Good Luck! Some of these questions may not have answers, and some questions/answers may be controversial! If anything is imprecise or incorrect, of if you wish to explain something in more detail, please contribute. (click on a question to show/hide answers)

- 1. Ghost surfaces are the “shadows” of an abitrary manifold projected onto a lower-dimensional subspace.

2. Ghost surfaces are surfaces to which the reduced dynamical motion of guiding centers is tangential.

3. Ghost surfaces are a class of surfaces that are “almost-invariant” under the magnetic fieldline dynamics.

4. Ghost surfaces are surfaces through which ghosts manifest. They are an underlying theme of children's literature.[#q16, Dr. S.R. Hudson, PPPL]

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The correct answer is 3

The action for magnetic fields is a functional of an arbitrary “trial” curve, ${\cal C}$, and is defined as the line integral of the magnetic vector potential, $S[{\cal C}] \equiv \int_{\cal C}{\bf A}\cdot d{\bf l}$, where ${\bf B}=\nabla\times{\bf A}$. Periodic fieldlines extremize the action. Ghost surfaces are defined by an “action-gradient flow” between the minimax (usually stable) and minimizing (unstable) periodic fieldlines, and were shown to be equivalent to quadratic-flux-minimizing surfaces [1]. By taking suitable limits, ghost surfaces “fill-in-the-gaps” in cantori; consequently, they may intuitively be considered as*pseudo*invariant (flux) surfaces in regions of phase space where true invariant surfaces are destroyed by chaos. Dr. S.R. Hudson & Dr. J. Breslau [2,3,4] showed that ghost surfaces are intimately related to anisotropic transport in non-integrable magnetic fields.

[1] S.R. Hudson & R.L. Dewar, Phys. Lett. A**373**, 4409 (2009)

[2] S.R. Hudson & J. Breslau, Phys. Rev. Lett.**100**, 095001 (2008)

[3] S.R. Hudson, #highlight23

[4] S.R. Hudson, 22nd IAEA Fusion Energy Conference, Invited Talk

- 1. If the magnetic field is integrable (i.e. there is a family of nested flux surfaces), then physically acceptable and mathematically valid solutions with smooth, non-trivial pressure must have discontinuous rotational-transform [1,2,3].
The rotational-transform must be almost-everywhere irrational.

[1] J. Loizu, S.R. Hudson*et al.*, Phys. Plasmas**22**, 090704 (2015)

[2] S.R. Hudson, #highlight20

[3] S.R. Hudson, 26th IAEA Fusion Energy Conference, Invited Talk

2. Three-dimensional magnetic fields are analogous to $1 \frac{1}{2}$ dimensional Hamiltonian systems [1,2], which are generally non-integrable [3]. Magnetic islands arise where the rotational-transform is rational, and irregular “chaotic” fieldlines emerge where the islands overlap. By virtue of the KAM theorem [3], a finite measure of invariant flux surfaces with irrational rotational-transform exist (if the system is not too far perturbed from an integrable state). Force-balance, $\nabla p = {\bf j}\times{\bf B}$, implies ${\bf B}\cdot\nabla p=0$, so the pressure is constant along a fieldline; and the pressure gradient is a fractal function of position [4].

[1] J.R. Cary & R.G. Littlejohn, Ann. Phys.**151**, 1 (1983)

[2] A.H. Boozer, Phys. Fluids**26**, 1288 (1983)

[3] J.D. Meiss, Rev. Mod. Phys.**64**, 795 (1992)

[4] H. Grad, Phys. Fluids**10**, 137 (1967)

3. Mathematical theorems guaranteeing the existence of well-defined solutions to the three-dimensional MHD equilibrium problem, with non-trivial pressure, were introduced by Bruno & Laurence [1], in which the pressure is piecewise constant and the magnetic field is allowed to be “partially” chaotic. The Stepped Pressure Equilibrium Code (SPEC) [2,3] is a numerical implementation of this class of equilibria.

[1] Oscar P. Bruno & Peter Laurence, Commun. Pure Appl. Math.**49**, 717 (1996)

[2] S.R. Hudson, R.L. Dewar*et al.*, Phys. Plasmas**19**, 112502 (2012)

[3] S.R. Hudson, #highlight5

4. In equilibria with integrable magnetic fields, for which the magnetic field is continuous (with a continuous rotational-transform), and smooth, non-trivial pressure profiles, the pressure-driven “Pfirsch-Schlüter” $1/x$ current-densities that arise near rational rotational-transform flux surfaces are non-integrable: cross sectional surfaces may be constructed through which the current is infinite!

5. All of the above.[#q26, Dr. S.R. Hudson, PPPL]

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The correct answer is 5

For more information regarding answers (1-3), please follow the provided links. Regarding the Pfirsch-Schlüter currents, if the magnetic field is integrable, straight fieldline flux coordinates can be introduced that allow the magnetic field to be written ${\bf B} = \nabla \psi \times \nabla \theta + \iota(\psi) \nabla \zeta \times \nabla \psi$, where $\psi$ is the toroidal flux and $\theta$ and $\zeta$ are straight-fieldline poloidal and toroidal angles. The directional derivative along the magnetic field is ${\bf B}\cdot\nabla = \sqrt g^{-1} ( \partial_\zeta + \iota \partial_\theta)$. The current density may be decomposed into parallel and perpendicular components, ${\bf j} = \sigma {\bf B} + {\bf j}_\perp$, and the equation $\nabla p = {\bf j}\times{\bf B}$ gives ${\bf j}_\perp = {\bf B}\times \nabla p / B^2$. Using the representation, $\sigma=\sum \mu_{m,n}\exp(im\theta-in\zeta)$, the Fourier harmonics of the parallel current density are $\sigma_{m,n} = (\nabla\cdot{\bf j}_\perp)_{m,n}/(m\iota - n)$, which gives a logarithmic singularity for the total current passing through a cross-sectional region bounded by the rational flux surface, $\iota = n/m$, and this is not physical. Except for very special geometries (e.g. axisymmetry), the only way to avoid this non-physical parallel current is to ensure that the pressure gradient is zero in a finite neighborhood of each rational surface. Given that the rational surfaces are dense, for any continuous magnetic field (with shear) the pressure profile, if it is smooth, must be trivial.

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