Surface heat loads on the ITER divertor vertical targets

For a complete description of the ITER divertor targets along with the motivation behind the proposed shaping solution the reader is referred to [10, 11]. Here we briefly evoke only those elements needed for our analysis. In what follows, the term 'poloidal gap' refers to gaps between plasma-facing units (PFUs) running parallel to the poloidal plane (figure 1). The term 'toroidal gap' refers to gaps which extend in the toroidal direction separating monoblocks on a given PFU. The toroidal component of the tokamak magnetic field is perpendicular to the sides of a PG formed by the short MB edges, while it is parallel to the sides of a TG (see figure 1). These definitions are commonly used in the scientific literature on the topic [12].

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Unshaped MBs are characterized in this study by toroidal length L_MB  =  28 mm, poloidal width W_MB  =  12 mm, and radial height H_MB  =  28 mm. The relative misalignments between MBs on a given PFU, and between PFUs on a given divertor cassette must respect specified tolerances. The gap widths between MBs and their tolerances are listed in table 1. The TG between MBs on the same PFU is of width g_MB  =  0.4 mm. There are three types of PGs between adjacent PFUs at the straight parts of the targets. The inter-cassette gaps between each of the 54 cassettes are of width g_PFU  =  20 mm. The relatively wide dimension of the latter is imposed by remote handling requirements. As a result, component tilting is required in order to protect the LE of each cassette which inevitably arises due to tolerance build-up during the manufacture and installation of these massive objects together with assembly tolerances of the vacuum vessel. The magnetic shadowing is complete, sometimes extending beyond the leading PFU, so analysis of this gap will not be discussed very much here. The precise tilting angles proposed in the current design are approximated here by a rotation of Δθ  =  0.5° about the cooling tube axis at both IVT and OVT.

Each VT is divided into two halves of steel support structure by a 2 mm vacuum gap in order to reduce the torque due to eddy currents (see figure 2). Eight (eleven) PFUs are mounted on each of the IVT (OVT) halves. The intra-cassette PG widths between PFUs on the half-targets are specified to be wider than the gap between their steel support structures in order that in the event of deformation the support structures touch each other before the MBs do. The OVT surface is purely vertical so the intra-cassette gaps are all of constant width g_PFU  =  2.8  ±  1 mm. The inter-PFU gaps on each half-target are g_PFU  =  0.5  ±  0.2 mm wide. Unlike the OVT, the inner divertor surface envelope is conical. For reasons of economy, the actual reference design calls for four families of increasing MB widths running from the bottom of the straight part of the IVT to the top. This results in variations of the gap widths. The IVT intra-cassette gap varies from 2.7  ±  1.0 mm at the bottom to 3.5  ±  1.0 mm at the top, while the inter-PFU gaps can be between 0.5  ±  0.2 and 1.0  ±  0.2 mm wide.

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Poloidal alignment, that is, a relative displacement along the cooling tube axis of neighbouring PFUs, plays a role at gap crossings in that it determines whether magnetic field lines that penetrate through TGs intersect the short side, long side, or corner of a given MB, forming an OHS (section 3.4.3). Most TGs in the current design should be aligned between PFUs. An exception is at the left hand (downstream) side of the OVT where the last two PFUs are successively displaced by about 3 mm downwards (see figure 24). This comes about from particular shaping requirements at the baffle region, where protection must be provided against vertical displacement events [31]. Poloidal alignment is not specified in this paper, nor is it specified in the design, and even though we mostly assume they fall on MB corners (the worst case), it must be assumed that OHSs can appear anywhere on the poloidal LEs.

The radial step Δr from one MB to the next is measured from the plasma-facing surface to the neighbouring edge across a PG or a TG. The radial step must respect the nominal design value within the tolerance m_rad, designated as 'radial misalignment'. It should be emphasized that m_rad is a not a tolerance defined in the usual way, that is, applied independently to each PFU with respect to the support plate. If that were true, then it would be possible to find radial steps up to 0.6 mm between neighbouring PFUs on the same half-target, which would allow exposed LEs even of MBs with a 0.5 mm toroidal bevel. Instead, the reference points on all MBs, for instance, the radial positions of all the leading or all the trailing poloidal edges, must be contained within a surface profile envelope not exceeding m_rad in height. This is schematized in figure 3 for MBs with a toroidal bevel of depth h_tor  =  0.5 mm.

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As viewed by an observer standing facing the target, the right hand (left hand) edges of each MB are designated as leading (trailing) edges, or upstream (downstream) edges in this paper. The upper and lower bands in figure 3 delimit the envelopes of allowed radial positions of the trailing and LEs, respectively, of each MB. The radial misalignment of a given MB is defined here as the difference in radial position between its trailing edge and the trailing edge of the preceding MB. For example, moving from the fourth to the third MB at the right in figure 5, m_rad  =  +0.3 mm. Between the third and second MBs, m_rad  =  −0.3 mm. Magnetic field lines which are connected to trailing edges of MBs are shown; they define flux tubes whose cross-sectional areas in the perpendicular direction determine the total plasma power flowing to each MB. This illustration shows how the toroidal wetted fraction is related to the MB toroidal bevel and radial misalignment. The cross-sectional area S₁ of the flux tube defined by the protruding third MB is larger than the nominal one, an example of which is connected to the first MB. The second MB receives less total power than the others because it is partially shadowed by the third MB. The effect of partial magnetic shadowing of the main loaded surface is analyzed in section 4.2, where it will be demonstrated that it is not the total power to the MB which counts, but the local heat flux.

If the MBs were cuboids without any beveling, there would potentially be LEs directly exposed to parallel plasma flux; there will more than enough power flowing through the SOL in ITER to melt them [14, 15]. Due to toroidal beveling, within the specified tolerances, the recession of a LE with respect to the trailing edge of the preceding MB can be as little as  −0.2 (−0.5) mm, and as much as  −0.8 (−2.5) mm at inter-PFU (intra-cassette) gaps, respectively.

Table 1 contains the most up-to-date design values available at the time of writing. The design values of radial steps, gap widths, and all the various tolerances are evolving as ITER Organization receives feedback from the industrial suppliers who are assembling full scale divertor prototypes. Therefore, some of the worst case reference values chosen for this work may differ from the final design values that will be ultimately specified. To render this work pertinent for future reference, we have therefore attempted to make scans over the widest reasonable range of expected dimensions for problems that depend critically on tolerances (such as edge melting during ELMs, section 6).

The magnetic field orientation is defined with respect to a nominal axisymmetric divertor target without local shaping or component tilting (figure 4). The additional angular increments due to these geometric factors must be included when calculating heat fluxes to particular surfaces.

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For the purpose of local MB analysis it is reasonable to neglect toroidal curvature, to assume that the nominal surface of a given vertical target is planar, and that the magnetic field is uniform in space. At each target a local Cartesian coordinate system is defined with respect to the toroidal component of the parallel heat flux vector approaching either vertical target. The x-direction points in the negative toroidal direction at the IVT (figure 1(a)) and in the positive toroidal direction at the OVT (figure 1(b); see also figure 5). The y-direction lies in the poloidal plane, parallel to the PFU cooling tube axis, and points downwards at both VTs. The z-direction is the normal vector of the nominal VT surface (neglecting local shaping and component tilting).

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A convenient feature of the geometry is that when viewed along the plasma flow direction, the target environments look nearly identical; apart from the slight variations in angle of incidence, the only major difference is that the ions rotate about B in opposite directions (figure 5). An observer sitting on the dome between the two VTs, whether facing inwards looking at the IVT, or outwards looking at the OVT, sees the plasma flowing from right to left, and slightly downwards from the X-point to the target. The bottom, long (toroidal) edges of the MBs, and the left-hand, short (poloidal) edges are magnetically shadowed. Therefore, if the heat flux were purely optical, one would expect to see the top, long edges and the right-hand, short edges overheating (if those edges were not protected by MB shaping). This simple picture is valid at the OVT. However, at the IVT, the bottom, long edges can also overheat because the Larmor gyration of incoming ions results in an upward vertical motion at the instant of impact.

The projection of the magnetic field onto the nominal VT surface, when viewed along the normal vector, makes an angle θ_// (so named because the projection lies in the plane that is parallel to the surface) with respect to the toroidal direction (figure 1). This angle arises because the magnetic flux surfaces are tilted with respect to the nominal VT surface normal. If the magnetic flux surfaces were to intercept the VT surface perpendicularly (θ_ψ  =  0°), then θ_//  =  0°. The projected angle in the plane perpendicular to the cooling tube axis is θ_⊥. The angles are related by

$$\begin{eqnarray}&&\tan {{\theta}_{\psi}}=\frac{\tan {{\theta}_{\parallel}}}{\tan {{\theta}_{\bot}}}\end{eqnarray} \tag{ 1 }$$

In this report we consider representative cases near the strike points at the bottom of the IVT and OVT corresponding to the reference 15 MA/5.3 T magnetic equilibrium (table 2). The angle α between the magnetic field vector and the nominal VT surface is related to θ_⊥ and θ_// through

$$\begin{eqnarray}&&\tan \alpha =\frac{\tan {{\theta}_{\bot}}}{\sqrt{1+{{\tan}^{2}}{{\theta}_{\parallel}}}}\approx \tan {{\theta}_{\bot}}.\end{eqnarray} \tag{ 2 }$$

It is helpful to describe the dynamics of ion flow to a planar surface in the presence of an inclined magnetic field not only to be able to interpret the results of the full 3D calculations, but also to appreciate the error potentially introduced by ignoring electric fields. We consider an infinite, planar, absorbing surface lying in the x–y plane towards which ions move (figure 7) and assume their trajectories are influenced only by the Lorentz force due to a uniform magnetic field $\vec{B}=\left({{B}_{x}},0,{{B}_{z}}\right)$ which intersects the surface with an angle $\alpha$. $\vec{B}$ points towards the surface with ${{B}_{x}}>0$, ${{B}_{z}}<0$, and $\alpha <0$. Far above the surface, the ion velocities are described by the distribution $f\left({{v}_{//}},{{v}_{\bot}},\phi \right)$ where ${{v}_{//}}>0$ is the speed parallel to $\vec{B}$, ${{v}_{\bot}}>0$ is the magnitude of the perpendicular velocity, and $0\leqslant \phi <2\pi$ is the uniformly distributed gyrophase angle describing the orientation of the perpendicular velocity vector with respect to the field line. The ion trajectories are helices of radius $r={{v}_{\bot}}/{{\omega}_{ci}}$ and pitch $l=2\pi {{v}_{//}}/{{\omega}_{ci}}$ where ${{\omega}_{ci}}$ is the Larmor frequency. The perpendicular and parallel speeds are uncorrelated, so the distribution function can be written

$$\begin{eqnarray}&&f\left({{v}_{//}},{{v}_{\bot}},\phi \right)=\frac{{{n}_{0}}}{2\pi}{{f}_{//}}\left({{v}_{//}}\right){{f}_{\bot}}\left({{v}_{\bot}}\right)\end{eqnarray} \tag{ 5 }$$

where ${{n}_{0}}$ is the ion density sufficiently far above the surface, $z\gg r\cos \alpha$, and ${{f}_{//}}$ and ${{f}_{\bot}}$ are both normalized to yield unity when integrated over all speeds.

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The distribution function at a specific point on the surface will be depleted of the population of ions whose gyromotion caused them to strike the target further upstream. Analysis of the helical orbits provides a means to calculate combinations of allowed velocity and gyrophase at the surface [17]. The velocity component normal to the surface is the sum of the projections of the parallel and perpendicular speeds

$$\begin{eqnarray}&&{{v}_{z}}={{v}_{//}}\sin \alpha -{{v}_{\bot}}\cos \left(\phi -{{\omega}_{ci}}t\right)\cos \alpha \end{eqnarray} \tag{ 6 }$$

where $\phi =0$ is defined when the perpendicular velocity vector is parallel to the x–z plane and points towards the surface. Gyrophase angles which correspond to ions leaving the surface with ${{v}_{z}}>0$ are not allowed due to the assumption of full absorption. The critical angle for which the ion grazes the surface at the time of impact (t  =  0) is found by setting the vertical speed to ${{v}_{z}}=0$

$$\begin{eqnarray}&&\cos {{\phi}_{1}}=\frac{{{v}_{//}}}{{{v}_{\bot}}}\tan \alpha.\end{eqnarray} \tag{ 7 }$$

The solution has roots in the second and third quadrants, which identify the extrema of the oscillatory vertical motion. The correct root for our analysis corresponds to the instant when the ion begins to move away from the surface with its vertical speed increasing from zero

$$\begin{eqnarray}&&\frac{\text{d}{{v}_{z}}}{\text{d}t}=-{{v}_{\bot}}{{\omega}_{ci}}\sin {{\phi}_{1}}\cos \alpha >0\end{eqnarray} \tag{ 8 }$$

which imposes $\pi <{{\phi}_{1}}<2\pi$, therefore the critical angle lies in the third quadrant (figure 8(a)).

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Some of the trajectories that approach the surface from above must also be excluded because, due to their gyromotion, the ions would have impacted upstream locations earlier in time (e.g. the red orbit in figure 7). Ions strike the target with increasing angle as the gyrophase increases above ${{\phi}_{1}}$ until the second critical angle ${{\phi}_{2}}$ is encountered, which corresponds to an ion that would have just grazed the surface somewhere upstream, but escaped, and gyrated one last time over the target before its impact (figure 8(b)). To evaluate ${{\phi}_{2}}$ consider the trajectory of such an ion that just escapes hitting the target at z  =  0 with the critical angle ${{\phi}_{1}}$ at time t  =  0,

$$\begin{eqnarray}&&z(t)=\frac{{{v}_{\bot}}}{{{\omega}_{ci}}}\left[\sin \left({{\phi}_{1}}-{{\omega}_{ci}}t\right)-\sin {{\phi}_{1}}\right]\,\cos \alpha +{{v}_{//}}t~\sin \alpha.\end{eqnarray} \tag{ 9 }$$

The time $0<{{t}_{2}}<2\pi /{{\omega}_{ci}}$ at which the ion finally strikes the target defines the second critical angle ${{\phi}_{2}}={{\phi}_{1}}-{{\omega}_{ci}}{{t}_{2}}+2\pi$. Combining equations (7) and (9), we obtain a transcendental equation relating ${{\phi}_{1}}$ and ${{\phi}_{2}}$

$$\begin{eqnarray}&&\sin {{\phi}_{2}}-\sin {{\phi}_{1}}=-\left({{\phi}_{1}}-{{\phi}_{2}}+2\pi \right)\,\cos {{\phi}_{1}}.\end{eqnarray} \tag{ 10 }$$

The critical angles must be calculated numerically as in figure 9(a).

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Despite the lack of an analytical expression for the critical angles, we can obtain an expression for the particle flux normal to the surface

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}{{\Gamma}_{z}}=n\langle {{v}_{z}}\rangle =\frac{{{n}_{0}}}{2\pi}{\int}_{0}^{\infty}\text{d}{{v}_{//}}{\int}_{0}^{\infty}\text{d}{{v}_{\bot}}\\ \quad {\int}_{{{\phi}_{1}}}^{{{\phi}_{2}}}\text{d}\phi ~\left({{v}_{//}}\sin \alpha -{{v}_{\bot}}\cos \alpha ~\cos \phi \right)~{{f}_{//}}~{{f}_{\bot}}\end{array}\end{eqnarray} \tag{ 11 }$$

by employing equation (10) to evaluate the integral over the gyrophase angle. The flux simplifies to

$$\begin{eqnarray}&&{{\Gamma}_{z}}={{n}_{0}}{\int}_{0}^{\infty}\text{d}{{v}_{//}}{\int}_{0}^{\infty}\text{d}{{v}_{\bot}}{{v}_{//}}\,\sin \alpha ~{{f}_{//}}~{{f}_{\bot}}={{n}_{0}}~\left\langle {{v}_{//}}\,\right\rangle \sin \alpha \end{eqnarray} \tag{ 12 }$$

which is the result expected from flux continuity. To calculate the density at the surface, we are not so lucky, and must resort to numerical evaluation of the integral. For a given magnetic field inclination, the critical angles depend only on the ratio of parallel to perpendicular speeds (figure 9(a)). If the distribution consists of a single value of speed ratio, the ion density at the surface is

$$\begin{eqnarray}&&{{n}_{i}}={{n}_{0}}\frac{{{\varphi}_{2}}-{{\varphi}_{1}}}{2\pi}.\end{eqnarray} \tag{ 13 }$$

In the limit of large Larmor radius ${{v}_{//}}/{{v}_{\bot}}\ll 1$, the ions can only strike the surface at grazing incidence, ${{\phi}_{1}}\approx {{\phi}_{2}}\approx -\pi /2$, and the ion density approaches zero at the surface (figure 9(b)). The normal component of the velocity increases as 1/n maintaining constant particle flux. In the opposite limit of vanishing Larmor radius, specifically for ${{v}_{//}}/{{v}_{\bot}}>1/\tan \alpha$, the critical angles are not defined and 100% of the gyrophases are allowed at the surface; the distribution functions at the surface and in the plasma are identical. To calculate the target ion density for an arbitrary distribution function, the curves of figure 9 must be convoluted into the integral

$$\begin{eqnarray}&&{{n}_{i}}=\frac{{{n}_{0}}}{2\pi}{\int}_{0}^{\infty}\text{d}{{v}_{//}}{\int}_{0}^{\infty}\text{d}{{v}_{\bot}}{\int}_{{{\varphi}_{1}}}^{{{\varphi}_{2}}}\text{d}\varphi ~{{f}_{//}}~{{f}_{\bot}}.\end{eqnarray} \tag{ 14 }$$

In partially detached baseline operation on ITER, the ion and electron temperatures are similar (table 3, section 5)

$$\begin{eqnarray}&&\frac{\langle {{v}_{//}}\rangle}{\langle {{v}_{\bot}}\rangle}\approx \frac{{{c}_{\text{s}}}}{\sqrt{2}{{v}_{T,\text{i}}}}\approx \sqrt{\frac{{{T}_{\text{e}}}+{{T}_{\text{i}}}}{2{{T}_{\text{i}}}}}\approx 1\end{eqnarray} \tag{ 15 }$$

so at the sheath entrance, a realistic ion distribution function is dominated by speed ratios of order unity. Reading from figure 9 it can be seen that such ions strike the target with gyrophase angles around $\phi$  ≈  270°, corresponding to near grazing angles of incidence of ion impacts with the surface. The physical angle of impact is obtained from the combination of ${{v}_{//}}$ and ${{v}_{\bot}}$ (calculations of a kinetic distribution function typical of the tokamak SOL yield, for example, a mean angle of impact of about 13° [18]). In the field of plasma surface interactions in fusion devices, 'grazing incidence' usually refers to the magnetic field angle, or the plasma flux in the context of the guiding center approximation. When considering, in addition, the Larmor gyration which allows charged particles to move perpendicular to the magnetic field, it could be supposed that this extra degree of freedom would allow a fraction of the particles to strike the surface at much larger angles of incidence. The results of this analysis demonstrate that this is not possible. Even when considering Larmor gyration, individual particles also strike the surface at grazing incidence. The consequences of this feature will become evident in the analysis of heat flux deposition on PG and TG edges.

The ion orbit model for estimating heat deposition at sharp edges is an improvement over the optical approximation. Nonetheless, some important physics is missing. The ions are launched towards the surface with a distribution function that corresponds to the sheath edge, as given by a published kinetic model that will be discussed in section 3.3 Since we ignore electric fields, the Debye sheath is absent, and there is no modification of the ion energies as they approach the wall. An important feature of the ion orbit model appears to be the grazing incidence of impacting ions (see figure 9) and so the question may be asked: how much does a self-consistent electric field modify the distribution of impact angles?

In the sheath the ions accelerate and the electrons decelerate in the intense electric field. As they approach the surface, the ions gain the energy that is lost by the electrons. Analysis of energy balance across the sheath is the basis for the derivation of the heat transmission factor [19]. Should one add all the kinetic energy gained in the sheath to the parallel component of ion velocity, or do the Larmor radii increase? Most papers treating the magnetized sheath consider ensemble-averaged quantities such as density or flow velocity. For example, the well-known paradigm of the magnetized sheath refers to sonic ion flow parallel to the field lines at the entrance of the quasi-neutral magnetic presheath, which deflects to become sonic normal to the surface at the transition to the positively charged Debye sheath [20]. The flow being discussed in those treatments is the fluid flow averaged over all ions, but, as such, no information can be deduced about how individual particle orbits behave. Do the ions maintain their Larmor gyration superimposed on an E  ×  B drift motion, or do they accelerate freely across the magnetic field as implied in figure 2.22 of [21]?

In a study of microscopic erosion patterns on a tokamak limiter, a comparison of ion impact angles calculated with and without the sheath electric field [18] suggests that there is a modest steepening of the ion impact angle of around a mere 1°. To understand why there is almost no change, it is helpful to visualize individual ion orbits that are the essential ingredient to describing how energy is spread over minute surface features. This is also a key ingredient to provide a quantitative basis for evaluating the validity of the ion orbit model.

To examine the ion orbits in the magnetized sheath, the electric potential is calculated using a self-consistent, collisionless 1D–3V PIC simulation of the sheath [22] for plasma parameters typical of an ELM at the IVT. The electron density at the sheath edge is n_e  =  5.4  ×  10¹⁹ m⁻³ and the ion and electron temperatures are T_i  =  T_e  =  3400 eV. It is assumed, as discussed in section 6, that during an ELM, magnetic flux tubes connect the H-mode pedestal directly to the divertor targets, and that the plasma flowing from the pedestal reaches the surface without collisions, with no change in perpendicular temperature, and thus no change of Larmor radius. Ion test particles are launched towards the target from the quasi-neutral region above the sheath, with parallel speed equal to the sheath-edge sound speed, and perpendicular speed equal to the mean thermal speed. In figure 10 the trajectories calculated with and without the electric force are compared. The electric field drives an E  ×  B drift parallel to the surface, but the ions maintain their Larmor gyration. The apparent fluid flow which deviates strongly from the magnetic field direction to become aligned with the surface normal direction is largely an ion orbit effect: approaching the surface within a few Larmor radii, there are fewer and fewer ions with upward components of perpendicular velocity because they are scraped off by the surface.

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Just like in the ion orbit model, in the self-consistent PIC solution there is still a narrow range of allowed grazing impact angles. The precise values of critical gyrophase angle will be modified depending on how the parallel and perpendicular velocity components change with respect to their initial values. Detailed analysis of this problem is beyond the scope of this paper. The essential result is that the mechanism of ion transport to the surface is similar to the zero electric field approximation. The ion orbit model gives a reasonable first estimate of heat flux spreading at exposed LEs which is sufficient for engineering scoping studies.

During ELMs there are suggestions that T_e at the VTs could be much less than the pedestal temperature, with T_e $\ll$ T_i. Some experimental observations have been interpreted as evidence for relatively cool electrons during ELM bursts [16, 23]. Modelling of ELM bursts as the collisionless, free expansion of a quasineutral plasma into vacuum indicate that the thermal energy of the electrons is quickly converted into ion kinetic energy [24], although 1D PIC simulations using the BIT1 code, presented in the same paper, show that electrons can carry an appreciable amount of energy to the target in the event that collisions are included. Earlier BIT1 simulations of ELMs in the JET tokamak predicted that about 30% of the ELM energy is carried to the target by electrons [25]. A clear consensus on how the electrons evolve during an ELM is lacking. If T_e $\ll$ T_i were to hold, then the effect of the sheath electric field would be much weaker and the ion orbits would tend towards unperturbed helices, making our ion orbit model even more applicable.

The incoming plasma is modelled as ions having a distribution of parallel speeds determined by a kinetic calculation of the presheath, and a Maxwellian distribution of perpendicular speeds. Analysis of the burning plasma scenario assumes equal concentrations of deuterium and tritium ions. For simplicity, we replace the two species by a single, fictional ion species having mass number A  =  2.5 for the calculation of quantities such as thermal velocity, sound speed, and Larmor radius. The electrons are assumed to be well described by the optical approximation due to their small Larmor radii. Only the Larmor gyration of ions due to the Lorentz force is included; electric fields which would arise in the magnetized Debye sheath near the surface are not considered. In order to represent in a heuristic way the filtering effect of the sheath [19], the ion and electron contributions to the heat flux are normalized so that they carry respectively 5/7 and 2/7 of the launched parallel heat flux, which corresponds to the assumption of ambipolarity. There is recent experimental evidence from Langmuir probe measurements and infra-red imagery in the COMPASS tokamak that when local non-ambipolar currents enter the target, the heat flux can be dominated by the electron component, making the power deposition profile close to optical [26]. Although beyond the scope of this paper, the effect of local target currents should be retained as a phenomenon of potential importance that merits further investigation.

The distribution of parallel speeds f_//(v_//) several Larmor radii away from the surface is approximated by a square function having a mean parallel speed V_// and thermal width V_T corresponding to the distribution function at the sheath entrance calculated by a collisionless kinetic model [27] (figure 11). The mean parallel speed is the kinetic sound speed

$$\begin{eqnarray}&&{{V}_{//}}={{c}_{\text{s}}}=1.3979\sqrt{\frac{Zk{{T}_{\text{e}}}}{{{m}_{\text{i}}}}}\end{eqnarray} \tag{ 16 }$$

and the thermal width from the model in [27] is

$$\begin{eqnarray}&&{{V}_{\text{T}}}=0.536\sqrt{\frac{Zk{{T}_{\text{e}}}}{{{m}_{\text{i}}}}}\end{eqnarray} \tag{ 17 }$$

with T_e  =  T_i in the source plasma far upstream. The kinetic width of the distribution is smaller than the Maxwellian thermal speed because the ions undergo adiabatic cooling as they accelerate down the presheath to the target (the thermal width of the parallel speed distribution is converted to net convective flow). The perpendicular speeds are described by a Maxwellian distribution of temperature T_i (which in ITER can be ~eV for inter-ELM loads, or ~keV for ELMs)

$$\begin{eqnarray}&&{{f}_{\bot}}\left({{v}_{\bot}}\right)=\frac{{{m}_{\text{i}}}{{v}_{\bot}}}{2\pi k{{T}_{\text{i}}}}\exp \left(-\frac{{{m}_{\text{i}}}v_{\bot}^{2}}{2k{{T}_{\text{i}}}}\right).\end{eqnarray} \tag{ 18 }$$

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The perpendicular velocity vector is assumed to have a random orientation in the plane perpendicular to $\overset{\scriptscriptstyle\rightharpoonup}{{B}}$, described by a phase angle 0  ⩽  ϕ  ⩽  2π.

The approximate form of the distribution is sufficient for this first scoping study. Originally, a simple monoenergetic beam was assumed for the parallel distribution with speed equal to the sound speed, but this produced unphysical, localized spikes of power distribution when the helical pitch was resonant with surface features. Taking a distribution with some thermal width is more physically appropriate, and produces smoother surface distributions. More sophisticated numerical distributions from kinetic modeling of ELMs [25] could be used instead, which also vary in time during the ELM, but the essential results of this analysis would not change.

The heat flux normal to any point on the MB surface is given by the integral

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}{{q}_{\text{surf}}}=C{\int}_{0}^{2\pi}{\int}_{0}^{\infty}{\int}_{0}^{\infty}\frac{1}{2}{{m}_{i}}(v_{\bot}^{2}+v_{//}^{2})\,(\overset{\scriptscriptstyle\rightharpoonup}{{v}}\cdot \widehat{n})\,{{f}_{\varphi}}\,{{f}_{\bot}}\,{{f}_{//}}\\ \quad \,{{R}_{E}}({{v}_{//}},{{v}_{\bot}},\phi )\,\,H({{v}_{//}},{{v}_{\bot}},\phi )\,\,\frac{\text{d}\phi}{2\pi}\,\text{d}{{v}_{\bot}}\text{d}{{v}_{//}}\end{array}\end{eqnarray} \tag{ 19 }$$

where f are the distribution functions at the sheath entrance and C is a normalization constant chosen so that the integral yields the parallel heat flux q_// for the case of an unshadowed surface perpendicular to the magnetic field. Each combination of v_//, v_⊥, and ϕ describes a helical orbit that is followed backwards in time from the moment of impact until the distance between the guiding center and the highest surface feature exceeds the Larmor radius. If an intersection occurs, it will be on one of the five nearest neighbouring MBs (figure 12). The mask function H  =  1 if the orbit extends back to the plasma without intersecting any other surface, and H  =  0 if it does not (i.e. the orbit is unpopulated because the particle would have struck another surface earlier in time).

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Ions and electrons have a probability to reflect from the target [28], carrying away some of their incident energy, resulting in a reduction of the net heat flux that is actually absorbed by the surface. The energy and incidence angle of each ion is used to calculate the energy reflection coefficients R_E by the analytical expression equation (2.13) on p 20 of [28], which are folded into the numerical integral (equation (19)) of the heat flux. For the electron component of the heat flux, given by the optical approximation, we assume the value 0.4, which is a reasonable value for reflection from tungsten in the energy range of interest (p 94 in [28]).

In this section the results of 3D heat flux calculations at IVT and OVT MBs using both the optical approximation and the ion orbit model are given. It is interesting to compare the two to appreciate the importance of Larmor radius effects. Figure 15 illustrates the locations of slices on a typical MB over which heat flux profiles will be plotted in the following sub-sections.

3.4.1. Poloidal gaps.

The purpose of the h_tor  =  0.5 mm MB toroidal bevel is to protect poloidal LEs from direct parallel heat flux at nearly normal incidence for the worst case radial misalignment m_rad  =  ±0.3 mm. For steady state and slow transients, the ion Larmor radii are so small (10  <  r_DT  <  100 µm) that this is fully accomplished; the optical approximation describes well enough the heat flux profile on a toroidal slice through the center of the MB far from gap crossings (figure 14). What happens at gap crossings will be discussed in section 3.4.3. The slight Larmor smoothing over a characteristic scale of 0.2–0.3 mm at the magnetic shadow boundary will not affect the thermal response of the MB significantly. On the other hand, we will see in the next section that the optical approximation fails to correctly describe the power deposition inside TGs, making the ion orbit calculation indispensable.

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The normalized heat flux on the top surface is 50% greater than q_tg because of the 1.5° increase of magnetic field angle resulting from 1° toroidal bevel combined with 0.5° component tilting. The difference in magnetic shadow position between IVT and OVT is due to the different nominal angles of incidence (2.7° and 3.2° respectively). The magnetic shadow at the intra-cassette gap is longer than at the inter-PFU gap because of the deeper half target radial step protecting the MBs on the downstream side of that gap. The heat flux on the downstream face of the MB (s_tor  >  28 mm) is zero because the model does not allow for particles to flow backwards along field lines.

Toroidal profiles at IVT MBs are shown in figure 15 during ELMs in the 15 MA/5.3 T burning reference scenario and the pre-nuclear, 7.5 MA/2.65 T scenario. The profiles are similar at the OVT so they are not shown here. During ELMs, the ion Larmor radius is large compared to the height of surface relief (figure 6). A large fraction of the ions penetrates into the magnetic shadow such that the heat flux to the shadowed part of the top surface is not zero as in the inter-ELM case (figure 14), but comparable to that on the wetted zone. In addition, there is a strong concentration of heat flux to the LE that can approach several times that on the top surface, which as we will see in section 6, can cause melting. At the intra-cassette gap (upper panel), 5 keV (r_DT  =  2.0 mm) ions are able to reach the LE, penetrating more than 0.5 mm down the gap. For T_i  =  2.5 keV (r_DT  =  2.9 mm) they penetrate nearly 1 mm into the gap. At narrower inter-PFU gaps (lower panel), the heat flux distributions at full and half field are similar.

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The grazing incidence of the ion orbits to an infinite flat surface is crucial to understanding how the incoming ion flux is distributed to a poloidal LE. It is tempting to assume that ions can penetrate into the PG to a characteristic depth equal to the Larmor radius, but for this to be possible, the ion would have to enter the gap with a large vertical component of velocity. Such ion orbits are not populated because they are removed from the distribution function when they strike the MB surface further upstream. Ions can penetrate into the gap deeper than predicted by optical shadowing, but less than a full Larmor radius because of the shallow angle of attack at the top of the gap (figure 16). The maximum possible penetration depth is associated with the steepest allowed incidence angle

$$\begin{eqnarray}&&\delta =\left({{g}_{\text{PFU}}}+{{m}_{\text{tor}}}\right)\tan \alpha -\frac{{{r}_{L}}\left(1+\sin {{\phi}_{2}}\right)}{\cos \alpha}.\end{eqnarray} \tag{ 20 }$$

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The first term is the depth of the magnetic shadow (39 µm at IVT inter-PFU gaps), where g_PFU is the nominal PG width and m_tor is its specified assembly tolerance (table 1). The second term comes from the Larmor gyration into the shadow. Recalling that ${{\phi}_{2}}$ is in the fourth quadrant for the most probable speed ratios (equation (15)), the second term is typically a fraction of a Larmor radius. For the maximum Larmor radius expected during an ELM (r_L  =  2 mm for T_i  =  5000 eV) the ions can penetrate down to 740 µm (figure 17). This is an important result for our problem; it means that even a 0.5 mm toroidal bevel may not be sufficient to fully protect poloidal LEs from ELM ions. The precise surface heat flux distribution will depend on the pitch of the helical ion orbits, the speed ratio ${{v}_{//}}/{{v}_{\bot}}$, etc.

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3.4.2. Toroidal gaps.

Shadowing of the ~28 mm long edges delimiting the TGs by an additional poloidal bevel is not presently foreseen. Nonetheless, the power entering a TG can be up to 5% (g_MB/(g_MB  +  W_MB)  =  0.6 mm/12.6 mm) of the total power impinging on each MB. Moreover, since it is deposited on a toroidal edge which is already hot due to loading of the main plasma-facing surface, it seems prudent to include its contribution in a full 3D thermal simulation. From the point of view of heat conduction, there is a significant difference between edges which are heated on only one versus both facets.

Consider a long TG between unshaped monoblocks on a nominal, untilted divertor cassette (figure 18). The field lines penetrate a radial depth $h={{g}_{\text{MB}}}/\tan {{\theta}_{\psi}}$ into the gap, creating a plasma-wetted strip along the toroidal LE that would be seen by an observer looking downward into the divertor (the lower edges, hidden from the observer, are magnetically shadowed, figure 5). Under the optical approximation, the plasma heat flux q_side to the toroidal strip is given by power balance, assuming that 100% of the power that enters the TG q_tg is distributed uniformly over the strip:

$$\begin{eqnarray}&&{{q}_{\text{side}}}h={{q}_{\text{tg}}}{{g}_{\text{MB}}}\end{eqnarray} \tag{ 21 }$$

which can be written

$$\begin{eqnarray}&&{{q}_{\text{side}}}={{q}_{\text{tg}}}\tan {{\theta}_{\psi}}.\end{eqnarray} \tag{ 22 }$$

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The corresponding expressions for shaped MBs are similar in magnitude but slightly more complicated to write down and bring nothing to this illustrative discussion.

According to the optical approximation only the side of each MB that faces upwards towards the X-point should receive plasma. It is not at all evident that this is realistic because the Larmor gyration of ions is almost purely poloidal. This means that there is a possibility that power can be deposited even on the magnetically shadowed side of a TG if the helicity of the ion orbits is in the correct sense.

Ion orbit calculations were made at the IVT and OVT assuming T_i  =  10 eV (figure 19). At the IVT unshadowed side, the heat flux decays with distance inside the gap, but its integral is less than the optical prediction. This means that some of the incident power goes elsewhere. This occurs for two reasons. Firstly, due to their Larmor gyration, some of the ions strike the shadowed toroidal edge of the upper MB and are thus lost before entering the gap (see the upper orbit in figure 20). The power deposition profile on the shadowed side is strongly peaked. Secondly, not all the ions that enter the gap strike its sides. Some of them strike the top surface of the MB due to their Larmor gyration (see the lower orbit on figure 20). The heat flux is about 4 times higher than q_surf at the center of the top. At the OVT the power deposition is entirely on the unshadowed side, but it is considerably more peaked than the optical prediction.

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Power loads at the edges of TGs have been calculated by the ion orbit model for ELM ions with T_i  =  5.0 keV at full field, and with T_i  =  2.5 keV at half field for IVT and OVT (figure 21). At the IVT the ions strike the magnetically shadowed side due to the helicity of their gyration, while at the OVT they strike the unshadowed side. The ELM results are different in that the ions only strike one side of the IVT TGs, whereas in the inter-ELM case they were distributed over both sides. During ELMs the Larmor radii are larger than the gap width, so the ions are scraped off on their first gyration into the gap. The normalized heat flux profiles at a given VT are nearly identical for the two ELM scenarios.

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3.4.3. Optical hot spots.

Despite the 0.5 mm toroidal bevel which hides the poloidal LEs and protects them from inter-ELM heat loads, magnetic field lines can penetrate down the long TGs and strike a poloidal LE or even a MB corner at a gap crossing, depending on the precise angle and alignment of the MBs. The projection of the TG along the magnetic field lines can thus form a triangular 'optical hot spot' on the LE. The minimum radial step necessary to avoid an OHS

$$\begin{eqnarray} &&\Delta r=-\frac{{{g}_{\text{MB}}}}{\tan {{\theta}_{//}}}\left(\tan \left({{\theta}_{\bot}}+ \Delta \theta \right)+\frac{{{h}_{\text{tor}}}}{L}\right)-{{g}_{\text{PFU}}}\tan \left({{\theta}_{\bot}}+ \Delta \theta \right)\end{eqnarray} \tag{ 23 }$$

depends principally on the intra-PFU gap width g_MB (i.e. the width of gaps between MBs on a given PFU). Such hotspots appear everywhere in the current ITER target design. They could be suppressed by introducing slightly deeper radial steps, or closing down the TGs (figure 22). For instance, OHSs would be completely hidden at OVT inter-PFU PGs if the radial step were guaranteed to be at least  −0.5 mm. Presently it is at least  −0.2 mm (−0.5 mm due to toroidal bevel plus 0.3 mm worst case radial misalignment).

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Figure 23 illustrates an OHS at the IVT. The magnetic field lines intersect the OHS at nearly normal incidence angle. Their surface area can be up to ~0.2 mm² in the worst case. The heat load can be represented as a point source that delivers a tiny fraction of the total MB power, striking the coldest zone of the front face of the MB at the edge of the magnetic shadow. The ion component of the heat flux is attenuated with respect to the optical approximation due to Larmor orbit losses to the sides of the TG through which they must pass to attain the OHS. For example, during inter-ELM periods taking T_i  =  10 eV, figure 23 shows that the heat flux carried by ions is reduced about a factor 2. We will see in section 5 that the additional LE heating at the OHS due to inter-ELM loads is of no concern, even assuming the full parallel flux without Larmor radius attenuation.

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Depending on their gyrophase angle, some of the hot ions released from the pedestal during ELM events also penetrate down TGs and locally contribute to the transient heat pulse at poloidal LEs. The deposition pattern is poloidally shifted with respect to the OHS, due to the preferred gyration direction which is upwards at the IVT and downwards at the OVT. The profile is also broadened; the ions 'spray' from the gap crossing rather than forming a well-defined plasma column. On the other hand, the Larmor radius of 5 keV electrons is 30 µm at the IVT, about three times smaller than that of 10 eV inter-ELM ions. Larmor radius losses of ELM electrons to the side of the TG will therefore not be as important as for ions; it is reasonable to assume that the electron heat flux follows the optical approximation. As opposed to inter-ELM loads, which are small enough to allow the system to come to equilibrium with a minimal temperature increase at the OHS, we will see in section 6 that transient heating due to ELM electrons will cause flash melting.

For the purposes of engineering design of the divertor targets, peak heat flux densities on the targets (for both steady state and slow transient) are specified, along with envelope profiles encompassing a wide range of operating conditions. This surface power deposition comprises both thermal plasma and photonic/CX particle loads. The relative fractions of power contained in these various load types is of no consequence for engineering qualification tests, but it does become important when the detailed edge loading studies described in this paper are performed. For the baseline, partially detached burning plasma operating condition, radiative dissipation is an extremely important component of the energy balance and without it, the necessary reduction of thermal plasma fluxes in the target strike point vicinity would not be possible. In fact, in the baseline, some 50–60 MW of the total ~100 MW scrape-off layer power is radiated volumetrically in the divertor plasma. At the strike points, where the radiation is found to concentrate in SOLPS simulations, this can lead to several MW m⁻² of photonic power deposition and is a major fraction of the peak heat flux density (CX loads are non-negligible, but considerably lower). Such high heat flux densities can in fact be an issue for loading of non-HHF components in the ITER divertor. This has been studied in detail using optical ray tracing based on SOLPS simulations of the expected radiation distributions [29].

For the SOLPS cases considered here (section 5, table 3), the radiant heat flux on the target varies considerably, as would be expected for a partially detached versus more attached plasma. Thus, the majority of the target load originates in thermal plasma fluxes for the more attached case and the contributions between radiant and plasma heat fluxes become comparable under partially detached conditions. In section 5, however, we compute for completeness calculations of MB loading for all possible radiative fractions. In this way, a set of reference calculations are available for the ITER burning plasma from the point of view purely of radiative loading.

Slight variations of the irradiation profile due to the toroidal bevel at the top plasma-wetted surface are ignored. The radiant heat flux profiles inside gaps depend only on the geometry of the gap. Therefore, we only need to calculate a normalized profile for each gap geometry and then scale it by the particular radiated power for each case. Five classes of MBs are defined according to their location with respect to gaps (figure 24). The plasma contribution to the heat flux is identical for classes A, C, and E since all those MBs are downstream of inter-PFU gaps. The plasma component to MBs situated downstream of intra-cassette gaps (class B) must be calculated separately due to the wider gap width, and the radial half target step, which result in different magnetic shadowing than the majority. The same is true for MBs downstream of the 20 mm wide inter-cassette gaps (class D), which are protected by target tilting that produces a radial step of about  −4 mm and which may often be completely shadowed from thermal plasma flux. The radiant heat flux to the sides of the gaps depends on the width of the gap, and which side of the gap (upstream or downstream) is considered. MBs that are upstream of an inter-cassette gap (class E) receive about 40 times more radiated power than those at inter-PFU gaps (class A), while those at intra-cassette gaps (class C) receive about 6 times more.

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The irradiation of the sides and bottom of a PG at the location where the radiated power peaks has been shown to be sufficiently well described by a simple analytical model representing the power source as a linear radiating filament [30]. This model was subsequently validated by 3D simulations of the real environment using the LightTools^TM software [29]. For the purposes of the analysis here, we will therefore assume that each divertor surface is heated by a single radiating filament. The expressions given in [30] are applied to PGs, after a trivial extension to account for radial MB misalignments.

The radiant heat flux to the sides of MBs at inter-PFU and inter-cassette gaps are shown in figure 25. In all cases, the MB at the upstream side of the gap is more exposed, and thus receives more power because it is essentially unshadowed by its neighbour over a distance equal to the radial step, minus the radial misalignment. The characteristic scale of the radial heat flux profiles is roughly equal to the gap width, which is why MBs at inter-cassette gaps receive substantial heat flux along their entire height.

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In the case of TGs, the single filament model is a poor choice because it yields results that are extremely sensitive to the precise poloidal position of the filament with respect to the gap. Therefore, we model the source as an infinite ribbon extending in the toroidal (x) direction, with a finite poloidal width of 2Δy  >  d centered above the TG, with the gap width d  =  g_MB. The distance between the ribbon and the target is b. The ribbon's radiant intensity is

$$\begin{eqnarray}&&{{I}_{0}}=\frac{{{P}_{\text{rad}}}}{2\pi R\cdot 2 \Delta y}\end{eqnarray} \tag{ 24 }$$

where P_rad is the radiated power above the divertor (e.g. 30 MW), and R is the major radius of the ribbon (e.g. 4 m). The origin of the coordinate system is at the top of the gap, midway between the two edges (figure 26).

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The heat flux density to the left side of the gap is

$$\begin{eqnarray}&&q=\frac{{{I}_{0}}}{4\pi}{\int}_{-\infty}^{\infty}\text{d}x{\int}_{-d/2}^{{{y}_{1}}}\text{d}y\frac{\overset{\scriptscriptstyle\rightharpoonup}{{r}}\cdot \widehat{n}}{{{r}^{3}}}\end{eqnarray} \tag{ 25 }$$

where $\overset{\scriptscriptstyle\rightharpoonup}{{r}}$ is the vector between a point on the surface and a point on the ribbon and $\widehat{n}$ is the surface normal vector. The upper limit of the poloidal integration depends on whether the extreme edge of the ribbon is shadowed by the opposite edge or not:

$$\begin{eqnarray}&&{{y}_{1}}= \Delta y;~z\geqslant \frac{bd}{\frac{d}{2}- \Delta y}\end{eqnarray} \tag{ 26 }$$

or

$$\begin{eqnarray}&&{{y}_{1}}=\frac{bd+\frac{d}{2}z}{z};~z<\frac{bd}{\frac{d}{2}- \Delta y}.\end{eqnarray} \tag{ 27 }$$

The preceding integration limits typically occur for points near the top of the gap, or deep inside the gap, respectively. The local heat flux density is

$$\begin{eqnarray}&&{{q}_{\text{gap}}}=\frac{{{I}_{0}}}{4\pi}\ln \frac{{{\left({{y}_{1}}(z)-\frac{d}{s}\right)}^{2}}+{{(b-z)}^{2}}}{{{d}^{2}}+{{(b-z)}^{2}}}.\end{eqnarray} \tag{ 28 }$$

The heat flux at the plasma-facing surface of the target directly under the ribbon is

$$\begin{eqnarray}&&{{q}_{\text{top}}}=\frac{{{I}_{0}}}{\pi}{{\tan}^{-1}}\frac{\Delta y}{b}.\end{eqnarray} \tag{ 29 }$$

Taking a 5 cm wide ribbon situated 5 cm from the divertor surface, we obtain the heat flux profile inside the gap, normalized to the heat flux at the divertor surface (figure 27). Within the first mm from the top of the gap the heat flux is nearly constant because all points see the same radiating surface. The sharp break at ~1 mm corresponds to the beginning of shadowing by the opposite gap edge. The maximum value at the top of the gap is only about 10% of that on the divertor surface. We note that at the top of PGs, the heat flux is typically 50% of that on the divertor surface. The reason for the difference is because in TGs the rays strike the surface at highly oblique angles, whereas in PGs, a significant fraction of the power arrives at angles that are closer to perpendicular incidence. We can therefore justify neglecting radiation into TGs for our thermal analysis.

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A reasonable estimate of the thermal response of shaped MBs can be obtained by resolving the 3D heat conduction equation in cuboid geometry using the explicit finite difference method. Shaped MBs are replaced by rectilinear cuboids that are rotated to obtain the same magnetic field incidence angle at the top surface, and displaced with respect to one another to obtain the same magnetic shadowing as the real geometry (figure 28). The circular cooling tube (12 mm inner diameter) is replaced by a jagged shape of roughly the same cross-sectional area whose perimeter is defined by the nodes of the rectangular mesh. For simplicity the MB is composed of pure W with no copper pipe. In reality the thickness of the CuCrZr cooling pipe plus the OHFC copper interlayer foreseen for ITER MBs is 2.5 mm. To obtain a similar temperature gradient as the case of 6 mm W armour thickness above the copper interlayer, we add 1 mm additional thickness of W (the ratio of the thermal conductivities of those two metals is 2.4 at room temperature). Explicitly, in our simulations there are 7 mm of W between the top of the MB and the highest point of the cooling channel, whereas in reality, there should be 8.5 mm of metal: W (6 mm), copper (1 mm), and CuCrZr (1.5 mm).

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The temperature dependencies of W thermal conductivity and heat capacity used by the ITER Organization are adopted, as well as the heat transfer coefficient between the cooling pipe and the cooling water (figure 29). For the latter, the sudden increase just before 300 °C corresponds to the onset of sub-nucleate boiling. On the other hand, the density is always assumed to be equal to the density at room temperature. It was verified for a few representative cases that this simplification does not significantly alter the thermal response compared to the differences that already arise due to the other approximations. The water temperature is 100 °C. The radiation of heat from hot surfaces is included, but only becomes comparable to the incident heat fluxes above 2500 °C (for example radiation removes about 3 MW m⁻² near the melting temperature. The emissivity of W is taken to be 0.3.

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The errors resulting from the simplifications of the finite difference model were quantified by comparing with calculations by the ANSYS^™ software (figure 30), which implements a finite element model of the heat conduction equation. Uniform heat flux was applied to the top surface of an unshaped MB, without any heat flux to the sides, as is assumed when simulating qualification tests in a HHF facility. Identical temperature dependencies of the thermal parameters were also assumed as in the finite difference simulations. The maximum temperature occurs at the short edges of the MB due to the longer path to the cooling channel with respect to the center of the top surface (see figure 31). Both the minimum and maximum temperatures at the top surface agree with the ANSYS^TM simulation results within 5% over the entire range, which for the purposes of scoping studies is acceptable, especially given the experimental uncertainty on the measured thermal properties of W in the high temperature range.

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At the scale of an individual MB, a toroidal bevel leads to the total power that is collected by the MB being concentrated onto a smaller area (see figure 3). For example, consider a normal heat flux of 20 MW m⁻² striking the nominal OVT during a slow transient. Target tilting increases the local magnetic field line incidence angle from 2.7° to 3.2°, and a MB toroidal bevel with h_tor  =  0.5 mm further increases the angle to 4.2°. The normal heat flux at the plasma-wetted surface is then 31 MW m⁻² which is well above the heat load specification for which the technology has been tested. However, as opposed to a full target composed of PFUs separated by vacuum gaps, the heat impinging on a MB can diffuse in the toroidal direction through the bulk W. Is it the total power or the local heat flux density which determines the peak surface temperature? It would be the former if heat conduction resulted in an efficient redistribution of the incident power that smoothed out surface temperature inhomogeneity. Unfortunately, the MB design does not allow this. The path from a point on the surface to the cooling channel is shorter than the toroidal length of the MB. Since the incident heat flows along the shortest path to the cooling channel, there is little interaction between opposite ends of the MB.

In order to assess the effect of shadowing on peak surface temperature, one of the simple cases discussed above (figure 30) was taken as a reference. The heat flux profile was truncated to vary the toroidal wetted fraction (figure 31). In practice this could be achieved by varying the radial alignment of the MB with respect to its upstream neighbour. The temperature at the trailing edge (x  =  28 mm) hardly changes even when half the surface is shadowed.

The toroidal wetted fraction of the top surface of a particular MB depends on where it is situated on the target, and on alignment relative to its neighbours. At inter-PFU gaps, MB toroidal beveling leads to magnetic shadows up to ~10 mm long at the LE. Downstream of a wider intra-cassette gap, the magnetic shadow can be even longer. Therefore, for the majority of cases, the temperature at the plasma-wetted trailing edge depends on the local normal heat flux alone and is not much affected by magnetic shadowing at the LE. The peak temperature does not depend on radial misalignment. The only exception is the leading PFU of each target, downstream of the 20 mm inter-cassette gap, which can be almost entirely shadowed due to target tilting. Indeed, the temperature of the MB with 25% toroidal wetted fraction is significantly cooler than the others.

The ion orbit model predicts peaked heat flux deposition profiles at TG edges during inter-ELM periods (section 3.4.2). A key issue for the ITER divertor is the thermal response to this concentrated heat flux. If the temperature dependence of the material properties can be neglected, the heat conduction equation, ${{\nabla}^{2}}T=0,$ is linear (volumetric neutron heating is not treated here). By the principle of superposition, the thermal response to a heat flux profile having a complicated form is given by the sum of the responses to an arbitrary number of simpler profiles having known solutions that, when summed, gives the temperature profile of interest. Let us consider a 2D domain representing the poloidal cross-section of a MB immediately above the cooling tube. The coordinate system is the one used throughout this paper; y is the direction parallel to the cooling tube axis and z is the direction normal to the top surface of the MB. The domain dimensions are W_MB  =  12 mm in the y-direction, and H_MB  =  6 mm in the z-direction. Mixed boundary conditions are applied to represent the heating of a cooled W block. The bottom surface z  =  0 is held at a fixed temperature T_b. The side surfaces y  =  0 and y  =  W_MB are insulating, ∂T/∂y  =  0. An incident heat flux which can have an arbitrary spatial profile is applied to the top surface

$$\begin{eqnarray}&&\frac{\partial T}{\partial z}=-{{q}_{\text{surf}}}(y)/\kappa \end{eqnarray} \tag{ 30 }$$

where κ is the thermal conductivity.

The ion orbit model predicts that the heat flux is often given by the sum of the optical heat flux spread uniformly over the top surface, plus a peaked heat flux profile near one or both edges due to Larmor radius focusing of charges streaming into the TG. To calculate the additional heating due to a narrow feature, we only need to model the local peaked heat flux profile, subtracting away the uniform component predicted by the optical approximation (whose solution is of course $T(z)={{T}_{\text{b}}}-zq/\kappa$). Let us model the narrow feature as a uniform power load q₀ applied on a thin toroidal strip of poloidal width ΔW at the upper left edge of the domain. The analytic solution for the temperature increase ΔT due to the narrow feature is given by:

$$\begin{eqnarray} &&\begin{array}{*{20}{l}}\Delta T(y,z)=\frac{-{{q}_{0}}}{\kappa}\left[\frac{\Delta W}{{{W}_{\text{MB}}}}z+\underset{n=1}{\overset{\infty}{\sum}}\frac{2{{W}_{\text{MB}}}}{{{\left(n\pi \right)}^{2}}\cosh \frac{n\pi {{H}_{\text{MB}}}}{{{W}_{\text{MB}}}}}\sin \frac{n\pi \Delta W}{{{W}_{\text{MB}}}}\right. \\ \left. \qquad \qquad \ \ \ \times \sinh \frac{n\pi z}{{{W}_{\text{MB}}}}\cos \frac{n\pi y}{{{W}_{\text{MB}}}}\right].\end{array}\end{eqnarray} \tag{ 31 }$$

Figure 32 shows the maximum temperature increase at the MB edge as a function of the profile width, keeping the total power flow constant, q₀  =  5000 W m⁻¹, corresponding to 10 MW m⁻² deposited on a 0.5 mm wide strip. The temperature increases as the strip narrows, but eventually saturates for ΔW  <  0.03 mm to the value resulting from a Dirac delta function heat flux profile in the limit of ΔW  →  0 (the 2D solution in that limit is the Green's function for this problem). The 2D solution for ΔW  =  0.5 mm (figure 33) makes clear how quickly information about the exact heat flux profile is lost when following streamlines from the loaded strip into the bulk; already 1 mm from the corner, the contours of constant temperature are nearly circular. The temperature sufficiently far from the hot toroidal strip does not depend on its microscopic profile, but only on the total deposited power, and the bulk dimensions H_MB and W_MB of the domain. The shape of the temperature profile in the bulk depends on the size of the surface heat flux profile with respect to domain, and not on the thermal properties of the material.

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Under the optical approximation, the heat flux (for example 10 MW m⁻²) entering a 0.5 mm TG is deposited on a thin strip whose width is determined by the inclination of the magnetic flux surfaces with respect to the target surface. What this simple analysis tells us first is that the temperature increase of the MB edge due to this heat flux is of the order of 100 °C. Secondly, the exact temperature depends on how the heat flux is distributed (e.g. due to different θ_ψ at IVT and OVT), but it will not vary by more than a few tens of °C. The ion orbit model predicts heat flux profiles that can be strongly peaked, with characteristic widths on the Larmor radius scale. The different thermal responses for the two heat flux models are not significant, especially when the mean surface temperature already exceeds 1000 or 2000 °C. Furthermore, to resolve the differences with a numerical model (finite difference or finite element) requires grid spacing finer than 0.1 mm; whether the edges of real machined MBs will be perfectly sharp at such scales is debatable. Therefore, for stationary inter-ELM periods, we conclude that modelling the exact shape of the heat flux profile is not necessary.

Power balance, on the other hand, must be respected. All the power entering the gap must be redistributed somehow over the nodes of the simulation grid in order to obtain an accurate estimate of the temperature increase. For the purposes of engineering scoping studies of MB edges with narrow heat flux deposition profiles, it is acceptable to use coarse grid spacing as long as it is at most equal to the width of the heat flux profile.

The above analysis of the linear heat conduction equation provides useful insight into the solutions that will be obtained using the 3D model described in the next section. We will use temperature-dependent heat capacity and heat conductivity, making the equations non-linear, so the superposition principle is no longer valid, and we must use the complete heat flux profile as input. For example, if we were to calculate the response due to top heating, and add to it the response due to gap heating only, the result would be slightly different than the response due to the total heating. Nonetheless, the variations of the thermal properties are quite gentle (less than a factor of 2 between room temperature and melting), so the generic results from this section remain valid.

Depending on their alignment, MBs can have an OHS at the corner which is visible along magnetic field lines through the gap crossing (figure 23). The full 3D heat flux profile calculated under the optical approximation provides the worst case scenario for the OHS because there is no attenuation of the heat flux due to Larmor radius losses in the preceding TG. The temperature increment due to the OHS can be evaluated by comparing with a case for which the corner heat flux is artificially suppressed. To accomplish this we set the surface heat flux on all nodes with x  <  2 mm to zero. The corner temperature increase due to the OHS is 210 °C (figure 34). The temperature remains lower than that at the top surface above the cooling tube. Taking into account the Larmor smoothing predicted by the ion orbit model, the temperature will be even lower.

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MBs experience three main types of heat deposition. The equilibrium temperature of a MB is dominated by the first type, the more-or-less uniform heat flux to the top surface. The problem is quasi-1D in the sense that the heat flux streamlines enter the bulk at near normal incidence at the near-surface region. The order of magnitude of the heat flux is constant along streamlines, and the temperature gradient is essentially determined by the distance between a given point on the surface and the cooling tube. More precisely, the top heat flux is focused purely by geometry at the top of the cooling tube, but increases only a factor ~2.5. This is an important element in the analysis of critical heat flux, discussed in section 5.

In section 4.3 we saw that the thermal response to the second type of profile, a long narrow strip of heat flux, is of secondary importance. That problem is 2D; the heat flux streamlines spread significantly in the plane perpendicular to the cooling tube axis and the toroidal and radial divergence terms contribute to reducing the heat flux magnitude along the streamlines, leading to a small temperature increase in comparison to the full surface heat load.

The third type of heat flux profile is an OHS, which can be represented as a point heat source at the surface. The heat flux streamlines have three dimensions into which they expand, making the temperature increase at the optical spot minimal. This beneficial effect of geometry wins out over the magnitude of the OHS heat flux which is about 20 times higher than that at the top surface, so that the temperature increase due to the OHS is of the same order of magnitude as that due to heating at gap edges. The OHS is thus of no concern for steady state and slow transient heat loads. Unfortunately, we will see in section 6.1.3 that transient heat loads during ELMs result in flash melting at the OHS.

Convex features such as edges and corners exposed to surface irradiation heat up more than flat surfaces. Linear analysis assuming temperature-independent thermal properties illustrates the problem well. Consider the sharp edge of a body delimited by the space x  ⩾  0 and z  ⩽  0. Uniform heat fluxes q_side and q_top are applied normal to the side and top surfaces, respectively. The temperature throughout the body is given by the linear superposition of the solutions of two simpler problems. For the case of heat flux applied only to the side surface, the solution is identical to that of the 1D, semi-infinite problem:

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}{{T}_{\text{side}}}(x,t)\\ \quad =\frac{\sqrt{{{\alpha}_{h}}}{{q}_{\text{side}}}}{\kappa}\left(2\sqrt{\frac{t}{\pi}}\exp \left(-\frac{{{x}^{2}}}{4{{\alpha}_{h}}t}\right)-\frac{x}{\sqrt{{{\alpha}_{h}}}}\text{erfc}\frac{x}{2\sqrt{{{\alpha}_{h}}t}}\right)\end{array}\end{eqnarray} \tag{ 35 }$$

and for the case of heat flux applied only to the top surface, the solution is

$$\begin{eqnarray}&&{{T}_{\text{top}}}(z,t)=\frac{\sqrt{{{\alpha}_{h}}}{{q}_{\text{top}}}}{\kappa}\left(2\sqrt{\frac{t}{\pi}}\exp \left(-\frac{{{z}^{2}}}{4{{\alpha}_{h}}t}\right)+\frac{z}{\sqrt{{{\alpha}_{h}}}}\text{erfc}\frac{-z}{2\sqrt{{{\alpha}_{h}}t}}\right)\end{eqnarray} \tag{ 36 }$$

where α_h is the heat diffusivity. The 2D solution is

$$\begin{eqnarray}&&T(x,z,t)={{T}_{\text{top}}}(z,t)+{{T}_{\text{side}}}(x,t)\end{eqnarray} \tag{ 37 }$$

which at the sharp edge (x  =  0, z  =  0) can be written in terms of heat flux factors

$$\begin{eqnarray}&&{{T}_{\text{edge}}}(t)=~T(0,0,t)=\frac{2}{\kappa}\sqrt{\frac{{{\alpha}_{h}}}{\pi}}\left({{F}_{\text{HF,top}}}+{{F}_{\text{HF,side}}}\right).\end{eqnarray} \tag{ 38 }$$

A heat flux factor can be defined at a sharp edge as the sum of the individual heat flux factors on each facet. A general expression that takes account of this geometrical effect and which applies to a flat surface, to sharp edges formed by the intersection of two planes, but equally to sharp corners formed by the intersection of three planes, is

$$\begin{eqnarray}&&{{F}_{\text{HF}}}=\underset{i=1}{\overset{N}{\sum}}{{F}_{\text{HF},i}}=\underset{i=1}{\overset{N}{\sum}}\frac{{{\varepsilon}_{\text{surf},i}}}{\sqrt{\Delta {{t}_{\text{ELM}}}}}\end{eqnarray} \tag{ 39 }$$

where N  =  1 at the center of a flat surface far from the edges, N  =  2 at a long edge far from a corner, N  =  3 at a corner, and ε_surf,i is the uniform energy fluence on each facet. For the more general case of an arbitrary pulse shape, the temperature increase at the sharp edge or corner is equal to the sum of the increases calculated separately on the individual facets.

Regarding experimental evidence for edge melting, mention was made in [50] of HHF tests [43, 44] in a quasi-stationary plasma accelerator (QSPA) [45] installed at SRC RF TRINITI, Russia. The edges of W macrobrush PFCs that were heated to temperature T  =  500 °C and irradiated by trapezoidal pulses for a duration Δt  =  500 µs at an incidence angle of α  =  30° were observed to just begin melting for incident surface energy fluences above ε_surf  =  0.4 MJ m⁻², while full surface melting began when the energy fluence exceeded ε_surf  =  1.0 MJ m⁻². Combined with results of PIC simulations of ELM heat fluxes to poloidal LEs of unshaped MBs that predicted values a factor 2–3 higher than at the top surface [46], it was decided that the maximum tolerable ε_surf to unshaped MBs should be specified as 0.5 MJ m⁻² in ITER because it provides a factor two margin against full surface W melting [3, 47, 50], while at the same time not driving deep edge melting.

The gaps between macrobrush elements in the QSPA experiments were 0.5 mm wide, which under the optical approximation leads to the exposure of the edge to a depth of 0.29 mm with a heat load which is a factor 1/tanα  =  1.73 larger than on the top surface. The heat flux factor at the edge is thus (1  +  1.73)  =  2.73 times larger than on top, and should melt for heat loads 0.37 times lower than the observed threshold of ε_surf  =  1.0 MJ m⁻². The simple estimate is in good agreement with experiment and 2D thermal calculations (figure 41). Sophisticated modelling of the melt layer dynamics and droplet formation using the MEMOS code [48] successfully reproduced many of the experimental observations. A helpful rule of thumb can be proposed: If a certain heat pulse provides a factor two margin against full surface melting of a given PFC, any sharp edges that receive at least the same load will melt.

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The heat flux profiles near MB edges are not spatially uniform, but can have characteristic scale lengths of a few tenths of a millimeter. An approximate procedure has been developed to calculate the temperature increase at an edge using arbitrary heat flux profiles. The local heat flux is averaged over one heat diffusion length along the surface around the point of interest. The heat diffusion length is defined as the characteristic depth of the heated layer

$$\begin{eqnarray} &&\Delta s=\sqrt{2{{\alpha}_{h}}~{{\tau}_{\text{IR}}}}.\end{eqnarray} \tag{ 40 }$$

Taking high temperature values of the thermal properties of W, c_p  =  180 J/kg/K and κ  =  100 W/m/K, the depth of the heated region is about Δs ~ 160 µm for the triangular pulse. A specified ELM heat flux q_tg to a nominal target would produce a certain peak temperature (which can be read from figure 40 or estimated using equation (34)), referred to as ΔT_1D. The local peak temperature at a MB edge can be expressed in terms of ΔT_1D

$$\begin{eqnarray}&&\frac{\Delta T}{\Delta {{T}_{1\text{D}}}}={{f}_{\text{geo}}}\frac{{\int}_{{{s}_{\text{edge}}}- \Delta s}^{{{s}_{\text{edge}}}+ \Delta s}{{q}_{\text{surf}}}\text{d}s}{2~ \Delta s~{{q}_{\text{tg}}}}\end{eqnarray} \tag{ 41 }$$

where the heat flux profile is integrated along a surface coordinate s running over the edge centered at s_edge, and f_geo is a numerical factor related to the edge geometry. At a sharp edge with temperature-independent material properties, the factor is exactly f_geo  =  2, and at a corner it is f_geo  =  3. For non-linear material properties, the factor at sharp edges is f_geo  =  2.1. The non-linearity is weak because the increase of heat capacity with temperature nearly compensates the decrease of heat conductivity, making the dependence of thermal effusivity on temperature weak. For other shapes not considered here such as chamfered or filleted edges, f_geo can also be calculated [49]. This simple procedure avoids the need to make a dynamic finite element calculation for each case.

Comparison with a full 2D heat conduction calculation using heat flux profiles given by the ion orbit model as input shows that the accuracy of the predicted edge temperature increase is of the order of 1%, which is more than acceptable for scoping studies such as this. The heat flux calculated by the ion orbit model at the lower toroidal edge of an IVT MB for the 15 MA/5.3 T burning plasma scenario (see figure 21) results in the temperature increase shown in figure 42. The ratio of the peak temperature increase at the sharp edge to the prediction of the 1D heat equation (for ε_tg  =  0.15 MJ m⁻² applied during a triangular pulse) is ΔT/ΔT_1D  =  3.60. The prediction of the simple procedure (equation (41)) is ΔT/ΔT_1D  =  3.56. The peak temperature increase is 1174 °C. If ε_tg  =  0.30 MJ m⁻² were absorbed, the peak temperature would be only 100 °C less than the W melting temperature. Note also that the temperature increase on the MB top surface far from the edge, for s_tor  <  11.6 mm, is 1.5 times higher than ΔT_1D due to the 1.5° inclination of the surface with respect to nominal.

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6.1.1. ELM-induced melting at poloidal gap edges.

The ELM temperature spike normalized to the nominal 1D value (ΔT/ΔT_1D) at IVT and OVT poloidal LEs on toroidally bevelled MBs is shown in figure 43 for 5 keV ions at full field operation (15 MA/5.3 T burning plasma scenario). The ELM heat pulse assumed here is triangular, but it has been verified that these results can be applied with ~5% accuracy to any pulse shape because the only dependent parameter is the heat diffusion length which varies as the square root of the pulse's rise time. These calculations do not account for the OHS which will be treated in section 6.1.3. The MB edge peaking factor is similar at both VTs. It is higher at the intra-cassette gaps than the narrower inter-PFU gaps. For worst case misalignment at the intra-cassette gap the temperature peaks at 4.1 times ΔT_1D. We assume as an example the usual value for the maximum allowed ELM surface energy fluence ε_tg  =  0.5 MJ m⁻², and that the OVT is operating in the baseline partially detached regime (q_tg  =  10 MW m⁻², figure 38) with a mean LE edge temperature of T_init  =  1100 °C. The nominal 1D temperature spike is around ΔT_1D  =  1075 °C (figure 40). The LE would rise to (T_init  +  4.1 ΔT_1D), about 1.6 times the W melting temperature. Obviously this is not what will happen, because we do not model the liquid phase of W, but it is clear that each ELM that delivers ε_tg  =  0.5 MJ m⁻² will melt the poloidal LEs. It should be noted that in the case of a misaligned MB with no shaping, the temperature excursion at the exposed LE has been calculated to be up to 5 times the melting temperature. Therefore, even if toroidal bevelling does not fully avoid LE melting, it does mitigate it appreciably.

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Empirical scaling laws based on observations in current tokamaks predict the energy released from the pedestal per uncontrolled ELM in the 15 MA burning plasma scenario will be ΔW_ELM ~ 20 MJ at a frequency of 1–2 Hz, evacuating 20–40% of P_SOL [2]. The remaining 60–80% is deposited during inter-ELM periods. The original specifications [50] for ELM energy flux deposition to the targets are now recognized as too pessimistic. An upper limit of the energy fluence to the targets had been obtained by assuming the ELM energy deposition footprint would not be broader than that during the inter-ELM phases [51] striking a toroidally symmetric ring of total surface area A_div ~ 1 m² at the targets (~2πRλ_q//OMP corrected for flux expansion factor ~7, with λ_q//OMP ~5 mm the ELM power decay length at the outer midplane, assumed equal to the inter-ELM heat flux width). The assumption that up to 1/2 of the ELM energy ΔW_ELM  =  20 MJ can be deposited at the OVT on a surface of 1.4 m², and up to 2/3 at the IVT on a surface of 0.9 m², leads to ε_tg  ⩽  7 MJ m⁻² at the OVT and ε_tg  ⩽  15 MJ m⁻² at the IVT. The latter number leads to the specification for ELM mitigation techniques, namely that the ELM size must be reduced at least a factor 30 to ensure ΔW_ELM  <  0.7 MJ in order not to exceed ε_tg  =  0.5 MJ m⁻². Inherent in this specification is the essentially unjustified assumption of no ELM broadening with respect to the inter-ELM heat flux decay length.

Experimental indications of ELM broadening were measured by Langmuir probes in JET [52] and confirmed by IR thermography following advances in the performance of the diagnostic, notably due to improved spatial resolution. Infra-red measurements in DIII-D [53] and JET [54, 55] extended the database considerably. The ELM wetted area was observed to increase up to a factor six above the inter-ELM wetted area for the largest ELMs. If similar broadening occurs in ITER, H-mode operation could be possible up to I_p  =  9.5 MA without the need for ELM mitigation [47], although it should be specified that this evaluation was made for the case of a nominal, axisymmetric VT with no component tilting, gaps, or MB shaping.

A multi-machine scaling supported by a simple model has recently been proposed for the peak parallel ELM energy fluence at the outer targets of tokamaks running H-mode with uncontrolled Type I ELMs [4]. Rather than scaling with relative ELM size as before, correlations have been found between the measured ELM energy fluence and the pedestal plasma properties. The essence of those findings is that the peak ELM energy fluence is proportional to the pedestal pressure and the tokamak major radius (or equivalently, the thermal energy content of a magnetic flux tube that connects the pedestal to the divertor target). It also contains a weaker scaling (E_ELM/W_plasma)^0.5 with the normalized ELM size. The poloidal width of the ELM footprint is found to increase with ELM size, which amounts to saying that a radially wider region of the pedestal that magnetically connects to the wall is what makes for larger ELM losses. For the half current, half field scenario (I_p  =  7.5 MA, B  =  2.65 T) representative of the pre-nuclear experimental campaigns, the prediction is 0.125  <  ε_tg  <  0.375 MJ m⁻², while for the burning plasma scenario (I_p  =  15 MA, B  =  5.3 T), it is 0.5  <  ε_tg  <  1.5 MJ m⁻² based on estimates for the pedestal density and temperature with a relative ELM energy loss of ΔW_ELM/W_th  =  5.4%, and assuming an incidence angle of α  =  3° at the nominal OVT rather than α  =  2.7° as assumed here.

If the new scaling is correct, poloidal LE melting would definitely occur at every uncontrolled ELM in the burning plasma scenario. For the pre-nuclear half-field scenario in D or He with A/Z  =  2, an ELM rise time of Δt_ELM  =  320 µs, and T_i  =  2.5 keV, the nominal surface temperature increase (equation (34)) is in the range 200  <  ΔT_1D  <  700°. Multiplying the highest expected temperature increase by the calculated edge enhancement factors (not shown here, but which are nearly identical to those in figure 43), the temperature increase at the magnetically shadowed OVT poloidal LE has been calculated (figure 44). SOLPS simulations of 7.5 MA/2.65 T pre-nuclear He plasmas [56] predict that the peak heat flux to the nominal divertor can be as high as q_tg  =  5 MW m⁻², for which the stationary surface temperature of the magnetically shadowed LE is ~600 °C (figure 38). For the worst case misalignment the ELM causes the temperature to jump about ΔT  =  2800 °C at the LE, enough to start melting.

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6.1.2. ELM-induced melting at toroidal gap edges.

In contrast to PGs which receive plasma only on the side that faces upstream towards the SOL, TGs can be irradiated on the magnetically shadowed side if the ion Larmor gyration has the correct helicity (see section 3.4.2). At the IVT, the direction of rotation about the ion guiding center is such that ions strike the lower, magnetically shadowed MB edges, whereas the opposite occurs at the OVT. At both VTs it is assumed that the electron component of the heat flux strikes the upper edge as predicted by the optical approximation. The MB edge peaking factors at the upper (lower) edges of IVT and OVT MBs are compiled in figure 45 (figure 46), respectively, for the 15 MA/5.3 T burning plasma scenario with T_i  =  5 keV.

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The upper IVT MB edges heat up typically twice as much as a nominal axisymmetric target, even though the ion gyromotion favours the lower edges, because there is still electron flux inside the gap which adds to the flux to the top surface (thin curve on upper panel of figure 21). At the upper OVT MB edges, in addition to the electrons, the ion gyromotion brings intensely focussed ion heat flux into the gap, resulting in peaking factors up to ΔT/ΔT_1D ~ 6. At the magnetically shadowed OVT lower toroidal edges (figure 46) there is no electron flux. At the IVT, the heating by the full plasma flux at the top surface is supplemented by the ion gyromotion into the gap leading to peaking factors in the range ΔT/ΔT_1D ~ 3.5  ±  1. At the OVT lower edges, there is no contribution from inside the gap, only from the top surface.

Even though the allowed radial misalignment between neighbouring MBs on a given PFU is specified as m_rad  =  ±0.3 mm (table 1), it is expected that the real radial step will not exceed  ±0.1 mm [13]. Depending on whether we adhere strictly to the specified radial tolerances or take a value that we believe to be more realistic, ELM energy fluences would need to be limited to avoid melting the MB toroidal edges. The long toroidal edges are thus just as critical as the poloidal edges, if not more so, because they are also subject to the inter-ELM heat loads.

6.1.3. Transient ELM heating at the optical hot spot.

During ELM events, strongly magnetized electrons shine directly through gap crossings, carrying 2/7 (or about 30%) of the raw parallel heat flux to the triangular OHS on poloidal LEs (section 3.4.3) in the ambipolar case, but could carry much more than that if there were net electron current to the target, as has recently been deduced in the COMPASS tokamak [26]. The ions with their large Larmor radii twist through the gap crossing such that their impact points are poloidally shifted with respect to the electron flux. The ion heat flux is diluted due to Larmor radius losses inside the TG, and spreading of their trajectories as they spray out of the gap crossing. The peak heat flux for 5 keV ions at the IVT is reduced to 11% of the pure parallel flux. Therefore, it is the electrons, carrying three times more power density than the ions, which pose the greatest threat to LEs. The ELM energy fluence due to electrons only is ε_OHS  =  2/7(ε_tg/tanα) ~ 6ε_tg, to which must be added the smaller contribution from ions (which is already enough to melt LEs as seen in section 6.1.1).

The ion and electron heat flux profiles have been applied separately to study their influence on OHS heating. The initial MB temperature is 100 °C. Melting starts at 79 µs due to ions alone, and at 19 µs due to electrons alone (figure 47). The calculations were performed with a poloidal shift of 6 mm between adjacent PFUs to ensure that the OHS lies in the middle of the LE for simplicity. Most often, the OHS will occur at the proximity of a MB corner. There is thus a risk of repeated flash melting at the corners of many MBs due to ELMs. Although the ion orbit model gives fairly good agreement with self-consistent 2D PIC calculations at PGs and TGs far from gap crossings, a 3D PIC solution of the OHS is essential to verify the results found in the present analysis. This has already been performed [57] for a perfectly aligned gap crossing with magnetic flux surfaces oriented perpendicularly to the target surface (i.e. θ_//  =  0° and no OHS), so it is technically feasible.

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To summarize the calculations of the three previous sections, the peaking factors at all points of interest on IVT and OVT MBs downstream of an intra-cassette gap are plotted as a function of I_p in figure 48. Worst case gap widths and radial steps across PGs, and zero radial step across TGs are assumed. Both B and T_i, respectively 5.3 T and 5 keV in the burning plasma scenario, were taken to vary linearly with I_p. The pedestal temperature and density are expected to be proportional to I_p [50] consistent with ELM losses scaling as ΔW_ELM ~ $I_{\text{p}}^{2}$. Constant B/I_p means constant magnetic field angles at the target (table 2). Appropriate ion species were selected for each scenario (see table 4 below).

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Since the heat flux to edges is principally governed by the ion orbit pitch angle in velocity space v_///v_⊥, which is independent of ion species and magnetic field (section 3.1), an isotope effect does not play a first-order role. The temperature peaking factors are essentially invariant with scenario. At both VTs the upper MB corner takes a hard hit from ions penetrating into the magnetic shadow to strike the top surface and the poloidal LE, combined with electrons to the OHS. The IVT lower corner also heats up considerably despite the absence of an OHS because of a flux of ions gyrating upward onto all three facets that form the corner. Most of the heat flux profiles shown in this paper (as a function of s_pol or s_tor) were taken far from gap crossings where the dimension running parallel to the gap is ignorable and the model dimensions can be reduced from 3D to 2D to save computation time. At the gap crossings, the ion orbits are much less constrained and can access the corner with greater facility, leading to higher local fluxes than further down the gap. At the OVT, the lower corner peaking factor is identical to the poloidal LE because it is at the downstream end of the PG (the ion gyromotion is directed downwards).

In table 4 the data needed to evaluate the ELM energy fluence ε_tg which causes melting at the OVT are compiled for the three operation scenarios. The predictions of the new scaling [4] are indicated for reference. The IVT is not treated here since there is yet no scaling available using the new approach. When two numbers appear on a line in the table, the first one refers to the MB top surface at the thinnest part of the W armour above the cooling tube, while the second one refers to the upper toroidal edge at the downstream extremity of the MB. Three energy fluences are calculated. The first is a limit that seeks to provide a factor two margin against full surface melting (thus following the original philosophy of the ITER ELM Heat Load Specification [50]) which implies that edge melting is not considered to be detrimental to tokamak operation. The remaining two values apply again to the top surface or to the upper toroidal edge, but correspond to the energy fluence needed to reach the W melting temperature. The required value in each case is given by

$$\begin{eqnarray}&&\varepsilon =\frac{\left({{T}_{\text{limit}}}-{{T}_{\text{init}}}\right)}{\frac{\Delta T}{\Delta {{T}_{1\text{D}}}}\frac{\Delta {{T}_{1\text{D}}}}{{{\varepsilon}_{\text{tg}}}}}\end{eqnarray} \tag{ 42 }$$

with T_limit  =  T_melt/2 when seeking a factor two margin against melting, or T_limit  =  T_melt when the surface temperature rises to the melting temperature. The temperature peaking factor ΔT/ΔT_1D and the surface temperature increase per ELM ΔT_1D/ε_tg come from the ion orbit calculations (figure 48) and the solution of the 1D heat conduction equation (equation (34)), respectively.

The relative peaking of the heat flux to edges, as well as the ELM rise time, are practically independent of the scenario because the ratio ZT_i/A is roughly constant. In the H and D/He scenarios the estimated energy fluences are nearly the same. The margin against full surface melting is reduced with respect to 0.5 MJ m⁻² as shown in table 4 because of the increased tilt of the MB top surfaces, and the limit to avoid edge melting is reduced a further four times lower due to the focusing of power onto the edges. It is important to note that a factor two margin against edge melting was not sought in the original ELM specifications; the numbers are calculated here as an exercise only in order to demonstrate the high probability that it could occur. It is evident that there is no point in making the calculation for corners because the predicted heat fluxes are so high that the only solution to avoid melting them is to suppress ELMs completely.

In the burning DT scenario, the limits are even lower than the two pre-nuclear scenarios because the surface temperature due to inter-ELM heat loads is higher, leaving less margin against melting. In fact, under the assumption of q_tg  =  10 MW m⁻², the toroidal edges already exceed half the melting temperature in steady state, meaning that it is impossible to obtain a factor two margin. Even on the flat top surface, the energy fluence must be limited to 60 kJ m⁻² per ELM, which is comparable to ELM deposition that has been observed in existing devices. Earlier, based on arguments related only to geometry, it was stated that the limit of ε_tg  =  0.5 MJ m⁻² decreases to ε_tg  =  0.3 MJ m⁻², but that conclusion did not account for the initial temperature of the MBs. These limits are compared with the new scaling in figure 49 and summarized in table 5 below.

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