An effect of an axial current on the propagation of low-frequency axisymmetric magnetohydroclynamic waves in a radially non-uniform plasma column was investigated theoretically and experimentally. It was found that the axial current and the density gradient cause a coupling between the torsional and compressional wave

Vaclavik, J.

Weise, E.

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J. Plasma Physics (1974), vol. 12, part 1, pp. 61-69 61Printed in Great BritainMagnetohydrodynamic waves in a non-uniformcurrent-carrying plasma columnBy J. VACLAVIK AND E. WEISEDepartment of Physics, University of Fribourg, Switzerland(Received 11 February 1974)An effect of an axial current on the propagation of low-frequency axisymmetricmagnetohydrodynamic waves in a radially non-uniform plasma column wasinvestigated theoretically and experimentally. It was found that the axialcurrent and the density gradient cause a coupling between the torsional andcompressional waves.1. IntroductionThe propagation of low-frequency magnetohydrodynamic waves in a non-uniform plasma column has recently been the subject of numerous investigations,both theoretical and experimental. The first theoretical investigation was carriedout by Pneuman (1965). Since then, there have been many others (McPherson &Pridmore-Brown 1966; Cross & Lehane 1968; Davies 1969; Morrow & Brennan1971).Under the usual laboratory conditions, where the gas pressure is negligiblecompared with the magnetic pressure, there are just two waves that can pro-pagate at frequencies below the ion cyclotron frequency; these are usually calledthe torsional (or Alfv6n) wave and the compressional (or fast magneto-acoustic)wave. Experimental observations of the axisymmetric torsional and compres-sional waves, propagating in a radially non-uniform plasma, were reported byCross & Lehane (1968), Morrow & Brennan (1971) and Muller, Rauchle &Schiiller (1973). The non-axisymmetric waves were investigated by McPherson& Pridmore-Brown (1966), who found that the density gradient introduces acoupling between the torsional and compressional waves. As a result, an axialcomponent of the magnetic field perturbation, which is normally associated onlywith the compressional wave, is now carried by the torsional wave as well, andwas in fact detected at frequencies well below the lowest waveguide cut-off of thecompressional wave.In this paper, we investigate an effect of an axial current on the propagationof low-frequency axisymmetric waves in a radially non-uniform plasma column.The plasma is immersed in a uniform axial magnetic field, and the region outsidethe plasma is assumed to be a vacuum. It turns out that, even in the axisym-metric case, the axial current and the density gradient cause a coupling betweenthe torsional and compressional waves. This coupling is similar to that of thenon-axisymmetric waves propagating in a non-uniform plasma with no current(McPherson & Pridmore-Brown 1966).use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms of62 J. Vaclavik and E. WeiseIn § 2 a simple theory of the phenomenon to be investigated is developed,assuming that the plasma is fully ionized and ideally conducting. A descriptionof the experimental apparatus and a presentation of the experimental resultsare given in § 3. Measurements of signal amplitudes and wave speeds are therediscussed in relation to the theory.2. TheoryLet p0, p0, Bo represent the equilibrium values of the mass density, pressureand magnetic field. The linearized equations of ideal single-fluid magneto-hydrodynamics can then be written in the form (Kadomtsev 1966)5 ^ ^ o ] , (1)•^• + (v.V)po + ypodivv = O, (2)flTJ— = curl[v,B0], (3)where v is the plasma velocity and y the adiabaticity index.In what follows, we adopt a cylindrical co-ordinate system r, <f>, z, and assumethat d\d<$> = 0 throughout. For the equilibrium quantities of the plasma, we canthen use the simplified model of pinched plasma column. Thus,(r < R),where R is the radius of the column.For the first-order quantitaties v, p, B, we next introduce the ansatzexp{i(kz — cot)}in (l)-(3). Since the quantities of interest in the measurements are the magneticfield components Br, B^, Bz we eliminate the other variables in favour of them.On making use of (4), (l)-(3) can then be reduced to one equation involving thequantity Br. Moreover, if we assume, in agreement with the experimental data,thatwe obtain the wave equationwhereuse, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms ofWaves on a column 63The quantities Bs and B^ are expressed explicitly in terms of Br asFinally, (5) must be supplemented by boundary conditions. Consistently with(4), we assume that no current flows at the plasma-vacuum interface. Theappropriate boundary conditions are then given by the requirement that Br anddBrjdr are continuous across the plasma-vacuum interface at r = R (Kadomtsev1966) and BT vanishes at infinity.Now, if the equilibrium current were absent, so that B^ = 0, we shouldimmediately obtain two uncoupled waves from (5) and (7):- - -NB+ = 0. (9)These equations describe, respectively, the compressional and torsional waves.The compressional wave will exhibit a cut-off frequency, below which only thetorsional wave should be found.In general, (5) cannot be solved in terms of elementary functions; but, if thefunctions B0^(r) and po(r) are given, it is easy to get solutions by numerical inte-gration. The Br, Bz and B^ profiles, together with the corresponding eigenvaluesof k and the value of the lowest cut-off frequency, were computed for the experi-mentally determined functions B0^(r) and po{r). The results are presented in § 3.From these computations, we conclude that, even in the case when Bo$ 4= 0,(5) possesses a discrete spectrum of eigenvalues of k at all frequencies above alowest cut-off frequency. We continue to associate these eigenvalues with thecompressional wave.We now show that, in addition to a discrete spectrum, (5) possesses a continuousspectrum of eigenvalues,minat all frequencies down to zero. We associate these eigenvalues with the torsionalwave. The existence of modes corresponding to these eigenvalues is allowed bythe singularity N(r = r') = Oof (5). If we suppose that this is a regular singularity,we find that two independent solutions have series expansions of the formBr2 = r-r' + O(\og(r-r')).Thus, we can construct a singular solution which satisfies the boundary conditionsby superposing different linear combinations of these independent solutions touse, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms of64 J. Vaclavik and E. WeiseFIGURE 1. Schematic drawing of the experimental apparatus.the left and to the right of r = r'.f Since we can construct such a solution for eachvalue of k in the range given in (10), we have a continuous spectrum of eigenvalues.At first glance, it might seem that, at a point where D = 0, (5) also has asingularity. However, this singularity is merely apparent. In fact, by means of(7), it can be shown that (5) is equivalent to a system of two first-order equationswith no singularity at D = 0.Thus, we have the following distribution of eigenvalues of k. At a frequencyabove the lowest cut-off frequency, discrete modes and a continuous spectrum ofmodes should propagate, while below this frequency only continuous spectrumshould be found. These modes carry all three components of the magnetic field.3. Experiment3.1. Experimental apparatusThe experimental apparatus consists of a linear pyrex glass tube 110 cm long and9-4 cm dia., coaxial with a pulsed uniform 4-7 kG magnetic field. The tube isfilled with helium at a pressure 0-445 torr. Figure 1 is a schematic drawing of theapparatus. The magnetic field solenoid consists of 8 series-connected 32 cm dia.coils separated by insulating spacers. Switching a 12 kJ capacitor bank throughthe coils produces the magnetic field which has a period 4-4 ms.The plasma is prepared by discharging a 185/iF capacitor bank, charged tof Mathematical details of the matching of singular solutions across a singularity weregiven by Barston (1964).use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms ofWaves on a column 651-5I1•30-5t I I0 0-5 1T/RFIGURE 2. Electron density distribution.8 kV, between the two electrodes, 80/is before the magnetic field reaches its peakvalue. The circuit is critically damped by a resistor. The current pulse rises to77kAin 15/is and is 55/is wide. 60 /is after the plasma preparation was initiated,a current of 30 kA, stationary during 50 /is, is produced by switching on a 3kVunmatched pulse line.The magnetohydrodynamic waves are generated, at peak magnetic field, bydischarging a capacitor through a single-loop antenna positioned in one end ofthe tube. The plane of the loop is perpendicular to the axial magnetic field.A variable number of capacitors, connected to the antenna by a coaxial con-ductor, is used to provide the excitation frequencies over the range 0- 3-1 • 1 Me s"1.A 20 cm dia. brass shield, of suitable ratio of inductance to resistance, is positionedinside the solenoid to minimize the transformer-like coupling between theantenna and the solenoid. For the theoretical computations this shield deter-mines the boundary condition at 'infinity' (i.e. Br vanishes on the shield).The waves are detected by small magnetic probes which enter the plasmaacross a diameter of the discharge tube. The probes consist of three mutuallyperpendicular coils, the output from the two ends being subtracted using adifferential amplifier. The velocity of the waves were determined by comparingthe received signals from the two probes situated at 13 cm and 26 cm from theexciter, respectively. The same probes were also used for the determination of thevalues of the stationary magnetic field components.Electron densities were determined using He-Ne laser interferometer(A = 6328 A), according to the method of Ashby & Jephcott (1963). The measure-ments were performed at different radii along the lines parallel to the axis of thedischarge tube.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms of66 J. Vaclavik and E. Weise1 -1 I I I I0-5rjRFIGURE 3. Profile of the azimuthal component of the stationary magnetic field.3.2. Results and discussionIn order that the experimental results may be compared with the conclusions ofthe theory, it is necessary to determine the radial distribution of the plasmadensity and the profile of the azimuthal component of the stationary magneticfield. The results of the measurements are shown in figures 2 and 3. The solid lines,plotted in these figures, are the polynomial fits used for the computations.The dispersion relation of the excited waves was determined by measuring thevelocity of the waves. For these measurements the probes were positioned on theaxis of the discharge tube, and the axial component Bz of the wave field wasdetected. The phase velocity is plotted as a function of frequency in figure 4.Here the solid line represents the theoretical curve corresponding to the lowestdiscrete mode of the compressional wave, and the dashed lines show the lowerand upper limits of the continuous spectrum of modes of the torsional wave. Fromthe experimental data we conclude that, above the cut-off frequency, the propa-use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms ofWaves on a column 67io2 -5 * .a1010-S 5_L _L_L _L3 4 5 6w(106s-1)FIGUBB 4. Variation of phase velocity with frequency.j i I I I i I0-2 0-80-4 0-6r/RFIGTJBE 5. Variation of phase velocity with radius.gating signal is predominately made of the discrete modes, while below thisfrequency the signal is only carried by the continuous spectrum of modes.A significant property of the torsional wave is that the phase velocity varieswith radius. Figure 5 gives the wave velocity against radius at a frequency ofO 33 Me s"1, together with a graph of the function Gz defined in (5). We see that thewave velocity does vary with radius, though the agreement with the theoreticalvelocity variation is not particularly close.The radial profiles of the magnetic field components of the compressional wavewere measured for the frequency of 0-45 Me s"1 at a distance of 13 cm from the5-2use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms of68 J. Vaclavik and E. Weise0-2 0-80-4 0-6r/RFIGURE 6. Profile of the axial component of the wave magnetic field.0-60-2, 0-8 10-4 0-6r/RPIGTJBE 7. Profile of the radial component of the wave magnetic field.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms ofWaves on a column 690-6 r0-40-20-2r/RFIGURE 8. Profile of the azimuthal component of the wave magnetic field.antenna. The results of these measurements are shown in figures 6-8, togetherwith the theoretical profiles. Since the theoretical cut-off frequency occursabove the experimental one, the theoretical field profiles correspond to thetheoretical frequency at which the wavelength was the same as that measuredat 0-45 Me s-1. We note, in particular, that the compressional mode carries anazimuthal component B$ of the wave field, as was also observed by Miilleret al. (1973).Thus our results show that, in a radially non-uniform plasma column anaxial current and a density gradient cause a coupling between the axisymmetriccompressional and torsional waves.The authors are grateful to Prof. O.Huber and Prof. H.Schneider for theircontinued interest in this work. The work was supported by the Swiss NationalScience Foundation.REFERENCESASHBY, D. E. & JEPHCOTT, D. F. 1963 Appl. Phys. Lett. 3, 13.BABSTON, E. M. 1964 Ann. Phys. (N.Y.) 29, 282.CROSS, R. C. & LEHANE, J. A. 1968 Phys. Fluids, 11, 2621.DAVTES, B. 1969 Phys. Rev. Lett. 22, 1246.KADOMTSEV, B. B. 1966 Reviews of Plasma Physics (ed. M. A. Leontovich), p. 2. NewYork: Consultants Bureau.MCPHERSON, D. A. & PRIDMOBE-BROWN, T>. C. 1966 Phys. Fluids, 9, 2033.MORROW, R. & BRENNAN, M. H. 1971 Plasma Phys. 13, 75.MUXLER, G., RAUCHLE, E. & SCHULLER, P. G. 1973 Plasma Phys. 15, 925.PNEUMAN, G. W. 1965 Phys. Fluids, 8, 507.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800024934Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:13, subject to the Cambridge Core terms of

Magnetohydrodynamic waves in a non-uniform current-carrying plasma column

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