Chapter
Boundary conditions and conservation laws
4 MHD equilibrium: general considerations
4.2 Basic equilibrium equations
4.4 The need for toroidicity
4.6 Surface quantities: basic plasma parameters and figures of merit
4.6.1 Fluxes and currents
4.6.2 Normalized plasma pressure,
β
4.6.4 Rotational transform, ι,
and the MHD safety factor, q
4.7 Equilibrium degrees of freedom
4.8 The basic problem of toroidal equilibrium
4.9 A single particle picture of toroidal equilibrium
5 Equilibrium: one-dimensional configurations
5.4 The general screw pinch
5.4.3 The perpendicular pinch
5.5 Inherently 1-D fusion configurations
5.5.1 The reversed field pinch
The equilibrium p and Bz profiles
5.5.2 The low β
ohmic tokamak
Simple transport profiles for p and
Bθ
Summary of the low β
tokamak
6 Equilibrium: two-dimensional configurations
6.2 Derivation of the Grad-Shafranov equation
6.2.1 The ∇‧
B = 0 equation
6.3 Plasma parameters and figures of merit
6.3.1 Simple flux coordinates
6.3.2 The volume of a flux surface
6.3.4 The kink safety factor
6.3.5 Rotational transform and the MHD safety factor
6.3.6 The MHD safety factor on axis
6.3.7 Alternate choices for
F(ψ)
6.4 Analytic solution in the limit
ε « 1 and βp ~ 1
6.4.1 The coordinate transformation
6.4.2 The asymptotic expansion
6.4.3 The ε0
equation: radial pressure balance
6.4.4 The ε1
equation: toroidal force balance
6.4.5 Application to early reversed field pinches (RFP)
6.4.6 Application to early ohmic tokamaks and modern reversed field pinches
Calculation of the vertical field
6.5 Analytic solution in the limit ε « 1 and βp ~ 1/ε (the high β
tokamak)
6.5.1 The high β
tokamak expansion
6.5.2 The circular high β
tokamak
Plasma parameters and figures of merit
The high β
tokamak equilibrium limit
Numerical results for a high β
tokamak
6.5.3 The flux conserving tokamak - avoiding the equilibrium β
limit
The basic idea behind flux conservation
Choosing p(ψ, t) and F(ψ,
t)
Implementing the flux conserving constraint
The flux conserving Solov'ev
equilibrium
6.5.4 The elliptic high β
tokamak
Plasma parameters and figures of merit
6.6 Exact solutions to the Grad-Shafranov equation (standard and spherical tokamaks)
6.6.1 Mathematical formulation
6.6.2 Examples: TFTR and JET
6.6.3 Example: the spherical tokamak (ST)
Description of the spherical tokamak
Example: the Mega Amp Spherical Tokamak (MAST)
6.6.4 The equilibrium β
limit
6.6.5 Up-down asymmetric solutions
Up-down asymmetric equilibria
X-point boundary constraints
6.6.6 Example: the International Thermonuclear Experimental Reactor (ITER)
6.6.7 Example: the National Spherical Torus Experiment (NSTX)
6.7 The helical Grad-Shafranov equation (the straight stellarator)
6.7.2 The helical Grad-Shafranov equation
6.7.3 Low β
analytic solution
6.7.4 The rotational transform
Derivation of the rotational transform
The loose helix approximation
The reversed field pinch (RFP)
The ohmically heated tokamak
The spherical tokamak (ST)
7 Equilibrium: three-dimensional configurations
7.2 The high β
stellarator expansion
7.2.2 The basic equations
7.2.3 The high β
stellarator expansion
7.2.4 Reduction of the equations
The J⊥ =
(B x ∇p)/B2 equation
Including a finite l = 0 mirror field
7.3 Relation of the high β
stellarator expansion to other models
7.3.2 The straight stellarator
7.4 The Greene-Johnson limit
7.4.1 Comparison of expansions
7.4.2 The Greene-Johnson limit of the HBS model
7.4.3 The Greene-Johnson model
7.5.1 Single helicity - the limiting helical field amplitude
7.5.2 Multiple helicity stellarators
The numerical formulation
7.6.1 Low β
single helicity solutions
7.6.2 Toroidal force balance in a current-free stellarator
Formulation of the toroidal force balance problem
The helical restoring force
7.6.3 How does a vertical field shift a stellarator with no net current?
The flux function in the presence of a vertical field
Applying the boundary conditions
The dipole surface current
The surface current body force
The dynamical equations of motion
7.6.4 The equilibrium β
limit in a stellarator
Simplification of the overlap equations
Relation of A and C to physical quantities
The flux function, the rotational transform, and the equilibrium β
limit
7.6.5 The flux conserving stellarator
7.6.6 Multiple helicity, finite β
stellarators
7.7 Neoclassical transport in stellarators
7.7.1 Review of transport in a tokamak
General neoclassical transport procedure
Neoclassical transport in a tokamak - overview
Neoclassical transport in a tokamak - the banana regime
Anomalous transport in a tokamak - gyro-Bohm diffusion
Transport in a tokamak - summary
7.7.2 The problem with neoclassical transport in a stellarator
7.7.3 One solution - the omnigenous stellarator
The constraint of ideal-omnigenity
The quasi-omnigenous stellarator
7.7.4 The isodynamic stellarator
The ideal-isodynamic constraint
The quasi-isodynamic stellarator
7.7.5 The symmetric stellarator
The ideal-symmetric stellarator - general velocity constraints
The ideal-symmetric stellarator - the large flow velocity constraint
The ideal-symmetric stellarator
What is symmetric in an ideal-symmetric stellarator?
Are ideal-symmetric stellarators omnigenous?
The quasi-symmetric stellarator
Relation of f&bar;1,f&bar;2,ḡ1,ḡ2 to physical quantities
Properties of Boozer coordinates
Applications of Boozer coordinates
7.7.7 Summary of neoclassical transport in a stellarator
7.8.1 The Large Helical Device (LHD)
7.8.2 The Wendelstein 7-X (W7-X) stellarator
MHD stellarator computation
8 MHD stability - general considerations
8.2 Definition of MHD stability
8.3 Waves in an infinite homogeneous plasma
8.3.1 The shear Alfven wave
8.3.2 The fast magnetosonic wave
8.3.3 The slow magnetosonic wave
8.4 General linearized stability equations
8.4.1 Initial value formulation
8.4.2 Normal mode formulation
8.5 Properties of the force operator
F(ξ)
8.5.1 Self-adjointness of
F(ξ)
8.5.2 The "standard form" of
δW
8.5.3 The "intuitive form" of
δW
8.5.4 The "intuitive self-adjoint form" of
δW
8.5.6 Orthogonality of the normal modes
8.6 Variational formulation
8.7.1 Statement of the Energy Principle
8.7.2 Proof of the Energy Principle
Proof assuming discrete normal modes
8.7.3 Advantages of the Energy Principle
8.8 The Extended Energy Principle
8.8.1 Statement of the problem
8.8.2 The boundary conditions
8.8.3 The natural boundary condition
8.8.6 Summary of the Extended Energy Principle
8.9.1 The general minimizing condition
8.9.3 Closed line systems
8.9.4 Summary and discussion
8.10 Vacuum versus force-free plasma
8.10.1 The nature of the problem
8.10.2 Vacuum vs. force-free plasma: the same results
8.10.3 Vacuum vs. force-free plasma: different results
8.10.4 The real situation: a resistive region
8.11 Classification of MHD instabilities
8.11.1 Internal/fixed boundary modes
8.11.2 External/free boundary modes
8.11.3 Pressure-driven modes
Interchange instabilities
8.11.4 Current-driven modes
9.2 Transition from collision dominated to collisionless regimes
9.5.2 Derivation of the kinetic MHD model
Coordinate transformations
Expressions for
ε˙, μ˙, α˙
Expressions for ε˙&bar;, μ˙&bar;,
α˙&bar;
The gyro radius expansion
9.6 The Chew, Goldberger, Low (CGL) double adiabatic model
9.6.1 Formulation of the problem
9.6.2 Derivation of the double adiabatic model
Closure of the moment equations
9.6.3 The modified double adiabatic model
9.7.3 The double adiabatic model
10 MHD stability comparison theorems
10.2 Ideal MHD equilibrium and stability
10.3 Double adiabatic MHD equilibrium and stability
10.3.1 Relation between the perturbed quantities and
ξ
10.3.2 The double adiabatic MHD Energy Principle
10.3.3 Summary of CGL stability
10.4 Kinetic MHD equilibrium and stability
10.4.2 Stability of a closed line cylindrical system
The linearized stability equations
Solution for the m = 0 mode in a Z-pinch
The Z-pinch Energy Relation
10.4.3 Stability of an ergodic cylindrical system
Solution for the perturbed distribution function in a general screw pinch
The cylindrical screw pinch Energy Relation
10.4.4 Stability of a general toroidal configuration
Derivation of the perturbed distribution function
Derivation of the general Energy Relation
10.5 Stability comparison theorems
10.5.1 Closed line cylindrical geometry
10.5.2 Ergodic cylindrical geometry
10.5.3 Closed line toroidal geometry
10.5.4 Ergodic toroidal geometry
11 Stability: one-dimensional configurations
11.2 The basic stability equations
11.2.1 The Energy Principle
11.2.2 The normal mode eigenvalue equations
11.2.3 Incompressible MHD
11.3 Stability of the
θ-pinch
11.3.1 Application of the Energy Principle to the
θ-pinch
11.3.2 Minimizing δWF for a
θ-pinch
11.3.3 Continuum damping in a "slab"
θ-pinch
Derivation of the differential equation
Mathematical solution for a linearly varying density profile
Physical implications of the solution
11.4 Stability of the Z-pinch
11.4.1 Energy Principle analysis of m
≠ 0 modes
Stability criterion for m≠
0 modes
11.4.2 Energy Principle analysis of the m = 0 mode
Stability criterion for the m = 0 mode
Validity of ideal MHD stability for the m = 0 mode in a Z-pinch
11.4.3 Double adiabatic Energy Principle analysis of the m = 0 mode
Stability criterion for the m = 0 mode
Summary of stability in a standard Z-pinch
11.4.4 The hard-core Z-pinch
Hard-core Z-pinch equilibria
The hard-core Z-pinch equilibrium limit
The hard-core Z-pinch m = 1 stability limit
The hard-core Z-pinch m = 0 stability limit
Summary of the hard-core Z-pinch
11.5 General stability properties of the screw pinch
11.5.1 Evaluation of δW
for a general screw pinch
Setting up the evaluation of
δWF
Evaluation of δWV and
δWS
11.5.2 Suydam's
criterion
11.5.3 Newcomb's
procedure
Internal mode stability when F(r)
≠ 0
Internal mode stability when F(rs) = 0
Minimizing the search over m and k
11.5.4 The normal mode eigenvalue equation
Formulation of the problem
Eliminating Q and p1 in terms of
ξ
Eliminating η and ξǁ
in terms of ξ
11.5.5 The oscillation theorem
Introducing cylindrical symmetry
The relation between δξ(rn) and
δω2
11.5.6 The resistive wall mode
The vacuum and wall solutions
The jump conditions across the wall
The resistive wall dispersion relation
11.6 The "straight"
tokamak
11.6.1 Reduction of δW
for the straight tokamak
The large aspect ratio expansion
11.6.2 Sawtooth oscillations - the internal m = 1 mode
Evaluation of δWF
for internal modes
11.6.3 Current-driven disruptions - external low m modes
Possible ranges of unstable wave numbers
The m = 1 Kruskal-Shafranov limit
The m ≥ 2
external kink modes
A flat J(r) profile with a jump at r = a
Linear J(r) edge gradient
Maximally flat J(r) edge gradient
11.6.4 Density-driven disruptions - external low m modes
A model problem for a turbulence-driven density limit
11.6.5 Resistive wall modes - external low m modes
11.6.6 Edge localized modes (ELMs) - external high m modes
Stability of the constant current density tokamak (reference case)
The high m stability equation
11.6.7 Summary of the straight tokamak
11.7 The reversed field pinch (RFP)
11.7.1 Physical parameters describing an RFP
11.7.2 Instability of an RFP without a Bz reversal
11.7.3 The m = 0 instability
11.7.4 Suydam's
criterion
Impossibility of Bz(r) reversal in a pure ohmic discharge
Taylor's
minimum energy formulation
The ideal MHD helicity constraints
Taylor's
constraints for a dissipative plasma
The minimum energy equations
Mathematical solution for the cylindrically symmetric minimum energy state
Mathematical solution for the mixed helical minimum energy state
Application to experiment
11.7.7 Ideal external modes
Theoretical/computational analysis
Experimental stabilization of external kink modes
11.7.8 Overview of the RFP
General screw pinch stability theory
12 Stability: multi-dimensional configurations
12.2 Ballooning and interchange instabilities
12.2.2 Shear, periodicity, and localization
Straightforward Fourier analysis
A simple attempt at localization
The ballooning mode formalism
Localizing the quasi-mode
12.2.3 General reduction of δW
for ballooning modes
12.3 The ballooning mode equations for tokamaks
12.4 The ballooning mode equation for stellarators
12.5 Stability of tokamaks - the Mercier criterion
12.5.2 Cylindrical limit: the Suydam criterion
12.5.3 Toroidal geometry: the Mercier criterion
12.5.4 Analytic limits of the Mercier criterion
12.6 Stability of tokamaks - ballooning modes
12.6.2 The s-α
model for ballooning modes
Reduction of the general ballooning mode equation
12.6.3 β
limits due to ballooning modes
Predictions from the s-α
diagram
Numerical studies of ballooning mode stability
12.7 Stability of tokamaks - low n internal modes
12.7.2 Low n internal modes with finite shear
12.7.3 Low n internal modes with small shear
12.8 Stability of tokamaks - low n external ballooning-kink modes
12.8.2 Simplification of δWF by the high β
tokamak expansion
12.8.3 High β
stability of the surface current model
Surface current equilibrium
12.8.4 Numerical studies of ballooning-kink instabilities
12.9 Stability of tokamaks - advanced tokamak (AT) operation
12.9.2 Bootstrap current profile - the βN
limit
12.9.3 Wall stabilization in an advanced tokamak
12.10 Stability of tokamaks - n = 0 Axisymmetric modes
12.10.2 Vertical instabilities in a circular plasma
12.10.3 Horizontal instabilities in a circular plasma
12.10.4 Vertical instabilities in an elongated plasma
The elliptic cylinder model
12.11 Overview of the tokamak
12.12 Stellarator stability
12.12.1 High n modes in a stellarator
12.12.2 The parallel current constraint
12.12.3 The relation between average curvature and magnetic well
12.12.4 Average curvature of a straight helix
12.12.5 Shear stabilization of a straight helix
12.12.6 Stabilization of a toroidal stellarator
The Greene-Johnson expansion
The helical field contribution
The vertical field contribution
Relation between σ
and BV
12.12.7 Numerical results
General ballooning mode equation
Tokamak and stellarator analytic theory
Stellarator numerical codes
Appendix A
Heuristic derivation of the kinetic equation
A.2 Heuristic derivation of the kinetic equation
A.3 Conservation of particles in 3-D physical space
A.4 Kinetic generalization to a 6-D phase space
A.7 The collision operater
A.8 Specific forms of the collision operators
The Fokker-Planck collision operator
Appendix B
The Braginskii transport coefficients
B.2 Collisional momentum transfer
B.4 Collisional energy transfer
Appendix C
Time derivatives in moving plasmas
C.2 Time derivatives in a moving volume
C.3 Flux change in a moving area
Appendix D
The curvature vector
Appendix E Overlap limit of the high β
and Greene-Johnson stellarator models
The (1/N1/2)0 contribution to the A equation
The (1/N1/2)1 contribution to the β
equation
The (1/N1/2)2 contribution to the β
equation
The (1/N1/2)2 contribution to the A equation
The (1/N1/2)3 contribution to the A equation
The final equation for loosely wound Greene-Johnson stellarator
E.3 Auxiliary analysis I: relation between G(β&bar;0) and
ψ
E.4 Auxiliary analysis II: relation between H(ψ) and
J(ψ)
Appendix F
General form for q(ψ)
Appendix G
Natural boundary conditions
G.1 The variational principle for simple boundary conditions
G.2 The variational principle for more complicated boundary conditions
Appendix H Upper and lower bounds on
δQKIN
Slight simplification of
I(τ)
Carrying out the l integration
H.2 A lower bound on
δQKIN
H.3 An upper bound on
δQKIN
Ideal MHD
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