Breakthroughs in Plasma Physics and Magnetohydrodynamics
============================================================


.. include:: /_generated/nav/plasma.rst

.. epigraph::

   It is not too much to say that we now live in a plasma era: the state
   Irving Langmuir named almost as an aside in 1928 turned out, within a
   few decades, to be the state of nearly everything -- a paraphrase of
   the spirit of his work rather than a verified quotation from it.

Plasma -- ionized gas dense enough that collective electromagnetic forces,
not two-body collisions, dominate its behavior -- is often called the
fourth state of matter, and it makes up more than 99% of the visible
universe: the solar wind, the ionosphere, lightning, fusion reactors, and
the interior of every star. This chronology traces the major conceptual
breakthroughs behind :mod:`physicskit.plasma`, from Debye and Huckel's
1923 theory of ionic shielding and Langmuir's naming of the plasma state
in 1928 to the gyrokinetic turbulence theory that
underlies modern fusion-reactor design. Every stop has a pointer to the
corresponding implementation in this package, a structural diagram of the
system it describes, and a short, runnable example reproducing the
milestone's signature observable.

.. contents:: Timeline
   :local:
   :depth: 1

1923 -- Debye and Huckel: Ionic Shielding and the Debye Length
------------------------------------------------------------------------

Peter Debye and Erich Huckel, working on the -- at the time entirely
unplasma-like -- problem of strong electrolyte solutions, asked how the
Coulomb field of a single ion is modified by the cloud of oppositely
charged ions it attracts and likewise charged ions it repels around
itself. Linearizing the self-consistent Poisson-Boltzmann equation for
that screening cloud, they found that the bare :math:`1/r` Coulomb
potential is replaced by one cut off exponentially beyond a single
characteristic length -- now called the Debye length,
:math:`\lambda_D=\sqrt{\varepsilon_0 k_BT/(nq^2)}` -- beyond which the
charge simply looks neutral to a distant observer. Plasma physicists
later adopted the identical mathematics essentially unchanged: a
plasma's own mobile charges screen any point charge or applied
perturbation over exactly this length, and the defining property of a
plasma -- quasineutrality on macroscopic scales, together with the
collective, wave-supporting behavior Langmuir would discover five years
later -- is really the statement that every length scale of physical
interest vastly exceeds :math:`\lambda_D`.

*Connection:* no function in :mod:`physicskit.plasma` computes the
Debye length as a standalone quantity, since throughout the package it
appears folded directly into other formulas rather than exposed on its
own: :func:`physicskit.plasma.kinetic.landau_damping_rate` forms it
explicitly as ``lambda_D = v_th / omega_pe`` on the way to the
weak-damping formula in the Landau entry below, and the "Debye-length
dispersion" that balances nonlinear steepening in
:func:`~physicskit.plasma.waves.ion_acoustic_soliton_evolve`
(Washimi-Taniuti entry below) is the same screening length entering
through the ion-acoustic dispersion relation.

*References:* P. Debye and E. Huckel, "Zur Theorie der Elektrolyte,"
Phys. Z. 24, 185-206 (1923), later adapted to plasmas.

.. minigallery:: ../../examples/plasma/kinetic/plot_01_landau_damping.py

1928 -- Langmuir Coins "Plasma" and Discovers Electron Oscillations
------------------------------------------------------------------------

Studying ionized gas in electrical discharge tubes, Irving Langmuir
noticed that the bulk of the ionized gas -- as opposed to the
charge-separated sheath near an electrode -- carried no net charge and
behaved collectively, reminding him of blood plasma carrying corpuscles;
the name stuck. Probing it with an oscillating electrode, he found the
electron gas resonates at a sharply defined natural frequency, the
plasma frequency, at which any local charge imbalance oscillates rather
than simply relaxing away: displace the electrons and their own
restoring electric field snaps them back, overshoots, and rings, exactly
like a mass on a spring built from nothing but the plasma's own charge
density.

*Implementation:* :func:`physicskit.plasma.waves.plasma_frequency`
computes :math:`\omega_p=\sqrt{nq^2/(\varepsilon_0 m)}` directly from
this restoring-force argument, and every cold-plasma dispersion relation
in :mod:`physicskit.plasma.waves` is built from it.

*References:* I. Langmuir, "Oscillations in Ionized Gases," Proc. Natl.
Acad. Sci. 14(8), 627-637 (1928); L. Tonks and I. Langmuir, "Oscillations
in Ionized Gases," Phys. Rev. 33, 195-210 (1929), and "A General Theory
of the Plasma of an Arc," Phys. Rev. 34, 876-922 (1929).

.. minigallery:: ../../examples/plasma/waves/plot_01_plasma_frequency.py

The oscillation itself -- rather than just its frequency -- is
reproduced from first principles by
:func:`physicskit.plasma.kinetic.langmuir_wave_ic`, which seeds a
particle-in-cell plasma with both the density ripple *and* the coherent
fluid-velocity perturbation linear theory predicts for a wave genuinely
oscillating at :math:`\omega_{pe}`; evolved with the same
:func:`~physicskit.plasma.kinetic.pic_simulate` used for Landau damping
and the two-stream instability, the density rings in place for many
periods rather than damping away, exactly the "mass on a spring" picture
above but built from particle orbits rather than assumed.
:func:`physicskit.plasma.visualizers.animate_langmuir_wave` animates the
oscillating density profile directly. See it in
:doc:`/api/gallery/plasma/kinetic/plot_04_langmuir_wave`.

1938 -- Vlasov's Collisionless Kinetic Equation
------------------------------------------------------------------------

Anatoly Vlasov challenged the standard collision-integral form of the
Boltzmann equation as applied to a plasma, arguing that because the
Coulomb force is long-range, a charged particle's motion is dominated
overwhelmingly by the smoothed-out, self-consistent field generated
collectively by every other particle at once, rather than by the
occasional close two-body encounters a collision integral counts -- and
that at plasma densities the mean free path for such close encounters is
typically many orders of magnitude larger than any length scale of
interest, so the collision term could simply be dropped. The resulting
equation,

.. math::

   \partial_t f + \mathbf{v}\cdot\nabla f +
   \frac{q}{m}(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\nabla_v f = 0,

-- the Vlasov equation, or "Vlasov-Poisson"/"Vlasov-Maxwell" once coupled
self-consistently to the field equations -- describes a plasma as a
continuous distribution :math:`f(\mathbf{x},\mathbf{v},t)` advected
through phase space along exact single-particle orbits, with no
dissipation anywhere in the equation. It is the single equation
underlying nearly every kinetic result in this chronology: Landau
damping below is its linearization about a Maxwellian, the two-stream
and Weibel instabilities are linearizations about other equilibria, and
gyrokinetics is its gyro-averaged reduction.

*Implementation:* :mod:`physicskit.plasma.kinetic` solves exactly this
equation by the method of characteristics -- "each particle *is* one
characteristic," as its module docstring puts it --
:func:`~physicskit.plasma.kinetic.pic_step` advancing every particle
along its characteristic while :func:`~physicskit.plasma.kinetic.solve_poisson_1d`
closes the system self-consistently each step, and
:func:`~physicskit.plasma.kinetic.pic_simulate` drives the whole loop
forward in time; :func:`~physicskit.plasma.kinetic.maxwellian_velocities`,
:func:`~physicskit.plasma.kinetic.landau_damping_ic`,
:func:`~physicskit.plasma.kinetic.langmuir_wave_ic`, and
:func:`~physicskit.plasma.kinetic.two_stream_ic` seed this one solver
with the different initial distributions examined individually in the
entries below.

*References:* A. A. Vlasov, "On Vibration Properties of Electron Gas,"
J. Exp. Theor. Phys. 8, 291-318 (1938).

.. minigallery:: ../../examples/plasma/kinetic/plot_01_landau_damping.py

1942 -- Alfven Waves and the Birth of Magnetohydrodynamics
------------------------------------------------------------------------

Hannes Alfven proposed that a magnetized, highly conducting fluid should
support a wave nobody had previously imagined: magnetic field lines,
frozen into the plasma by its high conductivity, behave like strings
under tension :math:`B^2/\mu_0`, plucked sideways by the fluid's own
inertia. The result -- the Alfven wave -- travels at
:math:`v_A=B/\sqrt{\mu_0\rho}`, purely transverse and non-compressive,
and founded magnetohydrodynamics (MHD) as a discipline treating the
plasma as a single conducting fluid rather than a swarm of individual
particles. The idea was greeted with skepticism -- as the story goes, a widely
repeated but unsourced anecdote has Enrico Fermi needing a blackboard
demonstration before he was convinced -- but earned Alfven the 1970
Nobel Prize in Physics.

*Implementation:* :func:`physicskit.plasma.mhd.alfven_speed` computes
:math:`v_A` directly; :func:`physicskit.plasma.mhd.magnetosonic_speeds`
extends the idea to the compressive fast and slow modes that appear once
plasma pressure is added back into the picture. See it in
:doc:`/api/gallery/plasma/mhd/plot_01_alfven_and_magnetosonic_waves`.

*References:* H. Alfven, "Existence of Electromagnetic-Hydrodynamic
Waves," Nature 150, 405-406 (1942).

Beyond the wave speed alone, :func:`physicskit.plasma.waves.alfven_wave_pulse_ic`
and :func:`~physicskit.plasma.waves.simulate_alfven_wave` propagate an
actual transverse-field pulse: a "plucked string" disturbance released
from rest, evolved under the linearized 1D ideal-MHD induction and
momentum equations, splits exactly in half and travels outward at
:math:`\pm v_A` in each direction -- the field-line-plucking picture
Alfven's own analogy describes, reproduced dynamically rather than
asserted. :func:`physicskit.plasma.visualizers.animate_alfven_wave`
animates the pulse splitting and propagating. See it in the same
gallery script,
:doc:`/api/gallery/plasma/mhd/plot_01_alfven_and_magnetosonic_waves`.

.. minigallery:: ../../examples/plasma/mhd/plot_01_alfven_and_magnetosonic_waves.py

1946 -- Landau's Collisionless Damping
------------------------------------------------------------------------

Lev Landau solved the linearized Vlasov equation for a wave on a
Maxwellian plasma and found something that seemed to violate reversible
mechanics: the wave damps, exponentially, with no collisions or
dissipation anywhere in the equations. The resolution is that particles
moving at very nearly the wave's phase velocity "surf" it -- and because a
thermal (Maxwellian) distribution always has slightly more slightly
slower particles being accelerated (gaining energy from the wave) than
slightly faster particles being decelerated (giving energy back) at any
finite temperature, the net exchange drains the wave. Experimentally
confirmed only in 1964 (Malmberg and Wharton, in a electron plasma
column), Landau damping is now the textbook example of dissipation
without collisions -- entropy generated by phase mixing in velocity
space alone.

*Implementation:* :func:`physicskit.plasma.kinetic.landau_damping_rate`
evaluates the analytic weak-damping formula, and
:func:`physicskit.plasma.kinetic.pic_simulate` reproduces the decay from
first principles with a particle-in-cell solution of the same Vlasov
equation Landau linearized -- no damping term appears anywhere in
:func:`~physicskit.plasma.kinetic.pic_step`; the decay is a pure
consequence of the particle orbits.

*References:* L. D. Landau, "On the Vibrations of the Electronic
Plasma," J. Phys. USSR 10, 25-34 (1946) [Zh. Eksp. Teor. Fiz. 16, 574
(1946)]; experimental confirmation J. H. Malmberg and C. B. Wharton,
Phys. Rev. Lett. 13, 184-186 (1964).

.. minigallery:: ../../examples/plasma/kinetic/plot_01_landau_damping.py

.. math::

   \gamma = -\omega_{pe}\sqrt{\frac{\pi}{8}}\,\frac{1}{(k\lambda_D)^3}\,
   \exp\!\left(-\frac{1}{2(k\lambda_D)^2} - \frac{3}{2}\right)

where :math:`\lambda_D=v_{th}/\omega_{pe}` is the Debye length and
:math:`\gamma<0` is the (negative, i.e. decaying) rate at which the wave's
electric-field energy amplitude falls off.

1948-1949 -- The Two-Stream Instability (Haeff, Pierce, Bohm & Gross)
------------------------------------------------------------------------

Studying velocity-modulated electron beams for microwave tubes, Andrew
Haeff (1948) and John Pierce (1948) independently found that a beam of
electrons streaming through a background plasma spontaneously breaks up
into density bunches that grow exponentially -- the beam gives up kinetic
energy to a plasma wave rather than propagating undisturbed. David Bohm
and Eugene Gross (1949) derived the general kinetic dispersion relation
for two interpenetrating electron populations and showed the instability
is the mirror image of Landau damping: instead of a single Maxwellian
with everywhere-negative velocity-space slope :math:`\partial f/\partial
v`, two counter-streaming beams create a region of *positive* slope
between their peaks, and it is precisely that positive slope -- formalized
a decade later as the Penrose criterion -- that pumps energy from the
particles into the wave instead of draining it. The two-stream
instability is now the archetypal kinetic (velocity-space) plasma
instability, underlying phenomena from auroral electron beams to the
saturation physics of particle accelerators and inertial-fusion hohlraums.

*Implementation:* :func:`physicskit.plasma.kinetic.two_stream_ic` seeds
exactly this doubly-peaked, counter-streaming initial condition; run
through the same :func:`~physicskit.plasma.kinetic.pic_simulate` used for
Landau damping above, it reproduces exponential *growth* instead of decay
from the identical Vlasov-Poisson machinery -- proof that both phenomena
are two faces of one kinetic mechanism.
:func:`physicskit.plasma.visualizers.animate_two_stream_phase_space`
redraws the full :math:`(x,v)` particle scatter frame by frame, so the
two initially separate, straight beams can be watched wrapping around
each other into the single saturated "cat's-eye" vortex that the static
before/after snapshot below can only hint at. See it in
:doc:`/api/gallery/plasma/kinetic/plot_02_two_stream_instability`.

*References:* Haeff's 1948 announcement appears to be the short
communication A. V. Haeff, "Space Charge Wave Amplification Effects,"
Phys. Rev. 74, 1532 (1948) (abstract only); his fuller treatment is "The
Electron-Wave Tube -- A Novel Method of Generation and Amplification of
Microwave Energy," Proc. IRE 37, 4-10 (1949). Both are cited here since
it is not certain from secondary sources alone which paper the "1948"
date above is meant to designate. J. R. Pierce, "Possible Fluctuations
in Electron Streams Due to Ions," J. Appl. Phys. 19, 231-236 (1948);
D. Bohm and E. P. Gross, "Theory of Plasma Oscillations," Phys. Rev. 75,
1851-1864 and 1864-1876 (1949); O. Penrose, Phys. Fluids 3, 258-265
(1960).

.. minigallery:: ../../examples/plasma/kinetic/plot_02_two_stream_instability.py

1940-1963 -- Guiding-Center Theory and the Adiabatic Invariants
------------------------------------------------------------------------

Rather than integrate every fast gyration of a charged particle around a
field line, plasma theorists -- beginning with Alfven's foundational
1940 and 1950 treatments and systematized over the following two decades
into a rigorous asymptotic theory by Northrop in 1963 -- showed that the
orbit separates cleanly into a rapid gyration plus a slow drift of the
gyration's center -- the guiding center -- driven by any force or field
gradient perpendicular to :math:`\mathbf{B}`. The gyration itself
conserves an adiabatic invariant, the magnetic moment
:math:`\mu=mv_\perp^2/2B`, to very high accuracy whenever the field
changes slowly compared to the gyration period; as a particle's guiding
center drifts into stronger field, :math:`\mu` conservation forces
:math:`v_\perp` to grow at the expense of :math:`v_\parallel`, the
mechanism behind magnetic mirror confinement.

*Implementation:* :func:`physicskit.plasma.single_particle.exb_drift`,
:func:`~physicskit.plasma.single_particle.grad_b_drift`, and
:func:`~physicskit.plasma.single_particle.curvature_drift` give the
guiding-center drift velocities directly;
:func:`~physicskit.plasma.single_particle.magnetic_moment` and
:func:`~physicskit.plasma.single_particle.mirror_force` implement the
invariant and the resulting bounce motion in
:func:`~physicskit.plasma.single_particle.magnetic_mirror_bounce`.

*References:* H. Alfven, Arkiv f. Mat. Astr. Fysik 27A, No. 22 (1940),
and Cosmical Electrodynamics (Oxford, 1950); systematized in T. G.
Northrop, The Adiabatic Motion of Charged Particles (Interscience,
1963).

.. minigallery:: ../../examples/plasma/single_particle/plot_01_guiding_center_drifts_and_mirror.py

1957-1958 -- The Sweet-Parker Model of Magnetic Reconnection
------------------------------------------------------------------------

Peter Sweet and Eugene Parker proposed the first quantitative model of
magnetic reconnection: oppositely directed field lines are driven
together into a long, thin resistive current sheet, where finite
conductivity finally lets them break and reconnect, converting stored
magnetic energy into plasma flow and heat. Mass conservation through the
sheet's narrow exit throttles the whole process to a reconnection rate
scaling as :math:`S^{-1/2}` in the Lundquist number -- for solar-flare
conditions (:math:`S\sim10^{12}`), millions of times slower than flares
are observed to release their energy, a discrepancy that stood as an
open problem until Petschek's 1964 revision.

*Implementation:* :func:`physicskit.plasma.mhd.lundquist_number` and
:func:`~physicskit.plasma.mhd.sweet_parker_rate` /
:func:`~physicskit.plasma.mhd.sweet_parker_layer_width` reproduce the
model's scaling laws directly. See it in
:doc:`/api/gallery/plasma/mhd/plot_02_sweet_parker_reconnection`.

*References:* E. N. Parker, "Sweet's Mechanism for Merging Magnetic
Fields in Conducting Fluids," J. Geophys. Res. 62, 509-520 (1957); P. A.
Sweet, in Electromagnetic Phenomena in Cosmical Physics, IAU Symp. 6,
ed. B. Lehnert (Cambridge Univ. Press, 1958), p. 123.

Beyond the steady-state scaling laws, :func:`physicskit.plasma.instabilities.reconnection_harris_ic`
builds an actual perturbed Harris current sheet -- the antiparallel
:math:`B_x=B_0\tanh(y/L)` reversal Sweet and Parker's picture presumes,
seeded with the small ripple needed to break the sheet's translational
symmetry and localize reconnection to one X-point -- and
:func:`~physicskit.plasma.instabilities.simulate_reconnection` advances
the kinematic resistive-induction equation
:math:`\partial_t\psi=\eta\nabla^2\psi-\mathbf{v}\cdot\nabla\psi` forward
under a prescribed stagnation-point inflow. This generic, spatially
extended resistive-diffusion model -- flux transported in by a bulk flow
and broken only where resistivity finally acts -- sits closer to
Sweet-Parker's diffusion-throttled picture than to Petschek's localized
standing-shock geometry below, even though (being purely kinematic
rather than solving the coupled momentum equation) it reproduces neither
rate law quantitatively; :func:`physicskit.plasma.visualizers.animate_reconnection`
animates the flux function merging at the X-point and the reconnected
flux squeezing out as outflow jets. See the extended animation in the
same gallery script,
:doc:`/api/gallery/plasma/mhd/plot_02_sweet_parker_reconnection`.

.. minigallery:: ../../examples/plasma/mhd/plot_02_sweet_parker_reconnection.py

1958 -- The Grad-Shafranov Equation
------------------------------------------------------------------------

Harold Grad and Hanan Rubin, and independently Vitalii Shafranov, derived
the equation governing any static, axisymmetric magnetized-plasma
equilibrium: force balance between the pressure gradient and the
:math:`\mathbf{J}\times\mathbf{B}` force reduces, in toroidal geometry,
to a single nonlinear elliptic PDE for the poloidal flux function
:math:`\psi(R,Z)`,
:math:`\Delta^*\psi=-\mu_0R^2p'(\psi)-FF'(\psi)`. Every tokamak's
magnetic geometry -- the nested flux surfaces confining its plasma -- is,
to leading order, a numerical solution of this one equation, making it
the starting point of essentially all magnetic fusion equilibrium and
stability analysis since.

*Implementation:* :func:`physicskit.plasma.mhd.solve_grad_shafranov`
solves the equation by successive over-relaxation for the linear
(Solov'ev) source term, validated against the exact closed-form solution
in :func:`~physicskit.plasma.mhd.solovev_particular_solution`;
:func:`~physicskit.plasma.mhd.safety_factor_large_aspect_ratio` extracts
the resulting field line pitch.

*References:* H. Grad and H. Rubin, Proc. 2nd UN Conf. Peaceful Uses of
Atomic Energy, Geneva, Vol. 31, p. 190 (1958); V. D. Shafranov, Zh.
Eksp. Teor. Fiz. 33, 710-722 (1957) [Sov. Phys. JETP 6, 545-554 (1958)].

.. minigallery:: ../../examples/plasma/mhd/plot_03_grad_shafranov_equilibrium.py

1959 -- Weibel and Fried: The Filamentation Instability
------------------------------------------------------------------------

Erich Weibel, analyzing the linearized Vlasov-Maxwell equations for a
plasma whose velocity distribution is anisotropic (hotter in one
direction than another, with no bulk streaming at all), found a purely
growing transverse electromagnetic mode that needs no beam, no density
gradient, and no external field to destabilize itself -- only the
anisotropy. Burton Fried, in a companion paper the same year, supplied
the missing physical picture: with :math:`T_\perp>T_\parallel`, a
random transverse magnetic fluctuation curves the orbits of the
otherwise-undeflected transversely-moving particles enough to bunch
them into current channels, and the growing current in each channel
reinforces exactly the magnetic fluctuation that curved the orbits into
existence in the first place -- a self-sustaining feedback loop with no
threshold. The resulting filamentation instability is now central to
understanding collisionless shocks in astrophysical and laser-plasma
settings alike: whenever counter-streaming plasma flows collide without
colliding *particles* (the shock is too fast, the plasma too hot, for
two-body collisions to mediate it), Weibel-generated magnetic turbulence
does the job instead, isotropizing the flow and generating the
seed fields later amplified into the microgauss-scale fields observed
in gamma-ray-burst afterglows and supernova-remnant shocks.

.. math::

   \gamma(k)^2 = \omega_{pe}^2\left(\frac{T_\perp}{T_\parallel}-1\right) - k^2c^2

growing (:math:`\gamma>0`) for every wavenumber below the cutoff
:math:`k_{max}=(\omega_{pe}/c)\sqrt{T_\perp/T_\parallel-1}`, fastest at
:math:`k=0` where the stabilizing magnetic tension vanishes entirely.

*Implementation:* :func:`physicskit.plasma.instabilities.weibel_growth_rate`
evaluates :math:`\gamma(k)` directly, and
:func:`~physicskit.plasma.instabilities.weibel_fastest_growing_mode`
extracts the long-wavelength growth rate that sets the overall
filamentation timescale;
:func:`~physicskit.plasma.instabilities.simulate_weibel_filamentation`
superposes a spectrum of independently growing transverse-current
Fourier modes -- exact linear theory, standing in for a full
2D-in-velocity electromagnetic PIC code -- to show the real-space
current filaments forming and sharpening as the fastest modes overtake
the rest.
:func:`physicskit.plasma.visualizers.animate_weibel_filamentation`
animates the growing current pattern. See it in
:doc:`/api/gallery/plasma/kinetic/plot_03_weibel_filamentation`.

*References:* E. S. Weibel, Phys. Rev. Lett. 2, 83-84 (1959); B. D.
Fried, Phys. Fluids 2, 337-337 (1959).

.. minigallery:: ../../examples/plasma/kinetic/plot_03_weibel_filamentation.py

1962 -- Stix and the Cold-Plasma Dielectric Tensor
------------------------------------------------------------------------

Thomas Stix's *The Theory of Plasma Waves* organized the bewildering
zoo of magnetized-plasma wave modes -- Alfven waves, whistlers, ion and
electron cyclotron waves, the ordinary and extraordinary modes of
magnetized-plasma radio propagation -- into a single unified framework:
a 3x3 dielectric tensor with components :math:`S`, :math:`D`, :math:`P`
built from each species' plasma and cyclotron frequencies, from which
every cold-plasma dispersion relation follows as one quartic equation in
the refractive index. The Clemmow-Mullaly-Allis (CMA) diagram maps the
entire parameter space of that quartic onto one two-dimensional plot.

*Implementation:* :func:`physicskit.plasma.waves.stix_parameters` builds
:math:`S`, :math:`D`, :math:`P` for an arbitrary multi-species plasma;
:func:`~physicskit.plasma.waves.cold_plasma_dispersion` solves the
resulting quartic at any propagation angle, and
:func:`~physicskit.plasma.waves.cma_coordinates` supplies the CMA
diagram's axes.

*References:* T. H. Stix, The Theory of Plasma Waves (McGraw-Hill,
1962).

.. minigallery:: ../../examples/plasma/waves/plot_02_cma_diagram.py

1964 -- Petschek's Fast Reconnection
------------------------------------------------------------------------

Harry Petschek proposed a resolution to the Sweet-Parker rate problem: if
the resistive diffusion region shrinks to a small X-point rather than
extending the full length of the current sheet, four standing slow-mode
shocks radiating from that X-point can carry away most of the inflowing
magnetic flux and energy. The resulting reconnection rate falls only
*logarithmically* with the Lundquist number,
:math:`v_{in}/v_A\approx\pi/(8\ln S)`, fast enough to plausibly explain
solar flare and magnetospheric substorm energy-release timescales that
Sweet-Parker reconnection could not.

*Implementation:* :func:`physicskit.plasma.mhd.petschek_rate` -- compare
directly against :func:`~physicskit.plasma.mhd.sweet_parker_rate` at the
same Lundquist number to see the difference explode as :math:`S` grows.

*References:* H. E. Petschek, "Magnetic Field Annihilation," AAS-NASA
Symposium on the Physics of Solar Flares, NASA SP-50 (1964), p. 425.

.. minigallery:: ../../examples/plasma/mhd/plot_04_petschek_reconnection.py

(The time-evolving X-point simulation of
:func:`~physicskit.plasma.instabilities.simulate_reconnection`, animated
in the Sweet-Parker entry above, is a generic resistive current-sheet
model rather than a localized-shock one, so it is presented there rather
than duplicated here.)

1966 -- Washimi and Taniuti: Ion-Acoustic Solitons
------------------------------------------------------------------------

Ion-acoustic waves -- the plasma analogue of ordinary sound, ions
providing the inertia and Boltzmann-distributed electrons providing the
restoring pressure -- are dispersive at short wavelength, since a
plasma's finite Debye length lets nearby wavelengths propagate at
slightly different speeds. Haruichi Washimi and Tosiya Taniuti, at
Nagoya University, applied the "reductive perturbation" method to the
coupled cold-ion-fluid/Boltzmann-electron equations in the weakly
nonlinear, weakly dispersive limit and showed the result reduces
exactly to the Korteweg-de Vries equation -- the same normal form
governing shallow-water solitary waves -- with the ion-acoustic physics
fixing only the physical units of its terms. A KdV equation supports
an exact traveling-wave solution that never disperses: nonlinear
steepening (which would otherwise shock the wave) is balanced, term for
term, against linear dispersion (which would otherwise spread it), so a
single density bump of just the right width-to-amplitude ratio
propagates forever at constant speed and shape -- an ion-acoustic
soliton, later observed directly in laboratory double-plasma
experiments and now understood as one member of the same integrable
soliton family whose most famous representative is the KdV equation's
own 1965 numerical rediscovery by Zabusky and Kruskal.

.. math::

   \partial_t u + 6u\,\partial_\xi u + \partial_\xi^3 u = 0, \qquad
   u(\xi,0) = \frac{c}{2}\,\mathrm{sech}^2\!\left(\frac{\sqrt{c}}{2}\xi\right)

*Implementation:* :func:`physicskit.plasma.waves.ion_acoustic_soliton_profile`
builds the exact single-soliton solution of speed :math:`c` (amplitude
:math:`c/2`) directly from this formula, and
:func:`~physicskit.plasma.waves.ion_acoustic_soliton_evolve`
time-steps it with a pseudo-spectral, Strang-split scheme -- the stiff
:math:`\partial_\xi^3` dispersion advanced exactly in Fourier space, the
nonlinear advection by RK4 -- so the soliton's unchanging shape as it
propagates is a genuine numerical result, not built in by construction.
:func:`physicskit.plasma.visualizers.animate_ion_acoustic_soliton`
animates the traveling pulse. See it in
:doc:`/api/gallery/plasma/waves/plot_03_ion_acoustic_soliton`.

*References:* H. Washimi and T. Taniuti, Phys. Rev. Lett. 17, 996-998
(1966); N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240-243
(1965).

.. minigallery:: ../../examples/plasma/waves/plot_03_ion_acoustic_soliton.py

1970 -- The Boris Particle Pusher
------------------------------------------------------------------------

Jay Boris, developing early particle-in-cell plasma codes, needed a
numerical integrator for the Lorentz force
:math:`m\dot{\mathbf{v}}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})` that
would not spuriously heat or cool a particle over millions of simulated
gyro-orbits. His scheme splits each leapfrog step into an electric
half-kick, an *exact* rotation about :math:`\mathbf{B}` computed without
any trigonometry (the "Boris rotation" trick), and a second electric
half-kick; the rotation step conserves speed to machine precision in a
pure magnetic field, regardless of step size. Half a century later it
remains the default particle integrator in essentially every
electromagnetic and electrostatic PIC plasma code written.

*Implementation:* :func:`physicskit.plasma.single_particle.boris_push`
and :func:`~physicskit.plasma.single_particle.boris_integrate`
implement the scheme with Numba-compiled kernels;
:mod:`physicskit.plasma.kinetic` reuses the same leapfrog philosophy for
its particle-in-cell push step. The closed-form
:func:`~physicskit.plasma.single_particle.larmor_radius` gives the exact
gyroradius the rotation step traces out, unchanged orbit after orbit.

*References:* J. P. Boris, "Relativistic Plasma Simulation --
Optimization of a Hybrid Code," Proc. 4th Conf. on Numerical Simulation
of Plasmas, Naval Research Laboratory (1970), pp. 3-67.

.. minigallery:: ../../examples/plasma/single_particle/plot_02_boris_pusher.py

1977-1978 -- Hasegawa and Mima: The Drift-Wave Turbulence Equation
------------------------------------------------------------------------

Akira Hasegawa and Kunioki Mima, working at Bell Laboratories, asked
what the simplest possible nonlinear equation is that still captures
how a magnetized plasma's density-gradient-driven drift waves saturate
into turbulence rather than growing forever. Their answer -- published
in two 1977-1978 papers on strong turbulence in a magnetized
nonuniform plasma -- reduces the full gyrokinetic problem to a single
scalar equation for the electrostatic potential, retaining just enough
finite-Larmor-radius physics (through the :math:`\nabla^2\phi` term) to
let :math:`\mathbf{E}\times\mathbf{B}` advection nonlinearly saturate
the wave, while assuming adiabatic (Boltzmann) electrons and dropping
parallel dynamics entirely. First derived independently by Jule Charney
in 1948 as a model for Rossby waves in a rotating atmosphere, the
identical equation in its plasma incarnation is now the standard
minimal model for the self-organization of drift-wave turbulence into
long-lived coherent vortices ("blobs") that dominate cross-field
particle and heat transport in a magnetic-confinement fusion device --
the same phenomenon that has made turbulent transport, rather than
classical collisional diffusion, the central obstacle in fusion reactor
design.

.. math::

   \partial_t(\phi-\nabla^2\phi) + \{\phi,\nabla^2\phi\} + \partial_y\phi = 0

where :math:`\{\phi,\zeta\}=(\partial_x\phi)(\partial_y\zeta)-(\partial_y\phi)(\partial_x\zeta)`
is the :math:`\mathbf{E}\times\mathbf{B}` advection of potential
vorticity :math:`q=\nabla^2\phi-\phi`, and :math:`\partial_y\phi` is the
linear drift wave sourced by the background density gradient.

*Implementation:* :func:`physicskit.plasma.turbulence.drift_wave_noise_ic`
seeds small-amplitude, featureless potential noise -- the "let it find
its own structure" initial condition appropriate for a turbulence that,
in a real device, is continuously driven by many unstable modes at
once rather than growing from one clean seed; and
:func:`~physicskit.plasma.turbulence.simulate_hasegawa_mima` advances
the potential-vorticity equation pseudo-spectrally with RK4, Strang-split
around an exactly-integrated dissipative term
(:func:`~physicskit.plasma.turbulence.hasegawa_mima_rhs` supplies the
non-dissipative right-hand side), reproducing the characteristic
nonlinear cascade of small-scale drift-wave noise into large-scale
coherent vortices -- the same inverse-cascade phenomenology 2D
Navier-Stokes turbulence shows in :mod:`physicskit.fluids`, arrived at
here from magnetized-plasma drift-wave physics rather than incompressible
hydrodynamics. :func:`physicskit.plasma.visualizers.animate_drift_wave_turbulence`
animates the potential field self-organizing. See it in
:doc:`/api/gallery/plasma/turbulence/plot_01_drift_wave_turbulence`.

*References:* A. Hasegawa and K. Mima, Phys. Rev. Lett. 39, 205-208
(1977), and Phys. Fluids 21, 87-92 (1978); precursor J. G. Charney,
Geofysiske Publikasjoner 17(2), 1-17 (1948).

.. minigallery:: ../../examples/plasma/turbulence/plot_01_drift_wave_turbulence.py

1979 -- Tajima and Dawson: The Laser Wakefield Accelerator
------------------------------------------------------------------------

Toshiki Tajima and John Dawson proposed a radically more compact
alternative to the kilometers-long radio-frequency cavities of
conventional accelerators: fire an intense, short laser pulse into a
plasma, let its ponderomotive force (the pulse's own intensity gradient
pushing electrons out of its way) excite a large-amplitude plasma
wave in its wake, and inject a second bunch of electrons to "surf" that
wake's longitudinal electric field. Because a plasma wave's electric
field is not limited by material breakdown the way a copper cavity's
is, the accelerating gradient in a laser-driven wakefield can exceed
conventional accelerators' by three to four orders of magnitude --
Tajima and Dawson's original estimate was gigaelectronvolts of energy
gain per centimeter of plasma, at a time when the largest
radio-frequency accelerators gained on the order of tens of
megaelectronvolts per meter. Their proposal, verified in one-dimensional
particle-in-cell simulation in the same paper, launched the field of
laser (and later beam-driven) plasma wakefield acceleration, which by
the 2020s had demonstrated multi-GeV energy gains over a few
centimeters of plasma in real experiments.

*Implementation:* :func:`physicskit.plasma.acceleration.wakefield_e_field`
prescribes exactly the traveling accelerating structure a driver
pulse or beam excites, :math:`E_z(x,t)=E_0\cos[k(x-v_{ph}t)]`, standing
in for the self-consistent field a full particle-in-cell wakefield
simulation would derive; and
:func:`~physicskit.plasma.acceleration.simulate_wakefield_acceleration`
injects a single test charge into it with the same energy-conserving
:func:`physicskit.plasma.single_particle.boris_push` used for gyro-orbits
elsewhere in this package, showing the particle gain kinetic energy by
riding the wake's accelerating phase for as long as it stays within the
"bucket" -- the qualitative surfing picture of wakefield acceleration,
without the driver's own self-consistent field generation.
:func:`physicskit.plasma.visualizers.animate_wakefield_acceleration`
animates the particle surfing the wake alongside its energy-gain trace.
See it in :doc:`/api/gallery/plasma/acceleration/plot_01_wakefield_acceleration`.

*References:* T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267-270
(1979).

.. minigallery:: ../../examples/plasma/acceleration/plot_01_wakefield_acceleration.py

1980s-Present -- Gyrokinetics and Fusion Turbulence
------------------------------------------------------------------------

Modern magnetic-confinement fusion research is dominated by
*gyrokinetic* theory: rather than track a particle's full gyration (as
the Boris pusher does) or reduce all the way to a fluid (as MHD does),
gyrokinetics analytically averages the Vlasov equation over the fast
gyro-angle while retaining the finite-Larmor-radius physics needed for
turbulence, producing a reduced kinetic equation for the guiding-center
distribution alone. This closes the gap between the guiding-center drift
theory of 1940-1963 above and full kinetic simulation, and is the theoretical
foundation of every modern turbulent-transport code (GENE, GYRO, GS2)
used to predict a fusion reactor's confinement performance before it is
built.

*Connection:* the guiding-center machinery in
:mod:`physicskit.plasma.single_particle` -- drifts, the adiabatic
invariant, mirror bounce motion -- is precisely the single-particle
foundation gyrokinetic theory builds on; the electrostatic
particle-in-cell solver in :mod:`physicskit.plasma.kinetic` is the same
method (sample the distribution with particles, evolve their orbits,
solve self-consistently for the field) that full gyrokinetic turbulence
codes use, one gyro-average away from the reactor-scale simulations
gyrokinetics enabled.

*References:* gyrokinetics developed gradually with no single milestone
paper; E. A. Frieman and L. Chen, Phys. Fluids 25, 502-508 (1982) is the
key formalizing reference for the nonlinear gyrokinetic equation used
throughout modern turbulence codes.

.. minigallery:: ../../examples/plasma/single_particle/plot_03_exb_drift_gyroaverage.py