Time-dependent theory and Landau damping of the stable oscillations of the kinetic screw pinch (Chandrasekhar, Kaufman and Watson, 1958)

In 1958 Chandrasekhar, Kaufman and Watson derived, from the linearized collisionless Boltzmann and Maxwell equations, the stability of a sharp-boundary cylindrical screw pinch — a current-carrying plasma column with axial magnetic fields inside and outside the plasma and a surrounding conducting wall — with bi-Maxwellian (anisotropic-pressure) ion and electron distributions. They obtained the marginal-stability equation, proved that unstable modes grow without oscillating (no overstability), and computed marginal stability curves; but they left the stable-oscillation branch incomplete. For stable modes the velocity-space integrals defining the plasma response become singular at the resonant particle velocities, and require the Laplace-transform initial-value treatment with analytic continuation introduced by Landau in 1946 for spatially uniform plasma oscillations. No completion of this deferred problem is known in the literature: the only direct follow-up of the 1958 paper in the pinch-stability lineage (Jaggi 1962) redoes the marginal state with the fluid double-adiabatic (Chew–Goldberger–Low) equations, and the adjacent literatures — homogeneous hydromagnetic-wave damping, kinetic damping of surface waves at a plane plasma–vacuum interface, Vlasov-fluid and gyrokinetic screw-pinch analyses (growth side and finite-Larmor-radius effects), and electrostatic cylindrical and dipole problems — treat different models or geometries. The core remains open: the causal initial-value formulation of the 1958 problem, the damped normal-mode spectrum and damping rates of its stable surface modes as functions of wavenumber, azimuthal mode number, pressure anisotropy, and wall position, together with the paper's stated extensions to non-bi-Maxwellian equilibria, weak collisions, and unequal ion/electron temperature ratios.

Background

A pinch is a cylindrical plasma column that carries an axial electric current and is radially constricted by the azimuthal magnetic field that the current itself produces; in the 1950s it was the leading candidate configuration for the magnetic confinement of a fusion plasma. A screw pinch adds an axial magnetic field, produced by external coils, to the current's own azimuthal field, so that field lines spiral about the axis; a surrounding perfectly conducting wall at radius $R_0$ stabilizes long-wavelength perturbations by excluding field disturbances. The classical stability question is whether small perturbations of the equilibrium column grow or oscillate.

Chandrasekhar, Kaufman and Watson (1958) treated this configuration kinetically. Instead of the magnetohydrodynamic (MHD) fluid equations they started from the collisionless Boltzmann equation (also called the Vlasov equation) for the ion and electron distribution functions coupled to Maxwell's equations — appropriate for a hot, effectively collisionless plasma — applying the general perturbation theory for the collisionless Boltzmann equation in a magnetic field that the same three authors had developed the year before (Chandrasekhar, Kaufman and Watson 1957). The equilibrium distribution of each species is taken bi-Maxwellian: Gaussian in the velocity component $q$ parallel to the local magnetic field and in the perpendicular speed $s$, but with different widths, so that the parallel and perpendicular kinetic pressures $p_\parallel$ and $p_\perp$ differ; the anisotropy ratio is $\eta=p_\perp/p_\parallel$ (taken equal for ions and electrons in the case analyzed, which makes a coefficient $\bar R$, multiplying the induced parallel electric field, vanish). The analysis is linear: each perturbation is proportional to $\exp(\Omega t+ikz+im\theta)$, where $z$ is the axial coordinate, $k$ the axial wavenumber, $m$ the azimuthal mode number ($m=0$ is the sausage mode, $|m|=1$ the kink mode), and $\Omega$ a complex growth rate; $\mathrm{Re}\,\Omega>0$ means instability. The resulting characteristic equation, written $\Phi(\Omega)=0$ for the squared growth rate $\Omega^2$, is transcendental: the dispersion function $\Phi$ is built from modified Bessel functions $I_m$ and $K_m$ through combinations the paper denotes $P_m$, $Q_m$ and $G_{m,\zeta}$, with dimensionless parameters $y=kr_0$ (normalized axial wavenumber), $\zeta=R_0/r_0$ (normalized wall radius), $\alpha_P$ and $\alpha_V$ (axial-field strengths inside the plasma and in the vacuum, in units of the azimuthal field $B_\theta$ at the plasma surface), and $\beta=4\pi P_\perp^0/B_\theta^2$ (normalized perpendicular pressure, fixed by radial force balance $2\beta=1+\alpha_V^2-\alpha_P^2$).

The paper's published results concern the growth side and the marginal state: the marginal-stability equation obtained by setting $\Omega^2=0$ (which the abstract notes differs slightly from a criterion derived by M. N. Rosenbluth from an analysis of the particle orbits); a proof, from the analytic properties of the function $\Phi(\Omega)$ of the growth rate, that no mode is overstable (no mode oscillates while growing): when $\Phi(0)>0$ and the supplementary condition $\Gamma^2(0)>0$ holds, $\Phi(\Omega)$ has no zeros in the right half $\Omega$-plane, so that the marginal state indeed bounds the unstable domain; numerical marginal curves for the kink mode $m=-1$, for anisotropies $\eta=1/2$ and $\eta=5$ at $\alpha_V=0$, $0.1$, $0.25$ and $0.5$; and, in an appendix, the homogeneous-plasma plane-wave dispersion relation from which the firehose and mirror instabilities follow (the two pressure-anisotropy instabilities of a uniform magnetized plasma: excess parallel pressure destabilizes transverse bending of field lines, excess perpendicular pressure destabilizes field compression). For unstable modes the velocity-space integrals defining the plasma response are nonsingular. For stable oscillations, $\Omega=\mathrm{i}\omega$ with real frequency $\omega$ (that is, $\Omega^2<0$ in the paper's $\exp(\Omega t)$ convention), the same integrals are singular at the resonant particle velocities $v_\parallel=\pm\omega/k$, where particles keep pace with the wave. Landau (1946), for spatially uniform electrostatic plasma oscillations, showed that the physically correct solution comes from posing the problem as an initial-value problem and transforming in time (a Laplace transform), which prescribes how the velocity integral is to be continued past its resonant singularity; the outcome is that modes that appear neutrally oscillating in the naive treatment are in fact damped (Landau damping), a collisionless damping caused by resonant particles. The kinetic stability theory of the pinch on the stable-oscillation side — the analog, for this bounded, inhomogeneous, electromagnetic surface-mode problem, of Landau's uniform electrostatic analysis — was not carried out in the paper.

Problem Statement

Model, fixed by the source paper. A sharp-boundary cylindrical plasma occupies $r<r_0$ with a uniform axial magnetic field $B_P^0$; the vacuum region $r_0<r<R_0$ contains a uniform axial field $B_V^0$ and the azimuthal field $B_\theta(r_0/r)$ of the axial current; the outer boundary $r=R_0$ is a perfectly conducting wall. The ion and electron equilibrium distributions are bi-Maxwellian with common anisotropy $\eta=p_\perp/p_\parallel$ (so the paper's coefficient $\bar R$ — the moment of the equilibrium distributions that multiplies the induced electric field parallel to the equilibrium magnetic field — vanishes), and the equilibrium satisfies $2\beta=1+\alpha_V^2-\alpha_P^2$ with $\alpha_P=B_P^0/B_\theta$, $\alpha_V=B_V^0/B_\theta$, $\beta=4\pi P_\perp^0/B_\theta^2$ ($P_\perp^0$ the total perpendicular pressure). Perturbations vary as $\exp(\Omega t+ikz+im\theta)$. The linearized collisionless Boltzmann and Maxwell equations reduce to a characteristic equation $\Phi(\Omega)=0$ for $\Omega^2$; the dispersion function $\Phi$ is built from modified Bessel functions $I_m$ and $K_m$ through combinations the paper denotes $P_m$, $Q_m$ and $G_{m,\zeta}$, and contains velocity-space integrals over the parallel velocity $q$ and perpendicular speed $s$ (through the paper's coefficients $\bar S$ and $\bar R$; for the stated equilibrium $\bar S=2(1-\eta)$ and $\bar R=0$) whose denominators $\Omega^2+k^2q^2$ vanish at the resonant velocities. The radial structure of the plasma response is set by the parameter $\Gamma$, defined by the paper through

$$\Gamma^2=\frac{P_\perp^0-P_\parallel^0+\alpha_P^2B_\theta^2/4\pi+\rho^0\Omega^2/k^2}{(\bar S+\bar R)P_\perp^0+\alpha_P^2B_\theta^2/4\pi},$$

with $P_\parallel^0$ the total parallel pressure and $\rho^0$ the equilibrium mass density; stability requires the supplementary condition $\Gamma^2>0$ (when $\Gamma^2<0$ the combination $P_m(\Gamma y)$ acquires poles, which the paper argues implies instability).

Completed portion (published in 1958). The marginal equation $\Phi(0)=0$ with its supplementary inequalities; the proof that when $\Phi(0)>0$ and $\Gamma^2(0)>0$ the function $\Phi(\Omega)$ has no zeros in the right half $\Omega$-plane (no overstability, so the marginal state bounds the unstable domain); the numerical marginal curves (kink mode $m=-1$) for $\eta=1/2$ and $\eta=5$ at $\alpha_V=0$, $0.1$, $0.25$, $0.5$; the homogeneous-plasma plane-wave limit of the appendix.

Open portion (deferred in the paper). (i) Time-dependent theory of the stable oscillations: determine the causal formulation of the initial-value problem for the stable modes and the analytic structure of the continued dispersion function — the disposition of its zeros and of any branch points and poles associated with the resonant velocities $v_\parallel=\pm\omega/k$ and with the vanishing of $\Gamma^2$ on the imaginary axis — and thereby obtain the damped normal-mode spectrum: the complex eigenfrequencies of the stable branch and their damping rates as functions of $(y,m,\eta,\alpha_P,\alpha_V,\zeta)$, including how the discrete surface modes are completed by, or dissolve into, the resonant-particle continuum, and the late-time behavior of an arbitrary initial perturbation. (ii) The stated generalizations, within the same framework: how the spectrum and criteria change for non-bi-Maxwellian equilibrium distributions; for a small number of collisions; and for unequal ion/electron temperature ratios ($\bar R\neq 0$, the induced parallel electric field retained — the case the source most explicitly defers). (iii) The extension of the numerical marginal-curve survey to representative further values of $\eta$ and $\zeta$.

Research target. Determine the causal time-dependent theory of the stable oscillations of the stated equilibrium: the continued dispersion relation together with the analytic structure of the resulting dispersion function (the disposition of its zeros and of the branch points and poles associated with the resonant velocities and with the vanishing of $\Gamma^2$ on the imaginary axis, together with how the discrete surface modes are completed by, or dissolve into, the resonant-particle continuum and the late-time behavior of an initial perturbation); the damped roots for the published cases ($\eta=1/2$ and $5$; $\alpha_V=0$, $0.1$, $0.25$, $0.5$) and for further values of $\eta$ and of the wall position $\zeta$; the marginal limits that reproduce the published marginal curves as the damping tends to zero; and the behavior under the stated extensions — a non-bi-Maxwellian equilibrium, weak collisions, and unequal ion/electron temperature ratios ($\bar R\neq 0$) — within the same framework.

References

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