Conclusion

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 𝑞 = 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 𝑞 -profile is formed within 𝑞 = 1 surface which drives quasi-interchange instability due to very small or no magnetic shear in the flat 𝑞-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 𝑞 = 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 − 𝑞 > 𝜀 and Wesson’s model is satisfied only if 1 − 𝑞 ≪ 𝜀. A new eigenvalue, 𝛏, has to be introduced when 1 − 𝑞 ~ 𝜀. This thesis may lead to better explanations of the exact physical phenomena involved in sawtooth oscailltion. The instability will be analyzed by Fourier analysis 𝛏 ~ 𝑒𝑖(𝑚𝜃−𝑛𝜙−𝜔𝑡). By obtaining 𝜔, the plasma rotation speed will be expressed in terms of the real part of 𝜔, 𝜔_𝑟, and the growth rate of magnetic perturbation will be expressed in terms of the imaginary part of 𝜔, 𝜔_𝑖. Using the Fourier analysis, adding a time-dependent term to MHD equations, the instability will be understood. By coupling the ideal MHD equations combining 𝑒𝑖(𝑚𝜃−𝑛𝜙−𝜔𝑡) term, the plasma velocity distribution, 𝐮(𝐫, 𝑡), will be obtained. The plasma current density, 𝐣(𝐫, 𝑡), and magnetic field, 𝐁(𝐫, 𝑡) 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.

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 𝑒 1.60217662×10⁻¹⁹ C

Vacuum permeability 𝜇₀ 4𝜋×10⁻⁷ T∙m/A(H/A)

Speed of light 𝑐 2.99792458×10⁸ m/s

Vacuum permittivity 𝜀₀ 1/(𝜇₀𝑐²) F/m

Plank’s constant ℎ 6.62607004 ×10⁻³⁴ J∙s

ℏ ℎ/2𝜋

Boltzmann’s constant 𝑘_𝐵 1.38063853×10⁻²³ J/K

Electron mass 𝑚 9.1095611×10⁻³¹ kg

Ion mass 𝑀 1.66053904020×10⁻³¹ kg

Mass ratio 𝑀/𝑚 1837

Electron volt 1 eV = 1. 6021766×10⁻¹⁹ J

Temperature at 1 eV 11,604.60602240 K

Appendix B. Vector relations

B.1 Vector formulas

𝐀 ∙ (𝐁×𝐂) = 𝐁 ∙ (𝐂×𝐀) = 𝐂 ∙ (𝐀×𝐁) (B.1) 𝐀×(𝐁×𝐂) = 𝐁(𝐀 ∙ 𝐂) − 𝐂(𝐀 ∙ 𝐁) (B.2) (𝐀×𝐁) ∙ (𝐂×𝐃) = (𝐀 ∙ 𝐁)(𝐂 ∙ 𝐃) − (𝐀 ∙ 𝐃)(𝐁 ∙ 𝐂) (B.3) (𝐀×𝐁)×(𝐂×𝐃) = (𝐀𝐁𝐃)𝐂 − (𝐀𝐁𝐂)𝐃 = (𝐀𝐂𝐃)𝐁 − (𝐁𝐂𝐃)𝐀 (B.4)

(𝐀 ∙ 𝛁)𝐀 = 𝛁 (𝟏

𝟐𝐀^𝟐) − 𝐀×(𝛁×𝐀) (B.5)

𝛁×{(𝐀 ∙ 𝛁)𝐀} = (𝐀 ∙ 𝛁)(𝛁×𝐀) + (𝛁 ∙ 𝐀)(𝛁×𝐀) − {(𝛁×𝐀) ∙ 𝛁}𝐀 (B.6)

𝛁 ∙ (𝜙𝐀) = 𝜙𝛁 ∙ 𝐀 + 𝐀 ∙ (𝛁𝜙) (B.7)

𝛁×(𝜙𝐀) = 𝜙𝛁×𝐀 + (𝛁𝜙)×𝐀 (B.8)

𝛁 ∙ (𝐀×𝐁) = 𝐁 ∙ 𝛁×𝐀 − 𝐀 ∙ 𝛁×𝐁 (B.9)

𝛁(𝐀 ∙ 𝐁) = 𝐀×(𝛁×𝐁) + (𝛁 ∙ 𝐀)𝐁 + 𝐁×(𝛁×𝐀) + (𝐁 ∙ 𝛁)𝐀 (B.10)

𝛁 ∙ (𝐀×𝐁) = 𝐁 ∙ (𝛁×𝐀) − 𝐀 ∙ (𝛁×𝐁) (B.11)

𝛁×(𝐀×𝐁) = 𝐀(𝛁 ∙ 𝐁) − 𝐁(𝛁 ∙ 𝐀) − (𝐀 ∙ 𝛁)𝐁 + (𝐁 ∙ 𝛁)𝐀 (B.12)

𝛁×(𝛁×𝐀) = 𝛁(𝛁 ∙ 𝐀) − ∇²𝐀 (B.13)

𝛁×(𝛁𝜙) = 0 (B.14)

𝛁 ∙ (𝛁×𝐀) = 0 (B.15)

B.2 Applications to the cylindrical coordinates (𝒓, 𝜽, 𝒛)

∇²𝜙 =1 𝑟

𝜕

𝜕𝑟(𝑟𝜕𝜙

𝜕𝑟) +1 𝑟

𝜕²𝜙

𝜕𝜃²+𝜕²𝜙

𝜕𝑧² (B.16)

𝛁 ∙ 𝐀 =1 𝑟

𝜕

𝜕𝑟(𝑟𝐴_𝑟) +1 𝑟

𝜕𝐴_𝜃

𝜕𝜃 +𝜕𝐴_𝑧

𝜕𝑧 (B.17)

𝛁×𝐀 = (1 𝑟

𝜕𝐴_𝑧

𝜕𝜃 −𝜕𝐴_𝜃

𝜕𝑧 ) 𝐫̂ + (𝜕𝐴_𝑟

𝜕𝑧 −𝜕𝐴_𝑧

𝜕𝑟) 𝛉̂ + (1 𝑟

𝜕

𝜕𝑟(𝑟𝐴_𝜃) −1 𝑟

𝜕𝐴_𝑟

𝜕𝜃) 𝐳̂ (B.18)

∇²𝐀 = {∇²𝐴_𝑟− 1

𝑟²(𝐴_𝑟 + 2𝜕𝐴_𝜃

𝜕𝜃)} 𝐫̂ + {∇²𝐴_𝜃− 1

𝑟²(𝐴_𝜃+ 2𝜕𝐴_𝑟

𝜕𝜃)} 𝛉̂ + ∇²𝐴_𝑧𝐳̂ (B.19) (𝐀 ∙ 𝛁)𝐁 = (𝐴_𝑟𝜕𝐵_𝑟

𝜕𝑟 + 𝐴_𝜃1 𝑟

𝜕𝐵_𝑟

𝜕𝜃 + 𝐴_𝑧𝜕𝐵_𝑟

𝜕𝑧 −1

𝑟𝐴_𝜃𝐵_𝜃) 𝐫̂

+ (𝐴_𝑟𝜕𝐵_𝜃

𝜕𝑟 + 𝐴_𝜃1 𝑟

𝜕𝐵_𝜃

𝜕𝜃 + 𝐴_𝑧𝜕𝐵_𝜃

𝜕𝑧 −1

𝑟𝐴_𝜃𝐵_𝑟) 𝛉̂ + (𝐴_𝑟𝜕𝐵_𝑧

𝜕𝑟 + 𝐴_𝜃1 𝑟

𝜕𝐵_𝑧

𝜕𝜃 + 𝐴_𝑧𝜕𝐵_𝑧

𝜕𝑟) 𝐳̂

(B.20)

Appendix C. Detail derivations

C.1 Integration of trigonometric function in section 2.1.4 To show

𝐴 = ∫ d𝑡 𝑎 + 𝑏 cos 𝑡

2𝜋 0

= 2𝜋

√𝑎²− 𝑏² where 𝑎 > 𝑏 > 0, substitution integral is useful.

First, let 𝑥 = tan₂^𝑡 , then cos 𝑡 = (1 − 𝑥²)/(1 + 𝑥²) and d𝑥 =¹₂sec^{2 𝑡}₂d𝑡 =¹₂(1 + 𝑥²)d𝑡 , then d𝑡 = ^2d𝑥

1+𝑥² . And 𝑥 → 0 as 𝑡 → 0 , 𝑥 → 0 as 𝑡 → 2𝜋 . Since inside of integral term is symmetry by 𝑡 = 𝜋 and 𝑥 → ∞ as 𝑡 → 𝜋,

𝐴 = ∫ d𝑡 𝑎 + 𝑏 cos 𝑡

2𝜋 0

= ∫ d𝑡

𝑎 + 𝑏 cos 𝑡

𝜋 0

+ ∫ d𝑡

𝑎 + 𝑏 cos 𝑡

2𝜋 𝜋

= 2 ∫ d𝑡 𝑎 + 𝑏 cos 𝑡

𝜋 0

= 2 ∫ 2d𝑥/(1 + 𝑥²) 𝑎 + 𝑏(1 − 𝑥²)/(1 + 𝑥²)

∞ 0

= ∫ 4d𝑥

𝑎 + 𝑏 + (𝑎 − 𝑏)𝑥²

∞ 0

.

Let 𝑥 = √^𝑎+𝑏

𝑎−𝑏tan 𝑦, d𝑥 = √^𝑎+𝑏

𝑎−𝑏sec²𝑦 d𝑦, 𝑦 → 0 as 𝑥 → 0 and 𝑦 →^𝜋

2 as 𝑥 → ∞,

𝐴 = ∫ 4d𝑥

𝑎 + 𝑏 + (𝑎 − 𝑏)𝑥²

∞ 0

= 4 ∫

√𝑎 + 𝑏

𝑎 − 𝑏sec²𝑦 d𝑦 𝑎 + 𝑏 + (𝑎 − 𝑏) (√𝑎 + 𝑏

𝑎 − 𝑏tan 𝑦)

2 𝜋

2 0

= 4 ∫

√𝑎 + 𝑏

𝑎 − 𝑏sec²𝑦 d𝑦 𝑎 + 𝑏 + (𝑎 + 𝑏) tan²𝑦

𝜋/2 0

= 4 ∫ d𝑦

√(𝑎 + 𝑏)(𝑎 − 𝑏)

𝜋/2 0

= 4

√(𝑎 + 𝑏)(𝑎 − 𝑏) ∫ d𝑦

𝜋/2 0

= 4

√(𝑎 + 𝑏)(𝑎 − 𝑏) 𝜋

2= 2𝜋

√(𝑎 + 𝑏)(𝑎 − 𝑏).

∴ The Integration of trigonometric function is derived as

C.2 Kadomtsev’s collapse time calculation process

1. Calculation of resistivity, 𝜂 The resistive is given as

𝜂 = 0.51 𝑚_𝑒

𝑛_𝑒𝑒²𝜏_𝑒 (C.1)

where 𝜏_𝑒 is the electron collision time, given as

𝜏_𝑒 = 3(2𝜋)^3/2𝜀₀²𝑚_𝑒^1/2𝑇_𝑒^3/2

𝑛_𝑖𝑍²𝑒⁴ln Λ (C.2)

where 𝑍 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

𝑛 𝑇

100 eV 1 keV 10 keV

10¹⁹m⁻³ 2.0199 𝜇s 63.874 𝜇s 2.0199 ms

10²⁰m⁻³ 201.99 ns 6.3874 𝜇s 201.99 𝜇s

Based on Table C.1, the electron resistivity is calculated as Table C.2.

Table C.2 Calculated resistivity

𝑛 𝑇

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

2. Calculation of mass density, 𝜌 The mass density is given as

𝜌 = 𝑚𝑖𝑛𝑖+ 𝑚𝑒𝑛𝑒 (C.3)

and known as 𝑚_𝑖 ≫ 𝑛_𝑖, and using quasi-neutrality of plasma,

𝜌 = 𝑚_𝑖𝑛 (C.4)

3. Alfvénic time

The Alfvénic time is given as

𝜏_𝐴= 𝑟₁ 𝐵^∗/√𝜌𝜇0

where 𝐵^∗= 𝐵𝜃(1 − 𝑞), and 𝑟1 is the radius of 𝑞 = 1 surface. At 𝑞 = 1, 𝑞 =𝑟₁𝐵_𝜙0

𝑅0𝐵𝜃

= 1 Then, Alfvénic time is expressed in terms of 𝐵_𝜙0 as

𝜏_𝐴= 𝑅₀√𝜌𝜇0

𝐵_𝜙0(1 − 𝑞). (C.5)

4. Resistive diffusion time

Resistive diffusion time is given as

𝜏_𝑅=𝜇₀

𝜂 𝑟₁². (C.6)

5. Kadomtsev’s collapse time

Finally, Kadomtsev’s collapse time is given as

𝜏_𝐾 ~ (𝜏_𝐴𝜏_𝑅)^1/2 (C.7)

or

𝜏_𝐾 ~ √ 𝑅₀𝜌𝜇₀³/𝜂

𝐵_𝜙0(1 − 𝑞)𝑟₁. (C.8)

REFERENCES

1. United Nations, ‘World Population Prospects: The 2015 Revision, Key Finding and Advanced Tables’, United Nations, New York (2015)

2. Garry McCracken and Peter Stott, ‘Fusion, the energy of the universe’ (2^nd edition), Burlington:

Elsevier Science (2012)

3. 차동형 외 18명, ‘리더들이 꼭 알아야 할 에너지 기술: 에너지 위기의 시대’,

서울

:

지오북

(2015)

4. OECD; IEA; International Energy Agency, ‘World Energy Outlook 2016’, Paris, France: IEA Publication (2016)

5. KEEI, ‘2015 KEEI Annual Report’, Korea Energy Economics Institute (2015)

6. Vyacheslav P. Smirov, ‘Tokamak foundation in USSR/Russia 1950-1990’, Nuclear Fusion, Volume 50, Number 1 (2009)

7. Christian Ngo, and Joseph B. Natowitz, ‘Our Energy Future: Resources, Alternatives and the Environment’, Hoboken, N.J.: Wiley (2009)

8. IAEA, ‘Summary of the ITER final design report’ International Atomic Energy Agency, Vienna (2001)

9. Weston M. Stacey, ‘The Quest for a Fusion Energy Reactor: An Insider’s Account of the INTOR Workshop’, New York: Oxford University Press (2010)

10. W. Patrick Mccray, ‘‘Globalization with hardware’: ITER’s fusion of technology, policy, and politics’, History and Technology Volume 26, Number 4 (2010)

11. Giuseppe Ciullo, et al., ‘Nuclear Fusion with Polarized Fuel’, Cham: Springer International Publishing: Imprint: Springer (2016)

12. Hartmut Zhom, ‘Magnetohydrodynamics Stability of Tokamaks’, Wiley-VCH (2015)

13. S. Von Goeler, et al., ‘Studies of Internal Disruptions and 𝑚 = 1 Oscillations in Tokamak Discharges with Soft-X-Ray Techniques’, Physical Review Letters, Volume 33, Number 20 (1974)

15. ITER Physics Expert Group on Disruption, Plasma Control, and MHD and ITER Physics Editors

‘ITER Physics Basis’, Nuclear fusion, Volume 39, Number 12 (1999)

16. Vitaly D. Shafranov, ‘Hydromagnetic Stability of a Current-Carrying Pinch in a Strong Longitudinal Magnetic Field’, Zhurnal Teknicheskoi Fiziki, Volume 4, Number 2 (1970)

17. M. N. Busac, R. Pellat, D. Edery and J. L. Soule, ‘Internal Kink Mode in Toroidal Plasmas with Circular Cross Section’, Physical Review Letter, Volume 35, Number 24 (1975)

18. Boris B. Kadomtsev, ‘Disruptive instability in tokamaks’, Fizika Plasmy, Volume 1, p. 710-175 (1975)

19. John A. Wesson, ‘Sawtooth Oscillation’, Plasma Physics and Controlled Fusion, Volume 28, Number 1A (1986)

20. Francis F. Chen, ‘Introduction to Plasma Physics and Controlled Fusion; Volume 1: Plasma Physics’

(2^nd edition), New York and London: Plenum Press (1974)

21. Paul M. Bellan, ‘Fundamentals of Plasma Physics’, Cambridge: Cambridge University Press (2006) 22. David K. Cheng, ‘Fundamentals of Engineering Electromagnetics’, Pearson Education Limited

(2014)

23. Jearl Walker, David Halliday, Robert Robert, ‘Principles of Physics’ (10^th edition), John Wiley &

Sons Singapore Pte. Ltd (2014)

24. Kenro Miyamoto, ‘Fundamentals of Plasma Physics and Controlled Fusion’ (2^nd edition), NIFS- PROC-48 (2000)

25. John Wesson, ‘Tokamak’ (3^rd edition), Oxford University Press (2005)

26. Nishikawa and Masahiro Wakatani, ‘Plasma Physics: Basic Theory with Fusion Application’ (3^rd edition), Berlin, Heidelberg: Springer Berlin Heidelberg: Imprint Springer (2000)

27. Alan Jeffrey, Hui H. Dai, ‘Handbook of Mathematical Formulas and Integrals’ (4^th edition), Burlington, MA: Academic Press/Elsevier (2008)

28. Thomas, George B., ‘Thomas’ Calculus: early transcendentals’ (12^th edition), United States:

Pearson Education (2010)

31. Y. Nam, ‘Validation of 𝑞-profile variation for the sawtoothing KSTAR plasmas by comparative study of measurement and simulation’, (doctoral dissertation) Pohang University of Science and Technology (2016)

32. A. Sykes and John A. Wesson, ‘Relaxation Instability in Tokamak’, Physical Review Letter, Volume 37, Number 3 (1976)

33. G. Lee et al., ‘Design and construction of the KSTAR tokamak’, Nuclear Fusion, Volume 41, Number 10 (2001)

34. O. Neubauer, et al., ‘Design Features of the Tokamak TEXTOR’, Fusion Science and Technoloigy, Volume 47, Number 2 (2005)

35. C. Ryu, et al., ‘Observation and Analysis of a High Frequency MHD Activity during Sawteeth in KSTAR Tokamak’, 23^rd IAEA Fusion Energy Conference, EXW/P2-09 (2010)

36. H. Liu, et al., ‘Operational region and sawteeth oscillation in the EAST tokamak’, Plasma Physics and Controlled Fusion, Volume 49, Number 7 (2007)

37. D. Campbell, et al., ‘Stabilization of Sawteeth with Additional Heating in the JET Tokamak’, Physical Review Letter, Volume 60, Number 21 (1988)

38. V. Beyakov, et al., ‘The T-15 Tokamak. Basic Characteristics and Research Program’, Physical Review Letter Volume 52, Number 2 (1982)

39. H. Park, ‘Comparison Study of 2D Images of Temperature Fluctuation during Sawtooth Oscillation with Theoretical Models’, Physical Review Letters, Volume 96, Issue 19, 195004 (2006)

40. A. W. Edwards, et al., ‘Rapid Collapse of a Plasma Sawtooth Oscillation in the JET Tokamak’, Soviet Atomic Energy, Volume 57, Number 2, (1986)

41. Rybicki, Geroge B. Lightman, Alan P, ‘Radiative processes in astrophysics’, Weinheim: Wiley- VCH (2004)

42. I. H. Hutchinson Author, ‘Principles of Plasma Diagnostics’ (2^nd edition), Cambridge: Cambridge University Press (2002)

43. Marc L. Kutner, ‘Astronomy: A Physical Perspective’ (2^nd edition), Cambridge: Cambridge University Press (2003)

of Scientific Instruments Volume 85, Number 11 (2014)

46. Y. Oh et al., ‘Completion of the KSTAR construction and its role as ITER pilot device’, Fusion Engineering and Design, Volume 83, Number 7 (2008)

47. C. Choe, ‘Dynamics of multiple flux tubes in sawtoothing KSTAR plasmas heated by electron cyclotron waves: 1. Experimental analysis of the tube structure’ Nuclear fusion, Volume 55, Number 1 (2015)

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

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