Rotation profile flattening and toroidal flow shear reversal due to the coupling

Rotation profile flattening and toroidal flow shear reversal due to the coupling of magnetic islands in tokamaks B. Tobias 1, M. Chen 2, I.G.J. Classen 3, C.W. Domier 2, R. Fitzpatrick 4, B.A. Grierson 1, N.C. Luhmann, Jr. 2, C.M. Muscatello 2, M. Okabayashi 1, K.E.J. Olofsson 5, C. Paz-Soldan 5 1 Princeton Plasma Physics Laboratory, Princeton, NJ 2 University of California at Davis, Davis, CA 3 Dutch Institute for Fundamental Fusion Energy Research, DIFFER, Rhinjuizen, Netherlands 4 University of Texas at Austin, Austin, TX 5 General Atomics, San Diego, CA Abstract The electromagnetic coupling of helical modes, even those having different toroidal mode number, modifies the distribution of toroidal angular momentum in tokamak discharges. This can have deleterious effects on other transport channels as well as on magnetohydrodynamic (MHD) stability and disruptivity. At low levels of externally injected momentum, the coupling of corelocalized modes initiates a chain of events whereby flattening of the core rotation profile inside successive rational surfaces leads to the onset of a large m/n=2/1 tearing mode and locked-mode disruption. With increased torque from neutral beam injection (NBI), neoclassical tearing modes (NTMs) in the core may phase-lock to each other without locking to external fields or structures that are stationary in the laboratory frame. The dynamic processes observed in these cases are in general agreement with theory, and detailed diagnosis allows for momentum transport analysis to be performed, revealing a significant torque density that peaks near the 2/1 rational surface. However, as the coupled rational surfaces are brought closer together by reducing q 95, additional momentum transport in excess of that required to attain a phase-locked state is sometimes

observed. Rather than maintaining zero differential rotation (as is predicted to be dynamically stable by single-fluid, resistive MHD theory), these discharges develop hollow toroidal plasma fluid rotation profiles with reversed plasma flow shear in the region between the m/n = 3/2 and 2/1 islands. The additional forces expressed in this state are not readily accounted for, and therefore analysis of these data highlights the impact of mode coupling on torque balance and the challenges associated with predicting the rotation dynamics of a fusion reactor a key issue for ITER. 1. Introduction Tearing modes, whether classically unstable [1] or driven unstable by the perturbed bootstrap current [2], are characterized by local reconnection at a rational surface (predominantly m/n = 4/3, 3/2, 2/1, or 3/1) that produces a new region of enclosed flux a magnetic island. High-β N operating scenarios, such as those under development for ITER, are often subject to multiple simultaneous instabilities. Near ITER relevant values of q 95 (approximately 3.2 on DIII-D), the coupling of helical modes and an associated collapse of core rotation often precede 2/1 NTM onset and locked-mode disruption. Our work strives to better understand this coupling by comparing experimental data from DIII-D to single-fluid resistive MHD theory [3]. By observing NTM coupling under non-disruptive conditions, we are able to proceed with empirical fitting of the associated momentum transport using the TRANSP code [4] and provide quantitative estimates of the MHD-induced torques. Resistive single-fluid MHD theory asserts that the mutual orientation of the helical currents at neighboring rational surfaces modifies the tearing stability of these layers through a form of driven reconnection, and that linear and nonlinear torques are generated to encourage the least stable orientation [5,6]. Modes of the same toroidal mode number, n, couple linearly through the

J B forces that are obtained as the overlap of their mode structures is integrated [7,8,9], but modes of different n-number are independent in this respect: toroidal mode number is quantized in the axisymmetric geometry used to describe tokamak equilibrium. Nonetheless, nonlinearity in the MHD response of the plasma to a multi-mode perturbation naturally gives rise to quadratic mixing products, ω 2 = ω 1 ± ω 0. If marginally stable modes exist at these three frequencies, they may be able to exchange energy through 3-wave mixing [10]. Furthermore, while a mode having n-number n 2 is linearly independent from modes having n-numbers n 0 and n 1, it is not necessarily independent of their mixing product. This couples the dynamics of all helical modes at some order, and provides a framework for describing bifurcations amongst rotating modes from regimes of high to low slip frequency. We will show that mode coupling and bifurcations from weak to strongly coupled regimes are accompanied by a potentially irreversible collapse of toroidal angular momentum in the core. The resulting modification of the momentum transport represents a destabilizing step on the path to many disruptions. The reduction of flow shear has previously been shown to correlate with reduced classical stability [11,12]. The constructive interference of perturbations near the outboard midplane also allows the modes to couple more strongly with external fields and conducting structures, dissipating the total angular momentum of the discharge [13,14]. Furthermore, reducing the rotational screening associated with a peaked rotation profile allows error fields to penetrate further into the confined plasma [15]. These effects have been documented in reversed-field pinches (RFPs) as giving rise to the so-called slinky mode [16,17,18]. While nonlinear mode coupling in tokamaks has been previously explored [19,20,21], only recently have internal, 2D measurements from ECE-Imaging [22] made the

positive identification of localized 3-wave coupling and the subsequent phase-locking of saturated tearing modes of different toroidal periodicity possible [23]. In this paper, we compare experimental data from DIII-D with a quasi-cylindrical model of nonlinear, 3-wave island coupling in which the nonlinear interaction of two NTMs is mediated by a global, n=0 mode, namely the axisymmetric toroidal shaping of the plasma including the Shafranov shift [3]. The coupling of two rotating magnetic islands to a third non-rotating perturbation inhibits the poloidal rotation of the resulting composite structure. Constraining the rotation of this composite structure to be directed toroidally produces phase-locking, a harmonic relationship between the Mirnov frequencies of the modes. Section 2 presents the nominal phenomenology of mode coupling and perturbation phase-locking as it is observed in some lowtorque, ITER scenario development discharges on DIII-D that end in disruption. In Section 3 we explore similar phenomena under non-disruptive conditions. The hybrid scenario explored on DIII-D [24,25] routinely exhibits multi-helicity instability, but increased torque injection and elevated safety factor allow the discharges to be sustained until normal shutdown. The behavior of these discharges is in qualitative agreement with the model presented in Ref. [3]. However, anomalous cases are found in which mode coupling does not result in phase-locking. Rather, the modes maintain different toroidal angular phase velocities in the laboratory frame and the dynamically stable state exhibits a hollow toroidal rotation profile with regions of reversed toroidal flow shear. Analysis presented in Section 4 quantifies the momentum transport observed as a localized torque density. The distribution of this coupling torque density, peaking near the q=2 rational surface, remains similar as q 95 is decreased in 3 steps from 5.8 to 4.3 and the rotation profile is flattened, then made hollow. In Section 5 we describe convective momentum transport observed at the lowest values of q 95 that cannot be accounted for by viscous diffusion or a phase-

locking torque, motivating further analytical and computational modeling, perhaps new theoretical development. 2. Momentum collapse in low-torque ITER baseline scenario development discharges Momentum transport, by any mechanism, can wreak havoc in slowly rotating discharges and low-torque ITER demonstration discharges. While many different mechanisms may contribute to a redistribution of momentum, one mechanism we have identified is mode-mode MHD coupling. An instance of mode coupling and an associated collapse of ion fluid rotation in the core is presented in Figure 1. Sawteeth appear early in the discharge, and each sawtooth crash temporarily flattens the core toroidal rotation inside the q=1 surface. Beginning around 2 s, the core rotation decreases further, ultimately synchronizing the toroidal rotation of the 1/1 kink with the q=3/2 surface. A m/n=3/2 mode grows, and in response to the drop in observed β N, NBI power is increased without providing additional torque. This is typical of so-called β-feedback operation on DIII-D. As the mode continues to grow, rotation inside the 3/2 surface continues to decay. By ~2400 ms, differential rotation between the m/n=3/2 and 2/1 surfaces has been lost and a large 2/1 NTM is triggered. The plasma then decelerates as a rigid body and locks to the wall. Thermal quench begins at 2800 ms. A spectrogram illustrating the evolution of mode activity in this discharge is shown in Figure 2. In part (a), the EIGSPEC code [26] is used to identify and follow the evolution of coherent modes observed by Mirnov coil array [27,28]. As is often observed, modes are destabilized in descending rational order beginning with those nearest the magnetic axis [29]: sawteeth are observed first, then a m/n = 4/3 island, then 3/2, then finally 2/1. Dividing the Mirnov frequency by the toroidal mode number of each fluctuation produces the toroidal angular phase velocities shown in (b). Note that the sawtooth crash near 2040 ms, the first to synchronize rotation at the

magnetic axis with that of the 3/2 surfaces (c.f. Figure 1), corresponds with the beginning of a bulk deceleration indicated by rollover of the 4/3 mode s toroidal velocity. As the 3/2 island begins to grow, it quickly locks in phase with the 4/3 island and a n=4 fluctuation. These modes then decelerate together, though rotation near the q=2 surface varies little. At 2410 ms, the rotation profile is essentially flat inside the q=2 surface. At this time, a 2/1 mode grows rapidly and in phase with the existing islands. Disruption follows the simultaneous locking of these coupled modes to the wall. The data shown in Figure 2 describe a discharge in which differential toroidal rotation is progressively lost from the core outward, one rational surface at a time. The synchronization of toroidal phase velocity is a ubiquitous, readily identified characteristic of these discharges and consistent with a nonlinear, electromagnetic coupling that holds the islands in a fixed mutual orientation as a rigid structure. Drag forces then impose this rigidity on the plasma fluid [30]. The loss of total angular momentum that results from the composite structure locking to an external error field is followed in each case by disruption. 3. Phase-locking of coupled rotating modes Further investigation of tearing mode phase-locking is carried out in the so-called hybrid operating regime, a mode of operation under development as an option for ITER [24,25]. This scenario exhibits attractive confinement properties at values of q 95 somewhat higher than the ITER baseline prescription. Core NTMs play an important role by redistributing the plasma current to elevate the central safety factor and prevent sawteeth by way of a flux pumping mechanism [31]. Although the preferred scenario exhibits only a solitary m/n=3/2 or 4/3 core NTM, hybrid discharges on DIII-D and JET are predisposed to developing multiple saturated NTMs of different n-number [23,32]. While this is clearly undesirable with respect to

performance and stability, it makes them ideal for the study of coupled mode interactions. The modes couple dynamically, but disruption is avoided by increasing the level of torque produced by neutral beam injection. Of the 32 discharges successfully generated in the high β N, hybrid scenario development experiment selected for this study of mode coupling, 26 developed multiple NTMs. In each of the 18 discharges subject to detailed analysis for this study, a 3/2 mode is destabilized early after the L-H transition, limiting the evolution of β in the core. As current in the solenoidal coil, or E- coil, reaches flat-top, NBI power is introduced in a modified β-feedback mode and the plasma is allowed to evolve through 4/3, 3/2, then 2/1 instability. Figure 3 shows that when the 2/1 NTM is triggered in discharge 155570, both NBI power and torque are increased in response to the degraded confinement. Nonetheless, core rotation continues to decay until all three modes rotate toroidally as a rigid body. Figure 4 further illustrates the evolution of shot 155570 toward rigid-body mode and plasma rotation. At 2100 ms, the 3/2 mode propagates with a toroidal angular phase velocity of 25 khz. For comparison, the 2/1 NTM and its harmonics propagate at approximately 17 khz. Note the n=1 oscillation at higher phase velocity. This bicoherent fluctuation [33] is a quadratic mixing product corresponding to f 3/2 f 2/1 and is strong evidence of nonlinear coupling [10]. By 2500 ms, the 3/2 mode rotation has decreased significantly. At 2560 ms, the 3/2 and 2/1 modes become phase-locked and co-rotate toroidally. The n=1 spectral mixing product becomes degenerate after this bifurcation and serves to reinforce mode coupling. In those shots that phase-lock to rigid-body toroidal rotation, ECE-Imaging has been used to verify that magnetic islands persist at both surfaces, evident from tearing parity, a 180 phase shift in the radiated electron temperature response across these radii. ECE-Imaging data also

reveals that the internal mode structures relax such that the local, poloidal wavenumbers associated with the m/n=3/2 island and the 4/2 mode (a harmonic of the 2/1 mode) are matched on interior flux surfaces. Changes of mode structure may indicate relaxation of the current profile, though direct measurement of the changing q-profile by way of motional Stark effect (MSE) spectroscopy [34] involves additional uncertainties and is therefore not presented here. The result of this evolution is an agreement in not only toroidal phase velocity, but also poloidal phase velocity near the outboard midplane [23]. This is shown in Figure 5, where the 2D power spectral density [35] provides internal measurement of the local poloidal wavenumber at ρ=0.4, a radius between the two rational surfaces. The qualitative behavior of these discharges is in keeping with the predictions of the quasicylindrical model described in Ref. [3]. The mode frequency and differential fluid rotation evolve gradually to reduce the slip frequency, ω, which is measured as the difference between the angular Mirnov frequencies of the m/n=3/2 and 4/2 fluctuations. When this slip is reduced to half its initial value, ω 0 /2, a bifurcation to phase-locking occurs. After bifurcation, the modes generally remain phase-locked until the end of the discharge, although brief instances of unlocking are observed near the times of edge-localized mode behavior (ELMs) and other transient MHD events. Repeat discharges under similar conditions exhibit the same behavior. 4. Momentum transport analysis Momentum transport analysis reveals that mode coupling generates forces comparable to external sources of momentum such as NBI. The coupling torque is not calculated directly. Instead, the TRANSP code is used to obtain an effective momentum diffusivity that reproduces the rotation profile observed in experiment. The resulting momentum flux is equivalent to a localized torque density, and therefore provides an estimate of the NTM-induced electromagnetic

torque that is not explicitly accounted for in the TRANSP model. In discharges with q 95 of 5.8 and 5.0 (exemplified by 155570 and 155577), mode structures evolve to become well matched in poloidal wavenumber and the rotation profile evolves such that the Mirnov frequency of the m/n=3/2 mode becomes exactly twice that of the 2/1 mode. In steady state, the simplified form of the momentum balance equation (neglecting convection terms V i V ) may be written as [4], T col + T jxb + T bth + T iz = n i m i R 2 Ω 1 τ CX Vol ρ 1 Vol ρ ρ n i m i R 2 ( ρ) 2 Ω χ φ ρ, (1) where Ω is the toroidal rotation frequency of the plasma and brackets denote flux surface averaged quantities. Momentum source terms on the left represent estimates made by the TRANSP code for collisional torque due to the injection of high energy neutral particles, JxB forces associated with the injected current, torque generated by the thermalization of injected beam ions, and that due to the ionization of circulating neutrals. Momentum flux terms are on the right hand side, where the constant τ C X represents momentum loss due to charge exchange. A similar term accounting for field ripple effects, τ φδ, is often included elsewhere but neglected in the present analysis. The viscous diffusion of momentum is encapsulated by the term at right. The TRANSP model does not include electromagnetic torques generated by the coupling of tearing modes. Therefore, assuming quasi-neutrality and constraining this equation with experimentally measured values of toroidal rotation, Z eff, ion temperature, and electron density, the free parameter χϕ, nominally the momentum diffusivity, is fit so as to balance the equality. Comparison of the momentum diffusion coefficient fit during phases of MHD activity to that obtained in earlier phases of each discharge provides an estimate of the impact of MHD coupling

on momentum transport and torque balance. It can also be used to estimate an equivalent torque due to mode coupling. Figure 9 compares the carbon impurity rotation profile measured after phase-locking or bifurcation with an expectation produced using the TRANSP code, revealing a discrepancy inside the 2/1 surface that increases as q 95 is reduced. To produce this expectation, the measured toroidal rotation provides a boundary condition at ρ = 0.8, source torques are estimated with contributions calculated by NUBEAM [39], and viscous transport is estimated by taking the momentum diffusivity, χϕ, to be equal to the ion thermal conductivity, χ i, which is obtained by solving power balance. This ad hoc approximation, equivalent to assuming a uniform effective Prandtl number ( Pr χ φ χ i ) inside ρ = 0.8, is motivated by historical observations [40] and found to be adequate for describing the rotation profiles observed in our experiment at times prior to the onset of large tearing modes. However, when multiple tearing modes are present, their coupling acts to flatten, even invert the rotation profile in the region between the rational surfaces. For coupling of m/n=3/2 and 2/1 modes, the toroidal rotation of the ion fluid is slowed in the core and at the 3/2 rational surface while the 2/1 surface is accelerated. The impact on momentum transport is much greater than the impact on energy confinement, making the choice of a uniform effective Prandtl number no longer appropriate. Flattened regions of the rotation profiles shown in Figure 6 (a-c) correspond to peaking of the fitted momentum diffusivity. These peaks do not occur at the rational surfaces, but at intervening radii. As the local toroidal flow shear approaches zero, momentum diffusivity becomes undefined. This is not observed for the case of an individual, uncoupled tearing mode. Within the context of a visco-diffusive empirical model, momentum transport that reverses the flow shear requires a convective term, i.e. an outward directed momentum pinch. However, from earlier

physical arguments, we believe that the dominant effect on toroidal force balance is properly encapsulated as a differential electromagnetic torque rather than a diffusive or convective transport term. For comparison to other sources of torque, such as those produced by NBI, the transport quantified by fitting to the TRANSP model may be re-cast as a local torque density using, T MHD = Vol ρ 1 Vol ρ ρ n i m i R 2 ( ρ) 2 ( χ φ χ i ) Ω ρ, (2) where χ i is again the solved for ion thermal conductivity and the anomalous, MHD-induced momentum diffusivity is given by χ φ MHD = χ φ χ i. Torque profiles obtained by evaluating Eqn. (2) are given in Figure 6 (d-f). The broad profile of negative torque density spanning the q=3/2 surface and a large positive peak near the 2/1 surface are ascribed to the presence and coupling of tearing modes at those surfaces. Additional excursions of the torque density profile at larger radii are beyond the scope of this paper and associated with physics of the edge transport barrier that produces the high-confinement mode. They indicate differences between rotation and ion temperature profiles manifest near the plasma edge. Large differential torque densities are required to account for the momentum flux obtained by this analysis a strong coupling that is easily capable of resisting modification with rotation actuators such as NBI. Static torque densities that we attribute to coupling of the m/n=3/2 and 2/1 islands are much greater than the neutral beam induced torque density and apply the equivalent of 2 neutral beam sources inside the q=3/2 surface in shot 155570 (q 95 =5.8). In shot 155577 (q 95 =5.0) the rational surfaces are moved further out in plasma radius and the volume integrated torque increases. In each discharge, momentum is conserved for much of the core, as indicated by the volume integrated torque approaching zero near where q=2. Interestingly,

decreasing q 95 to 4.3 while maintaining similarity in other parameters produces additional momentum transport between the 3/2 and 2/1 surfaces. In shots exemplified by 155587 shown in Figure 6 (c, f), the bifurcated rotation profile is hollow, with the 2/1 surface rotating faster than the 3/2. This additional momentum transport is anomalous in that it cannot be accounted for by the simple phase-locking model we have thus far described. 5. Discharges exhibiting momentum transport in excess of that required to attain a phaselocked state Of the 18 multi-helicity discharges subject to detailed analysis, 6 exhibited a clearly identifiable bifurcation event not obscured by other MHD transients such as ELMs. Each of these bifurcated to a regime of strong coupling at approximately half the initial slip frequency, a value consistent with a simple analytical model [3]. For discharges at q 95 of 5.8 and 5.0, the highly coupled state is one of phase-locking and rigid-body toroidal mode rotation as predicted by single fluid resistive MHD. However, in discharges at q 95 of 4.3, the difference in Mirnov frequency between the 3/2 mode and the 4/2 fluctuation becomes negative after bifurcation. The toroidal plasma rotation profile also becomes hollow inside the 2/1 surface, implying large anomalous torque densities producing convective momentum transport. This data is difficult to reconcile with the theory used to describe phase-locking and therefore represents a challenge to the assumptions contained therein. The evolution of mode structure observed by toroidal and poloidal Mirnov arrays as coupling proceeds is investigated by using the EIGSPEC code to calculate the so-called shape coherence [26] of the fluctuating poloidal magnetic field. This technique quantifies the similarity of the 2D mode structure observed at the vessel wall to a reference taken at a particular frequency and time. Figure 7 illustrates the evolution of mode frequency and shape coherence within shot

155587 with respect to the reference indicated by a red square. As in other shots described earlier, a bifurcation event occurs after 2/1 mode onset (approximately 250 ms in this discharge) as the slip frequency approaches half its initial value. However, phase-locking does not occur. The toroidal rotation of the 3/2 island, whose mode structure is identified by the red trace in Figure 7 b), slows to generate a magnetic fluctuation with a frequency that is less than that of the 4/2 fluctuation (light blue). This fluctuation frequency is also less than twice that generated by the 2/1 island, making for a lesser toroidal phase velocity. A negative differential toroidal phase velocity, Δv pφ = v 3/2 2/1 pφ v pφ, corresponds with a negative differential toroidal fluid rotation. Figure 8 compares the difference between the 3/2 and 4/2 Mirnov frequencies with the differential toroidal fluid rotation of carbon impurities at these rational surfaces. Within the limits of diagnostic time resolution, mode frequency and fluid rotation appear to transition simultaneously, precluding a definitive causal relation. This hollow rotation profile was sustained to shutdown in 6 similar discharges with I p of 1.4 MA (although 155587 was terminated by a false control coil over-current fault near 4.2 s). Noting that phase-locking is the result of holding the island x-points aligned at the outboard midplane [3], and that poloidal rotation of the composite structure would interrupt phase-locking, we have also compared the poloidal fluid rotation in this shot to that in phase-locking discharges. The vertical line-of-sight velocity of the carbon impurity fluid at the outboard midplane is obtained by analysis of CER data from vertical viewing chords [36] (with corrections determined from both vertical and tangential views). The estimates shown in Figure 9 indicate that the poloidal rotation profiles obtained after phase locking are similar at q 95 of 5.8 and 5.0. In shot 155587, however, the region of upward flow disappears, replaced by a weak (< 1 km/s) downward flow across the outboard midplane. Neoclassical theory and other experimental

observations would suggest a main ion poloidal rotation of slightly greater magnitude [37,38]. However, while this change in the poloidal rotation profile is interesting, it does not appear to be sufficient to account for the difference in the mode frequencies that we observe. It is also important to note that our analysis neglects coupling of the modes to external structures, which is likely to be justified in discharges such as 155570 where the loss of total angular momentum after mode onset is minimal. In the case of 155587, where a significant external drag force is apparent, we have assumed that those drag forces are stronger at the q=2 surface than at the more interior 3/2. Although the external 2/1 perturbation field is larger, and the uncorrected error field on DIII-D is predominantly n=1, this may prove a poor assumption in cases where error field correction is applied. Furthermore, our model asserts that 3-wave coupling is dominantly mediated by toroidicity and plasma shaping, neglecting the possibility of coupling to another independently rotating mode (e.g. m/n=1/1). No evidence of such a mode was found in shot 155587, though this does not absolutely preclude its presence [40]. Finally, the estimation of source torques in TRANSP assumes axisymmetry, which may not be adequate. This will be explored in future work alongside investigation of 2-fluid and neoclassical effects, such as island-induced neoclassical toroidal viscosity. 6. Summary and discussion The phenomena of mode coupling amongst concomitant m/n=3/2 and 2/1 tearing modes on DIII-D has been compared to the qualitative features of a simplified model of NTM phaselocking in cylindrical geometry. In addition to the generation of bicoherent spectral mixing products and beat wave fluctuations that indicate nonlinear coupling, the differential rotation of these modes evolves dynamically toward a bifurcation in torque balance. This bifurcation occurs

at half of the initial slip frequency and often results in a set of coupled modes that rotate with the same toroidal angular phase velocity. For all cases in which the local poloidal wavenumbers of the two modes become harmonic, this leads to phase-locking consistent with a model of 3-wave coupling mediated by a non-rotating m/n=1/0 mode. However, shot 155587 and similar discharges present a challenge to the simplifying assumptions made in that analysis. Momentum flux observed during the penultimate phases of these discharges is equivalent to a static torque density far greater than that available from NBI. Though the volume integrated torque approaches zero near the q=2 surface, the profile of this torque density is asymmetric, with a broad distribution around the 3/2 surface and a large peak centered outside the 2/1 island. In contrast, single-fluid, resistive MHD theory asserts that the electromagnetic torque exerted on the modes is applied to the fluid only at the rational surface, resulting in a torque density profile that is symmetric and peaks within the separatrix of each island. However, the empirical method that we have applied for this analysis is under constrained. Therefore, electromagnetic torques, diffusive momentum transport, and convective mechanisms are all likely to contribute, and more sophisticated experimental techniques are required to determine their weighting. Many uncertainties remain with respect to the rotation dynamics and momentum transport properties of large tokamaks. This is particularly true of ITER, where contemporary actuators such as NBI are not expected to impart significant rotation and the dominant forces may be intrinsic [42,43]. And yet, there is much evidence that rotation and MHD stability are intrinsically interdependent in relevant regimes [44,45]. This inevitably impacts our approach to disruption prediction, avoidance and mitigation. Recognizing that rotation can respond more sensitively than transport in other channels to the presence of low level, core-localized mode activity, we assert that it is valuable to consider subtle changes in core rotation as early indicators

of processes contributing to a disruptive chain of events. The empirical evidence presented here also suggests that flattening of the rotation profile contributes to the destabilization of disruptive NTMs, which we believe warrants continued exploration of the many ways in which contemporary actuators can be used to maintain peaked rotation profiles. If these methods prove beneficial for expanding the safe operating space around ITER-relevant values of q 95 and plasma β, they provide further motivation to develop innovative techniques for imparting significant angular momentum and controlling its distribution in fusion devices. Acknowledgements This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Awards DE-AC02-09CH11466, DE-FG02-99ER54531, DE-FC02-04ER54698, and DE-SC0012551. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/d3d_dmp. The authors would like to thank the DIII-D team for their support of these experiments.

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List of Figures and Captions Figure 1. (color online) a) Evolution of β N, and neutral beam power, P inj. b) RMS amplitude of magnetic fluctuations measured by the toroidal Mirnov coil array for n=1 (red), n=2 (blue), and n=3 (green). c) Toroidal carbon impurity fluid velocity measured by charge exchange recombination spectroscopy (CER) at the magnetic axis (green), the 3/2 (blue) and 2/1 (red) rational surfaces (blue). Torque from neutral beam injection, T inj, is also plotted Modes decelerate and lock to the wall at 2.5 s, thermal quench begins at 2.8 s. B T = -1.6 T, I P = 1.2 MA, n e = 5.3x10 19 m -1.

Figure 2. (color online) a) The magnetic fluctuation spectrum obtained by the toroidal Mirnov coil array in shot 147029. Colors differentiate n=1 (red), n=2 (yellow), n=3 (green), and n=4 (blue) mode activity. b) Toroidal angular phase velocity (frequency divided by toroidal mode number) of the Mirnov fluctuation spectrum is plotted for the time spanning the L-H transition, mode locking, and the deceleration of a coupled mode structure to the laboratory frame. c) Toroidal rotation profiles obtained by CER at 1930 ms (mid sawtooth period), 2230 ms (after sawtooth crash), and 2410 ms (immediately preceding 2/1 onset).

Figure 3. (color online) a) The evolution of β N, and neutral beam power, P inj. b) RMS amplitude of magnetic fluctuations measured by the toroidal Mirnov coil array for n=1 (red), n=2 (blue), and n=3 (green). c) Toroidal carbon impurity fluid velocity measured by CER at the magnetic 4/3 (green), 3/2 (blue), and 2/1 (red) rational surfaces (blue). Torque from neutral beam injection, T inj, is also plotted. B T = -2.05 T, I P = 1.1 MA, n e = 4.0x10 19 m -1.

Figure 4. (color online) a) The magnetic fluctuation spectrum obtained by the toroidal Mirnov coil array in shot 147029. Colors differentiate n=1 (red), n=2 (yellow), n=3 (green), and n=4 (blue) mode activity. b) Toroidal angular phase velocity (frequency divided by toroidal mode number) of the Mirnov fluctuation spectrum is plotted for the time spanning onset and phaselocking of the 3/2 and 2/1 NTMs. c) Toroidal rotation profiles (CER) obtained at 2100 ms (after 3/2 onset, green), 2500 ms (after 2/1 onset, blue), and 2800 ms (after phase-locking, red). Profiles at 2500 and 2800 ms indicate rigid-body toroidal rotation of the 4/3+3/2 and 3/2+2/1 surfaces, respectively.

Figure 5. (color online) 2D power spectral density estimates obtained by ECE-Imaging near ρ=0.4, a region between the 3/2 and 2/1 rational surfaces. a) Mode frequency and poloidal wavenumber of the 3/2 and 2/1 modes approx. 100 ms before phase-locking. Poloidal phase velocity is fit with dashed lines. b) After phase-locking, the 3/2 mode becomes indistinguishable from the 4/2 harmonic fluctuation. Differential poloidal phase velocity is reduced below the level of confidence for this method the value fitted to both modes is shown.

Figure 6. Carbon impurity rotation profiles observed after bifurcation (diamonds) are compared with expectations produced by assuming a uniform Prandtl number within ρ = 0.8 for shots (a) 155570, (b) 155577, and (c) 155587. (d-f) Torque densities ascribed to MHD activity are compared with the estimated torque from neutral beam injection. Solid lines represent plasma volume integrated torque within the given flux surface, dashed lines represent localized torque densities.

Figure 7. (color online) a) Time-frequency spectra of the Mirnov array signal. Differences between 3/2 and 4/2 fluctuation frequencies at 2/1 mode onset and just before torque bifurcation are identified with arrows. b) Shape coherence analysis of the 3/2 and 4/2 mode evolution. Red box indicates time-frequency reference.

Figure 8. The difference in toroidal angular mode velocity, i.e. the Mirnov frequency between 3/2 and 4/2 fluctuations divided by n-number, 2, and the differential toroidal rotation of the carbon impurity ion fluid measured by CER near the 3/2 and 2/1 rational surfaces. Scatter in the CER data is attributed in part to interference from ELMs.

Figure 9. The vertical line of sight velocity of the carbon impurity ion species estimated for times after phase-locking. Positive values are directed upward through the outboard midplane. As q 95 is reduced, upward flow of carbon impurities outside the 2/1 rational surface is reduced, becoming a downward flow in 155587.

147029 t=2230 ms (km/s) (Gauss) 2.5 2.0 1.5 1.0 0.5 30 25 20 15 10 5 0 200 150 100 (a) β N P inj = 2.4 MW (b) n1 amplitude n2 Disruption n3 Mode locking Sawteeth (c) v φ (q=2) v φ (q=1.5) v φ (axis) 50 T inj = 2.3 MW 0 1000 1500 2000 Time (ms) 5.4 MW 2500 3000

Frequency (khz) Toroidal phase velocity, f/n (khz) Toroidal rotation, Ω (khz) 35 30 25 20 15 10 5 0 20 15 10 147029 t1 4/3 Island Sawteeth 1/1 kink 3/2 Island 5 4/3 Island 3/2 Island 0 1600 1800 2000 2200 2400 2600 20 Time (ms) t1 = 1930 ms (c) 15 t2 = 2230 ms t3 = 2410 ms 10 5 Sawtooth q = 3/2 2/1 0 inversion radius 0.0 0.2 0.4 0.6 0.8 1.0 ρ N t2 2/1 Island onset n=4 t3 (a) (b) n=4 Wall locking

155570 t=2340 ms (km/s) (Gauss) 3.5 3.0 2.5 2.0 1.5 1.0 20 15 10 5 0 500 400 300 200 100 (a) β N P inj = 6 MW (b) n1 amplitude 4x n2 amplitude 8x n3 (c) T inj = 5.4 Nm 1000 1500 2000 Time (ms) v φ (q=2) v φ (q=1.5) v φ (axis) 6.6 Nm 7.8 MW 2500 3000

Frequency (khz) Toroidal phase velocity, f/n (khz) Toroidal rotation, Ω (khz) 155570 t1 t2 t3 80 3/2 Island (a) 70 4/3 and 6/3 60 4/3 6/3 50 3/2 40 4/2 3/2 and 4/2 30 3/2-2/1 2/1 20 10 0 2/1 Island onset 40 3/2-2/1 (b) 35 (Mixing product) 30 3/2 Island 25 20 15 10 2/1 Island 2/1 + 3/2 5 onset phase-locking 1500 1750 2000 2250 2500 2750 3000 40 Time (ms) t1 = 2100 ms (c) 30 20 t2 = 2500 ms t3 = 2800 ms 10 q = 4/3 3/2 2/1 0 0.0 0.2 0.4 0.6 0.8 1.0 ρ N

Frequency (khz) Frequency (khz) 155570, 2458-2478 ms 60 (a) 50 3/2 Island 40 30 20 10 0 2560-2580 ms 60 (b) 50 40 30 20 10 33.3 km/s 2/1 Island 2/1 20.8 km/s 4/2 (harmonic) 27.3 km/s 3/2+4/2 0 0.0 0.5 1.0 1.5 2.0 2.5 Poloidal wavenumber /2π (m -1 ) 0-10 -20-30 0-10 -20-30 Array signal power (db) Array signal power (db)

Torque rotation, Ω (krad/s) Torque density (N-m/cm3) vol. int. torque (N-m) 200 155570, 2.85 s, q 95=5.8 155577, 3.92 s, q 95 =5.0 155587, 2.8 s, q 95 =4.3 (a) (b) (c) 150 B.C. 100 50 0 15 10 5 0-5 (d) 3/2 T NBI T MHD 2/1 (e) (f) 0.0 0.2 0.4 ρn 0.6 0.8 1.0 0.0 0.2 0.4 ρn 0.6 0.8 1.0 0.0 0.2 0.4 ρn 0.6 0.8 1.0

f (khz) f (khz) 155587 60 50 (a) 40 30 20 10 0 60 50 40 30 20 (b) f 0 f0 /2 3/2 Island 4/2 Harmonic 10 2/1 0 2000 2100 2200 2300 Time (ms) 4/2 Fluctuation 3/2 Island 2400 2500 0-1 -2-3 -4-5 -6 1.0 0.8 0.6 0.4 0.2 0.0 Mirnov array signal power log 10 (T 2 /s 2 ) Mirnov shape coherence (a.u.)

Diff. toroidal angular velocity (khz) 155587 15 10 5 0 CER data -5 2100 2200 2300 Time (ms) Mirnov diff. frequency/2 ELM times 2400 2500

vz, outboard midplane (km/s) 4 155570, q 95 = 5.8 2 155577, q 95 = 5.0 0-2 155587, q 95 = 4.3-4 0.0 0.2 0.4 0.6 0.8 1.0 ρ N