Magnetic Feedback Control of 2/1 Locked Modes in Tokamaks

Magnetic feedback control of 2/1 locked modes in tokamaks

Wilkie Choi

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

Magnetic feedback control of 2/1 locked modes in tokamaks

Wilkie Choi

This thesis presents simulation and experimental work on feedback control of the phase of non-rotating magnetic islands (locked modes) in the DIII-D tokamak, as well as its application to synchronized modulated current drive, for stability studies and control of the locked mode amplitude. A numerical model has been developed to predict mode dynamics under the effect of various electromagnetic torques, due to the interaction with induced currents in the wall, error fields, and applied resonant magnetic perturbations (RMPs). This model was adapted to predict entrainment capabilities on ITER, suggesting that small (5 cm) islands can be entrained in the sub-10 Hz frequency range. Simulations and subsequent experiments on DIII-D demonstrated a novel technique to prevent locked modes. Preemptive entrainment applies a rotating RMP before a neoclassical tearing mode fully decelerates such that it will be entrained by the RMP and mode rotation can be sustained. A feedback control algorithm was designed and implemented on DIII-D to offer the ability to prescribe any toroidal phase to the mode and to allow for smoother entrainment. Experimental results confirmed simulation predictions of successful entrainment, and demonstrated one possible application to electron cyclotron current drive (ECCD). Feedback-controlled mode rotation and pre-programmed ECCD modulation were synchronized at DIII-D. This allowed a fine control of the ECCD deposition relative to the island O-point. Experiments exhibited a modulation of the saturated island width, in agreement with time-dependent modeling of the modified Rutherford equation. This work contributes to control and suppression of locked modes in future devices, including ITER. Contents

List of Figures iii

List of Tables xi

Acknowledgements xiii

1 Introduction 1

1.1 Background ...... 1

1.2 Tearing modes ...... 3

1.3 Measuring the mode ...... 8

1.4 Mode position control ...... 14

1.5 Outline of the thesis ...... 16

2 Simulation of mode dynamics 17

2.1 Numerical model ...... 17

2.2 2/1 Mode behaviour in DIII-D ...... 22

2.3 Comparison of 2/1 mode entrainment across machines ...... 30

2.4 Predicted entrainment capabilities for ITER ...... 37

3 Feedforward Control 42

i 3.1 Preemptive entrainment ...... 43

3.2 Experimental result ...... 45

3.3 Simulation of improved entrainment ...... 47

4 Feedback Control 51

4.1 Control system ...... 51

4.2 Controller design ...... 53

4.3 Simulation of controller ...... 56

4.4 Experimental results ...... 61

5 Mode Suppression 69

5.1 ECCD in DIII-D ...... 69

5.2 Experimental Results ...... 72

5.3 Effect of radial misalignment ...... 75

5.4 Mode amplitude suppression ...... 78

5.5 Comparison with theory ...... 79

6 Conclusions and future work 84

6.1 Summary and Conclusions ...... 84

6.2 Future work ...... 86

Bibliography 89

ii List of Figures

1.1 (a) A single magnetic field line (red) on the m/n = 2/1 surface closes upon itself

after 2 toroidal transits and 1 poloidal transit. (b) Particles traveling along field

lines on rational surfaces cannot cross from one line to another (red to blue).

The wire frame (black) depicts the flux surface...... 4

1.2 (a) shows equilibrium field lines (black) around the rational surface (red). (b)

shows the existence of a tearing mode. (c) indicates the flattening of the pres-

sure profile due to the mode...... 6

1.3 m/n = 2/1 island (a) surface current pattern in 3D on at the rational surface and

(b) radial magnetic field in 2D measured at the wall...... 7

1.4 Schematic of common magnetic sensors [17] that picks up changes in magnetic

flux through the loop, and integrates in time to get the magnetic field...... 9

1.5 Arrays of poloidal and radial magnetic sensors installed on DIII-D. Figure from

[18]...... 10

1.6 Comparison of temperature profiles for O- and X-points passing in the view of

ECE diagnostic. Figure from [21]...... 13

1.7 Island O-points with lower temperature can be observed as the mode rotates in

front of ECE. Figure from [21]...... 14

iii 1.8 Internal I-coils and external C-coils built on DIII-D...... 15

1.9 Magnetic field of RMP applied by 3D coils on DIII-D, calculated on the q = 2

surface...... 16

2.1 The actual cross-section of a typical DIII-D plasma (right) and its circular

approximation (left). Depicted are the magnetic axis (cross), mode surface

(green), plasma separatrix (red), and the wall (black)...... 18

2.2 Example of mode locking to an aribitrary position, with no error field and RMP

applied. (a) shows the magnetic field at the wall is well-shielded until the mode

is nearly locked. (b) shows in more detail the phase of the mode around locking.

[23] ...... 23

2.3 Initially fast rotating modes slow around 20 ms, then aligns with error fields

of different amplitudes. Alignment of mode phase with error field is faster for

increasing strengths of EF...... 24

2.4 The relation between initial and final mode positions, for a given EF strength.

Standard deviation measures how much each EF strength affects the final mode

position...... 25

2.5 Standard deviation of the collection of final mode position, as described in

figure 2.4. It is decreased monotonically for increasing error field strength,

implying faster alignment...... 27

iv 2.6 Examples of successful (blue) and failed (red) entrainment cases, where a 5 cm

island comes to a full locking at approximately 335 ms. Dynamics begins to

differ at 400 ms, where a 3 kA (blue) and 0.5 kA (red) RMP is turned on and

start to ramp up in frequency. Plotted are (a) the frequency and (b) of the mode

and the RMP...... 29

2.7 (a) Critical entrainment frequency contour calculated from torque balance be-

tween wall drag and applied RMP, as a function of driven current and mode

width. (b) Examples of successful entrainment below the limit and failed en-

trainment above. [23] ...... 31

2.8 Similar to figure 2.7, the contour of critical entrainment for 2/1 islands using

DIII-D’s external correction coils...... 33

2.9 Torque contour of strength of steady state wall torque on a rotating island in

DIII-D...... 33

2.10 3D coils in J-TEXT...... 34

2.11 (a) Critical entrainment frequency contour and (b) steady state wall torque con-

tour for J-TEXT...... 34

2.12 3D coils in KSTAR...... 35

2.13 (a) Critical entrainment frequency contour and (b) steady state wall torque con-

tour for KSTAR...... 35

2.14 3D coils in NSTX-U...... 36

2.15 (a) Critical entrainment frequency contour and (b) steady state wall torque con-

tour for NSTX-U...... 36

v 2.16 Planned 3D coils in ITER, with 3 sets of 6 external coils (green) and 3 sets of

9 internal coils (blue)...... 37

2.17 Simulation for ITER of a 5 cm island (blue) following a 10 kA/turn static RMP

(red), advancing in steps of 60◦ every 3 seconds. There is an offset at each time

due to the residual error field of 1 G. The initial oscillations in mode phase are

simply an artifact of the magnetic field from the mode trying to penetrate the

wall, which was initialized at zero...... 38

2.18 Simulated preemptive feedforward entrainment at 5 Hz of a 5 cm island in

ITER, with the rotating 10 kA RMP in the external coils turned on from the

beginning. (a) shows the rotation frequency decrease and settle to 5 Hz, (b)

gives the details of mode phase behaviour around locking...... 40

2.19 Predicted critical entrainment frequency contours for ITER using (a) internal

ELM coils and (b) external correction coils...... 41

3.1 Comparison of (a) frequency and (b) phase of island dynamics with (blue) and

without (green) a preemptively applied rotating RMP of 3 kA. It can be seen

that, apart from some oscillations, the phase of the entrained mode (blue) fol-

lows that of the RMP (black) without locking...... 44

3.2 Comparison of (a) frequency and (b) phase of island dynamics with preemp-

tively applied rotating RMP of 0.5 kA, too weak to be able to entrain the mode.. 46

vi 3.3 Amplitude, frequency, and phase of the NTM, before and after locking. Note

that instead of dropping to zero, the frequency after locking is around 70 Hz

(with some noise), showing the island is locked to the applied RMP rotation

frequency...... 48

3.4 Nonuniform island rotation (black, solid) in response to an applied RMP rotat-

ing uniformly (red, dashed), within a single period. (a)-(b) Measurements in

the case of maximum and mean nonuniformity, respectively, as measured by

∆Φ˜. (c) Simulation illustrating reduced nonuniformity when switching from a

feedforward scheme (blue) to feedback control (black)...... 49

4.1 Basic schematic of a feedback controller depicting the controller, actuator,

plant, and sensor. An example for each in the mode control has been given. . . 52

4.2 A schematic diagram of the proportional-integral controller implemented on

DIII-D. The controller compares the inputs of requested ϕref and measured

ϕLM to determine the a correction phase ϕcorr. This ϕcorr is the angle by which

the phase of the applied RMP ϕRMP is advanced with respect to the mode phase

◦ ϕLM . The applied RMP phase ϕRMP has been limited to ϕLM 90 for maxi-

mum torque...... 57

4.3 Simulink model used to simulate mode response to various control schemes.

Labeled are the numerical model as plant, extraction of mode phase as observer,

error angle calculation and PI-algorithm as controller, and I-coil inputs as actuator. 58

vii 4.4 Simulink results of proportional-only control of mode phase for three different

Kp values of 0.5 (green), 1 (blue) and 2 (red). (a) shows the mode phase follow

the reference (dashed), with the lowest Kp=0.5 case following at a larger error

than the Kp = 1 case. A p-gain of 2 results in unstable behaviour. (b) plots

the error angle between reference and mode, it can be seen that the Kp=1 has

lowest error throughout the simulation...... 59

4.5 Simulink results of proportional-integral control, with Kp = 1,Ki = 4. Align-

ment of mode phase to reference phase improves as the integral terms comes

into effect, as can be seen by decreasing average error angle...... 60

4.6 The measured (black) and reference (red) phase of a locked mode, as gain is

varied every 100 ms. In both fixed and rotating reference cases, the mode phase

roughly follows the reference, with higher gain resulting in better alignment as

expected...... 63

4.7 Experimental and simulated tracking error (defined as discrepancy between ref-

erence phase and actual mode phase averaged over each 200 ms period) for (a)

fixed phase and (b) rotating phase references respectively...... 65

4.8 Simulated tracking error in a proportional only controller, as dependent on the

gain used, the requested frequency, as well as (a) the island width and (b) posi-

tion of the q = 2 surface. Three contours are presented in each panel, where the

colours indicate levels of tracking error in each case and thickness differentiates

the scenarios...... 68

viii 5.1 Deposition location is where the EC beams intersects the 2nd harmonic of the

electron cyclotron frequency, shown in a cross-section of DIII-D, ...... 70

5.2 Schematic illustration of a population of electrons moving in velocity space

due to EC heating by (a) the Fisch-Boozer and (b) Ohkawa effects. Steady state

electron density in velocity space are shown in colour for (a) Fisch-Boozer and

(b) Ohkawa, where warmer colour indicate higher density. Figure from [54]. . . 71

5.3 Plots of measured mode amplitude, in black, and electron cyclotron power, in

blue. A coherent average over 25 periods is shown for each shot on the right,

where the dotted lines mark one standard deviation away from mean black line.

A sinusoidal fit to each averaged mode amplitude is shown in red. (a) Mode

amplitude is increased when ECCD is thought to be deposited in the O-point,

against expectation. (b) X-point deposition correlates with mode amplitude

decrease. (c) Mode appears to first grow, then be suppressed as deposition

transits from O- to X-point, while (d) the deposition transition from X- to O-

point has oppositve effect...... 75

5.4 The radial and toroidal deposition location of ECCD in shots (a) 166566 and

(b) 166567, based on post-shot analysis from EFIT (mode location), TORAY-

GA (ECCD location), and magnetics (island width). The solid box shows the

locations near the island where EC power was deposited, which spreads over

the flux surface due to parallel transport, as depicted by the shaded region.(c)

shows the current drive distribution in terms of perturbed flux surface for 50%

duty cycle deposition centered on the O-point toroidally...... 77

ix 5.5 Experimental and calculated MRE growth rates as a function of island width,

where the MRE includes a term for ECCD with actual misalignment and ex-

pected result assuming perfect alignment...... 78

5.6 Contours of ECCD deposition efficiency in terms of relative island size, radial

misalignment, and duty cycle being (a) full-on, (b) 50% centered on O-point,

and (c) 50% centered on X-point. Here, wisland is the island half-width, wECCD

is the FWHM of the gaussian current drive profile, and misalignment is the

absolute difference between peak CD location and the q = 2 surface. The white

regions in panels (b) and (c) mark approximately the experimental values for

shot 166566 and 166567 respectively...... 81

5.7 (a) shows the instantaneous efficiency K1 for current deposited at a particular

helical angle, for both radially aligned and misaligned case. (b) and (c) depicts

a toroidally localized slice of current deposited in the O- and X-points of the

island, respectively, where the solid box shows actual deposition location (5%

duty cycle used for clarity), and the shaded region shows the effect of current

being spread over the associated flux surfaces by fast parallel transport. . . . . 83

x List of Tables

2.1 Parameters of other devices studied, where wall time and toroidal field factor

into wall torque, control coils determine RMP torque, and typical density gives

the mass of the island...... 32

xi Acknowledgements

First and foremost, I would like to thank my advisor, Francesco Volpe, who has been under- standing and supportive through difficult times. His guidance has proven to be invaluable in my growth as a scientist, and I have learned a lot in both physics and life. I would also like to thank him for sponsoring my attendence to the 2015 Carolus Magnus Summer School which were an enjoyable two weeks in Leuven, Belgium, as well as a US-Japan workshop and ITPA working group meeting for an unforgettable week in Japan.

I would like to thank Kenneth Hammond and Michel Doumet, who were both supportive and helpful as I adapted to a busy first year in grad school. A further thank you to Ken and other Columbia plasma students (and one staff) for helping me prepare for oral qualifiers taken earlier than usual.

Many thanks to Ryan Sweeney, with whom I studied for the written qualifiers in New

York, then helped me settle into San Diego. We shared an office for 2.5 years there, during which various discussions often led to impromptu digressions, but occasionally did offer new insights.

I thank Erik Olofsson, whose work I have inherited, for his time spent explaining even the most minute aspects to me.

I would also like to thank Rob La Haye and Edward Strait for their patience and men-

xii toring during my time at General Atomics. I have become a better scientist with their assistance.

Lastly, I give thanks to all my friends, family, and colleagues, both local and long distance, who have supported me in many ways during these years. Especially my girlfriend,

Joy, who have accompanied me through difficult times, always encouraging me to be the best I can be. Thank you.

xiii Chapter 1

Introduction to locked modes

1.1 Background

Magnetic confinement is one of the most promising methods to achieve sustained thermonu- clear fusion for energy production. A strong magnetic field can bind the charged particles of a plasma to travel along the field line, effectively confining the particles to a small area in two of the three spatial dimensions. Successful confinement is achieved by maintaining closed, nested flux surfaces that have layers of plasma with increasing temperature and pressure towards the center. This gives sufficient time for some of the ions to attempt the fusion reaction before they can escape.

A tokamak is a magnetic confinement device characterized by a strong toroidal plasma current that generates a poloidal magnetic field. The combination of the poloidal field with an externally imposed toroidal field results in a net helical magnetic field, which is necessary for stable operation. The requirement of a plasma current in order to maintain confinement also implies that, if the current is disturbed, confinement may be lost.

Disruptions—the sudden and complete loss of plasma current and confinement— have long been a challenge for tokamak devices [1, 2]. The plasma current is a source of free energy for instabilities in the plasma, yet is required to maintain confinement. This can

1 lead to the uncontrolled release of stored energy in a disruption event, typically ending the discharge. These events violently exhaust the stored energy into the surrounding vessel, potentially causing significant damage. While there are different causes of disruptions, a study has found that a large majority of disruptions at JET are immediately preceded by a locked mode [3]. Another database study at DIII-D found more than 18% of disruptions were caused by 2/1 locked modes with rotating precursors [4], with an even greater fraction if born locked modes were included.

This thesis focuses on controlling these locked modes, with the goal of preventing them from causing a disruption.

Bootstrap current

A figure of merit to measure confinement is the ratio of plasma pressure to the magnetic pressure < p > β = 2 B /2µ0 where < p > is the volume-averaged plasma pressure and B is the total field strength [5].

As both the toroidal field created by external coils and poloidal field resulting from driven plasma current are sustained by external input into the system, they can be considered the

“cost”. The fusion power produced is dependent on the density and temperature of the confined plasma, so the pressure is a measure of the “benefit”. Thus, a high value of β is desired for peak fusion reactor performance.

A large gradient in pressure must exist within the plasma somewhere between its edge, where the pressure is close to zero, and its core, where a high pressure is maintained. This

2 pressure gradient results in the bootstrap current as follows. First, recall the magnetic mirror concept in which particles can be bounced back when moving from an area of low field to high field. When the magnetic field is enclosed into a torus, its strength decays like

1/R, with R being the major radius. Thus the outboard side of the tokamak has a weaker magnetic field, and is called the low field side (LFS). As particles move along the field line towards the inboard high field side (HFS), some of them do not have sufficient parallel kinetic energy to overcome the magnetic mirror effect. These trapped particles are bounced back, and a banana orbit is the result of the combination of particle trapping by changing field strength and vertical drifts by field curvature. Particles in a banana orbit on a pressure gradient will preferentially collide and transfer momentum in one direction, creating an additional current in the co-Ip direction. This is the bootstrap current that is generated by high pressure gradient.

1.2 Tearing modes

Nature has a tendency to seek the lowest energy state in a system. In the case of hot, dense plasmas with considerable energy confined within, various instabilities may arise, usually causing a degradation in confinement [6]. In particular, a non-ideal magnetohydrodynamic

(MHD) phenomenon known as tearing modes are prevalent in high energy plasmas. This section introduces their possible origins and expected behaviours, why we wish to avoid or suppress them, and some basic methods of measuring and controlling them.

3 (a) (b)

Figure 1.1: (a) A single magnetic field line (red) on the m/n = 2/1 surface closes upon itself after 2 toroidal transits and 1 poloidal transit. (b) Particles traveling along field lines on rational surfaces cannot cross from one line to another (red to blue). The wire frame (black) depicts the flux surface.

Origin of m/n = 2/1 tearing modes

Neglecting second order effects of drifts and diffusion, charged particles are bound to travel along magnetic field lines. Successful confinement is achieved by maintaining closed, nested flux surfaces upon which pressure and current are constant by fast parallel transport. Due to the existence of both toroidal and poloidal magnetic fields, all of the surfaces have helical field lines, allowing particles to simultaneously traverse in both directions as they move along the equilibrium field lines. The majority of surfaces have field lines that are irrational: from any initial position, a single magnetic field line will pass arbitrarily close to all other points on the surface if traced indefinitely. Thus, particles on these surfaces will eventually reach all points on the surface by simply traveling along the magnetic field. However, there exists a set of surfaces where the field lines will close upon themselves, “biting their own tails”, after m toroidal transits and n poloidal transits, where m and n are integer numbers. These are known as rational surfaces.

Particles traveling along field lines on rational surfaces are not able to reach the entire

2D surface, but are instead confined to the helical path. Figure 1.1(a) shows a magnetic

4 field line on the m/n = 2/1 surface, where m and n are also the poloidal and toroidal mode numbers respectively, named for how many periods exist in each direction. Rational surfaces can be labeled by the ratio of toroidal and poloidal transits of the magnetic field lines: m aB q = = T n RBP

where a and R are the local minor and major radii, and BT and BP are the local toroidal and poloidal magnetic field strengths. Only surfaces for which q can be written as a rational fraction have the property of closed magnetic field lines.

A tearing mode (TM) is a perturbation that can arise on internal rational surfaces. Con- sider the two coloured filaments in figure 1.1(b), where it is possible for each field line to carry different currents. If this occurs, the perturbation can grow if the free energy in the equilibrium current profile is sufficiently large, known as a classical tearing mode [7, 8, 9], or if bootstrap current is significant, a small seed island can result in a neoclassical tearing mode (NTM) [10, 11, 12].

The perturbations in current carried by the TM modifies the equilibrium magnetic field to “tear open” a magnetic island, as shown in figure 1.2. Particles can then travel along the perturbed flux surfaces depicted in figure 1.2(b). This can equalize the temperature, density, and pressure across the island, causing a degradation in confinement, shown in figure 1.2(c).

With typical modern tokamaks being stable to classical TM [13], neoclassical tearing modes (NTMs) are the focus of this study. To form a NTM, a seed island (on order of cm) is required to create a local flattening in the pressure profile. This removes the bootstrap

5 Figure 1.2: (a) shows equilibrium field lines (black) around the rational surface (red). (b) shows the existence of a tearing mode. (c) indicates the flattening of the pressure profile due to the mode. current, which reinforces the original current perturbation, further destabilizing the island.

Once formed, NTMs can grow to be large, causing significant transport of heat and particles out of the core, sometimes leading to a disruption [14] as described earlier.

Thus, it is important to have means of avoiding NTMs, suppressing their growth, or preventing them from causing disruptions in present and future devices.

Mode locking

In addition to growing in amplitude, NTMs will also decrease in rotation velocity. At the time of formation, NTMs typically rotate toroidally at a frequency near that of the local plasma rotation. However, the non-axisymmetric current generates a non-axisymmetric magnetic field as depicted in figure 1.3, As the island rotates, so does this field, thus induc- ing a current in the conductive vacuum vessel wall by Faraday’s law. This induced current in the wall has a magnetic field that drags upon mode, siphoning angular momentum from the NTM. The interaction between the induced currents and the mode results in a decrease

6 (b) Radial Field at Wall [G] 3 2 2 θ 1 1

0 0

−1 −1 Poloidal Angle −2 −2

−3 0 2 4 6 Toroidal Angle φ

Figure 1.3: m/n = 2/1 island (a) surface current pattern in 3D on at the rational surface and (b) radial magnetic field in 2D measured at the wall.

in mode rotation and eventually completely locking to rest in the lab frame. This is one origin of a locked mode (LM) [15] and is the type of LM studied in this thesis. Another source of LMs is for them to be born-locked, where the perturbation is already non-rotating when it first appears. Whatever the cause, these LMs can be treated equally once they appear.

Locked modes often appear in high-β plasmas, and is the cause of a significant fraction of disruptions in DIII-D [4] and JET [3]. It is also predicted to be a challenge for ITER

7 [16].

In most cases, it is preferable to force an island to maintain its rotation than allowing it to lock. In terms of stability, a rotating mode can benefit from stabilizing effects of plasma rotation and conducting wall. As for position, modes near locking tend to align themselves with the error field (EF) as a compass needle aligns with an external magnet.

This may prevent the use of toroidally localized current drive or diagnostics, which are essential to studying LMs. Lastly, a disruption caused by a LM would have a toroidally localized exhaust of heat and particles, possibly damaging the first wall or the divertor.

Based on a study of locked modes with rotating precursors, m/n = 2/1 NTMs almost always become locked or nearly locked before some can grow in amplitude and cause a disruption [4]. The study of controlling LMs is essential to continue development in the tokamak path to fusion, whether it is to restore confinement of the plasma, prevent a disruption or, at a minimum, mitigate any possible damage to the machine.

1.3 Measuring the mode

A tearing mode, whether locked or rotating, is a helical perturbation in local current, temperature, and pressure profiles. Attempts to characterize the mode must then look for these perturbations.

Magnetic measurement

Magnetic fields can be either be measured directly by a Hall probe, or indirectly through flux loops. The latter is more commonly used in fusion applications as they are less susceptible

8 Figure 1.4: Schematic of common magnetic sensors [17] that picks up changes in magnetic flux through the loop, and integrates in time to get the magnetic field. to neutron damage. When a closed loop of wire experiences a change in magnetic flux, an electromotive force is generated in the loop according to Faraday’s law:

∫ d V = −N B · dA loop dt

where Vloop is the voltage measured in the loop, N is the number of turns in the loop, and the integral is of the dot product between magnetic field passing through the loop and its area. Integrating the voltage in time provides a magnetic field strength reading relative to a baseline, to be discussed later. Figure 1.4 shows a few possible ways a flux loop can be constructed, the field measured is dependent on the orientation of the loop.

The m/n = 2/1 perturbed current produces a 2/1 magnetic field that can be measured by sensors external to the plasma. DIII-D has a large suite of magnetic sensors, as shown in figure 1.5, which has been used throughout this work. The external saddle loops, which measure the radial field, are used primarily to measure features with low n, as is the case here. The poloidal field measurements come a n = 1 fit of a toroidal array of Mirnov probes,

9 Figure 1.5: Arrays of poloidal and radial magnetic sensors installed on DIII-D. Figure from [18].

located inside the vessel. The measured amplitude and phase of the mode were corroborated between measurements of the perturbed poloidal and radial fields. Further discussion of the magnetic sensors on DIII-D and their capabilities to measure 3D phenomenon can be found in [18, 19].

The amplitude and phase of the n = 1 signal can be extracted from the set of six saddle loops as follows. First, signals from saddle loops that are 180◦ apart are subtracted to remove all even n measurements. This gives three signals of ESLD079, ESLD139, and

ESLD199, where ESLD stands for external saddle loops differenced, and the numbers indicate the toroidal position of the center of each loop in degrees. Let ⃗x be sum of perturbations,

10 such that

⃗x = x1,ccos(ϕ) + x1,ssin(ϕ) + x3,ccos(3ϕ) + x3,ssin(3ϕ) + x5,ccos(5ϕ) + ...

where numerical subscripts indicate toroidal mode number n, and subscripts s or c refer to the sine or cosine components respectively. The set of three signals can determine both the sine and cosine components of the n = 1 perturbation, as well as one component of

◦ the n = 3. Let us choose ϕ0 = 139 as the center of coordinate system such that the three

ESLD signals are located at −60◦, 0◦, and 60◦ respectively, and the sine component of the n = 3 signal is not measured in this coordinate system. With the assumption that the n = 5 component, with a period of 72◦, will have a negligible averaged signal on a sensor with toroidal spread of 60◦, the measured signal on ESLD079 can be written as

◦ ◦ ◦ y079 = x1,ccos(−60 ) + x1,ssin(−60 ) + x3,ccos(3(−60 )).

Similarly, the set of three signals can be written as

      y  cos(−60◦) sin(−60◦) cos(3(−60◦)) x   079    1,c             y  =  cos(0◦) sin(0◦) cos(3(0◦))  x  .  139    1,s       ◦ ◦ ◦ y199 cos(60 ) sin(60 ) cos(3(60 )) x3,c

11 Then the components can be obtained by simple inverse matrix

      √ −1 x  0.5 − 3/2 −1 y   1,c    079             x  =  1 0 1  y  .  1,s    139    √    x3,c 0.5 3/2 −1 y199

Lastly, the amplitude and phase of the n = 1 radial magnetic field is given by

√ 2 2 ◦ B1 = x1,c + x1,s, ϕ1 = arctan2(x1,s, x1,c) + 139

where arctan2 is the arctangent function with output between (−π, π], and the addition of

139◦ is to put the measured phase back into machine coordinates.

Temperature measurement

The perturbations of the magnetic island to the temperature profile can be measured by the electron cyclotron emission (ECE) diagnostic. As the name suggests, ECE consists of electromagnetic radiation emitted by the electrons when undergoing gyro-motion around the magnetic field. The fundamental frequency of the emission is equal to the cyclotron frequency ωce = eB/me where e is the elementary charge and me is the mass of the electron. The local magnetic field strength B varies inversely to the major radius for different radial positions, allowing a multi-channel system tuned to different frequencies to collect cyclotron emission from different points and create a radial profile. The spatial resolution of ECE are limited by the selected frequencies of the diagnostic, sensitive to local magnetic field strength, and are typically on the order of 2 cm. The local electron temperature can

12 Figure 1.6: Comparison of temperature profiles for O- and X-points passing in the view of ECE diagnostic. Figure from [21]. be calculated from the intensity of the radiation recorded in each channel.

The ECE diagnostic on DIII-D gives a toroidally localized measurement of the electron temperature profile across the midplane [20]. Digitized at 20 kHz, it provides a reliable passive measurement that can detect the existence of NTMs.

If the O-point of the island is located in view of the diagnostic, a flattening of the temperature profile near the rational surface can be observed. Figure 1.6 shows a comparison of the O- and X-point profiles as measured by the ECE. For rotating islands, the ECE can detect spots of slightly lower temperature associated with the island O-point, as seen in figure 1.7.

While DIII-D can also measure temperature through the Thomson Scattering (TS) diagnostic, it can often be unreliable for the purposes of detecting locked modes. This is due to the underlying principles of TS, which takes a measurement by actively firing a laser through the plasma, and observing the frequency broadening of the light scattered off

13 Figure 1.7: Island O-points with lower temperature can be observed as the mode rotates in front of ECE. Figure from [21]. the beam. The ensemble of lasers is fired in bursts with long waiting periods in between.

Thus, while each measurement is temporally resolved to the laser pulse length, the off-time between pulses prevents detailed analysis of the time-evolution of these profiles.

1.4 Mode position control

After successfully measuring the position of LMs, the next step is to be able to control it.

The radial location of the mode is centered on the rational-q surface, which is dependent on the global current profile. On the other hand, the toroidal phase of a locked island is determined by the error field. Thus, if one were to artificially create an error field in the plasma, it can be used to guide the position of LMs.

A resonant magnetic perturbation (RMP) can be generated by driving current through non-axisymmetric coils, also known as 3D coils. DIII-D has two sets of such coils: two rows of six coils internal to the vacuum vessel (I-coils) and one row of six correction coils

14 Figure 1.8: Internal I-coils and external C-coils built on DIII-D. external to the vessel (C-coils), as depicted in figure 1.8.

For maximum effectiveness, the RMP should attempt to match the decomposed Fourier components of the island, as torque between the two can be calculated through a surface integral of cross-product of island current and local RMP field. The C-coils can wired in a simple n = 1 scheme, with coils on opposite sides of the machine carrying opposite current.

While each row of I-coils is also wired in n = 1, the relative phasing between the two rows has a big effect on the final RMP applied. It was found that a relative phasing of 240◦ best couples to left-handed plasmas on DIII-D [22] due to helicity.

When locked or nearly locked, the magnetic island will always attempt to align itself with the net sum of the EF and RMP. If static RMP was applied, the mode would simply lie somewhere between the two. However, in the case of a rotating RMP, the mode may follow the rotation and be entrained, if the necessary conditions discussed in the next chapter are met. Thus, rotating RMP can be used to entrain LMs, partially regaining the benefit of

15 (a) I−coils magnetic field [G] (b) C−coils magnetic field [G] 3 4 3 8 6 2 2

θ 2 θ 4 1 1 2 0 0 0 0 −2 −1 −1 −2 −4 Poloidal Angle Poloidal Angle −2 −2 −6 −4 −8 −3 −3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Toroidal Angle φ Toroidal Angle φ

Figure 1.9: Magnetic field of RMP applied by 3D coils on DIII-D, calculated on the q = 2 surface. rotation on mode stability and control.

1.5 Outline of the thesis

The thesis is organized as follows. Chapter 1 has presented the challenge of a particular type MHD instability, the locked mode, and some of its features. Chapter 2 introduces the numerical model used to predict mode rotation dynamics, as well as select simulation results. The remaining three chapters combine simulation with experimental results in different areas, starting with feedforward position control in chapter 3. The development of finer control through a feedback algorithm, and its implementation on DIII-D, are discussed in chapter 4. Lastly, chapter 5 gives an example of application of position control: the synchronization with local current deposition for mode amplitude suppression.

16 Chapter 2

Simulation of mode dynamics

This portion of the thesis succeeds the work of Olofsson [23] regarding a numerical model used to describe the dynamics of a tearing mode. More specifically, this nonlinear model calculates the interactions between (1) a fixed-width tearing mode, (2) resonant magnetic perturbations (RMPs) applied by 3D coils, as described in the previous chapter, and (3) a conducting wall. While the model was written for arbitrary poloidal and toroidal mode numbers m/n, only 2/1 modes will be studied in DIII-D and other devices, as it is ubiquitous in many of the machines presented here, and is expected to be a major concern for ITER.

2.1 Numerical model

Let us first lay out the groundwork of the numerical model and the assumptions used. The experimental cross-section of the plasma, with elongation and triangularity, in toroidal geometry has been reduced to a circular cross-section in linear periodic geometry, as depicted by figure 2.1. That is, the torus has been “unwrapped” into a periodic cylinder with length

L = 2πR where R is the major radius of the tokamak, and toroidal angle ϕ is now measured by the linear coordinate z along the axis of the cylinder, where ϕ = z/R. This geometry allows analytical solutions to represent the perturbed quantities.

The minor radii are ordered as follows: 0 < s < a < b < w < c, where, starting from

17 2

0 Z (m)

Mag. Axis −1 q = 2 surface Separatrix Vacuum Vessel

−2

−3. 0 −1. 5 0. 0 1. 5 3. 0 R (m)

Figure 2.1: The actual cross-section of a typical DIII-D plasma (right) and its circular approximation (left). Depicted are the magnetic axis (cross), mode surface (green), plasma separatrix (red), and the wall (black). the magnetic axis at r =0, s is the minor radius of the q = 2 surface, a is the minor radius of the plasma separatrix, b is the location of the internal coils, w is the radius of the conducting wall, and c is the location of the external coils. The values of s, a, and w in circular cross- section were evaluated by a poloidal average of the actual cross-section. The values of b and c were obtained by solving for the locations of the internal and external coils in this geometry with the magnetic axis as the origin, then evaluating average effective radius of the coil sets. The n = 1 wiring of the coils was discussed in the previous chapter, and for the purposes of this section, it is assumed that the power supplies are capable of delivering the maximum rated current at all frequencies of interest.

18 As described in the previous chapter, the 2/1 mode can be considered as a thin sheet of current on the mode surface. The distribution of current in this sheet is sinusoidal in the helical angle ξ = mθ + nϕ, where θ and ϕ are the poloidal and toroidal angles.

When adapted to this geometry, this current density can be written as Js = |Js|exp(iξ) =

|Js|exp(i(mθ + kz)). Here k = n/R is the toroidal wave number. The mode current density Js is a vector quantity representing the amplitude and phase of the mode. Other currents and magnetic fields will also be introduced in this phasor notation later. The perturbed current carried by the mode is one component of the J × B torque for rotation (or equivalently, linear acceleration in z direction), where B is the magnetic field at the mode location. The inertia associated with the mode is provided by the mass of the plasma partly frozen-in within the island.

The plasma response to the mode rotation, induced currents in the wall, and applied

RMPs have been neglected for now. The vacuum solutions of the perturbed and applied magnetic fields are assumed for everywhere outside the surfaces of interest. In this cylindrical geometry, modified Bessel functions appear in these solutions.

The wall of the vacuum vessel is modeled under a thin wall approximation: the magnetic field is constant throughout the radial extent of the wall at any given time. While the effect of the graphite tiles is significant when the mode is rotating in the kHz range, it can be neglected for entrainment studies of interest at frequencies of up to a few hundred Hz [24].

With the above information, a time-dependent model that predicts how each quantity will change as a function of the present state was created. The system of ordinary differential equations [23] describing (1) the perturbed radial field at the wall, (2) mode rotation, and

19 (3) mode amplitude and phase can be written as:

˙ τBw = −Bw + αsJs + αbJb + αcJc (2.1a)

ω˙ = β1(Js × Y ) (2.1b)

˙ Js = iωJs (2.1c)

and

Y = γ1Bw + γ2Jb + BEF . (2.2)

Equation 2.1a describes the changes to the magnetic field at the wall Bw due to resistive decay of induced shielding currents, as well as how they are induced by the mode and 3D coils. Here τ is the wall time for the 2/1 mode, the α are geometric parameters given later, and Jb and Jc are the currents from the internal and external coils, respectively. Equation

2.1b is the angular acceleration equation, which describes the change in rotation frequency

ω, where β1 is inversely proportional to the moment of inertia, and Y , given by equation

2.2, is the effective magnetic field at the q = 2 surface. Lastly, equation 2.1c rotates the

iωt mode, and is the consequence of Js ∝ e , where island width w evolves negligibly during a single rotation period 2π/ω.

Equation 2.2 sums the contributions of the magnetic field at the wall, contribution from the internal coils, and the error field BEF at the mode surface.

Analytical expressions for the parameters in equation 2.1 are given below, where kx =

| | | | ′ · ′ · k x = n x/R, and Im( ) and Km( ) are modified Bessel functions of the first and second

20 kind, respectively, differentiated with respect to their arguments.

− 2 ′ ′ αs = k sIm(ks)Km(kw) (2.3a)

− 2 ′ ′ αb = k bIm(kb)Km(kw) (2.3b)

− 2 ′ ′ αs = k cIm(kw)Km(kc) (2.3c)

′ ′ − Im(kw)Km(kw) τ = τw 2 (2.3d) 1 + (m/kw)

′ Im(ks) γ1 = ′ (2.3e) Km(kw) I′ (k ) − 2 m s × ′ ′ − ′ ′ γ2 = k b ′ (Im(kw)Km(kb) Im(kb)Km(kw)) (2.3f) Im(kw) n β1 = 2 (2.3g) 2µ0R wρ˜

| | × 2 Js = C1 wisland (2.3h)

−(1/16) |m|Bθ(s) C1 = 2 ′ ′ (2.3i) k sIm(ks)Km(ks) sLq(s)

τw = µ0σdw (2.3j)

Here, γ1 and γ2 are geometric factors needed to account for magnetic field decay. ρ˜ is the local mass density, wisland is the island width, Bθ(s) is the local poloidal magnetic field, and

Lq(s) = q/(dq/dr)|s is the lengthscale of the q-profile evaluated at the rational surface. The nominal expression for the wall-time τw includes the conductivity of the material σ and wall thickness d. However, a prescribed value of τw = 12 ms better matches the experimental observations of the 2/1 mode wall time closer to τ ≈ 3 ms [24].

21 The viscous effects between the plasma and the island contribute a torque that has been neglected from this model. During fast rotation, this viscous torque plays an important role in the bifurcation of island dynamics in mode locking (low-slip) and unlocking (high-slip) behaviour [25]. Here “slip” refers to the difference in frequency between mode and induced or applied currents. However, this thesis focuses on the last stage before locking, long after the bifurcation has occured and mode rotation is low. Studies of mode entrainment with applied RMPs are at modest rotation frequencies (compared to the inverse wall time), also in the slow-slip branch. At these low frequencies, the viscous torque tends to be small and can be neglected [24] when compared to the electromagnetic torques described below.

The torque contribution by neutral beam injection to the island was also excluded from this model. An estimate of this torque, based on the cross-sectional area of the island, found it to be at least an order of magnitude smaller than the expected electromagnetic torques from the wall or applied RMPs.

2.2 2/1 Mode behaviour in DIII-D

With this simple model, we can begin to predict the behaviour of a 2/1 island. As described in the previous chapter, one source of 2/1 locked modes (LM) is a NTM initially rotating at approximately the rotation frequency of the plasma. This precursor will induce currents in the wall, interact with them, slow down as a result, and become locked, as is shown in figure 2.2.

22 Figure 2.2: Example of mode locking to an aribitrary position, with no error field and RMP applied. (a) shows the magnetic field at the wall is well-shielded until the mode is nearly locked. (b) shows in more detail the phase of the mode around locking. [23]

This motion can be described by:

dω I = T (2.4) dt wall

where I is the moment of inertia and ω is the rotation frequency of the island. Twall is the wall torque on the mode, due to its interaction with the eddy currents that it induces in the wall. Figure 2.2(a) shows the result of the mode slowing down from its initial rotation frequency of 5 kHz, according to equation 2.4. At a sufficiently low frequency (around the

23 Modes aligning to Error Fields 180 No EF 0.1 G EF 120 0.5 G EF EF Phase

−60 Mode Phase [deg]

−120

−180 0 100 200 300 400 500 600 Time [ms]

Figure 2.3: Initially fast rotating modes slow around 20 ms, then aligns with error fields of different amplitudes. Alignment of mode phase with error field is faster for increasing strengths of EF.

inverse wall time of roughly 50 Hz), the radial field penetrates the thin wall. Figure 2.2(b) shows that the mode locking to an arbitrary position, as there is no error field (EF) in this case.

Adding the error field torque TEF to the equation above, we have:

dω I = T + T (2.5) dt wall EF where the error field torque is only dependent on the position of the EF and the present position of the mode, regardless of its history.

Figure 2.3 shows the results of another simulation, now based on equation 2.5. The initially rotating modes slow due to wall drag, and eventually become aligned to EFs of

24 phase of 0◦ and varying amplitudes. In the case of no error field (black trace) and applied

RMP, the mode has no preferential location of locking. When there is an EF (blue and green traces) and no other perturbations, however, the mode will always align towards the position of the EF over time.

The time required for a given island to align with the EF is dependent on the strength of the EF. If infinite time is given, all modes will eventually align with the existing error field.

However, it would be unrealistic to allow the simulation to continue for durations longer than the plasma discharge itself, so a “final” time has to be dictated. Here, let the effect of error fields on mode locking be measured through the modes’ final positions, taken at 30 ms

(10 wall-times) after the island phase stops oscillating and moves monotonically towards the EF phase.

Sample Standard Deviations of final mode position 180 No EF, St. Dev. = 108◦ 0.1 G EF, St. Dev. = 82.6◦ 0.5 G EF, St. Dev. = 38.9◦ 120 EF Phase

−60 Final Mode Phase [deg]

−120

−180 −180 −120 −60 0 60 120 180 Initial Mode Phase [deg]

Figure 2.4: The relation between initial and final mode positions, for a given EF strength. Standard deviation measures how much each EF strength affects the final mode position.

25 Figure 2.4 shows the relation between initial and final mode positions, for different

EF strengths. In a simulation without error field, the slowing of the mode is completely axisymetric, and the initial and final positions have a linear relationship. However, the existence of even a small EF changes the rate of kinetic energy removal from the mode by the wall, due to the periodic acceleration and deceleration of the mode as it passes the

EF. Finally, when sufficient energy has been resistively dissipated in the wall, such that the mode can no longer rotate past the potential well of the EF, it would slowly align itself with the EF.

For a given error field strength, the mode phase was initialized at different toroidal positions in figure 2.4, in 10◦ steps. A sufficiently strong EF will always align the mode to itself, while a weaker EF will allow some deviation away from its position (defined as 0◦).

The standard deviation of final mode phase was chosen to represent this effect. For larger

EF, one would expect a quicker alignment and therefore a smaller standard deviation, as observed in figure 2.5.

The field of an applied RMP at the q = 2 surface, for all intents and purposes, can be considered as another error field that can be controlled in amplitude and phase by the inputs. Thus, in the case of a static RMP with no intrinsic error fields present, one would expect the mode to align itself to the RMP over time. Finally, when both an EF and a static applied RMP exists, the mode will align with the total EF and RMP field. This is equivalent to a balance between the torques imparted by the EF and the RMP respectively:

0 = TEF + TRMP .

The more interseting cases arise when the applied RMP is no longer static, but rotating.

26 120

100

40 Standard Deviation [deg] 20

0 0 0.5 1 1.5 Error Field Amplitude [G]

Figure 2.5: Standard deviation of the collection of final mode position, as described in figure 2.4. It is decreased monotonically for increasing error field strength, implying faster alignment.

This then gives the equation of motion used in all time-dependent simulations:

dω I = T + T + T (2.6) dt wall EF RMP with

2 ′ ′ Twall ∝ |Js| F(ϕLM (t |0 ≤ t ≤ t)) (2.7a)

TEF ∝ |BEF ||Js|sin(ϕLM (t) − ϕEF ) (2.7b)

TRMP ∝ |BRMP ||Js|sin(ϕLM (t) − ϕRMP (t)) (2.7c)

These simplified equations emphasize the important relationships between the mode and each of the torques: the wall torque is proportional to the square of the perturbed current,

27 and is dependent on the history of mode rotation. The EF and RMP torques are very similar, in that each is proportional to the mode amplitude and the sine of the angle between them.

Let us now examine the behaviour of a locked mode in the presence of a rotating applied

RMP, and some intrinsic or residual EF. There are now two shielding currents induced in the wall, one from the rotation of the mode, and the other from a.c. currents driven in the control coils. These are correctly accounted for in the more detailed equation 2.1a. One of two results is possible: either the applied RMP is sufficiently strong to drag the mode to complete rotations at the applied frequency (successful entrainment), or the combination of wall drag on the mode and weakened RMP due to shielding prevents the mode from being entrained, usually resulting in some erratic behaviour.

Figure 2.6 shows two such cases, for a 5 cm island with no background EF. Two RMPs, of amplitudes 3 kA and 0.5 kA driven through the I-coils, are turned on at 400 ms and ramped up to rotate at 100 Hz. It can be seen that the low amplitude case (red) initially has the mode following the RMP (black dashed), but is unable to maintain entrainment as frequency is increased. Starting at roughly 520 ms, the mode continuously attempts to align with the RMP, fails to do so, and slips back to the next period. The successful entrainment scenario is shown in blue, where the frequency and phase of the mode closely matches that of the applied RMP.

28 (a) 800 Successful Entrainment 700 Failed Entrainment RMP frequency 600

500 RMP turns on

400

300

Frequency f [Hz] 200

100

−100 300 400 500 600 700 800 900 1000 Time t [ms]

(b) 180

120

60 [deg]

φ 0 Phase −60 RMP turns on

−120 Successful Entrainment Failed Entrainment RMP phase −180 400 450 500 550 600 Time t [ms]

Figure 2.6: Examples of successful (blue) and failed (red) entrainment cases, where a 5 cm island comes to a full locking at approximately 335 ms. Dynamics begins to differ at 400 ms, where a 3 kA (blue) and 0.5 kA (red) RMP is turned on and start to ramp up in frequency. Plotted are (a) the frequency and (b) of the mode and the RMP.

To quantify the entrainment capabilities of DIII-D, let us consider the case of a well corrected error field, such that the rotation of the mode is at a constant frequency [26]. This

29 steady-state scenario can then be written as:

0 = Twall + TRMP (2.8)

The balance of the wall torque and the RMP torque can then be used to predict the maximum frequency at which a mode can be entrained. The contour of this critical frequency is shown in figure 2.7(a), as a function of the current driven in the I-coils and the mode width. The shape of this contour comes from wall torque being quadratic in the perturbed current, while the RMP torque is linear. Figure 2.7(b) shows the successful (red) and failed (green) entrainment just below and above this limit, respectively. It is interesting to note that this limit can be transiently exceeded for a sufficiently fast ramp in frequency, at the cost of destabilized behaviour later (blue).

2.3 Comparison of 2/1 mode entrainment across

machines

This predictive model has been extended to estimate the entrainment capabilities for 2/1 islands on other machines. In particular, J-TEXT [27], KSTAR [28], and NSTX-U [29] were selected in an initial study. Table 2.1 records relevant parameters of these machines for comparison.

In each case, several machine-specific parameters in the code were replaced:

• 3D coils: Their geometry, number of turns, and proximity to the plasma or conductive

wall affects the applied RMP on the mode surface. Note that it has been assumed that

30 Figure 2.7: (a) Critical entrainment frequency contour calculated from torque balance between wall drag and applied RMP, as a function of driven current and mode width. (b) Examples of successful entrainment below the limit and failed entrainment above. [23]

the power supplies are sufficient to provide the currents at frequencies of interest.

• Vacuum vessel: Its location, thickness, and conductivity all factor into calculation

of wall torque and RMP shielding

• Equilibrium magnetic field: This determines the conversion factor between is-

land width and strength of the perturbed field, which factors into all electromagnetic

torque calculations.

• Typical density: This gives the mass associated with the island, and thus changes

the moment of inertia of the island.

31 Device Control coils used in simu- Major Wall Typical Typical density lation radius time toroidal [m−3] [m] [ms] field [T] DIII-D 1x6 external or 2x6 internal 1.72 3 1.86 2.2 × 1019 J-TEXT 3x4 internal 1.05 3.1 2.2 1 × 1019 KSTAR 3x4 internal 1.8 20 3.5 1 × 1020 NSTX-U 1x6 external 0.86 5 0.18 3 × 1019

Table 2.1: Parameters of other devices studied, where wall time and toroidal field factor into wall torque, control coils determine RMP torque, and typical density gives the mass of the island.

DIII-D

In figure 2.7, the contours of critical entrainment frequency were plotted as a function of island width and driven current, for the internal I-coils at DIII-D. This device also has a set of external C-coils with which n = 1 RMPs can be applied. The corresponding contours are shown in figure 2.8. When compared to the internal coils contours, the maximum frequencies are just slightly lower, for the same island width and coil currents. While the longer distance from the plasma and wall shielding reduces the effectiveness of the external coils, their larger size and higher turn number help to overcome these challenges, resulting in similar entrainment capabilities.

So far, it has been assumed that the rotation at constant frequency is a result of the dragging wall torque balanced by torque from the applied RMP. In reality, there can be other sources of torque, such as neutral beam injection, that serve to rotate the island, as long as the torque is received directly by the island. Figure 2.9 shows a contour of torque required to rotate an island at a given frequency. As expected, there is a peak at

32 C−coils Critical Entrainment Frequency [Hz] 5 175 130 175 130 140 4.5 150 120 110 140

4 120 150 100 3.5 90 110 150 130 140 120 80 3 130 70 110 140 60 2.5 100 50 120 130 90 2 120 80 40 110 70 30 1.5 110 60

Coil current [kA/turn] 100 100 50 20 90 40 1 90 80 70 30 80 60 20 0.5 50 70 3040 10 60 2010 2 4 6 8 10 NTM width [cm]

Figure 2.8: Similar to figure 2.7, the contour of critical entrainment for 2/1 islands using DIII-D’s external correction coils.

the inverse wall time, the frequency at which the drag from the wall torque is maximum. DIII−D Steady State Wall Torque [Nm] 200

1 180

0.05

0.5 160 2.5

140 0.25 1.5 0.01 120 0.15

1 3.5 100 3

0.05 80 0.5 2.5 2

0.15

1.5 60 0.25

0.01

Entrainment Frequency [Hz] 40 1 3.5 3 20 0.05 0.5 2.5 1.52 0.150.25 1 0.01 0.010.05 0.150.250.050.5 1 2 3 4 5 6 7 8 9 10 NTM width [cm]

Figure 2.9: Torque contour of strength of steady state wall torque on a rotating island in DIII-D.

33 J-TEXT

J-TEXT has three sets of four internal coils with small angular widths, as seen in figure

2.10. Compared with DIII-D, J-TEXT has a smaller major radius and a larger aspect ratio, resulting in a much smaller minor radius. As a result, the close proximity of the coils allows for a higher frequency of entrainment for similar steady state wall torques.

Figure 2.10: 3D coils in J-TEXT.

J−TEXT Critical Entrainment Frequency [Hz] J−TEXT Steady State Wall Torque [Nm] (a) (b)

5 450 700 600 550 500 0.5 500 600 400 0.01 550 0.25 350 600 0.15 4 400 550 300 500 250 0.05 500 1 450 3 350 450 500 200 400 400 350 0.5

0.01 0.25 400 450 300 150 0.15 250 300 2 2 1.5 400 200 100 1 350 350 200 0.05 150 Coil current [kA/turn] 300 3 1 300 250 50 0.5 2.5 200 100 100 Entrainment Frequency [Hz] 0.25 2 0.15 1.5 250 150 50 0.01 100 1 3.54 200 50 0.050.01 0.150.250.050.5 0.150.251.52.5132 2 4 6 8 10 2 4 6 8 10 NTM width [cm] NTM width [cm] Figure 2.11: (a) Critical entrainment frequency contour and (b) steady state wall torque contour for J-TEXT.

34 KSTAR

KSTAR is similar in size to DIII-D, it has a much longer wall time, leading to a peak in the wall torque at lower frequencies, but with similar values. The non-axisymmetric coils in

KSTAR [30] are a bit unusual in that each filament has an independent lead into and out of the vacuum vessel. For the purposes of this model, it is treated as three sets of four internal coils, as seen in figure 2.12. The numerous coils results in a similar entrainment contour as

DIII-D.

Figure 2.12: 3D coils in KSTAR.

K−STAR Critical Entrainment Frequency [Hz] K−STAR Steady State Wall Torque [Nm] (a) (b)

5 150 200

170 190 170 130 190

4 110 0.05 0.5

130 150 190 90 170 150 0.15 130 3 150 170 0.01 70 0.25

110 100 1

150 50 0.5 90 0.05 2 130 130 70 30 110 0.15 110 50 50 1.5 2 Coil current [kA/turn] 90 0.25 1 0.01 1 90 70 30 10 Entrainment Frequency [Hz] 0.5 4 3 3.5 70 50 2.5 30 10 0.05 1.52 50 10 0.01 0.15 0.250.05 0.50.151 0.25 2.51.50.53.53124 2 4 6 8 10 2 4 6 8 10 NTM width [cm] NTM width [cm] Figure 2.13: (a) Critical entrainment frequency contour and (b) steady state wall torque contour for KSTAR.

35 NSTX-U

NSTX-U is a spherical tokamak with a much smaller aspect ratio compared to the other tokamaks presented here, and typically operates with the q = 2 surface closer to the core.

As a result, the steady state wall torque on 2/1 islands on NSTX-U are smaller than in other machines (figure 2.15(b)). This implies that even the external coils can easily entrain the mode (figure 2.15(a)).

Figure 2.14: 3D coils in NSTX-U.

NSTX−U Critical Entrainment Frequency [Hz] NSTX Steady State Wall Torque [Nm] (a) (b) 5 400 450 500

0.01

0.001 0.025 0.015 4 300 400 350 400

0.0001 400 0.005 300 0.05 3 250

350 0.01 300 350 0.025 0.001 0.015 2 200 300 200 300

250 0.005 0.05 0.0001

Coil current [kA/turn] 250 250 100 1 200 0.01 0.025 150 Entrainment Frequency [Hz] 200 200 100 0.015 150 0.001 0.05 150 100 0.0001 0.0050.001 0.010.0150.0250.005 0.010.050.015 2 4 6 8 10 2 4 6 8 10 NTM width [cm] NTM width [cm] Figure 2.15: (a) Critical entrainment frequency contour and (b) steady state wall torque contour for NSTX-U.

36 2.4 Predicted entrainment capabilities for ITER

The predictive model for mode dynamics was adapted to ITER to estimate the entrainment capabilities for 2/1 islands. In its present design, ITER will have two sets of 3D coils: the internal three rows of nine ELM control coils [31] and the external three rows of six correction coils [32], shown in figure 2.16.

Figure 2.16: Planned 3D coils in ITER, with 3 sets of 6 external coils (green) and 3 sets of 9 internal coils (blue).

Continuing the representation used by Olofsson [23], the field generated by each set of coils was modeled by means of a phasor of appropriate, optimized for maximum coupling

(in vacuum) to the 2/1 island. For now, the simulation assumed that the power supplies used for these 3D coils are capable of delivering the necessary currents at low frequencies

(< 20 Hz).

ITER will also have a beryllium first wall and two layers of stainless steel vacuum vessel walls [33], treated here as the source of the braking torque on the island. Using a thin wall

37 180 Mode Applied RMP 120

0 Phase [deg] −60

−120

−180 0 3 6 9 12 15 18 Time [s]

Figure 2.17: Simulation for ITER of a 5 cm island (blue) following a 10 kA/turn static RMP (red), advancing in steps of 60◦ every 3 seconds. There is an offset at each time due to the residual error field of 1 G. The initial oscillations in mode phase are simply an artifact of the magnetic field from the mode trying to penetrate the wall, which was initialized at zero.

approximation, they are reduced to a simple time constant that affects the mode dynamics and applied RMP penetration. Other factors such as test blanket modules and divertor were not included at this stage.

A basic step reference simulation was performed as a first check, using a fixed-width

5 cm island, with peak current of 10 kA/turn driven in the external coils as the RMP, and a

1 G residual error field. The phase of the RMP is advanced by 60◦ every 3 seconds. Figure

2.17 shows the mode aligning to the RMP as expected, with an offset caused by the error field.

Feedforward preemptive entrainment (described in the next chapter) was simulated using the same conditions as above, except now with a rotating RMP that is applied before the

38 mode appears, meant to prevent locking by entraining the mode as it slows down. Figure

2.18 shows the mode slowing down and locking to the 5 Hz entraining RMP. As entrainment is approached, the phase of the mode follows that of RMP with some oscillations caused by the induced currents in the wall. After these currents have fully decayed, smooth entrainment is observed for t ≳ 10 s. Equation 2.6 was used in both time domain simulations, which includes the effects of error field and slow decay of shielding currents in the wall.

To predict the maximum entrainment frequencies possible in ITER, we searched for a steady-state torque balance between the wall drag and the applied RMP, neglecting error fields and possible time history effects. Equation 2.8 was used in this simplified scenario to calculate the critical entrainment frequency, dependent on island width and applied current.

Figure 2.19 suggests that small 2/1 islands can be entrained in the sub-10 Hz range.

Although an externally applied RMP must penetrate through the walls in order to affect the mode, the higher number of turns in the external coils balances out the disadvantage in location. As a result, external coils exhibit entrainment capabilities (figure 2.19(b)) similar to the internal coils (figure 2.19(a)).

39 (a) Simulated ITER preemptive entrainment 500

400

300

200

Frequency [Hz] Frequency 100

−100 0 5 10 15 Time [s] (b)

180 Mode Applied RMP 120

0 Phase[deg] −60

−120

−180

8 8.5 9 9.5 10 Time [s]

Figure 2.18: Simulated preemptive feedforward entrainment at 5 Hz of a 5 cm island in ITER, with the rotating 10 kA RMP in the external coils turned on from the beginning. (a) shows the rotation frequency decrease and settle to 5 Hz, (b) gives the details of mode phase behaviour around locking.

40 (a) ITER ELM Coils Critical Entrainment Frequency [Hz] 10 14

8 12

10 6 8

6 4

Coil Current [kA/turn] 4

2 2

0 5 10 15 20 25 30 35 40 NTM width [cm]

(b) ITER External Coils Critical Entrainment Frequency [Hz] 10 14

8 12

10 6 8

4 6

Coil current [kA/turn] 4 2 2

5 10 15 20 25 30 35 40 NTM width [cm]

Figure 2.19: Predicted critical entrainment frequency contours for ITER using (a) internal ELM coils and (b) external correction coils.

41 Chapter 3

Feedforward Control

The predicted behaviour of locked modes (LMs) aligning themselves with the applied resonant magnetic perturbation (RMP) has been studied extensively in both theory [34, 35, 25] and experiment on various devices including COMPASS-C [36], DIII-D [37], JET [38], and TEXTOR [39]. These early works described how a sufficiently strong applied RMP would drive an island within the plasma and examined the phase relation between the two.

More recent work at DIII-D deployed a rotating RMP to steer the phase of a m/n = 2/1

LM as a means of error field detection [40]. Regardless of the origin of magnetic islands, whether driven by RMP or resulting from internal instabilities in the current or pressure profile, they are expected to interact with the applied RMP in a similar manner.

In these instances, the currents in the non-axisymmetric coils were driven in feedforward. That is to say, the amplitude and phase of the applied RMP were programmed in advance, without having to consider how the plasma or the LM might respond. The advan- tage of a feedfoward scheme is simplicity: any applied RMP will always be able to affect the island’s position, though the degree to which this effect is possible is dependent on several factors including strength of RMP, mode amplitude, possible existence of error field, and whether the applied perturbation is static or rotating. For typical island sizes in DIII-D, a requirement on the amount of current drive can be estimated to ensure that the RMP is

42 the dominant electromagnetic torque influencing LM phase.

3.1 Preemptive entrainment

Preemptive entrainment is a novel feedforward strategy able to prevent complete mode locking in the lab frame. This technique is accomplished by applying a rotating RMP early in the current flattop period, before the mode appears. The amplitude of the applied RMP should be below the threshold to drive a tearing mode, yet be sufficiently large to entrain the naturally-occurring LM. When the m/n=2/1 rotating precursor forms, initially in the kHz range, and decelerates due to wall drag, the mode is expected to lock onto the existing rotating RMP instead of locking to the wall. In theory, the prevention of complete mode locking would limit the island growth, as it is partially stabilized by rotation in the presence of a conducting wall [41] and by rotation shear [42, 43, 44].

Mode behaviour in the preemptive entrainment scenario was simulated with the model described in the previous chapter. A saturated island with a fixed width of 5 cm was initialized to be rotating at 2 kHz. Also initialized at time zero is a 70 Hz rotating RMP, with

3 kA driven in the internal I-coils. Figure 3.1 shows a simulation of successful preemptive entrainment, where the phase of the mode locks onto that of the RMP when sufficiently decelerated. While the preemptively applied rotating mode does not change the slow-down time, the time taken for a mode to slow from 2 kHz to locked, it does alter the rotation on a sub-period timescale, causing an oscillation in the frequency during the first 100 ms of the simulation. This behaviour is not expected in actual experiment, where there would be some viscous-damping on the mode.

43 (a) 2000 3 kA Preemp. RMP 1800 No RMP RMP frequency 1600

1400

1200

1000

800

Frequency f [Hz] 600

400

200

0 50 100 150 200 250 300 350 400 Time t [ms]

(b) 180

120

60 [deg]

φ 0 Phase −60

−120 3 kA Preemp. RMP No RMP RMP phase −180 100 150 200 250 Time t [ms]

Figure 3.1: Comparison of (a) frequency and (b) phase of island dynamics with (blue) and without (green) a preemptively applied rotating RMP of 3 kA. It can be seen that, apart from some oscillations, the phase of the entrained mode (blue) follows that of the RMP (black) without locking.

44 The critical entrainment frequency calculated by torque balance, described in the last chapter, can be applied here to predict if steady-state entrainment is possible after the transient deceleration has passed. Figure 3.2 shows a case of failed entrainment, where the applied RMP of 0.5 kA is unable to sustain mode rotation. The mode attempts to align itself with the passing RMP, lagging too far each time due to drag from the wall, then repeats its attempt in the next period.

3.2 Experimental result

An experiment was performed with the preemptive entrainment technique at DIII-D. The discharges had a flat top plasma current of 0.9 MA and a toroidal field of 1.7 T, giving the plasma a safety factor of q95 between 4.8 to 5.8 during the flat top. This high safety factor was chosen to allow sufficient time to study the entrained island before it disrupts the plasma. After current flat top was reached at 700 ms, a RMP rotating at 70 Hz was applied starting at 1300 ms, with 4.3 kA of current in the internal or I-coils. An average of

3 MW of neutral beam power pushed the plasma into H-mode at approximately 1760 ms, after which NBI power was increased up to a peak of 9 MW. In the discharge shown in figure 3.3, the 2/1 NTM appeared at around 2392 ms, initially rotating at about 7 kHz.

Neutral beam power was reduced to roughly 5.8 MW near this time, which directly injected approximately 0.04 Nm of torque to the island, which can be neglected when compared to the electromagnetic torques on the order of 1 Nm. The neutral beam torque on the island was estimated based on ratio of the poloidal cross-sectional areas of the island and the plasma.

As the island rotation was reduced to near zero at 2514 ms, it locked to the 70 Hz rotat-

45 (a) 2000 0.5 kA Preemp. RMP 1800 RMP frequency 1600

1400

1200

1000

800

Frequency f [Hz] 600

400

200

0 50 100 150 200 250 300 350 400 Time t [ms]

(b) 180

120

60 [deg]

φ 0 Phase −60

−120 0.5 kA Preemp. RMP RMP phase −180 100 150 200 250 Time t [ms]

Figure 3.2: Comparison of (a) frequency and (b) phase of island dynamics with preemptively applied rotating RMP of 0.5 kA, too weak to be able to entrain the mode..

ing RMP without locking to the DC error field. To be more precise, the island tends to align to the vector sum of the rotating RMP and static error field. This explains why its rotation is non-uniform (figure 3.3(c)) and why the frequency oscillates around 70 Hz on a sub-period timescale (figure 3.3(b)). Note that the instantaneous rotation frequency plotted in figure

46 3.3(b) is defined as the time-derivative of the toroidal phase signal in figure 3.3(c). Also, the frequency oscillations can be so ample that the frequency can become negative (although only positive values are plotted here), i.e. the mode rotation can temporarily change direction (figure 3.3(c)) [45], and the rotation frequency can temporarily vanish, for a small fraction of the 14.3 ms rotation period. Four instances of vanishing rotation frequency can be noticed in figure 3.3(b), but they should not be interpreted as mode locking. Other shots from the same experiment exhibit similar behaviour.

As the applied RMP and the mode rotate at the same frequency (except for the oscillations just discussed), it was important to isolate the actual mode measurement and remove any coil-sensor pickup of the applied fields. This was realized by a.c. compensation [46].

In summary, this preemptive entrainment technique traded complete mode locking for a slowly rotating entrained island. This technique requires a rough prediction of when the mode might appear, and with what amplitude, based on previous discharges. This allows the rotating RMP to be applied only when needed, and with an amplitude just sufficient for entrainment, so that the RMP’s negative impact on confinement can be minimized.

3.3 Simulation of improved entrainment

By definition, a preemptive rotating RMP is applied before the rotating mode has appeared.

Therefore, preemptive entrainment can only be applied in feedforward as there is no signal to feed back on.

Even if the applied fields rotate uniformly, the island will very likely rotate non- uniformly, due to electromagnetic torques from residual error fields, fluctuating island

47 (a) 162991 40 40

30 Locking 30

20 20 (G) of 2/1 LM 10 10 r B (G) of 2/1 rotating NTM

B 0 0 2400 2450 2500 2550 2600 (b) 10 100 80 8 Locking 60 6 40 20 4 0 2 −20 Frequency (kHz) −40 −1*Frequency (Hz) 0 2400 2450 2500 2550 2600 Time (ms) (c) 180 180

90 90

0 0

−90 −90 RMP Phase (deg) Mode Phase (deg) −180 −180 2500 2520 2540 2560 Time (ms)

Figure 3.3: Amplitude, frequency, and phase of the NTM, before and after locking. Note that instead of dropping to zero, the frequency after locking is around 70 Hz (with some noise), showing the island is locked to the applied RMP rotation frequency.

widths, and changing plasma conditions. An extreme case and a typical example are illustrated in figures 3.4(a) and 3.4(b), respectively. Both cases are taken from the same experiment reported in figure 3.3, although at different times. Here the amount of deviation from uniform rotation is quantified by the root-mean-square of the difference between

48 Max ∆Φ˜ = 19.1 ◦ for period 41 Simulated Feedforward and Feedback Behaviour (a) (c) 180 180 Feedback ∆Φ˜ of 5.3 ◦ 90 Feedforward ∆Φ˜ of 10.2 ◦ 120 0 Ideal constant frequency

−90 Phase (deg) 60 −180 3095 3100 3105 3110 Time (ms) 0 ◦ (b) Mean ∆Φ˜ = 9.5 for period 26

180 Exp. Mode Phase Phase (deg) Ideal const. Freq. −60 90

0 −120 −90 Phase (deg)

−180 2880 2890 2900 −180 Time (ms) 0 2 4 6 8 10 12 14 Time (ms) Figure 3.4: Nonuniform island rotation (black, solid) in response to an applied RMP rotating uniformly (red, dashed), within a single period. (a)-(b) Measurements in the case of maximum and mean nonuniformity, respectively, as measured by ∆Φ˜. (c) Simulation illustrating reduced nonuniformity when switching from a feedforward scheme (blue) to feedback control (black).

the actual mode phase and a fixed-frequency trajectory. These r.m.s. phase-differences are evaluated over one period, and are denoted by ∆Φ˜. For the 76 periods in which the mode was entrained, the values of ∆Φ˜ ranged between 3.5◦ and 19◦ (figure 3.4(a)), with a mean of 9.5◦ (figure 3.4(b)).

It is interesting to note that, in the experiment, the phase deviation appears to be a mix of n = 1 and n = 2 perturbations, exemplified by the asymmetric deviations of the measured phase in black from the ideal red dashed trace. As both the mode and applied RMP are n = 1, this behaviour might be the result of an n = 2 error field interacting with the mode.

In the time-dependent simulation above, the dynamics of the island is governed by the equation of motion:

49 d2ϕ I = T + T + T (3.1) dt2 wall EF RMP where I and ϕ are the moment of inertia and toroidal phase of the island, respectively.

On the right-hand-side of the equation are the electromagnetic torques exerted by the wall, error field, and applied RMP onto the island. A few simplifying assumptions have been made: (1) no other tearing modes exist or can interact with the 2/1 mode of interest, (2) the plasma around the island has low rotation and is not viscously dragging on the island, and (3) torque from the neutral beams on the island is negligible. The assumption is valid for either balanced co- and counter-injection, or for beam deposition away from the island location.

In simulations of the preemptive feedforward entrainment, an uncorrected n = 1 error field of 0.5 G at the rational surface causes an n = 1 perturbation to the rotation with an average phase deviation of ∆Φ˜ = 10.2◦. This error field amplitude was chosen to reconcile ∆Φ˜ values from experiments and simulations. Simulations performed under the same plasma conditions and for the same error field indicate that the deviation is reduced to ∆Φ˜ = 5.3◦ when feedback is used (figure 3.4(c)).

50 Chapter 4

Feedback Control

Feedback control is a widely applicable concept used throughout multiple fields in both industry and academia. From basic control of water level to complex control systems used in industrial manufacturing, feedback controllers can come in various forms (mechanical systems and analog or digital circuits) to adapt to different requirements. With drastic decrease in cost of processing power, modern control systems are often based on computer chips with digitized inputs and outputs. A feedback controller for the toroidal phase of m/n

= 2/1 islands has been developed and implemented on DIII-D as a part of this thesis work.

4.1 Control system

A feedback control system must consists of at least four basic components: controller, actuator, plant, and sensor [47]. Figure 4.1 shows the relation between these components.

The controller is the unit that decides what action should be taken based on available information. In this case, the controller takes the form of an algorithm running on a CPU, a small program within the larger plasma control system (PCS) on DIII-D [48].

An actuator is the device that can effect some change in the object being controlled.

The obvious choice to control LM position would be the non-axisymmetric coils available on DIII-D. Both amplitude and phase of the RMP can be prescribed as an output of the

51 Figure 4.1: Basic schematic of a feedback controller depicting the controller, actuator, plant, and sensor. An example for each in the mode control has been given.

controller.

The plant refers to the physical system one is attempting to control. This would describe how the wall and the mode reacts to the applied RMP. While the plant should also include a description of the capabilities of the power supplies driving the 3D coils, they have been verified to be able to meet the current and frequency requirements of interest well and are assumed to be ideal [49].

Lastly, the sensor measures the result of the control, ie. the phase of the locked mode, calculated from the perturbed radial field measurements. This is accomplished through the external saddle loops differenced (ESLD) described in chapter 2, with compensation for coil pickup. The measured mode phase is compared to the requested reference phase, and the difference is used as input to the controller.

52 4.2 Controller design

The control algorithm is responsible for translating the input error signal into output com- mands to the actuators. While detailed knowledge of system response to the actuators would allow for better control , the basic types of controllers adopted in this thesis, and described below, only require that the relation between command and response to be monotonic. A commonplace example is the cruise control on a car: opening the gas throttle will accelerate the vehicle, though the amount of acceleration also depends on other factors, such as slope or wind.

As a first step, a single-input-single-output (SISO) controller was designed for DIII-D, where the input is the error ϕerr between reference phase ϕref and measured mode phase

ϕLM , and the output is a correction phase ϕcorr. A RMP of pre-determined amplitude and phase ϕRMP (t) is then applied in order to change the mode position, where

ϕRMP (t) = ϕLM (t) + ϕcorr. (4.1)

The torque exerted by the applied RMP on the mode is calculated through a surface integral of J × B, and in this model [23] can be simplified to:

TRMP = α|BRMP ||Js|sin(ϕRMP (t) − ϕLM (t)) = α|BRMP ||Js|sin(ϕcorr) (4.2)

where α is a geometric factor, BRMP is the strength of the applied field, and Js is the perturbed current in the mode. The sine term satisfies the requirement for monotonic relation

53 between command and response, for the condition that

◦ |ϕcorr(t)| = |ϕRMP (t) − ϕLM (t)| ≤ 90 . (4.3)

This limit is imposed to ensure that maximum torque is achieved in the high ϕcorr cases. As

◦ this is a periodic system, an unbounded increase in ϕcorr above 90 would lead to a decrease and eventual reversal in the torque applied by the RMP to the mode.

Proportional control

Proportional control is one of the simplest control schemes, where one simply sets the command to be proportional to the error:

ϕcorr = Kpϕerr (4.4)

and the proportional gain Kp determines the response of the system. If the mode happens to be aligned with the requested reference, we would have

ϕerr = ϕref − ϕLM = 0 (4.5)

ϕcorr = Kpϕerr = 0 (4.6)

and no RMP torque would be applied to the mode. If the mode is lagging behind the reference phase (ϕerr > 0), both ϕcorr and TRMP would be positive, and accelerate the mode towards the requested position.

54 The trade off for the simplicity of proportional control is steady-state error. In the case of an intrinsic or residual error field (EF) attracting the mode, a static reference phase would result in a finite steady-state error. In this model, the final mode position would be determined by torque balance between RMP and EF. Except in the special case of error field phase aligned with reference ϕEF = ϕref , there would be some EF torque pulling the mode away from the reference. This can only be counteracted by the RMP torque, which would require a non-zero ϕcorr and thus a non-zero ϕerr.

In the case of no error field, a rotating reference phase would also result in a finite steady-state error. In this instance, the rotation is a torque balance between the steady-state wall drag and the applied RMP. In order to maintain rotation, ϕRMP must lead ϕLM , which also requires a non-zero ϕcorr and ϕerr.

Proportional-integral control

A proportional-integral control scheme can alleviate the steady-state error problem in both cases. This strategy adds an integral term that accounts for the past history of ϕerr:

∫ t ′ ′ ϕcorr(t) = Kpϕerr(t) + Ki ϕerr(t )dt (4.7) 0

where Ki is the integral gain. For the static reference with error field scenario, the system would behave as follows: initially, the integral term is zero and the proportional term attempts to align the mode to the reference, but a finite phase error ϕerr exists due to the EF.

As time passes, this finite phase error ϕerr is accumulated by the integral term, which slowly grows in amplitude. The contribution of the integral term to ϕcorr continues to increase as

55 long as ϕerr ≠ 0, reducing ϕerr over time and slowly replacing the proportional contribution. Eventually, the mode approaches and aligns with the reference, at which point ϕerr becomes zero and the proportional term is eliminated. As ϕerr remains zero for times into the future, the integral term will remain constant to balance the torque from the error field.

The steady-state error in the rotating reference case is eliminated in a similar manner.

The initial response of proportional gain leaves a finite error, which accumulates to increase the integral term. After some time, the integral term dominates ϕcorr, aligns the mode with the reference and replaces the proportional term entirely. Examples of these behaviours are shown in the following section.

4.3 Simulation of controller

Simulink [50] is a programming environment developed by MathWorks (known for Mat- lab) able to simulate behaviour of complex dynamics systems and test response to different control schemes. The numerical model described in Chapter 3 is used to predict mode rotation in this code, acting as the plant. Its output of mode position is given directly to the controller to compare with the reference. This is equivalent to assuming a perfect observer. The controller takes the inputs of measured mode phase ϕLM and reference phase

ϕref to give an output of RMP phase ϕRMP , as shown in figure 4.2. Finally, a RMP of pre-determined amplitude and real-time calculated phase ϕRMP is applied to the numerical model to simulate the mode response This assumes that the power supplies are capable of delivering the requested currents at the frequencies of interest.

The Simulink model, shown in figure 4.3, was used to predict the system response

56 Figure 4.2: A schematic diagram of the proportional-integral controller implemented on DIII-D. The controller compares the inputs of requested ϕref and measured ϕLM to determine the a correction phase ϕcorr. This ϕcorr is the angle by which the phase of the applied RMP ϕRMP is advanced with respect to the mode phase ϕLM . The applied RMP phase ◦ ϕRMP has been limited to ϕLM 90 for maximum torque. to proportional-only (P-only) and proportional-integral (PI) control methods as described above. In addition to comparing the capabilities of these different algorithms, Simulink also assists in optimizing the values of the gains to balance between quick response and stability.

57 Figure 4.3: Simulink model used to simulate mode response to various control schemes. Labeled are the numerical model as plant, extraction of mode phase as observer, error angle calculation and PI-algorithm as controller, and I-coil inputs as actuator.

Starting with P-only control, figure 4.4(a) illustrates how the mode phase tracks, in time, the requested reference phase that is static for the first 100 ms and rotates afterwards.

Related to that, figure 4.4(b) shows the discrepancy between the mode phase ϕLM and reference phase ϕref . This error angle ϕerr decreases as a result of increasing the proportional gain from Kp = 0.5 to Kp = 1.

However, a further increase to Kp = 2 exceeds a stability threshold, and the system will result in erratic and unstable motion, shown by the red trace. For the rotating reference scenario, simulation predicts the entrainment will be slightly perturbed due to the existence

58 of the error field. The effect of increasing the proportional gain maintains its effect of first reducing the error angle, then causing unstable response. (a) Proportional−only control mode phase 180 Kp = 2 Kp = 1 120 Kp = 0.5 Ref. Phase

60 [deg]

φ 0 Phase −60

−120

−180 0 100 200 300 400 500 600 700 Time t [ms]

(b) Proportional−only control error 90

30 [deg]

φ 0 Phase −30

−60 Kp = 2 Kp = 1 Kp = 0.5 −90 0 100 200 300 400 500 600 700 Time t [ms]

Figure 4.4: Simulink results of proportional-only control of mode phase for three different Kp values of 0.5 (green), 1 (blue) and 2 (red). (a) shows the mode phase follow the reference (dashed), with the lowest Kp=0.5 case following at a larger error than the Kp = 1 case. A p-gain of 2 results in unstable behaviour. (b) plots the error angle between reference and mode, it can be seen that the Kp=1 has lowest error throughout the simulation.

59 Proportional−integral control 180

120

60 [deg]

φ 0 Phase −60

−120 Mode Phase φLM Error angle φe rr Ref. Phase φre f −180 0 100 200 300 400 500 600 700 Time t [ms]

Figure 4.5: Simulink results of proportional-integral control, with Kp = 1,Ki = 4. Align- ment of mode phase to reference phase improves as the integral terms comes into effect, as can be seen by decreasing average error angle.

As for PI control, figure 4.5 shows the elimination of steady-state error for reasonable gain values. Using the same reference and error field as the previous simulation, the slow decrease of error angle during the rotating reference case can be observed.

In order to have an effective controller, it is important for the gains to be optimized.

Lower gain values lead to slower responses, while higher gain values may cause the system to be unstable. Thus, higher gains up to the stability threshold are desired. However, the optimized gains must also be robust in order to account for different island widths, radial position, applied RMP strength, and plasma parameters. The optimization performed in

[23] has been replicated here, and is summarized as follows.

A family of parameters was created for this optimization. This included various island widths and phase relations between the applied RMP and EF. The reason for this is that both

60 mode amplitude and EF location are uncontrolled quantities in the experiment, whereas the

RMP strength and mode radial position are predetermined during a shot. In each case, a linear parametrization of mode dynamics was created around torque balance equilibria to be used in controller response analysis. For a step change in reference phase, the response of each parametrization was calculated in terms of total phase error between mode and reference. Gains were selected based on criteria of minimal phase error and high robustness

[47, 51].

4.4 Experimental results

We implemented a feedback controller for the 2/1 mode phase on the DIII-D plasma control system, as shown in figure 4.2. Using the external saddle loops differenced (ESLD) signal as input, the algorithm calculates the amplitude and phase of the n = 1 locked or slowly rotating mode. Time-integrated signals suffer from instrumental (non-physical) drifts. The issue is addressed here by subtracting from each measured value a baseline, taken 50 ms before mode locking. At that time, the mode is rotating rapidly, at frequencies above 100 Hz, that make it undetectable by the ESLDs.

The calculated phase of the island, ϕLM is compared to a pre-determined reference phase, ϕref and the error angle between the two, ϕerr, is used as input to a proportional- integral (PI) controller. The controller’s output is the optimal correction ϕcorr by which the phase of the applied perturbation, ϕRMP , needs to be advanced with respect to ϕLM .

Ultimately, a.c. currents of constant amplitude and proper frequency and phases are delivered to the internal 3D field coils (I-coils) as to apply an n=1 RMP of the desired

61 ◦ phase ϕRMP (t), after being limited to be within 90 of the mode position, for maximum torque. This algorithm was implemented on the CPU with a cycle time of 50 µs, much faster than the physical timescale of this low frequency entrainment, related to the inverse wall time.

The present approach is satisfactory at frequencies well below the inverse wall-time.

For faster entrainment the amplitudes of the coil-currents should be increased as a function of the rotation frequency to overcome the effect of the increasing wall-torque, as predicted in [23]. This is left as a future improvement.

We performed an experiment on DIII-D with the goal of using this feedback algorithm to achieve smooth entrainment of the mode, combined with synchronized deposition of

ECCD to suppress the island amplitude. Here smooth refers to uniform rotation, both within a single rotation period, as well as from period to period.

The DIII-D discharges presented in the the next two chapters have the following parameters: a plasma current of 1.0 MA and toroidal field of 1.7 T gave a safety factor between

4.3 and 4.5, allowing some response time for intervention in an otherwise disruptive mode.

H-mode was obtained at approximately 1500 ms and subsequently lost around 2200 ms, when an initially rotating 2/1 NTM appears, quickly decelerates and becomes locked within

200 ms. The locking event triggers a response in the phase controller and, for later shots discussed in chapter 6, gyrotron power.

We first tested the capability of the controller to follow the pre-programmed reference phase for fixed-position and 20 Hz rotation. Figure 4.6 shows the results of a shot where only proportional feedback was used, with the proportional gain Kp stepped over four different values in each type of request.

62 (a) Shot 166560 Phase Match 180 φ LM φ re f 120

Phase (deg) −60

−120

−180 2600 2800 3000 3200 3400 (b)

p 2 K

1 Prop. Gain 0 2600 2800 3000 3200 3400 Time (ms)

Figure 4.6: The measured (black) and reference (red) phase of a locked mode, as gain is varied every 100 ms. In both fixed and rotating reference cases, the mode phase roughly follows the reference, with higher gain resulting in better alignment as expected.

The agreement between the requested LM phase ϕref and the actual, magnetically measured LM phase ϕLM is qualitatively evident from fig. 4.6(a). The discrepancy between the two, ϕerr, averaged over the duration of each fixed proportional gain, is used as a quantita- tive figure of merit. This averaged discrepancy < ϕerr > is calculated for both the experimental measurements (black symbols in figure 4.7) and the equivalent simulated scenario

(blue symbols). All simulations in this section used the same parameters as in shot 166560, including the same radial position of the island, and same RMP strength, corresponding to

63 2.7 kA of current in the I-coils.

In the fixed-position case, it is clear from figure 4.6(a) that the controller can control the LM phase as desired, as long as the proportional gain Kp exceeds a threshold located in the interval 0.5 < Kp < 1.

The entrainment tests exhibit larger errors. In fact, the lowest proportional gain,

Kp = 0.5, was unable to entrain the mode and, at t ≃ 3040ms, it allowed the mode to

“slip” backwards and catch the next rotation (figure 4.6(a)). This slippage resulted in a brief period in which ϕLM (t) matched ϕref (t), which may have reduced the apparent tracking error for several periods that followed (into the next proportional gain value). Higher gain, Kp ≥ 1, yielded continuous entrainment without such slipping events, but with some significant discrepancy ϕerr. In reality, finite discrepancies were actually expected (see blue symbols in figure 4.7(b)) as a result of how the phase-controller operating in proportional- only configuration. Additionally, the controller’s measurement of the mode phase has not completely accounted for the currents induced in the wall, which is another source of error when the mode is entrained and rotating. As a matter of fact, experiments exhibited even larger tracking errors (black symbols in figure 4.7(b)), but this was also reasonable, as it is well-known that a proportional-only controller cannot track a ramped reference without error [47].

64 Fixed Phase Tracking Error (a) 90 Exp. Fixed Sim. Fixed [deg] >

e rr 60 φ <

30 Tracking Error

0 0.5 1 1.5 2 Proportional Gain K p

Rotating Phase Tracking Error (b) 150 Exp. Rot. Sim. Rot.

[deg] 120 > e rr φ 90 <

Tracking Error30

0 0.5 1 1.5 2 Proportional Gain K p

Figure 4.7: Experimental and simulated tracking error (defined as discrepancy between reference phase and actual mode phase averaged over each 200 ms period) for (a) fixed phase and (b) rotating phase references respectively.

In addition to gain and entrainment frequency, other major factors contributing to the

65 tracking error in this proportional-only scheme are the island width and its proximity to the wall, or equivalently, the position of the q = 2 surface in normalized minor radius ρ.

For these steady-state, constant frequency, simulations, the effect of the error field must be dropped from equation 3.1, to prevent it from adding a sub-period perturbation to an otherwise uniform entrainment. Experimentally, this can be achieved if the error field is well- corrected, made negligible compared with the applied, slowly rotating RMP. The resultant motion will be a balance between the wall torque and the applied RMP torque only:

0 = Twall + TRMP (4.8)

Figure 4.8 shows the toroidal tracking errors expected from simulations, as described by equation 4.8, where the phase of the RMP is determined by the proportional control.

Each panel presents three colour contours with different thicknesses for changing (a) island width or (b) normalized radial position of the q = 2 surface. For example, the contour with medium thickness lines in panel (a) shows that a 5 cm island entrained at 30 Hz with

◦ Kp of 2 has a predicted tracking error of 20 , whereas a 4 cm (thin lines) or 6 cm (thick lines) island using the same parameters would have errors of 17◦ and 22◦ respectively. The expected phase lag increases for larger islands or shorter distance from the wall, due to the different dependencies of the RMP and wall torques on these parameters. A higher entrainment frequency also increases the error due to stronger wall shielding of the applied

RMP and increased wall torque (peaks at the inverse wall time) on the mode. Simulations show that increasing the gain reduces the tracking error, but at the risk of overall system stability in transient evolution[47].

66 As it currently stands, using as sensors saddle loops external to the vessel prevents the controller from detecting modes rotating much faster than the inverse wall time. Even at frequencies yielding measurable signals (comparable with the inverse wall time, or ap- proaching it), shielding presents issues: the measured mode phase lags behind the true mode phase. A future combination of internal sensors (whether Mirnov or saddle loops) and real-time a.c. compensation for pickup between driving coil and magnetic sensors will contribute to deploying the controller at frequencies above the present limit.

67 Figure 4.8: Simulated tracking error in a proportional only controller, as dependent on the gain used, the requested frequency, as well as (a) the island width and (b) position of the q = 2 surface. Three contours are presented in each panel, where the colours indicate levels of tracking error in each case and thickness differentiates the scenarios.

68 Chapter 5

Mode Suppression

With successful control of mode position, we would now like to also control or suppress its amplitude. Chapter 2 had described the origin of neoclassical tearing modes (NTMs) as a localized deficit of bootstrap current on a rational-q surface. Theoretically, replacement of this missing bootstrap current should eliminate the mode and restore high confinement to the plasma [52]. One common method of current deposition is electron cyclotron current drive.

5.1 ECCD in DIII-D

Electron cyclotron heating (ECH) and electron cyclotron current drive (ECCD) are well established techniques to heat plasmas and drive current in tokamak devices [53, 54]. It employs electromagnetic waves near the electron cyclotron frequency or its low harmonics to provide a highly localized and controllable deposition of energy. When the frequency of the wave matches a harmonic (integer multiple) of the electron cyclotron frequency, a phenomenon known as cyclotron damping transfers energy from the oscillating field into kinetic energy of electrons gyrating around the magnetic field. The localization arises from the equilibrium magnetic field strength varying across the major radius R like 1/R. Control of deposition location is achieved by steering the beam in the vertical direction, as seen in

69 Figure 5.1: Deposition location is where the EC beams intersects the 2nd harmonic of the electron cyclotron frequency, shown in a cross-section of DIII-D, figure 5.1. The path of the electromagnetic wave can be calculated by a ray-tracing code,

TORAY-GA [55].

Electron cyclotron current drive is achieved when localized obliquely aimed ECH power creates an asymmetry in velocity space. Two possible mechanisms due to Ohkawa

[56] and Fisch and Boozer [57] are illustrated in figure 5.2. In the Fisch-Boozer effect,

EC-heated electrons have increased perpendicular velocity, thus increased temperature and reduced collisionality. As collision rate is proportional to v−3, the decrease of electrons at low perpendicular velocity is replaced faster than the excess of electrons at high velocity is removed. The resulting electron distribution function (5.2(c)) has a net ensemble-averaged

70 Figure 5.2: Schematic illustration of a population of electrons moving in velocity space due to EC heating by (a) the Fisch-Boozer and (b) Ohkawa effects. Steady state electron density in velocity space are shown in colour for (a) Fisch-Boozer and (b) Ohkawa, where warmer colour indicate higher density. Figure from [54].

parallel velocity v∥ (parallel to the magnetic field), implying a net current.

The Ohkawa effect also moves a population of electron into higher perpendicular velocity v⊥, but in this case across the trapping-passing boundary due to the magnetic mirror effect as described in Chapter 1. The equalization of the high perpendicular velocity population is now achieved by parallel transport bouncing on the magnetic mirror, which is much faster than the collisional relaxation of the lower perpendicular velocity population.

Thus, there is an overall decrease in number of electrons with positive parallel velocity, resulting in a positive current.

Six gyrotrons are available in DIII-D to inject up to approximately 3.5 MW of EC power into the plasma [58]. Launchers are located on the LFS of the machine, above the mid-plane, where toroidal steering of mirrors allow for current drive ranging between co- to counter-

71 plasma current direction. The gyrotrons can be modulated up to the kHz frequency range, with their effect on heating and current drive established on a collisional timescale (on the order of tens of microseconds for a plasma with an electron temperature of 1 keV).

5.2 Experimental Results

After mode entrainment was achieved, modulated ECCD with a 50% duty cycle was deposited in synchronization with the mode rotation at 20 Hz. Note that in earlier experiments at DIII-D and AUG [59], ECCD was modulated in feedback with island phase measurements, though it is a challenge to do this in real-time with available diagnostics. More simply, here the island phase is prescribed by a feedback controller as a function of time, and the ECCD modulation is simply programmed in advance. Six gyrotrons were used in this experiment to provide up to 3.5 MW of heating and drive about 22.5 kA of current in the co-plasma current direction [58].

A coherent average over 25 periods was performed for each shot, shown on the right subpanels in figure 5.3. This averaging process reduces the noise in the signal such that the data can be fitted with a sinusoid of fixed 20 Hz frequency, in red, equal to that of the entrainment and synchronized depostion The thin dotted black lines show one standard deviation away from the mean mode amplitude.

The observed mode amplitude response contradicted the expectation that O-point deposition suppresses the mode. It was actually observed that the mode amplitude had increased when the O-point was believed to be in phase with the deposition location (figure

5.3(a)). Similarly, current nominally driven in the X-point was correlated with a decrease

72 in mode amplitude (figure 5.3(b)). These peculiar results were further corroborated by two additional shots where the gyrotrons were turned on half-way between the island O- and

X-points, as in figure 5.3(c) and (d).

73 (a) Nominal O−point Deposition 166566 Coherently Averaged 10 10 Exp. Mode Amp Fitted Mode Amp 8 EC Power 8

6 6

4 4

2 2 EC on EC off Mode Amp [G], EC Power [MW]

0 0 3050 3100 3150 3200 0 25 50 Time [ms] Time in period [ms]

(b) Nominal X−point Deposition 166567 Coherently Averaged 10 10 Exp. Mode Amp Fitted Mode Amp 8 EC Power 8

6 6

4 4

2 2 EC on EC off Mode Amp [G], EC Power [MW]

0 0 3050 3100 3150 3200 0 25 50 Time [ms] Time in period [ms]

6 6

4 4

2 2 EC on EC off Mode Amp [G], EC Power [MW]

0 0 3050 3100 3150 3200 0 25 50 Time [ms] Time in period [ms]

74 (d) Nominal X− to O−point Transition 166569 Coherently Averaged 12 12 Exp. Mode Amp Fitted Mode Amp 10 EC Power 10

8 8

6 6

4 4

Mode Amp [G], EC Power [MW] 2 2 EC on EC off

0 0 3050 3100 3150 3200 0 25 50 Time [ms] Time in period [ms]

Figure 5.3: Plots of measured mode amplitude, in black, and electron cyclotron power, in blue. A coherent average over 25 periods is shown for each shot on the right, where the dotted lines mark one standard deviation away from mean black line. A sinusoidal fit to each averaged mode amplitude is shown in red. (a) Mode amplitude is increased when ECCD is thought to be deposited in the O-point, against expectation. (b) X-point deposition correlates with mode amplitude decrease. (c) Mode appears to first grow, then be suppressed as deposition transits from O- to X-point, while (d) the deposition transition from X- to O-point has oppositve effect.

5.3 Effect of radial misalignment

Upon closer analysis, it was found that the radial location of ECCD deposition, r, differed from the rational surface location rs by an amount |r−rs|/wECCD = 1.66, where wECCD is the ECCD deposition full width at half maximum (FWHM). This was due to a poor between- shot equilibrium reconstruct based on magnetics alone and gyrotron mirrors that could only be aimed in feedforward, lacking real-time tracking of the radial position of the rational surface. The current was driven outside of the island (solid color box in figure 5.4(a)) and redistributed, due to parallel transport, over the “intercepted” flux-surfaces, external to the island (shaded lighter color). X-point phasing yields a figure similar to 5.4(b), except that current is driven in flux-surfaces farther away from the island separatrix.

75 In other words, when modulated ECCD is entirely deposited outside the island, it does not result in a helical current-filament that can compensate for the bootstrap current-deficit in the island. Rather, it results in a current-sheet or toroidal annulus being driven in a range of minor radii, but at all poloidal locations. At most, the current-sheet will be kink- deformed because so are the unperturbed flux-surfaces, due to proximity with the island. In these cases, scanning the phase of the time-modulation does not affect the helical phase of

ECCD deposition relative to the island O-point. Rather, it is equivalent to radially scanning the ECCD deposition relative to the island separatrix.

Here the experimental deposition location (modeled as a Gaussian) is reconstructed with the aid of the TORAY-GA code [55] (right-hand-side of figure 5.4(a)) In addition, the toroidally O-point centered 50% duty cycle and parallel transport are taken into account. As a result, one obtains the current-density profile plotted in figure 5.4(c). This is a function of the normalized flux-surface coordinate, perturbed by the presence of the island, introduced in [60]. Here Ψ = −1 corresponds to the center of the island, Ψ = 0 to its separatrix, and Ψ > 0 to the region outside the island. Zero (dark blue) to small (light blue) radial misalignment still results in the majority of the current being driven inside the island O- point as desired. However, increased radial misalignments places a large fraction of the current either onto separatrix (vertical dashed line), which is brought near the island X-point by parallel transport, or outside of the island altogether. Current driven at these locations reinforces the initial perturbation, causing the mode to grow rather than be suppressed. This reversal of effectiveness has been previously predicted by [60] at sufficiently large radial misalignment, |r − rs|/wECCD ≳ 0.75.

76 ECCD Deposition (a) 2 2

is l and 1.5 1.5 /w )

s 1 1 r −

r 0.5 0.5

0 0

−0.5 −0.5

−1 −1

Radial−1.5 position ( −1.5

−2 −2 −3 −2 −1 0 1 2 3 0 0.5 1 Helical Angle ξ Norm. Current Density J (r ) ECCD Deposition (b) 2 2

is l and 1.5 1.5 /w )

s 1 1 r −

r 0.5 0.5

0 0

−0.5 −0.5

−1 −1

Radial−1.5 position ( −1.5

−2 −2 −3 −2 −1 0 1 2 3 0 0.5 1 Helical Angle ξ Norm. Current Density J (r )

(c) Current Distribution for various Radial Misalignment |r − r s|/w ECCD 1 Inside 0 Outside island 0.9 island 0.5 1 0.8 1.66 )

Ψ 0.7

0.6

0.5

0.4

Current0.3 Density J(

0.2

0.1

0 −1 0 1 2 3 4 5 Location Ψ

Figure 5.4: The radial and toroidal deposition location of ECCD in shots (a) 166566 and (b) 166567, based on post-shot analysis from EFIT (mode location), TORAY-GA (ECCD location), and magnetics (island width). The solid box shows the locations near the island where EC power was deposited, which spreads over the flux surface due to parallel transport, as depicted by the shaded region.(c) shows the current drive distribution in terms of perturbed flux surface for 50% duty cycle deposition centered on the O-point toroidally.

77 Comparison with MRE 166566 2 Exp EC On Exp EC Off (low power) 1.5 MRE EC well−aligned MRE EC misaligned 1 MRE EC off (low power) dw/dt

∗ 0.5 /r R τ 1 0 − 22 . 1 −0.5

−1

−1.5 0 0.02 0.04 0.06 0.08 0.1 Island width w is l and[m]

Figure 5.5: Experimental and calculated MRE growth rates as a function of island width, where the MRE includes a term for ECCD with actual misalignment and expected result assuming perfect alignment.

5.4 Mode amplitude suppression

The coherently averaged mode amplitude of shot 166566, shown in figure 5.3(a), was compared with theoretical predictions in figure 5.5. The sinusoidal fit of prescribed frequency was used to completely eliminate noise in the time derivative of the signal. A multiplica- tive factor calculated by [4] was used to convert from measured radial field to island width w, verified to be within error of ECE measurements. The time derivative dw/dt is plotted against w in figure 5.5.

Figure 5.5 also plots the expected growth rates calculated from the MRE given in equation 5.1, where ∆′ is estimated from a saturated island in an earlier shot without current drive. The other terms on the right-hand-side of equation 5.1 are calculated on the basis of experimental quantities. For instance, the lengthscales Lq and Lp are derived from the

78 experimental profiles of safety factor q and pressure p.

The experimental trajectory of the island width is also plotted for comparison. There is qualitative agreement between the behaviour of island and the predicted saturated widths with and without ECCD. It can be seen that, for good alignment, ECCD is expected to move the dw/dt curve (in blue in the figure) to the lower, stable half-plane where dw/dt<0, and fully suppress the mode. However, with the large radial misalignment at which the experiment was carried out, the mode is actually expected to grow in size (red curve) as indeed observed in the experiment (figure 5.3(a)).

5.5 Comparison with theory

The original model of the growth of tearing modes was introduced by Rutherford in [9], which accounts for the free energy in the equilibrium current profile in a term we now call the classical stability index ∆′. Over time, various terms have been added to the equation to include neoclassical effects, and the modified Rutherford equation (MRE) used in this thesis is given below [61]:

( ) [ ( )] τ dw L rw rw2 j −1 R ′ 1/2 q − pol − ECCD 1.22 = ∆ (w)r + ϵ βθ 2 2 3 K1 (5.1) r dt Lp w + wd w jBS

where τR is the local resistive time and r is the minor radius of the rational surface. The right hand side includes the classical stability index ∆′(w) and neoclassical terms for small island effects wd and polarization wpol [62]. These small island terms modify the solution of the MRE and now requires a seed island in order for the mode to grow. The last term

79 describes the effect of electron cyclotron current drive on the growth of the island dependent on an efficiency K1 and the ratio of ECCD and bootstrap currents.

An early model by Perkins [60] calculates the efficiency K1 of ECCD mode suppression by direct replacement of the missing bootstrap current, averaged over the modulation period.

( ) ∫ w ∆R ∞ K1 , = dΨW (Ψ)jECCD(Ψ, wECCD, ∆R) (5.2) wECCD wECCD −1 where the efficiency is a function of island size (affects perturbed flux Ψ), a weighting function W (Ψ), deposition width wECCD, radial misalignment ∆R, as well as duty cycle and toroidal phasing. Figure 5.6 shows contours of this efficiency, where the white marks in panels (b) and (c) show estimated experimental values. In both the cw and O-point centered modulation cases, the highest efficiency occurs for well-aligned deposition, as expected.

O-point deposition yields higher efficiencies everywhere, compared to cw deposition. In the 50% X-point modulation case (Figure 5.6(c)), a well-aligned deposition results in a moderately negative effect, which diminishes for larger misalignment or island size.

More recent work by Westerhof [63] also includes the effects of current deposited outside the island that changes the local current profile, thus modifying the classical stability index. However, the current profile can only be perturbed on an L/R timescale, where L and R are the local inductance and resistance respectively. In this particular experiment, the L/R time of approximately 60 ms allows us to assume current profile just outside the island does not change significantly during each 25 ms half-period of the ECCD turning on or off.

80 (a) K1 No Modulation 4 1

3.5 0.1

0.2

ECCD 3

/w 0.3 0.5 2.5 0.1

is l and 0 w 2

0.4 −0.1 1.5 0.2 0 0.1 0 0.3 Island Size 1 −0.1 −0.1 0.5 0.5 0.4 0.2 0 −0.1 0.3 0.1 −0.5 0 0.5 1 1.5 2 Misalignment |r ECCD− r s|/w ECCD (b) K1 50% Modulation O−point 4 1

3.5 0.3

ECCD 3 0.3 0.2 /w 0.5 2.5 0.4 0.1 is l and 0 w

2 0.4 0.5 0.3 0.2 1.5 0.6 0.1 0 0

Island Size 1

0.5 0.2 0.7 0.5 0.1 0.8 0.3 0 0.6 0.4 −0.5 0 0.5 1 1.5 2 Misalignment |r ECCD− r s|/w ECCD (c) K1 50% Modulation X−point 4 1

3.5 −0.05

ECCD 3 −0.05

/w 0.5 2.5 −0.1 is l and w 2 −0.1

−0.1 1.5 0 Island Size 1 −0.05 −0.2 −0.1

−0.2 0.5 −0.3 −0.4 −0.05 −0.5 0 0.5 1 1.5 2 Misalignment |r ECCD− r s|/w ECCD

Figure 5.6: Contours of ECCD deposition efficiency in terms of relative island size, radial misalignment, and duty cycle being (a) full-on, (b) 50% centered on O-point, and (c) 50% centered on X-point. Here, wisland is the island half-width, wECCD is the FWHM of the gaussian current drive profile, and misalignment is the absolute difference between peak CD location and the q = 2 surface. The white regions in panels (b) and (c) mark approximately the experimental values for shot 166566 and 166567 respectively.

81 De Lazzari [64] further refined the model by including the effect of local heating on the local resistivity, and thus on the local current. The relative strengths of this term depend on the ratio of island and deposition widths, and on local temperature, bootstrap current, and perpendicular heat conducitivity. This was evaluated to be roughly an order of magnitude smaller than the direct current replacement effect in this experiment. Lastly, the change in local pressure profile gradient produces another perturbed current, but is numerically evaluated to be yet another order of magnitude smaller and therefore neglected in further analysis. Thus, the strongest contributing term of current replacement became the focus of further analysis, neglecting the other effects.

These works have been extended to calculating the change in saturated island width by using the MRE and accounting for the time-evolution of the ECCD term in the equation, due to the power being modulated and the island being entrained, changing the deposition location. This is accomplished by evaluating the effect on the efficiency K1 (figure 5.7(a)) of a thin slice of current deposited in a toroidally localized position. The K1 contours in figure 5.6 can be interpreted as the weighted averages over one period. As the deposition location is moved by rotating the island, the effect of the EC suppression will change in time, resulting in a gradual shift of the saturated island width. Figure 5.7(b) and (c) depict a radially well-aligned blip of current in the O- and X-points of the island, and panel (a) shows the drastic difference between suppression efficiencies for radially well-aligned and misaligned depositions. This highlights a need for robust real-time aiming of the EC mirrors towards the q = 2 surface in order to achieve maximum efficiency.

82 (a) Effect of instantaneous current 2.5 |r − r s|/w cd = 0 |r − r s|/w cd = 1.66 2

1.5 1 K 1

ciency 0.5 ﬃ E

−0.5

−1 −3 −2 −1 0 1 2 3 Helical Angle ξ

ECCD Deposition (b) 2 2

is l and 1.5 1.5 /w )

s 1 1 r −

r 0.5 0.5

0 0

−0.5 −0.5

−1 −1

Radial−1.5 position ( −1.5

−2 −2 −3 −2 −1 0 1 2 3 0 0.5 1 Helical Angle ξ Norm. Current Density J (r ) ECCD Deposition (c) 2 2

is l and 1.5 1.5 /w )

s 1 1 r −

r 0.5 0.5

0 0

−0.5 −0.5

−1 −1

Radial−1.5 position ( −1.5

−2 −2 −3 −2 −1 0 1 2 3 0 0.5 1 Helical Angle ξ Norm. Current Density J (r )

Figure 5.7: (a) shows the instantaneous efficiency K1 for current deposited at a particular helical angle, for both radially aligned and misaligned case. (b) and (c) depicts a toroidally localized slice of current deposited in the O- and X-points of the island, respectively, where the solid box shows actual deposition location (5% duty cycle used for clarity), and the shaded region shows the effect of current being spread over the associated flux surfaces by fast parallel transport.

83 Chapter 6

Conclusions and future work

6.1 Summary and Conclusions

This thesis studies the usage of applied resonant magnetic perturbations (RMPs) to control the position of locked modes (LMs). These are instabilities that can form on rational q surfaces internal to the plasma, leading to degradation of plasma performance or even a complete loss of confinement. Locked modes can either be born-locked or, in the cases studied here, be the result of an initially fast rotating neoclassical tearing mode (NTM) slowed by an electromagnetic torque from a resistive wall and becoming locked. As LMs tend to align themselves with imperfections in the magnetic field known as error fields (EFs), the toroidal phase of the LM can be controlled by intentionally creating a non-axisymmetric magnetic field through applied RMPs. This allows for localized deposition of electron cyclotron current drive (ECCD) to suppress LMs and prevent a catastrophic loss of plasma confinement known as a disruption.

This work, realized at the DIII-D tokamak, is built upon the numerical mode described in chapter 2 that was initially created by Olofsson, a former post-doc of this research group.

The author has since expanded its capabilities to simulate mode dynamics for LMs on other machines, including ITER, and tested system response to new control schemes using

84 Simulink software. Preemptive feedfoward control of NTMs is a novel technique capable of entraining the mode before it is fully decelerated by the wall, avoiding degradation of plasma confinement associated with complete locking in the lab frame. Simulation and experimental results on DIII-D regarding this technique were shown in chapter 3.

The main result presented in this work is the design and implementation of an active feedback control algorithm for the phase of n = 1 LMs in DIII-D. Chapter 4 presented the design of the control system, which uses the set of internal 3D coils (I-coils) as actuator and external saddle loops as observer. Simulation results illustrate the capabilities of proportional-only and proportional-integral control schemes, while experiments on DIII-

D successfully demonstrated the controller’s ability to follow a fixed-phase or rotating reference. An important application is the synchronization of mode rotation with modulated ECCD, discussed in chapter 5. This differs from previous experiments of modulated

ECCD as they required real-time measurements of mode phase to control ECCD pulses, whereas the controller is able to dictate the phase of the LM, and ECCD timing can be prescribed a priori. The initially counter-intuitive result of current deposition at nominal

O-point phasing destabilizing the island was later explained by a radial misalignment of deposition location. This led to time-dependent modeling of the direct current replacement term in the modified Rutherford equation (MRE), which showed the oscillation of saturated island width during a single period in qualitative agreement with experiment.

85 6.2 Future work

A numerical simulation capable of predicting the mode behaviour can be used to plan experiments or even guide future tokamak design. The model presented here has assumed a saturated island of fixed width, thus avoiding the need to account for varying mode amplitude or moment of inertia. This can be generalized by including a model which predicts tearing stability, allowing the saturated width to change due to various effects such as rotation or ECCD. The rotation has been modeled with the assumption that electromagnetic torques are the dominant contribution to dynamics, and the resonant component of the applied field is largely unaffected by plasma response. Each of these effects can be addressed by adding correctional terms to the model in chapter 2. However, accurate assessment of these terms will likely require resistive MHD codes.

Preemptively applying an RMP is a useful method to avoid mode locking, but there can be concerns of the applied RMP itself causing a degradation in plasma confinement before the mode appears or, for sufficiently high amplitude, the RMP creating an error- driven tearing mode. Further studies can establish a lower bound as to the current required for preemptive entrainment. Another possibility is to apply the RMP only after the rotating

NTM is detected to be decelerating.

The feedback control algorithm as presently implemented on DIII-D can be improved in several areas. The observer limits operation to frequencies below the inverse wall time for two reasons. The observer assumes the measured n = 1 signal is entirely contributed by the m/n = 2/1 mode, ignoring other possible tearing mode as well as the 2/1 shielding currents created in the wall for finite frequency. This leads to a lag between the phase measured

86 by magnetics and the true phase of the mode. To correct this lag, a real-time frequency measurement is required, which is difficult to obtain due to high noise associated with time- derivatives on a short (50 µs) CPU cycle. One possible scheme is to calculate the frequency, at any given time, based on thousands of cycles from earlier times. An additional issue with the observer is that the saddle loop signals are presently only DC compensated for currents driven in the 3D coils. While this is a small effect at low frequencies, AC compensation is required in order to expand the operating range of the controller to frequencies above the inverse wall time. This single-input single-output (SISO) controller uses measured phase of the mode to determine an optimal phase of a fixed-amplitude RMP to be applied. One can also envision a control scheme where the output is the amplitude of the RMP, with its phase fixed to be 90◦ ahead for maximum torque. This approach would eliminate the component of the RMP in phase with the mode that does not produce torque, reducing the amplitude required.

Here it was demonstrated that, by simply entraining the mode at a given frequency and modulating the ECCD at the same frequency, it is straightforward to synchronize the

ECCD deposition with the transit of the island O-point. However, the deposited current did not fully suppress the mode as expected, due to a radial misalignment. In a possible future experiment, after optimizing the radial and toroidal deposition location, both the gyrotrons and the RMP should be turned off to restore pre-NTM confinement. Additionally, the minimum required EC power can be predicted by MRE and should be tested in the experiment.

Tearing modes and subsequent locked modes are expected to be a major challenge in

ITER. This thesis offers a means of mode phase control in feedback, which allows for

87 efficient suppression of mode amplitude by modulated ECCD.

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