Nernst-Wienecke equilibrium in linear magnetized plasmas

BOYE AHLBORN Dc~por~irrrerri yf'Nr~cleor. Errgirreerirrg, Ut~i'rr..siiy c(f'Wri.shit~,qior~. Srtriilc. WA 98/95, U . S . A

Received March 7. 1979

A cylindrical hydrogen plasma imbedded in a strong axial magnetic field, in which a tempera- ture maximum is maintained by axial heating and r i~di i~l heat transfer. has a pressure maximum and a density minimum on axis. The pressure in dynlcm' and density in cm-' can be approxi- mated asp r 1.8 x 10-'BTHJ and r r , r 6.4 x 101'BT-'I4 (B in gauss and T i n kelvin).

Un plasma cylindrique d'hydrogkne soumis i~ un champ magnitique axial intense, dans lequel un maximum d e temperature est maintenu p;l~.chauff>~ge axial e t trnnsfert d e chaleu~. ~.adial, a un maximum d c pression et un minimum d e densite \ur I'axe. La pression en dynlcm2 et la densite en cm-' peuvent 6t1.e exprimees app~.oxim;ltivement pal./) 1 1.8 x 10-3BP'4 et t r , 2 6.4 x I O I Z B T ' l J ( B e n gauss et 7'cn kelvin).

( T ~ x d u i t pal. le journal] Can. J . Phys . .57 . l O Y O ( I Y 7 Y )

Cylindrical plasmas in strong axial magnetic fields are encountered in many plasma physics ex- periments such as arcs, laser sparks, and passive liners. Radial diffusion in such devices can be driven either by temperature gradients (ther- modiffusion) or by gradients of the number density, which are particularly strong in regions where ioni- zation takes place (ambipolar diffusion). This radial diffusion in the presence of a strong magnetic field B, causes azimuthal CUI-rents j+, as explained later. The Lorentz force B, x j+ points in the - r direction and it can suppol-t a radial pressure gradient, so that in cylindrical devices the pressure will increase towards the high temperature regions of the plasma column.

The effects of thel-modiffusion (Nernst effect), and the ambipolar diffusion which was first pre- dicted by Wienecke (I) , lead to an equilibrium which may be very important for a steady linear fusion plasma imbedded in a strong axial magnetic field. They can maintain an electron density minimum in a hydrogen plasma and may provide means of radial refuelling. The purpose of this arti- cle is to give an approximate but simple analytical formula which combines the Nernst and the Wienecke effects to yield the pressure and density as function of the applied magnetic field B A and the local temperature T in the plasma. Such a function is of value if one tries to obtain simple scaling laws, for instance for laser solenoid plasmas or fusion plasmas in passive liners. In order to appreciate the nature of the approximation, results of the Wienecke and the Nernst effects are first sum- marized.

'On sabbatical leave from the Department of Physics, Univer- sity of British Columbia, Vancouver, B.C., Canada V6T 1W5.

The production of a current at right angle to a magnetic field and a temperature gradient has been known in metals as the Nernst effect for quite a long time. More recently it has been predicted by Braginsky (2) and observed and analyzed by Wulff (3), Kliiber (4), and Grassmann ( 5 ) in cylindrical steady arc plasma in axial magnetic fields and has been found to be responsible for a substantial pres- sure increase towards the axis. Typical parameters of these experiments are ni = 1016, T = 10 eV, B = 60 kG. The pressure increase in regions of am- bipolar diffusion was predicted first by Wienecke (1) and later observed in steady high temperature arcs by Mahn et ol. (6. 7) and PotoEnik and Meyer (8). In all cases very good agreement was found between experiments and calculations.

Wienecke ( I ) considers the cylindrical column of a high current arc surrounded by neutral gas and imbedded in an axial magnetic field B,. In equilib- rium neutrals diffuse inwards into regions of higher temperature where they are ionized, and ions and electrons diffuse outwards into cooler regions where they recombine. If no axial magnetic field is present the diffusion cannot modify the total pres- sure distribution and the pressure must be constant across the radius. However, in the presence of a magnetic field the outward motion of the ions is retarded while the inward motion of the neutrals is not. An equilibrium is then established in which a higher total particle pressure is found in the hot region. This pressure is calculated by Wienecke in general terms as function of temperature, magnetic field, and outside applied pressure pA. To this end he describes the ambipolar radial diffusion by Schliiter's momentum equations (9) for ions, elec- trons, and neutrals, and imposes radial pressure

0008-4204179108 1090-04SO I .0O/0 0 1 9 7 9 National R e s e x c h Council of Canada/Conseil national d e recherches du Canada

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AHLBOKN 1091

balance by the use of B2/2po + p = constant. Ioni- zation equilibrium is included via the Saha equa- tion. The system of equations is numerically inte- grated, and as a result the pressure inside the dis- charge is given in graphical form as function of the local temperature T, the applied magnetic field BA, and the pressure in the cold gas surrounding the arc, p,. The pressure modification due to the pinch effect is not included in these calculations since it is negligible for the range of currents encountered in the associated experiments (6, 7). Therefore, Wienecke's results can be applied to any steady cylindrical plasma, with no axial current, imbedded in a magnetic field B,. Some of his results are re- plotted in Fig. lct. An external pressure of p, = lo4 dyn/cm2 corresponds to about lo1' hydrogen atoms in the cold regions outside the ~ l a s m a . It mav - be noted that the pressure levels off at temperatures where the gas is fully ionized, namely for T 2 i?;, = 20000 K in hydrogen and T 2 GI, = 30000 K in h e l i ~ m . ~ The maximum PI-essul.e reached,fi, is I-el- atively independent ofp,. This may be seen in Fig. lb, where p - p, is plotted as a function of the magnetic field. Helium and hydrogen with the same outside pressure yield PI-actically an identical pres- sure increment. In the range where data are calcu- lated one can approximately give the pressure as function of the magnetic field as

[ l ] f i - p A z 3 B

where p is measured in dynlcm%and B in gauss. The number density 11, can then be calculated at the temperature T where the maximum pressure @ is just reached using the equation of statefi = 2n,kT to yield for hydrogen with T = 20 000 K,

where 11 is measured in cm-' and B in gauss. p >> p, is assumed in this relation. Kliiber (4) concludes that the pressure increase due to the thermodiffu- sion is small compared to the pressure increment due to ambipolar diffusion, essentially because the temperature gradients are small in the ionization region. (The effective thermal conductivity X, is very large in this temperature range (lo).)

It is interesting to notice that the radial inward diffusion of neutral particles which was observed by Grassmann (5) will persist as long as a grad 11,

exists. Since the concentration of neutrals on the axis of a cylindrical discharge with fusion temper- atures is very small, even a low neutral gas density of, say, 10" neutl-als/cm' on the outside will estab-

'These calculations agree vel-y well with experlmentb (6).

FIG. 1. ( ( I ) Pressure inside a discharge imbedded in an axial magnetic field as function of temperature and applied magnetic field B,. for the outside pressures p, = lo4 and IOS dyn/cm2. Solid curves, hydrogen; broken curve, helium. (11) Pressure incrementp - p, as function of the applied magnetic field B, for hydrogen and helium.

lish a grad no and hence facilitate radial motion of hydrogen atoms. This mechanism could be used for refuelling of a steady fusion reactor simply by bleeding DT fuel gas of low density into the reactor vessel. The exact amount of radial diffusion veloc- ity of the neutrals can be calculated using Wienecke's ( 1 ) analysis.

In a cylindrical plasma column where the tem- perature rises above the minimum temperature of full ionization, T , there is a second effect which causes the pressure to continue to rise towards the axis (4). This is the thermofol-ce (2) which has the azimuthal component

where o,, = eB/~n,. The thermal force is related to the azimuthal current as R+ = en,j+/o, with o = e2n,~/m,. Hence one finds

j+ = $(n,klB,)(a Tla~.)

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1092 CAN. J . PHYS. VOL. 57, 1979

Notice that j, does not depend on the collision I-ate T,. This may be understood by Braginsky's (2) argument that adjacent orbits in a temperature gra- dient embedded in the magnetic field carry different electron fluxes through a control su~face giving rise to a net current. Equilibrium between the Lorentz force and a pressure gradient in radial dii-ection then yields

(aplat.) = j4B, = $n,k(a Tlat.)

The pressure p is given by

p = nekT(Z + 1)/Z

where Z is the charge, assumed constant. Differ- en t i a t i ng~ with respect to I . and entering the I-esult on the left-hand side of the pressure balance equa- tion one finds

This differential equation is solved by

[3] n, = const ~(z-')/'(Zf 1 )

Kliiber uses this result to predict a constant elec- tron number density over the radius of his helium discharge plasma for Z = 2, n, # f T). This is very well confirmed by Grassmann's (5) Faraday rota- tion measurements on the same arc in helium. For hydrogen with Z = 1 one has the density distribu- tion

[4] n, = const T-'I4 (hydrogen)

and the pressure p = 2nekT must then vary a s p = const T3I4. Notice that the density should decrease in regions of higher ternperature in a hydrogen dis- charge. This density decrease is in fact noticeable observed from side-on, Abel unfolded spectros- copic measurements (6).

For the description of the pressure distribution of the entire discharge column it is important to notice that the Nernst effect takes over where the Wienecke effect leaves off. Hence we use the maximum pressure p produced by the Wienecke effect at a temperature 7 to determine the constants for the pressure and density relations in the Nernst regime. Using T = 20 000 Kfor hydrogen we obtain

[5a] p Z 1.8 10-3BT314

and

[5t71 11, z 6.4

where p is measured in dynlcm2, B is the local magnetic field given in gauss, T is measured in K , and the assumption was made that f i >> PA. In plasma configurations where the pressure is un-

checked at the ends of the cylindrical plasma col- umn (5, 8), the magnetic field can start a circulation which brings cold gas inwards in the radial direction at or near the midplane and ejects hot plasma at the ends in the axial direction. Such a mechanism may provide an easy way of refueling continuously operating fusion reactors with a similar geometry.

Equations [5cl] and [50] represent an 'equation of state' p = f (B, T) or n = f (B, T) for a (cylindrical) plasma imbedded in an axial magnetic field, in which an unspecified source of energy (such as Joule heat, inverse bremsstrahlung absol-ption, a particle production) maintains a temperature distribution with a maximum on axis. If one combines [5] with the momentum equation one can obtain p = f (T) and B = f(T). If one further adds the complete energy equation for the particular problem at hand, T(r) can be found. Here we only want to show how p and B vary with the ternperature. For this purpose we employ the radial pressure balance

P + BZ/2po = PA + B A ~ / ~ ~ o

wherep, and BA are the pressure and magnetic field strength applied outside the plasma column. En- tering p from [ 5 ~ ] we have a quadratic equation with one positive root for B:

FIG. 2. Magnetic pressure B2/2p0, plasma pressure p, and number density r l , in a hydrogen plasma which is in radial pressure equilibrium with an applied magnetic field B, = 50kT (solid line) and lOOkT (broken line).

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AHLBORN I093

where T is measured in electronvolts. Equation [6] was evaluated to give B as a function of T for BA = 5, 10 T , shown in Fig. 2. Also drawn is the pressure p (obtained from [Sa] and t z , = p/2kT). The curves for BA = 10 T are dashed to acknowledge the fact that the scaling laws [5rl] and [50] have been ex- trapolated into a magnetic field regime where no numerical values for the Wienecke effect have been calculated. The magnetic pressure B2/2p, drops, as one can see, only very slowly as function of tem- perature. Obviously the Nernst effect is not very efficient in expelling magnetic flux from hot regions so that p remains fairly low, up to kiloelectronvolt temperatures. When the reduction of magnetic field with temperature is taken into account, the density scaling law (4) is slightly modified and the density ratio between the cold and the hot regions is in- creased. For instance, [4] yields a ratio iiln % 8.7 for the tem~erature ratio TIT = 1.72 eV/10 keV.

columns (12, 13), which suggests that the Wienec- ke-Nernst effect can cause the density minimum, established by some other means such as laser or shock heating. to persist for the lifetime of the plasma.

Acknowledgments The stimulating atmosphere in the plasma

physics group at the University of Washington led to the pursuit of this topic. It is a pleasure to thank Profs. George C. Vlases and Adam Pietrzyk for many clarifying discussions, and Prof. Vlases for valuable comments on the manuscript.

I. R. WIENECKE. Z. N:~tulforsch., Teil A. 18. 1151 (1963). 2. S. I . BRAGINSKY. SOV. Phys. JETP, 6, (1958); D. L. BOOK.

Revised plasma formulary. Naval Research L;tboratory, Washington, DC.

3. H. WULFF. Proceedings of the International Conference on Phenomena in Ionized Gases. Vol. I. 1965. p. 829.

The detaileh calculations give instead a' density 4. 0. K L ~ ~ B E R . Z. N;ttulforsch., Teil A, 22, 1599 (1967).

I.atio of 14 at B, = 5 T which reduces to 9.8 when B, 5. P. H . GRASSMANN. Z. Natul-forsch., Teil A. 23.25 1 (1968). 6. C. MAHN, H. RINGLER, R. WIENECKE. S. WITKOWSKI, and

is increased to 20 T. G. ZANKL. Z. Naturforsch., Teil A , 19, 1202 (1964). Conclusions 7. C. MAHN, H. RINGLER, and G. ZANKL. Z. Naturforsch.,

Tell A. 23.867 (1968). The in the of a magnetized hy- 8. V. P O T O ~ N ~ K and J. MEYER. Can. J . Phys. 53,2485 (1975).

drogen plasma column is depressed due to the 9. A. SCHL"TER. Z. ~;ltulforsch.. TeilA,5,72(1950). Nernst effect; the depression is quite pronounced 10. R. W ~ E N E C K F . Z. N;~tulfo~.sch., Teil A , 19.675(1964). at T % lo4 eV. This steadv state ~henolnenon is due 11. L. STLINHAUER and G. C. V L A S ~ S . Nucl. Fusion. To be

to the thermal diffusion effect, ahd it might be quite published. 12. H . L. RUTKOWSKI, D. W. SCUDDER. Z. A. PIETRZYK. and

useful for laser heated solenoid applications (1 I). It G , C, VLASES. Appl. Phys, Lett, 26, 421 (1975), is interesting to note that this depression is similar 13, H , A , B,,,, ;,,d G, c. phys. Fluids, 18. 982 to that found on a transient basis for laser heated (1975).

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