Plasmas and Magnetohydrodynamics
Almost all the visible matter in the universe is plasma: an ionized gas in which the long-range Coulomb interaction, rather than binary collisions, organizes the motion. The name is Langmuir's, and so is the first quantitative study of the collective oscillation that distinguishes the state [Langmuir:1928] [Tonks:1929]. This chapter is where the field theory of The Maxwell Equations and Generalized Classical Field Theory meets the many-body mechanics of Kinetic Theory of Gases and the continuum dynamics of Fluid Dynamics: the fields move the charges, the charges are the sources of the fields, and the closure of that loop produces behaviour that neither theory shows alone — screening [Debye:1923], collective oscillation at the plasma frequency, collisionless damping [Landau:1946], and a fluid that carries its own magnetic field frozen into it [Alfven:1942].
The chapter is placed at the end of Part VII because it is applied classical electrodynamics, and it is placed before the quantum parts because nothing in it needs \(\hbar\). Its evidence base is unusually broad and unusually direct: laboratory measurements of Landau damping [Malmberg:1964] and of Alfvén waves in liquid metal [Lundquist:1949]; spacecraft in the solar wind whose existence Parker predicted from a hydrodynamic argument [Parker:1958] and Mariner II confirmed [Neugebauer:1962]; magnetometers inside a reconnecting current sheet [Burch:2016]; a laboratory dynamo [Gailitis:2000]; and fusion devices measured against a criterion written down in 1957 [Lawson:1957] and finally exceeded in inertial confinement [AbuShawareb:2024]. Standard monographs are [Alfven:1950] [Spitzer:1962].
Plasmas and Magnetohydrodynamics: all derivations of this chapter are pending.
The plasma state
Ionization equilibrium
[Reserved: the Saha equation for the degree of thermal ionization [Saha:1920], derived from the chemical equilibrium of Statistical Mechanics; the sharpness of the transition from neutral gas to plasma at temperatures well below the ionization potential; non-thermal ionization by radiation and by impact; the worked cases this treatise needs — the solar photosphere and chromosphere (Stellar Structure and Nucleosynthesis), the terrestrial ionosphere, and a laboratory discharge.]
Debye screening and the plasma parameter
[Reserved: the screened potential \(\varphi\propto\ee^{-r/ \lambda_{D}}/r\) with \(\lambda_{D}=\sqrt{\epsilon_{0}k_{B}T/nq^{2}}\), transferred from the electrolyte theory of Debye and Hückel [Debye:1923]; the plasma parameter \(\Lambda=n\lambda_{D}^{3}\) and the definition of a plasma as a medium with \(\Lambda\gg1\), so that collective effects dominate binary collisions; numerical values — \(\lambda_{D}\approx74\,\mu\mathrm{m}\) for a fusion plasma at \(10^{20}\,/\mathrm{m}^{3}\) and \(10\,\mathrm{keV}\), and metres in the solar wind; the Coulomb logarithm \(\ln\Lambda\) that will appear in every transport coefficient of Section 66.3.4.]
Quasi-neutrality and the plasma frequency
[Reserved: quasi-neutrality as a consequence of screening rather than an assumption, valid on scales above \(\lambda_{D}\); the restoring force of a rigid charge displacement and the electron plasma frequency \(\omega_{p}=\sqrt{nq_{e}^{2}/\epsilon_{0}m_{e}}\), i.e. \(f_{p}\approx8.98\sqrt{n}\) hertz with \(n\) in inverse cubic metres, identified experimentally by Langmuir [Langmuir:1928] and analysed with Tonks [Tonks:1929]; the three defining conditions of the plasma state (many particles per Debye sphere, system size above \(\lambda_{D}\), collision time above \(1/\omega_{p}\)).]
An ionized gas does not respond to a disturbance the way a neutral gas does. Probed in a discharge, the electrons are found to oscillate collectively at a frequency fixed by their number density alone — not by the wavelength of the disturbance, not by the gas species, not by the geometry of the vessel, and, at long wavelength, not by the temperature [Langmuir:1928] [Tonks:1929]. The frequency is
and it is this collective response, rather than any property of the individual collisions, that makes the ionized gas a distinct state of matter.
Derivation. Consider a slab of plasma and displace every electron rigidly through a small distance \(x\) along the slab normal, leaving the much heavier ions in place. The displacement uncovers a layer of positive charge on one face and deposits a layer of negative charge on the other, of surface density \(\pm nq_{e}x\), and a pair of such sheets produces a uniform field between them of magnitude
directed so as to pull the electrons back. The equation of motion of each electron is then
which is simple harmonic motion at exactly the frequency Equation (66.1). Three features of the observation are already in this one line. The frequency does not depend on the amplitude, because the restoring force is linear in the displacement; it does not depend on the wavelength, because no gradient of anything entered — the correction that does introduce one comes from the thermal pressure and is the Bohm–Gross term of Section 66.2.1; and it depends on the ion mass not at all, the ions serving only as a stationary neutralizing background.
The same \(\omega_{p}\) governs the transparency of metals and the reflection of radio waves treated in Electrodynamics in Matter, and evaluating Equation (66.1) with the CODATA values of the electron charge and mass [Tiesinga:2021] gives the working form \(f_{p}=\omega_{p}/2\pi\approx8.98\sqrt{n}\) hertz with \(n\) in inverse cubic metres.
∎Where plasmas are found
[Reserved: the parameter map in density and temperature, all in SI, spanning the interstellar medium, the solar corona and wind, the ionosphere, laboratory discharges, tokamak and inertial-fusion plasmas, and the degenerate interiors of Compact Stars and Relativistic Astrophysics; the direct evidence for each — the ionospheric reflection of radio waves demonstrated by Appleton and Barnett [Appleton:1925], and the Zeeman splitting that showed sunspots to be strongly magnetized [Hale:1908], which is the founding observation of astrophysical magnetism.]
Spectral lines formed in a sunspot are split into components whose separation is measurable with a grating spectrograph, and whose polarization is the signature that identifies the splitting as magnetic: viewed along the field the outer components are circularly polarized with opposite handedness, viewed across it they are linearly polarized. The splitting is absent in the surrounding photosphere and largest at the darkest part of the spot. The inferred field strengths are of order a tenth of a tesla, thousands of times the field at the Earth's surface [Hale:1908]. This is the first measurement of a magnetic field outside the Earth, and it is the observation on which the whole of astrophysical magnetism rests.
Derivation. The inference from a measured splitting to a field strength needs only the classical theory of a bound electron in a magnetic field. Let the electron be bound isotropically with natural frequency \(\omega_{0}\) and place the field along \(z\). Writing the transverse displacement as the complex combination \(u=x+\ii y\), the equation of motion \(m_{e}\ddot{\vect{r}}=-m_{e}\omega_{0}^{2}\vect{r} +q_{e}\dot{\vect{r}}\times\vect{B}\) becomes
and substituting \(u=\ee^{-\ii\omega t}\) gives \(\omega^{2}-(q_{e}B/m_{e})\,\omega-\omega_{0}^{2}=0\). For fields weak enough that \(\abs{q_{e}}B/m_{e}\ll\omega_{0}\) — which holds by many orders of magnitude for optical lines in any attainable field — the two roots are
The two transverse modes are circular motions of opposite sense, shifted up and down by \(\abs{\Delta\omega}\); the third mode, along \(z\), feels no magnetic force and is unshifted. Hence a triplet, with the outer components circularly polarized when the line of sight is along \(\vect{B}\) and linearly polarized when it is across — which is exactly the polarization pattern that identifies the effect and, with it, the direction of the field. Inverting Equation (66.5),
so a splitting measured in metres of wavelength returns a field in tesla. The classical account is not the whole story — real lines split into patterns richer than a triplet, for reasons that are quantum and belong to the atomic chapters — but the magnitude, and therefore the inferred field, is the classical one.
∎Waves in a plasma
Langmuir waves
[Reserved: electrostatic electron oscillations, their near-dispersionless character at long wavelength, and the Bohm–Gross correction \(\omega^{2}=\omega_{p}^{2}+3k^{2}v_{\text{th}}^{2}\) from thermal pressure [Bohm:1949]; the standing puzzle that the fluid treatment gives no damping at all, resolved kinetically in Section 66.3.2; the observation of these waves as the electron-plasma oscillations first reported in [Langmuir:1928] [Tonks:1929].]
Electromagnetic waves and the cutoff
[Reserved: the transverse dispersion relation \(\omega^{2}= \omega_{p}^{2}+c^{2}k^{2}\), the cutoff at \(\omega_{p}\) below which the plasma reflects, and the resulting refractive index below unity, whose consequences for phase and group velocity connect to Electrodynamics in Matter; the ionospheric application, with \(f_{p}\) of order \(9\,\mathrm{MHz}\) for \(n\approx10^{12}\,/\mathrm{m}^{3}\), which is why long-distance short-wave radio works and why satellite links must exceed the cutoff [Appleton:1925]; interferometry and cutoff reflectometry as plasma density diagnostics.]
Radio waves directed upwards return to the ground if their frequency is below a critical value and are lost into space if it is above. The returning wave was identified as such by beating it against the ground wave and varying the transmitter frequency: the resulting fringes give the difference in path length, and hence the height of the reflecting layer, which is found to lie of order a hundred kilometres above the ground and to change between day and night [Appleton:1925]. The critical frequency is not a property of the apparatus but of the medium, and it varies with the hour, the season and the solar cycle. There is therefore a permanent ionized layer in the upper atmosphere, and the critical frequency measures its electron density.
Derivation. The reflection itself is the free-carrier cutoff derived in Electrodynamics in Matter: a wave of angular frequency \(\omega\) in a medium of free electrons obeys \(\omega^{2}=\omega_{p}^{2}+c^{2}k^{2}\), so \(k\) is imaginary and the wave cannot propagate wherever \(\omega<\omega_{p}\). A wave sent upwards into a layer whose density increases with height therefore travels freely until it reaches the level at which \(\omega_{p}\) has risen to \(\omega\), and is returned from there — so the reflection height depends on the frequency, and the highest frequency returned at vertical incidence is the plasma frequency of the densest part of the layer.
What makes this a measurement is the numerical size of Equation (66.1). Evaluating it with the CODATA electron charge and mass [Tiesinga:2021],
with \(n\) in inverse cubic metres; so a critical frequency of a few megahertz corresponds to an electron density of order \(10^{12}\,/\mathrm{m}^{3}\), which is what the ionosphere has. The inverse statement is the diagnostic: sweeping the transmitter frequency and recording the echo delay returns the density as a function of height, and the same principle, applied at much higher frequency, is cutoff reflectometry in a laboratory plasma.
∎Magnetized waves and ion-acoustic waves
[Reserved: propagation across and along a background field, the ordinary and extraordinary modes, cyclotron resonances at \(\omega_{c}=qB/m\), Faraday rotation as a measurement of the line integral of \(n_{e}B_{\parallel}\) used throughout radio astronomy; the ion-acoustic mode with speed \(\sqrt{k_{B}T_{e}/m_{i}}\), sustained by electron pressure and ion inertia, and its Landau damping when \(T_{e}\sim T_{i}\); the classification of the many modes as a warning that “the plasma dispersion relation” is a family, not a formula [Spitzer:1962].]
Kinetic theory and Landau damping
The Vlasov equation
[Reserved: the collisionless kinetic equation for \(f(\vect{x},\vect{v},t)\) with the self-consistent field [Vlasov:1938], its standing as the leading term of the BBGKY hierarchy of Kinetic Theory of Gases, and the sense in which it is a mean-field rather than an exact description; its conservation laws and the infinity of Casimir invariants; the Jeans-type equilibria that any function of the constants of motion provides.]
Landau damping
[Reserved: Landau's solution of the initial-value problem by Laplace transform and the contour prescription that produces a damping rate for a wave in a collisionless medium [Landau:1946]; the damping proportional to \(\pp f/\pp v\) at the phase velocity, and the resonant-particle reading (particles slightly slower than the wave gain energy, slightly faster ones lose it); the inverse effect — growth on an inverted distribution — as the bump-on-tail instability; why this is not thermodynamically irreversible, the entropy being carried into fine velocity-space structure, with echo experiments as the evidence.]
The observation
[Reserved: Malmberg and Wharton's measurement of the spatial damping of electron plasma waves in a collisionless column, with the measured damping length agreeing with Landau's prediction over more than a decade in wavenumber [Malmberg:1964]; the plasma-wave echo as the direct demonstration that the information is not lost; and the methodological point that a purely mathematical prescription for a contour was confirmed as physics eighteen years after it was written.]
An electron plasma wave launched into a column so tenuous that a particle crosses it many times between collisions is nevertheless damped, and damped strongly. The damping is spatial and exponential, its rate depends on the wavenumber, and it persists as the collision frequency is reduced — so it is not a collisional effect at all. The measured damping length agrees with the collisionless prediction over more than a decade in wavenumber [Malmberg:1964], confirming an analysis that had been produced eighteen years earlier from a contour prescription in a Laplace transform [Landau:1946]. A dissipative process therefore occurs in a system whose equations of motion are exactly time-reversible and conserve entropy.
Landau damping: solve the linearized Vlasov–Poisson initial-value problem by Laplace transform, analytically continue the dielectric function below the real axis to obtain the correct prescription for the singularity at the phase velocity, and derive the damping rate proportional to the slope of the distribution function there. Then give the resonant-particle reading — particles slightly slower than the wave gain energy and slightly faster ones lose it, and a Maxwellian has more of the former — and explain why the process is not thermodynamically irreversible, the information being carried into fine structure in velocity space and recoverable as a plasma-wave echo.
Collisions and transport
[Reserved: the Fokker–Planck description of small-angle Coulomb scattering with the Rosenbluth potentials [Rosenbluth:1957]; the \(T^{3/2}\) scaling and the Spitzer resistivity [Spitzer:1953], from which a fusion-grade plasma is a better conductor than copper; collisional relaxation times and their ordering \(\tau_{ee}<\tau_{ii}<\tau_{ie}\); the full transport coefficients — resistivity, viscosity, thermal conduction, all anisotropic in a magnetic field — as tabulated by Braginskii [Braginskii:1965], connecting to Nonequilibrium Thermodynamics and Transport.]
Single-particle motion
Gyration and the guiding centre
[Reserved: the helical orbit in a uniform field, the gyrofrequency and gyroradius in SI, and the guiding-centre expansion in the small parameter \(\rho_{L}/L\) [Alfven:1950]; the separation of fast gyration from slow drift as an averaging procedure of the kind used in Nonlinear Dynamics and Chaos; the limits of the expansion where the field varies on the gyroradius scale.]
Drifts
[Reserved: the \(\vect{E}\times\vect{B}\) drift, charge- and mass-independent and therefore not a current; the grad-\(B\), curvature and polarization drifts, which are charge dependent and do drive currents; the resulting ring current and the diamagnetism of a plasma; worked numbers for the magnetosphere; the general statement that a drift is a force divided by \(qB\) crossed with \(\hat{\vect{b}}\) [Alfven:1950] [Spitzer:1962].]
Adiabatic invariants and magnetic mirrors
[Reserved: the magnetic moment \(\mu=mv_{\perp}^{2}/2B\) as an adiabatic invariant in the exact sense of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, together with the longitudinal and flux invariants; magnetic mirroring, the loss cone and its opening angle; the mirror machine as a confinement scheme and its failure modes; the observational confirmation in the trapped radiation belts discovered by Van Allen [VanAllen:1959], whose particle populations are the loss-cone distribution made visible.]
Geiger counters carried above the atmosphere found, instead of the expected slow decline of the cosmic-ray rate, counting rates rising by orders of magnitude in two toroidal regions encircling the Earth [VanAllen:1959]. The particles are not passing through: they are trapped, bouncing along the field lines between turning points in the two hemispheres on time scales of a second, and the belts persist indefinitely although nothing material confines them. What confines them is the geometry of the field, and the confinement is incomplete in a definite way — particles whose velocity is too nearly parallel to the field are not turned back and are lost into the atmosphere at the foot of the line, which is why the trapped distributions are empty inside a cone in velocity space.
Derivation. A particle gyrating in a field that varies slowly over one gyroradius and one gyroperiod conserves its magnetic moment
as an adiabatic invariant. A static magnetic field does no work, so the speed \(v\) is constant as well. Writing the pitch angle \(\alpha\) by \(v_{\perp}=v\sin\alpha\), constancy of \(\mu\) and of \(v\) gives
As the particle moves into a region of stronger field, \(\sin^{2}\alpha\) must rise in proportion; when it reaches unity the velocity is entirely perpendicular, the parallel motion has stopped, and the particle is reflected. Starting from a field \(B_{0}\) with pitch angle \(\alpha_{0}\), the mirror point is therefore where
A dipole field grows towards its poles, so every particle whose \(B_{m}\) lies below the field at the top of the atmosphere turns around before reaching it and is trapped, bouncing between the conjugate mirror points — which is the observed configuration.
Equation (66.10) also fixes what escapes. If \(B_{\max}\) is the field at the atmospheric foot of the line, the particles lost are those with
a cone about the field direction whose half-angle depends only on the ratio of the two field strengths and not at all on the particle's charge, mass or energy. That is the loss cone, and its emptiness in the measured distributions is the signature that the trapping mechanism is this one and no other. The same argument, with \(B_{\max}\) set by a coil rather than by the Earth, is the magnetic mirror as a laboratory confinement scheme — and Equation (66.11) is equally the statement of why that scheme leaks.
∎Fermi acceleration
[Reserved: stochastic energy gain from collisions with moving magnetic irregularities [Fermi:1949], the second-order \((v/c)^{2}\) mechanism and its first-order diffusive-shock refinement; the power-law spectrum produced, and its match to the observed cosmic-ray spectrum treated in Cosmic Rays and Astroparticle Physics; the same mechanism as the origin of the relativistic electrons whose synchrotron emission is discussed in Radiation and Scattering of Electromagnetic Waves.]
Magnetohydrodynamics
The MHD equations and their validity
[Reserved: the one-fluid equations obtained by taking moments of the kinetic equation and eliminating the displacement current — mass continuity, the momentum equation with the \(\vect{J}\times\vect{B}\) force, an Ohm's law, and the induction equation [Alfven:1950] [Braginskii:1965]; the ordering assumptions (slow compared with \(\omega_{p}\) and \(\omega_{c}\), large compared with \(\lambda_{D}\) and \(\rho_{L}\), non-relativistic); the closure problem and the standard ideal, resistive and Hall variants; the decomposition of the Lorentz force into magnetic pressure \(B^{2}/2\mu_{0}\) and tension.]
Flux freezing and the magnetic Reynolds number
[Reserved: Alfvén's theorem that in a perfectly conducting fluid the magnetic flux through any comoving surface is constant, so field lines are material lines [Alfven:1942]; the induction equation as advection plus diffusion, with the magnetic Reynolds number \(R_{m}=UL/\eta_{m}\) measuring their ratio; the enormous values of \(R_{m}\) in astrophysics and the consequence that field topology is conserved except in thin layers — which is the whole subject of Section 66.6.1.]
Alfvén and magnetosonic waves
[Reserved: the transverse wave on a field line with speed \(v_{A}=B/\sqrt{\mu_{0}\rho}\), predicted by Alfvén [Alfven:1942] against considerable scepticism; the fast and slow magnetosonic modes and the Friedrichs diagram of their anisotropic phase speeds; the first laboratory detection, in liquid mercury [Lundquist:1949]; and the ubiquitous detection in the solar wind and the magnetosphere, where Alfvénic fluctuations dominate the turbulent spectrum.]
A conducting fluid threaded by a magnetic field supports a transverse wave that ordinary hydrodynamics does not have: the field lines behave as stretched strings, and a displacement propagates along them at
independent of the wavelength and of the amplitude, with the fluid velocity and the field perturbation transverse to the background field and correlated with one another. The mode was predicted before it was seen [Alfven:1942], first detected in liquid mercury in a strong magnet [Lundquist:1949], and is now measured continuously by spacecraft in the solar wind, where fluctuations of exactly this character dominate the turbulent spectrum.
Alfvén waves: linearize the ideal magnetohydrodynamic equations about a uniform field and a uniform density, retaining the tension part of the Lorentz force, and obtain the transverse mode propagating along the field at the speed above, together with the fast and slow magnetosonic modes and their anisotropic phase speeds. The tension reading should be derived as well — the magnetic force decomposed into a pressure and a tension along the field lines, whence the wave speed is the tension divided by the mass per unit length in exactly the form of a wave on a string.
Equilibria and the pinch
[Reserved: magnetostatic equilibrium \(\vect{J}\times\vect{B}= \vect{\nabla}p\), force-free fields, and the plasma beta as the ratio of kinetic to magnetic pressure; the Bennett relation for a self-pinching current channel [Bennett:1934]; the Grad–Shafranov equation for axisymmetric equilibria [Grad:1958] and the flux-surface geometry of a tokamak; the safety factor and the equilibria that Section 66.7 will test for stability.]
Magnetic reconnection
The Sweet–Parker layer
[Reserved: the problem stated — ideal MHD conserves topology, yet solar flares release magnetic energy in minutes; Sweet's neutral point [Sweet:1958] and Parker's quantitative current-sheet model [Parker:1957], giving a reconnection rate scaling as \(R_{m}^{-1/2}\); the arithmetic showing that this rate is still orders of magnitude too slow for a flare, which is the reason the subject did not end in 1958.]
Petschek and fast reconnection
[Reserved: Petschek's slow-shock geometry with a much shorter diffusion region and a rate scaling only as \(1/\ln R_{m}\) [Petschek:1964]; the objection that the configuration is not sustained with uniform resistivity; the modern resolutions — plasmoid instability of thin sheets, and collisionless Hall and kinetic terms in Ohm's law; the honest statement that the rate problem is settled observationally before it is settled theoretically.]
Reconnection observed
[Reserved: Dungey's open magnetosphere, in which dayside reconnection with the interplanetary field drives the whole convection cycle and the aurora [Dungey:1961]; in situ measurement of the electron diffusion region by the four Magnetospheric Multiscale spacecraft, resolving the non-ideal electric field and the energy conversion directly [Burch:2016]; solar flares and coronal mass ejections as the large-scale case; laboratory reconnection experiments; the quantities to tabulate are the reconnection rate, the layer thickness and the partition of released energy.]
Magnetic field topology, which ideal magnetohydrodynamics conserves exactly, is observed to change. The evidence is of three kinds. Solar flares release in minutes an amount of magnetic energy that took days to accumulate, and do so in configurations where oppositely directed fields have been pushed together. The whole convection cycle of the magnetosphere, and with it the aurora, switches on when the interplanetary field turns southward and off when it turns northward — a dependence on the direction of the external field that makes sense only if the two field systems join [Dungey:1961]. And four spacecraft flying in formation through the dayside boundary traversed an electron diffusion region directly, measuring there an electric field that no ideal Ohm's law permits, together with the crescent-shaped electron distributions and the conversion of electromagnetic energy into particle energy that accompany it [Burch:2016]. The ideal description therefore fails, and it fails in thin layers while remaining excellent everywhere else.
Magnetic reconnection: derive the Sweet–Parker current sheet from mass and flux conservation across a resistive layer, obtaining a reconnection rate scaling as the inverse square root of the magnetic Reynolds number, and show by substituting coronal values that this rate is orders of magnitude too slow for a flare. Then derive Petschek's slow-shock geometry with its much weaker logarithmic dependence, and state honestly which of the modern resolutions — plasmoid instability of thin sheets, or the collisionless Hall and pressure-tensor terms in Ohm's law — the measurements support.
Instabilities and turbulence
Ideal MHD instabilities
[Reserved: the interchange or flute mode as the magnetic analogue of the Rayleigh–Taylor instability [Rayleigh:1883]; kink and sausage modes of a current channel and the Kruskal–Schwarzschild analysis [Kruskal:1954]; the Kruskal–Shafranov limit on the safety factor; the energy principle that decides linear stability from the sign of a quadratic form [Bernstein:1958], which is the Hamiltonian stability criterion of Hamiltonian Mechanics in field form.]
Microinstabilities and turbulence
[Reserved: velocity-space free energy driving the two-stream and Buneman instabilities [Buneman:1958] and the drift-wave family; anomalous transport as the practical consequence, since observed confinement times fall far below the collisional predictions of Section 66.3.4; MHD turbulence, its anisotropic cascade and the measured spectra in the solar wind, connecting to the general turbulence discussion of Fluid Dynamics.]
Fusion and confinement
The Lawson criterion
[Reserved: Lawson's power balance between fusion output, bremsstrahlung loss and the energy invested in heating, giving a condition on \(n\tau_{E}\) [Lawson:1957]; the modern triple-product form \(nT\tau_{E}\) and the ignition condition when alpha heating suffices; numerical targets for deuterium–tritium, with \(n\tau_{E}\gtrsim 10^{20}\,/\mathrm{m}^{3}\,\mathrm{s}\) at about \(10\,\mathrm{keV}\); the input cross-sections come from Nuclear Forces and Nuclear Structure, and the bremsstrahlung loss from Radiation and Scattering of Electromagnetic Waves.]
Magnetic confinement
[Reserved: the tokamak and stellarator as the two solutions to the rotational-transform problem posed by Section 66.5.4; confinement scaling laws as empirical relations rather than theory; heating and current drive; the deuterium–tritium campaigns of JET, with \(16\,\mathrm{MW}\) of fusion power and a measured gain below unity [Keilhacker:1999]; what remains between that and a reactor, stated without advocacy.]
Inertial confinement and ignition
[Reserved: compression of a fuel capsule by laser ablation, the \(\rho R\) criterion as the inertial form of Lawson's condition, and the hydrodynamic instabilities that limit compression; the National Ignition Facility result of target gain above unity — fusion yield exceeding the laser energy delivered to the target [AbuShawareb:2024] — with the precise statement of what is and is not included in that ratio, since the distinction between target gain and wall-plug gain is where most public confusion lives.]
Space and astrophysical plasmas
The solar wind
[Reserved: Biermann's inference from the anti-solar direction of comet ion tails that a continuous corpuscular stream must exist [Biermann:1951]; Parker's hydrodynamic solution showing that a hot corona cannot be in hydrostatic equilibrium and must expand supersonically [Parker:1958], with the transonic critical point; the confirmation by Mariner II, which measured a continuous flow at \(400\,\mathrm{km}/\mathrm{s}\) [Neugebauer:1962]; the Parker spiral geometry of the interplanetary field; the quantities to tabulate — speed, density, temperature and field strength at \(1\) astronomical unit.]
The Sun's atmosphere does not stay bound to it. A continuous supersonic stream of ionized gas flows outward past the Earth and beyond. Three independent observations establish it. The ion tails of comets point almost directly away from the Sun whatever the comet's own motion, and respond to solar activity within about a day — which radiation pressure alone cannot produce and a corpuscular stream can [Biermann:1951]. A corona at the observed temperature cannot be in hydrostatic equilibrium against the low pressure of the interstellar medium, and must instead expand, passing through a critical point to supersonic speed [Parker:1958]. And a spacecraft carrying a plasma analyser on the way to Venus measured the flow directly and continuously — never absent over months — at speeds of order \(400\,\mathrm{km}/\mathrm{s}\) [Neugebauer:1962].
The solar wind: derive Parker's transonic solution. Integrate the steady, spherically symmetric momentum equation for an isothermal corona in the Sun's gravitational field, exhibit the critical point at which the flow speed equals the sound speed, show that of the family of solutions only the transonic branch matches both a subsonic base and a vanishing pressure at large distance, and obtain the asymptotic speed. The contrasting hydrostatic solution must be shown to leave a finite pressure at infinity, which is the argument that rules it out.
The magnetosphere
[Reserved: the bow shock, magnetopause and magnetotail as the solution of the solar wind's encounter with a dipole; Dungey convection [Dungey:1961]; the trapped belts [VanAllen:1959]; substorms and the aurora as the visible end of the chain; geomagnetic storms as a measured, dated phenomenon with technological consequences.]
The dynamo
[Reserved: the question posed by Larmor — how a rotating body can maintain a magnetic field against ohmic decay [Larmor:1919]; Cowling's anti-dynamo theorem forbidding a strictly axisymmetric solution [Cowling:1933], which is why the subject is hard; Parker's cyclonic \(\alpha\)-effect and mean-field dynamo [Parker:1955]; the solar cycle and its polarity reversal as the observational anchor, resting on Hale's sunspot magnetograms [Hale:1908]; the geodynamo, its numerical demonstration with a simulated reversal [Glatzmaier:1995], and the decisive laboratory evidence from the Riga liquid-sodium experiment, where a self-sustaining field eigenmode grew from a purely hydrodynamic flow [Gailitis:2000].]