Discriminating the role of rotation and its gradient in determining ion stiffness mitigation in JET

At JET it is possible to vary the B_T ripple amplitude δ, defined as $\delta =(B_{T_{\max}} -B_{T_{\min}} )/(B_{T_{\max}} +B_{T_{\min}} )$. For standard JET operations, which are carried out with a set of 32 B_T coils carrying equal current, δ = 0.08%. Because odd and even coils are powered independently, the imbalance current between the two coils sets can be changed increasing the B_T ripple up to δ = 3%. These values refer to the maximum B_T ripple values in the plasma, taken at the outer mid-plane at R = 3.8 m and z = 0. Energetic particles can be toroidally trapped in the magnetic ripple and then drifted out of the plasma because of the field curvature. In addition, the trajectory of banana-orbits trapped particles can be altered by B_T ripple, which causes them to drift radially outwards. In the presence of a large B_T ripple other ion losses, possibly those of thermal ions, may be involved [18]. The friction between circulating particles and those locally trapped in B_T ripple tends to reduce the toroidal rotation [19]. However, the dominant effect on the toroidal rotation is given by the outward lost ion flow, which induces a radial return current j in order to preserve neutrality. The resulting j × B torque is always in the counter-current direction. It leads to a significant reduction of both the edge and core rotation, but affects less the core spatial gradient [10, 11, 20, 21].

Also the external perturbation fields applied through the EFCC system influence the JET plasma rotation and rotation shear. The EFCC set consists of four square shaped coils mounted outside the vacuum vessel. Depending on the wiring of the EFCCs either n = 1 or n = 2 fields can be created. Low n perturbation fields break the toroidal symmetry. The plasma flows then along distorted flux surfaces and it is subjected to a drag toroidal force, due to the NTV, that influences the plasma rotation and its gradient [13, 22]. A reduction in toroidal rotation at different radii by the same factor (gradv_φ/v_φ = constant) is observed over the whole plasma core and stronger rotation braking is found near the edge pedestal [12, 23, 24].

The presence of a 3D distortion of the magnetic field may yield to the non-axisymmetry of ion temperature and rotation profiles, thus affecting the determination of ion thermal transport within a 1D approach [25]. However, studies of these effects are out of the scope of this investigation, which concerns core transport, i.e. in a region where the 3D field perturbations are very small, particularly in the case of enhanced B_T ripple [17, 25], which turned out to be the most useful for these studies. In other words we can correctly study in a core region without 3D field perturbations the effects of rotation profiles changes driven by edge 3D perturbations.

The rotation braking experiments are performed in H-mode plasmas at B_T = 2 T for shots with varying B_T ripple and at B_T = 1.85 T for shots with EFCC, with I_p = 1.5 MA, low triangularity, q₉₅ = ∼5, n_e0 ∼ 3 × 10¹⁹ m⁻³, 0.9 < T_e/T_i < 1.1. Ion cyclotron resonance heating (ICRH) and NBI are used as heating systems. Keeping the total injected power fixed, the ICRH and NBI powers are varied to obtain a sufficiently wide ion heat flux scan and to observe different rotation profiles. In order to perform two basic steps of core ion heat flux scan, ICRH (0–2 MW) is applied varying deposition between on- and off-axis at ρ_tor ∼ 0.7. It is used in a (H)-D scheme (33 MHz on-axis, 42 MHz off-axis) with n_H/n_e ∼ 8% and 20–30% of the ICRH core power delivered to thermal ions. ICRH power deposition profiles are obtained by the PION code [14] calculations, for NBI power deposition a first analysis is carried out using the code PENCIL [15]. This code is computationally faster; however, it omits the ion orbit physics and does not take into account the ripple effect. Then three representative discharges (one with enhanced B_T ripple, one with standard ripple and one with the application of EFCCs fields) are reprocessed with ASCOT [16], an orbit-following Monte Carlo code that includes in the calculations fast ion losses due to the enhanced B_T ripple [17]. The ripple effect is found negligible for ρ_tor < 0.4, allowing to calculate the NBI heat flux in the core—which is what is required for the stiffness study—without any ripple correction. However, the NBI ion heat flux obtained by ASCOT is found 20% higher than that provided by the PENCIL code, with the same multiplication factor between the two code calculations for all the discharges. Since ASCOT features a more refined treatment of NBI deposition, the PENCIL NBI ion heat fluxes obtained for all shots are then corrected by this multiplication factor, in order to minimize the systematic error. Different NBI powers from 4 to 7 MW are injected in the plasma to produce a large spectrum of high rotation profiles at standard ripple. For each case, enhanced B_T ripple or EFCCs are then applied.

In figure 1 the rotation as a function of the rotation gradient is shown for the investigated shots. Triangles refer to standard B_T ripple (0.08%) discharges, squares and diamonds to enhanced B_T ripple discharges (respectively 1% and 1.5%) and EFCCs shots are indicated by circles. The values of rotation, measured by means of charge exchange recombination spectroscopy (CXRS), are averaged in time over a stationary interval and taken at the radial position ρ_tor = 0.25: it encloses the on-axis power but not the off-axis one, giving the maximum ion heat flux scan, useful in order to investigate the stiffness level of the analysed shots. The gradient of rotation is calculated from the channels 2–5 of the CX diagnostic, with centre at ρ_tor = 0.25.

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Discharges without methods of rotation braking present higher values of rotation and rotation gradient with respect to discharges in which enhanced B_T ripple or EFCCs are applied. The enhanced B_T ripple shots reach higher values of rotation gradient relative to the rotation value if compared with the trend proper for the standard B_T discharges. Shots where EFCCs are switched on are in line with the standard discharge trend. Only enhanced B_T ripple seems to be able to decouple rotation and rotation gradient for discharges characterized by the above-reported physical values and with the above-described heating powers. It is then possible to investigate the role that rotation and rotation gradient have separately on the ion stiffness.

To this aim, a first analysis is carried out by comparing T_i profiles of plasmas characterized by similar gradients of rotation and different rotation values, as in the example shown in figure 2. It refers to one of the shots with enhanced B_T ripple characterized by reduced rotation with respect to rotation gradient and a standard B_T ripple shot with similar power level and ion heat deposition and plasma parameters except for the ripple level, with a similar rotation gradient but larger rotation value all along the plasma radius. As shown in figure 2(b), the T_i profiles for the two shots and their gradients are identical, even if the values of rotation differ significantly. This is verified not only around the radial position ρ_tor = 0.25 but along all the temperature profiles (the perfect match of the two profiles has to be considered a coincidence).

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The method of enhancing the B_T ripple or applying the EFCCs fields does not allow us to obtain discharges with a reduced rotation gradient with respect to the rotation value. Therefore, limiting the analysis to shots of this braking experiment, a comparison between plasmas with similar values of rotation and different rotation gradients would not be feasible. However, using as auxiliary heating counter-NBI in reverse B_T and I_p configuration is known to produce plasmas with flatter toroidal rotation with respect to the corresponding co-NBI plasmas because of off-axis torque deposition [7, 26]. A discharge that belongs to such a set of experiments (the pulse number 59630 at 14 s), characterized by similar parameters and standard B_T ripple and heated by counter-NBI with a power level nearly equal to the shots of the rotation braking experiments, was then taken into account for the comparison. In figure 3 it is compared with an enhanced ripple discharge, which has similar values of rotation and higher rotation gradient around the radial position ρ_tor = 0.25 (figure 3(a)). The associated T_i profiles shown in figure 3(b) are then very different, indicating that the similar absolute value of Ω does not lead to similar core T_i peaking if ∇Ω is very different. As we can see in figure 3(c), although the power deposition profiles are different for the two discharges (in the case of counter-NBI the distribution of power to ions and electrons is less peaked in the central part of the plasma), inside the considered radius ρ_tor = 0.25 the counter-NBI shot does not have significantly less ion power than the co-NBI shot (3.8 MW against 4.1, see also [7], where an exhaustive analysis about the comparison between co- and counter-NBI plasmas is reported). In contrast, in terms of normalized ion heat flux, one can see in figure 4 that the counter-NBI shot is the one with the highest normalized ion heat flux.

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We conclude that from the simple comparison of experimental profiles, the T_i profile peaking seems related more to ∇Ω than to Ω.

One important limitation of simple profile analysis is that it cannot determine if larger ∇T_i are due to higher ion threshold values or to lower ion stiffness values. In order to clarify this point a second type of analysis is carried out, taking into account the value of the ion heat flux in all the discharges. Although this point was already clarified in [7], a further confirmation was searched in this set of discharges, where in addition we try to separate the different roles of ∇Ω and Ω.

The ion heat fluxes q_i at the radial position ρ_tor = 0.25 are plotted in figure 4 as functions of the logarithmic gradient of the ion temperature $R/L_{T_{\rm i}}$ calculated at the same radius. As in [7], q_i is calculated by spatial integration of ion power density profiles and by determining the collisional electron–ion transfer by means of interpretative transport simulations. The heat flux is expressed in gyro-Bohm units. Gyro-Bohm scaling is predicted to characterize plasmas dominated by anomalous transport [2]. In the following formula the gyro-Bohm normalization is made explicit

$$\begin{equation} q_{\rm i}^{{\rm GB}} =\frac{q_{\rm i}}{(\rho _{\rm i} /R)^2v_{{\rm ith}} n_{\rm i} T_{\rm i}}, \end{equation} \tag{ 1 }$$

where ρ_i is the ion Larmor radius, R is the major radius of the JET tokamak, $v_{{\rm ith}} =\sqrt {(T_{\rm i} /m_{\rm i} )}$, n_i is the ion density and T_i is the ion temperature. $R/L_{T_{\rm i}}$ is obtained by an exponential best-fit over 4 CX channels, with centre in ρ_tor = 0.25 and averaged in time over a stationary interval. The ITG is calculated with respect to the flux surface minor radius, ρ = (R_out − R_in)/2, where R_out (R_in) is the outer (inner) boundary of the flux surface on the magnetic axis plane. In figure 4 circles and squares represent pulses characterized by coupled rotation and rotation gradient values: circles are low rotation shots with ∇Ω < 24 krad ms⁻¹ and Ω < 25 krad s⁻¹; squares are high rotation shots with ∇Ω > 24 krad ms⁻¹ and Ω > 25 krad s⁻¹. The two sets of points form two distinct curves: the one formed by circular points is characterized by high stiffness level, and the one formed by square points is characterized by low stiffness. We can see that the behaviour shown by the experimental results described in [7] is confirmed. Points with decoupled rotation gradient and rotation are shown with diamonds. They are characterized by ∇Ω > 24 krad m s⁻¹ and Ω < 25 krad s⁻¹. These points sit well in the low stiffness curve, although they are characterized by low rotation values. Then we can conclude that it is the rotation gradient that matters in reducing the ion stiffness level, and not the absolute value of rotation. With respect to the results represented in [7] in this experiment the excursion of $q_{\rm i}^{{\rm GB}}$ and $R/L_{T_{\rm i}}$ is limited by the impossibility of using ³He ion heating due to the absence of suitable ICRH frequencies at the low B_T values needed for enhanced ripple. Therefore the highest values of $R/L_{T_{\rm i}}$ are not achievable in the enhanced ripple discharges. Also very low q_i values, which would be relevant for the determination of the ion threshold, are not feasible because they require off-axis ICRH, which is too risky under conditions of high B_T ripple due to losses of highly energetic ICRH ions in the plasma periphery.

The analysis refers to L-mode plasmas, characterized by B_T = 3.3 T, I_P = 1.8 MA, n_e0 = 3.5 × 10¹⁹ m⁻³. The heating is provided by RF power, with a frequency of 33 MHz and a power of 7 MW. The power deposition is varied from dominant electron in 3% (H)-D minority to dominant ion in 4–8% (³He)-D to dominant electron in 20% (³He)-D where mode conversion takes place. Other discharges characterized by the parameters reported above but NBI heated are included in our analysis, in order to have high negative rotation gradient plasmas to compare with the positive ones obtained from hollow rotation profiles. The power deposition and then the ion heat flux are derived using the codes PION and ASCOT. Physical quantities are measured and obtained as described in the previous section.

In figure 5, rotation profiles and corresponding T_i profiles of one pulse at two different times (and different ³He concentration) are shown. The rotation profile is hollow for ρ_tor < 0.3 at t = 7.5 s, with rotation values still in the co-I_P direction. The rotation becomes flat at t = 11.2 s. As shown in figure 5(b), we find different T_i profiles in the plasma region within ρ_tor = 0.3. In this zone higher values of the T_i gradient are observed for the case of hollow rotation where the rotation gradient is rather high and positive, even if the rotation value is lower with respect to the case of flat rotation.

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In order to compare properly the different shots, they are plotted again in the $q_{\rm i}^{{\rm gB}}$ versus $R/L_{T_{\rm i}}$ plot in figure 6. Here circles, characterized by low Ω and ∇Ω, belong to the high stiffness curve, triangles have medium Ω and ∇Ω and sit in the medium stiffness curve and squares, which are characterized by high Ω and ∇Ω, form the low stiffness curve, consistently with the graph of figure 4. The points obtained from the profiles shown in figure 5 are encircled. The one with flat rotation has a low value of Ω and ∇Ω, so it is shown with a circle. It is found to sit among the other circles and clearly belongs to high stiffness family. The point calculated at the time in which the rotation is hollow is represented by a diamond and is characterized by medium positive ∇Ω and low Ω. The normalized ion heat flux in the hollow rotation cases is comparable to that of other flat cases, whilst the ratio T_e/T_i is a bit higher in the case of hollow rotation, which should even decrease $R/L_{T_{\rm i}}$ in this case due to a threshold decrease [5]. Instead, a systematic high value of $R/L_{T_{\rm i}}$ is seen in hollow cases, which we ascribe to reduced stiffness due to the (positive) rotation gradient. We can see that the points with medium rotation gradient belong to the medium stiffness curve independently of the sign of the gradient. It is then the absolute value of the rotation gradient that influences the ion stiffness level, no matter what sign it has.

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