An ionized gas or plasma possesses two obvious general classes of dynamical modes as a consequence of its being a mixture of two electrically coupled gases. In the electron gas the velocity of sound is much higher than in the other constituent, the ion gas, so that if the plasma is deformed sufficiently rapidly, the ion gas will lag behind the electron gas; this will result in separation of charge and in large electrostatic forces that tend to restore the initial relative configuration. The resulting oscillations of this "electric" mode are called plasma oscillations. In the absence of magnetic fields, the oscillations have a fundamental angular frequency
&>,=(4wieTi/m)>*, (1)
where e is the charge of an electron in electro- static units, m is the electron mass in grams, and n is the number of electrons per cm8. The frequency defined in equation (1) is called the plasma frequency, and we might add that in the presence of magnetic fields there are three possible frequencies that are algebraic combina- tions of wp with the electron gyrofrequency Be/me (Westfold, 1949).
If the disturbances initiated in a plasma are slow compared to the plasma frequency, then obviously the electron and ion gases will move together. In this case the plasma is well approximated by a conducting classical fluid (Elsasser, 1954, 1955a). Since the motions of an electrically conducting fluid interact with magnetic fields, and in astronomical phenomena are so generally associated with magnetic fields, this plasma mode may be termed the "magnetic" mode. The study of the dynamics of conducting fluids in the
1 Assisted In part by the Office of Scientific Research and the Oeo- Phystcs Research Directorate, Air Force Cambridge Research Center, Air Research and Development Command, V. 8. Air Force.
'Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, m.
presence of magnetic fields is variously referred to as hydromagnetics, magnetohydrodynamics, and fluid electrodynamics.
Hydromagnetics
The importance of hydromagnetics in astro- physical phenomena may be seen from the fact that much of the region within the galaxy, including stellar atmospheres and interiors, is occupied by gas in various degrees of ionization and readily transports electric charge. Both direct observation and theoretical inference now generally support the belief that much of interstellar space, together with a large fraction of the stars, is permeated by magnetic fields with an energy density comparable to the kinetic energy density of many of the gas motions. Thus, the magnetic field stresses are comparable to the Reynolds stresses. The high conductivity of the matter indicates that it is strongly coupled to the pervading field, so that in treating most fluid motions of astro- physical interest, instead of the hydrodynamic equation,
(2) one must use the hydromagnetic equations,
dv
df (VXB)XB. (4)
Here c represents the conductivity of the fluid and M the permeability in mks units.
The complexity of the nonlinear hydro- magnetic equations indicates that the general problem must be divided into the several basic physical processes; consequently, the study of hydromagnetics involves the investigation of individual effects, one at a time. Instead of a
60 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS V O L . 1
formal general solution of equations (3) and (4), we will ultimately achieve a general under- standing based on quantitative understanding of the many basic concepts (Elsasser, 1955b).
To mention but a few of the physical prin- ciples, let us consider first a fluid of very high conductivity. Then equation (3) reduces to
f=VX(vXB).
(5)From equation (5) it is readily shown that the lines of force move exactly with the fluid.
Formally, if C represents a contour moving with the fluid, and S the area enclosed by the contour, then
(6) The lines of force retain their identity, since any two elements of fluid threaded by the same line of force will remain so connected. Equation (5) is the same as the Helmholtz vorticity equation for an inviscid fluid. Cauchy's in- tegration gives B as an algebraic function of the derivatives of the Lagrangian coordinates of the fluid particles (Brand, 1947).
If the conductivity is not essentially infinite, then of course the lines of force slip relative to the fluid. Besides the slipping, however, they lose their identity by continually changing their pattern of connection. Thus, for instance, a loop in a long tube of flux may detach itself from the tube, giving a magnetic ring and the original, more or less straight tube. This detaching process forms one of the basic steps in the regenerative hydromagnetic dynamo discussed below.
The forces exerted on the fluid by the mag- netic field are derivable from the magnetic stress tensor, BtB}/n, (i, j=l, 2, 3), representing tension along the lines of force, and the isotropic magnetic pressure, B2/2fi (Cowling, 1953), in mks units. Thus the total pressure isp-\-B2/2pi.
Within a region of magnetic field the gas pres- sure p will tend to be less than it is outside by the amount B2/2n. The gas density tends to be less within the field, resulting in the phe- nomenon of magnetic buoyancy (Parker, 1955a).
The tension in the lines of force and the mass of the fluid clinging to them indicate that the
lines are analogous to stretched strings, allowing the propagation of transverse waves with a velocity whose square is equal to the stress density B2\\i divided by the material density (Alfv6n, 1950). These hydromagnetic waves are transverse, and for this reason are generated more copiously by the transverse fluid motions of subsonic hydrodynamics than are the longitudinal acoustical waves. As the mag- netic energy density becomes large compared to the kinetic energy density of the fluid, all hydromagnetic motions reduce to hydromag- netic waves (Parker, 1955b). I t is interesting to note, however, that the presence of the magnetic field may enhance the generation of acoustical waves (Kulsrud, 1955).
The fundamental problem of the convective stability of a rotating conducting fluid in the presence of a field has been solved by Chan- drasekhar (1953b; see also 1952, 1953a, 1954a) with remarkable results. Simple Benard cells do not occur. Instead, competition develops between the rotational and magnetic effects and results in two unstable regimes. One might say, very roughly, that the magnetic field tends to stiffen the fluid, producing stability, and the Coriolis forces tend to pro- duce instability. The results of Chandrasek- har's analytical formulation of this very dif- ficult problem have been given experimental confirmation (Nakagawa, 1955).
Chandrasekhar (1955) has also treated the problem of hydromagnetic turbulence and finds that there are two statistical modes, with either the fluid motions or the magnetic field dominat- ing at large wave numbers. The analysis shows that there is not a close approach to equipartition of energy between the fluid mo- tion and the magnetic field, but, on the other hand, it does indicate that K pv2 and B2/2n will not differ by an order of magnitude.
The origin of the stationary dipole field of earth and that of the migrating solar fields resulting in sunspot formation are beginning to be understood on the basis of hydromagnetic dynamo effects whereby two magnetic fields in a region are each generated from the other by fluid motions. The so-called toroidal field is produced by simple shearing from the poloidal field; the latter is then regenerated by C37clonic
NEW HORIZONS TN ASTRONOMY 61 motions in the fluid, forming loops from the
toroidal field (Parker, 1955c; Elsasser, 1955b).
Presumably a sunspot is formed from the solar toroidal field through a combination of mag- netic buoyancy, hydrostatic equilibrium (Par- ker, 1955a), and the observed cooling of the interior of the spot, though most of the details remain to be treated quantitatively.
Probably nowhere does one find more phe- nomena whose origin so obviously cannot be explained on hydrodynamic grounds than in the violent phenomena of solar activity, such as spicules, active prominences, flares and surge prominences, the ejection of ion clouds, and cosmic ray bursts. The origins of these phenomena are today not at all understood, but one presumes that at least some of them admit of a hydromagnetic origin. The ter- restrial consequences make solar activity a particularly pressing challenge to hydromag- netic research.
Understanding the terrestrial effects is im- portant, but it has often been suggested that solar activity may prove also to be the "little brother" to some of the grander phenomena that result in the Wolf-Rayet stars, flare stars, and Babcock's magnetic variables where prom- inence activity, flares, and sunspots have devel- oped to pathological proportions.
Solar activity apparently ejects field-bearing clouds of gas into interplanetary space. At the earth the clouds are somehow responsible for the aurorae, magnetic storms, certain iono- spheric disturbances, the configuration of the terrestrial weather pattern, and the observed interplanetary gas density of as much as 500 hydrogen atoms per cm3 (Storey, 1953; Sieden- topf, Behr, and Elsasser, 1953; Behr and Siedentopf, 1953). Further, cosmic ray in- tensity variations not directly due to the immediate emission of new particles by the sun are now generally regarded (Neher, 1955) as being due to the antics of interplanetary clouds. It becomes important, therefore, to investigate the dynamics of such clouds. If at the same time the dynamics of ejection from the sun can be expounded, we will be able to check our ideas by both solar and terrestrial observations.
Cosmic rays are presumed to be accelerated by interaction with the magnetic fields carried
by moving matter. The acceleration is pre- sumed to take place in the galaxy and solar flares. One consequence of the high conduc- tivity of most matter is that the electric field in the frame of reference moving with the fluid is very small. Thus, it is readily shown that
E = - Y X B , (7)
where v is the fluid velocity, and there can be no large electrostatic acceleration effects. The general solution of the motion of a charged particle in slowing varying hydromagnetic fields, as well as calculations of several cases of abruptly varying fields, indicate that there are no hydromagnetic accelerating mechanisms other than Fermi's mechanism (Fermi, 1949) and the betatron effect (Swann, 1953; Riddiford and Butler, 1952). Since both mechanisms require that the particles initially have energies of the order of 0.5 Bev per nucleon to avoid the enormous ionization losses at low energies, one presumes that there is some more vigorous mechanism capable of accelerating particles from thermal energies in spite of the tremendous energy losses. Finding such a mechanism is obvious^ of fundamental importance.
The assumption that cosmic rays are accel- erated throughout the galaxy (Morrison, Olbert, and Rossi, 1954; Unsdld, 1955; Parker, 1955b;
Elsasser, 1955a) presents energetic difficulties.
Davis (1955b) has suggested that the gas ejected from the sun forms a cavity in the magnetic field of the galaxy capable of trapping particles in the solar system, though the mechanism apparently cannot account for the cosmic ray cutoff with the attendant correlation with the sunspot cycle. A galactic origin of cosmic rays and the sweeping away of galactic cosmic rays by outgoing magnetic interplanetary clouds seems to give a suitable cutoff, but so little is known about the nature of interplanetary clouds and Davis' cavity that the details of the mech- anism are entirely speculative.
Besides the evidence from the polarization of starlight (Davis and Greenstein, 1951), it has been shown (Chandrasekhar and Fermi, 1953) that the observed properties of the spiral arms of the galaxy require a magnetic flux density of about 10"5 gauss along each spiral arm. A magnetic field of this strength permeating the
381014—56 «
62 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 1
interstellar medium will be important in deter- mining the characteristics of stars formed by gravitational collapse of the interstellar medium, for, although it does not directly affect Jeans' instability criterion (Chandrasekhar, 1954b), the field will strongly influence the nature of the fluid motions against which work the gravita- tional forces of collapse. Hydromagnetics evi- dently plays an important role in the formation, structure, and evolution of galaxies, though little has been done on the integrated problem.
The pulsations of some variable stars suggest (Chandrasekhar and Fermi, 1953; Chandrase- khar and Limber, 1954; Ferraro, 1954) that the general magnetic fields of the individual stars play an important role in determining the form and period of the oscillations of the stellar body. The tremendous general stellar field that is required implies a hydromagnetic dynamo of immense power.
Plasma problem
The extraordinary number of modes of oscil- lation of the electron gas in a plasma is exhibited in Bailey's (1950, 1951) general electromagneto- ionic theory. Starting with a plasma in the presence of a magnetic field, and using Max- well's equations, the equation of continuity, etc., neglecting damping by collisions, and con- sidering small sinusoidal deviations from uni- formity, Bailey finds a number of steady state modes with frequencies that are simple alge- braic combinations of the plasma frequency uP and the gyrofrequency <ae, which is the angular velocity of an electron in the given magnetic field. Besides these usual modes, however, there are modes which grow or decay exponentially by energy transfer to the over-all drift velocity of the ions. The presence of the magnetic field enhances the exponential growth effects. Since the theory neglects the second order terms, one does not know to what extent the exponential growth is limited.
The response of a plasma to electromagnetic waves results in an index of refraction that can be real, imaginary, or complex, and the propa- gation of electromagnetic waves through a non- uniform plasma is complicated and of funda- mental importance (see Pawsey and Smerd, 1953). Consider the solar atmosphere in the
absence of magnetic fields. The plasma fre- quency decreases outward so that escaping radio waves of low frequency can only have originated in the outer corona. Higher fre- quencies may originate farther into the atmos- phere. The observed cutoff frequency at the low end of the frequency spectrum affords an indication of the level of the source in the solar atmosphere; the time rate of change of the cut- off frequency is an indication of the velocity of the source. But, at the same time, the re- fraction of the waves in the solar atmosphere complicates the problem of determining the heliographic position of the source.
The presence of a magnetic field further complicates the propagation, producing, among other things, a tendency of the transverse electromagnetic waves to propagate parallel to the magnetic field, as a consequence of the electrons' circling around the field. This effect appears in the terrestrial phenomena of atmos- pheric whistlers investigated by Storey (1953).
The propagation depends upon the interplane- tary gas density near the earth and so gives information on the interplanetary medium.
The anomalous sources of radio noise in the sun and elsewhere are often associated with high velocity gas or particle streams. The rate of decrease with time of the observed frequency of a given solar burst suggests that the excitation might be due to clouds that are moving out through the solar atmosphere with velocities of the order of 600 km/sec (Pawsey and Smerd, 1953); intense galactic sources have been identi- fied with galaxies in the process of passing through each other (Baade and Minkowski, 1954). The radio emission presumably arises from the collisions of the interstellar gas clouds contained in the interpenetrating galaxies (Greenstein and Minkowski, 1954). The pres- ence of high velocity streaming and the general difficulty with thermal mechanisms (requiring temperatures up to 1O150 K) in the more intense sources suggest the investigation of vigorous macroscopic phenomena such as the exponenti- ally growing plasma oscillation (Bailey, 1950, 1951; Haeff, 1948, 1949; Pierce, 1948; Bohm and Gross, 1949, 1950; Buneman, 1950; Malm- fors, 1950; Blum, Denisse, and Steinberg, 1951;
Twiss, 1951). To establish whether plasma
NEW HORIZONS IN ASTRONOMY 63 oscillations are responsible for the intense
sources, one will have to obtain a detailed knowledge of the streaming which supposedly excites the oscillations. Then one must in- vestigate the effects of the nonlinear terms and the actual boundary conditions found in nature in order to determine to what extent the oscil- lations will grow and whether the electromag- netic radiation generated by the radiations can escape. The question of escape has already been investigated but with conflicting results.
Perhaps the acceleration of cosmic rays is closely associated with the origin of radio noise; both might seem to require, or are observed to have, association with violent fluid motions. Hydromagnetic processes seem unable to furnish the injection energies needed for Fermi's mechanism and for the betatron effect. Thus, obviously, investigations of vio- lent plasma phenomena should be carried out with cosmic rays as well as radio noise in mind.
But in contrast to this point of view is the recent discovery that the visible and radio spectrum of the Crab Nebula can only be ex- plained as synchrotron radiation from electrons with energies of the order of 1 Bev, which by definition are cosmic ray particles (Shklovsky, 1953). The radio emission from the galactic halo seems to require a similar mechanism.
Thus, because we are ignorant of the "basic"
energy source, cosmic rays become the apparent prime mover.
References
ALTVBN, H.
1950. Coemical electrodynamics. Oxford Uni- versity Press.
BAADE, W., AND MINKOWBKI, R.
1964. Astrophys. Journ., vol. 119, pp. 206, 215.
BAILET, V. A.
1950. Phys. Rev., vol. 78, p. 428.
1951. Phys. Rev., vol. 83, p. 439.
BEHB, A., AND SIEDENTOPF, H.
1953. Zeitschr. Astrophys., vol. 32, p. 19.
BLUM, E. J., DENISSE, J. F., AND STEINBERG, J. L.
1951. Comptee Rendus Acad. Sci., Paris, vol. 232, p. 483.
BOHU, D., AND GROSS, E. P.
1949. Phys. Rev., vol. 75, p. 1851.
1950. Phys. Rev., vol. 79, p. 992.
BBAND, L.
1947. Vector and tensor analysis. John Wiley and Sons, New York.
BUNEMAN, O.
1950. Nature, vol. 165, p. 474.
CHANDRASEKHAR, S.
1952. Philos. Mag., vol. 43, pp. 501, 1317.
1953a. Proc. Roy. Soc. London, vol. 216, p. 293;
vol. 217, p. 306.
1953b. Monthly Notices Roy. Astron. Soc. Lon- don, vol. 113, p. 667.
1954a. Philos. Mag., vol. 45, p. 1177.
1954b. Astrophys. Journ., vol. 119, p. 7.
1955. Proc. Roy. Soc. London, vol. 233, p. 330.
CHANDRASEKHAR, S., AND FERMI, E.
1953. Astrophys. Journ., vol. 118, pp. 113, 116.
CHANDRASEKHAR, S., AND LIMBER, D. N.
1954. Astrophys. Journ., vol. 119, p. 10.
COWLING, T. G.
1953. In Kuiper, ed., The sun, p. 532. University of Chicago Press.
DAVIS, L., JR.
1955a. Summer meeting of the American Physical Society, Mexico City, Mexico, August.
1955b. Phys. Rev., vol. 100, p. 1440.
DAVIS, L., JR., AND GREENSTEIN, J. L.
1951. Astrophys. Journ., vol. 114, p. 206.
ELSASSER, W. M.
1954. Phys. Rev., vol. 95, p. 1.
1955a. Amer. Journ. Phys., vol. 23, p. 590; vol.
24, p. 85.
1955b. Summer meeting of the American Physical Society, Mexico City, Mexico, August.
FERMI, E.
1949. Phys. Rev., vol. 75, p. 1169.
FERRARO, V. C. A.
1954. Astrophys. Journ., vol. 119, p. 407.
GREENSTEIN, J. L., AND MINKOWBKI, R.
1954. Astrophys. Journ., vol. 119, p. 238.
HABTT, A. V.
1948. Phys. Rev., vol. 74, p. 1532.
1949. Phys. Rev., vol. 75, p. 1546.
KuLBRUD, R.
1955. Astrophys. Journ., vol. 121, p. 461.
MALMFOBS, K. G.
1950. Ark. Fys., vol. 1, p. 569.
MORRISON, P., OLBERT, 8., AND ROSSI, B.
1954. Phys. Rev., vol. 94, p. 440.
NAKAOAWA, Y.
1955. Nature, vol. 175, p. 417.
NEHER, H. V.
1955. Spring meeting of the American Physical Society, Washington, D. C , Apr. 28-30.
PARKER, E. N.
1955a. Astrophys. Journ., vol. 121, p. 491.
1955b. Phys. Rev., vol. 99, p. 241.
1955c. Astrophys. Journ., vol. 122, p. 293.
PAWSET, J. L., AND SHERD, 8. F.
1953. In Kuiper, ed., The sun, p. 466. Uni- versity of Chicago Press.
PIERCE, J. R.
1948. Bell Syst. Tech. Journ., vol. 28, p. 33.
64 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS RlDDIFOBD, L., AND BtJTLEB, S. T . SWANN, W . F . G.
1952. Phuos. Mag., vol. 43, p. 447. 1933. Phys. Rev., vol. 43, p. 217.
SlBDENTOPF, H . , BBHB, A., AND E L S A S S E B , H . T w i 8 8 R Q
1953. Nature, vol. 171, p. 1066. ^ p h R L g 4 4 4 g
SHKLOVSKY, I. S. A
1953. Dokl. Akad. Nauk S.S.S.R., vol. 90, p. 983. UNSOLD, A.
9TOBET. L. R. O. 1 9 5 5- Zeitschr. Phys., vol. 141, p. 70.
1953. Philos. Trans. Roy. Soc. London, ser. A, WBSTFOLD, K. C.
vol. 246, p. 113. 1949. Australian Journ. Sci. Res., A., vol. 2, p. 169.