• Plasma Physics I (2023/1)


    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.



    The lectures will take place in the classroom 2009:

    • Wednesdays: from 16h00 to 18h00
    • Thursdays    : from 16h00 to 18h00



    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, Wiley-VCH, 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



    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 pre-distributed 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.

    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 pre-distributed list with 15 problems
    • Group Interview (29/06): Theoretical and practical aspects associated to the development of the miniaturized tokamak

  • 1. Basic concepts in plasma physics

    • 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: quasi-neutrality, Debye shielding, plasma oscillations, the plasma definition criteria

    • 29/03: Thermodynamic equilibrium states; Degree of ionization and the Saha equation
  • 2. Single particle orbits

    • 30/03: Motion of charged particles in electromagnetic fields: static electric and magnetic fields, non-uniform magnetic field, non-uniform electric field

    • 12/04: Non-uniform and time-dependent electric and magnetic fields; Adiabatic invariants; Particle orbits in a tokamak; Trapped and passing particles
  • 3. Nuclear fusion and tokamak physics

    • 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
  • 4. Fundamentals of plasma kinetic theory

    • 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
  • 5. Plasma as a fluid

    • 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 multi-fluid model: the cold and warm plasma models

    • 27/04: The two-fluid model; The single-fluid 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

  • 6. MHD equilibrium

    • 03/05: MHD equilibrium in fusion plasmas: the theta-pinch, the z-pinch, the screw pinch, the tokamak; The Grad-Shafranov 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

  • 7. The boundary of tokamak plasmas

    • 10/05: Particle recycling; The scrape-off 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 sheath-limited (low recycling) regime; The conduction-limited (high recycling) regime; The two-point model; The extended two-point model; The detached regime
  • 8. MHD waves and instabilities

    • 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 Hain-Lüst 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, non-linear 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
  • Video-lectures by Prof. J. D. Callen

    As an additional resource, I strongly recommend the video-lectures by Prof. J. D. Callen, from the University of Wisconsin-Madison, USA.