Contains reports on four research projects

East, D. A.

Lewis, A. T.

Mawardi, O. K.

Stekly, Z. J. J.

Woodson, H. H.

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III. PLASMA DYNAMICSProf. S. C. Brown Prof. H. H. Woodson A. T. LewisProf. O. K. Mawardi D. A. East Z. J. J. SteklyA. HYDROMAGNETIC WAVEGUIDEThe purpose of this investigation is to study the dynamics of wave propagation in ahydromagnetic waveguide. This waveguide consists of an electrically conducting conduitinserted in the field of a steady magnetic field, and it is assumed to be filled with plasma.Suppose, for the simplest case, that the applied field is parallel to the axis of the tube.When the plasma moves with a fluctuating velocity in a direction normal to the axis of thewaveguide, the lines of force are shaken to and fro in the direction of the applied velocity.A transverse wave is thereby made to travel along the lines of force. More complexsituations will also be discussed.It is well known to workers in hydromagnetics that the governing relations for themotion of a plasma in a magnetic field are analogous to those describing the behavior ofan ideally conducting fluid in the presence of a magnetic field. Hence, our discussionbegins with the relations for the conservation of momentum and matter and with the equa-tion of state. As a result of linearization, we find that these equations arepa = -Vp + jX (1)o at oap (2)+ p V* v= 02 (3)p=a pIn these expressions, zero subscripts indicate quiescent values, and lower-caseletters fluctuating variables. The velocity is denoted by V, the pressure by p, the den-sity by p, the velocity of sound waves in free space by a, the fluctuating local currentdensity by j, and the applied steady magnetic field by B o .The set of corresponding Maxwellian relations, corrected for relativistic effects, isVX = - at (4)VX b = 1j (5)e+vX B =0 (6)The fluctuating magnetic field is denoted by b, the electric field by e, and the permeabil-ity of the medium by t. Relations 4, 5, and 6 are valid when the plasma is quasineutral,and when the characteristic dimension of the apparatus is large compared with both the(III. PLASMA DYNAMICS)mean free path of the gas and the Debye shielding distance.It can be readily found that the velocity satisfies the equation2- 2 -k2 + P V(V. v) + [V X V X (-X ib)] X ib = 0 (7)in which k = w/c, the wave number for the Alfv6n wave velocity in free space. Thisvelocity c equals Bo/(p)/2. The parameter p = a/c is the ratio of the two velocitiesof wave propagation, and ib indicates a unit vector in the direction of the magnetic field.Similar relations for the other field variables can be obtained by manipulating the set ofEqs. 1-6. The appropriate boundary conditions for the problem require the vanishingof the normal components of the oscillating velocity and the magnetic field at the wallsof the waveguide. If we define the velocity by the identityv = V + V X M (8)it can be shown by substituting Eq. 8 in Eq. 7 that we obtain two simultaneous equationsin the velocity potentials c and M. These equations for the general case, when ib is atan arbitrary angle with the axis of the waveguide, are quite complicated and have to besolved approximately. Two extreme cases, however, allow the equations to be solvedexactly. The results for these two cases will now be briefly indicated.Case 1. The magnetic field is aligned with the axis of the waveguide. Then the twowave modes propagate along the axis of the waveguide. One of these modes displays thecharacter of a longitudinal, or compressive, wave and is called, in this report, the"acoustic wave." The other mode is of transverse character and represents the hydro-magnetic mode. Several interesting alternatives may occur that depend upon whetherp is less than or greater than unity.When P is less than unity, the acoustic mode has no cutoff for all orders of the waveeigen numbers. This is quite different from the conventional acoustic wave propagationthat takes place in a pipe. The hydromagnetic mode does, however, have a cutoff thatdepends upon the order of the eigen number. It can be checked that whenever p < 1, thepressure from collisions, p, is considerably lower than the hydromagnetic pressureBo/2L. This means, of course, that the collective behavior of the electrons is controlled,in large part, by the electromagnetic forces. When P > 1, we find that the hydromagneticmode is then the mode that suffers no cutoff for all orders of the wave eigen number. Theacoustic mode, on the other hand, has a cutoff frequency that depends on the order of thewave eigen number. The case of p > 1 indicates that the density of the plasma is high,and is probably more representative of the density of a liquid metal than of the densityof a plasma.An interpretation of the reversal of the noncutoff property of the two waves forp 1 can be given by visualizing the behavior of the plasma as it is squeezed by the(III. PLASMA DYNAMICS)lines of forces during their transverse motion. For stronger magnetic forces with P < 1,a side distortion of the lines is always accompanied by a longitudinal forward motion ofthe plasma, hence the acoustic wave suffers no cutoff. A similar explanation can begiven for the behavior with P > 1.The expression for the component of the velocity transverse to the axis of the wave-guide is given byv = (1+ K) eiKI y (n + 1)  k n + 1)rx (2m + 1)Tz (9)x 2L cos 2L 2L1 1 22 iKy (2m + 1)r sin (2n + 1) rx (2m + l)z (10)vz = (1 + K ) e 2L sin -L I 2L (10)z 2L 2L 2L 2in which n, m = 0, ±1, ±2 ... ; and 2L 1 2L2 are the width and height of the waveguidesection. In expressions 9 and 10, K is the propagation constant for the waves. The func-tional relation of K on k, the wave number, is shown in graphical form in Fig. III-1 forp 1 and p = 1.It is obvious that for p = 1, it is not possible to identify the particular wave associatedACOUSTIC MODE9- 98 8 HYDROMAGNETICH B> MODE7 76 6- _ HYDOROMAGNETIC 5MODE4 43 3 -ACOUSTIC MODE2 2 -n2 (I +2) 1+'90 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 12(k LL , )2  (k LL2 )2(a) (b)Fig. III-1. Dependence of the propagation constant on the frequency.(a) The cutoff for the hydromagnetic mode is given as afunction of2(2n + 1) 2 L (2m + 1 ) 2 L2 n 2 m2+mn 4 L L z(b) The cutoff for the acoustic mode is given as a function ofS(2n + 1) L (2m + 1) Z L2mn Lmn = 4\ L 1 L 2(III. PLASMA DYNAMICS)with the two branches of the function K = f(k).Case 2. The magnetic field direction is at right angles to the axis of the waveguide.In this case, the analysis shows that no hydromagnetic wave propagates along the axisof the waveguide. Indeed, consideration of this situation leads us to conclude that thehydromagnetic wave appears as a standing wave along the lines of force, and hence itis trapped between the walls of the waveguide.The alignment of the magnetic field in another direction besides the two that havebeen mentioned gives rise to intermediate situations which, however, cannot be obtainedas a superposition of the two waves indicated in Eqs. 1 and 2 because Eq. 7 is not linearin the vector ib.The analysis that has been given cannot be extended to frequencies higher than the ioncyclotron frequency, without taking into account the necessary correction, because theplasma is now composed of two fluids interacting with the magnetic field. This correctionis easily made, and it can be shown that the symmetry of the eigenfunctions in the positiveand negative values is lost.Work is now being done to check these speculations experimentally.O. K. MawardiB. HYDROMAGNETIC INSTABILITIESA series of investigations has been started to study the onset and the growth rateof the instability in a cylinder of plasma under different conditions. In the preliminaryexperimental phase, the work will be done on liquid metals.The situation that has just been investigated is concerned with an instability similarto that produced by the "pinch effect." Here, however, the electromagnetic forces areLIQUIDMETALFig. 11-2. Schematic representationof apparatus used to studyinstabilities.ACCOILDCCOILcaused by the interaction of an applied steady magnetic field parallel and external to acolumn of fluid with azimuthal currents induced in the column. The column with freesurfaces is simulated by allowing liquid metal to fall unimpeded from a reservoir. The(III. PLASMA DYNAMICS)forced current is induced by a coil that is coaxial to the column, as illustrated in Fig. III-2.The theoretical prediction for the dynamical behavior of the column is being studiedby means of a normal-mode technique.O. K. Mawardi, D. A. EastC. STUDIES OF MAGNETOHYDRODYNAMIC SHOCKSThe work reported here was started for the purpose of investigating the dynamicsand the structure of hydromagnetic shocks. In particular, the parameters of the shockthat have to be estimated are its thickness, pressure ratio, magnetic-field ratio, andthe corresponding density ratio. The preliminary theoretical work was carried out onthe basis of a continuum theory.The calculations follow conventional techniques for studies on shock waves, i. e.,the discussion begins with the equation for the conservation of momentum and mass. Anappropriate equation of state is also introduced. The hydromagnetic interaction is takeninto account by means of a well-known relation for the magnetic field,a VX (v X H) V H (1)where 11 is the intensity of the magnetic field, - is the velocity, N is the permeability,and o- is the conductivity.It can be shown that a one-dimensional dependence for the variables leads to expres-sions 2 and 3 which relate the value of the upstream parameters of the shock to its down-stream parameters. The relations are valid for distances that are large compared withthe thickness of the shock. Manipulation of all of the equations mentioned in Sec. III-Aleads to a pair of simultaneous expressions, the first of which isS1 + I + M (2)2 1 02The symbolo,1= 0,1 p, 1stands for the ratio of the square of the Alfvn velocity to the square of the velocity ofsound; Mo, 1 is the appropriate hydromagnetic Mach number, defined as the ratio of thelocal velocity to the Alfvn velocity.For the second relation, we have(III. PLASMA DYNAMICS)2 (y-1) 2 21 1+ 2 1 1+2p + _ o + (3)Equations 2 and 3 are sufficient to define completely the state of the gas downstreamof the shock.The experimental verification of this discussion will be carried out by means of anapparatus that will allow a magnetically driven shock to travel in an externally applieduniform magnetic field. This apparatus is now being constructed.O. K. Mawardi, Z. J. J. SteklyD. CONSTRUCTION AND INSTRUMENTATION OF A T TUBE FOR MAGNETICALLYDRIVEN SHOCKSA T tube with a backstrap for magnetic driving of a shock has been constructed andis in operation. The tube is of conventional design, as illustrated in Fig. 111-3. Theimmediate purpose of this tube is to reproduce results of earlier workers (1, 2, 3) in thisfield so that instrumentation can be checked and perfected.O-RING SEALS ALUMINUM CYLINDRICALELECTRODES PYREX GLASSALUMINUMBACKSTRAPI" WIDETO AC MI" DIAMETER SYSTEMPOLYETHYLENEINSULATIONTRANSMISSIONLINEFig. 111-3. T tube construction.The transmission line passes through a triggered spark-gap switch to a 2.5- pfdcapacitor bank. A 5-kv power supply is being used, but a 35-kv power supply isavailable and it will be used to reach voltages higher than 5 kv. Tank hydrogen atpressures in the range 1-2 mm Hg is used in the tube. The current waveform hasbeen observed, both conductively and inductively, and it is characterized by an under-damped oscillation, the frequency of which is 250 kcps, and of damping in which theoscillations decrease by a factor of 3/5 per cycle. The current amplitude is meas-ured by a calibrated shunt.(III. PLASMA DYNAMICS)Photomultiplier circuitry is being assembled so that three tubes will be available formeasuring the average velocity of the luminous front. Data obtained from a single 1P28photomultiplier tube, located 10 cm from the spark, show an abrupt rise in light intensity3 psec after initial breakdown of the gap when the hydrogen pressure is 0.7 mm Hg, andthe voltage is 5000 volts. The observed waveform is shown in Fig. 111-4. The initial peakis caused by stray pickup of the capacitor discharge. The peak at approximately 15 1psecmay arise from a shock reflected from the wall just in front of the backstrap. These areonly preliminary results; however, the sudden increase in light intensity at 3 iLsecindicates that a fairly strong shock has formed in the gas and is propagating at anaverage speed of approximately 3 cm/psec.SWEEP: 5FL SEC/CMSWEEP: 2,/SEC/CMFig. 111-4. Oscillogram showing the response of a 1PZ8 photomultipliertube at 10 cm from the discharge. A 2. 5-Rfd capacitor bankat 5000 volts is discharged into hydrogen gas at a pressure of0.7 mm Hg. The oscilloscope is triggered by the time rate ofchange of the arc current.The picture taken with a slow sweep shows a light output of approximately 40 Rsecduration. Photoexcitation of the gas by the light from the initial arc and excitationfrom incident and reflected shock waves offer the most logical explanation for this.The relaxation time for the gas that is excited in this manner is of the correctorder of magnitude for explaining the observed phenomenon. An instrumentationapparatus has been perfected for measuring the properties of that part of the gaswith only ordered motion.The pressure in the tube before the discharge of the capacitor bank is measured bya thermocouple gauge at pressures lower than 1 mm Hg, and a mercury-filled manometeris used at higher pressures. Unfortunately, both of these devices are inaccurate in thevicinity of 1 mm Hg pressure; consequently, improved pressure-measuring apparatuswill have to be assembled.Some difficulty was experienced, initially, in photographing the single-shot sweeps- -C- -- ~F - --- ----- ---- ~----(III. PLASMA DYNAMICS)obtained from the experimental apparatus. At present, Polaroid Type 44 film (Polapan 400)produces adequate pictures down to a sweep rate of 1 cm/[sec with a P-ll tube phosphorand with the 10, 000-volt accelerating potential of the oscilloscope (Tektronix Model 541with a 53/54-k plug-in amplifier).A. T. Lewis, H. H. WoodsonReferences1. R. G. Fowler, J. S. Goldstein, and Beryl E. Clotfelter, Luminous fronts in pulsedgas discharges, Phys. Rev. 82, 879 (1951).2. R. G. Fowler, W. R. Atkinson, and L. W. Marks, Ion concentrations and recom-binations in expanding low pressure sparks, Phys. Rev. 87, 966 (1952).3. A. C. Kolb, Production of high-energy plasmas by magnetically driven shock waves,Phys. Rev. 107, 345 (1957).

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