Lecture 14 Equilibrium Reconstruction and Properties

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Lecture 14
Equilibrium Reconstruction and Properties

Overview

This lecture is about what an equilibrium really gives you, and how experimentalists infer it.
1.
It collects the geometric quantities attached to an axisymmetric equilibrium: flux coordinates, safety factor, flux-surface averages, shaping, and current profiles.
2.
It shows how the Grad–Shafranov equation (12.28) becomes both a forward problem and an inverse problem.
3.
It treats two genuinely classic examples: Solov’ev equilibria as a semi-analytic family, and EFIT-style reconstruction as the moment when equilibrium solving became routine experimental analysis.

Magnetic equilibrium reconstruction lies at the heart of experimental magnetic-fusion research. Nearly every quantitative comparison between theory and experiment—stability, transport, confinement, current drive, energetic particles, and wave physics—requires a map of the flux surfaces and the current density that generated them. The previous equilibrium lectures established the force-balance logic and the Grad–Shafranov equation. The present lecture asks a different question: once an axisymmetric equilibrium exists, how do we describe it compactly, and how do we infer it from diagnostics?

14.1 Topology and geometry of axisymmetric equilibria

Closed surfaces, magnetic axis, and separatrix structure. In an axisymmetric toroidal plasma the generic integrable topology consists of nested flux surfaces surrounding a magnetic axis. In a poloidal cross-section the magnetic axis appears as an \(O\)-point, that is, an elliptic fixed point of the field-line map. Divertor equilibria introduce \(X\)-points, which are hyperbolic fixed points where the poloidal field vanishes and the separatrix branches. This distinction matters physically: closed surfaces confine the core plasma, whereas open field lines in the scrape-off layer intersect material surfaces.

Figure 14.1: Tokamak start-up involves a topological change from an open-field lines like on the left to one with closed flux surfaces and a magnetic axis like on the right. Forest et al. [1992, 1994a].

Historical Perspective

There are really three historical steps behind this lecture. First came the recognition, due to Grad and Shafranov, that isotropic axisymmetric equilibrium reduces to a single elliptic equation for \(\psi (R,Z)\). Second came analytic families, especially Solov’ev equilibria, that made shaping and free-boundary thinking concrete. Third came the marriage of free-boundary equilibrium solvers with internal magnetic-pitch measurements from motional Stark effect (MSE) polarimetry. Levinton’s PBX-M paper showed that the internal pitch angle could be measured locally, while the DIII-D work of Rice and collaborators made the EFIT+MSE combination a practical route to routine \(q(\psi )\) reconstruction Lao et al. [1985a], Levinton et al. [1989], Rice [1997], Cerfon and Freidberg [2010].

Field-line topology and rational surfaces. Away from separatrices, each flux surface may be labeled by the safety factor \(q(\psi )\), the ratio of toroidal to poloidal winding. When \(q\) is irrational, a field line never closes and densely fills the surface. When \(q=m/n\) is rational, the field line closes after finitely many turns; those rational surfaces are where magnetic islands arise most naturally once axisymmetry is weakly broken. In that sense the equilibrium geometry already contains the seeds of later tearing and resonant-MHD physics.

Choose a flux-surface label. Several equivalent radial labels are common:

\psi , \qquad \Phi (\psi )\equiv \mbox {toroidal flux}, \qquad V(\psi )\equiv \mbox {enclosed volume}, \qquad \rho \equiv \sqrt {\frac {\Phi }{\pi B_{T,0}}}.

No single choice is sacred. The important point is that in an axisymmetric equilibrium all thermodynamic flux functions are most naturally regarded as functions of one surface label.

Figure 14.2: Characteristic geometric shape parameters used in modern tokamak equilibria.

Plasma shape and the last closed flux surface. In a fixed-boundary calculation the last closed flux surface is given in advance. In a free-boundary calculation it emerges from force balance plus the external coil set. A widely used parameterization of a shaped plasma boundary is

\begin{aligned}R(\tau ) &=R_0 + a \cos \!\left (\tau +\delta \sin \tau \right ), \\ Z(\tau ) &= \kappa a \sin \tau , \qquad 0\le \tau \le 2\pi ,\end{aligned} \tag{14.3}

where \(a\) is the minor radius, \(\kappa \) is the elongation, and \(\delta \) is the triangularity. These shape parameters enter almost every equilibrium data set produced for tokamak analysis.

Figure 14.3: (a) an inboard wall limited plasma, (b) a positive triangularity double null plasma and (c) a negative triangularity single null plasma.

Derive the safety factor. Starting from the axisymmetric field representation

\B = \grad \psi \times \grad \phi + F(\psi )\grad \phi = \B _p+\B _\phi , \tag{14.4}

the safety factor is

q(\psi ) = \frac {1}{2\pi } \oint \frac {B_\phi }{R B_p}\,d\ell , \tag{14.5}

where the line integral is taken around the poloidal cross-section of one flux surface.

Derive \(V'(\psi )\) and a useful alternative form of \(q\). Consider the thin shell between \(\psi \) and \(\psi +\delta \psi \). Let \(\Delta (\ell )\) be the distance between the two surfaces measured normal to the contour in the poloidal plane. Since \(R B_p \Delta = \delta \psi \),

\delta V = \oint 2\pi R\,\Delta \,d\ell = 2\pi \delta \psi \oint \frac {d\ell }{B_p},

\boxed { \pp {V}{\psi } = 2\pi \oint \frac {d\ell }{B_p}. } \tag{14.7}

Likewise, the toroidal flux between the two surfaces is

\delta \Phi = \oint B_\phi \Delta \,d\ell = \delta \psi \oint \frac {B_\phi }{R B_p}\,d\ell .

Therefore

\pp {\Phi }{\psi } = \oint \frac {B_\phi }{R B_p}\,d\ell = 2\pi q(\psi ). \tag{14.9}

Use \(B_\phi =F/R\) and Eq. (14.7):

\begin{aligned}2\pi q(\psi ) &= F(\psi )\oint \frac {1}{R^2 B_p}\,d\ell \\ &= F(\psi )\, \frac {1}{2\pi }\pp {V}{\psi } \, \frac {\oint R^{-2} d\ell /B_p}{\oint d\ell /B_p}.\end{aligned}

Define the flux-surface average

\langle f\rangle = \frac {\oint f\,d\ell /B_p}{\oint d\ell /B_p}. \tag{14.12}

Figure 14.4: The relation between \(q\), toroidal flux, and shell geometry.

Then

\boxed { q(\psi ) = \frac {F(\psi )}{4\pi ^2}\pp {V}{\psi }\left \langle \frac {1}{R^2}\right \rangle . } \tag{14.13}

This form is especially convenient when equilibrium solvers already provide \(V'(\psi )\) and surface averages.

Large-aspect-ratio estimate. For a circular large-aspect-ratio tokamak, \(R\simeq R_0\) and \(d\ell \simeq r\,d\theta \), so Eq. (14.5) reduces to

q \sim \frac {r B_\phi }{R_0 B_p},

which is the familiar cylindrical estimate used in the previous lecture.

Useful figures of merit. Once \(V'(\psi )\) is known, the volume-averaged pressure is

\langle p\rangle _V = \frac {1}{V_p} \int p(\psi )\,\pp {V}{\psi }\,d\psi , \qquad V_p = \int \pp {V}{\psi }\,d\psi .

This leads to

\beta _t = \frac {2\muo \langle p\rangle _V}{B_0^2}, \qquad \beta _p = \frac {4\pi ^2 a^2(1+\kappa ^2)\langle p\rangle _V}{\muo I_p^2}.

For shaped plasmas it is also useful to define the cylindrical-edge estimate

q_{\rm cyl} = \frac {2\pi a^2 B_0}{\muo R_0 I_p}, \qquad q_* \equiv q_{\rm cyl}\frac {1+\kappa ^2}{2}. \tag{14.17}

These are not exact invariants, but they are widely used engineering summaries of an equilibrium. The previous lecture showed that the particular combination of \(\beta _p\), \(\ell _i\), and aspect ratio enters toroidal force balance through Eq. (13.75); full reconstruction generalizes that asymptotic lesson to shaped, finite-aspect-ratio equilibria.

14.2 Current structure and flux-surface-averaged Ohm’s law

Decompose the current in axisymmetry. Apply Ampère’s law, Eq. (1.11), to Eq. (14.4). One finds

\muo \J = -\Delta ^\star \psi \,\grad \phi + F'(\psi )\,\grad \psi \times \grad \phi , \tag{14.18}

where \(F'(\psi )=\pp {F}{\psi }\). The toroidal part is set by \(-\Delta ^\star \psi \), while the poloidal part is set by \(F'(\psi )\).

Project the current along the magnetic field. Dot Eq. (14.18) with Eq. (14.4):

\begin{aligned}\muo \J \cdot \B &= -\Delta ^\star \psi \,F\,(\grad \phi )^2 + F'(\psi )\,B_p^2 \\ &= -\Delta ^\star \psi \,\frac {F}{R^2} + F'(\psi )\,B_p^2.\end{aligned}

Now use the Grad–Shafranov equation (12.28),

\Delta ^\star \psi = -\muo R^2 p'(\psi )-F(\psi )F'(\psi ),

to eliminate \(\Delta ^\star \psi \):

\begin{aligned}\muo \J \cdot \B &= \left [ \muo R^2 p'(\psi )+F F' \right ]\frac {F}{R^2} + F' B_p^2 \nonumber \\ &= \muo p'(\psi )F(\psi ) + F'(\psi )\left (B_\phi ^2+B_p^2\right ) \nonumber \\ &= \boxed { \muo \J \cdot \B = \muo p'(\psi )F(\psi ) + F'(\psi )B^2. }\end{aligned} \tag{14.22}

Take the flux-surface average. Because \(p'(\psi )\) and \(F'(\psi )\) are flux functions,

\muo \left \langle \J \cdot \B \right \rangle = \muo p'F + F'\left \langle B^2\right \rangle .

Hence

F' = \frac {\muo \left \langle \J \cdot \B \right \rangle - \muo p'F}{\left \langle B^2\right \rangle }. \tag{14.24}

Insert Eq. (14.24) back into Eq. (14.22):

\frac {\muo J_\parallel }{B} = \frac {\muo \left \langle \J \cdot \B \right \rangle }{\left \langle B^2\right \rangle } + \muo p'F \left ( \frac {1}{B^2} - \frac {1}{\left \langle B^2\right \rangle } \right ). \tag{14.25}

The second term is the Pfirsch–Schlüter variation of the parallel current within a flux surface. Even though the flux-surface average is a flux function, \(J_\parallel \) itself is generally not uniform around the poloidal contour.

Average Ohm’s law over a flux surface. Write the electric field as

\E = \pp {\psi }{t}\grad \phi - \grad \Phi _e, \tag{14.26}

where \(\Phi _e\) is an electrostatic potential. Dot with \(\B \):

\begin{aligned}\E \cdot \B &= F\pp {\psi }{t}(\grad \phi )^2 - \B \cdot \grad \Phi _e \\ &= F\pp {\psi }{t}\frac {1}{R^2} - \B \cdot \grad \Phi _e.\end{aligned}

If \(\Phi _e\) is single-valued on a closed flux surface, the surface average of the field-line derivative vanishes, so

\boxed { \left \langle \E \cdot \B \right \rangle = F\pp {\psi }{t} \left \langle \frac {1}{R^2}\right \rangle . } \tag{14.29}

A simple surface-averaged Ohm’s law then reads

\left \langle \J \cdot \B \right \rangle = \sigma _\parallel \left \langle \E \cdot \B \right \rangle + \left \langle \J _{\rm NI}\cdot \B \right \rangle , \tag{14.30}

where \(\J _{\rm NI}\) represents non-inductive sources such as bootstrap current, neutral beams, RF current drive, or electron-cyclotron current drive.

Why internal measurements are essential for \(q(\psi )\). External magnetic data determine the total plasma current and the boundary shape rather well, but they do not uniquely determine how that current is distributed on flux surfaces. Since the safety factor is a functional of the internal current distribution, \(q(\psi )\) remains weakly constrained unless one brings in internal measurements. MSE is powerful precisely because it measures the local magnetic pitch angle, and those internal pitch-angle constraints can then be folded into EFIT or related Grad–Shafranov solvers to determine the current profile in a way that external magnetics alone cannot Lao et al. [1985a], Levinton et al. [1989], Rice [1997].

Figure 14.5: Time sequences of equilibrium reconstructions, together with internal measurements of the magnetic field from motional Stark effect polarimetry on DIII-D, allowed direct inference of \(\langle \J \cdot \B \rangle \), the inductive electric field through \(\pp {\psi }{t}\), and the non-inductive current contributions Forest et al. [1994b].

Experimental perspective. This is one of the reasons equilibrium reconstruction became so important experimentally. EFIT and MSE are not competing ideas; they are complementary pieces of the same diagnostic chain. EFIT enforces the global free-boundary force balance and coil geometry, while MSE supplies the internal magnetic-pitch information required to pin down \(q(\psi )\). That combined strategy was first established on PBX-M and matured into routine profile analysis on DIII-D Levinton et al. [1989], Rice [1997]. In the DIII-D work shown in Fig. 14.5, time-dependent reconstructions together with internal MSE constraints made it possible to infer the inductive electric field directly and to separate non-inductive current sources. Those measurements were then used to verify and validate models of bootstrap current, neutral-beam current drive, fast-wave current drive, and electron-cyclotron current drive Forest et al. [1994b, 1997, 1996], Petty et al. [1995], Oikawa et al. [2000].

What does it take to “measure” the \(q\)-profile?

Strictly speaking, the \(q\)-profile is inferred, not directly measured: a reconstruction is not a direct measurement. Because \(q(\psi )\sim B_\phi /B_p\), reconstructing \(q(\psi )\) requires the poloidal field \(B_p(\psi )\), and therefore the toroidal current density \(j_\phi (\psi )\), throughout the plasma.
Writing \(f(\psi )\equiv R B_\phi \), the Grad–Shafranov source is \[ -\Delta ^\star \psi = R\,p'(\psi ) + \frac {1}{\mu _0 R}\,f f'(\psi ). \] For a circular plasma with medium-to-high aspect ratio, \(R=R_0(1+\epsilon \cos \theta )\) with \(\epsilon \ll 1\), so \[ R\,p' + \frac {f f'}{\mu _0 R} = \left (R_0 p' + \frac {f f'}{\mu _0 R_0}\right ) +\epsilon \cos \theta \left (R_0 p' - \frac {f f'}{\mu _0 R_0}\right ) +O(\epsilon ^2). \] Thus external magnetics are mainly sensitive to the first combination, while the part that separates \(p'(\psi )\) from \(f f'(\psi )\) appears only through small geometric corrections. This is the origin of the near-degeneracy of \(p'\) and \(f f'\) in circular, higher-aspect-ratio plasmas.
Historically, this is why external magnetic data determine the plasma boundary and a few global moments, but not a unique \(j(r)\) or \(q(r)\). In the circular high-aspect-ratio limit one essentially recovers \(I_p\) and the combination \(\beta _p+l_i/2\), not the full current profile. Elongation, triangularity, up–down asymmetry, and especially low aspect ratio increase the sensitivity of the external field to the internal current distribution and partially break this degeneracy.
In practice, “measuring” \(q(\psi )\) means combining external magnetics with at least one internal or topological constraint: internal pitch-angle data (MSE, Li-beam, polarimetry/Faraday rotation), kinetic pressure profiles, or flux-surface-shape information. The last category includes diagnostics whose contours approximately label flux surfaces, such as TS/ECE isotherms or tangential soft-X-ray imaging when the emissivity is sufficiently flux-function-like.

External magnetics determine boundary shape and global moments; internal/topological data make \(q(\psi )\) measurable.

Already in the classical equilibrium literature it was clear that, for circular medium- to high-aspect-ratio tokamaks, external magnetic data do not uniquely determine \(j_\phi (r)\) because \(p'(\psi )\) and \(ff'(\psi )\) are nearly degenerate; noncircular shaping and lower aspect ratio help break this degeneracy, but genuinely reconstructing \(q(\psi )\) requires additional internal or topological information, such as MSE or flux-surface-shape constraints from soft-x-ray imaging Christiansen and Taylor [1982], Braams [1991], Lao et al. [1985b, 1990], Blum et al. [2012], Zakharov et al. [2008], Hirshman et al. [1994], Tritz et al. [2003], Qian et al. [2009].

14.3 Why reconstruction is an inverse problem

What is known and what is unknown? Once the plasma boundary is specified—either directly, or implicitly through external coil currents and magnetic measurements—the Grad–Shafranov problem reduces to determining the source functions on the right-hand side of Eq. (12.28), namely

p'(\psi ) \qquad \mbox {and} \qquad F(\psi )F'(\psi ).

These functions determine the pressure profile and the toroidal-current distribution. In experiment, however, they are not measured directly. What is measured are magnetic probe signals, flux loops, Rogowski coils, coil currents, and, when available, internal constraints such as MSE, interferometry, polarimetry, Thomson scattering, or soft x-ray emissivity.

Why the problem is ill posed. If only external magnetic measurements are available, then recovering completely arbitrary functions \(p'(\psi )\) and \(F F'(\psi )\) is fundamentally underdetermined. The unknowns are continuous functions; the data are finite in number. Worse, different internal current profiles can produce very similar external fields, especially for circular plasmas. This is why equilibrium reconstruction is an inverse problem rather than a direct inversion formula.

Parameterize the profile functions. A common strategy is to represent the unknown functions in a finite basis:

p'(\hat \psi ) = \sum _{n=0}^{N_a} a_n P_n(\hat \psi ), \qquad F F'(\hat \psi ) = \sum _{n=0}^{N_b} b_n Q_n(\hat \psi ), \tag{14.32}

where the normalized flux is

\hat \psi = \frac {\psi -\psi _{\rm lim}}{\psi _0-\psi _{\rm lim}}. \tag{14.33}

Here \(\psi _0\) denotes the flux at the magnetic axis and \(\psi _{\rm lim}\) the value at the boundary or last closed flux surface. Low-order polynomials are common, but splines, piecewise linear profiles, or other regularized basis sets are also used.

Turn the inverse problem into optimization. For a given set of coefficients \(\{a_n,b_n\}\), one solves the Grad–Shafranov equation and computes the predicted diagnostic signals. This is the forward problem. One then adjusts the coefficients to minimize a goodness-of-fit function, typically of \(\chi ^2\) form,

\chi ^2 = \sum _m \frac {\left (d_m^{\rm meas}-d_m^{\rm model}\right )^2}{\sigma _m^2},

possibly augmented by smoothness constraints or Bayesian priors. Reconstruction therefore lives at the intersection of equilibrium theory, numerical PDEs, and inverse methods.

Historical Perspective

A classic problem: EFIT plus internal constraints. The work of Lao and collaborators turned Grad–Shafranov reconstruction into a practical experimental tool Lao et al. [1985a]. But the deeper lesson of the next decade was that the most important equilibrium quantities are not all equally constrained by boundary magnetics. In particular, the \(q\) profile becomes a true experimental observable only when the free-boundary solver is paired with internal measurements such as MSE. That is why EFIT and MSE together, rather than either one alone, deserve to be treated as a classic tokamak equilibrium problem Levinton et al. [1989], Rice [1997].

14.4 Numerical Techniques

Fixed-boundary finite-difference discretization. For numerical work, Eq. (12.28) is often solved on a rectangular \((R,Z)\) grid. Write

R_i = R_{\min }+i\Delta R, \qquad Z_j = Z_{\min }+j\Delta Z, \qquad \psi _{i,j}\equiv \psi (R_i,Z_j),

and define the staggered radii

R_{i\pm 1/2}=R_i\pm \frac {\Delta R}{2}.

Assemble the five-point stencil. Using the conservative form

\Delta ^\star \psi = R\pp {}{R}\!\left (\frac {1}{R}\pp {\psi }{R}\right ) + \pp {^2\psi }{Z^2},

a standard second-order discretization yields

\begin{aligned}A_i^{R+} &= \frac {R_i}{\Delta R^2 R_{i+1/2}}, & A_i^{R-} &= \frac {R_i}{\Delta R^2 R_{i-1/2}}, \\ A_i^{R0} &= -\frac {R_i}{\Delta R^2} \left ( \frac {1}{R_{i+1/2}}+\frac {1}{R_{i-1/2}} \right ), & A^Z &= \frac {1}{\Delta Z^2}.\end{aligned}

The discrete equation at node \((i,j)\) is

\left (A_i^{R0}-\frac {2}{\Delta Z^2}\right )\psi _{i,j} + A_i^{R+}\psi _{i+1,j} + A_i^{R-}\psi _{i-1,j} + \frac {1}{\Delta Z^2}\left (\psi _{i,j+1}+\psi _{i,j-1}\right ) = b_{i,j}(\psi _{i,j}), \tag{14.40}

with nonlinear source

b_{i,j}(\psi _{i,j}) = -\muo R_i^2\left .\pp {p}{\psi }\right |_{\psi _{i,j}} - \left .F\pp {F}{\psi }\right |_{\psi _{i,j}}.

Thus one obtains a sparse nonlinear algebraic system.

Picard iteration. A common strategy is to lag the nonlinear source:

\vect {A}\,\vect {\psi }^{(k+1)} = \vect {b}\!\left (\vect {\psi }^{(k)}\right ). \tag{14.42}

Written componentwise,

\begin{aligned}\left (A_i^{R0}-\frac {2}{\Delta Z^2}\right )\psi _{i,j}^{(k+1)} &+ A_i^{R+}\psi _{i+1,j}^{(k+1)} + A_i^{R-}\psi _{i-1,j}^{(k+1)} \nonumber \\ &+ \frac {1}{\Delta Z^2} \left ( \psi _{i,j+1}^{(k+1)}+\psi _{i,j-1}^{(k+1)} \right ) = -\muo R_i^2\left .\pp {p}{\psi }\right |_{\psi _{i,j}^{(k)}} - \left .F\pp {F}{\psi }\right |_{\psi _{i,j}^{(k)}}.\end{aligned}

This is usually robust when the profiles are not too stiff.

Green’s-function free-boundary methods. An alternative is to treat the plasma and external coils as toroidal current filaments. If the toroidal current density is represented by discrete current elements, then the poloidal flux is

\psi (R,Z) = \muo \sum _n I_n\,G(R,Z;R_n,Z_n). \tag{14.44}

For a filament at \((R',Z')\), the standard Green’s function is

G(R,Z;R',Z') = \frac {1}{\pi }\sqrt {R R'}\; \frac {(2-k^2)K(k)-2E(k)}{k}, \tag{14.45}

with

k^2 = \frac {4RR'}{(R+R')^2+(Z-Z')^2}. \tag{14.46}

This representation is especially attractive for free-boundary problems, because the same Green’s-function table can be used for both plasma currents and coil currents.

Vacuum and external coils are naturally handled with Green’s functions. In a vacuum region the source is purely toroidal current, so one solves

\Delta ^\star \psi = -\muo R J_\phi (R,Z). \tag{14.47}

Formally one may write the solution as

\psi (R,Z) = \muo \int G(R,Z;R',Z')\,R'J_\phi (R',Z')\,dR'\,dZ', \tag{14.48}

where \(G\) is the Green’s function of \(\Delta ^\star \). For a current filament, the result is expressible in terms of complete elliptic integrals; deriving that kernel is a good exercise because it connects the abstract PDE back to the magnetic field of real coils.

Nonlinear profiles are usually solved iteratively. Given an iterate \(\psi ^{(n)}\), define the source

S^{(n)}(R,Z) = -\muo R^2 p'\!\left (\psi ^{(n)}\right ) - F\!\left (\psi ^{(n)}\right )F'\!\left (\psi ^{(n)}\right ). \tag{14.49}

Then solve the linear problem

\Delta ^\star \psi ^{(n+1)} = S^{(n)} \tag{14.50}

with the chosen boundary conditions, and under-relax if necessary. This Picard viewpoint is conceptually simple and already captures the structure of many practical equilibrium solvers.

Experimental perspective. For a fusion experimentalist the Grad–Shafranov equation is not an abstract exercise. It is the daily language of equilibrium reconstruction. Magnetic probes, flux loops, Rogowski coils, diamagnetic measurements, and sometimes motional-Stark-effect constraints are combined with a Grad–Shafranov solver to infer the plasma boundary, current profile, safety-factor profile, X-point position, and Shafranov shift. In that sense this lecture is one of the clearest examples of a classic analytical result becoming a routine diagnostic tool.

Figure 14.6: Logical flow of a free-boundary equilibrium solver for an axisymmetric tokamak.

An EFIT-style reconstruction loop. A representative workflow is:

1.: guess \(\psi ^{(0)}\) and choose profile parameterizations for \(p'(\hat \psi )\) and \(F F'(\hat \psi )\);
2.: compute the plasma current density implied by the current iterate;
3.: solve the forward Grad–Shafranov problem, either by finite difference or by Green’s functions, including the known external coil currents;
4.: compare the predicted magnetic and internal diagnostic signals against the measured ones;
5.: update the profile coefficients and repeat until the residuals and regularization criteria are satisfactory.

This is the practical form in which equilibrium reconstruction entered experimental fusion.

Takeaways

1.
The equilibrium is more than a contour plot of \(\psi \): it furnishes \(q(\psi )\), \(V'(\psi )\), flux-surface averages, current profiles, and shaping parameters that feed directly into stability and transport theory.
2.
EFIT-style reconstruction is a classic experimental problem because it turned the Grad–Shafranov equation into an everyday analysis tool, and MSE made the internal \(q\) profile part of the experimentally constrained state rather than a weakly constrained model output.

Bibliography

Sydney Chapman and T. G. Cowling. The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, Cambridge, 2 edition, 1952.

Jr. Spitzer, Lyman and Richard Härm. Transport phenomena in a completely ionized gas. Physical Review, 89(5):977–981, 1953. doi:10.1103/PhysRev.89.977.

S. I. Braginskii. Transport phenomena in a completely ionized two-temperature plasma. Soviet Physics JETP, 6(2):358–369, 1958. English translation of Zh. Eksp. Teor. Fiz. 33, 459–472 (1957).

Allan N. Kaufman. Plasma viscosity in a magnetic field. Physics of Fluids, 3(4):610–616, 1960. doi:10.1063/1.1706096.

W. B. Thompson. The dynamics of high temperature plasmas. Reports on Progress in Physics, 24(1):363–424, 1961. doi:10.1088/0034-4885/24/1/308.

A. Simon and W. B. Thompson. Hydromagnetic equations with viscosity for a collisionless plasma. Journal of Nuclear Energy, Part C: Plasma Physics, Accelerators, Thermonuclear Research, 8(4):373, 1966. doi:10.1088/0368-3281/8/4/302.

G. F. Chew, M. L. Goldberger, and F. E. Low. The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 236(1204):112–118, 1956. doi:10.1098/rspa.1956.0116.

Russell M. Kulsrud. Plasma Physics for Astrophysics. Princeton University Press, Princeton, NJ, 2005. ISBN 9780691120737.

Problems

Problem 14.1. Solov’ev equilibrium and shaping parameters

Starting from Eq. (12.49), show how the truncated form may be obtained from a suitable choice of coefficients \(c_j\). Identify which combinations of coefficients control the magnetic-axis position, the effective minor radius, and the elongation.

Problem 14.2. Green’s function for a toroidal filament

Derive the Green’s function (14.45) for the Grad–Shafranov operator and show how \(B_R\) and \(B_Z\) may be recovered from \(\psi (R,Z)\). Then implement a numerical routine that computes contours of \(\psi \) and \(B_Z(R,Z=0)\) for

(a): a single current loop,
(b): a quadrupole pair of loops,
(c): an octupole arrangement of four loops.

Use your plots to explain how external shaping coils build up multipole vacuum fields.

Problem 14.3. Reconstruction as an inverse problem

Choose a simple fixed-boundary geometry and profile parameterization, generate synthetic magnetic-diagnostic data from a known equilibrium, and then reconstruct the equilibrium by fitting the coefficients in Eq. (14.32). Which quantities are recovered robustly, and which are sensitive to the chosen basis and regularization?