Energy Engineering

Lecture 1 (3 H). Introduction to the class. Introduction to plasmas and controlled fusion. Ordinary fluids: Navier-Stokes equation. Ordinary gases: Boltzmann equation. Plasmas: both kinetic and fluid description. Magnetic confinement: close and open configurations, toroidal configurations. Toroidal field, poloidal field, vertical field. Tokamaks and stellarators. Inertial confinement: Laser-driven inertial confinement. Direct and indirect drive. Brief history on the research on controlled fusion energy. Lecture 2 (2 H). Definition of plasma. Saha equation. Individual and collective effects. Debye shielding and quasi-neutrality and Debye length. Lecture 3 (3 H). Plasma oscillations and plasma frequency: double layer model and fluid model. Characteristic parameters of a fusion plasma. Charged particle motion in external electric and magnetic fields, particle drifts, fluid drifts. Uniform B: Larmor radius, cyclotron frequency. Uniform E and B fields: ExB drift. Non-uniform B field: grad B perp to B (grad B drift), curvature drift. Lecture 4 (2 H). Grad B parallel to B: first adiabatic invariant. Examples: magnetic mirror, need for ”rotational transform” in toroidal configurations. Lecture 5 (3 H). Coulomb collisions. Collision parameter, scattering angle. Rutherford cross-section. Collision frequencies, mean-free-paths. Relaxation times and energy equilibration times. Lecture 6 (2 H). Electrical resistivity: collisional model, runaway electrons. Radiation emission. General aspects. Larmor formula. Cyclotron radiation. Bremsstrahlung radiation. Lecture 7 (3 H). Plasmas and controlled nuclear fusion. Principles. Fusion reactions, cross sections, reactivity. Main reactions between hydrogen isotopes (DD,DT) Thermonuclear fusion. Steady-state power balance of a thermonuclear plasma: ideal ignition temperature. Lawson criterion, n-tauE-T criterion, critical p-tauE curve for ignition vs T. Lecture 8 (2 H). The physics Q factor. Thermal stability. The dW/dt vs T curve. The role of auxiliary heating. Heating to ignition. Presentation of the class project. The tokamak. Its components and its principle of operation. Central solenoid, toroidal magnetic field coils, poloidal magnetic field coils. Induced current, bootstrap current, current drive by external means. Reactor parameters (ITER case). Lecture 9 (3 H). From kinetic models to fluid models an MHD model. Boltzmann equation and Vlasov equation. Moments of Boltzmann equation, fluid equations. Lecture 10 (2 H). Two-fluid plasma model. Single fluid plasma models: MHD. Ideal and resistive MHD. Qualitative discussion of the range of application of MHD. Lecture 11 (2 H). Simple applications of MHD. Magnetic field diffusion. B-lines freezing. Fluid drifts perp to B: diamagnetic drift and current, ExB drift Lecture 12 (3 H). Particle and energy transport. Classical treatment of diffusion coefficient. Diffusion coefficient for magnetized and unmagnetized plasma. Discussion to anomalous transport. Lecture 13 (2 H). Plasma waves. Wave equation and wave solutions, group and phase velocity. Linear perturbation theory and dispersion relation. Lecture 14 (3 H). Waves in a fluid plasma. Linear theory, electrostatic and electromagnetic waves. Electron plasma waves (Bohm-Gross waves). Electromagnetic waves in unmagnetized plasmas: dispersion relation, cut-off and critical density, collisional absorption. Lecture 15 (2 H). Alfven waves. MHD equilibrium. General remarks: flux surfaces, magnetic pressure, magnetic tension. Lecture 16 (3 H). Radial equilibrium in the zeta-pinch, theta-pinch, screw-pinch. The beta factor. Toroidal force balance. The three forces, formula for the vertical magnetic field. Rotational transform, the q-factor. Lecture 17 (2 H). General concepts of magnetized plasma stability. Interchange instability, kink instability, concept of favorable and unfavorable curvature. Qualitative examples. Lecture 18 (3 H). The general formulation of the ideal MHD stability problem. The method of linearization. The linearized momentum equation: the eigenvalue problem. Stability properties of the zeta and theta pinch. Lecture 19 (3 H). Stability of the circular tokamak: surface current model, pressure balance matching condition. Equilibrium beta-limit. Lecture 20 (2 H). Stability criterion to current-driven (kink) modes. Stability criterion to pressure-driven (kink-ballooning) modes. Troyon beta-limit. Lecture 21 (3 H). Stability of the elongated tokamak. Transport phenomena in tokamaks. Classical transport, neo-classical transport, turbulent transport. Lecture 22 (2 H). Scaling laws of magnetic confinement. The L (low) and H (high) confinement mode. Advance tokamak regime with high bootstrap current fraction. Lecture 23 (3 H). Introduction to plasma kinetic theory: drift- and gyro-kinetic models. Lecture 24 (2 H). Introduction to plasma kinetic theory: drift- and gyro-kinetic models. Lecture 25 (3 H). Overview of computer codes to study the equilibrium, stability and temporal evolution of magnetized plasmas. Lecture 26 (2 H). Overview of computer codes to study the equilibrium, stability and temporal evolution of magnetized plasmas.

1st level degree (Laurea Triennale) in Engineering or Physics.

"Plasma physics and fusion energy" by J. Freidberg, and additional notes distributed by the Instructor.

The assessment will be based on the student's ability to describe qualitatively (in words) and quantitatively (with mathematical formulas) the physical phenomena underlying equilibrium, stability, and transport in fusion magnetized plasmas. The student must also present and comment the results of the assigned project, demonstrating mastery of the mathematical and numerical procedures used.

Lecture 1 (3 H). Introduction to the class. Introduction to plasmas and controlled fusion. Ordinary fluids: Navier-Stokes equation. Ordinary gases: Boltzmann equation. Plasmas: both kinetic and fluid description. Magnetic confinement: close and open configurations, toroidal configurations. Toroidal field, poloidal field, vertical field. Tokamaks and stellarators. Inertial confinement: Laser-driven inertial confinement. Direct and indirect drive. Brief history on the research on controlled fusion energy. Lecture 2 (2 H). Definition of plasma. Saha equation. Individual and collective effects. Debye shielding and quasi-neutrality and Debye length. Lecture 3 (3 H). Plasma oscillations and plasma frequency: double layer model and fluid model. Characteristic parameters of a fusion plasma. Charged particle motion in external electric and magnetic fields, particle drifts, fluid drifts. Uniform B: Larmor radius, cyclotron frequency. Uniform E and B fields: ExB drift. Non-uniform B field: grad B perp to B (grad B drift), curvature drift. Lecture 4 (2 H). Grad B parallel to B: first adiabatic invariant. Examples: magnetic mirror, need for ”rotational transform” in toroidal configurations. Lecture 5 (3 H). Coulomb collisions. Collision parameter, scattering angle. Rutherford cross-section. Collision frequencies, mean-free-paths. Relaxation times and energy equilibration times. Lecture 6 (2 H). Electrical resistivity: collisional model, runaway electrons. Radiation emission. General aspects. Larmor formula. Cyclotron radiation. Bremsstrahlung radiation. Lecture 7 (3 H). Plasmas and controlled nuclear fusion. Principles. Fusion reactions, cross sections, reactivity. Main reactions between hydrogen isotopes (DD,DT) Thermonuclear fusion. Steady-state power balance of a thermonuclear plasma: ideal ignition temperature. Lawson criterion, n-tauE-T criterion, critical p-tauE curve for ignition vs T. Lecture 8 (2 H). The physics Q factor. Thermal stability. The dW/dt vs T curve. The role of auxiliary heating. Heating to ignition. Presentation of the class project. The tokamak. Its components and its principle of operation. Central solenoid, toroidal magnetic field coils, poloidal magnetic field coils. Induced current, bootstrap current, current drive by external means. Reactor parameters (ITER case). Lecture 9 (3 H). From kinetic models to fluid models an MHD model. Boltzmann equation and Vlasov equation. Moments of Boltzmann equation, fluid equations. Lecture 10 (2 H). Two-fluid plasma model. Single fluid plasma models: MHD. Ideal and resistive MHD. Qualitative discussion of the range of application of MHD. Lecture 11 (2 H). Simple applications of MHD. Magnetic field diffusion. B-lines freezing. Fluid drifts perp to B: diamagnetic drift and current, ExB drift Lecture 12 (3 H). Particle and energy transport. Classical treatment of diffusion coefficient. Diffusion coefficient for magnetized and unmagnetized plasma. Discussion to anomalous transport. Lecture 13 (2 H). Plasma waves. Wave equation and wave solutions, group and phase velocity. Linear perturbation theory and dispersion relation. Lecture 14 (3 H). Waves in a fluid plasma. Linear theory, electrostatic and electromagnetic waves. Electron plasma waves (Bohm-Gross waves). Electromagnetic waves in unmagnetized plasmas: dispersion relation, cut-off and critical density, collisional absorption. Lecture 15 (2 H). Alfven waves. MHD equilibrium. General remarks: flux surfaces, magnetic pressure, magnetic tension. Lecture 16 (3 H). Radial equilibrium in the zeta-pinch, theta-pinch, screw-pinch. The beta factor. Toroidal force balance. The three forces, formula for the vertical magnetic field. Rotational transform, the q-factor. Lecture 17 (2 H). General concepts of magnetized plasma stability. Interchange instability, kink instability, concept of favorable and unfavorable curvature. Qualitative examples. Lecture 18 (3 H). The general formulation of the ideal MHD stability problem. The method of linearization. The linearized momentum equation: the eigenvalue problem. Stability properties of the zeta and theta pinch. Lecture 19 (3 H). Stability of the circular tokamak: surface current model, pressure balance matching condition. Equilibrium beta-limit. Lecture 20 (2 H). Stability criterion to current-driven (kink) modes. Stability criterion to pressure-driven (kink-ballooning) modes. Troyon beta-limit. Lecture 21 (3 H). Stability of the elongated tokamak. Transport phenomena in tokamaks. Classical transport, neo-classical transport, turbulent transport. Lecture 22 (2 H). Scaling laws of magnetic confinement. The L (low) and H (high) confinement mode. Advance tokamak regime with high bootstrap current fraction. Lecture 23 (3 H). Introduction to plasma kinetic theory: drift- and gyro-kinetic models. Lecture 24 (2 H). Introduction to plasma kinetic theory: drift- and gyro-kinetic models. Lecture 25 (3 H). Overview of computer codes to study the equilibrium, stability and temporal evolution of magnetized plasmas. Lecture 26 (2 H). Overview of computer codes to study the equilibrium, stability and temporal evolution of magnetized plasmas.

1st level degree (Laurea Triennale) in Engineering or Physics.

"Plasma physics and fusion energy" by J. Freidberg, and additional notes distributed by the Instructor.

The assessment will be based on the student's ability to describe qualitatively (in words) and quantitatively (with mathematical formulas) the physical phenomena underlying equilibrium, stability, and transport in fusion magnetized plasmas. The student must also present and comment the results of the assigned project, demonstrating mastery of the mathematical and numerical procedures used.