Russell M. Kulsrud

The Protogalactic origin for cosmic magnetic fields

It is demonstrated that strong magnetic fields are produced from a zero initial magnetic field during the pregalactic era, when the galaxy is first forming. Their development proceeds in three phases. In the first phase, weak magnetic fields are created by the Biermann battery mechanism. During the second phase, results from a numerical simulation make it appear likely that homogenous isotropic Kolmogorov turbulence develops that is associated with gravitational structure formation of galaxies. Assuming that this turbulence is real, then these weak magnetic fields will be amplified to strong magnetic fields by this Kolmogorov turbulence. During this second phase, the magnetic fields reach saturation with the turbulent power, but they are coherent only on the scale of the smallest eddy. During the third phase, which follows this saturation, it is expected that the magnetic field strength will increase to equipartition with the turbulent energy and that the coherence length of the magnetic fields will increase to the scale of the largest turbulent eddy, comparable to the scale of the entire galaxy. The resulting magnetic field represents a galactic magnetic field of primordial origin. No further dynamo action after the galaxy forms is necessary to explain the origin of magnetic fields. However, the magnetic field will certainly be altered by dynamo action once the galaxy and the galactic disk have formed. It is first shown by direct numerical simulations that thermoelectric currents associated with the Biermann battery build the field up from zero to 10-21 G in the regions about to collapse into galaxies, by z ~ 3. For weak fields, in the absence of dissipation, the cyclotron frequency -ωcyc = eB/mH c and ω/(1 + χ), where ∇ × v is the vorticity and χ is the degree of ionization, satisfy the same equations, and initial conditions ωcyc = ω = 0, so that, globally, -ωcyc(r, t) = ω(r, t)/(1 + χ). The vorticity grows rapidly after caustics (extreme nonlinearities) develop in the cosmic fluid. At this time, it is made plausible that turbulence has developed into Kolmogorov turbulence. Numerical simulations do not yet have the resolution to demonstrate that, during the second phase, the magnetic fields are amplified by the dynamo action of the turbulence. Instead, an analytic theory of the turbulent amplification of magnetic fields is employed to explore this phase of the magnetic field development. From this theory, it is shown that, assuming the turbulence is really Kolmogorov turbulence, the dynamo action of this protogalactic turbulence is able to amplify the magnetic fields by such a large factor during the collapse of the protogalaxy that the power into the magnetic field must reach saturation with the turbulent power. For the third phase, there is as yet no analytic theory capable of describing this phase. However, preliminary turbulence calculations currently in progress seem to confirm that the magnetic fields may proceed to equipartition with the turbulent energy, and that the coherence length may increase to the largest scales. Simple physical arguments are presented that show that this may be the case. Such an equipartition field is actually too strong to allow immediate collapse to a disk. Possible ways around this difficulty are discussed.

Equilibrium of a Magnetically Confined Plasma in a Toroid

A variety of properties are derived satisfied by any static equilibrium of a plasma governed by the well‐known magnetostatic equations. Some of these are local and quite trivial. Others involve integrals over surfaces of constant pressure, which are shown to be topologically toroidal under fairly general assumptions.A variational principle for such equilibria is derived. One of its consequences is to provide a characterization of equilibria by their values of certain invariants.Finally, conditions are obtained additional to the magnetostatic equations appropriate to the steady state of a plasma slowly diffusing across a magnetic field out of a topologically toroidal region.

The spectrum of random magnetic fields in the mean field dynamo theory of the Galactic magnetic field

The fluctuation spectrum that must arise in a mean field dynamo generation of galactic fields if the initial field is weak is considered. A kinetic equation for its evolution is derived and solved. The spectrum evolves by transfer of energy from one magnetic mode to another by interaction with turbulent velocity modes. This kinetic equation is valid in the limit that the rate of evolution of the magnetic modes is slower than the reciprocal decorrelation time of the turbulent modes. This turns out to be the case by a factor greater than 3. Most of the fluctuation energy concentrates on small scales, shorter than the hydrodynamic turbulent scales. The fluctuation energy builds up to equipartition with the turbulent energy in times that are short compared to the e-folding time of the mean field. The turbulence becomes strongly modified before the dynamo amplification starts. Thus, the kinematic assumption of the mean dynamo theory is invalid. Thus, the galactic field must have a primordial origin, although it may subsequently be modified by dynamo action.

On the origin of cosmic magnetic fields

We review the extensive and controversial literature concerning how the cosmic magnetic fields pervading nearly all galaxies and clusters of galaxies actually got started. Some observational evidence supports a hypothesis that the field is already moderately strong at the beginning of the life of a galaxy and its disc. One argument involves the chemical abundance of the light elements Be and B, while a second one is based on the detection of strong magnetic fields in very young high red shift galaxies.Since this problem of initial amplification of cosmic magnetic fields involves important plasma problems it is obvious that one must know the plasma in which the amplification occurs. Most of this review is devoted to this basic problem and for this it is necessary to devote ourselves to reviewing studies that take place in environments in which the plasma properties are most clearly understood. For this reason the authors have chosen to restrict themselves almost completely to studies of dynamos in our Galaxy. It is true that one can get a much better idea of the grand scope of galactic fields in extragalactic systems. However, most mature galaxies share the same dilemma as ours of overcoming important plasma problems. Since the authors are both trained in plasma physics we may be biased in pursuing this approach, but we feel it is justified by the above argument. In addition we feel we can produce a better review by staying close to that which we know best.In addition we have chosen not to consider the saturation problem of the galactic magnetic field since if the original dynamo amplification fails the saturation question does not arise.It is generally accepted that seed fields, whose strength is of order 10−20 G, easily spring up in the era preceding galaxy formation. Several mechanisms have been proposed to amplify these seed magnetic fields to a coherent structure with the microgauss strengths of the currently observed galactic magnetic fields.The standard and most popular mechanism is the α–Ω mean field dynamo theory developed by a number of people in the late sixties. This theory and its application to galactic magnetic fields is discussed in considerable detail in this review. We point out certain difficulties with this theory that make it seem unlikely that this is the whole story. The main difficulty with this as the only such amplification mechanism is rooted in the fact that, on galactic scales, flux is constant and is frozen in the interstellar medium. This implies that flux must be removed from the galactic discs, as is well recognized by the standard theory.For our Galaxy this turns out to be a major problem, since unless the flux and the interstellar mass are somehow separated, some interstellar mass must also be removed from the deep galactic gravitational well. This is very difficult. It is pointed out that unless the field has a substantial field strength, much larger than that of the seed fields, this separation can hardly happen. And of course, it must if the α–Ω dynamo is to start from the ultra weak seed field. (It is our philosophy, expressed in this review, that if an origin theory is unable to create the magnetic field in our Galaxy it is essentially incomplete.)Thus, it is more reasonable for the first and largest amplification to occur before the Galaxy forms, and the matter embedded in the field is gravitationally trapped. Two such mechanisms are discussed for such a pregalactic origin; (1) they are generated in the turbulence of the protogalaxy and (2) the fields come from giant radio jets. Several arguments against a primordial origin are also discussed, as are ways around them.Our conclusion as to the most likely origin of cosmic magnetic fields is that they are first produced at moderate field strengths by primordial mechanisms and then changed and their strength increased to their present value and structure by a galactic disc dynamo. The primordial mechanisms have not yet been seriously developed, and this preliminary amplification of the magnetic fields is still very open. If a convincing case can be made that these primordial mechanisms are necessary, more effort will of course be devoted to their study.

Forced magnetic reconnection

By studying a simple model problem, the time evolution of magnetic field islands which are induced by perturbing the boundary surrounding an incompressible plasma with a resonant surface inside is examined. The reconnection and island formation process for sufficiently small boundary perturbations occurs on the tearing mode time scale defined by Furth, Killeen, and Rosenbluth [Phys. Fluids 6, 459 (1963)]. For larger perturbations the time scale is that defined by Rutherford [Phys. Fluids 16, 1903 (1973)]. The resulting asymptotic equilibrium is such that surface currents in the resonant region vanish. A detailed analytical picture of this reconnection process is presented.

Electromagnetic Fluctuations during Fast Reconnection in a Laboratory Plasma

Experimental evidence for a positive correlation is established between the magnitude of electromagnetic fluctuations up to the lower-hybrid frequency range and enhancement of reconnection rates in a well-controlled laboratory plasma. The fluctuations belong to the right-hand polarized whistler wave branch, propagating obliquely to the reconnecting magnetic field, with a phase velocity comparable to the relative drift velocity between electrons and ions. The measured short coherence lengths indicate their strongly nonlinear nature.

Nonlinear evolution of parallel‐propagating hydromagnetic waves

The nonlinear evolution of plane hydromagnetic fluctuations propagating along the unperturbed magnetic field direction is considered. From an expansion of the ideal magnetohydrodynamic equations and the hydromagnetic shock jump conditions, it is shown that a wave in which the magnitude of the magnetic field is nonconstant steepens into a shock and subsequently evolves toward a purely Alfvenic fluctuations of lower mean energy density. Explicit expressions are derived for the asymptotic state and for the characteristic lines which describe the evolution toward that state. A class of fluctuations which includes linearly polarized waves is shown to evolve into rotational discontinuities. The results are applied to observations of hydromagnetic fluctuations in the solar wind.

Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies

We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., beta>Re^{1/2}. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10^{-18} G to dynamically important strengths of ~10 microG on cosmologically trivial time scales (~10^8 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., β Re1/2. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10-18 G to dynamically important strengths of ~10 μG on cosmologically trivial timescales (~108 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.

Some Stable Hydromagnetic Equilibria

Hydromagnetic equilibria are obtained for a variety of situations which differ little from that of a zero pressure uniform axial magnetic field. Criteria for ascertaining the stability of these equilibria are found by means of an energy principle. In particular, if helically invariant fields are present, stable equilibria with nonzero pressure and net axial current can be found.

Experimental study of two-fluid effects on magnetic reconnection in a laboratory plasma with variable collisionality

This article describes the recent findings on two-fluid effects on magnetic reconnection in plasmas with variable collisionality in the magnetic reconnection experiment (MRX) [M. Yamada et al., Phys. Plasmas 4, 1936 (1997)]. The MRX device has been upgraded to accommodate a variety of reconnection operation modes and high energy density experiments by increasing its capacitor bank energy and extending the discharge duration. As our experimental operation regime has moved from the collisional to the collision-free, two-fluid effects have become more evident. It is observed that the two-dimensional profile of the neutral sheet is changed significantly from the rectangular shape of the familiar Sweet-Parker type to a double wedge shape as the collisionality is reduced and the reconnection rate increases. The recent evolution of our experimental research from the magnetohydrodynamics (MHD) to the two-fluid analysis is presented to illuminate the physics of Hall MHD in a collision-free reconnection layer. In p...