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It is known that most of our universe is in the plasma state, that is, in the form of an ionized gas whose atoms are dissociated into positive ions and electrons. Plasma physics is the area of knowledge that studies the behavior of such ionized gases. Research in plasma physics started with Irving Langmuir (1932 Nobel Prize) in the 1920s. Modern plasma physics, however, started around 1952, when it was first proposed that nuclear fusion reactions produced during the explosion of a hydrogen bomb could be controlled to produce energy in a reactor. Since then, research in plasma physics has advanced significantly and this advance has been crucial to the development of controlled thermonuclear fusion as a potential new source of energy.
This course will provide the students with a broad view on various physics phenomena occurring in tokamak plasmas. Practical aspects related to tokamak control and operation will also be provided. In this course, the students will be introduced to the physical models that are most used to describe the behavior of both natural and artificial plasmas, namely the theory of single particle orbits, the plasma kinetic theory and the fluid models, including the magnetohydrodynamic (MHD) approximation. A significant part of the course is devoted to study the physics issues associated to the MHD equilibrium and stability of tokamak plasmas.
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Lectures
The lectures will take place in the classroom 2009: Wednesdays: from 16h00 to 18h00
 Thursdays : from 16h00 to 18h00
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Bibliography
Below is a list of the books that will be used during this course. The specific books/chapters used in each lecture are indicated in the slides of that particular lecture.
 Introduction to Plasma Physics and Controlled Fusion, 3a Edição, F. F. Chen, Springer, 2016
 Fundamentals of Plasma Physics, 3a Edição, J. A. Bittencourt (INPE), Springer, 2004
 Industrial Plasma Engineering, Volume 1: Principles, J. Reece Roth, Institute of Physics Publishing, 1995
 Fundamentals of Plasma Physics, V.E. Golant, A.P. Zhilinsky and I.E. Sakharov, Wiley Series in Plasma Physics, 1980
 Ideal MHD, J. P. Freidberg, Cambridge University Press, 2014
 Plasma Physics and Fusion Energy, J.P. Friedberg, Cambridge University Press, 2007
 Teoria Magnetohidrodinâmica  Equilíbrio e Estabilidade, Prof. Ricardo M. O. Galvão (IFUSP), 1989
 Tokamaks, 3a Edição, J. Wesson, Clarendon Press, 2004
 Magnetohydrodynamic Stability of Tokamaks, H. Zohm, WileyVCH, 2015
 The Plasma Boundary of Magnetic Fusion Devices, P. C. Stangeby, Institute of Physics Publishing, 2000
 Principles of Plasma Discharges and Materials Processing, 2a Edição, J. A. Lieberman and A. Lichtenberg, Wiley, 2005
 Plasma Physics: An Introduction, R. Fitzpatrick, CRC Press, 2014
 Magnetic Fields, H. E. Knoepfel, John Willey & Sons, 2000
 Fundamentals of Magnetic Thermonuclear Reactor Design, Woodhead Publishing, 2018
 Plasma Physics for Controlled Fusion, K. Miyamoto, Springer, 2016
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Examination
In this course, the students will have the option of choosing between two examination methods: the ordinary and the extraordinary. The extraordinary examination method was introduced to allow for students coming from engineering courses to take advantage of their practical skills. Of course, the students will be free to choose the examination method they prefer.
 The ordinary evaluation method
 The students will be evaluated via 3 written exams
 Each written exam will be composed by 10 problems and each student will receive a different set of problems
 The students will have 5 days to finish the written exam at home
 The absence in a written exam implies in zero grade for that particular exam
 The student's score will be \(M = (P_1 + P_2 + P_3)/3\)
 Students with \(M \geqslant 5 \) are Approved while students with \(M < 5 \) are Not Approved.
 The final grade for those who are approved will be: (A) if \(M \geqslant 9 \), (B) if \( 7 \leqslant M < 9 \) and (C) if \( 5 \leqslant M < 7 \)
 The extraordinary evaluation method
 Students that select this evaluation method will form a single group that will have to design, build (3D printing) and assemble a miniaturized tokamak (30 cm maximum diameter viewed from the top)
 The miniaturized tokamak will be composed by toroidal field coils, poloidal field coils, a central solenoid, a metal ring (mimicking the presence of the plasma) and a simple plasma control system to control the current in the metal ring induced by the current in the central solenoid
 All the expenses involved in the fabrication of this miniaturized tokamak will be provided by Prof. Canal
 Although the design activities will be carried out by the students, they will have constant support from Prof. Canal
 Students that select this evaluation method will be able to contribute with theoretical and/or technical support to the project
 The students will be evaluated via 1 individual oral exam at the end of the semester, regarding the solution of 5 problems sorted out from a predistributed list with 15 problems, and via a group interview, regarding the theoretical and practical aspects associated to the development of the miniaturized tokamak
 During the group interview, one or more students will present the physics/engineering criteria used in the design of each part of the miniaturized tokamak. The students must emphasize their contributions to the project.
 After the individual oral examination and group interview, a final score will be attributed to the student
I strongly encourage the students to choose the extraordinary examination method as, in my opinion, it will provide a broader view about the various aspects associated to tokamak physics and operation. However, as already pointed out, the students will be free to choose the examination method they prefer.
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Written Exams: Dates and Topics P\(_1\) (20/04): Basic concepts in plasma physics (Topic 1), Single particle orbits (Topic 2), Nuclear fusion and tokamak physics (Topic 3) and Fundamentals of plasma kinetic theory (Topic 4)
 P\(_2\) (11/05): Plasma as a fluid (Topic 5), MHD equilibrium (Topic 6) and The boundary of tokamak plasmas (Topic 7)
 P\(_3\) (28/06): MHD waves and instabilities (Topic 8)
Individual Oral Exams and Group Interview: Dates and Topics Individual Oral Exams (29/06): Discussion about 5 problems sorted out from a predistributed list with 15 problems
 Group Interview (29/06): Theoretical and practical aspects associated to the development of the miniaturized tokamak

 22/03: Introduction; Theoretical descriptions of plasma phenomena; Review of basic concepts in kinetic theory of gases; Collision cross section; The Rutherford cross section; Collision parameters; Collisional processes; Particle detailed balance
 23/03: Low pressure electrical discharges: the Townsend avalanche, the effect of secondary electrons, the Townsend criterion for breakdown; Macroscopic features of plasmas: quasineutrality, Debye shielding, plasma oscillations, the plasma definition criteria
 29/03: Thermodynamic equilibrium states; Degree of ionization and the Saha equation

 30/03: Motion of charged particles in electromagnetic fields: static electric and magnetic fields, nonuniform magnetic field, nonuniform electric field
 12/04: Nonuniform and timedependent electric and magnetic fields; Adiabatic invariants; Particle orbits in a tokamak; Trapped and passing particles

 13/04: Basic aspects of nuclear fusion, tokamak design and tokamak operation: fusion reactions, thermonuclear fusion, the Lawson criterion and the triple product, the ignition condition, scaling laws
 19/04: The hoop and tire tube forces; Toroidal field coils; The central solenoid; Poloidal field coils; The vertical plasma instability; The need for a plasma control system; Tokamak engineering and the RZIP model

 20/04: Introduction; The phase space; Distribution function; Number density and average velocity; The Boltzmann equation; The Vlasov equation; The Klimontovich distribution function; Example: Debye shielding using the Vlasov equation
 20/04: Introduction; The phase space; Distribution function; Number density and average velocity; The Boltzmann equation; The Vlasov equation; The Klimontovich distribution function; Example: Debye shielding using the Vlasov equation

 26/04: Macroscopic transport equations; Macroscopic plasma quantities; Moments of the Boltzmann equation; General transport equation; Conservation of mass; Conservation of momentum; Conservation of energy; The multifluid model: the cold and warm plasma models
 27/04: The twofluid model; The singlefluid model; The magnetohydrodynamic (MHD) model; General aspects of the MHD equations; The validity of the MHD model; Boundary conditions; Consequences of the MHD equations: magnetic flux conservation and MHD equilibrium

 03/05: MHD equilibrium in fusion plasmas: the thetapinch, the zpinch, the screw pinch, the tokamak; The GradShafranov equation; Boundary conditions; Equilibrium in a conducting shell; Equilibrium held by external currents
 04/05: Plasmas of circular cross section and large aspect ratio: equilibrium held by a vertical field; The Solov’ev solution: wall limited and diverted plasma configurations

 10/05: Particle recycling; The scrapeoff layer (SOL): limiter SOL and diverted SOL; The role and properties of the sheath; Mapping of upstream plasma quantities to the divertor targets
 11/05: The sheathlimited (low recycling) regime; The conductionlimited (high recycling) regime; The twopoint model; The extended twopoint model; The detached regime

 17/05: MHD waves in an homogeneous plasma: shear (Alfvén) waves and (fast and slow) magnetosonic waves; Linear ideal MHD stability; The energy principle of ideal MHD; The eigenmode formalism; The standard form and the intuitive form of \(\delta W\); The energy principle for the tokamak
 18/05: The linear ideal MHD stability of a screw pinch; The HainLüst equation
 24/05: The Newcomb equation; The tokamak ordering; The intuitive form of \(\delta W\) in the tokamak ordering;
 25/05: Current driven ideal MHD (global) modes in tokamaks: internal kink modes and external kink modes
 31/05: Pressure driven ideal MHD (localized) modes in a screw pinch: interchange modes, the Suydam criterion; Pressure driven ideal MHD (localized) modes in a tokamak: interchange modes, the Mercier criterion, ballooning modes, the second stability region
 01/06: Combined pressure gradient and current driven (localized) modes in a tokamak: the low and high confinement operational regimes, Edge Localized Modes (ELMs), the linear ideal MHD stability of the pedestal, the peeling mode
 07/06: The physics of the plasma pedestal evolution; the ELM cycle; Magnitude and timescale of ELM crashes; Active ELM control
 14/06: Combined pressure gradient and current driven (global) modes in a tokamak: plasma operational scenarios, external kink modes with finite \(\beta\); The effect of a conducting wall on external kink modes: ideally conducting wall, resistive wall; The Resistive Wall Mode (RWM); The Troyon limit
 21/06: Resistive MHD stability: general comments, stability of a current sheet, magnetic islands, tearing modes in a screw pinch, the Rutherford equation
 22/06: Classical tearing modes in tokamaks: the effect of tearing modes on kinetic profiles and confinement, nonlinear saturation, rotation and locking; Neoclassical Tearing Modes (NTMs) in tokamaks: the modified Rutherford equation, the onset criteria, NTM active control
 28/06: Disruptions: phenomenology, the density limit, consequences of disruptions, generation of runaway electrons, disruption avoidance and mitigation

As an additional resource, I strongly recommend the videolectures by Prof. J. D. Callen, from the University of WisconsinMadison, USA.