Lecture 43 The 2-D Toroidal Energy Principle and the High-\(n\) Ballooning Limit

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Lecture 43
The 2-D Toroidal Energy Principle and the High-\(n\) Ballooning Limit

Overview

This appendix carries the ballooning reduction all the way from the exact two-dimensional tokamak energy functional to the local one-dimensional field-line equation. The chain is \[ \begin {aligned} \delta W[\xi _\psi ,\xi _\chi ,\xi _\parallel ] &\longrightarrow \delta W[X,U,Z] \longrightarrow \text {eliminate }Z \equiv \text {eliminate }\xi _\parallel \\ &\longrightarrow \text {high-}n\text { eliminate }U \longrightarrow \delta W[X] \longrightarrow \widehat X(\Psi ,y) \longrightarrow F_0(y;\Psi ,y_0). \end {aligned} \] The two central ideas are:
1.
the first minimization is exact and removes the parallel degree of freedom by enforcing \(\nabla \cdot \vect {\xi }=0\);
2.
the ballooning transform removes the awkward periodicity constraint, so the large-\(n\) mode can be treated as a slowly varying envelope on an unwrapped field line.

This appendix is written as a working derivation rather than a summary. I keep the intermediate algebra visible, especially in the two minimizations and in the ordering steps that turn the exact \(X\)-only functional into the local field-line equation.

Historical Perspective

The conceptual obstacle solved by Connor, Hastie, and Taylor was not merely “doing large-\(n\) MHD more carefully.” The real obstacle was geometric: ballooning modes want short perpendicular wavelength and long parallel wavelength, but a sheared torus insists on periodicity in both the poloidal and toroidal directions. Their transform to an infinite field-line domain resolves that tension cleanly. Once that is done, the high-\(n\) problem becomes a systematic asymptotic expansion, and the leading-order stability problem collapses to an ordinary differential equation on a single unwrapped field line.

Caution

Notation dictionary. Connor–Hastie–Taylor write the toroidal-field function as \[ I(\Psi )=RB_\phi , \] and they write the poloidal field as \(B_\chi \). In the notation of these notes, \[ I \;\longrightarrow \; \muo I_{\mathrm {pol}}, \qquad B_\chi \;\longrightarrow \; B_p. \] I keep the Connor–Hastie–Taylor notation inside most of the derivation because it makes the operator algebra shorter. Also, to avoid confusion with the displacement variable \(X\), I denote the fast radial scale by \[ s \equiv \sqrt {n}\,(\Psi -\Psi _0), \] rather than by \(x\).

43.1 Geometry, Variables, and the Exact 2-D Functional

We work in orthogonal axisymmetric coordinates \((\Psi ,\chi ,\phi )\), where \(\Psi \) labels flux surfaces, \(\chi \) is a poloidal angle-like coordinate on each surface, and \(\phi \) is the ignorable toroidal angle. The metric is taken in the Connor–Hastie–Taylor form

ds^2 = \frac {d\Psi ^2}{R^2B_\chi ^2} + (JB_\chi )^2 d\chi ^2 + R^2 d\phi ^2, \qquad dV = J\,d\Psi \,d\chi \,d\phi . \tag{I.1}

Thus the scale factors are

h_\Psi = \frac {1}{RB_\chi }, \qquad h_\chi = JB_\chi , \qquad h_\phi = R. \tag{I.2}

The magnetic field is written as

\B = \nabla \Psi \times \nabla \phi + I(\Psi )\,\nabla \phi = B_\chi \,\hat {\vect {e}}_\chi + \frac {I}{R}\,\hat {\vect {e}}_\phi . \tag{I.3}

It is convenient to introduce

\nu (\Psi ,\chi ) \equiv \frac {IJ}{R^2}, \qquad \B \cdot \nabla = \frac {1}{J}\left (\pp {}{\chi }+\nu \pp {}{\phi }\right ). \tag{I.4}

For a single toroidal Fourier harmonic

\vect {\xi }(\Psi ,\chi ,\phi ) = \vect {\xi }(\Psi ,\chi )e^{in\phi }, \tag{I.5}

we therefore have

\B \cdot \nabla \;\longrightarrow \; \frac {1}{J}\left (\pp {}{\chi }+in\nu \right ), \qquad ik_\parallel = \frac {1}{JB}\left (\pp {}{\chi }+in\nu \right ). \tag{I.6}

The safety factor is

q(\Psi )=\frac {1}{2\pi }\oint \nu (\Psi ,\chi )\,d\chi . \tag{I.7}

The equilibrium current components needed below are

J_\chi = B_\chi I', \qquad j_\phi \equiv J_\phi = -\frac {R}{J}\pp {}{\Psi }(JB_\chi ^2), \qquad \frac {j_\phi }{R}=p'+\frac {II'}{R^2}, \tag{I.8}

where the prime denotes \(d/d\Psi \) and the last identity is the Grad–Shafranov force-balance relation in this notation.

Displacement components and the variables \(X,U,Z\)

Write the displacement in the orthonormal basis associated with Eq. (I.1) as

\vect {\xi } = \xi _\psi \,\hat {\vect {e}}_\Psi + \xi _\chi \,\hat {\vect {e}}_\chi + \xi _\phi \,\hat {\vect {e}}_\phi . \tag{I.9}

Connor–Hastie–Taylor then package these components into the variables

X \equiv RB_\chi \,\xi _\psi , \qquad U \equiv \frac {\xi _\phi }{R}-\frac {I}{R^2B_\chi }\,\xi _\chi , \qquad Z \equiv \frac {\xi _\phi }{B_\chi }. \tag{I.10}

The variable \(X\) is the normal displacement written in a metric-weighted form. The variable \(U\) measures the displacement inside the flux surface and perpendicular to the magnetic field. The variable \(Z\) is the combination in which the parallel degree of freedom enters the compressional term.

Relation to the user-facing variable \(\xi _\parallel \). The actual parallel displacement is

\xi _\parallel \equiv \bvec \cdot \vect {\xi } = \frac {B_\chi }{B}\,\xi _\chi + \frac {I}{RB}\,\xi _\phi . \tag{I.11}

Substituting Eq. (I.10) into this gives

\begin{aligned}\xi _\parallel &= \frac {B_\chi }{B}(B_\chi Z) + \frac {I}{RB}\left (RU+\frac {I}{R}Z\right ) \nonumber \\ &= \frac {B_\chi ^2+I^2/R^2}{B}\,Z + \frac {I}{B}\,U \nonumber \\ &= BZ + \frac {I}{B}U.\end{aligned} \tag{I.12}

Equivalently,

Z = \frac {\xi _\parallel }{B}-\frac {I}{B^2}U. \tag{I.13}

This is the important point for the first minimization: at fixed \(U\) and \(X\), varying \(Z\) is exactly the same as varying \(\xi _\parallel \).

Three basic identities

The reduction of the energy functional rests on three simple coordinate identities.

The pressure-gradient term Because \(p=p(\Psi )\),

\nabla p = p'(\Psi )\,\nabla \Psi = p' RB_\chi \,\hat {\vect {e}}_\Psi . \tag{I.14}

Hence

\vect {\xi }\cdot \nabla p = p' RB_\chi \,\xi _\psi = p'X. \tag{I.15}

The compressibility operator In orthogonal coordinates, the divergence of a vector with physical components \(A_\Psi \), \(A_\chi \), \(A_\phi \) is

\nabla \cdot \vect {A} = \frac {1}{h_\Psi h_\chi h_\phi } \left [ \pp {}{\Psi }(h_\chi h_\phi A_\Psi ) + \pp {}{\chi }(h_\phi h_\Psi A_\chi ) + \pp {}{\phi }(h_\Psi h_\chi A_\phi ) \right ]. \tag{I.16}

Applying this to \(\vect {\xi }\) and using Eq. (I.2) gives

\begin{aligned}\nabla \cdot \vect {\xi } &= \frac {1}{J} \left [ \pp {}{\Psi }(JRB_\chi \xi _\psi ) + \pp {}{\chi }\left (\frac {\xi _\chi }{B_\chi }\right ) + \pp {}{\phi }\left (\frac {J\xi _\phi }{R}\right ) \right ] \nonumber \\ &= \frac {1}{J}\pp {}{\Psi }(JX) + \frac {1}{J}\pp {Z}{\chi } + \frac {1}{J}\pp {}{\phi }\left [J\left (U+\frac {IZ}{R^2}\right )\right ] \nonumber \\ &= \frac {1}{J}\pp {}{\Psi }(JX) + inU + \frac {1}{J}\left (\pp {}{\chi }+in\nu \right )Z \nonumber \\ &= \frac {1}{J}\pp {}{\Psi }(JX) + inU + iBk_\parallel Z.\end{aligned} \tag{I.17}

This is exactly the combination that appears in the last square of the exact functional.

The perturbed magnetic field Define

\vect {Q} \equiv \B _1 = \nabla \times (\vect {\xi }\times \B ). \tag{I.18}

The cross product \(\vect {\xi }\times \B \) is already suggestive. Using Eq. (I.3) and the definitions in Eq. (I.10), one finds

\begin{aligned}\vect {\xi }\times \B &= \left (\frac {\xi _\phi }{R}-\frac {I}{R^2B_\chi }\xi _\chi \right ) RB_\chi \,\hat {\vect {e}}_\Psi + \frac {IX}{R^2B_\chi }\,\hat {\vect {e}}_\chi - \frac {X}{R}\,\hat {\vect {e}}_\phi \nonumber \\ &= RB_\chi U\,\hat {\vect {e}}_\Psi + \frac {IX}{R^2B_\chi }\,\hat {\vect {e}}_\chi - \frac {X}{R}\,\hat {\vect {e}}_\phi .\end{aligned} \tag{I.19}

Now apply the standard orthogonal-coordinate curl formula. A direct calculation shows that the physical components of \(\vect {Q}\) are

\begin{aligned}Q_\Psi &= \frac {1}{JRB_\chi }\left (\pp {}{\chi }+in\nu \right )X = \frac {B}{RB_\chi }\,ik_\parallel X, \\[0.4em] Q_\phi &= \frac {R}{J} \left [ \pp {U}{\chi }-\pp {}{\Psi }\left (\frac {IJX}{R^2}\right ) \right ], \\[0.4em] Q_\chi &= -B_\chi \left (inU+\pp {X}{\Psi }\right ).\end{aligned} \tag{I.20}

It is convenient to peel the \(I'\) piece out of \(Q_\phi \) by writing

Q_\phi = \frac {R}{J} \left [ \pp {U}{\chi }-I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) \right ] - \frac {I'}{R}X. \tag{I.23}

Equations (I.20)–(I.23) are the raw algebraic identities behind the two field-line-bending squares.

The exact 2-D functional before eliminating \(\xi _\parallel \)

Start from the Bernstein energy principle in the form used by Connor–Hastie–Taylor,

\delta W = \frac 12\int dV\left [ |\vect {Q}|^2 - \vect {J}\cdot (\vect {Q}\times \vect {\xi }^*) + (\vect {\xi }\cdot \nabla p)(\nabla \cdot \vect {\xi }^*) + \gamma p\,|\nabla \cdot \vect {\xi }|^2 \right ]. \tag{I.24}

Here and below the \(e^{in\phi }\) dependence is understood, so the \(\phi \) integral simply produces a factor \(2\pi \); consequently

\delta W = \pi \int J\,d\Psi \,d\chi \;\mathcal {L}. \tag{I.25}

Substituting Eqs. (I.15), (I.17), (I.20), (I.22), and (I.23), and then using the equilibrium relations in Eq. (I.8), one can regroup the density into the exact Connor–Hastie–Taylor form

\boxed { \begin {aligned} \delta W =\pi \int J\,d\Psi \,d\chi \Biggl [ &\frac {B^2}{R^2B_\chi ^2}|k_\parallel X|^2 +\frac {R^2}{J^2}\left |\pp {U}{\chi }-I\pp {}{\Psi }\left (\frac {JX}{R^2}\right )\right |^2 \\[0.4em] &\qquad +B_\chi ^2\left |inU+\pp {X}{\Psi }+\frac {j_\phi }{RB_\chi ^2}X\right |^2 -2K|X|^2 \\[0.4em] &\qquad +\gamma p\left |\frac {1}{J}\pp {}{\Psi }(JX)+inU+iBk_\parallel Z\right |^2 \Biggr ]. \end {aligned}} \tag{I.26}

The coefficient \(K\) is

K = \frac {II'}{R^2}\pp {\ln R}{\Psi } - \frac {j_\phi }{R}\pp {}{\Psi }\ln (JB_\chi ). \tag{I.27}

Equation (I.26) is the exact 2-D tokamak functional before the parallel degree of freedom is eliminated.

How the exact 2-D pieces map onto the intuitive form. This is the bookkeeping bridge back to the intuitive energy-principle language.

\begin{aligned}\text {field-line bending} &\longleftrightarrow \frac {B^2}{R^2B_\chi ^2}|k_\parallel X|^2, \\ \text {sheared in-surface bending} &\longleftrightarrow \frac {R^2}{J^2}\left |\pp {U}{\chi }-I\pp {}{\Psi }\left (\frac {JX}{R^2}\right )\right |^2, \\ \text {magnetic compression + current/pressure shift} &\longleftrightarrow B_\chi ^2\left |inU+\pp {X}{\Psi }+\frac {j_\phi }{RB_\chi ^2}X\right |^2, \\ \text {bad-curvature remainder} &\longleftrightarrow -2K|X|^2, \\ \text {plasma compression} &\longleftrightarrow \gamma p\left |\frac {1}{J}\pp {}{\Psi }(JX)+inU+iBk_\parallel Z\right |^2.\end{aligned} \tag{I.28}

The exact functional is therefore the coordinate version of the intuitive energy principle, not a different stability principle.

43.2 Eliminating \(\xi _\parallel \) and \(U\)

The point of the variables \(U\) and \(Z\) is now clear:

\(U\) contains the in-surface displacement perpendicular to \(\B \),
\(Z\) contains the parallel degree of freedom through Eq. (I.12).

Hence the first minimization with respect to \(\xi _\parallel \) is the same thing as minimization with respect to \(Z\) at fixed \(X\) and \(U\).

Introduce the shorthand

\mathcal {C}[X,U,Z] \equiv \frac {1}{J}\pp {}{\Psi }(JX)+inU+iBk_\parallel Z. \tag{I.33}

Then the \(Z\)-dependent part of Eq. (I.26) is simply

\delta W_Z = \pi \int J\,d\Psi \,d\chi \;\gamma p\,|\mathcal {C}|^2. \tag{I.34}

Because \(\gamma p>0\), this contribution is manifestly non-negative. Therefore its absolute minimum is zero, and the minimizing choice of \(Z\) is obtained by setting

\mathcal {C}=0. \tag{I.35}

Written out explicitly, this is

\frac {1}{J}\pp {}{\Psi }(JX)+inU+iBk_\parallel Z = 0, \tag{I.36}

or, using Eq. (I.6),

\left (\pp {}{\chi }+in\nu \right )Z = -\pp {}{\Psi }(JX)-inJU. \tag{I.37}

But Eq. (I.17) shows that Eq. (I.36) is exactly

\nabla \cdot \vect {\xi } = 0. \tag{I.38}

So the first minimization is not an ad hoc assumption of incompressibility. It is the actual variational elimination of the parallel displacement.

Explicit field-line solution for \(Z\). Equation (I.37) is a first-order ODE along the field line. Define

\alpha (\Psi ,\chi )\equiv \int _{\chi _0}^{\chi }\nu (\Psi ,\bar \chi )\,d\bar \chi . \tag{I.39}

Multiplying Eq. (I.37) by \(e^{in\alpha }\) gives

\pp {}{\chi }\left (e^{in\alpha }Z\right ) = -e^{in\alpha }\left [\pp {}{\Psi }(JX)+inJU\right ]. \tag{I.40}

Integrating once along the field line,

Z(\Psi ,\chi ) = e^{-in\alpha (\Psi ,\chi )} \left [ C(\Psi ) - \int _{\chi _0}^{\chi } e^{in\alpha (\Psi ,\chi ')} \left (\pp {}{\Psi }(JX)+inJU\right )(\Psi ,\chi ')\,d\chi ' \right ], \tag{I.41}

with \(C(\Psi )\) fixed by the periodicity or ballooning boundary condition. This is the precise statement behind the comment in Connor–Hastie–Taylor that the last square may be made to vanish.

Variational check. If one varies Eq. (I.34) directly, then

\delta _Z\delta W_Z = \pi \int J\,d\Psi \,d\chi \;\gamma p \left [ \mathcal {C}^*(iBk_\parallel \delta Z) + \mathcal {C}(-iBk_\parallel \delta Z^*) \right ]. \tag{I.42}

Integrating \(k_\parallel \) by parts along the field line gives the stationarity condition

k_\parallel (\gamma p\,\mathcal {C})=0. \tag{I.43}

However the stronger condition \(\mathcal {C}=0\) is available and gives the absolute minimum \(\delta W_Z=0\). That is why the minimizer is precisely the incompressible choice.

The reduced exact 2-D functional

Once Eq. (I.35) is imposed, the compressional square in Eq. (I.26) vanishes identically. The functional reduces to

\boxed { \begin {aligned} \delta W =\pi \int J\,d\Psi \,d\chi \Biggl [ &\frac {B^2}{R^2B_\chi ^2}|k_\parallel X|^2 +\frac {R^2}{J^2}\left |\pp {U}{\chi }-I\pp {}{\Psi }\left (\frac {JX}{R^2}\right )\right |^2 \\[0.4em] &\qquad +B_\chi ^2\left |inU+\pp {X}{\Psi }+\frac {j_\phi }{RB_\chi ^2}X\right |^2 -2K|X|^2 \Biggr ]. \end {aligned}} \tag{I.44}

This is the exact 2-D tokamak functional after elimination of the parallel displacement. In the notation of these notes, it is exactly the reduced two-dimensional tokamak functional quoted in the ballooning lecture.

Second minimization: eliminating \(U\) in the high-\(n\) limit

We now take the next step and minimize Eq. (I.44) with respect to the in-surface displacement \(U\). Unlike the elimination of \(Z\), this is not an exact algebraic minimization because \(\partial U/\partial \chi \) appears explicitly. The simplification comes from the ballooning ordering at large toroidal mode number \(n\).

Ballooning ordering For high-\(n\) ballooning modes one assumes

k_\parallel X = O(1), \qquad k_\parallel U = O(1), \qquad \frac {1}{n}\pp {X}{\Psi }=O(1). \tag{I.45}

The last statement says that \(X\) varies across flux surfaces on the short perpendicular scale \(k_\perp \sim n\), so that \(\partial X/\partial \Psi \) is itself an \(O(n)\) quantity. By contrast, the combinations \(k_\parallel X\) and \(k_\parallel U\) remain \(O(1)\) because ballooning modes keep a long wavelength along the field.

The \(U\)-dependent part of Eq. (I.44) is

\mathcal {L}_U = \frac {R^2}{J^2} \left | \pp {U}{\chi } - I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) \right |^2 + B_\chi ^2 \left | inU+\pp {X}{\Psi }+\frac {j_\phi }{RB_\chi ^2}X \right |^2. \tag{I.46}

Using Eq. (I.6),

\pp {U}{\chi } = -in\nu U + iJBk_\parallel U. \tag{I.47}

Also,

\begin{aligned}I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) &= I\left ( \frac {J}{R^2}\pp {X}{\Psi } + X\pp {}{\Psi }\frac {J}{R^2} \right ) \nonumber \\ &= \nu \pp {X}{\Psi } + IX\pp {}{\Psi }\frac {J}{R^2},\end{aligned} \tag{I.48}

because \(\nu =IJ/R^2\). Therefore the first square in Eq. (I.46) becomes

\begin{aligned}\pp {U}{\chi } - I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) &= -in\nu U +iJBk_\parallel U - \nu \pp {X}{\Psi } - IX\pp {}{\Psi }\frac {J}{R^2} \nonumber \\ &= -\nu \left (inU+\pp {X}{\Psi }\right ) - IX\pp {}{\Psi }\frac {J}{R^2} + iJBk_\parallel U.\end{aligned} \tag{I.49}

Choosing the combination that diagonalizes the leading quadratic form Introduce

a \equiv \frac {j_\phi }{RB_\chi ^2}, \tag{I.50}

so that the second square in Eq. (I.46) is simply

B_\chi ^2\left |inU+\pp {X}{\Psi }+aX\right |^2. \tag{I.51}

The right combination to isolate is

S \equiv inU+\pp {X}{\Psi }+cX, \tag{I.52}

with \(c(\Psi ,\chi )\) chosen so that the two large squares combine into \(B^2|S|^2\) at leading order. The required choice is

c \equiv \frac {1}{B^2} \left [ \frac {j_\phi }{R} + \frac {I^2}{J}\pp {}{\Psi }\left (\frac {J}{R^2}\right ) \right ]. \tag{I.53}

Let us now show that this is the same coefficient written by Connor–Hastie–Taylor.

First expand the second term:

\begin{aligned}\frac {I^2}{J}\pp {}{\Psi }\left (\frac {J}{R^2}\right ) &= \frac {I^2}{R^2} \left ( \frac {1}{J}\pp {J}{\Psi } - \frac {2}{R}\pp {R}{\Psi } \right ).\end{aligned} \tag{I.54}

Next use the equilibrium relation

\frac {j_\phi }{R}=p'+\frac {II'}{R^2}, \tag{I.55}

so Eq. (I.53) becomes

\begin{aligned}c &= \frac {1}{B^2} \left [ p' + \frac {II'}{R^2} + \frac {I^2}{R^2} \left ( \frac {1}{J}\pp {J}{\Psi } - \frac {2}{R}\pp {R}{\Psi } \right ) \right ] \nonumber \\ &= \frac {p'}{B^2} + \frac {I^2}{R^2B^2} \left [ \frac {I'}{I} + \frac {1}{J}\pp {J}{\Psi } - \frac {2}{R}\pp {R}{\Psi } \right ].\end{aligned} \tag{I.56}

But

\frac {\nu '}{\nu } = \frac {I'}{I} + \frac {1}{J}\pp {J}{\Psi } - \frac {2}{R}\pp {R}{\Psi }, \tag{I.57}

since \(\nu =IJ/R^2\). Hence

\boxed { c = \frac {p'}{B^2} + \frac {I^2}{R^2B^2}\frac {\nu '}{\nu }.} \tag{I.58}

Now rewrite the two large squares in terms of \(S\). From Eq. (I.52),

inU+\pp {X}{\Psi }=S-cX, \tag{I.59}

so Eq. (I.49) becomes

\pp {U}{\chi } - I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) = -\nu S + \left [ \nu c - I\pp {}{\Psi }\left (\frac {J}{R^2}\right ) \right ]X + iJBk_\parallel U, \tag{I.60}

while Eq. (I.51) becomes

B_\chi ^2\left |S+(a-c)X\right |^2. \tag{I.61}

At leading order we may ignore \(iJBk_\parallel U\) inside the first square because it is down by one power of \(1/n\) relative to the pieces containing \(inU+\partial X/\partial \Psi \). Expanding the two squares then produces a term linear in \(S\) of the form

2\Re \left \{ S^* \left [ -\frac {R^2}{J^2}\nu \left ( \nu c - I\pp {}{\Psi }\frac {J}{R^2} \right ) + B_\chi ^2(a-c) \right ]X \right \}. \tag{I.62}

Now simplify the coefficient:

\begin{aligned}&-\frac {R^2}{J^2}\nu \left ( \nu c - I\pp {}{\Psi }\frac {J}{R^2} \right ) + B_\chi ^2(a-c) \nonumber \\ &\qquad = -\frac {R^2}{J^2}\nu ^2 c + \frac {R^2}{J^2}\nu I\pp {}{\Psi }\frac {J}{R^2} + \frac {j_\phi }{R} - B_\chi ^2 c \nonumber \\ &\qquad = -\frac {I^2}{R^2}c + \frac {I^2}{J}\pp {}{\Psi }\frac {J}{R^2} + \frac {j_\phi }{R} - B_\chi ^2 c \nonumber \\ &\qquad = -B^2 c + \frac {I^2}{J}\pp {}{\Psi }\frac {J}{R^2} + \frac {j_\phi }{R} \nonumber \\ &\qquad =0,\end{aligned} \tag{I.63}

where the last step is exactly the definition of \(c\) in Eq. (I.53). Therefore, to leading order in the large-\(n\) expansion, the two big squares collapse to

\mathcal {L}_U = B^2|S|^2 + (\text {$X$-only terms}) + O(1/n). \tag{I.64}

So the leading-order minimizer is simply \(S=0\), i.e.

inU+\pp {X}{\Psi } + X \left ( \frac {p'}{B^2} + \frac {I^2}{R^2B^2}\frac {\nu '}{\nu } \right ) =0. \tag{I.65}

The first correction in \(1/n\) To obtain the accuracy retained by Connor–Hastie–Taylor we must also keep the first correction coming from \(iJBk_\parallel U\). From Eq. (I.52),

inU = S-\pp {X}{\Psi }-cX, \qquad U=\frac {i}{n}\left (\pp {X}{\Psi }+cX-S\right ). \tag{I.66}

Now use the ordering (I.45):

\(\partial X/\partial \Psi = O(n)\),
\(X=O(1)\),
\(S=O(1)\).

Therefore

U = \frac {i}{n}\pp {X}{\Psi } + O(1/n), \tag{I.67}

and hence

iJBk_\parallel U = iJBk_\parallel \left (\frac {i}{n}\pp {X}{\Psi }\right ) + O(1/n) = - JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) + O(1/n). \tag{I.68}

Substituting this into Eq. (I.60) gives

\pp {U}{\chi } - I\pp {}{\Psi }\left (\frac {JX}{R^2}\right ) = -\nu S + \left [ \nu c - I\pp {}{\Psi }\left (\frac {J}{R^2}\right ) \right ]X - JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) + O(1/n). \tag{I.69}

When Eq. (I.69) is substituted into Eq. (I.46), the \(SX^*\) cross term still cancels by Eq. (I.63); the only new term linear in \(S\) is the one involving the parallel-gradient correction:

\begin{aligned}\mathcal {L}_U &= B^2|S|^2 + 2\frac {R^2\nu }{J^2} \Re \left \{ S^*\,JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) \right \} + (\text {$X$-only terms}) + O(1/n) \nonumber \\ &= B^2|S|^2 + 2\frac {I^2}{\nu R^2} \Re \left \{ S^*\,JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) \right \} + (\text {$X$-only terms}) + O(1/n),\end{aligned} \tag{I.70}

because

\frac {R^2\nu }{J^2} = \frac {I}{J} = \frac {I^2}{\nu R^2}. \tag{I.71}

Now complete the square:

\begin{aligned}\mathcal {L}_U &= B^2 \left | S + \frac {I^2}{\nu R^2B^2} JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) \right |^2 \nonumber \\ &\qquad - \frac {I^4}{\nu ^2R^4B^2} \left | JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) \right |^2 + (\text {$X$-only terms}) + O(1/n).\end{aligned} \tag{I.72}

The minimizing choice of \(U\) is therefore obtained by setting the square to zero:

\boxed { inU+\pp {X}{\Psi } + X\left ( \frac {p'}{B^2} + \frac {I^2}{R^2B^2}\frac {\nu '}{\nu } \right ) + \frac {I^2}{\nu R^2B^2} JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) =0.} \tag{I.73}

This is exactly Connor–Hastie–Taylor Eq. (12), now written out with the intermediate algebra visible.

Equivalently,

U = \frac {i}{n} \left [ \pp {X}{\Psi } + X\left ( \frac {p'}{B^2} + \frac {I^2}{R^2B^2}\frac {\nu '}{\nu } \right ) + \frac {I^2}{\nu R^2B^2} JBk_\parallel \left (\frac {1}{n}\pp {X}{\Psi }\right ) \right ] + O(1/n^2). \tag{I.74}

So in the high-\(n\) ballooning limit the in-surface displacement is no longer an independent variable: it is slaved to the radial displacement \(X\) and its radial derivative.

Result of substituting the minimizing \(U\) back into the functional At this point the remaining algebra is long but straightforward: one substitutes Eq. (I.73) into Eq. (I.44), keeps all contributions through \(O(1/n)\), rewrites every occurrence of \(JBk_\parallel (\partial X/\partial \Psi )\) in terms of \(\partial (JBk_\parallel X)/\partial \Psi \), and then groups the leftover pieces into the three combinations

P = X\sigma - \frac {B_\chi ^2}{\nu B^2} \frac {I}{n} \pp {}{\Psi }(JBk_\parallel X), \qquad Q = \frac {Xp'}{B^2} + \frac {I^2}{\nu R^2B^2} \frac {1}{n} \pp {}{\Psi }(JBk_\parallel X), \qquad \sigma = \frac {Ip'}{B^2}+I'. \tag{I.75}

After also moving the outside Jacobian \(J\) inside the braces, the result is

\boxed { \begin {aligned} \delta W = \pi \int d\Psi \,d\chi \Biggl \{ &\frac {JB^2}{R^2B_\chi ^2}|k_\parallel X|^2 + \frac {R^2B_\chi ^2}{JB^2} \left | \frac {1}{n}\pp {}{\Psi }(JBk_\parallel X) \right |^2 \\[0.4em] &\qquad - \frac {2J}{B^2}p' \left [ |X|^2\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {iI}{JB^2}\pp {}{\chi }\left (\frac {B^2}{2}\right )\frac {X^*}{n}\pp {X}{\Psi } \right ] \\[0.4em] &\qquad + \frac {X^*}{n}JBk_\parallel (X\sigma ') - \frac {1}{n} \left [ P^*JBk_\parallel Q + P\,JBk_\parallel ^*Q^* \right ] \Biggr \}, \end {aligned}} \tag{I.76}

which is Connor–Hastie–Taylor Eq. (13).

For the purposes of the notes, the crucial structural point is already contained in Eq. (I.73): the \(U\) minimization is another square completion, but now only after the large-\(n\) ordering has converted the \(\chi \)-derivative of \(U\) into an algebraic expression plus a controlled \(1/n\) correction.

43.3 Ballooning Transform and the Local High-\(n\) Functional

The remaining task is to turn the exact periodic problem for \(X(\Psi ,\chi )\) into a local problem on an unwrapped field line. Let

\chi _0 \equiv \oint d\chi \tag{I.77}

denote one poloidal period. The ballooning transform writes the periodic function \(X(\Psi ,\chi )\) as

X(\Psi ,\chi ) = \sum _{m=-\infty }^{\infty } \exp \!\left (-\frac {2\pi i m\chi }{\chi _0}\right ) \int _{-\infty }^{\infty } dy\, \exp \!\left (\frac {2\pi i my}{\chi _0}\right ) \widehat X(\Psi ,y). \tag{I.78}

The transformed function \(\widehat X(\Psi ,y)\) lives on the infinite interval \(-\infty <y<\infty \) and need not be periodic. That is the whole point: the rapid phase variation responsible for large perpendicular wavenumber is now carried by an eikonal factor, while the envelope is free to localize where the curvature drive is most unfavorable.

Eikonal representation. Choose

\widehat X(\Psi ,y) = F(\Psi ,y)\, \exp \!\left [-in\alpha (\Psi ,y;y_0)\right ], \qquad \alpha (\Psi ,y;y_0)\equiv \int _{y_0}^{y}\nu (\Psi ,\bar y)\,d\bar y. \tag{I.79}

Here \(y_0\) is the ballooning origin. Its value is not yet fixed. The point of the phase choice (I.79) is that

\pp {\alpha }{y}=\nu , \qquad JBk_\parallel = \frac {1}{i}\left (\pp {}{y}+in\nu \right ), \tag{I.80}

so the large \(in\nu \) contribution cancels immediately:

\begin{aligned}JBk_\parallel \widehat X &= \frac {1}{i}\left (\pp {}{y}+in\nu \right ) \left (F e^{-in\alpha }\right ) \nonumber \\ &= \frac {1}{i}e^{-in\alpha }\pp {F}{y} = -i\,e^{-in\alpha }\pp {F}{y}.\end{aligned} \tag{I.81}

Therefore

|k_\parallel \widehat X|^2 = \frac {1}{J^2B^2}\left |\pp {F}{y}\right |^2. \tag{I.82}

Large-\(n\) ordering after the ballooning transform

To make the asymptotics explicit, introduce a fast radial scale

s \equiv \sqrt {n}\,(\Psi -\Psi _0). \tag{I.83}

Then any slowly varying envelope is regarded as a function \(F=F(\Psi ,s,y)\) with

\pp {}{\Psi } = \sqrt {n}\,\pp {}{s} + \left .\pp {}{\Psi }\right |_{s}. \tag{I.84}

Consequently,

\frac {1}{n}\pp {F}{\Psi } = \frac {1}{\sqrt {n}}\pp {F}{s} + \frac {1}{n}\left .\pp {F}{\Psi }\right |_{s} = O(n^{-1/2}). \tag{I.85}

The other key quantity is the radial derivative of the phase:

\Theta (\Psi ,y;y_0) \equiv \pp {\alpha }{\Psi } = \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y, \tag{I.86}

where, at this order, \(y_0\) is treated as independent of \(\Psi \). Now differentiate (I.79) with respect to \(\Psi \):

\begin{aligned}\frac {1}{n}\pp {\widehat X}{\Psi } &= \frac {1}{n}\pp {}{\Psi }\left (F e^{-in\alpha }\right ) \nonumber \\ &= e^{-in\alpha } \left [ \frac {1}{n}\pp {F}{\Psi } -i\Theta F \right ] \nonumber \\ &= e^{-in\alpha } \left [ -i\Theta F + O(n^{-1/2}) \right ].\end{aligned} \tag{I.87}

This one line is the essential large-\(n\) bookkeeping step: the explicit \(1/n\) sitting in front of \(\partial /\partial \Psi \) is cancelled by the \(n\) hidden in the eikonal phase, so radial derivatives of the phase survive at leading order while radial derivatives of the envelope are smaller by \(n^{-1/2}\).

The same logic gives

\begin{aligned}\frac {1}{n}\pp {}{\Psi }\!\left (JBk_\parallel \widehat X\right ) &= \frac {1}{n}\pp {}{\Psi } \left ( -i e^{-in\alpha }\pp {F}{y} \right ) \nonumber \\ &= -e^{-in\alpha }\Theta \,\pp {F}{y} + O(n^{-1/2}), \\[0.4em] \widehat X^*\,\frac {1}{n}\pp {\widehat X}{\Psi } &= -i\Theta |F|^2 + O(n^{-1/2}).\end{aligned} \tag{I.88}

Leading behavior of the auxiliary combinations \(P\) and \(Q\). Using (I.88) in the definitions (I.75) gives

\begin{aligned}P &= e^{-in\alpha } \left [ \sigma F + \frac {IB_\chi ^2}{\nu B^2}\, \Theta \,\pp {F}{y} \right ] + O(n^{-1/2}), \\[0.4em] Q &= e^{-in\alpha } \left [ \frac {p'}{B^2}F - \frac {I^2}{\nu R^2B^2}\, \Theta \,\pp {F}{y} \right ] + O(n^{-1/2}).\end{aligned} \tag{I.90}

Because the last two terms in (I.76) are already multiplied by an explicit factor \(1/n\), they contribute only at \(O(1/n)\) and hence drop out of the lowest-order local problem.

Term-by-term reduction of the \(X\)-only functional

We now substitute the transformed field (I.79) into the exact \(X\)-only functional (I.76) and keep the leading \(O(1)\) terms.

The two bending terms From (I.82),

\frac {JB^2}{R^2B_\chi ^2}|k_\parallel \widehat X|^2 = \frac {1}{JR^2B_\chi ^2} \left |\pp {F}{y}\right |^2. \tag{I.92}

From (I.88),

\frac {R^2B_\chi ^2}{JB^2} \left | \frac {1}{n}\pp {}{\Psi }(JBk_\parallel \widehat X) \right |^2 = \frac {R^2B_\chi ^2}{JB^2}\, \Theta ^2 \left |\pp {F}{y}\right |^2 + O(n^{-1/2}). \tag{I.93}

Adding (I.92) and (I.93),

\begin{aligned}&\frac {JB^2}{R^2B_\chi ^2}|k_\parallel \widehat X|^2 + \frac {R^2B_\chi ^2}{JB^2} \left | \frac {1}{n}\pp {}{\Psi }(JBk_\parallel \widehat X) \right |^2 \nonumber \\[0.4em] &\qquad = \frac {1}{JR^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B}\,\Theta \right )^2 \right ] \left |\pp {F}{y}\right |^2 + O(n^{-1/2}).\end{aligned} \tag{I.94}

The pressure-curvature term The term proportional to \(p'\) in (I.76) is

-\frac {2J}{B^2}p' \left [ |X|^2\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {iI}{JB^2}\pp {}{\chi }\left (\frac {B^2}{2}\right ) \frac {X^*}{n}\pp {X}{\Psi } \right ]. \tag{I.95}

After the ballooning transform the periodic coordinate \(\chi \) is replaced by \(y\), and (I.89) gives

\begin{aligned}\frac {X^*}{n}\pp {X}{\Psi } &= -i\Theta |F|^2 + O(n^{-1/2}).\end{aligned} \tag{I.96}

Substituting this into (I.95) yields

\begin{aligned}&-\frac {2J}{B^2}p' \left [ |X|^2\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {iI}{JB^2}\pp {}{\chi }\left (\frac {B^2}{2}\right ) \frac {X^*}{n}\pp {X}{\Psi } \right ] \nonumber \\[0.4em] &\qquad = -\frac {2J}{B^2}p' \left [ |F|^2\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {iI}{JB^2}\pp {}{y}\left (\frac {B^2}{2}\right ) \left (-i\Theta |F|^2\right ) \right ] + O(n^{-1/2}) \nonumber \\[0.4em] &\qquad = - \left [ \frac {2Jp'}{B^2}\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {Ip'}{B^4}\Theta \,\pp {B^2}{y} \right ] |F|^2 + O(n^{-1/2}).\end{aligned} \tag{I.97}

Terms that are subleading at lowest order The term

\frac {X^*}{n}\,JBk_\parallel (X\sigma ') \tag{I.98}

is explicitly \(O(1/n)\) because \(JBk_\parallel (X\sigma ')=O(1)\) for ballooning orderings. The final bracket in (I.76) is also \(O(1/n)\) because the leading parts of \(P\) and \(Q\) are \(O(1)\), while \(JBk_\parallel P\) and \(JBk_\parallel Q\) are likewise \(O(1)\). Therefore none of those terms contribute to the lowest-order local field-line problem.

The leading-order one-dimensional functional

At lowest order the exact \(X\)-only functional becomes

\delta W_0[F;\Psi ,y_0] = \pi \int _{-\infty }^{\infty }dy \left \{ A(\Psi ,y;y_0)\left |\pp {F}{y}\right |^2 - V(\Psi ,y;y_0)|F|^2 \right \}, \tag{I.99}

with

\begin{aligned}A(\Psi ,y;y_0) &= \frac {1}{JR^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B}\,\Theta \right )^2 \right ], \\[0.4em] V(\Psi ,y;y_0) &= \frac {2Jp'}{B^2}\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {Ip'}{B^4}\Theta \,\pp {B^2}{y}.\end{aligned} \tag{I.100}

The corresponding leading-order kinetic normalization follows from (I.87):

\begin{aligned}\mathcal N_0[F;\Psi ,y_0] &= \pi \int _{-\infty }^{\infty }dy\; \frac {J}{R^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B}\,\Theta \right )^2 \right ] |F|^2 \nonumber \\ &\equiv \pi \int _{-\infty }^{\infty }dy\;M(\Psi ,y;y_0)|F|^2,\end{aligned} \tag{I.102}

with

M(\Psi ,y;y_0) = \frac {J}{R^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B}\,\Theta \right )^2 \right ]. \tag{I.103}

What has happened physically? At this order each flux surface has decoupled from its neighbors. The local problem depends on \(\Psi \) and on the ballooning origin \(y_0\) only as parameters. The field-line structure along the unwrapped coordinate \(y\) is determined by the one-dimensional functional (I.99), while the transverse radial envelope is left for higher order.

Variation of the local functional: the field-line ODE

Define the local eigenvalue problem by extremizing

\mathcal F_0[F] \equiv \delta W_0[F;\Psi ,y_0] - \omega ^2(\Psi ,y_0)\,\mathcal N_0[F;\Psi ,y_0]. \tag{I.104}

Let \(F\rightarrow F+\epsilon \eta \) with \(\eta (\pm \infty )=0\), and vary with respect to \(F^*\). Using (I.99) and (I.102),

\begin{aligned}\delta \mathcal F_0 &= \pi \epsilon \int _{-\infty }^{\infty }dy \left [ A\,\pp {F}{y}\,\pp {\eta ^*}{y} - \left (V+\omega ^2 M\right )F\eta ^* \right ] + \text {c.c.} \nonumber \\ &= -\pi \epsilon \int _{-\infty }^{\infty }dy\; \eta ^* \left [ \pp {}{y}\!\left (A\,\pp {F}{y}\right ) + \left (V+\omega ^2 M\right )F \right ] + \text {c.c.},\end{aligned} \tag{I.105}

where the boundary term vanishes because \(\eta (\pm \infty )=0\). Since \(\eta \) is arbitrary, the coefficient of \(\eta ^*\) must vanish:

\pp {}{y}\!\left (A\,\pp {F_0}{y}\right ) + \left (V+\omega ^2 M\right )F_0 = 0. \tag{I.106}

Now substitute (I.100), (I.101), and (I.103). The result is

\boxed { \begin {aligned} \pp {}{y} \Biggl \{ \frac {1}{JR^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B} \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right )^2 \right ] \pp {F_0}{y} \Biggr \} \\[0.5em] \qquad + F_0 \Biggl [ \frac {2Jp'}{B^2}\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {Ip'}{B^4} \left ( \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right ) \pp {B^2}{y} \Biggr ] \\[0.5em] \qquad + \omega ^2(\Psi ,y_0)\, \frac {J}{R^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B} \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right )^2 \right ] F_0 = 0. \end {aligned}} \tag{I.107}

This is the local field-line equation. It is the lowest-order Connor–Hastie–Taylor ballooning equation before any large-aspect-ratio or circular-cross-section specialization.

If one divides (I.107) by \(J\), it takes the printed form of Connor–Hastie–Taylor Eq. (24):

\boxed { \begin {aligned} \frac {1}{J}\pp {}{y} \Biggl \{ \frac {1}{JR^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B} \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right )^2 \right ] \pp {F_0}{y} \Biggr \} \\[0.5em] \qquad + 2\frac {F_0p'}{B^2} \left [ \pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {I}{B^2} \left ( \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right ) \frac {1}{J}\pp {}{y}\left (\frac {B^2}{2}\right ) \right ] \\[0.5em] \qquad + \omega ^2(\Psi ,y_0)\, \frac {1}{R^2B_\chi ^2} \left [ 1+ \left ( \frac {R^2B_\chi ^2}{B} \int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y \right )^2 \right ] F_0 = 0. \end {aligned}} \tag{I.108}

Caution

Boundary condition on the unwrapped field line. For unstable modes, \(\omega ^2<0\), the acceptable solution of (I.107) is the one that decays as \(|y|\rightarrow \infty \). At marginality, \(\omega ^2=0\), the large-\(|y|\) asymptotics are more subtle and one recovers the Mercier index from the indicial equation. So the transform to the infinite domain is not just a formal trick; it changes the stability problem into a well-posed boundary-value problem on an unwrapped field line.

43.4 Bridge to the Standard High-\(n\) Ballooning Equation

Note

Equation (I.108) is the general axisymmetric high-\(n\) ballooning equation. To recover the familiar lecture-level \(\hat s\)–\(\alpha \) equation, we now make one more specialization: large aspect ratio, circular flux surfaces, and the standard local shifted-circle model for the integrated shear. Once those geometric assumptions are inserted, all of the remaining algebra is just bookkeeping.

From this point onward I switch to the notation used in the ballooning lecture: \[ B_\chi \to B_p, \qquad y \to \theta , \qquad F_0(y) \to X(\theta ). \] We also place the ballooning origin at the outboard midplane,

y_0=0, \tag{I.109}

which is the usual symmetric choice for the most unstable local mode in the simple large-aspect-ratio model.

Caution

What is being assumed here. The reduction from Eq. (I.108) to the standard \(\hat s\)–\(\alpha \) equation uses more than just the large-\(n\) asymptotics. It also assumes a specific local equilibrium model. So the logical structure is \[ \text {general axisymmetric high-}n\text { equation} \Longrightarrow \text {large-aspect-ratio local model} \Longrightarrow \hat s\text {--}\alpha . \] That is why Connor–Hastie–Taylor is more general than the textbook \(\hat s\)–\(\alpha \) equation.

Large-aspect-ratio circular identities Take a circular flux surface of minor radius \(r\) in a tokamak of major radius \(R_0\), with geometric poloidal angle \(\theta \):

R(\theta )=R_0+r\cos \theta , \qquad Z(\theta )=r\sin \theta , \qquad \epsilon \equiv \frac {r}{R_0}\ll 1. \tag{I.110}

To leading order in \(\epsilon \), the toroidal field dominates,

B(\theta ) \simeq B\left (1-\frac {r}{R_0}\cos \theta \right ), \tag{I.111}

so that

\pp {}{\theta }\left (\frac {B^2}{2}\right ) \simeq \frac {B^2 r}{R_0}\sin \theta , \qquad \pp {}{r}\left (\frac {B^2}{2}\right ) \simeq -\frac {B^2}{R_0}\cos \theta . \tag{I.112}

The poloidal field and Jacobian are approximated by

B_p \simeq \frac {rB}{qR_0}, \qquad J \simeq \frac {qR_0}{B}, \qquad JB_p \simeq r, \tag{I.113}

so that

\frac {d\Psi }{dr}=RB_p\simeq \frac {rB}{q}, \qquad \pp {}{\Psi } \simeq \frac {q}{rB}\pp {}{r}. \tag{I.114}

Two combinations that occur repeatedly are

JR^2B_p^2 \simeq \frac {BR_0r^2}{q}, \qquad J^2R^2B_p^2 \simeq R_0^2r^2. \tag{I.115}

The toroidal-field function is

I=RB_\phi \simeq R_0B. \tag{I.116}

The general shear integral becomes \(\Lambda (\theta )\) In the general local equation the quantity that multiplies the derivative term is

1+ \left ( \frac {R^2B_p^2}{B} \int _0^{\theta }\nu '(r,\bar \theta )\,d\bar \theta \right )^2. \tag{I.117}

Define the standard large-aspect-ratio shear combination

\Lambda (\theta ) \equiv \frac {R^2B_p^2}{B} \int _0^{\theta }\nu '(r,\bar \theta )\,d\bar \theta . \tag{I.118}

Then the derivative and inertial weights become simply \(1+\Lambda ^2\).

The ordinary magnetic-shear contribution is easy to identify explicitly. If we keep only the \(\theta \)-independent part of \(\nu \), namely \(\nu \simeq q(r)\), then

\nu '\simeq q' =\pp {q}{\Psi } =\frac {dq/dr}{d\Psi /dr} \simeq \frac {q}{rB}\dd {q}{r}. \tag{I.119}

Hence

\begin{aligned}\frac {R^2B_p^2}{B}\int _0^{\theta } q'\,d\bar \theta &\simeq \frac {R_0^2}{B} \left (\frac {rB}{qR_0}\right )^2 \left (\frac {q}{rB}\dd {q}{r}\right )\theta \nonumber \\ &= \frac {r}{q}\dd {q}{r}\,\theta \nonumber \\ &\equiv \hat s\,\theta ,\end{aligned} \tag{I.120}

with

\hat s\equiv \frac {r}{q}\dd {q}{r}. \tag{I.121}

In the standard shifted-circle local equilibrium one keeps one additional toroidal correction, the pressure-gradient contribution to the local magnetic shear. It is written as

\Lambda (\theta ) \simeq \hat s\,\theta -\alpha \sin \theta , \tag{I.122}

where

\alpha \equiv -\frac {2\muo R_0 q^2}{B^2}\dd {p}{r}. \tag{I.123}

Equivalently, if one keeps the original Connor–Hastie–Taylor convention \(\muo =1\) inside the appendix algebra, the same definition reads \[ \alpha = -\frac {2R_0 q^2}{B^2}\dd {p}{r}. \] Equation (I.122) is exactly the \(\Lambda (\theta )\) that appears in the reduced variational problem of the ballooning lecture.

Reducing the derivative term Start from Eq. (I.108) and rewrite it using \(\Lambda (\theta )\):

\frac {1}{J}\pp {}{\theta } \left \{ \frac {1}{JR^2B_p^2}(1+\Lambda ^2)\pp {X}{\theta } \right \} + 2\frac {Xp'}{B^2} \left [ \pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {I}{B^2}\Theta \frac {1}{J}\pp {}{\theta }\left (\frac {B^2}{2}\right ) \right ] + \omega ^2\frac {1}{R^2B_p^2}(1+\Lambda ^2)X = 0, \tag{I.124}

where

\Theta (\theta ) \equiv \int _0^{\theta }\nu '(r,\bar \theta )\,d\bar \theta = \frac {B}{R^2B_p^2}\,\Lambda (\theta ). \tag{I.125}

At leading order in \(\epsilon \), the prefactor \(1/(JR^2B_p^2)\) is independent of \(\theta \), so the first term becomes

\frac {1}{J}\pp {}{\theta } \left \{ \frac {1}{JR^2B_p^2}(1+\Lambda ^2)\pp {X}{\theta } \right \} \simeq \frac {1}{J^2R^2B_p^2} \pp {}{\theta } \left [(1+\Lambda ^2)\pp {X}{\theta }\right ]. \tag{I.126}

Now multiply the whole equation by \(J^2R^2B_p^2\). Using Eq. (I.115), the derivative term becomes simply

\pp {}{\theta } \left [(1+\Lambda ^2)\pp {X}{\theta }\right ]. \tag{I.127}

Reducing the curvature-drive term The remaining task is to simplify

\mathcal {D}(\theta ) \equiv J^2R^2B_p^2\,2\frac {p'}{B^2} \left [ \pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) - \frac {I}{B^2}\Theta \frac {1}{J}\pp {}{\theta }\left (\frac {B^2}{2}\right ) \right ]. \tag{I.128}

First convert \(p'\) from a \(\Psi \) derivative to an \(r\) derivative:

p' = \pp {p}{\Psi } \simeq \frac {q}{rB}\dd {p}{r}. \tag{I.129}

The first term inside the square bracket is

\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) = p' + \pp {}{\Psi }\left (\frac {B^2}{2}\right ). \tag{I.130}

In the standard \(\hat s\)–\(\alpha \) reduction one drops the \(p'^2\) piece, because it is higher order than the toroidal-field-curvature term that one wants to keep. Therefore

\pp {}{\Psi }\left (p+\frac {B^2}{2}\right ) \simeq \pp {}{\Psi }\left (\frac {B^2}{2}\right ) \simeq \frac {q}{rB}\pp {}{r}\left (\frac {B^2}{2}\right ) \simeq -\frac {qB}{rR_0}\cos \theta . \tag{I.131}

Substituting Eqs. (I.115), (I.129), and (I.131) into the first part of Eq. (I.128) gives

\begin{aligned}\mathcal {D}_1(\theta ) &\equiv J^2R^2B_p^2\,2\frac {p'}{B^2} \pp {}{\Psi }\left (\frac {B^2}{2}\right )\nonumber \\ &\simeq R_0^2r^2\times 2\left (\frac {q}{rB}\dd {p}{r}\right )\frac {1}{B^2} \left (-\frac {qB}{rR_0}\cos \theta \right )\nonumber \\ &= -\frac {2R_0q^2}{B^2}\dd {p}{r}\cos \theta \nonumber \\ &= \alpha \cos \theta ,\end{aligned} \tag{I.132}

where the last step uses Eq. (I.123).

Now turn to the second part of Eq. (I.128). Using Eqs. (I.116), (I.125), and (I.112),

\begin{aligned}\frac {I}{B^2}\Theta \frac {1}{J}\pp {}{\theta }\left (\frac {B^2}{2}\right ) &\simeq \frac {R_0B}{B^2} \left (\frac {B}{R^2B_p^2}\Lambda \right ) \left (\frac {B}{qR_0}\right ) \left (\frac {B^2r}{R_0}\sin \theta \right )\nonumber \\ &= \frac {qB}{rR_0}\,\Lambda \sin \theta .\end{aligned} \tag{I.133}

Therefore

\begin{aligned}\mathcal {D}_2(\theta ) &\equiv -J^2R^2B_p^2\,2\frac {p'}{B^2} \frac {I}{B^2}\Theta \frac {1}{J}\pp {}{\theta }\left (\frac {B^2}{2}\right )\nonumber \\ &\simeq -R_0^2r^2\times 2\left (\frac {q}{rB}\dd {p}{r}\right )\frac {1}{B^2} \left (\frac {qB}{rR_0}\Lambda \sin \theta \right )\nonumber \\ &= -\frac {2R_0q^2}{B^2}\dd {p}{r}\,\Lambda \sin \theta \nonumber \\ &= \alpha \Lambda \sin \theta .\end{aligned} \tag{I.134}

Adding Eqs. (I.132) and (I.134), the total drive coefficient is

\mathcal {D}(\theta ) = \alpha \bigl (\cos \theta +\Lambda \sin \theta \bigr ). \tag{I.135}

The final \(\hat s\)–\(\alpha \) equation After multiplying Eq. (I.124) by \(J^2R^2B_p^2\), using Eq. (I.135), and absorbing the remaining positive constant multiplying the inertial term into a dimensionless frequency \(\hat \omega ^2\), we obtain

\boxed { \dd {}{\theta } \left [ \left (1+\Lambda ^2\right )\dd {X}{\theta } \right ] + \alpha \left (\cos \theta +\Lambda \sin \theta \right )X = -\hat \omega ^2\left (1+\Lambda ^2\right )X,} \tag{I.136}

with

\Lambda (\theta )=\hat s\,\theta -\alpha \sin \theta . \tag{I.137}

This is exactly the \(\hat s\)–\(\alpha \) ballooning equation written in the main ballooning lecture.

Because the same approximations can be applied at the variational level, the local functional (I.99) reduces to

\delta W[X] \propto \int _{-\infty }^{\infty }d\theta \left [ \left (1+\Lambda ^2\right ) \left |\dd {X}{\theta }\right |^2 - \alpha \left (\cos \theta +\Lambda \sin \theta \right )|X|^2 \right ], \tag{I.138}

which is the one-dimensional energy functional quoted in the lecture. So the lecture-level \(\hat s\)–\(\alpha \) model is not a separate derivation; it is the large-aspect-ratio circular specialization of the general axisymmetric local ballooning equation.

Takeaways

1.
The general coefficient \[ \frac {R^2B_p^2}{B}\int _0^{\theta }\nu '\,d\bar \theta \] becomes the familiar \[ \Lambda (\theta )=\hat s\,\theta -\alpha \sin \theta . \]
2.
After multiplying the general local equation by \(J^2R^2B_p^2\), the derivative term becomes \[ \dd {}{\theta }\left [(1+\Lambda ^2)\dd {X}{\theta }\right ]. \]
3.
The two pieces of the curvature-drive coefficient reduce separately to \[ \alpha \cos \theta , \qquad \alpha \Lambda \sin \theta , \] and therefore combine into \[ \alpha \bigl (\cos \theta +\Lambda \sin \theta \bigr ). \]
4.
Those two steps are the whole bridge from the general axisymmetric Connor–Hastie–Taylor equation to the standard large-aspect-ratio circular \(\hat s\)–\(\alpha \) ballooning equation.

What the lowest-order equation fixes, and what it leaves for higher order

The field-line ODE (I.107) determines, for each flux surface \(\Psi \) and each ballooning origin \(y_0\), a local eigenvalue \(\omega ^2(\Psi ,y_0)\) and an eigenfunction along the extended field-line coordinate. Because \(\Psi \) enters only parametrically, the lowest-order solution has the form

F_0(\Psi ,s,y) = A(s)\,f_0(y;\Psi ,y_0), \tag{I.139}

where the function \(f_0\) is fixed by the one-dimensional problem in \(y\), but the radial envelope \(A(s)\) is not yet determined.

That remaining freedom is resolved one order higher in the asymptotic expansion. The solvability condition gives

\pp {\omega ^2(\Psi ,y_0)}{y_0}=0, \tag{I.140}

so the ballooning origin must sit at an extremum of the local eigenvalue, and the next-order radial equation localizes \(A(s)\) near the surface where that local eigenvalue is smallest. This is the precise sense in which the leading-order ballooning theory fixes the structure along the field line first and only afterwards determines how the mode is assembled across neighboring flux surfaces.

In the large-aspect-ratio circular limit, the general local equation (I.107) reduces to the familiar \(\hat s\)–\(\alpha \) ballooning equation used in the main ballooning lecture. So the \(\hat s\)–\(\alpha \) model is not a different theory; it is a specialization of the general axisymmetric high-\(n\) field-line equation derived here.

Takeaways

1.
The first minimization is exact: \[ \text {vary }Z \Longleftrightarrow \text {vary }\xi _\parallel \Longleftrightarrow \nabla \cdot \vect {\xi }=0. \]
2.
The second minimization is asymptotic: \[ \text {high-}n\text { vary }U \Longrightarrow U \text { is slaved to }X \text { by Eq.~\eqref {eq:balloon:app:U_minimizer}}. \]
3.
After those two steps, the exact periodic problem is encoded in the \(X\)-only functional (I.76).
4.
The ballooning transform (I.78) removes periodicity without discarding shear, and the eikonal phase (I.79) converts the large parallel gradient into the shear integral \[ \Theta (\Psi ,y;y_0)=\int _{y_0}^{y}\nu '(\Psi ,\bar y)\,d\bar y. \]
5.
The lowest-order local stability problem is the ordinary differential equation (I.107). That equation fixes the mode structure along a single unwrapped field line; higher order chooses \(y_0\) and determines the radial envelope.