Chapter
Some closing comments on fusion power generation
3.5.1 Overview of radiation losses
3.5.2 Calculation of W, the energy lost per particle per Coulomb collision
3.5.4 The effect of multiple ion species
The nuclear physics of fusion reactions
4 Power balance in a fusion reactor
4.2 The 0-D conservation of energy relation
4.3 General power balance in magnetic fusion
4.4 Steady state 0-D power balance
4.5 Power balance in the plasma
4.6 Power balance in a reactor
4.6.1 The physics gain factor Q
4.6.2 The engineering gain factor Q
4.7 Time dependent power balance in a fusion reactor
4.7.1 Time dependent 0-D power balance relation
4.7.3 The minimum external power
4.8 Summary of magnetic fusion power balance
Power balance in a fusion reactor
5 Design of a simple magnetic fusion reactor
5.2 A generic magnetic fusion reactor
5.3 The critical reactor design parameters to be calculated
5.4 Design goals, and basic engineering and nuclear physics constraints
5.4.2 Engineering constraints
5.4.3 Nuclear physics constraints
5.5 Design of the reactor
5.5.1 Outline of the design calculation
5.5.2 The blanket-and-shield thickness
5.5.3 The plasma radius and coil thickness
The minimum coil thickness
5.5.4 The major radius and plasma surface area and volume
5.5.5 Power density and plasma pressure
5.5.6 The plasma physics quantities beta and…
Specific designs for ignition experiments and fusion reactors
Part II The plasma physics of fusion energy
6 Overview of magnetic fusion
6.2 Basic description of a plasma
6.3 Single-particle behavior
6.4 Self-consistent models
6.5 MHD equilibrium and stability
6.6 Magnetic fusion concepts
6.8 Heating and current drive
6.9 The future of fusion research
7 Definition of a fusion plasma
7.2 Shielding DC electric fields in a plasma – the Debye length
7.2.1 A physical picture of Debye shielding
7.2.2 Derivation of the Debye length
7.3 Shielding AC electric fields in a plasma – the plasma frequency
7.3.1 A physical picture of the screening of AC fields
7.3.2 Derivation of the electron plasma frequency
7.4 Low collisionality and collective effects
7.4.1 A statistical picture of long-range collective effects
7.4.2 The inter-particle spacing versus the Coulomb interactive distance
7.4.3 The plasma frequency vs. the Coulomb collision frequency
7.5 Additional constraints for a magnetic fusion plasma
7.6 Macroscopic behavior vs. collisions
8 Single-particle motion in a plasma
8.2 General properties of single-particle motion
8.2.1 Exact equations of motion
8.2.2 General conservation relations
8.3 Motion in a constant B field
8.3.2 Perpendicular motion
8.3.3 Consequences of gyro motion
8.4 Motion in constant B and E fields: the drift
8.4.1 Effect of a parallel electric field
8.4.2 Effect of a perpendicular electric field
8.5 Motion in fields with perpendicular gradients: the ∇B drift
8.5.1 Perpendicular gradient in B with E = 0
8.5.2 Perpendicular gradient in E with uniform B
8.6 Motion in a curved magnetic field: the curvature drift
8.7 Combined V and V drifts in a vacuum magnetic fieldh
8.8 Motion in time varying E and B fields: the polarization drift
8.8.1 The polarization drift for…
8.8.2 The polarization drift for…
8.9 Motion in fields with parallel gradients: the magnetic moment and mirroring
8.9.1 The mathematical formulation
8.9.2 Solution to the equations
8.9.3 The mirror effect and the mirror machine
A qualitative picture of the mirror effect
The quantitative conditions for mirroring
The simple mirror machine
8.10 Summary – putting all the pieces together
9 Single-particle motion – Coulomb collisions
9.2 Coulomb collisions – mathematical derivation
9.2.1 Formulation of the problem
9.2.2 Solution to the problem
9.3 The test particle collision frequencies
9.3.1 The general formulation
9.3.2 Evaluation of… and the integration over the impact parameter
9.3.3 Integration over target velocities
9.3.4 Properties of… and other collision frequencies
9.4 The mirror machine revisited
9.4.2 Power balance in a simple mirror machine
9.5 The slowing down of high-energy ions
9.5.1 The high-energy-ion slowing down model
9.5.2 Which species is preferentially heated?
9.5.3 The alpha particle slowing down time
9.5.4 The fraction of alpha energy transferred to electrons and to ions
9.5.5 Discussion of beam heating
9.6.1 The threshold condition for runaway electrons
9.6.2 Properties of runaway electrons
9.7 Net exchange collisions
9.7.1 Formulation of the problem
9.7.2 The net momentum exchange collision rate
9.7.3 The net energy exchange collision rate
10 A self-consistent two-fluid model
10.2 Properties of a fluid model
10.2.1 Macroscopic averages
10.2.2 Size of a fluid element
10.2.3 When is a plasma fluid model useful?
10.3 Conservation of mass
10.4 Conservation of momentum
10.4.1 The basic principle
10.4.2 The inertial force
10.4.3 The electric field force
10.4.4 The magnetic field force
10.4.5 The pressure gradient force
10.4.6 The collisional friction force
10.4.7 The conservation of momentum equations
10.5 Conservation of energy
10.5.1 The basic principle
10.5.2 The rate of change of internal energy
10.5.3 The compression work
10.5.4 Thermal conduction
10.5.6 The external auxiliary heating power
10.5.7 The ohmic heating power
10.5.8 Bremsstrahlung radiation
10.5.9 Energy equilibration
10.5.10 The conservation of energy equations
10.6 Summary of the two-fluid model
11 MHD – macroscopic equilibrium
11.1 The basic issues of macroscopic equilibrium and stability
11.2 Derivation of MHD from the two-fluid model
11.2.1 Basic scaling relations for MHD
11.2.2 The “obvious” simplifications
11.2.3 The single-fluid variables
11.2.4 The conservation of mass equations
11.2.5 The conservation of momentum equations
11.2.6 The conservation of energy equations
11.2.7 Summary of the MHD equations
11.3 Derivation of MHD from guiding center theory
11.3.2 The guiding center drift current and Ohm’s law
11.3.3 The magnetization current
11.3.4 The perpendicular MHD momentum equation
11.4 MHD equilibrium – a qualitative description
11.5 Basic properties of the MHD equilibrium model
11.5.1 The MHD equilibrium model
11.5.2 General properties – flux surfaces
11.5.3 General properties – current surfaces
11.5.4 General properties – magnetic pressure and tension
11.6 Radial pressure balance
11.6.4 General definition of beta in a screw pinch
11.7 Toroidal force balance
11.7.3 The tire tube force
11.7.5 The restoring force due to a perfectly conducting wall
11.7.6 The restoring force due to a vertical field
11.7.7 Analytic derivation of toroidal force balance
The vertical field force FV
The vertical field for toroidal force balance
Why bending a θ-pinch into a torus doesn’t work
11.7.8 Single particle picture of toroidal force balance
The pure toroidal θ-pinch
The Z-pinch and the screw pinch
11.7.9 Calculating the rotational transform
Rotational transform in a straight cylinder
Rotational transform in an axisymmetric torus
11.7.10 Toroidal force balance in configurations without toroidal current
The second order solution
11.8 Summary of MHD equilibrium
12 MHD – macroscopic stability
12.2 General concepts of stability
12.2.1 Instabilities in physical systems
12.2.2 The frozen-in-field-line concept
12.2.3 Classifications of MHD instabilities
Internal and external modes
Pressure-driven and current-driven modes
Conducting wall vs. no wall configurations
12.3 A physical picture of MHD instabilities
12.3.3 Current-driven instabilities
12.3.4 Single-particle picture of favorable and unfavorable curvature
12.4 The general formulation of the ideal MHD stability problem
12.4.1 The concept of linear stability
12.4.2 The MHD linear stability equations
12.4.3 A general property of linear MHD stability
12.5 The infinite homogeneous plasma – MHD waves
12.5.1 General derivation of MHD waves
12.5.2 The shear Alfvén wave
12.5.3 The compressional Alfvén wave
12.6 The linear Phi-pinch
12.6.1 The equilibrium and perturbation
12.6.2 The radial differential equation
12.6.3 Stability of the Phi-pinch
12.7 The m = 0 mode in a linear Z-pinch
12.7.1 Derivation of the differential equation
12.7.2 Stability of the m = 0 mode
12.7.3 Profile implications for stabilizing the m = 0 mode
12.8 The m = 1 mode in a linear Z-pinch
12.8.1 The Phi component of the momentum equation
12.8.2 The z component of the momentum equation
12.8.3 The r component of the momentum equation
12.8.4 Solution to the m = 1 eigenvalue equation
12.9 Summary of stability
13 Magnetic fusion concepts
13.2 The levitated dipole (LDX)
13.2.1 Overview of the LDX
The global radial pressure balance relation
The poloidal magnetic field
The equilibrium beta limit
13.2.5 Summary of the levitated dipole
13.3 The field reversed configuration (FRC)
13.3.1 Overview of the FRC
13.3.3 The FRC as a source of fusion energy
13.3.4 Summary of the FRC
13.4 The surface current model
13.4.2 The 2-D surface current equilibrium
13.4.3 The perturbed magnetic field in the plasma
13.4.4 The perturbed magnetic field in the vacuum
13.4.5 The pressure balance matching condition
13.4.6 Summary of the surface current model
13.5 The reversed field pinch (RFP)
13.5.1 Overview of the RFP
13.5.2 RFP surface current equilibrium
13.5.3 RFP surface current stability
The pressure balance matching condition
13.5.6 The resistive wall mode
The vacuum and resistive wall magnetic fields
The resistive wall stability boundary
13.6.1 Overview of the spheromak
13.6.2 Spheromak surface current equilibrium
13.6.3 The m = 1 tilt instability
13.6.4 Summary of the spheromak
13.7.1 Overview of the tokamak
13.7.2 The circular cross section tokamak – equilibrium
The aspect ratio expansion
The surface current pressure balance relation
The equilibrium beta limit
13.7.3 The circular cross section tokamak – stability
The pressure balance matching condition
The… ballooning-kink instability
13.7.4 The non-circular cross section tokamak
The n = 0 axisymmetric instability
A more realistic n = 0 wire model
Stabilization of the n = 0 mode by a conducting wall
The n = 1 ballooning-kink instability
13.7.4 The advanced tokamak (AT)
The effect of a wall on the kink current limit
The effect of a wall on the ballooning-kink beta limit
13.7.5 The spherical tokamak (ST)
MHD beta limit in a spherical torus
Relation between beta and pressure in tokamaks
13.7.6 Summary of the tokamak
13.8.1 Overview of the stellarator
13.8.2 The Large Helical Device (LHD)
13.8.3 Guiding center particle orbits in a stellarator
The guiding center orbits in Boozer coordinates
13.8.4 The Wendelstein 7-X (W7-X)
13.8.5 The National Compact Stellarator Experiment (NCSX)
13.9 Revisiting the simple fusion reactor
13.9.1 Goal of the analysis
14.2 Transport in a 1-D cylindrical plasma
14.2.2 Calculating transport coefficients from the random walk model
14.2.3 Particle diffusion in a magnetized plasma
Comparison with the fluid model and numerical values
14.2.4 Thermal conductivity of a magnetized plasma
14.3 Solving the transport equations
14.3.1 Temperature equilibration
14.3.2 Effect of the heating profile on the central temperature
14.3.3 Ohmic heating to ignition
Approximate solution to the problem
Physical properties of the solution
Irony – too much confinement can be a disadvantage
14.4 Neoclassical transport
14.4.2 Neoclassical transport due to passing particles
The transport coefficients
14.4.3 Neoclassical transport due to trapped particles
The fraction of trapped particles
The effective collision frequency
The trapped particle neoclassical transport coefficients
14.4.4 The bootstrap current
The trapped electron magnetization current
The passing electron magnetization current
The collision-driven bootstrap current
14.5 Empirical scaling relations
14.5.2 Edge transport phenomena in a tokamak
Edge localized modes (ELMs)
Internal transport barriers
14.5.3 Empirical fit for E
14.6 Applications of transport theory to a fusion ignition experiment
14.6.2 A superconducting ignition experiment
The minimum volume experiment
14.6.3 Heating to ignition
The minimum power for ignition
14.6.4 The bootstrap fraction
Derivation of the bootstrap fraction
Standard monotonic profiles
Internal transport barriers
15 Heating and current drive
15.2.1 The ohmic heating model
15.2.2 Ohmic power balance
15.2.3 Thermal conduction losses
15.2.5 Ohmic power balance
15.3 Neutral beam heating
15.3.2 How is a neutral beam produced?
15.3.3 The physics problem – energy required for beam penetration
15.3.4 The technology problem – conversion efficiency in the neutralizer
15.4 Basic principles of RF heating and current drive
15.4.2 RF sources and launching structures
15.4.3 Principles of electromagnetic wave propagation in a plasma
Cutoffs and wave resonances
Reflection, transmission, absorption, and mode conversion
15.4.4 Analysis of electromagnetic wave propagation in a plasma
15.5 The cold plasma dispersion relation
15.6 Collisionless damping
15.6.2 X-mode cyclotron damping
15.6.3 O-mode cyclotron damping and generalized Landau damping
15.7 Electron cyclotron heating (ECH)
15.7.1 O-mode accessibility
15.7.3 X-mode accessibility
15.8 Ion cyclotron heating (ICH)
15.8.1 X-mode fundamental accessibility and polarization
15.8.2 Fast mode second harmonic accessibility
15.8.3 Fast mode second harmonic absorption
15.8.4 Fast wave minority accessibility
15.8.5 Fast wave minority absorption
15.9 Lower hybrid current drive (LHCD)
15.9.2 Lower hybrid accessibility
15.9.3 Slow wave power absorption
Lower hybrid heating and current drive
16 The future of fusion research
16.2 Current status of plasma physics research
16.2.1 Macroscopic equilibrium and stability
16.2.3 Heating and current drive
16.2.4 Alpha particle plasma physics
16.2.5 Fusion technology issues
16.4 A demonstration power plant (DEMO)
Appendix A Analytic derivation of sigma
Appendix B Radiation from an accelerating charge
B.1 Definition of the radiation field
B.2 Calculation of A and φ from a time dependent source
B.3 Application to a single accelerating charge
B.4 Calculation of E and B
B.5 Calculation of the power radiated
Appendix C Derivation of Boozer coordinates
C.1 General coordinate transformation
C.2 The partial simplification to the cross-product form of Boozer coordinates
C.3 The partial simplification to the gradient form of Boozer coordinates
C.4 Elimination of the free functions…
C.5 Introduction of physical quantities into the Boozer coordinates
C.6 The guiding center orbits in Boozer coordinates
Appendix D Poynting’s theorem
Plasma Physics and Fusion Energy
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