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Master's Thesis
Theoretical Review of m/n=1/1 Sawtooth Instability and Comparison with Experimental Observations in KSTAR
Soomin Lee
Department of Physics
Graduate School of UNIST
2017
Theoretical Review of m/n=1/1 Sawtooth Instability and Comparison with Experimental
Observations in KSTAR
Soomin Lee
Department of Physics
Graduate School of UNIST
Theoretical Review of m/n=1/1 Sawtooth Instability and Comparison with Experimental
Observations in KSTAR
A thesis/dissertation
submitted to the Graduate School of UNIST
in partial fulfillment of the
requirements for the degree of
Master of Science
Soomin Lee
06. 09. 2017
Approved by
______
Advisor
Hyeon-Keo Park
Theoretical Review of m/n=1/1 Sawtooth Oscillation and Comparison with Experimental
Result in KSTAR
Soomin Lee
This certifies that the thesis/dissertation of Soomin Lee is approved.
06. 09. 2017
signature ______Advisor: Hyeon-Keo Park
signature ______Kyujin Kwak: Thesis Committee Member #1
signature ______Min Sup Hur: Thesis Committee Member #2
signature
Abstract
One of the main problems in nuclear fusion is plasma instability. This instability causes the loss of energy in tokamak, and results in shorter confinement time. By studying plasma instability, it can be predicted in more detail, and therefore, better controlled. In this thesis, sawtooth instability at the core of the tokamak is discussed.
At the core of tokamak, the plasma density and temperature shows periodic temporal behavior. When the plasma is heated, the temperature increases linearly. As the core temperature increases and reaches a critical value, the core density and temperature rapidly drop and start to increase again. The periodic relaxation of the core plasma is observed as the sawtooth-like signals of the density and temperature.
In this thesis, two famous models by Kadomtsev and Wesson to explain the “sawtooth instability” are introduced. Kadomtsev’s model explains that the fast crash is due to the magnetic reconnection. This model is called ‘the fast reconnection model’. Due to the internal kink instability, the plasma is concentrated near 푞 = 1 surface, and pressure increases in this region. Then the magnetic reconnection takes place near 푞 = 1 surface, and the core heat and temperature collapses through this region. Wesson’s model explains the sawtooth instability assuming a flat 푞-profile inside the 푞 = 1 surface. Due to the flat 푞-profile, magnetic shear near the core is very small, and that results in the formation of ‘hot crescent, cold bubble’ inside the 푞 = 1 surface. This model is called ‘the quasi-interchange’ model. In order to explain both models theoretically, the change of potential energy, 훿푊 for a finite plasma displacement 훏, is derived to determine stability. When 훿푊 > 0, the plasma equilibrium is determined to be stable whereas when 훿푊 < 0, the plasma equilibrium is determined to be unstable.
In 푚/푛 = 1/1 internal kink mode, 훿푊 is composed of a second order term, 훿푊2, and a fourth order term, 훿푊4 . Kadomtsev’s model assumes d휉/훿푟 → 0 except 푞 = 1 surface, and Wesson’s model assumes 푞 ~ 1 inside the 푞 = 1 surface. These models have different approaches for explaining the fast disruption in the core. To validate the theoretical models, experimental observation is conducted by measuring several parameters such as temperature or density.
In tokamak, it is impossible to measure the temperature, because there is no material to measure the plasma temperature at that high temperature. To measure the temperature, an indirect method by I
measuring radiation intensity emitted by gyrating electrons is used in which the radiation is proportional to electron temperature. The ECEI system is a diagnostic system which can directly visualize the MHD instabilities by measuring electron temperature fluctuation. The KSTAR ECEI system measures 2- D/quasi 3-D electron temperature fluctuation which is proportional to the intensity of radiation emitted by electrons. According to Rayleigh-Jean’s law, the optical depth of the plasma is thick enough, the plasma temperature is proportional to the intensity of radiation.
Sawtooth oscillations were observed in this study using ECEI system by measuring electron temperature fluctuations in 3D. The sawtooth period was observed as 휏sawtooth ~ 10 ms and collapse time was observed as 휏푐 ~ 100 휇s. The observed spatial structure at the core of the sawtoothing plasma resembles the 푚/푛 = 1/1 internal kink mode, which concludes that the Kadomtsev’s model gives a more relevant explanation of the sawtooth instability in the experiment than Wesson’s model.
Keywords: Sawtooth Oscillation, MHD instability, KSTAR ECEI system
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III
Contents
I. Introduction ------1
1.1 Motivation ------1
1.2 Fusion energy and tokamak ------2
1.3 Background of this study ------6
1.4 Thesis Outline ------9
II. Theoretical & Mathematical Development ------10
2.1 MHD Physics Review in Tokamak------10
2.1.1 MHD equations ------10
2.1.2 Mathematical Description in Tokamak ------13
2.1.3 The 푧-pinch configuration ------15
2.1.4 The Screw Pinch Configuration------17
2.1.5 The Safety factor, 푞 ------19
2.1.6 The Beta, 훽 ------22
2.1.7 Properties of large aspect ratio in tokamak ------24
2.2 Reviews of MHD instability in tokamak ------27
2.2.1 Basic idea of MHD stability ------27
2.2.2 Energy principle ------28
2.3 Theories of sawtooth instability ------39
2.3.1 Internal kink, 푚/푛 = 1/1 mode ------39
2.3.2 Kadomtsev’s model ------41
2.3.3 Wesson’s model ------46
2.3.4 Comparison of the two models ------49
IV
III. Experimental Methods ------50
3.1 KSTAR ECEI System ------50
3.1.1 Physical principle of ECE ------50
3.1.2 KSTAR ECEI System ------52
3.2 Experimental Set-up ------53
IV. Results & Discussions ------55
4.1 Experimental results ------55
4.1.1 Observation of periodic behavior ------55
4.1.2 Observation of precursor phase ------59
4.1.3 Observation of fast crash ------62
4.2 Discussions ------63
V. Conclusion ------65
Appendix A. Plasma Parameters ------66
Appendix B. Vector Relations ------67
Appendix B1. Vector formulas ------67
Appendix B2. Applications to the cylindrical coordinates (푟, 휃, 푧) ------68
Appendix C. Detail derivations ------69
Appendix C1. Integration of trigonometric function in section 2.1.4 ------69
Appendix C2. Kadomtsev’s collapse time calculation process ------70
V
References ------73
Acknowledgement ------77
VI
List of figures
Figure 1.1 Schematics of ITER tokamak (a) poloidal cross section of ITER structure (b) cutaway view.
Figure 1.2 The picture of PF coil, one of the constructing ITER device.
Figure 1.3 The pictures of KSTAR. (a) The picture taken from blew (b) the picture taken from above.
Figure 1.4 Schematics of magnetic confinement in tokamak (poloidal cross-section of tokamak).
Figure 1.5 (a) The periodic behavior of electron density at JET tokamak, and (b) the ion temperature description of sawtooth plasma.
Figure 1.6 The comparison of safety factor and temperature distribution at before crash and after crash. (a) The safety factor with respect to radius (b) The safety factor with respect to time (c) The temperature with respect to radius (d) The temperature with respect to time.
Figure 2.1 The schematics of tokamak. (a) The schematic description of tokamak (b) The poloidal cross section of tokamak.
Figure 2.2 Schematics of instability on 푧-pinch (a) sausage instability (b) kink instability. The arrow direction indicates the magnetic pressure.
Figure 2.3 The magnetic field line by each component. (a) the toroidal plasma current induces the poloidal magnetic field at the 푧-pinch and (b) The external coil current induces the toroidal magnetic field at the 휃-pinch.
Figure 2.4 The basic idea of MHD stability. (a) Stable equilibrium, (b) unstable equilibrium.
Figure 2.5 The displacement distribution, 휉(푟), based on Kadomtsev’s model.
Figure 2.6 The 푚/푛 = 1/1 internal kink within 푞 = 1 surface. (a) total view of tokamak (b) poloidal view of tokamak.
Figure 2.7 The flow pattern of plasma in Kadomtsev’s model. The red region is hot region, and the blue region is cold region.
Figure 2.8 The whole process of sawtooth oscillation in Kadomtsev’s model.
Figure 2.9 The displacement distribution, 휉(푟), based on Wesson’s model.
Figure 2.10 The flow pattern of plasma in Wesson’s model. The red region is hot region, and the blue region is cold region.
VII
Figure 3.1 Gyrating electron around magnetic field and its emitted radiation.
Figure 3.2 The schematics of the ECEI system. The first system is in H-port, and the second system is in G-port.
Figure 4.1 1-D time trace of 훿푇푒/〈푇푒〉 where 훿푇푒 is electron temperature fluctuation and 〈푇푒〉 is the averaged electron temperature over time. (a) 훿푇푒/〈푇푒〉 at central region (observed in high field side of
H-port), (b) 훿푇푒/〈푇푒〉 at outer region (observed in G-port).
Figure 4.2 1-D time trace of 훿푇푒/〈푇푒〉 in one period (a) at the central region (b) at the outer region.
Figure 4.3 The periodic behavior of sawtooth oscillation. The result represents 훿푇푒/〈푇푒〉.
Figure 4.4 1-D time trace of 훿푇푒/〈푇푒〉 in precursor phase and fast collapse phase (a) at the central region (b) at the outer region.
Figure 4.5 The process of precursor phase. The result represents 훿푇푒/〈푇푒〉.
Figure 4.6 The process of fast collapse phase. The result represents 훿푇푒/〈푇푒〉.
VIII
List of tables
Table 1.1 Three important nuclear fusion reaction.
Table 2.1 The MHD equations.
Table 2.2 Leading order relations in large aspect ratio tokamak.
Table 2.3 Calculated Kadomtsev’s collapse time according to several tokamaks.
Table 2.4 Comparison between Kadomtsev’s model and Wesson’s model.
Table 3.1 Parameters of KSTAR.
Table 3.2 Experiment set-up for # 11264.
Table 4.1 Sawtooth period, and collapse times.
Table 4.2 The time lengths of the three phases in Figure 4.2.
Table A.1 Plasma parameters.
Table C.1 Calculated electron collision time
Table C.2 Calculated resistivity.
IX
List of abbreviations
COMPASS Compact Assembly
DEMO Demonstration power station
DIII-D Doublet III - D shaped plasma
EAST Experimental Advanced Superconducting Tokamak
ECE Electron Cyclotron Emission
ECEI Electron Cyclotron Emission Imaging
HFS High Field Side
ITER International Thermonuclear Experimental Reactor
JT-60SA Japan Torus - 60 Super Advanced
KEEI Korea Energy Economics Institute
KSTAR Korea Superconducting Tokamak Advanced Research
LFS Low Field Side
MHD Magneto-Hydro Dynamics
NSTX-U National Spherical Torus Experiment – Upgrade
OECD Organization for Economic Co-operation and Development
PF Poloidal Field
SST-1 Steady-state Superconducting Tokamak -1
T-15U Tokamak - 1 Upgrade
TCV Tokamak á Configuration Variable
Tokamak Toroidal Chamber Magnetic Coil
WEST Tungsten (W) Environment in Steady-state Tokamak
X
XI
1. Introduction
1.1 Motivation
Nowadays, news about disasters, conflicts, famines and war around the world can be heard through the media more easily than before. Although science and technology have been developed a great deal, these radical problems have not yet been solved. In addition, the world population has been growing rapidly. By the end of the 21st century, it is expected to be over 10 billion [1, 2]. In the face of rising energy demands around the world, fundamental and permanent alternatives must be implemented.
Most of the world population depends on fossil fuel energy. Recently, the statistics say that the proportion of fossil energy is over 80% of the total global energy resources [3, 4]. The problem is that fossil fuels burn and generate CO2, which causes global warming and climate change. A more pressing problem is that fossil energy is expected to run out in decades. According to the 2015 KEEI Annual Report, the situation in the Republic of Korea is even more serious. In the Republic of Korea, the dependence on energy imports is 96% and the dependence on energy by oil from the Middle East is 86% whereas the rate of renewable energy is 3.52% of the total energy. This is the lowest among OECD member countries [5]. This shows the need for alternative energy in the Republic of Korea. If renewable energy is expanded, the dependence on energy imports will be reduced significantly, and the money can be invested elsewhere. Korea can then become stronger by energy independence. Energy independence will provide Republic of Korea with greater economic wealth and, as a result, political influence. This could be crucial in its relationship with North Korea and possible reunification. Finally, oppressed North Koreans will be free by breaking down of the North Korean regime. Developing alternative energy not only prevents global warming, but also it is an opportunity to end this war and reduce unification costs though the economic wealth of Korea.
Renewable energy includes solar energy, wind power, biomass, nuclear fission energy and nuclear fusion energy. Nuclear fusion energy has overwhelmingly higher efficiency than the other alternative energy sources. For example, for D-T reaction in fusion, the emitted energy by fusion reaction is 17.6 MeV whereas ionization energy of hydrogen atom is 13.6 eV [6]. This shows that D-T fusion energy per reaction is over one million times that of ionization energy per reaction for the same amount of hydrogen. In addition, it emits less radiation than in nuclear fission. The resource is hydrogen peaked by water, so if fusion energy is commercialized, there is no need to use fossil energy and uranium.
To summarize, there are three motives for studying nuclear fusion energy. First, fusion energy contributes to Korea achieving economic independence. Second, fusion energy has high efficiency and
1
it can save enormous energy. Third, fusion energy is an eco-friendly energy without emitting CO2 and it emits less radiation than nuclear fission.
1.2 Fusion energy and its history
The nuclear fusion research for toroidal system began in 1951 [7] and in the early 1970’s, the tokamak (Toroidal Chamber Magnetic Coil) was newly installed by introducing heating technology [2]. Currently, the joint research on fusion reactors is conducted by several countries and is known as the ‘ITER (International Thermonuclear Experimental Reactor) project’. The picture of PF (Poloidal Field) coil which is device of ITER is shown in Figure 1.1 [8] and Figure 1.2. From 1978, ITER project began with collaborated research across nations, which are the United States of America, the Union of Soviet Socialist Republics (now it’s Russia), European Union and Japan [9, 10]. Now, participating researchers come from the Republic of Korea, United States of America, Japan, People’s Republic of China, Russian Federation, Republic of India and the European Union. At present, in Korea, nuclear fusion research is being conducted on KSTAR (Korea Superconducting Tokamak Advanced Research) tokamak. The pictures of KSTAR are shown in Figure 1.3. The purpose of the KSTAR study is a 300 second steady operation by the year 2020. After finishing the ITER project, DEMO (DEMOnstration Power Station) will be built upon ITER and verified, leading to the first fusion commercial power plant. Then, nuclear fusion will be commercialized.
Nuclear fusion is the reaction in which two atomic nuclei form one particle. A nuclear fusion reaction can be done in three ways as shown in Table 1.1.
Table 1.1 Three important nuclear fusion reaction [11].
Deuterium and tritium collide to create a fusion reaction D-T reaction D+ + T+ → He(3.5 MeV) + n(14.1 MeV) + 17.6 MeV
Deuterium ions collide to create a fusion reaction D-D reaction D+ + D+ → T+(1.01 MeV) + H+(3.02 MeV) + 4.03 MeV (50% of D-D) D+ + D+ → He+(0.82 MeV) + n+(2.45 MeV) + 3.27 MeV (50% of D-D)
Deuterium and 3He+ collide to create a fusion reaction D-He reaction D+ + 3He+ → He(3.6 MeV) + p+(14.7 MeV) + 18.3 MeV
2
(a)
(b) Figure 1.1 Schematics of ITER tokamak (a) poloidal cross section of ITER structure (b) cutaway view [8]. 3
Figure 1.2 The picture of PF coil, one of the constructing ITER device.
(a) (b) Figure 1.3 The pictures of KSTAR. (a) The picture taken from blew (b) the picture taken from above.
4
To implement this fusion reaction, the particles should be fully ionized. To be ionized, energy has to exceed the repulsive force between atomic nucleus and the electron should. This state is the plasma state and all particles should be over 108 K. At present, there is no material to manufacture a container that can stand this temperature. The plasma is ionized gas, and it is an assembly of charged particles. So, if the external magnetic field is applied, the plasma moves by Lorentz force, perpendicular to both current direction and magnetic field direction as
퐟 = 퐣×퐁 (1.1)
where 퐟 is Lorentz force density (force per volume), 퐣 is the current density (current of plasma per area) of the plasma and 퐁 is the external magnetic field (magnetic flux density).
Currently, using this property of plasma, magnetic pressure is used to be hold the plasma. To confine the plasma, tokamak is often used. Tokamak is a device for holding plasma by using magnetic field in torus, as in Figure 1.4.
But the problem is that confinement time in tokamak is not enough for commercialization. To increase confinement time, the plasma needs to be secured stably. A review of the relevant studies will be presented in the next section.
Magnetic pressure
Plasma pressure
Plasma
Vacuum
Figure 1.4 Schematics of magnetic confinement in tokamak (poloidal cross-section of tokamak).
5
1.3 Background of this study
To stabilize the plasma, instabilities in tokamak should be controlled. Plasma instabilities lead to energy loss of plasma by sudden discharge of plasma current escaping magnetic field line [2, 12].
In the core of tokamak, the current of plasma is increased as the temperature is increased [13]:
1 휎 ∝ ∝ 푇3/2 ∝ 푗 . (1.2) 휂 휙
As the core temperature reaches the critical boundary, the core temperature rapidly drops. This instability causes energy loss and reduces confinement time. The shape of the 1-D time trace of the core temperature over time looks similar to sawtooth. This periodic behavior is called “sawtooth oscillation”.
The sawtooth pattern of the density, 푛(푟, 푡), and the temperature, 푇(푟, 푡), is described in Figure. 1.5 [14]. As Figure 1.5b, the sawtooth periodic cycle can be divided into 3 phases:
i) Ramp phase: Plasma density, 푛(푟, 푡), and temperature, 푇(푟, 푡), is increasing approximately linear as over time.
ii) Precursor phase: Due to instability, helical magnetic perturbation is growing.
iii) Fast collapse phase: Plasma density, 푛(푟, 푡), and temperature, 푇(푟, 푡), fall rapidly with respect to time. It is called “fast crash”.
The issue of this study is the analysis of fast crash which occurs in the fast collapse phase. Due to the instability, heat on the core is transported to the outer region of the tokamak. After crash, the safety factor, 푞(푟, 푡) and the temperature, 푇(푟, 푡) is flattened inside the mixing radius as in Figure 1.6 [15].
An explanation of the safety factor will be discussed in section 2.1.4. The inversion radius, 푟inv is at radial position of 푞 = 1 , which means, inside of the inversion radius is 푞 < 1 and outside of the inversion radius, 푟1, has 푞 > 1. In addition, the radius is the mixing region affected by the crash at the core. Outside the mixing region, the temperature and the safety factor are not changed by the crash. To explain the crash, the studies regarding sawtooth instabilities have already been conducted.
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(a) (b) Figure 1.5 (a) The periodic behavior of electron density at JET tokamak, and (b) the ion temperature description of sawtooth plasma [14].
(a) (b) (c) (d) Figure 1.6 The comparison of safety factor and temperature distribution at before crash and after crash. (a) The safety factor with respect to radius (b) The safety factor with respect to time (c) The temperature with respect to radius (d) The temperature with respect to time [15].
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The sawtooth oscillation was discovered by S. Von. Geoler in 1974 [12] by observing sawtooth-like soft-x-ray intensity signal in ST tokamak core. Based on the theoretical studies already conducted [16, 17], the mode of sawtooth oscillation is predicted as 푚 = 1 internal kink [13].
An instability can be controlled by knowing the exact cause of the instability. To explain the fast crashes in sawtooth oscillation, many different models have been introduced. In 1975, B. B. Kadomtsev introduced “the fast reconnection” model [18]. Kadomtsev’s model describes the fast reconnection on the narrow resistive layer near 푞 = 1 surface with rigid body, 푚/푛 = 1/1 internal kink mode. In 1986, J. A. Wesson introduced “the quasi-interchange” model [19] to explain the inconsistencies between Kadomtsev’s theoretical model and experimental observation of JET tokamak. Wesson’s model describes quasi-interchange instability which results from the flat 푞-profile in the central region. These are the theoretical bases of the following thesis.
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1.4. Thesis Outline
The thesis will review and compare the two models. In addition, using experimental results in KSTAR, experimental observations and theoretical models will be compared. Then, the thesis will conclude which model is closest to the experimental results and analyze it.
In Chapter 2, a theoretical review of the sawtooth oscillation will be conducted. First, MHD physics used in tokamak will be reviewed briefly. Then, the basics of MHD instability in tokamak will be reviewed to analyze the sawtooth instability. Based on this knowledge, the two theoretical models for sawtooth instability will be reviewed. The Kadotmsev’s model is a fast reconnection model and Wesson’s model is a quasi-interchange model. Also, comparison of the two models will be introduced.
In Chapter 3, experimental observations of sawtooth oscillation will be conducted. The principle of ECEI (Emission Cyclotron Electron Imaging) System in KSTAR will be reviewed. Also, the experimental set-up will be presented.
In Chapter 4, the experimental results will be shown. The experimental results will be compared with the theoretical models. Then, the thesis will identify which theory is close to the experimental result. At the same time, the thesis will discuss how to create a new model through the comparison.
In Chapter 5, a summary and conclusion of the thesis will be discussed. Also, future study will be discussed.
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II. Theoretical & Mathematical Development
2.1 MHD Physics Review in Tokamak
Plasma is a fluid and can be deformed when shear stress is applied. Plasma is also an ensemble of charged particles, and thus the behavior is affected by applied magnetic field. These two properties of plasma, allows plasma behavior to be analyzed by MHD (Magneto-Hydro Dynamics) physics.
2.1.1 MHD equations
MHD provides seven commonly recognized governing equations: Two fluid equations, one thermodynamics equation and four Maxwell’s equations.
The continuity equation is given by
d휌 = −휌∇ ∙ 퐮 (2.1) d푡
where 휌 is the mass density of the plasma, and 퐮 is the velocity of the plasma. The continuity equation including partial derivative term can be expressed as
∂휌 + 퐮 ∙ ∇휌 = −휌∇ ∙ 퐮 ∂푡 or
∂휌 + ∇ ∙ (휌퐮) = 0. ∂푡
The momentum equation is given by
d퐮 휌 = 퐣×퐁 − 훁푝 (2.2) d푡
where 푝 is the pressure, 퐣 is the current density and 퐁 is magnetic field strength. The momentum equation including partial derivative term can be expressed as
∂퐮 휌 ( + 퐮 ∙ ∇퐮) = 퐣×퐁 − 훁푝. ∂푡
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The adiabatic equation is given by
d푝 = −훾푝훁 ∙ 퐮 (2.3) d푡
where 훾 is the heat capacity ratio, of which the value is often 5/3. The adiabatic equation including partial derivative term can be expressed as
휕푝 = −(퐮 ∙ 훻)푝 − 훾푝훻 ∙ 퐮 휕푡 or
휕푝 + 훻 ∙ (푝퐮) = −(훾 − 1)푝훻 ∙ 퐮. 휕푡
Four Maxwell’s equations are given as follows:
Ampere’s law
1 퐣 = (훁×퐁), (2.4) 휇0
Faraday’s law
휕퐁 = −훁×퐄, (2.5) 휕푡
Ohm’s law (ideal mode)
퐄 + 퐮×퐁 = 0. (2.6)
Gauss’ law (Magnetic Conservation law)
훁 ∙ 퐁 = 0. (2.7)
An ideal MHD model is used in this thesis, which the plasma to be a perfect conductor. In the resistive MHD model, plasma resistivity due to collision is taken into consideration. In the resistive MHD formalism, the adiabatic equation is given as
d푝 = −훾푝훁 ∙ 퐮 + (훾 − 1)휂푗2 d푡 and Ohm’s law is given as
퐄 + 퐮×퐁 = 휂퐣 where 휂 is the resistivity of the plasma.
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Summarizing section 2.1.1, the MHD equations are shown in Table 2.1. The governing equation of the thesis consists of 7 ideal MHD equations. Using these equation, the physical phenomena of tokamak and MHD stability will be reviewed.
Table 2.1 The MHD equations Equation Ideal Resistive Continuity d휌 = −휌∇ ∙ 퐮 equation d푡 Momentum d퐮 휌 = 퐣×퐁 − 훁푝 equation d푡 Adiabatic d푝 d푝 = −훾푝훁 ∙ 퐮 = −훾푝훁 ∙ 퐮 + (훾 − 1)휂푗2 equation d푡 d푡 Ampere’s 1 퐣 = (훁×퐁) law 휇0 Faraday’s 휕퐁 = −훁×퐄 law 휕푡
Gauss’ law 훁 ∙ 퐁 = 0
Ohm’s law 퐄 + 퐮×퐁 = 0 퐄 + 퐮×퐁 = 휂퐣
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2.1.2 Mathematical description in tokamak
The schematics of tokamak is shown in Figure 2.1.
Radial direction
X Poloidal direction 휃 휃Ԧ 푟Ԧ A 푟 B O 푅0
Major radius 휙ሬԦ Toroidal direction
(a)
푏
푎
Plasma
Vacuum
(b) Figure. 2.1 The schematics of tokamak. (a) The schematic description of tokamak (b) The poloidal cross section of tokamak.
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For the convenience of calculation, tokamak is assumed to be stretched from a torus shape into a cylinder. Then, the coordinate in the tokamak is a cylindrical coordinate (푟, 휃, 푧). The 휃-direction is the poloidal direction and 푧 direction is toroidal direction. It can be expressed another way as (푟, 휃,
휙) and the relation of these coordinates are 푧 = 푅0휙 where 푅0, called major radius is the distance from the center of the tokamak (point O) magnetic axis (point B, magnetic axis means the center of poloidal section of tokamak) (at 푟 = 0). The toroidal length of the tokamak is 2휋푅0. In the thesis, the plasma is assumed to be axisymmetric.
As described in Figure 2.1 (a), 푅 is the distance from the center of the tokamak to an arbitrary position (point X), and poloidal radial position, 푟, is the distance from the center of the poloidal cross section of the tokamak (푟 = 0) to an arbitrary point (point X). The relation between 푅 and 푟 at point X is expressed from the law of cosine as
2 2 푅 = √푅0 + 푟 + 2푅0푟 cos 휃 (2.8)
The angle 휃 is ∠ABX where point A is the outer edge of the tokamak.
As described in Figure 2.1 (b), the minor radius, 푎, is the plasma radius, and the poloidal radius, 푏 is the distance from the magnetic axis to the conducting wall. The plasma region is at 0 < 푟 ≤ 푎, and the vacuum region is at 푎 < 푟 ≤ 푏.
Based on this mathematical picture, MHD physics on tokamak is reviewed.
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2.1.3 The 푧-Pinch Configuration
The 푧-pinch is the configuration in which the direction of plasma current is the 푧-direction. The real tokamak is configured as a screw pinch. The plasma moves in the toroidal direction following the tokamak. Following the direction of plasma, the plasma current flows as [20, 21]
퐣휙 = 푛𝑖푞𝑖퐯𝑖 + 푛푒푞푒퐯푒 = 푒(푛𝑖퐯𝑖 − 푛푒퐯푒) ≈ 푒푛(퐯𝑖 − 퐯푒) where 퐯𝑖 is velocity of the ion, 퐯푒 is velocity of the electron and the density approximated as 푛 ≃
푛푒 ≃ 푛𝑖 according to the quasi-neutrality of the plasma. According to the momentum conservation,
푀퐯𝑖 = 푚퐯푒. where 푀 is mass of an ion and 푚 is mass of an electron. As given in Appendix A, 푀 ≫ 푚, so |퐯𝑖| ≪
|퐯푒|. Then, the plasma current is a result of the toroidal movement of an electron as
퐣휙 ≈ −푒푛퐯푒. (2.9)
According to Ampere’s law, ‘the poloidal magnetic field is induced by the plasma current’ as shown in Figure 2.3a [20, 21]. Using Eq. (2.4) and Eq. (B.17), the relation between the toroidal current and the poloidal magnetic field is expressed by
1 1 푑 푗휙 = (푟퐵휃). 휇0 푟 푑푟
Multiplying both sides by 푟 and taking the integral, poloidal magnetic field, 퐵휃 is expressed as
휇 퐼(푟) 퐵 (푟) = 0 (2.10) 휃 2휋푟
where
푟 ′ ′ ′ 퐼(푟) = ∫ 푗휙(푟 )2휋푟 d푟 (2.11) 0
is the toroidal current in tokamak, and 푗휙(푟) is the toroidal plasma current density.
15
There are several instabilities caused by current in 푧-pinch. When plasma becomes narrower, the 2 poloidal magnetic field density increases and the magnetic pressure, 푝 = 퐵휃/2휇0 increases in the direction of inward radial direction. Then, the plasma becomes narrower. This is the sausage instability, as shown in Figure 2.2 (a). Similarly, when plasma is kinking upward, the poloidal magnetic field density at the top is larger than the poloidal magnetic field density at the bottom. That is, the magnetic pressure at the top is larger than at the bottom. Then, net magnetic pressure goes upward. This is so called the kink instability as shown in Figure 2.2 (b).
푧
(a)
푧
(b) Figure 2.2 Schematics of instability on 푧-pinch (a) sausage instability (b) kink instability. The arrow direction indicates the magnetic pressure.
Due to these instabilities, self-confining fails without external force in the tokamak itself. So, to control the plasma, external force should be applied.
16
2.1.4 The Screw Pinch Configuration
The screw pinch configuration is the sum of 푧-pinch and 휃-pinch. As mentioned in section 2.1.2, the 푧-pinch is the direction of plasma current in the 푧-direction. Additionally, the 휃-pinch is the direction of plasma current in the 휃-direction. The real tokamak is configured as screw pinch.
As a solution to sausage instability and kink instability on 푧-pinch, an external coil is added to produce external force to control the instability. According to Ampere’s law, ‘the toroidal magnetic field is induced by external coil current’ [21, 22]. The relation between toroidal magnetic field and external field is obtained [22] as
휇 푁푖 퐵 (푅) = 0 coil (2.12) 휙 2휋푅
or
1 퐵휙(푅) = 퐵휙0 푟2 푟 (2.13) √(1 + 2) + cos 휃 푅0 푅0
where 휇0 is vacuum permeability, 푁 is the number of coils turns in tokamak, 푖coil is the electrical current of the coil and 퐵휙0 is the toroidal magnetic field at 푧-axis (푟 = 0) as 퐵휙0 = 휇0푁푖coil/2휋푅.
17
퐼coil
퐵휃
퐼(푟) 퐵휙
(a) (b) Figure. 2.3 The magnetic field line by each component. (a) the toroidal plasma current induces the poloidal magnetic field at the 푧-pinch and (b) The external coil current induces the toroidal magnetic field at the 휃-pinch.
The toroidal magnetic field strength is much stronger than the poloidal magnetic field strength. Then, the magnetic pressure is much higher than that in the 푧-pinch. This pressure endures kink instability and sausage instability. Because the pressure is high, it resists pressure in the radial direction or kinking. Although kinking and sausage instability occurs, the instability is reduced by external force. The external force should be large enough to be stabilized. The details about this problem will be covered in section 2.1.4.
To summarize, the magnetic field in tokamak is expressed in vector form as
퐁 = 퐵휙휙̂ + 퐵휃휃̂ (2.14)
where 퐵휙 is the toroidal magnetic field and 퐵휃 is the poloidal magnetic field. The movement of the plasma follows the movement of the magnetic field line. The pattern of the magnetic field of the line in tokamak is helical.
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2.1.5 Safety factor, 푞
The magnetic field in tokamak is in helical direction. To express the helicity numerically, rotational transformation, 휄 (iota) is introduced [23]. Rotational transformation, 휄 is defined as the poloidal angle when the magnetic field turns one cycle in the toroidal direction. For example, 휄 = 2휋 when the poloidal magnetic field turns one cycle while the toroidal magnetic field turns one cycle. Mathematically, 휄/2휋 is defined as [24]
휄 d휃 = 2휋 d휙
where d휃 is the change of angle in the poloidal magnetic field and d휙 is the corresponding change of angle in the toroidal magnetic field.
Another expression for helicity is the safety factor, 푞 is defined as 푞 = 1/휄. To explain in words, the safety factor is the ratio of the number of toroidal turns to corresponding poloidal turns when the magnetic field line travels in tokamak helically. The role of the safety factor is to roughly determine stability. The critical safety factor varies according to the system. A higher safety factor has a high probability of stability whereas a lower safety factor has a lower probability of stability. The safety factor is defined as
d휙 푞 = . (2.15) d휃
As mentioned in Section 2.1.3, ‘the magnitude of the toroidal magnetic field induced by an external coil current should be larger than the magnitude of the poloidal magnetic field induced by plasma current [12] to reduce the instabilities. That is why in the field of fusion research, researchers prefer to use a safety factor rather than the rotational transformation.
First, the safety factor, 푞 will be calculated only depending on the radial direction. If the poloidal magnetic field travels one turn while the toroidal magnetic field travels at the angle of ∆휙, the safety factor is
∆휙 푞 = . 2휋
The equation of field line is [25]
푅d휙 퐵휙 = . 푟d휃 퐵휃
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Then, 푞 is calculated as
2휋 2휋 ∆휙 1 1 푟퐵휙 푟퐵휙0 1 푞 = = ∮ 푑휙 = ∫ d휃 = ∫ 2 d휃. 2휋 2휋 2휋 0 푅퐵휃 2휋퐵휃푅0 0 푟 푟 (1 + 2) + 2 cos 휃 푅0 푅0
Using integration by substitution of trigonometric function, the integration relation is derived as [26]
2휋 푑푡 2휋 ∫ = , 푎 > 푏. (2.16) 2 2 0 푎 + 푏 cos 푡 √푎 − 푏
The detailed derivation of Eq. (2.16) is in Appendix A. Using Eq. (2.13), the safety factor, 푞(푟) is derived as
푟퐵 ( ) 휙0 푞 푟 = 2 2 . (2.17) 퐵휃푅0(1 − 푟 /푅0)
For a large aspect ratio tokamak, 푟 ≤ 푎 ≪ 푅0, then the safety factor, 푞(푟) approximated as
2 푟퐵휙0 푟 푟퐵휙0 푞(푟) ≈ (1 + 2) ≈ . (2.18) 퐵휃푅0 푅0 퐵휃푅0
Approximated Eq. (2.18) is often used rather than Eq. (2.17).
For further calculation of the safety factor, 푞(푟) at each radial position, the poloidal magnetic field,
퐵휃(푟) at any radial position should be obtained. For large aspect-ratio tokamak, the 휃-averaged current profile, arbitrarily defined as 푗휙 is as follows:
휈 푟2 푗 = 푗 (1 − ) (2.19) 휙 휙0 푎2
where 푗휙0 is the central plasma current, and 휈 is the peakdeness factor. Using integration by substitution to Eq. (2.19), 퐼(푟) in Eq. (2.11) is calculated as
2 2 휈+1 휋푎 푗휙0 푟 퐼(푟) = {1 − (1 − ) } (2.20) 휈 + 1 푎2
and 퐵휃(푟) in Eq. (2.10) is obtained as
2 2 휈+1 푎 휇0푗휙0 푟 퐵 (푟) = {1 − (1 − ) }. (2.21) 휃 2(휈 + 1)푟 푎2
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Then, the general expression for 푞(푟) is
2 2 2(휈 + 1)퐵휙0 푟 /푎 푞(푟) = 2 2 휈+1 . (2.22) 휇0푗휙0푅0 {1 − (1 − 푟 /푎 ) }
At the magnetic axis (푟 = 0), using ‘L’Hospital’s rule’ [27], 푞(0) is
2퐵휙0 푞(0) = (2.23) 휇0푗휙0푅0
and 푞(0) is often denoted as 푞0. At the edge (푟 = 푎), 푞(푎) is
2(휈 + 1)퐵휙0 푞(푎) = (2.24) 휇0푗휙0푅0
and 푞(푎) is often denoted as 푞(푎). In this model, the ratio, 푞(푎)/푞(0), is
푞(푎) = 휈 + 1 (2.25) 푞(0)
This is the definition and calculation process of the safety factor, 푞(푟) with arbitrary current profile, 2 2 휈 푗휙 = 푗휙0(1 − 푟 /푎 ) .
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2.1.6 The Beta, 훽
As described in chapter 1, plasma is confined by pressure in tokamak. The magnetically confined tokamak is in equilibrium. Equilibrium is the complete balance of force [28]. In a tokamak, the force pressure is balanced by the magnetic force as
∇푝 = 퐣×퐁 (2.26)
The beta value, 훽, is ‘the efficiency confinement of plasma pressure by magnetic pressure’ [29] or expressed as
푝 훽 = 2 . (2.27) 퐵 /2휇0
2 2 2 where 퐵 is the magnitude of the magnetic field, or 퐵 = 퐵휙 + 퐵휃. Starting from Ampere’s law,
1 휕퐵휙 1 휕 퐣 = [− 휃̂ + { (푟퐵휃)} 푧̂]. (2.28) 휇0 휕푟 푟 휕푟
The 푟-componenet of equilibrium equation, Eq. (2.26) becomes
2 d푝 1 휕퐵휙 1 휕 1 휕 1 2 2 퐵휃 = [− 퐵휙 − { (푟퐵휃)} 퐵휃] = − [ (퐵휙 + 퐵휃) + ]. (2.29) d푟 휇0 휕푟 푟 휕푟 휇0 휕푟 2 푟
The average pressure, 〈푝〉 is expressed as
푎 ∫ 2휋푟푝d푟 〈푝〉 = 0 휋푎2
Using integration by part, and using boundary condition, 푝(푎) = 0,
푎 1 2 d푝 〈푝〉 = − 2 ∫ 푟 d푟. (2.30) 푎 0 d푟
Substituting Eq. (2.29) into Eq. (2.30), 〈푝〉 is expressed in terms of 퐵휃 and 퐵휙 calculated as
푎 2 퐵2 2 푎 2 2 퐵2 1 2 휕 퐵휃 휙 퐵휃푟 1 휕 푟 퐵휃 2 휕 휙 〈푝〉 = 2 ∫ [푟 ( + ) + ] d푟 = 2 ∫ [ ( ) + 푟 ( )] d푟 푎 0 휕푟 2휇0 2휇0 휇0 푎 0 휕푟 2휇0 휕푟 2휇0 푎 2 2 2 2 2 2 2 푎 2 1 휕 푟 퐵휃 푟 퐵휙 퐵휙푟 퐵휃푎 퐵휙푎 1 퐵휙2휋푟 = 2 ∫ [ ( + ) − ] d푟 = + − 2 ∫ [ ] d푟 푎 0 휕푟 2휇0 2휇0 휇0 2휇0 2휇0 푎 휋 0 2휇0 where 퐵휃푎 is the poloidal magnetic component at 푟 = 푎 , and 퐵휙푎 is the toroidal magnetic component at 푟 = 푎.
22
Then, 〈푝〉 is expressed as
1 2 2 2 〈푝〉 = (퐵휃푎 + 퐵휙푎 − 〈퐵휙〉). (2.31) 2휇0 Finally, the averaged 훽 is obtained as
2 2 2 〈푝〉 퐵휃푎 + 퐵휙푎 − 〈퐵휙〉 〈훽〉 = 2 = 2 2 . (2.32) 퐵 /2휇0 퐵휃푎 + 퐵휙푎 and 〈훽〉 is denoted as 훽 from now on.
Other expressions for 훽 is the poloidal 훽, 훽푝. The toroidal 훽, 훽푡 is also introduced. The relation between 훽, 훽푝 and 훽푡 is [29]
1 1 1 = + . (2.33) 훽 훽푝 훽푡
where
2 2 2 〈푝〉 퐵휃푎 + 퐵휙푎 − 〈퐵휙〉 훽푝 = 2 = 2 (2.34) 퐵휃푎/2휇0 퐵휃푎
and
2 2 2 〈푝〉 퐵휃푎 + 퐵휙푎 − 〈퐵휙〉 훽푡 = 2 = 2 . (2.35) 퐵휙푎/2휇0 퐵휙푎
This is the introduction to 훽 value.
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2.1.7 Properties of large aspect ratio in tokamak
In tokamak, the aspect ratio, 푅0/푎, is assumed to be large. Introducing a small number, 휀, as an inverse large aspect ratio of tokamak, 휀 ≡ 푎/푅 ≪ 1 in tokamak.
The safety factor, 푞 is the order of 1 as
푟퐵휙0 퐵휙0 푞 = ~ 휀 ~ 1. 푅0퐵휃 퐵휃
So, 퐵휃 is the order of
퐵휃 ~ 휀퐵휙. (2.36)
For the equilibrium equation, ∇푝 = 퐣×퐁, each term of Eq. (2.26) is equivalent as
d푝 1 1 휕 ~ { (푟퐵휃)} 퐵휃 (2.37) d푟 휇0 푟 휕푟
and
d푝 1 휕퐵휙 ~ 퐵휙. (2.38) d푟 휇0 휕푟
From Eq. (2.37), 푝 is the order of
퐵2 푝 ~ 휃. (2.39) 휇0
So, poloidal beta, 훽푝, is the order of 1 as
훽푝 ~ 1 (2.40)
and toroidal beta, 훽푡, is the order equivalent as
2 퐵휃푎 2 훽푡 = 2 훽푝 ~ 휀 (2.41) 퐵휙푎
and beta, 훽, is the order equivalant as
2 2 퐵휃푎 퐵휃푎 2 훽 = 2 훽푝 ~ 2 훽푝 ~ 휀 (2.42) 퐵 퐵휙푎
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From Eq. (2.38), 휕퐵휙/휕푟 is the order of
휕퐵 2 퐵 휙 1 d푝 1 퐵휃 1 1 2 2 1 1 2 2 1 2 휙 ~ 휇0 ~ 휇0 ~ (휀 퐵휙) ~ (휀 퐵휙) ~ 휀 . 휕푟 퐵휙 d푟 퐵휙 휇0 푎 퐵휙 푎 퐵휙 푎 푎
From Ampere’s law, 퐣 = 1/휇0(∇×퐁) each componenets of current, 퐣 in Eq. (2.28) are equivalent as
휕퐵 퐵 1 휙 2 휙 2 퐵 푗휃 = ~ 휀 ~ 휀 , (2.43) 휇0 휕푟 휇0푎 휇0푎
1 1 휕 퐵휃 퐵휙 퐵 (2.44) 푗휙 = { (푟퐵휃)} ~ ~ 휀 ~ 휀 . 휇0 푟 휕푟 휇0푎 휇0푎 휇0푎
So, 푗휃 is the order of
푗휃 ~ 휀푗휙. (2.45)
Summarizing section 2.1.7, leading order relation is shown in Table 2.2.
Table 2.2 Leading order relations in large aspect ratio tokamak.
Safety factor 푞 ~ 1
Magnetic field 퐵휃 ~ 휀퐵휙
퐵2 Pressure 푝 ~ 휃 휇0
훽푝 ~ 1
2 Beta, 훽 훽푡 ~ 휀
훽 ~ 휀2
2 퐵 푗휃 ~ 휀 휇0푎 퐵 Current density 푗휙 ~ 휀 휇0푎
푗휃 ~ 휀푗휙
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These properties are used to approximate the equations. The MHD equations can be solved more simply by the approximation of large aspect ratio in tokamak.
This concludes the basic physical concept review of tokamak. In this section, the principle of magnetic fields induced by currents was reviewed. Based on the magnetic fields, the concept and the role of the safety factor and beta value were also reviewed. The properties of large aspect ratio in tokamak was also reviewed. Based on this knowledge, MHD instability in the tokamak will be reviewed to study the sawtooth instability.
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2.2 Review of MHD instability in tokamak
In this section, the basics of MHD instability will be reviewed.
2.2.1 Basic idea of MHD stability
Whether the equilibrium is stable or unstable depends on whether a small perturbation is growing or is damping [28]. When the perturbation is growing, the plasma is in an unstable condition whereas when the perturbation is damping, the plasma is in a stable condition. Sawtooth oscillation is the result of instability, which shows periodic behavior.
The basic idea about stability is shown in Figure 2.4.
Unstable Equilibrium Stable Equilibrium
푥0 푥0 + 휉
푥0 푥0 + 휉
(a) (b) Figure 2.4 The basic idea of MHD stability. (a) Stable equilibrium, (b) unstable equilibrium.
This can be illustrated by thinking about a ball, as shown in Figure 2.4. The ball tends to move in the direction which gives the potential energy to the outer system. In Figure 2.4 (a), the ball is perturbed and potential energy increases (훿푊 > 0). As time goes by, the ball will return to its original point. The ball is determined as stable. In Figure 2.4 (b), the ball is perturbed and potential energy deceases (훿푊 < 0). As time goes by, the ball will roll away.
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2.2.2 Energy principle
Based on the perturbation introduced in section 2.2.1, the energy principle is introduced here. The energy principle is used to determine stability by the sign of change of potential energy, 훿푊. By energy conservation, the summation of the potential energy change and the kinetic energy change is zero. The change of kinetic energy from the perturbation is
1 훿퐾 = ∫ 훏 ∙ 퐟 (훏)d푉 2 1 where 퐟1 is the first order perturbation of force density. According to the law of energy conservation, the potential energy comes from the result of the plasma displacement, 훏 as
1 훿푊 = − ∫ 훏 ∙ 퐟 (훏)d푉. (2.46) 2 1
The change of potential energy, 훿푊 is derived from ideal MHD equations and small displacement, 훏 . Using the perturbation theory, but neglecting the second order term and putting the equilibrium velocity to zero, u0(r) = 0, since the plasma is in static equilibrium. Thus
퐮(퐫, 푡) = 퐮1(퐫, 푡), (2.47)
휌(퐫, 푡) = 휌0(퐫) + 휌1(퐫, 푡), (2.48)
푝(퐫, 푡) = 푝0(퐫) + 푝1(퐫, 푡), (2.49)
퐁(퐫, 푡) = 퐁0(퐫) + 퐁1(퐫, 푡), (2.50)
The mass conservation, separating equilibrium term and first order perturbation term, is given as
휕휌 휕휌 0 + 1 + (퐮 ∙ 훁)휌 = −휌 훁 ∙ 퐮 휕푡 휕푡 1 0 0 1
The equilibrium term for mass conservation is
휕휌 0 = 0, 휕푡 and the first order perturbation term of mass conservation is
휕휌 1 = −퐮 ∙ 훁휌 − 휌 훁 ∙ 퐮 . (2.51) 휕푡 1 0 0 1
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For momentum conservation, the plasma is assumed to be in laminar flow, then (퐮 ∙ 훁)퐮 = 0 and separating equilibrium term and first order perturbation term, the momentum conservation is expressed as
휕퐮 휌 1 = 퐣 ×퐁 + 퐣 ×퐁 + 퐣 ×퐁 − 훁푝 − 훁푝 0 휕푡 0 0 0 1 1 0 0 1
The equilibrium term for momentum conservation is
훁푝0 = 퐣0×퐁0 and the first order perturbation term of momentum conservation is
휕퐮 휌 1 = 퐣 ×퐁 + 퐣 ×퐁 − 훁푝 . (2.52) 0 휕푡 0 1 1 0 1
The adiabatic equation, separating equilibrium term and first order perturbation term neglecting second order perturbation term is
휕푝 휕푝 0 + 1 + (퐮 ∙ 훁)푝 = −훾푝 훁 ∙ 퐮 . 휕푡 휕푡 1 0 0 1
The equilibrium term for the adiabatic equation is
휕푝 0 = 0 휕푡 and the first order perturbation term of the adiabatic equation is
휕푝 1 = −퐮 ∙ 훁푝 − 훾푝훁 ∙ 퐮 . (2.53) 휕푡 1 1
Combining Faraday’s law and Ohm’s law,
휕퐁 = −훁×(−퐮×퐁) = 훁×(퐮×퐁). 휕푡
Separating equilibrium term and first order perturbation term and neglecting second order perturbation term, the equation is
휕퐁 휕퐁 0 + 1 = 훁×(퐮 ×퐁 ). 휕푡 휕푡 1 0
The equilibrium term for the equation is
휕퐁 0 = 0 휕푡
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and the first order perturbation term of the equation is
휕퐁 1 = 훁×(퐮 ×퐁 ). (2.54) 휕푡 1 0
The first order perturbation of ideal MHD equations are as below.
휕휌 1 = −퐮 ∙ 훁휌 − 휌 훁 ∙ 퐮 (2.51) 휕푡 1 0 0 1 휕퐮 휌 1 = 퐣 ×퐁 + 퐣 ×퐁 − 훁푝 (2.52) 0 휕푡 0 1 1 0 1 휕푝 1 = −퐮 ∙ 훁푝 − 훾푝훁 ∙ 퐮 (2.53) 휕푡 1 1 휕퐁 1 = 훁×(퐮 ×퐁 ) (2.54) 휕푡 1 0
The displacement vector, 훏 is the integration of velocity perturbation as
푡 ′ 훏(퐫, 푡) = ∫ 퐮1(퐫, 푡)푑푡 . (2.55) 0
From now on, 훏 is used instead of 퐮1. Integrating Eq. (2.13), Eq. (2.15) and Eq. (2.16) using Eq. (2.55), the Eqs. (2.51) – (2.54) are expressed as
휌1(퐫, 푡) = −훏(퐫, 푡) ∙ 훁휌0 − 휌0훁 ∙ 훏(퐫, 푡) (2.56)
휕2훏 1 1 (2.57) 휌0 2 = (훁×퐁0)×퐁1 + (훁×퐁1)×퐁0 − 훁푝1 휕푡 휇0 휇0
푝1(퐫, 푡) = −훏(퐫, 푡) ∙ 훁푝0 − 훾푝0훁 ∙ 훏(퐫, 푡) (2.58)
퐁1(퐫, 푡) = 훁×{훏(퐫, 푡)×퐁0(퐫)} (2.59)
Substituting Eq. (2.58) and Eq. (2.59) into Eq. (2.57),
휕2훏 1 휌0 2 = 퐣0×{훁×(훏×퐁0)} + [훁×{훁×(훏×퐁0)}]×퐁0 + 훁(훏 ∙ 훁푝0 + 훾푝0훁 ∙ 훏) 휕푡 휇0 and the left hand side of this term is first order perturbation of force density. So, the force density perturbation is obtained as
1 퐟1(훏) = 퐣0×퐁1 + [훁×{훁×(훏×퐁0)}]×퐁0 + 훁(훏 ∙ 훁푝0 + 훾푝0훁 ∙ 훏). (2.60) 휇0
30
Substituting Eq. (2.60) into Eq. (2.46), the change of potential energy, 훿푊, is obtained as
1 1 훿푊 = − ∫ [훏 ∙ (퐣0×퐁1) + 훏 ∙ [(훁×퐁1)×퐁0] − 훏 ∙ 훁(훾푝0훁 ∙ 훏 − 훏 ∙ 훁푝0)] d푉. (2.61) 2 휇0
Using the vector relations Eq. (B.2), Eq. (B.7) and Eq. (B.10) to Eq. (2.61), 훿푊, is obtained as
1 1 1 훿푊 = − ∫ [훁 ∙ {( 퐁0 ∙ 퐁1) 훏} + 퐁1 ∙ {훁×(훏×퐁0)} + 퐣0 ∙ (퐁1×훏) + 훁 ∙ (푝1훏) + 푝1(훁 ∙ 훏)] d푉. 2 휇0 휇0
Using divergence theorem,
1 1 1 1 2 훿푊 = ∫ ( 퐁0 ∙ 퐁1 + 푝1) 훏 ∙ d퐀 + ∫ { 퐵1 − 퐣0 ∙ (퐁1×훏) − 푝1(훁 ∙ 훏)} d푉. (2.62) 2 휇0 2 휇0
The potential energy change, 훿푊, can be distinguished by three different regions: plasma region
(0 < 푟 < 푎), 훿푊푝, plasma surface region (푟 = 푎), 훿푊푆 and vacuum region (푎 < 푟 ≤ 푏), 훿푊푣. At the vacuum region, there is no displacement. So, the Eq. (2.52) is divided as
1 1 2 훿푊 = ∫ { 퐵1푝 − 퐣0 ∙ (퐁1×훏) − 푝1(훁 ∙ 훏)} d푉 2 plasma 휇0 (2.63) 1 1 1 2 + ∫ ( 퐁0 ∙ 퐁1 + 푝1) 훏 ∙ d퐀 + ∫ 퐵1푣d푉. 2 surface 휇0 2휇0 vacuum
where 퐁1푣 is perturbed magnetic field at vacuum and
1 퐣1푣 = (∇×퐁1푣) = 0. (2.64) 휇0
is satisfied. Each term 훿푊푝, 훿푊푠 and 훿푊푣 is
1 1 2 훿푊푝 = ∫ { 퐵1푝 − 퐣0 ∙ (퐁1×훏) − 푝1(훁 ∙ 훏)} d푉, (2.65) 2 plasma 휇0 1 1 (2.66) 훿푊푠 = ∫ ( 퐁0 ∙ 퐁1 + 푝1) 훏 ∙ d퐀, 2 surface 휇0 (2.67) 1 2 훿푊 = ∫ 퐵1푣d푉. 2휇0 vacuum
31
For steady state, the continuity equation, Eq. (2.1) is
∇ ∙ 퐮1 = 0. (2.68)
Integrating Eq. (2.68) by 푡
∇ ∙ 훏 = 0. (2.69)
The volume integral in tokamak is
∫ d푉 = ∫ 푟d푟d휃d푧 = ∫ 푟d푟d휃푅0d휙 = 푅0 ∫ 푟d푟d휃d휙
Substituting Eq. (2.69) into Eq. (2.63), the surface term is removed by taking volume integral, and assuming 휉푟 ~ 휉휃 ≫ 휉휙 and 퐵푟1 ~ 퐵휃1 ≫ 퐵휙1,
1 1 2 1 2 훿푊 = ∫ { 퐵1푝 − 퐣0 ∙ (퐁1푝×훏)} 푟d푟d휃d푧 + ∫ 퐵1푣푟d푟d휃d푧 2 휇0 2휇0 2휋 2휋 푎 2휋 2휋 푏 1 1 2 1 2 = ∫ ∫ ∫ { 퐵1푝 − 퐣0 ∙ (퐁1푝×훏)} 푟d푟 d휃 푅0d휙 + ∫ ∫ ∫ 퐵1푣푟d푟 d휃 푅0d휙 2 0 0 0 휇0 2휇0 0 0 푎 푎 2 푏 2 1 2 2휋 푅0 2 = 2휋 푅0 ∫ { 퐵1푝 − 퐣0 ∙ (퐁1푝×훏)} 푟d푟 + ∫ 퐵1푣푟d푟 0 휇0 휇0 푎 푎 퐵2 + 퐵2 2 푏 2 1푝,푟 1푝,휃 2휋 푅0 2 2 = 2휋 푅0 ∫ { − 푗휙 ∙ (퐵푟1휉휃 − 퐵휃1휉푟)} 푟d푟 + ∫ (퐵1푣,푟 + 퐵1푣,휃)푟d푟 0 휇0 휇0 푎
So, the change of potential energy, 훿푊, is
푎 퐵2 + 퐵2 2 1푝,푟 1푝,휃 훿푊 = 2휋 푅 ∫ { − 푗휙 ∙ (퐵1푝,푟휉휃 − 퐵1푝,휃휉푟)} 푟d푟 0 휇0 (2.70) 2 푏 2휋 푅0 2 2 + ∫ (퐵1푣,푟 + 퐵1푣,휃)푟d푟. 휇0 푎
where the first term is plasma term and the second term is vacuum term.
The plasma term, 훿푊푝 which is the first term of Eq. (2.70), is first calculated. The displacement, 훏, analyzed by Fourier analysis as 훏 ~ 푒𝑖(푚휃−푛휙), is given as
𝑖(푚휃−푛휙) 𝑖(푚휃−푛휙) 훏 = 훏0푒 = (휉푟푟̂ + 휉휃휃̂)푒 . (2.71)
Substituting Eq. (2.71) into Eq. (2.70) and using Eq. (B.17),
1 d 1 ∇ ∙ 훏 = { (푟휉 ) + 푖푚휉 } 푒𝑖(푚휃−푛휙) = 0. 푟 d푟 푟 푟 휃
32
Then, the relation between 휉푟 and 휉휃 is
푖 d 휉 = − (푟휉 ). (2.72) 휃 푚 d푟 푟
Substituting Eq. (2.72) into Eq. (2.59) and using Eq. (B.18),
푖푚퐵 퐵 휙0 푛 1 휙0 휕 푛 1 𝑖(푚휃−푛휙) 퐁1푝 = [− ( − ) 휉푟푟̂ + [ {( − ) 푟휉푟}] 휃̂] 푒 푅0 푚 푞 푅0 휕푟 푚 푞 where the safety factor, 푞 is
푟퐵휙0 푞 = . 푅0퐵휃 then, the components of perturbed magnetic field at plasma region are
푖푚퐵휙0 푛 1 퐵1푝,푟 = − ( − ) 휉푟, (2.73) 푅0 푚 푞
퐵휙0 휕 푛 1 (2.74) 퐵1푝,휃 = [ {( − ) 푟휉푟}]. 푅0 휕푟 푚 푞
Substituting Eq. (2.72), Eq. (2.73) and Eq. (2.74) into the first term of Eq. (2.70), the plasma term, 훿푊푝 is obtained as
푛 1 2 2 2 2 2 푎 2휋 푟 퐵 ( − ) d휉 휙0 푚 푞 푟 훿푊푝 = ∫ 2 2 ( ) 0 휇0푅0 푛 푟 d푟 1 + 2 2 [ 푚 푅0
2 2 2 2 ′ 2 2휋 퐵휙0 푛 1 4휋 푟푝 푛 1 + { ( − ) + ( ) } 휉2 푟푑푟 (2.75) 2 푚 푞 푅 푚 푛푟 2 푟 휇0푅0 0 1 + ( ) 푚푅0 ] 푛2 1 2 2 ( − ) 2휋 퐵휙0 푚2 푞2 + 푎2휉2. 휇 푅 푛푟 2 푎 0 0 1 + ( ) 푚푅0
where 휉푎 is the radial displacement, 휉푟, at 푟 = 푎.
33
Subsequently, the vacuum term, 푊푣 which is the second term of Eq. (2.70) is calculated. As in Eq.
(2.64), ∇×퐁1푣 = 0, and the flux function, 훙 satisfies 퐁1푣 = ∇×훙. Then
∇×(∇×훙) = ∇(∇ ∙ 훙) − ∇2훙 = 0 or
∇2훙 = 0 or
1 휕 휕휓 1 휕2휓 휕휓 (푟 ) + + = 0. 푟 휕푟 휕푟 푟2 휕휃2 휕푧2
𝑖(푚휃−푛휙) Then flux function 휓 can be expressed as 휓0푒 , and 휓 satisfies Bessel’s equation as
1 푑 푑휓 푛2 푚2 (푟 ) − ( 2 + 2 ) 휓 = 0, (2.76) 푟 푑푟 푑푟 푅0 푟
and the magnetic field components in the vacuum are expressed in terms of 휓 as
1 휕휓 푖푚 퐵 = − = − 휓, (2.77) 1푣,푟 푟 휕휃 푟 휕휓 (2.78) 퐵 = . 1푣,푟 휕푟 The solution of 휓 is
푛 푛 휓 = 퐶1퐼푚 ( 푟) + 퐶2퐾푚 ( 푟). (2.79) 푅0 푅0
At 푟 = 푎, 퐵1,푟 is continuous, so
푖푚 푖푚퐵휙0 푛 1 퐵1푣,푟(푎) = 퐵1푝,푟(푎) = 휓푎 = ( − ) 휉푎 푎 푅0 푚 푞푎
Then 휓(푎) is
푎퐵휙0 푛 1 푎퐵휙0 푛 1 휓(푎) = ( − ) 휉푎 = ( − ) 휉푎 푅0 푚 푞푎 푅0 푚 푞푎
At 푟 = 푏, 휓(푏) = 0, then the solution of 휓(푟) is
푛푏 푛 푛푎 푛 퐼푚 ( ) 퐾푚 ( 푟) − 퐾푚 ( ) 퐼푚 ( 푟) 푅0 푅0 푅0 푅0 푎퐵휙 푛 1 휓(푟) = ( − ) 휉푎. (2.80) 푛푏 푛푎 푛푏 푛푎 푅0 푚 푞푎 퐼푚 ( ) 퐾푚 ( ) − 퐾푚 ( ) 퐼푚 ( ) 푅0 푅0 푅0 푅0
34
Substituting Eq. (2.77) and Eq. (2.78) into the second term of Eq. (2.70), the vacuum term is calculated as
2 푏 2 푏 2 2 2휋 푅0 2 2 2휋 푅0 푚 d휓 훿푊푣 = ∫ (퐵1푣,푟 + 퐵1푣,휃)푟d푟 = ∫ {( 휓) + ( ) } 푟d푟 휇0 푎 휇0 푎 푟 d푟 2 푏 2 2휋 푅0 푚 2 d휓 d휓 = ∫ (푟 2 휓 + 푟 ) d푟 휇0 푎 푟 d푟 d푟 2 푏 2 2 푏 푏 2휋 푅0 푚 2 2휋 푅0 d휓 d d휓 = ∫ (푟 2 휓 ) d푟 + {[휓푟 ] − ∫ 휓 (푟 ) d푟} 휇0 푎 푟 휇0 d푟 푎 푎 d푟 d푟 2 푏 2 2 푏 2휋 푅0 푚 2 d d휓 2휋 푅0 d휓 = ∫ {푟 2 휓 − 휓 (푟 ) d푟} d푟 + [휓푟 ] 휇0 푎 푟 d푟 d푟 휇0 d푟 푎
Then,
2휋2푅 푏 푚2 1 d d휓 2휋2푅 d휓 푏 0 0 (2.81) 훿푊푣 ≈ ∫ 휓 ( 2 휓 − (푟 ) d푟) 푟d푟 + [휓푟 ] . 휇0 푎 푟 푟 d푟 d푟 휇0 d푟 푎
Substituting Eq. (2.76) into Eq. (2.81),
2 2휋 푅0 ′ 훿푊푣 = − 푎휓(푎)휓 (푎). (2.82) 휇0
Substituting Eq. (2.80) into Eq. (2.82), the vacuum term is calculated as
′ 푛푎 푛푏 ′ 푛푎 푛푎 2휋2푛퐵2 퐾푚 ( ) 퐼푚 ( ) − 퐼푚 ( ) 퐾푚 ( ) 2 휙0 푅0 푅0 푅0 푅0 푛 1 3 2 훿푊푣 = − 2 ( − ) 푎 휉푎. (2.83) 휇0푅0 푛푎 푛푏 푛푎 푛푏 푚 푞푎 퐾푚 ( ) 퐼푚 ( ) − 퐼푚 ( ) 퐾푚 ( ) 푅0 푅0 푅0 푅0
The properties of modified Bessel function given as
d 휈 (2.84) {퐼 (푥)} = 퐼 (푥) + 퐼 (푥), d푥 휈 푥 휈 휈+1 d 푛 (2.85) {퐾 (푥)} = 퐾 (푥) − 퐾 (푥). d푥 푛 푥 푛 푛+1
35
Using these properties, 훿푊푣 in Eq. (2.83) is calculated as
훿푊푣
푚 푅0 푛푎 푛푎 푛푏 푚 푅0 푛푎 푛푎 푛푎 2 2 { 퐾푚 ( ) − 퐾푚+1 ( )} 퐼푚 ( ) − { 퐼푚 ( ) + 퐼푚+1 ( )} 퐾푚 ( ) 2휋 푛퐵휙0 푛 푎 푅0 푅0 푅0 푛 푎 푅0 푅0 푅0 푛 = − 2 ( 휇0푅0 푛푎 푛푏 푛푎 푛푏 푚 퐾푚 ( ) 퐼푚 ( ) − 퐼푚 ( ) 퐾푚 ( ) 푅0 푅0 푅0 푅0 2 1 3 2 − ) 푎 휉푎 푞푎 2휋2퐵2 2 휙0 푛 1 2 2 = − 푚 ( − ) 푎 휉푎 휇0푅0 푚 푞푎 푛푎 푛푏 푛푎 푛푎 2휋2푛퐵2 퐾푚+1 ( ) 퐼푚 ( ) + 퐼푚+1 ( ) 퐾푚 ( ) 2 휙0 푅0 푅0 푅0 푅0 푛 1 3 2 + 2 ( − ) 푎 휉푎 휇0푅0 푛푎 푛푏 푛푎 푛푏 푚 푞푎 퐾푚 ( ) 퐼푚 ( ) − 퐼푚 ( ) 퐾푚 ( ) 푅0 푅0 푅0 푅0 푛푎 푛푏 푛푎 푛푎 2휋2퐵2 2 퐾푚+1 ( ) 퐼푚 ( ) + 퐼푚+1 ( ) 퐾푚 ( ) 휙0 푛 1 푛푎 푅0 푅0 푅0 푅0 2 2 = 푚 ( − ) { − 1} 푎 휉푎 휇0푅0 푚 푞푎 푚푅0 푛푎 푛푏 푛푎 푛푏 퐾푚 ( ) 퐼푚 ( ) − 퐼푚 ( ) 퐾푚 ( ) 푅0 푅0 푅0 푅0
Then, the vacuum term, 훿푊푣 is obtained as
푛푎 푛푎 퐼 ( ) 퐾 ( ) 푚+1 푅 푚 푅 1 + 0 0 푛푎 푛푏 푛푎 2 2 푛푎퐾 ( ) 퐼 ( ) 퐾푚+1 ( ) 2휋 퐵휙0 푚+1 푅 푚 푅 푅0 훿푊 = 푚 0 0 푣 휇 푅 푛푎 푛푎 푛푏 0 0 푚푅0퐾푚 ( ) 퐼 ( ) 퐾 ( ) 푅0 푚 푅 푚 푅 1 − 0 0 푛푏 푛푎 퐼 ( ) 퐾푚 ( ) { 푚 푅 푅0 0 (2.86)
2 푛 1 2 2 − 1 ( − ) 푎 휉푎 . 푚 푞푎
}
36
Finally, summing Eq. (2.75) and Eq. (2.86), the total change of potential energy is derived as [8]
푛2 1 푎 2 2 2 − 2 2 2휋 퐵휙0 푚 푞 푛 1 훿푊 = ∫ (푓휉′2 + 푔휉2)푟푑푟 + 푎2휉2 푎 + 푚훬 ( − ) . (2.87) 휇 푅 푎 푛푟 2 푚 푞 0 0 0 1 + ( ) 푎 { 푚푅0 } where
푛 1 2 푛푟 2 푛푟 2 2 2 2 ( − ) 2 2 2 2 {1 + ( ) } − ( ) 2휋 푟 퐵휙0 푚 푞 2휋 푟 퐵휙0 푛 1 푚푅0 푚푅0 푓 = 2 2 = ( − ) 2 휇0푅0 푛 푟 휇0푅0 푚 푞 푛푟 1 + 2 2 1 + ( ) 푚 푅0 푚푅0 푛푟 2 2 2 2 2 ( ) 2 2 2 2 2 2 2휋 푟 퐵휙0 푛 1 푚푅 2휋 푟 퐵휙0 푛 1 푟 푛 = ( − ) {1 − 0 } = ( − ) {1 − ( ) ( ) } 휇 푅 푚 푞 푛푟 2 휇 푅 푚 푞 푅 푚 0 0 1 + ( ) 0 0 0 푚푅0 and
2 ′ 2 2 2 2 4휋 푟푝 푛 1 2휋 퐵휙0 푛 1 1 푔 = ( ) + ( − ) {푚2 − } 푅 푚 푛푟 2 2 푚 푞 푛푟 2 0 1 + ( ) 휇0푅0 1 + ( ) 푚푅0 푚푅0 2 2 2 2 2 4휋 퐵휙 푛 푟 푛 1 1 + ( ) ( ) ( 2 − 2) 2 푅0 푚 푅0 푚 푞 푛푟 2 {1 + ( ) } 푚푅0 푛푟 2 푛푟 2 푛푟 2 푛푟 2 2 ′ 2 {1 + ( ) } − ( ) 2 2 2 {1 + ( ) } − ( ) 4휋 푟푝 푛 푚푅 푚푅 2휋 퐵휙0 푛 1 푚푅 푚푅 = ( ) 0 0 + ( − ) {푚2 − 0 0 } 푅 푚 푛푟 2 휇 푅 푚 푞 푛푟 2 0 1 + ( ) 0 0 1 + ( ) 푚푅0 푚푅0 푛푟 2 푛푟 2 2 2 2 2 2 {1 + ( ) } − ( ) 4휋 퐵휙0 푛 푟 푛 1 푚푅0 푚푅0 + ( ) ( ) ( 2 − 2) 2 휇0푅0 푚 푅0 푚 푞 푛푟 2 {1 + ( ) } 푚푅0 and
푛푎 푛푎 퐼 ( ) 퐾 ( ) 푚+1 푅 푚 푅 1 + 0 0 푛푎 푛푏 푛푎 푛푎퐾 ( ) 퐼 ( ) 퐾푚+1 ( ) 푚+1 푅 푚 푅 푅0 훬 = 0 0 − 1. 푛푎 푛푎 푛푏 푚푅0퐾푚 ( ) 퐼 ( ) 퐾 ( ) 푅0 푚 푅 푚 푅 1 − 0 0 푛푏 푛푎 퐼푚 ( ) 퐾푚 ( ) 푅0 푅0
37
The change of potential, 훿푊 consists of plasma term, 훿푊푝, and the vacuum term, 훿푊푣, as
푛2 1 푎 2 2 − 2휋 퐵휙0 푚2 푞2 훿푊 = ∫ (푓휉′2 + 푔휉2)푟푑푟 + 푎2휉2 , 푝 휇 푅 푎 푛푟 2 0 0 0 1 + ( ) 푚푅0
2휋2퐵2 2 휙0 2 2 푛 1 훿푊푣 = 푎 휉푎 푚훬 ( − ) . 휇0푅0 푚 푞
Applying large aspect ratio assumption, 푟/푅 ≪ 1, functions 푓 and 푔 are approximated as
2 2 2 2 2휋 푟 퐵휙0 푛 1 푓 ≈ ( − ) , 휇0푅0 푚 푞
2 2 2 2 2 2 2 2휋 퐵휙0 푛 1 2휋 푛 퐵휙0 푟 푛 1 3푛 1 푔 ≈ ( − ) (푚2 − 1) + ( ) {2푟푝′ + ( ) ( − ) ( + )}. 휇0푅0 푚 푞 푅0 푚 휇0 푅0 푚 푞 푚 푞
2 The beta values are equivalent to 훽 ~ 훽푡 ~ 휀 and 훽푝 ~ 1. Then, the change of potential can be divided into second order and fourth order terms as
2휋2퐵2 푎 2 2 휙0 푑휉 2 2 푛 1 훿푊2 = ∫ {(푟 ) + (푚 − 1)휉 } ( − ) 푟푑푟 휇0푅0 0 푑푟 푚 푞 푛2 1 2휋2퐵2 2 − 2 2 2 휙 2 2 푚 푞 푎 1 푛 + 휉푎 푚 2 2 + 훬 ( − ) , 휇0푅 푛 푚 푚 푞 푚 2 + 2 {푅0 푎 }
2 2 푎 퐵2 2휋 푛 ′ 휙0 푛 1 3푛 1 2 훿푊4 = ( ) ∫ {2푟푝 + ( − ) ( + )} 휉 푟푑푟 푅0 푚 0 휇0 푚 푞 푚 푞 and
퐵2 2 2 휙0 2 푟 훿푊2 ~ 2휋 푅0 휉 푂 ( ) , 휇0 푅0
퐵2 4 2 휙0 2 푟 훿푊4 ~ 2휋 푅0 휉 푂 ( ) 휇0 푅0
This is the whole derivation process for change of potential energy, 훿푊, in tokamak using the energy principle. Using the change of potential energy, sawtooth instability is reviewed.
38
2.3 Theories of sawtooth instability
2.3.1 Internal kink, 푚/푛 = 1/1 mode
As mentioned in section 1.3, sawtooth oscillation is a type of ‘푚 = 1 internal kink mode’ [9]. In the internal mode, plasma surface displacement is zero (휉푎 = 0) whereas the external kink mode is nonzero
(휉푎 ≠ 0). For the external mode, stability is effected by both the integral term and the other term of Eq. (2.87). So, the external kink mode both has 푚 = 1 mode and 푚 ≥ 2 mode. For the internal kink mode, with 휉푎 = 0, Eq. (2.87) becomes
2 2 푎 2 2 2휋 퐵휙 푑휉 푛 1 훿푊 = ∫ {(푟 ) + (푚2 − 1)휉2} ( − ) 푟푑푟 휇0푅0 0 푑푟 푚 푞 (2.88) 2 2 푎 2 2휋 푛 퐵휙 푛 1 3푛 1 + ( ) ∫ {2푟푝′ + ( − ) ( + )} 휉2푟푑푟. 푅0 푚 0 휇0 푚 푞 푚 푞
with
2휋2퐵2 푎 2 2 휙 푑휉 2 2 푛 1 훿푊2 = ∫ {(푟 ) + (푚 − 1)휉 } ( − ) 푟푑푟, 휇0푅0 0 푑푟 푚 푞
2 2 푎 퐵2 2휋 푛 ′ 휙 푛 1 3푛 1 2 훿푊4 = ( ) ∫ {2푟푝 + ( − ) ( + )} 휉 푟푑푟. 푅0 푚 0 휇0 푚 푞 푚 푞
Instability occurs when 훿푊 < 0, thus the plasma is likely to be unstable, when 훿푊2 is minimized. Then, the poloidal number, 푚, is always 푚 = 1 for internal kink mode. So,
2 2 푎 2 2 2휋 퐵휙 푑휉 1 훿푊 = ∫ (푟 ) (푛 − ) 푟푑푟 휇0푅0 0 푑푟 푞 (2.89) 2 2 푎 2 2휋 푛 퐵휙 1 1 + ∫ {2푟푝′ + (푛 − ) (3푛 + )} 휉2푟푑푟. 푅0 0 휇0 푞 푞
with
2 2 푎 2 2 2휋 퐵휙 푑휉 1 훿푊2 = ∫ (푟 ) (푛 − ) 푟푑푟, 휇0푅 0 푑푟 푞
2 2 푎 퐵2 2휋 푛 ′ 휙 1 1 2 훿푊4 = ∫ {2푟푝 + (푛 − ) (3푛 + )} 휉 푟푑푟. 푅0 0 휇0 푞 푞
According to Eq. (2.89), instability occurs at 푞 = 1/푛 surface. But the real tokamak has 푞0 > 0.5. Hnece, the only the possible toroidal number, 푛 , is 푛 = 1 . This concludes that the internal kink instability with 푚/푛 = 1/1 occurs if 푞 = 1 surface exists.
39
Finally, the change of potential energy, 훿푊, for 푚/푛 = 1/1 mode is derived as
2 2 푎 2 2 2휋 퐵휙 푑휉 1 훿푊 = ∫ (푟 ) (1 − ) 푟푑푟 휇0푅0 0 푑푟 푞 (2.90) 2 푎 2 2휋 퐵휙 1 1 + ∫ {2푟푝′ + (1 − ) (3 + )} 휉2푟푑푟 푅0 0 휇0 푞 푞
with
2 2 푎 2 2 2휋 퐵휙 푑휉 1 훿푊2 = ∫ (푟 ) (1 − ) 푟푑푟, 휇0푅 0 푑푟 푞
2 푎 2 2휋 ′ 퐵휃 2 훿푊4 = ∫ {2푟푝 − (1 − 푞)(1 + 3푞)} 휉 푟푑푟. 푅0 0 휇0
or
푎 2 2 2 퐵휃 2 d휉 훿푊 = 2휋 푅0 ∫ (1 − 푞) ( ) 푟푑푟 0 휇0 d푟 (2.91) 2휋2 푎 퐵2 + ∫ {2푟푝′ − 휃 (1 − 푞)(1 + 3푞)} 휉2푟푑푟 푅0 0 휇0
with
푎 2 2 2 퐵휃 2 푑휉 훿푊2 = 2휋 푅0 ∫ (1 − 푞) ( ) 푟푑푟, 0 휇0 푑푟
2 푎 2 2휋 ′ 퐵휃 2 훿푊4 = ∫ {2푟푝 − (1 − 푞)(1 + 3푞)} 휉 푟푑푟. 푅0 0 휇0
Two models for sawtooth instability are introduced to minimize 훿푊2 of Eq. (2.91). Kadomtsev’s model tries to explain instability by taking d휉/d푟 → 0 inside the 푞 = 1 surface [30], whereas Wesson’s model tries to explain instability by taking 푞 → 1 inside the 푞 = 1 surface.
40
2.3.2 Kadomtsev’s model
As mentioned in section 2.3.1, Kaomtsev’s model tries to minimize 훿푊2 by d휉/d푟 → 0 inside the
푞 = 1 surface. At the magnetic axis (푟 = 0), the displacement is 휉 = 휉0, and at the edge (푟 = 푎), 휉푎 =
0. To minimize 훿푊2, d휉/d푟 = 0 except near 푞 = 1 surface (∵ At 푞 = 1, d휉/d푟 cannot be zero). Based on this picture, the radial displacement distribution, 휉(푟), is described in Figure 2.5 as
휉
휉 = 휉0 as 푟 < 푟1 휉0
푞 = 1 surface
휉 = 0 as 푟 > 푟1
푟1 − 훿 푟1 푟
Figure 2.5 The displacement distribution, 휉(푟), based on Kadomtsev’s model.
The corresponding 훿푊 is
2 2 푟1 2 2휋 휉0 ′ 퐵휃 훿푊 = 2 ∫ {2푟푝 − (1 − 푞)(1 + 3푞)} 푟푑푟 (2.92) 푅 0 휇0
′ and 푝 < 0 (∵ 푝 = 푛푘퐵푇 and 푇 is decreased as 푟 increased as shown in Figure 1.3), then it is concluded as 훿푊 < 0.
41
The 푚/푛 = 1/1 internal kink mode has a radial displacement of 휉 ~ 푒𝑖(휃−휙).
① ②
① ③ ④
④ 휉 = 휉0 cos(휃 − 휙) ②
③
(a) (b) Figure 2.6 The 푚/푛 = 1/1 internal kink within 푞 = 1 surface. (a) total view of tokamak (b) poloidal view of tokamak.
The corresponding flow pattern of Figure 2.6 (b) - ② is shown in Figure 2.7 as
푣2
푣1 훿
푣2
Figure 2.7 The flow pattern of plasma in Kadomtsev’s model. The red region is hot region, and the blue region is cold region.
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The velocity, 푣1, is the hot core velocity and the velocity, 푣2, is the plasma velocity which comes from the narrow layer. The thickness, 훿, is the thickness of the narrow resistive layer.
The helical magnetic field line is
∗ 퐵 = 퐵휃 − (푟/푅)퐵휙 = 퐵휃(1 − 푞) (2.93)
and current, 푗 comes from the electric field as equivalent to
푣 퐵∗ 푗 ~ 휎푣 퐵∗ = 1 . (2.94) 1 휂
From Ampere’s law,
∗ 푗~퐵 /휇0훿. (2.95)
Combining Eq. (2.94) and Eq. (2.95), 푣1 is expressed as 휂 푣1 = . (2.96) 휇0훿
The magnetic pressure is equivalent to plasma pressure by 푣2 as
∗2 퐵 1 2 1 2 2 ~ 휌푣2 + 휌푣2 = 휌푣2 . 2휇0 2 2
Then, the velocity 푣2 is
퐵∗ 푣2~ . (2.97) √2휌휇0
From continuity equation
푣1푟1~푣2훿, then
푣1푟1 휂푟1√2휌휇0 훿 ~ = ∗ . 푣2 퐵 휇0훿 or
1/2 휏퐴 훿 ~ ( ) 푟1 (2.98) 휏푅
43
where
휇0 휏 = 푟2, (2.99) 푅 휂 1 푟 휏 = 1 . (2.100) 퐴 ∗ 퐵 /√휌휇0 The collapse time is given by
푟1 휏퐾 ~ . (2.101) 푣1
‘ Substituting (2.95) into (2.98), the collapse time is calculated as
2 훿 휇0 1/2 휏퐴 휇0 2 1/2 1/2 휏퐾 ~ 1/2 ~ (휏푅/휏퐴) ( ) 푟1 = (휏퐴/휏푅) 휏푅 = (휏퐴휏푅) (휏퐴/휏푅) 휂 휏푅 휂
Finally, the collapse time in Kadomtsev’s model is
1/2 휏퐾 ~ (휏퐴휏푅) . (2.102)
The whole process of sawtooth oscillation in Kadomtsev’s model is shown in Figure 2.9. Initially, 푞 > 1 everywhere as in Figure 2.9 (a). As the temperature at the core increases, 푞 = 1 surface appears and, instability occurs as in Figure 2.9 (b). The core starts to move by internal kink, and a new magnetic island appears on the other side, resulting in two apparent islands. One magnetic island by internal kink has 푞 < 1 whereas the magnetic island on the other side has 푞 > 1 . Then, magnetic reconnection occurs at the narrow resistive layer near 푞 = 1 surface as in Figure 2.9 (c). One magnetic island which has 푞 < 1 is annihilated by magnetic reconnection whereas the new magnetic island which has 푞 > 1 is expanded. After the old magnetic island is annihilated completely, it returns to the initial state as 푞 > 1 everywhere as in Figure 2.9 (d). Then the periodic behavior is repeated, starting from the state illustrated in Figure 2.9 (a). This is the theoretical review of Kadomtsev’s model to observe fast crash by magnetic reconnection.
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Core starts to move Two island systems (a) 푞 > 1 everywhere (b) Internal kink
푞 < 1 New magnetic island 푞 > 1 푞 < 1
(c) Magnetic reconnection (d) 푞 > 1 everywhere
Magnetic reconnection
Figure 2.8 The whole process of sawtooth oscillation in Kadomtsev’s model [31].
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2.3.3 Wesson’s model
Wesson’s model tries to explain the inconsistencies of Kadomtsev’s model. There are a few discrepancies in Kadomtsev’s model when compared to the experimental observations. One discrepancy is the crash time. The collapse times of several types of tokamak, using Kadomtsev’ model, 20 −3 were calculated as in Table 2.3 with 푇 = 1 keV, 푛 = 2×10 m , and 푞0 ~ 0.9 . The detailed calculation process is in Appendix C.2. According to Table 2.1, Kadomtsev’ collapse time is 휏퐾 ∝ −4 −3 −4 푟1/√퐵휙, and 휏퐾 ~ 2×10 -3×10 s. But the actual tokamaks have always 휏퐾 ~ 10 s and does not depend on the tokamak geometry [32].
To explain this inconsistency, Wesson assumed a flat 푞-profile inside the 푞 = 1 surface. By taking 푞 → 1 to Eq. (2.91),
2휋2 푎 퐵2 훿푊 → ∫ {2푟푝′ − 휃 (1 − 푞)(1 + 3푞)} 휉2푟푑푟 (2.103) 푅0 0 휇0
and the corresponding displacement distribution [18] is illustrated in Figure 2.10.
Kadomtsev’s collapse time calculation is not applicable for the 푞 → 1 case, since 휏퐾 becomes infinite. So, Wesson introduced a new model [18]. In this model, there is no magnetic shear stress inside the 푞 = 1 surface, because of the magnetic shear stress, given in Eq. (2.103) [12],
푟 d푞 푠 = (2.103) 푞 d푟
goes to zero as 푞 goes to 1. Due to the absence of magnetic shear, the magnetic surface deforms into a crescent shape, and cold plasma comes into the core as shown in Figure 2.10 [14, 33]. This flow pattern is called ‘hot crescent, cold bubble’. The magnetic reconnection takes place during the ramp phase and not in the fast collapse phase [18].
46
Table 2.3 Calculated Kadomtsev’s collapse time according to several tokamaks [34-39]. Major Toroidal Resistive Kadomtsev's q=1 surface Alfven Tokamak radius Magnetic diffusion collapse radius [m] time [s] [m] field [T] time [s] time [s] 0.1 DIII-D 1.67 2.2 2.26×10−1 4.91×10−6 1.05×10−3 (estimated)
JET 2.96 3.45 0.2 9.05×10−1 5.54×10−6 2.24×10−3
0.03 ADITYA 0.75 1.2 2.04×10−2 4.04×10−6 2.87×10−4 (estimated) 0.1 ASDEX 1.65 3.1 2.26×10−1 3.44×10−6 8.82×10−4 (estimated) 0.03 TCV 0.88 1.43 2.04×10−2 3.98×10−6 2.85×10−4 (estimated) 0.1 WEST 2.25 4.5 2.26×10−1 3.23×10−6 8.55×10−4 (estimated) 0.03 COMPASS 0.56 0.9 ~ 2.1 2.04×10−2 2.41×10−7 2.22×10−4 (estimated) 0.1 NSTX-U 0.85 1 2.26×10−1 5.49×10−6 1.11×10−3 (estimated)
TEXTOR 1.75 3 0.09±0.01 1.83×10−1 3.77×10−6 8.31×10−4
0.04 SST-1 1.1 3 3.62×10−2 2.36×10−7 2.93×10−3 (estimated)
EAST 1.75 3.5 0.05~0.09 1.11×10−1 3.23×10−6 5.98×10−4
KSTAR 1.8 3.5 0.1 2.26×10−1 3.32×10−6 8.67×10−4
0.15 T-15U 2.43 3.5 5.09×10−1 4.49×10−6 1.51×10−3 (estimated) 0.2 JT-60SA 2.96 2.25 9.05×10−1 8.50×10−6 2.77×10−3 (estimated)
47
휉
휉0
푞 = 1 surface
휉 = 0 as 푟 > 푟1
푟1 푟
Figure 2.9 The displacement distribution, 휉(푟), based on Wesson’s model [18].
Figure 2.10 The flow pattern of plasma in Wesson’s model. The red region is hot region, and the blue region is cold region.
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2.3.4 Comparison of the two models
This concludes the theoretical review of sawtooth instability. Summarizing section 2.3.2 and section 2.3.3, a comparison of Kadomtsev’s model and Wesson’s model is shown in Table 2.4.
Table 2.4 Comparison between Kadomtsev’s model and Wesson’s model. Kadomtsev’s model Wesson’s model
Model Fast reconnection model Quasi-interchange model
Safety factor, 1 − 푞 > 휀 1 − 푞 ≪ 휀 푞
To diminish d휉 → 0 except 푞 = 1 surface 푞 → 1 inside 푞 = 1 surface d푟 훿푊2 term
2휋2휉2 푎 퐵2 2휋2 푎 퐵2 0 ∫ {2푟푝′ − 휃 (1 − 푞)(1 + ∫ {2푟푝′ − 휃 (1 − 푞)(1 + Approximated 푅0 0 휇0 푅0 0 휇0 훿푊 3푞)} 푟푑푟 3푞)} 휉2푟푑푟
Flow structure Hot bubble, cold crescent Hot crescent, cold bubble
Magnetic Occurs at fast collapse phase Occurs at ramp phase reconnection
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III. Experimental Methods
The ECEI system is a diagnostic system that visualizes electron temperature fluctuation in 2-D/3-D spaces of tokamak plasmas. The ECEI system operates based on the principle of ECE (Electron Cyclotron Emission). First, the physical principle of ECE will be reviewed. Then, the ECEI (Electron Cyclotron Emission Imaging) system in KSTAR will be reviewed. Finally, the experimental set up for the measurement of sawtooth instability will be described.
3.1 ECE and KSTAR ECEI System
In tokamak, it is impossible to measure the plasma electron temperature with commercial thermometers, because the plasma electron temperature is too high to be measured. Instead of a thermometer, an ECE (electron cyclotron emission) radiometer or ECE imaging system can be used to measure electron temperature by measuring radiation intensity emitted from gyrating electrons.
3.1.1 Physical principle of ECE
The electron is gyrating around a magnetic field in tokamak as shown in Figure 3.1. The cyclotron radiation is emitted from gyrating electrons.
Trace of electron
Magnetic field line
Cyclotron radiation
Figure 3.1 Gyrating electron around magnetic field and corresponding cyclotron radiation.
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In this picture, the equation of motion is
푑퐯 푚 = 푞퐯×퐁 (3.1) 푑푡
and the solution of the equation is simple harmonic motion. The angular velocity of the solution is
푒퐵 휔 = . (3.2) 푚푒
where 푚푒 is the mass of the electron, and 퐵 is the magnitude of magnetic field [20].
In this gyration motion, the radioactive transfer equation is given as
푑퐼 휈 = 푗 − 훼퐼 . (3.3) 푑푠 휈 푣
where 퐼휈 is the intensity of radiation, 푠 is the radiation traveling distance, 푗휈 is the emission coefficient, and 훼 is the absorption coefficient [40]. The optical depth, 휏, is defined as
휏 = ∫ 훼푑푠. (3.4) 푠
If the radiation travel from 0 to 휏, the solution for the equation is given as
푗 (휏) 푗 (휏) 푗 (휏) 퐼 (휏) = 퐼 (0)푒−휏 + 휈 (1 − 푒−휏) = 휈 + {퐼 (0) − 휈 } 푒−휏. (3.5) 휈 휈 훼 훼 휈 훼
When the optical depth is thick enough, the radiation density is approximated as 퐼휈(휏) = 푗휈/훼 [41]. This means all radiation is absorbed. In this case, the plasma can be regarded as a blackbody, so it follows Rayleigh-Jean’s law as
2ℎ휈3/푐2 퐼휈 = . (3.6) 푒ℎ휈/푘퐵푇푒 − 1
where ℎ is the Planck constant and 푇푒 is the electron temperature [42, 43]. With low frequency approximation, ℎ휈 ≪ 푇푒, Eq. (3.6) approximated as
2ℎ휈2푘 푇 퐼 = 퐵 푒. (3.7) 휈 푐2
With 휈 = 휔/2휋, the radiation intensity is expressed in terms of 휔 as
ℎ휔2푘 푇 퐼 = 퐵 푒. (3.8) 휔 2휋2푐2
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Finally, for thick optical depths, the electron temperature is obtained as
2휋2푐2퐼 2휋2푐2푚2퐼 휔 휔 (3.9) 푇푒 = 2 = 2 2 . ℎ휔 푘퐵 ℎ푒 퐵 푘퐵
This shows that the electron temperature, 푇푒, is proportional to radiation intensity, 퐼휔, for thick optics. In order to satisfy this characteristic, the plasma should be optically thick.
3.1.2 KSTAR ECEI System
The KSTAR ECEI system is used to visualize MHD instability in 3D by measuring relative electron temperature, 훿푇푒/〈푇푒〉, where 훿푇푒 = 푇푒 − 〈푇푒〉, and 〈푇푒〉 is the time averaged temperature.
In KSTAR, there are 3 detector arrays for ECEI measurement. Two of them are placed in H-port and another is placed in G-port as shown in Figure 3.2. The angle difference of G-port and H-port is
22.5° toroidally. Each of the two detector arrays in G-port is placed on HFS (High Field Side, 푅 < 푅0) and LFS (Low Field Side, 푅 > 푅0 ) as shown in Figure 3.2. The magnetic field, 퐵 can be
2 2 approximated by 퐵휙 ( ∵ 퐵 = √퐵휃 + 퐵휙 and 퐵휙 ≫ 퐵휃 ), and according to Eq. (2.12), 퐵휙 is proportional to 1/푅. Thus, 퐵 ~ 1/푅.
Each detector array has 24 vertical and 8 radial detection channels. In other words, each detector has 192 detection channels, and at H-port, there are 384 channels and at G-port, there are 192 channels, so in total, there are 576 channels on the ECEI system. The vertical and radial resolution is ~1.5 cm per channel and the time resolution is 0.5, 1, 2 휇s [44].
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Figure 3.2 The schematics of the ECEI system. The first system is in H-port, and the second system is in G-port [45].
3.2 Experimental Set-up
The major and minor radii of KSTAR is shown in Table 3.1. These parameters are constant in every experiment.
Table 3.1 Parameters of KSTAR [32] Radius
Major radius 푅0 = 1.8 m Minor radius 푎 = 0.5 m
The intrinsic rotation experiment # 11264 is used to observe the sawtooth oscillation in 3D with the KSTAR ECEI system. The experimental set-up is shown in Table 3.2. In the next chapter, the experimental results will be shown and discussed by the results. Then, the results will be compared with the theoretical models.
53
Table 3.2 Experiment set-up for # 11264
Set-up Measurement duration 푡 = 0 ~ 10.5 s Observed time 푡 = 3.5 ~ 4.5 s Time resolution ∆푡 = 2 휇s Mode H-mode 19 −3 Average density of electron 〈푛푒〉 = 1.8×10 m 3 −3 Average density of electron at the core 〈푛푒0〉 = 3×10 m
Average electron temperature 〈푇푒0〉 = 1.5 ~ 2.7 keV
Average electron temperature at the core 〈푇푒0〉 = 2.5 keV
Sampling frequency 푓푠 = 500 kHz Harmonic extraordinary mode
Total plasma current 퐼푝 = 600 kA
Total driven current 퐼푡 = 20 kA Heating power 0.765 MW Elongation 휅 = 1.8
Safety factor at 푟 = 0.95푎 푞95 ≈ 5 Radial (푅) position / HFS, H-port 180 m - 190 m / 40 cm / 1.3 Vertical size / Zoom G-port 190 m - 200 m / 50 cm / 1.6 factor LFS, H-port 205 m - 215 m / 40 cm / 1.3 Vertical position 0.02 ~ 0.04 m
Toroidal magnetic field 퐵푡 = 3.5 T Radial spatial resolution ∆푅 = 1 cm Vertical spatial resolution ∆푧 = 2 cm Triangularity 훿 = 0.8
Poloidal beta 훽푝 = 0.4
Rotation speed 푣휙 = 110 km/s
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IV. Results & Discussion
4.1 Experimental results
4.1.1 Observation of periodic behavior
The relative temperature fluctuation was observed in 10.5 s intervals. Within this time interval, 3.4 - 4.6 s was selected as shown in Figure 4.1. The results show the time traces of the ratio of electron temperature fluctuation to average temperature, observed through two detection channels from the ECEI system.
(a) (b)
Figure 4.1 1-D time trace of 훿푇푒/〈푇푒〉 where 훿푇푒 is electron temperature fluctuation and 〈푇푒〉 is the averaged electron temperature over time. (a) 훿푇푒/〈푇푒〉 at central region (observed in high field side of H-port), (b) 훿푇푒/〈푇푒〉 at outer region (observed in G-port).
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The observed sawtooth period, theoretically calculated Kadomtsev’s collapse time, and the observed collapse time, are shown in Table 4.2.
Table 4.1 Sawtooth period, and collapse times. Times Symbol Result
Sawtooth period (observed) 휏sawtooth 28.2 ms Kadomtsev’s collapse time 휏퐾 867 휇s (calculated)
Collapse time (observed) 휏푐 150 휇s
According to the results, the observed collapse time is very short compared to the sawtooth period, −2 i.e., 휏푐/휏sawtooth ~ 10 . That is why this phenomenon is called fast collapse.
The whole periodic behavior of at the central region (observed in high field side of H-port) and the outer region (observed in G-port) is shown in Figure 4.2. Also, the corresponding three phases are shown in Table 4.2.
(a)
Ramp phase (b) Precursor phase Fast collapse phase
56
Figure 4.2 1-D time trace of 훿푇푒/〈푇푒〉 in one period (a) at the central region (b) at the outer region.
Table 4.2 The time lengths of the three phases in Figure 4.2. Times Time interval (s) Duration
Ramp phase 3.52482 ~ 3.54982 25 ms
Precursor phase 3.54982 ~ 3.55287 3.05 ms
Fast collapse phase 3.55287 ~ 3.55302 150 휇s
The ECE images of periodic behavior are shown in Figure 4.3. The observation time is from 3.52488 s to 3.55308 s. Initially, as shown in Figure 4.3 (a), the core is relatively cold and it is stable. The temperature of the core increases over time in Figure 4.3 (a) – Figure 4.3 (f). Then, the new magnetic island is growing in Figure 4.3 (g) – Figure 4.3 (k). A new magnetic island was first distinctly observed as shown in Figure 4.3 (i). The old magnetic island and new magnetic island were observed alternately, as plasma was noticed to be rotating. Finally, between Figure 4.3 (g) – Figure 4.3 (k), the crash occurred. After the crash occurred, the heat of the core spread to the outer region. The plasma returned to its initial conditions as shown in Figure 4.3 (l). This behavior of relative electron temperature fluctuation occurs repeatedly.
The stable ramp phase is shown in Figure 4.3 (b) – Figure 4.3 (f); the precursor phase is shown in Figure 4.3 (g) – Figure 4.3 (j); and the fast collapse phase is shown in Figure 4.3 (k) – Figure 4.3 (l). In the ramp phase, the core is heating. In the precursor phase, the temperature reached critical temperature, and the hot core started to move by internal kink. On the other side of the hot core plasma, a new magnetic island appeared, and a new magnetic island grew. The old magnetic island disappeared suddenly, and the electron temperature fluctuation distribution returned to initial sate, which is the first state of the stable ramp phase. This demonstrated that the experiment resembles Kadomtsev’s model more than Wesson’s model, which was described in section 2.3.2.
57
푡 = 3.52482 s 푡 = 3.552500 s 푡 = 3.55300 s 푡 = 3.55350 s
(a) (b) (c) (d) 푡 = 3.55400 s 푡 = 3.55450 s 푡 = 3.55500 s 푡 = 3.55200 s
(e) (f) (g) (h) 푡 = 3.55206 s 푡 = 3.55293 s 푡 = 3.55298 s 푡 = 3.55302 s
(i) (j) (k) (l)
Figure 4.3 The periodic behavior of sawtooth oscillation. The result represents 훿푇푒/〈푇푒〉.
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4.1.2 Observation of precursor phase
The precursor phase and fast collapse phase at the central region and the outer region are shown in Figure 4.4.
Fast collapse phase Precursor phase
Figure 4.4 1-D time trace of 훿푇푒/〈푇푒〉 in precursor phase and fast collapse phase (a) at the central region (b) at the outer region.
The plasma was observed to rotate in the poloidal direction, as illustrated in Figure 4.5. In the stable ramp phase, it seems that plasma at the core stopped rotating, but when the new magnetic island appeared, the hot core plasma appeared to be rotated in the poloidal direction. At the local maximum in Figure 4.4, the old magnetic island is shown, and at the local minimum in Figure 4.4, the new magnetic island is shown. The difference between local minimum and local maximum becomes larger in time. In other words, the perturbation is growing in time. As shown in Figure 4.5, the new magnetic island is growing. This is why the perturbation appeared to be growing.
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푡 = 3.54982 s 푡 = 3.55000 s 푡 = 3.55093 s 푡 = 3.55100 s
(a) (b) (c) (d) 푡 = 3.55200 s 푡 = 3.55206 s 푡 = 3.55213 s 푡 = 3.55218 s
(e) (f) (g) (h) 푡 = 3.55250 s 푡 = 3.55255 s 푡 = 3.55285 s 푡 = 3.55287 s
(i) (j) (k) (l)
Figure 4.5 The process of precursor phase. The result represents 훿푇푒/〈푇푒〉.
60
To calculate the perturbation period as illustrated in Figure 4.5, the poloidal angular velocity should be calculated. Because the plasma follows the magnetic field line, the plasma rotates the tokamak helically. The plasma rotation speed is 푣휙, so the toroidal angular velocity is
푣휙 휔휙 = . (4.1) 푅0
In order to find the poloidal angular velocity, 휔휃, the helicity is used. The safety factor, 푞, is already defined as Eq. (2.15),
d휙 d휙/d푡 휔휙 푞 = = = d휃 d휃/d푡 휔휃
Then, the poloidal angular velocity is
휔휙 푣휙 휔휃 = = . (4.2) 푞 푅0푞
Using Eqs. (4.1) – (4.2), the toroidal rotation period and poloidal rotation period are obtained as
2휋푅0 휏휙 = , (4.3) 푣휙
2휋푅0푞 (4.4) 휏휃 = . 푣휙
The plasma rotation speed, 푣휙 is given in Table 3.2, which outlines the experimental set-up. Using Eq. (4.4) with 푞 ~ 1, the plasma rotation period in the poloidal direction is
−4 휏휃 ≈ 10 s. (4.5)
According to the result shown in Figure 4.5, the poloidal rotation period is roughly 1.2~1.3 × 10−4 s. The poloidal rotation period of both the theoretically calculated value and the observed value are roughly matched.
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4.1.3 Observation of fast crash
The fast collapse occurs between Figure 4.3 (j) – Figure 4.2 (l). The detailed collapse process is shown in Figure 4.6.
The collapse occurred suddenly, and the heat was transported to the outer region. The collapse was observed as the magnetic reconnection drove the hot core to move outside. Then, the plasma became stable.
푡 = 3.55287 s 푡 = 3.55289 s 푡 = 3.55291 s 푡 = 3.55293 s
(a) (b) (c) (d) 푡 = 3.55295 s 푡 = 3.55297 s 푡 = 3.55299 s 푡 = 3.55303 s
(e) (f) (g) (h)
Figure 4.6 The process of fast collapse phase. The result represents 훿푇푒/〈푇푒〉.
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4.2 Discussion
The difference between Kadomtsev’s model and Wesson’s model is that they have different eigenvalue solutions, 훏 , of Eq. (2.91). Kadomtsev’s model tries to explain this by taking 1 − 푞 > 휀, whereas Wesson’s model takes 1 − 푞 ≪ 휀. According to the result, if the tokamak satisfies 1 − 푞 > 휀, the instability follows Kadomtsev’s model, and if the tokamak satisfies, 1 − 푞 ≪ 휀 the instability follows Wesson’s model.
According to the recent research [47], the central safety factor, 푞0 is about 0.84, so
1 − 푞 ~ 0.1, (4.6) and the aspect ratio, 휀 ~ 푟1/푅0 is 0.056, so
휀 ~ 0.05. (4.7)
In KSTAR, 1 − 푞 > 휀, so it resembles Kadomtsev’s model more than Wesson’s model. As observed, in the initial state, the core was heated linearly, and when the core reached the critical point, a new magnetic island appeared and grew. The old magnetic island (hot core) disappeared suddenly, and the distribution of relative electron temperature fluctuation returned to initial state.
A question about a new eigenvalue, 훏, when 1 − 푞 ~ 휀 becomes a motive for introducing a new model that encompasses both theoretical models. In addition, the existing study analyzes instability by Fourier analysis using 훏 ~ 푒𝑖(푚휃−푛휙) , but a new study could utilize the different formation 훏 ~ 푒𝑖(푚휃−푛휙−휔푡), adding a time-dependent term. Applying Fourier analysis, a time derivative and a differential operator applied to the perturbed term can be transformed as
휕 = −푖휔, (4.8) 휕푡 휕 푖푚 푖푛 (4.9) 훁 = 푟̂ + 휃̂ − 푧̂. 휕푟 푟 푅0
Using Eq. (4.6) and Eq. (4.7), the Eq. (2.57) is transformed as
휕퐵 2 1 푚 푛 휙 1 휕 −휔 휌0훏 = {푖 ( 퐵휙 + 퐵휃) 푟̂ − 휃̂ + (푟퐵휃)푧̂} ×퐁1 휇0 푟 푅0 휕푟 푟 휕푟
1 푖푛퐵1,휃 푖푛퐵1,푟 1 휕 푖푚퐵1,푟 (4.10) + { 푟̂ − 휃̂ + ( (푟퐵1,휃) − ) 푧̂} ×퐁0 휇0 푅0 푅0 푟 휕푟 푟
− 훁푝1,
Solving Eq. (4.10) by coupling the ideal MHD equations, the solutions for 휔 will be obtained.
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The real part of 휔, 휔푟, will represent the rotation speed of the plasma observed in section 4.1.2. The poloidal rotation period and the toroidal rotation period will be
2휋푛 휏휙 = , (4.11) 휔푟 2휋푚 휏휃 = . (4.12) 휔푟
The instability is 푚/푛 = 1/1 internal kink mode, then the displacement is expressed as 훏 ~ 푒𝑖(휃−휙−휔푡), then the poloidal rotation period and the toroidal rotation period will be
2휋 휏휙 = , (4.13) 휔푟 2휋 휏휃 = . (4.14) 휔푟
The imaginary part of 휔, 휔𝑖 will represent the growth rate. By obtaining the growth rate, the variables which can affect the growth rate will be identified.
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V. Conclusions
In this thesis, two theoretical models for sawtooth oscillation followed by a sudden crash has been reviewed by comparing them with experimental observations. Kadomtsev’s full reconnection model describes that the hot region at the core is “kinked” toward the 푞 = 1 surface due to the internal kink instability, and a narrow resistive layer is formed. Then, collapse of the pressure and temperature suddenly occurs on the narrow layer in a very short time due to magnetic reconnection. On the other hand, Wesson’s model describes that a flat 푞 - profile is formed within 푞 = 1 surface which drives quasi-interchange instability due to very small or no magnetic shear in the flat 푞-profile region.
The 2-D images of the electron temperature fluctuation, observed at the core of the sawtoothing plasma, shows that the spatial structure of the hot region inside the 푞 = 1 surface is circular. This indicates that the actual physical phenomena resemble Kadomtsev’s model, rather than Wesson’s model.
A future study will be conducted to develop a new model which encompasses both theoretical models. Kadomtsev’s model is satisfied only if 1 − 푞 > 휀 and Wesson’s model is satisfied only if 1 − 푞 ≪ 휀. A new eigenvalue, 훏, has to be introduced when 1 − 푞 ~ 휀. This thesis may lead to better explanations of the exact physical phenomena involved in sawtooth oscailltion. The instability will be analyzed by Fourier analysis 훏 ~ 푒𝑖(푚휃−푛휙−휔푡). By obtaining 휔, the plasma rotation speed will be expressed in terms of the real part of 휔, 휔푟, and the growth rate of magnetic perturbation will be expressed in terms of the imaginary part of 휔, 휔𝑖. Using the Fourier analysis, adding a time-dependent term to MHD equations, the instability will be understood. By coupling the ideal MHD equations combining 푒𝑖(푚휃−푛휙−휔푡) term, the plasma velocity distribution, 퐮(퐫, 푡), will be obtained. The plasma current density, 퐣(퐫, 푡), and magnetic field, 퐁(퐫, 푡) will be also obtained. By obtaining these analytical solutions, the exact cause of instability will be revealed, and the instability can be controlled based on this theoretical background. Therefore, it is hoped that this study will contribute to the commercialization of nuclear fusion.
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Appendix A. Plasma Parameters
The physical constants used in plasma physics are given in Table A.1.
Table A.1 Plasma parameters Plasma Parameter Symbol Constant (SI unit) Electron charge 푒 1.60217662×10−19 C −7 Vacuum permeability 휇0 4휋×10 T∙m/A(H/A) Speed of light 푐 2.99792458×108 m/s 2 Vacuum permittivity 휀0 1/(휇0푐 ) F/m ℎ 6.62607004 ×10−34 J∙s Plank’s constant ℏ ℎ/2휋 −23 Boltzmann’s constant 푘퐵 1.38063853×10 J/K Electron mass 푚 9.1095611×10−31 kg Ion mass 푀 1.66053904020×10−31 kg Mass ratio 푀/푚 1837 Electron volt 1 eV = 1. 6021766×10−19 J Temperature at 1 eV 11,604.60602240 K
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Appendix B. Vector relations
B.1 Vector formulas
퐀 ∙ (퐁×퐂) = 퐁 ∙ (퐂×퐀) = 퐂 ∙ (퐀×퐁) (B.1)
퐀×(퐁×퐂) = 퐁(퐀 ∙ 퐂) − 퐂(퐀 ∙ 퐁) (B.2)
(퐀×퐁) ∙ (퐂×퐃) = (퐀 ∙ 퐁)(퐂 ∙ 퐃) − (퐀 ∙ 퐃)(퐁 ∙ 퐂) (B.3)
(퐀×퐁)×(퐂×퐃) = (퐀퐁퐃)퐂 − (퐀퐁퐂)퐃 = (퐀퐂퐃)퐁 − (퐁퐂퐃)퐀 (B.4) ퟏ (퐀 ∙ 훁)퐀 = 훁 ( 퐀ퟐ) − 퐀×(훁×퐀) (B.5) ퟐ 훁×{(퐀 ∙ 훁)퐀} = (퐀 ∙ 훁)(훁×퐀) + (훁 ∙ 퐀)(훁×퐀) − {(훁×퐀) ∙ 훁}퐀 (B.6)
훁 ∙ (휙퐀) = 휙훁 ∙ 퐀 + 퐀 ∙ (훁휙) (B.7)
훁×(휙퐀) = 휙훁×퐀 + (훁휙)×퐀 (B.8)
훁 ∙ (퐀×퐁) = 퐁 ∙ 훁×퐀 − 퐀 ∙ 훁×퐁 (B.9)
훁(퐀 ∙ 퐁) = 퐀×(훁×퐁) + (훁 ∙ 퐀)퐁 + 퐁×(훁×퐀) + (퐁 ∙ 훁)퐀 (B.10)
훁 ∙ (퐀×퐁) = 퐁 ∙ (훁×퐀) − 퐀 ∙ (훁×퐁) (B.11)
훁×(퐀×퐁) = 퐀(훁 ∙ 퐁) − 퐁(훁 ∙ 퐀) − (퐀 ∙ 훁)퐁 + (퐁 ∙ 훁)퐀 (B.12)
훁×(훁×퐀) = 훁(훁 ∙ 퐀) − ∇2퐀 (B.13)
훁×(훁휙) = 0 (B.14)
훁 ∙ (훁×퐀) = 0 (B.15)
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B.2 Applications to the cylindrical coordinates (풓, 휽, 풛)
1 휕 휕휙 1 휕2휙 휕2휙 ∇2휙 = (푟 ) + + (B.16) 푟 휕푟 휕푟 푟 휕휃2 휕푧2 1 휕 1 휕퐴 휕퐴 훁 ∙ 퐀 = (푟퐴 ) + 휃 + 푧 (B.17) 푟 휕푟 푟 푟 휕휃 휕푧 1 휕퐴 휕퐴 휕퐴 휕퐴 1 휕 1 휕퐴 훁×퐀 = ( 푧 − 휃) 퐫̂ + ( 푟 − 푧) 훉̂ + ( (푟퐴 ) − 푟) 퐳̂ (B.18) 푟 휕휃 휕푧 휕푧 휕푟 푟 휕푟 휃 푟 휕휃 1 휕퐴 1 휕퐴 ∇2퐀 = {∇2퐴 − (퐴 + 2 휃)} 퐫̂ + {∇2퐴 − (퐴 + 2 푟)} 훉̂ + ∇2퐴 퐳̂ (B.19) 푟 푟2 푟 휕휃 휃 푟2 휃 휕휃 푧 휕퐵 1 휕퐵 휕퐵 1 (퐀 ∙ 훁)퐁 = (퐴 푟 + 퐴 푟 + 퐴 푟 − 퐴 퐵 ) 퐫̂ 푟 휕푟 휃 푟 휕휃 푧 휕푧 푟 휃 휃 휕퐵 1 휕퐵 휕퐵 1 + (퐴 휃 + 퐴 휃 + 퐴 휃 − 퐴 퐵 ) 훉̂ (B.20) 푟 휕푟 휃 푟 휕휃 푧 휕푧 푟 휃 푟 휕퐵 1 휕퐵 휕퐵 + (퐴 푧 + 퐴 푧 + 퐴 푧) 퐳̂ 푟 휕푟 휃 푟 휕휃 푧 휕푟
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Appendix C. Detail derivations
C.1 Integration of trigonometric function in section 2.1.4
To show
2휋 d푡 2휋 퐴 = ∫ = 2 2 0 푎 + 푏 cos 푡 √푎 − 푏 where 푎 > 푏 > 0, substitution integral is useful.
푡 1 푡 1 First, let 푥 = tan , then cos 푡 = (1 − 푥2)/(1 + 푥2) and d푥 = sec2 d푡 = (1 + 푥2)d푡 , then 2 2 2 2 2d푥 d푡 = . And 푥 → 0 as 푡 → 0 , 푥 → 0 as 푡 → 2휋 . Since inside of integral term is symmetry by 1+푥2 푡 = 휋 and 푥 → ∞ as 푡 → 휋,
2휋 d푡 휋 d푡 2휋 d푡 휋 d푡 퐴 = ∫ = ∫ + ∫ = 2 ∫ 0 푎 + 푏 cos 푡 0 푎 + 푏 cos 푡 휋 푎 + 푏 cos 푡 0 푎 + 푏 cos 푡 ∞ 2d푥/(1 + 푥2) ∞ 4d푥 = 2 ∫ 2 2 = ∫ 2. 0 푎 + 푏(1 − 푥 )/(1 + 푥 ) 0 푎 + 푏 + (푎 − 푏)푥
푎+푏 푎+푏 휋 Let 푥 = √ tan 푦, d푥 = √ sec2 푦 d푦, 푦 → 0 as 푥 → 0 and 푦 → as 푥 → ∞, 푎−푏 푎−푏 2
휋 푎 + 푏 ∞ √ sec2 푦 d푦 4d푥 2 푎 − 푏 퐴 = ∫ = 4 ∫ ( ) 2 2 0 푎 + 푏 + 푎 − 푏 푥 0 푎 + 푏 푎 + 푏 + (푎 − 푏) (√ tan 푦) 푎 − 푏
푎 + 푏 휋/2 √ sec2 푦 d푦 휋/2 푎 − 푏 d푦 = 4 ∫ 2 = 4 ∫ 0 푎 + 푏 + (푎 + 푏) tan 푦 0 √(푎 + 푏)(푎 − 푏) 4 휋/2 4 휋 2휋 = ∫ d푦 = = . √(푎 + 푏)(푎 − 푏) 0 √(푎 + 푏)(푎 − 푏) 2 √(푎 + 푏)(푎 − 푏)
∴ The Integration of trigonometric function is derived as
2휋 d푡 2휋 ∫ = . 2 2 0 푎 + 푏 cos 푡 √푎 − 푏
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C.2 Kadomtsev’s collapse time calculation process
1. Calculation of resistivity, 휂
The resistive is given as
푚푒 휂 = 0.51 2 (C.1) 푛푒푒 휏푒
where 휏푒 is the electron collision time, given as
휀2푚1/2푇3/2 3/2 0 푒 푒 (C.2) 휏푒 = 3(2휋) 2 4 푛𝑖푍 푒 ln Λ
where 푍 is charge of ions, and for electron-electron collision, ln Λ ≈ 17. Then, the electron collapse time is calculated as Table C.1.
Table C.1 Calculated electron collision time 푇 푛 100 eV 1 keV 10 keV 1019m−3 2.0199 휇s 63.874 휇s 2.0199 ms 1020m−3 201.99 ns 6.3874 휇s 201.99 휇s
Based on Table C.1, the electron resistivity is calculated as Table C.2.
Table C.2 Calculated resistivity 푇 푛 100 eV 1 keV 10 keV 1019m−3 1.7569×10−6 5.5558×10−8 1.7569×10−6 1020m−3 1.7569×10−6 5.5558×10−8 1.7569×10−6
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2. Calculation of mass density, 휌
The mass density is given as
휌 = 푚𝑖푛𝑖 + 푚푒푛푒 (C.3)
and known as 푚𝑖 ≫ 푛𝑖, and using quasi-neutrality of plasma,
휌 = 푚𝑖푛 (C.4)
3. Alfvé nic time
The Alfvé nic time is given as
푟 휏 = 1 퐴 ∗ 퐵 /√휌휇0
∗ where 퐵 = 퐵휃(1 − 푞), and 푟1 is the radius of 푞 = 1 surface. At 푞 = 1,
푟1퐵휙0 푞 = = 1 푅0퐵휃
Then, Alfvé nic time is expressed in terms of 퐵휙0 as
푅0√휌휇0 휏퐴 = . (C.5) 퐵휙0(1 − 푞)
4. Resistive diffusion time
Resistive diffusion time is given as
휇0 휏 = 푟2. (C.6) 푅 휂 1
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5. Kadomtsev’s collapse time
Finally, Kadomtsev’s collapse time is given as
1/2 휏퐾 ~ (휏퐴휏푅) (C.7)
or
3 푅0휌휇0/휂 휏퐾 ~ √ 푟1. (C.8) 퐵휙0(1 − 푞)
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Acknowledgements
This work was supported by NRF of Korea under contract no. NRF-2014M1A7A1A03029865.
석사과정 동안 많은 일들과 많은 변화들이 있었습니다. 그 동안 제가 공부할 때 방향
을 제시해주시고 상담과 조언을 해주시고 지도해주신 박현거 교수님께 머리 숙여 감사드
립니다. 이 부족한 제자가 교수님으로부터 받은 은혜를 평생 잊지 않겠습니다. 앞으로 박
사과정을 하고 박사 후 진로를 밟는데 지금 석사졸업이 끝이 아니라 시작인 줄 알고 더
욱 향상되고 발전된 모습으로 나아가겠습니다.
또한 시간을 내어주셔서 논문심사를 해주시는 곽규진 교수님, 허민섭 교수님께 감사
드립니다. 특별히 인생의 선배로써 조언을 가끔씩 해주신 곽규진 교수님께 감사드립니다.
랩생활을 하면서 제가 연구하는데 도와 주시고 조언을 해 주신 박사님들께 감사의 글을
올려드립니다.
랩을 이끌어 가시며 랩원들에게 한사람, 한사람 신경을 써 주신 김민우 박사님께 감
사드립니다. 또한 연구분야를 직접 지도를 해 주신 남윤범 박사님께 감사 드립니다. 또한
제가 물어볼 때 친절하게 답해주신 이재현 박사님께 감사 드립니다. I am grateful to Dr.
Pravesh Dhayani, for helpful discussions and advises about the researches. 또한 행정일을 한다고
수고하시는 정유진 선생님께 감사 드립니다. 이 외에도 ITER School을 가서 연수경험을
같이 하게 된 서울대생 서재민 학생에게 고맙다는 말을 전합니다. 또한 원자력과에 있을
때 힘들 때 같이 야이기를 나누고 어려울 때 상담을 해주신 동한이형, 상일이형께 감사
드립니다. I am grateful to teachers at Language Education Center in UNIST, Alan Stubbs and Danial
Croydon, who advised me to correct the order, paragraphs, sentences, and checked grammatic errors in the thesis. 77
석사과정을 2년 반 하면서 만약 신앙생활을 하지 않았으면 무너졌을 법한데 저의 신
앙생활이 저의 에너지를 공급하는데 원동력이 되었습니다.
먼저 대학원 입학할 때부터 2015년 11월까지 신앙생활을 하였던 빛의 교회! 교회 사
정으로 교회를 새로 옮겼지만 목사님은 여전히 기억합니다. 빛의 교회에 있을 때 정말
힘들 때마다 먹을 걸 주시고 격려를 해 주신 신광호 목사님께 감사 드립니다.
또한 신복교회에 신앙생활 하면서 도움을 주시고 응원을 하신 한 분 한 분께 감사 드
립니다. 항상 말씀을 전하시는 김규섭 목사님, 주재강 목사님, 박규태 목사님께 감사 드
립니다. 항상 따듯하게 맞아 주시고 챙겨 주시는 사모님께 감사 드립니다. 특별 새벽기도
때 새벽 일찍 학교에 오셔서 교회까지 차를 태워 주신 박정희 집사님, 청소년부 같은 반
하면서 맛있는 간식을 주시는 강수정 집사님, 3월 달 프랑스 갈 때 용돈 주시고 기도로
응원해주신 권사님, 집사님들, 한 살 동생이지만 친구 같은 신용이, 건민이, 이제 곧 할
결혼준비를 위하여 외모를 변신시키도록 도와준 하나님께서 보내신 두 천사 예진이, 소
이, 그 외에도 저를 위하여 기도하신 신복교회 성도님들께 감사 드립니다. 신복교회에서
감사하는 분들 한 분, 한 분 다 넣고 싶지만 다 넣으면 몇 페이지 될 것 같아서 대표하
는 몇 사람만 올렸습니다. 혹시 섭섭하다고 저에게 말씀하시면 맛있는 거 사드릴게요.
대학원 생활하면서 빼놓을 수 없는 분! 부모님께 감사드립니다. 부모님의 은혜를 통
하여 하나님의 은혜를 느낍니다. 비록 멀리 떨어졌지만 연락을 하면서 동고동락 하시면
서 저보다 저를 더 걱정하신 부모님, 부모님의 은혜는 평생 갚아도 못 갚을 겁니다. 부모
님께서 바라시는 대로 하나님에게서 크게 쓰임을 받는 하나님의 사람이 되겠습니다. 부
모님으로부터 신앙교육을 받은 그대로 제 자녀들에게도 그대로 전달하겠습니다. 그리고
어릴 때부터 함께 있었던 동생 수한이, 항상 고맙고 미안하다. 반 오십 년을 살아도 늘
항상 남아있는 사람은 가족 밖에 없네요. 78
마지막으로 이 모든 걸 인도하신 주신 살아 계신 하나님 아버지께 영광을 올려드립니
다. 제가 예측한대로 되지는 않았지만 항상 나의 삶을 선한 길로 이끌어 주신 하나님 아
버지께 감사드립니다. 이 죄인이 무엇이 관대 그의 아들 예수님을 보내셔서 저를 구원하
셨는지 생각하면 말로 표현할 수 없습니다. 이 은혜는 영원할 것이며 천국에 가서도 그
은혜를 기리며 찬양할 것입니다. 대학원 생활 하면서 내가 큰 위기와 고난을 당할 때도
하나님은 버리신 것 같았지만 함께 하셨고 나를 구원해주신 하나님께 찬양 드립니다. 하
나님은 그의 사랑하시는 자를 연단하시니 기쁘나 슬프나 늘 하나님께서 함께 하시는 줄
믿습니다. 성령으로 나의 마음을 이끄시고 나의 길을 이끄시는 하나님 아버지, 그 동안
지켜 주신 것 감사 드립니다. “진리를 알지니 진리가 너희를 자유롭게 하리라 (요한복음
8장 32절)”의 말씀과 같이 하나님의 뜻에 입각하여 어두움을 밝히고 진실을 밝힌다는 사
명감으로 학문에 임하도록 하겠습니다. 제가 이 학문을 하는 것은 단순히 잘 먹고 잘 살
기 위해서 하는 것이 아니라 하나님의 뜻을 이루고 다른 사람을 유익하게 하기 위함 임
을 확신합니다. 제가 이 학문을 함으로 하나님께서 온전히 드러나게 하소서. 오직 내가
한 것은 아무것도 없고 하나님께서 모든 것을 하셨다는 고백이 나오게 하여 주옵소서.
나는 드러나지 않고 오직 하나님만 드러나길 원합니다. 성령으로 나의 마음을 진리로 인
도하여 주옵소서.
이 석사논문과 이학석사 졸업장을 살아 계신 여호와 하나님께 바칩니다.
79