I. Lerche

Enhanced diffusion in strongly magnetized astrophysical plasmas

We demonstrate that the diffusion coefficient for low energy particles, tied to a magnetic field which random walks, may be considerably larger than previously estimated in a strongly magnetized system — like the solar wind or the Galaxy. This is of interest with respect to propagation and lifetime considerations of low energy cosmic rays in the solar wind and the Galaxy.

Initial‐Value Problem for Relativistic Plasma Oscillations

The solution of the collisionless, relativistic Vlasov‐Maxwell set of equations is given for the case where neither ambient electric nor magnetic fields are present and a smeared‐out negative charge background preserves over‐all space‐charge neutrality with a relativistic proton plasma. The method of solution is based upon the eigenfunction expansion method invented by van Kampen. It is shown that, besides the relativistically modified modes of Case and Zelazny, there exists a new discrete set of modes whose phase velocities are greater than the speed of light in vacuo. The solution also exhibits the coupling of electrostatic and electromagnetic modes and the existence of complex phase velocities. This initial‐value problem (or the problem with the electron and proton roles reversed) is of interest because of the existence of cosmic rays, relativistic electrons emitting synchrotron radiation in nonthermal radio sources, solar noise generation, where electron velocities may approach c (Bailey, 1951), and h...

Effect of interstellar density fluctuations on signal dispersion measure

The effect of electron number density fluctuations in the interstellar medium on signals from pulsars is studied in terms of the frequency dependent signal dispersion. It is shown that if the density fluctuations are representative of long wavelength (1∼100 pc) [or large scale length (1∼100 pc)] disturbances in the interstellar gas, then the observed signal dispersion is not a measure of the integral of the electron number density in the line of sight. Evidence has been presented elsewhere for believing that such long wavelength disturbances should exist in the interstellar gas, so this result indicated that some care must be exercised in the interpretation of signal dispersion measurements from pulsars.

The effect of turbulent fluctuations in the interstellar gas and magnetic field on Faraday rotation

The effect of fluctuations in both the interstellar electron number density and galactic magnetic field on the propagation of high frequency radio waves is discussed in terms of the frequency dependent Faraday rotation. It is shown that when the fluctuations are representative of large scale disturbances (1–102 pc) in the interstellar medium, then the observed Faraday rotation is not a measure of the line of sight integral of the product of the magnetic field with the electron number density.Since evidence has been presented elsewhere for believing that such large scale disturbances do exist in our galaxy, some care must be exercised in the physical interpretation of Faraday rotation measurements.

Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. II. Incompressible, helical velocity turbulence and a pair of coupled, singular, nonlinear integral equations

Using Kraichnans direct interaction approximation, we set up the equations governing the normal modes of the ensemble average magnetic field under incompressible, nonmirror symmetric velocity turbulence. We show that (i) the Greens stress tensor enjoys equipartition of its symmetric and antisymmetric parts at the normal mode frequencies of the ensemble average field, (ii) for static velocity turbulence, including helicity, there are no growing modes, (iii) the commonly used first order smoothing theory approximation is invalid when compared to the Kraichnan equations, for the Kraichnan equations do not satisfy Hammersteins theorem while first order smoothing theory requires the satisfaction of Hammersteins theorem, (iv) if there is to be any growth of the ensemble average magnetic field it must come from time dependent velocity turbulence, and when the velocity turbulence is time dependent we have so far been unable to solve the Kraichnan equations. We have done these calculations for two reasons. First to illustrate, by exact solution, the manner in which the normal modes of the ensemble average magnetic field depend on the helicity and Reynolds number of the turbulent velocity field. Second to show that approximate treatments of the hydromagnetic equation (like first order smoothing theory), rather than exact solution, are liable to give rise to substantial error in view of the fact that the Kraichnan equations do not satisfy Hammersteins theorem.

Random function theory revisited: Exact solutions versus the first order smoothing conjecture

We remark again that the mathematical conjecture known as first order smoothing or the quasilinear approximation does not give the correct dependence on correlation length (time) in many cases, although it gives the correct limit as the correlation length (time) goes to zero. In this sense, then, the method is unreliable.

ON THE PROPAGATION OF MAGNETIC DISTURBANCES IN THE SOLAR WIND

Under the geometrical optics approximation we discuss the propagation of a polarized magnetic profile, made up of Alfvén waves, in the solar wind. We show that (i) the profile propagates at an angle to the radial direction (the direction of the solar wind flow), (ii) the radial half-width of the profile stays essentially constant, or even diminishes a little, with distance from the Sun, (iii) the half-width in a direction transverse to the radial direction increases without limit as the magnetic profile moves outward from the Sun. Thus the profile stretches out into a ‘ribbon’ which could, of course, be experimentally identified as a discontinuity. We also give equations for the variation of polarization of the profile, and illustrate the behavior of polarization in a simple case.We have done these calculations to show that the production of ‘discontinuities’ in the solar wind can arise from propagation effects on irregularly shaped ‘blobs’ of magnetic field, as well as from other causes.

Concerning a criterion for the validity of the first order smoothing approximation

This paper presents some statistically exact variational principles for problems in random function theory for the purpose of obtaining a criterion which will test the validity of an approximate method known variously in the many different fields of its application as first order smoothing theory, first order cumulant discard, quasilinear theory, or the adiabatic approximation. The hydromagnetic dynamo equations are used here, as particular mathematical instances of general mathematical points. The calculations show that when the random equation under investigation is self‐adjoint, and when the quasilinear approximation to it is also self‐adjoint, then the exact and approximate solutions will, almost surely, agree. When either (or both) the true equation or the quasilinear approximate equation is not self‐adjoint, then the exact and approximate solutions will, almost surely, disagree.

On the application of Cramér's theorem to axisymmetric, incompressible turbulence

In the theory of homogeneous, stationary, axisymmetric, incompressible velocity turbulence there arise four scalar functions. The incompressibility condition provides two relations between these four functions.We will demonstrate here that application of Cramérs theorem imposes two additional constraints on the four functions. These constraints do not uniquely define the allowed functional form but they do provide very powerful criteria for limiting the class of functions which are permitted. In view of the growing use of velocity turbulence in kinematic dynamo theory and its importance in astrophysical situations (e.g., Earth, Sun, Galaxy) to maintain or regenerate a large scale magnetic field, we believe that the present constraints are of more than academic interest. In particular, application of the constraints to a form of velocity turbulence used by Steenbeck, Krause and Rädler when computing kinematic dynamo action, shows that their assumed turbulence is not physically realizable in nature.

Some Exact Statistical Properties of Turbulent Kinematic Dynamo Equations

The exact statistical properties of solutions to two restricted turbulent kinematic dynamo problems are given and discussed in some detail. In view of the fact that all discussion and solutions of the turbulent kinematic dynamo equations given so far in the literature are approximate, we believe that the present paper, containing two exactly soluble turbulent dynamo problems, is of more than academic interest. The method of solution is rather general and may, perhaps, be of basic interest. Further, the exact statistical solutions, which admit of regenerative dynamo action, allow approximate solutions to be compared and contrasted with the exact solutions, thus outlining the regime of applicability of the approximate solutions.