Dependence of neoclassical toroidal viscosity on the poloidal spectrum of applied nonaxisymmetric fields

The coupling of the NTV torque to an applied nonaxisymmetric spectra was measured in a quiescent high confinement scenario (QH-mode) in the DIII-D tokamak. The QH-mode is characterized by the high confinement of H-mode, an absence of edge localized modes (ELMs), and the presence of an edge harmonic oscillation (EHO) [23, 24]. The QH-mode is dependent on rotation shear in the edge but independent of its sign, allowing for either co-current or counter-current rotation [6, 25, 26]. Counter-current rotation was chosen for this work, taking advantage of the observation that nonaxisymmetric fields drive QH-mode plasmas in DIII-D toward a counter-current neoclassical offset rotation [6, 26]. This distinguishes the NTV momentum source from competing terms in the momentum balance such as resonant braking, which is a momentum sink independent of the sign of the rotation. Note that fast ion losses can produce some counter-current torque from the thermal ion return current. This is offset by loss of injected counter-current torque however, typically leading to a reduced effect of prompt ion losses on the angular momentum during counter NBI [27].

Perturbative nonaxisymmetric fields were applied to these plasmas using the three nonaxisymmetric coil arrays of six discrete coils each shown in figure 1. For the purposes of this work, the C-coil array external to the vessel was used for correction of the intrinsic $n\leqslant 2$ error field. The upper I-coil (IU) and lower I-coil (IL) arrays internal to the vessel were used to apply the maximum additional n  =  2 perturbation, corresponding to a peak current amplitude of 4.3 kA. The poloidal spectrum of the applied nonaxisymmetric field was varied by changing the relative toroidal phase of currents in the two I-coil arrays between discharges.

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Figure 2 shows the variation in n  =  2 poloidal spectra applied to the plasma surface obtained by adjusting the phasing between the IL and IU arrays. Consistent with the experiment, these spectra were obtained with the IU phase ${{\phi}_{\text{IU}}}={{90}^{{}^\circ}}$ held constant. Changes in the IL phase ${{\phi}_{\text{IL}}}$ thus produce changes in the upper–lower phasing, $\Delta {{\phi}_{\text{UL}}}={{\phi}_{\text{IU}}}-{{\phi}_{\text{IL}}}$. There is a sharp peak at m  =  −1 to 1 in all cases. The secondary peak in the positive m spectra (right-handed pitch, like the field lines) shifts from m  =  8 to 16. Also shown, is the IPEC calculated 'dominant mode' spectrum, which is discussed in detail in section 3.2 and peaks at m  =  15 in the QH-mode equilibrium.

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In establishing the initial equilibrium, neutral beam injected (NBI) torque was used to produce strong rotation in the counter-current direction of the ion neoclassical offset. The amount of NBI was then reduced such that the angular momentum ${{L}_{\phi}}$ evolved on a time scale that was slow compared to the momentum confinement time ${{\tau}_{\phi}}$ and the plasma remained in torque balance. The integral angular momentum measured in the reference plasma with no applied I-coil field follows a simplified angular momentum evolution,

$$\begin{eqnarray}&&\frac{\text{d}{{L}_{\phi}}}{\text{d}t}\approx 0=T_{\text{NBI}}^{\text{ref}}+{{T}_{\text{INT}}}-{{L}_{\phi}}/{{\tau}_{\phi}}.\end{eqnarray} \tag{ 1 }$$

The injected torque $T_{\text{NBI}}^{\text{ref}}$ balances the intrinsic ${{T}_{\text{INT}}}$ [28–30] and viscous torque approximated by $-{{L}_{\phi}}/{{\tau}_{\phi}}$. In the presence of nonaxisymmetric fields, additional terms corresponding to the NTV torque ${{T}_{\text{NTV}}}$, additional prompt ion loss, and resonant braking may appear in the right hand side of equation (1). The resonant braking is damped with rotation [31–35], and initial studies with codes such as SPIRAL [36] show prompt ion loss torques in DIII-D on the order of 10% the injected torque. Assuming that the NTV was thus the dominant effect of the nonaxisymmetry in rotating plasmas during the controlled NBI ramp-down, the evolution of each discharge in the presence of applied nonaxisymmetric fields is described by,

$$\begin{eqnarray}&&\frac{\text{d}{{L}_{\phi}}}{\text{d}t}\approx 0={{T}_{\text{NTV}}}+{{T}_{\text{NBI}}}+{{T}_{\text{INT}}}-{{L}_{\phi}}/{{\tau}_{\phi}},\end{eqnarray} \tag{ 2 }$$

Combining this with (1) gives,

$$\begin{eqnarray}&&{{T}_{\text{NTV}}}\left({{\omega}_{\phi}}\right)=-\left[{{T}_{\text{NBI}}}\left({{\omega}_{\phi}}\right)-T_{\text{NBI}}^{\text{ref}}\left({{\omega}_{\phi}}\right)\right],\end{eqnarray} \tag{ 3 }$$

Under the assumptions that the momentum confinement time, moment of inertia, and intrinsic torque are unchanged by the small nonaxisymmetry.

Equation (3) describes the NTV torque as the difference in NBI torque needed to obtain a given angular rotation with and without nonaxisymmetric fields. This was determined experimentally by comparing successive discharges with applied I-coil fields to a reference scenario with no applied I-coil fields. A comparison of two such discharges is given in figure 3. Large amounts (∼8 Nm) of NBI torque are used to produce fast rotation and ${{\beta}_{N}}$ of approximately 2. At 2250 ms the NBI torque is slowly ramped toward zero, the rotation evolves in torque balance until reaching a bifurcation, characterized by a sudden locking to the tokamak frame, a disruption, and the termination of the discharge. The presence of nonaxisymmetric fields, indicated here as the amplitude of currents in the lower I-coil array, introduces a NTV torque that accelerates the plasma and extends the discharge. Note for the analysis presented in this paper the rotation is obtained from the velocity of the minority impurity C⁶⁺ measured by charge exchange recombination (CER) spectroscopy [37] and assumed to be proportional to that of the majority plasma ion species D⁺ .

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Note that equations (1) and (2) describe of the evolution of the integral angular momentum when integral torques are used. In general the torque, as expressed in equation (3), may vary radially with the rotation profile and temporally with the profile evolution. Figure 4, however, shows the angular momentum $\left({{L}_{\varphi}}\right)$ evolution has a strong radial dependence that can be used to reduce the number of free parameters. In the shot with applied fields the NTV props up edge ${{L}_{\varphi}}$ compared to the reference discharge, while the core angular momentum decreases steadily with decreasing NBI torque similar to the reference. This is consistent with the expectation that the n  =  2 torque profile is concentrated in the plasma edge in DIII-D and modeling presented in section 3. This concentration of the torque profile suggests that the volume integrated NTV torque will be coupled to the modes of plasma response that dominate the perturbation in this specific region. It also enables the reduction of the NTV rotation dependence in equation (3) to a single variable: edge rotation. The edge rotation is defined here as ${{\omega}_{\varphi}}\left({{\psi}_{n}}=0.8\right)$, although the ultimate result is insensitive to the exact location in the outer plasma as long as it is well inside the pedestal for all the discharges. In the remainder of this paper, this rotation is used as the dependent variable in which to compare the NBI torque trajectories between multiple discharges with different applied spectra and a reference with no applied fields and calculate the integral NTV torque according to equation (3).

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The difference between the injected neutral beam torque required for equilibrium at a given rotation with and without applied nonaxisymmetric fields gives a direct measurement of the integral NTV torque according to equation (3). Thus, the NBI torque is mapped to a rotation variable as shown in figure 5. The NBI torques with and without applied fields as a function of the CER edge rotation are represented as third degree polynomial fits to points corresponding to the reconstructed profile times shown in figure 4, and the shaded error bars indicate propagation of errors determined by the fit accuracy. The temporal evolution of individual points is indicated in the reference shot, giving a sense of the scatter associated with the measurements. The two lines with $\Delta {{\phi}_{\text{UL}}}={{120}^{{}^\circ}}$ represent repeat discharges, showing the shot-to-shot reproducibility of the results. The vertical separation between the colored curves and black reference curve provides an estimate of the NTV for each applied field.

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The phasing dependence of the measured NTV torque is shown in figure 6. The torque is taken from the difference in NBI torques at the edge rotation ${{\omega}_{\varphi}}(0.8)=30$ krad s⁻¹; a rotation with large nonaxisymmetric effects well away from the discharge endpoints and possible influences of resonant braking at low rotation. At this rotation, the NTV measurements show a sinusoidal dependence on the poloidal spectrum. The envelope of a least squares fit error analysis is shown to provide a sense of the accuracy of this description. The five phasings obtained in the experiment are enough to constrain the amplitude, offset, and phase of the sinusoid fit, and the major trends are clear despite significant experimental uncertainty. The analysis at $\Delta {{\phi}_{\text{UL}}}={{240}^{{}^\circ}}$ as well as a sinusoidal fit to the collected data show that it is possible to eliminate, to the extent of experimental uncertainty, the coupling of the I-coil fields to NTV torque. This sinusoid with negligible offset is the mark of the applied fields coupling and decoupling to a single plasma mode that drives NTV. Nonlinear NTV modeling at this rotation also shown in figure 6 agrees with this spectral dependence and the corresponding single mode model is discussed in detail in the following section. The main results (a maximum at $15\pm {{19}^{{}^\circ}}$, and peak to valley ratio of order 10) are robust for edge rotations above 25 krad s⁻¹. Below this, the experimental scatter increases due in part to a growing tearing mode while the fitted torque becomes uncorrelated or insensitive to the phasing as evidenced by the convergence of many lines in figure 5 near 20 krad s⁻¹.

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An unanticipated observation from the experiment was a general decrease in the NTV torque as the edge rotation approaches zero. This can be seen in the convergence of the reference NBI torque with the phase scan trajectories in figure 5 near zero rotation, where all but one of the error bars overlap. Previous results in DIII-D QH-mode plasmas observed a NTV peak between the neoclassical offset and zero rotation [5, 38], but the associated bifurcation of the torque balance is not discernible here. The large error bars near zero rotation do not rule out this behavior, but prevent any conclusions from being drawn in this region. Instead, the focus on of this analysis is on the smooth rotation space evolution where there is clear separation of torques beyond the errors, away from low rotation where bifurcations (as well as resistive MHD modes) might appear. The previous results, however, were associated with the super-banana plateau regime [9, 39] in which the electric precession frequency is much smaller than the magnetic precession. The next section will discuss the cross regime NTV modeling of the plasmas used here, showing that their kinetic profiles make them susceptible to a different set of kinetic resonances than the previous studies.

The PENT code, in conjunction with IPEC, was used to model the dependence of the NTV torque in the QH-mode plasmas discussed above on the applied poloidal spectrum. The combined NTV theory modeled in PENT makes use of the bounce averaged drift kinetic equation, and describes the resulting torque in a general form [12],

$$\begin{eqnarray}&&{{T}_{\varphi}}=-\frac{{{n}^{2}}}{\sqrt{\pi}}\frac{{{R}_{0}}}{{{B}_{0}}}\mathop{\int}^{}\text{d}\psi NT\mathop{\int}^{}\text{d} \Lambda{{\bar{\omega}}_{b}}|\delta {{\bar{J}}_{\ell}}{{|}^{2}}\mathop{\int}^{}\text{d}\varepsilon {{\mathcal{R}}_{T\ell}},\end{eqnarray} \tag{ 4 }$$

with,

$$\begin{eqnarray}&&{{\mathcal{R}}_{T\ell}}=\frac{\left[{{\omega}_{\varphi}}+{{\omega}_{\ast T}}\left(\varepsilon -\frac{5}{2}\right)\right]{{\varepsilon}^{5/2}}{{\text{e}}^{-x}}}{\text{i}\left[(\ell -\sigma nq){{\omega}_{b}}+n\left({{\omega}_{E}}+{{\omega}_{D}}\right)\right]-{{\nu}_{i}}}.\end{eqnarray} \tag{ 5 }$$

Here, R₀ is the major radius of the magnetic axis, B₀ is the magnitude of the magnetic field on axis, N is the species density, T is the species temperature, ψ is the poloidal flux, $\Lambda=\mu {{B}_{0}}/E$ is a normalized magnetic moment, and ε is a temperature normalized energy. The resonance operator ${{\mathcal{R}}_{T\ell}}$, normalized bounce frequency ${{\bar{\omega}}_{b}}={{\omega}_{b}}{{R}_{0}}/\sqrt{2\varepsilon T/M}$ and normalized perturbed action $\delta \bar{J}_{\ell}^{2}=\delta J_{\ell}^{2}/2\varepsilon \text{TMR}_{0}^{2}$ are unit-less quantities.

This model is used for its generality, including many common features of the NTV torque that make its prediction difficult in tokamak plasmas. The model includes the treatment of multiple collisionality regimes, strong drift orbit resonances, and rotational scalings. These, however, are not the focus of this paper. More fundamentally, it includes the inherent nonlinearity of the NTV torque (by any model) in the quadratic dependence on the perturbed action (proportional to the perturbed field in a circular large-aspect ratio approximations). This results in poloidal mode coupling, means that the torque from two sources of perturbation in not the sum of the torque from each source individually, and suggests a sensitivity to perturbed mode structures far more complicated than commonly optimized resonant coupling. This section, however, shows that a single mode model can be used to effectively describe the NTV torque in the plasmas of interests without sacrificing the generality of the NTV physics.

Equilibria properties used in the model were taken from the middle of the torque ramp, corresponding to a period of rotation used in the experimental analysis of the phasing dependence. The input equilibrium was taken from shot 153276 (a representative case with roughly half the maximum observed torque) and the applied fields varied in the model as in the experiment. The QH-mode kinetic profiles during this period were characterized by strong rotation, with electric precession frequencies approaching the typical bounce frequency of trapped particles and much larger than the magnetic precession. The large aspect ratio approximations of these frequencies [11], and the effective Krook collision operator used in PENT are shown in figure 7. The kinetic precession resonance term is much larger than the collisionality, placing the majority of the plasma in the $\nu -\sqrt{\nu}$ regime [9, 40, 41]. The integral torque, also shown in figure 7, is dominated by large NTV in the edge region. This region contains the highest perturbed fields and largest displacements in the perturbed equilibrium. The sharp spikes in the NTV, are additionally amplified by a strong $\ell =\pm 1$ bounce harmonic resonance. These characteristics are important, as they emphasize the model is fully capturing the complex geometric dependencies and kinetic resonances, but are not explicitly tested here. Instead, we focus on the recovery of a simple dependence of the total torque on the applied field despite the presence or lack of complexity in the profile.

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The integral torque modeled by the PENT code has a sinusoidal dependence on the phasing between IU and IL arrays, shown in figure 8. The points represent modeled torque using the experimental n  =  2 current amplitudes and the 6 lower I-coil phases attempted in the experiment, fixing ${{\phi}_{\text{IU}}}={{90}^{{}^\circ}}$. These are then used in a least squares fit of a sinusoid with amplitude, offset, and phase as the free parameters. The quality of the fit for the input profiles is clear from figure 8, which includes a shaded band corresponding to the uncertainty of the fit function. Note that this result includes only the n  =  2 perturbations. While n  =  2 coil waveforms are known to produce n  =  4 sidebands in DIII-D [42], modeling found the torque produced by the sidebands to be well over an order of magnitude below the n  =  2 torque as has been found in forward-Ip EFC experiments [43]. Varying the rotation profile in the experimentally realized range above 25krad/s varies the amplitude of the modeled torque by  ±20%, remaining comparable to the experimental result within the experimental uncertainty. The main result is the qualitative, single mode, sinusoidal behavior that is recovered by the model, not the exact amplitude. The model shows that maximum n  =  2 torque can be obtained in these plasmas using ${{12}^{{}^\circ}}$ phasing, while the spectrum produced by ${{192}^{{}^\circ}}$ phasing has minimal coupling to the torque. The difference in extrema shows the importance of the poloidal spectra, as the applied torque can vary by an order of magnitude.

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The phasing scan can be mapped to a two dimensional phase space of the complex lower I-coil current $\mathbf{I}=I{{\text{e}}^{\text{i}{{\phi}_{\text{IL}}}}}$. As the scan was done at constant amplitude, the modeled points lie on a circular contour in this phase space and the sinusoidal dependence of the torque in ${{\phi}_{\text{IL}}}$ corresponds to the intersection of this contour with circular contours of constant torque quadratically approaching their centroid extremum.

Figure 8 shows IL phase space contours of the torque described by,

$$\begin{eqnarray}&&{{T}_{\varphi}}=\mathcal{A}|\mathbf{I}-{{\mathbf{I}}_{0}}{{|}^{2}}+\mathcal{B}.\end{eqnarray} \tag{ 6 }$$

The scan defined by a constant amplitude I and varied phase ϕ is described by the family of phase space vectors, $\mathbf{I}=I{{\text{e}}^{\text{i}n\phi}}$, while the extremum is given by constant amplitude and phase, ${{\mathbf{I}}_{0}}={{I}_{0}}{{\text{e}}^{\text{i}n{{\phi}_{0}}}}$. Inserting these into equation (6), the dependence is reduced to the simple sinusoid,

$$\begin{eqnarray}&&{{T}_{\varphi}}=A{{\text{e}}^{\text{i}n\left(\phi -{{\phi}_{0}}\right)}}+B,\end{eqnarray} \tag{ 7 }$$

with amplitude $A=-2\mathcal{A}I{{I}_{0}}$, phase $n{{\phi}_{0}}$ and offset $B=\mathcal{A}\left({{I}^{2}}+I_{0}^{2}\right)+\mathcal{B}$ shown in the corresponding plot on the left.

IPEC calculates the linear plasma response to external nonaxisymmetric fields, allowing superposition of the plasma response fields. Extensive sensitivity studies have been performed using the IPEC model for n  =  1 resonant locking thresholds [18, 19, 44]. Although the ideal model does not capture the field penetration physics, the resonant component suppressed by shielding currents prior to penetration is readily obtainable. The linear coupling between resonant components and spectra applied to a control surface (taken to be the plasma surface) in IPEC can be expressed by a linear matrix equation,

$$\begin{eqnarray}&&{{\mathbf{\Phi}}_{r}}=\mathbf{C}\centerdot {{\mathbf{\Phi}}_{x}},\end{eqnarray} \tag{ 8 }$$

where ${{\mathbf{\Phi}}_{r}}$ is a r-row vector containing the area normalized resonant field shielded by surface currents at each of the r resonant surfaces in the plasma, ${{\mathbf{\Phi}}_{x}}$ is a m-row vector representing the m applied spectral components on the control surface given by equation (9), and elements of the coupling matrix ${{\mathbf{C}}_{i,j}}$ are the linear coefficients relating applied poloidal spectrum to the resultant resonant components. Here, the resonant components contain both the applied vacuum field and the plasma response.

Fourier decomposition of the area weighted spectrum is defined as,

$$\begin{eqnarray}&&{{\Phi}_{x,m,n}}=\frac{1}{A}{{\mathop{\oint}^{}}_{S}}\text{d}\varphi \text{d}\vartheta \mathcal{J}\,|\nabla \psi |\left(\delta {{\mathbf{B}}_{x}}\centerdot \hat{n}\right){{\text{e}}^{\text{i}(m\vartheta -n\varphi )}},\end{eqnarray} \tag{ 9 }$$

where S is a flux surface with area A and unit normal $\hat{n}$, $\delta {{\mathbf{B}}_{x}}$ is the externally applied field. The spectral components ${{\Phi}_{x,m,n}}$ have units of flux and are independent of the choice of magnetic coordinates $(\psi,\vartheta,\varphi )$ corresponding to the Jacobian $\mathcal{J}$.

Singular value decomposition of the coupling matrix $\mathbf{C}$ gives eigenvectors ${{\mathbf{\Phi}}_{i}}$ and eigenvalues ${{\lambda}_{i}}$ of ${{\mathbf{C}}^{\dagger}}\mathbf{C}$, ranked by the energy content of the resonant response to the applied spectrum. The first eigenvalue is regularly an order of magnitude above the next highest [17, 45], indicating the dominance of the corresponding eigenspectrum ${{\mathbf{\Phi}}_{1}}$. The magnitude of the resonant response is then well defined by the extent to which an applied spectrum overlaps with the dominant mode. That is, the resonant coupling is proportional to the overlap parameter,

$$\begin{eqnarray}&&{{\mathcal{C}}_{i}}\equiv \sqrt{\mathop{\int}^{}\text{d}{{\varphi}^{\prime}}{{\left[\mathop{\int}^{}da\left(\delta {{\mathbf{B}}_{x}}\centerdot \hat{n}\right)\left(\delta {{\mathbf{B}}_{i}}\centerdot \hat{n}\right)\right]}^{2}}}=\,|{{\mathbf{\Phi}}_{x}}\centerdot {{\mathbf{\Phi}}_{i}}|.\end{eqnarray} \tag{ 10 }$$

The validity of the single dominant mode approximation is confirmed for the QH-mode plasmas used here, by a ratio of the first eigenvalues ${{\lambda}_{1}}/{{\lambda}_{2}}=41.0$. The poloidal spectrum of the dominant mode is shown in figure 2. Figure 9 shows the square of the overlap exhibits sinusoidal dependence on the phasing similar to the torque and thus a similar circular contour mapping in the phase space of the lower I-coil array. The sinusoid is a direct consequence of the quadratic definition (10) evaluated on the single control surface, and the near-zero minimum the mark of a single dominant mode. The phase of the overlap contour extremum is determined by the phasing of I-coils that best overlaps the IPEC ideal MHD dominant mode, and agrees with that of the NTV extremum despite no explicit rotation dependence or other neoclassical kinetic physics. This suggests that the ideal MHD dominant mode structure is well coupled to the nonambipolar transport and torque. The IPEC dominant mode is not exactly the mode structure best coupled to the NTV, as evidenced by the fact that the extremum I-coil current amplitude is slightly larger. However, the quadratic nature of equation (6) has produced a shallow and broad well near the single mode optima. The difference between the NTV calculated by PENT at the two optima is 1% of the phase scan sinusoidal amplitude, confirming that the ideal MHD calculation is a close proxy for the NTV dominating mode.

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The quality of the overlap as a metric parameterizing the nonresonant torque is apparent in the quality of a quadratic fit of the points in the phasing scan shown in figure 10. The nonlinear torque is well correlated with the square of the linear metric in these QH-mode plasmas. This is expected, as the high ${{\beta}_{N}}$ results in perturbations dominated by the plasma response. The dominant mode plasma response being proportional to the overlap, the torque can then be approximated as proportional to the square of the overlap through a common large aspect ratio simplification of the variation in the action $\delta J\sim \delta {{B}^{2}}$ [11]. Note that this approximation provides insight into the expected relationship, but the full perturbed action including arc-length change, energy and pitch angle dependences was used in the PENT modeling presented.

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The IPEC overlap parameter is sometimes used as a proxy for coupling to the least stable plasma Kink, a global MHD mode with predictable structure. This approximation is valid in the limit of high β (the ratio of plasma pressure to magnetic pressure) and strong shaping, in which the global plasma response is dominated by the least stable kink mode. In these plasmas, this mode is a pressure driven kink with a common large wavelength ballooning structure near the low field side (LFS) mid-plane that dominates the plasma response. Similar LFS structure has been found to be a common characteristic of the IPEC dominant mode in many high performance regimes and tokamaks [17, 45, 46].

Note that plasmas in DIII-D that have been observed to have a multi-modal response to n  =  2 fields have had less spacing in the SVD eigenvalues as well as significant High Field Side (HFS) plasma response [47, 48]. The poloidal field sensors used to measure the HFS response in multi-modal experiments measure signals at or near the noise floor of  ∼$5\times {{10}^{-5}}$ T in these plasmas, which is much lower than the measured LFS response maximum of  ∼$4\times {{10}^{-4}}$ T. This confirms the single modal description of these plasmas in this new metric as well as the largely LFS nature of the response.

The maximized resonant perturbations associated with the IPEC dominant mode are associated with the NTV torque as well. The first reason for this association is that this response excites a large plasma amplification of the applied fields, maximizing the total perturbed field. There are more subtle reasons, however, to expect the coupling of the particular kink excited by the IPEC dominant mode and NTV torque.

Although NTV is often associated with 'nonresonant' perturbations, it is the field-aligned modulations and not the perpendicular components that drive NTV torque [11]. As described above, the application of fields aligned with the dominant mode result in the largest resonant response at rational surfaces sampling the entirety of the plasma volume. This naturally results in the very pitch-aligned perturbations throughout the plasma volume that are responsible for NTV. Furthermore, both the reduced large-aspect-ratio approximation of PENT's semi-analytic model and full particle simulations using POCA have demonstrated that field-orbit alignment leads to peaking of the NTV near $m-nq\approx 0$ [49]. Particles near these rationals repeatedly sample the same field distortion without phase mixing, leading to rapid nonambipolar transport and large torques. That is, the fields maximized by the dominant mode also have maximum effectiveness in driving NTV torque.

Figure 11 shows the field-aligned modulation of the field modeled by IPEC for ${{0}^{{}^\circ}}$ and ${{180}^{{}^\circ}}$ phasing, corresponding to large and low overlap with the dominant mode respectively. The large resonant response at the rational surfaces in the ${{0}^{{}^\circ}}$ case clearly create a pitch-aligned kink throughout the plasma. The largest amplitudes are inside the inner most rational (q  =  3/2) and at the plasma edge, which is where figure 7 confirms the majority of the torque is. The ${{180}^{{}^\circ}}$ case, despite having the same energy in the coils, drives very little pitch aligned field and results in very low NTV torque.

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Note that the perturbations shown in figure 11 are weighted towards the LFS. In the common regimes, the torque is dominated by trapped particles that execute banana orbits bouncing across the field minimum on the LFS [8, 9, 11]. All these particles thus sample the perturbations near the LFS mid-plane, while only a subset of weakly trapped particles sample perturbations on the HFS. The deeply trapped particles are especially susceptible as they sample the same unbalanced perturbation many times at high frequency.

The combined NTV theory modeled in PENT makes use of the bounce averaged drift kinetic equation, and describes the resulting torque as a flux function. Some sense of the relative importance of poloidal positions can be obtained by plotting the torque flux density $\partial T/\partial \psi$ as a cumulative integral in pitch angle. From equation (19) in [12], this quantity can be written as,

$$\begin{eqnarray}&&\frac{\partial T}{\partial \psi}(\psi,\Lambda)=-\frac{{{n}^{2}}}{\sqrt{\pi}}\frac{{{R}_{0}}}{{{B}_{0}}}NT\int_{0}^{\Lambda}\text{d}{{\Lambda}^{\prime}}{{\bar{\omega}}_{b}}|\delta {{\bar{J}}_{\ell}}{{|}^{2}}\mathop{\int}^{}\text{d}\varepsilon {{\mathcal{R}}_{T\ell}}.\end{eqnarray} \tag{ 11 }$$

Figure 12 shows the this quantity for one of the cases modeled above mapped to $(\psi,\theta )$ using the poloidal bounce points determined by $\Lambda$. The color thus represents the amount of NTV torque coming from particles that have sampled the perturbations in that region. Radially, the contour shows torque concentrated inside the q  =  3/2 surface and outside the q  =  9/2 surface as was seen in figure 7 and is expected from the perturbations in figure 11. The additional poloidal information indicates that the LFS mid-plane is the most important region in the plasma for NTV. Note that this example is from a phasing that is nearly de-coupled from the dominant mode, which produced a plasma response concentrated more heavily away from the LFS extremum. The case of the maximum NTV coupling with a large LFS mid-plane plasma response looks similar, confirming the rigidity of the distribution expected in a single mode model. This LFS weighting helps facilitate the coupling of the applied field structure to the NTV, although the amplification of the parallel fields by the plasma response is expected to be of primary importance.

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