PPPL
QUIZ: ADVANCED TOPICS
Challenging theoretical and mathematical aspects of plasma physics are reviewed. This is graduate level material, and a background in physics and mathematics is required.

(click on a question to show/hide answers)
1. The Lyapunov exponent describes
  • 1.  the exponential efficiency increase of energy production achieved by placing wind-turbines constructed from solar panels on top of fusion power-stations in very windy, sunny and fusionishly locations.

    2.  the instability growth rate of the kinetic Alfvén mode (KAW).

    3.  the exponential rate at which initially nearby trajectories in the phase space of a dynamical system separate over time.

    [#q17]

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      The correct answer is 3
      Given a dynamical flow, $d{\bf x}/dt = {\bf V}({\bf x})$, and two initially nearby trajectories, ${\bf x}_0$ and ${\bf x}_0 + \delta {\bf x}$, the Lyapunov exponent quantifies how quickly (more precisely, the exponential rate at which) the two trajectories separate over time. This quantity is best computed in the “tangent space”, $d \delta{\bf x}/dt = \partial {\bf V} / \partial {\bf x} \cdot \delta {\bf x}$, and the Lyapunov exponent is given by \begin{eqnarray} \sigma({\bf x}_0,\delta{\bf x}) \equiv \lim_{t \rightarrow \infty}\frac{1}{t}\ln \frac{d({\bf x}_0,t)}{d({\bf x}_0,0)}. \end{eqnarray} So-called “chaotic” trajectories separate at an exponential rate, and so have a non-zero Lyapunov exponent. See, for example, Lichtenberg & Lieberman [1] for further details and references.
      [1] A.J. Lichtenberg & M.A. Lieberman, Regular and Chaotic Dynamics, 2nd ed., Springer-Verlag (1992)
2. The basic approximation of gyrokinetics is:
  • 1.  that the length scale of the variations in the magnetic field is much larger than the Larmor radius.

    2.  that collisions are less frequent than gyrations.

    3.  that the electric field is much less that then magnetic field.

    [#q4]

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      The correct answer is 1
      Taking the average over the fast gyromotion yields equations of motion for the guiding center that are an accurate approximation to the particle's true trajectory only when the Larmor radius is small compared to the “system size”. Formally, $\delta B \ll B$, where $\delta B$ is the variation in the magnetic field, $B$, over the Larmor radius.
3. The “Fundamental Theorem of Curves” states that:
  • 1.  any one-dimensional closed curve embedded in three-dimensional space admits a unique two-dimensional surface that has the curve as its boundary and has non-vanishing surface element.

    2.  the force on an object of unit mass traversing a curve at unit speed (with respect to arc-length) is inversely proportional to the local radius of curvature.

    3.  every regular curve in three-dimensional space, with non-zero curvature, has its shape and size completely determined by its curvature and torsion.

    [#q27]

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      The correct answer is 3
      The curvature and torsion completely describe the size and shape of a curve. To completely identify a unique curve, i.e. to eliminate the freedom of rigid shifts and rotations, so-called “starting point” information must also be provided. Given $\kappa(s)$ and $\tau(s)$, the curve may be constructed by integrating \begin{eqnarray} \left( \begin{array}{c} {\bf t}^\prime \\ {\bf n}^\prime \\ {\bf b}^\prime \end{array} \right) = \left( \begin{array}{c} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array} \right) \left( \begin{array}{c} {\bf t} \\ {\bf n} \\ {\bf b} \end{array} \right) \end{eqnarray} with initial conditions e.g. ${\bf t}(0)={\bf t}_0$, ${\bf n}(0)={\bf n}_0$, and ${\bf b}(0)={\bf t}_0 \times {\bf n}_0$. The position of the curve is obtained by further integrating the tangent, i.e. $d{\bf x}(s) = {\bf t}(s) \, ds$. (This representation is being exploited in a new approach to stellarator coil design by Caoxiang Zhu and Stuart Hudson, in which each “external” current-carrying coil is represented as a closed curve using the above representation, more details to be provided soon.)
4. What is magnetic reconnection?
  • 1.  Magnetic reconnection is the “breaking” and “rejoining” of magnetic field lines. Reconnection can only occur in vacuum, i.e. when an plasma is not present, because if the magnetic field is coupled to an infinitely conducting medium (such as a plasma, and all plasmas are infinitely conducting) reconnection would violate Alfvén's theorem [1].
    [1] T.J.M. Boyd & J.J. Sanderson, The Physics of Plasmas, Cambridge University Press (2003)

    2.  Magnetic reconnection is a physical process in which two initially separated, large-scale plasma columns, being driven by magnetic fields, collide and disintegrate into small-scale magnetic turbulent fluctuations.

    3.  Magnetic reconnection is a physical process whereby the magnetic field line connectivity is modified due to the presence of a localized diffusion region, allowing magnetic energy to be converted into plasma particle energy.

    [#q33, Dr. L. Comisso, PPPL]

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      The correct answer is 3
      To define magnetic reconnection one should first define magnetic field line connectivity. In a plasma that satisfies the ideal Ohm’s law, namely ${\bf E} + {\bf v}\times{\bf B} = 0$ where ${\bf E}$ is the electric field, ${\bf v}$ is the plasma velocity and ${\bf B}$ is the magnetic field, two plasma elements connected by a magnetic field line at a given point in time will remain connected by a field line for all subsequent times [1]. This concept can be generalized, with some subtleties, to relativistic plasmas [2].
      Magnetic reconnection in a non-ideal plasma, e.g. a plasma that satisfies ${\bf E} + {\bf v} \times {\bf B} = \eta {\bf j}$ where ${\bf j}$ is the plasma current and $\eta \ne 0$ is the resistivity, changes the magnetic field line connectivity. (Note that no plasmas are perfectly ideal.) The violation of the connectivity property itself is, however, not sufficient for defining magnetic reconnection, since a significant energy conversion is also necessary to distinguish reconnection from simple diffusion [3]. Magnetic reconnection has been posited as the key driver of some of the most spectacular and energetic phenomena in laboratory, space and astrophysical plasmas. The most prominent examples include sawtooth crashes [4], Earth magnetospheric substorms [5], solar and stellar flares [6], and non-thermal signatures of pulsar wind nebulae [7].
      [1] William A Newcomb, Ann. Phys. 3, 347 (1958)
      [2] Felipe A. Asenjo & Luca Comisso, Phys. Rev. Lett. 114, 15003 (2015)
      [3] Yi-Min Huang, A. Bhattacharjee & Allen H. Boozer, Astrophys. J. 793, 106 (2014)
      [4] A.W. Edwards, D.J. Campbell et al., Phys. Rev. Lett. 57, 210 (1986)
      [5] J.W. Dungey, Phys. Rev. Lett. 6, 47 (1961)
      [6] R.G. Giovanelli, Nature 158, 81 (1946)
      [7] Lorenzo Sironi & Anatoly Spitkovsky, Astrophys. J. Lett. 783, L21 (2014)