniform injection, the electron and ion temperatures after dilution would be 2333 eV.Ambient plasma ions played no role in the plasmoid dynamics on the sub-millisecond expansion timescale.Yet, the hot ion thermalization time ranges from a few to tens of ms in such a cold and dense electron background, which means that the electrons will eventually heat up.
Figure 6 shows the ultimate normalized ion energy W i and electron temperature T as functions of the pre-quench temperature and the amount of assimilated hydrogen N (where N/N a = 1 corresponds to doubling plasma density).The calculations end when the central plasmoid density drops to 10 times the ambient plasma density.This figure shows that the ions gain up to 90 percent of the pre-pellet electron thermal energy when the plasma density increases considerably due to injected deuterium.The post-pellet electron temperature is very low in that case.A thermodynamic equilibrium will then be established on the ion-electron collision timescale.
The red curve in figure 6 shows where the mean free path of the ambient particles equals the plasmoid length.Below that curve, the plasmoid is not transparent for the ambient plasma and experiences an external pressure during the expansion.The external pressure can slow the plasmoid expansion and reduce the ambipolar energy transfer.Our model does not describe this regime.Deep inside the short mean free path region, a hydrodynamic description is appropriate for both the plasmoid and the ambient plasma.

High Z plasmoid
Injection of high-Z pellets entails powerful line radiation.Figure 7 shows an evolution of the temperature, length and the energy balance for a pure neon plasmoid with N = 2.5 • 10 22 m −2 in an ambient plasma of T a = 7 keV, n a = 10 20 m −3 on a 1000 m field line.Unlike the pure deuterium case, the temperature does not grow initially in the neon plasmoid because line emission radiates all the incoming energy (even though the plasmoid is not transparent to line radiation).The temperature stays around 10 eV during the initial phase, which allows the plasmoid to expand to several meters in approximately 40 µs.
The radiation losses decrease considerably around 40 µs because the plasmoid density decreases during the initial expansion.The reduced radiation facilitates further heating and an accelerated expansion of the plasmoid.When the plasmoid heats up to over 100 eV, the ambipolar energy transfer to the ions becomes the dominant energy loss mechanism for the electrons.By 80 µs, the thermal energy of the ambient plasma is exhausted, and the heating stops.By this stage, 60% of the initial electron energy is radiated away.The remaining 40% splits approximately equally between the electron thermal energy and the ion kinetic energy.
During next phase, the plasmoid expands with roughly constant kinetic energy of the ions.The electrons thermal energy still goes down due to radiation.We stop the calculation when the electron temperature drops to 10 eV.The ions gain about 25% of the electron thermal energy stored on the flux surface.The other 75% are radiated away with most losses localized in space and time at the beginning of the expansion.
To illustrate the extreme sensitivity of the expansion scenario to radiation losses, we repeat the calculation assuming full transparency to the line radiation.Figure 8 shows the results.In this case, the plasmoid length increases from 0.5 m to only 2 m, and all the incoming thermal energy is radiated instantaneously.The electron temperature never exceeds 10 eV.This case represents an unrealistic scenario in which the plasma loses its thermal energy via radiation within one collisional time.
We observe a threshold feature in terms of plasmoid density or, more precisely, the radiation power.If the plasmoid can radiate all the incoming heating power, it will unlikely spread over the entire flux surface because its temperature remains low.On the other hand, if the heating overcomes the  line radiation and the plasmoid reaches several tens of eVs, the parallel expansion plays the dominant role during the assimilation.The radiation losses take place on a longer timescale in this case.
Figure 9 shows the fraction of the thermal energy converted to the ion kinetic energy and the radiated fraction depending on the plasmoid line integrated density.Note that the electron temperature never exceeds a few tens of eV for N/N a ≳ 0.25 despite the reduced line radiation.The timescale of radiation is very short at N/N a ≳ 0.25.For the smaller plasmoids with N/N a ≲ 0.25, a significant fraction of the initial thermal energy goes to the ions.The remainder is radiated away on a longer timescale.At very small N, the plasmoid density homogenizes quickly, after which one can describe the thermal quench (temperature equilibration and radiative losses of thermal energy) as in a homogeneous plasma [19].These calculations demonstrate that high-Z pellets may cause unacceptably fast thermal quenches in hot plasmas.The same concern comes from experimental observations that pure high-Z pellets facilitate runaway electron generation [20,21], presumably due to a much faster thermal quench than in the case of massive gas injection.

Mixture plasmoids
When the plasmoid consists of a hydrogen/neon mixture, the assimilation scenario combines the features of pure deuterium plasmoid and high-Z plasmoid.In the case of a high fraction of high-Z material, the central electron density increases in time due to a higher degree of ionization despite the fast expansion of the plasmoid.The radiated power, given by the first term in the second line of equation (17), increases rapidly in time in such a case.This produces a feature shown in figure 7, where the deposited power is radiated away by a slowly expanding plasmoid during the first 40 µs.
In the case of a small fraction of high-Z material, both the electron and the ion densities decrease with the expansion of the plasmoid so that the radiated power drops quickly.This changes the initial phase of assimilation qualitatively.Figure 10 shows assimilation of a plasmoid that consists of N H = 1.5 • 10 23 m −2 hydrogen and N Ne = 1.5 • 10 22 m −2 Neon.The total line-integrated electron density is thus 3 • 10 23 m −2 .The other parameters are the same as in figure 7. Despite the higher line integrated density, the radiation plays a less prominent role in the expansion dynamics.In particular, the electron temperature increases monotonically during the initial stage, and the radiation power drops quickly.As a result, a significant part of the energy goes to the ions.
One can vary the fraction of energy radiated quickly by varying impurity content of the plasmoid.Figure 11 presents the contours of the ultimate fraction of the pre-quench thermal energy transferred to the ions (black contours) and the radiated energy fraction (red) for a composite plasmoid with a total number of electrons N and the impurity fraction ZN i /N.The field line length is assumed to be 1000 m, the ambient plasma density n a = 10 20 m −3 and temperature T a = 7 keV.In areas where the two contours overlap and add up to unity, the entire pre-quench electron thermal energy is either radiated or transferred to the ions; the ultimate electron temperature is thus nearly zero.These cases represent a complete thermal quench within the timescale of plasmoid assimilation.Because of the low plasma temperature and low conductivity, these cases are particularly prone to runaway electron generation.In the areas where the two contours do not add up to unity, the remainder of the energy is still in the electron temperature when the plasmoid homogenizes.
As shown in these calculations, pellets with high impurity content are generally undesirable because they cause a too-fast thermal quench with a collisional timescale.Using lower concentrations of impurities is a less dangerous option, in which the electron thermal energy goes to the ions first and is later lost on a longer timescale.

Discussion and conclusions
Thermal quenches are often modeled under the assumption of a spatially uniform plasma [19,[22][23][24].In this work, we have demonstrated the essential role of the inherent nonuniformity for pellet-triggered disruptions.The nonuniformity affects the line radiation losses significantly because radiation is initially trapped within the plasmoid.This aspect calls for modeling of the line radiation and ionization balance, which generally involves radiation transport codes, as attempted recently in [3].The spatial nonuniformity also opens up a new energy loss channel for electrons.That is the ambipolar acceleration of ions.
Despite the differences in our model from [3], some of our results are qualitatively similar.In particular, the significance of line radiation is evident in both models, even for a small content of impurities.In [3], the plasmoid is treated as a solid object that experiences the full external plasma pressure, whereas, in our case, plasmoid transparency to ambient particles precludes the impact of plasma pressure.However, the plasmoid pressure is much higher than the ambient pressure during the expansion phase of interest, and the plasmoids thus expand almost freely in both models.On the other hand, the per-particle heating rate, limited by the ambipolar ambient fluxes, is much smaller in [3], which is why plasmoids in their model never exceed a few meters in length after the expansion.This aspect results in significant differences in predicted plasmoid dynamics and, therefore, in the thermal quench scenario.The two models describe two different regimes of the thermal quench.
The key points of our work can be summarized as follows: 1. We predict a significant (up to 90%) transfer of the prepellet thermal energy to the newly injected ions for deuterium or hydrogen plasmoids.This effect is similar to ion heating by fuelling pellets [8].Because a complete thermal quench is not achievable for pure H or D pellets, a thermal equilibrium will be established on the ion collisional timescales.Yet, a drastically different electron temperature evolution is expected within the plasmoid assimilation timescale compared to uniform injection.The latter can affect plasmoid drifts [25] and other plasma processes.2. For densities relevant to disruption mitigation pellets, the plasmoid impurity profile should follow that of the main species due to the strong friction on the latter.3. Ionization levels are close to CR equilibrium levels for plasmoids with line-integrated electron densities exceeding 10 23 m −2 .In smaller plasmoids, the ionization levels deviate from the CR equilibrium, which involves rate equations.4. The line radiation is initially trapped in the cold and dense plasmoids.This circumstance affects the expansion dynamics considerably because the total radiated power is lower than the one from a would-be transparent plasmoid by at least an order of magnitude.Line radiation trapping facilitates faster heating, and the plasmoid transitions more quickly into the regime where ambipolar ion heating dominates.5. Despite trapping, line radiation still plays a significant role in the energy balance initially.This very localized in space (∼1 m) and time (∼τ ee ) radiation can be detrimental for plasma-facing components.
Our scan over plasmoid density and composition demonstrates that, for a sufficiently low impurity content, only a minor fraction of the pre-quench plasma energy radiates away on a very short timescale.Yet, that low content of impurities is still sufficient to trigger a complete thermal quench.

Figure 2 .
Figure 2. Evolution of the mean ion charge < Z > (top panel) and normalized radiated power P radNNe in an expending plasmoid.Non-equilibrium charge distribution (blue curve) is computed according to equation(7).

Figure 3 .
Figure 3. Modified spectral intensity (per unit surface area) of a 10 cm slab of neon plasma with n i = 10 22 m −3 at 5 eV.

Figure 4 .
Figure 4. Power radiated from a unit surface of a slab of neon with n i = 10 22 m −3 .

Figure 6 .
Figure 6.The ultimate ion energy W i (top) and electron temperature T (bottom) as a function of pre-pellet temperature and the amount of assimilated hydrogen.

Figure 7 .
Figure 7. Same as figure 5 for a pure neon plasmoid with N = 2.5 • 10 22 m −2 .The additional red curve represents normalized radiation losses P rad .

Figure 8 .
Figure 8. Same as figure 7 assuming complete transparency for line radiation.

Figure 9 .
Figure 9.The ultimate fraction of pre-pellet thermal energy transferred to ions during assimilation of a pure Neon plasmoid with line integrated density N in an ambient plasma with Ta = 7 keV, na = 10 20 m −3 on a 1000 m field line.

Figure 11 .
Figure 11.2D isolines of ultimate fraction of the pre-quench thermal energy transferred to ion kinetic energy W i (black) and radiated fraction (P rad ) (red) after assimilation of a mixture plasmoid with Ni N fraction of high-Z atoms and normalized line integrated density of N Na .Ambient plasma density na = 10 20 m −3 and temperature Ta = 7 keV.Field line length is 1000 m.