NONLINEAR EFFECT OF THE IONIZATION WAVES ON THE TEMPERATURES CHARGED PARTICLES IN A MAGNETIZED POSITIVE COLUMN

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NONLINEAR EFFECT OF THE IONIZATION WAVES ON THE TEMPERATURES CHARGED

PARTICLES IN A MAGNETIZED POSITIVE COLUMN

S. Imazu, H. Fujita, K. Miura, T. Takamatsu

To cite this version:

S. Imazu, H. Fujita, K. Miura, T. Takamatsu. NONLINEAR EFFECT OF THE ION- IZATION WAVES ON THE TEMPERATURES CHARGED PARTICLES IN A MAGNE- TIZED POSITIVE COLUMN. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-153-C7-154.

�10.1051/jphyscol:1979775�. �jpa-00219481�

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JOURNAL DE PHYSIQUE CoZloque C7, suppZ6ment au n07, Tome 40, JuilZet 1979, page C7- 153

NONLINEAR EFFECT ff THE IOMZATXON WAVES ON THE TEMPERATURES CHARGED PARTICLES IN A MAGNETIZED PO!3TIVE COLUMN

S. Imazu, H. Fujita, K. Miura and T. Takamatsu.

Faculty of Engineering, Hiroshima University, Sendacho 3-8-2 Hiroshima Japan.

Introduction: The diffusion of charged particles across a magnetic field,especially the enhanced diffusion, is a problem of considerable interest.

In many experiments /1-2/ on the positive column, a longitudinal electric field E0,decreases according to the classical diffusion theory with increasing the magnetic field B. Furthermore, when B exceeds a certain critical value Bc

,

the helical instability starts, and then Eoz increases ab- ruptly due to the enhanced diffusion. However we have experimentally found the phenomenon that,when the ionization waves start at the magnetic field slightlybelow Bc

,

EOZ increases to exceed the values obtained from the classical diffusion theory.

In this report, the nonlinear effect of the ionization wave on the longitudinal electric field EOz is investigated for helium. Then it is found that the mean energies of the charged particles increase according to the nonlinear efcfect of the longitudinal perturbed electric field El and, con- sequently, the longitudinal electric field Eo, increases. The theoretical values obtained are compared with experimental data, and found to be in agreezent with present experiments.

Methods and results: A cylindrical symmetry is assumed in a coordinate system (r,+ ,z) with z axis along the column, and the column is situated in. a longitudinal magnetic field B. The theoretical equations are derived under the following as- sumptions: 1) The mean free paths of electrons and ions are small to the tube radius R. 2) The electron and ion gases have Maxwellian distributions.

3) As the metastable atoms of helium, 2'S.t is tak- en into account. 4) Molecular ions are neglected compared with atom ions. 5) The equivalent temperatures of ions at plasma edge are equal to Te/2.

As basic equations, the continuity equations for the number densities of electrons, ions and metastable atoms, respectively, and the conserva- tion equations for the energies of charged particles are used. Moreover Poisson's equation is used. (a) Steady state: We first derive the expressions for the mean energies of electrons Te and ions T; as a function of B for the steady state. Figure 1 shows the theoretical curves

@=o)

of Te-vs-B and Ti-VS-B for the steady state. As shown in Fig. 1, we confirm that Te and T. de- z crease together with increasing B, according to the classical diffusion theory. Further, we obtain the expressions for the longitudinal electric field Eo, as a function of B for steady state, and then show the theoretical curves of E0,-vs-B for

d

=O in Fig. 2, which corres ond to the curves of Te-vsrB and Ti-vs-B for 8 - 0 in Fig. 1. Figure 3 shows the comparison of the theoretical curve

(2

=o) with experiments. As shown in Fig. 3, up to B=B1 (1.25 kG) the theoretical values for $ =O are good agreement with experimental values.

Therefore we confirm the classical diffusion theory in the region of B from 0 to Bl. However Fig.3

shows that the difference between the theoretical and experimental values occur above B1 that the ionization waves develop. Although it is well known that an anomalous diffusion occurs above the critical magnetic field Bc(=2.25 kG) that the helical instabilities develop.

(b) Perturbes state: We can choose a perturvation of the form Fl.exp(iWt

-

^ikZ⁾

^,

^where

^k

^{is the}

wave number ( k =2 1C / h )

,

^W⁼^(A)r+i, UJi the angular frequency and h the wave length. is the perturbed quantity-of complex number. The perturbed electric field El in the longitudinal direction is derived by solving Poisson's equation, assuming the density distribution in the radial direction to be the zero order Bessel function Jo(U-r).

1Ahere CC

=m,

^Zis the ionization frequency and Da the ambipolar diffusion coefficient. Here we postulate that the perturbed part of dens'ity exists in addition to steady state density. In nonlinear calculation, we can not use e x p ( i ~ t - i k z) and then take its real part. Putting the determi- nant deduced from these expressions be equal to zero, we obtain the dispersion equation for (nl. We have clli<O, and then the amplitude of the structure exponentially with time.

(c) Effect of perturbed field El : The electric field term in the energy balance equation contains the second order perturbed electric field. In the nonlinear calculation, the zero harmonic term can be written as the sum of two terms, one of which is e ) 4 ~ & ~ for the equibrium state. What remains is Pe,UE1

.

^Here^U^, is the mobility. !The term

&e~~.?!f

gives cause to increase the energies of electrons and ions when the ionization wave exists

Figure 1 shows the curves of Te-VZ-B and Ti-vs -B for the variaus of i'

.

^:fiere

d

⁼

IE$

/..Eo. Eo is not equal to Eoz when the perturbation exists.

is obtained from the equation of the total ax-

:%

current. Here we take into account of ,the nonlinear effect of the electron temperature on the ionization frequency

.

The influence of ions on the growth of the ionization waves beczme to be greater than that of metastable atoms with increasing.B. Figure 2 shows the curves of Eoz-vs-B for the various values of

\r' .

In Fig. 3 the curve of Eoz-VS-B ( the broken line ,) obtained from the conditions of ionization waves and the perturbed state mentioned above is compared with the present exper'imental values. In our experiments-f3/

,

the discharge tube of 90 cm long and 1 cm radius, con- taining helium ranging from 0.5 to 1.0 Torr, was placed at the axis of a solenoid coil with the

length of 70 cm. To measure the electric field EOz two ring probes, P1 and P2,were introduced in- to the middle part of the tube. Two pairs of wall probes Pg and P4 were mounted in a planeherpen- dicular to the axis of thr tube. The mode m and the frequency f of the helical instability were measured with these wall probes. The wave length

h and the frequency f of the ionization waves were measured with the optical observation of the ^' side light from the discharge tube. For typical

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979775

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examples, the dependencies of A , f and E as a function of B are shown in Fig. 3. Here ?ge ionization waves occur above B1. When B is raised further, the ionization waves coexist with the helical instabilities that start above Bc, and that EOZ increases rapidly.

Discussion: The theoretical curve of Eoz-vs-B calculated from the classical diffusion theory is indicated by the full line for f=0 in Fig. 3. The theoretical curve below B1 shows a satisfactory agreement between the theoretical values and experiments. Above B1 the experimental values of Eoz exceed slightly the theoretical values for

d

=0, and'above Bc the great discrepancies between the theoretical values and experiments arise due to appearance of the helical instabilities. 'e( attend to the results that in the region between B1 and Bc the experimental values of Eo, exceed the theoretical values for $ =O. The theoretical values calculated from the nonlinear theory of ionization waves are indicated by the broken lines

This theoretical curve is in agreement with the experimental values in the region that the ionization waves appear.

le confirm that, when the ionization waves develop above B1, the perturbed electric field El causes to increase the temperatures of electrons and ions, and then the diffusion coefficients of charged particles become more than the classical diffusion theory in the equibrium state. There- fore the electric fleld EOZ in the axial direction increases.

We can draw the following conclusions: 1) The ionization wave near Bc is only a positive ion- guided ionization wave. 2) The ionization waves cause to increase the mean energies of electrons and ions according to nonlinear effects of waves.

RS1.0 an P20.67 torr

ry

1.5 2.0 2.5

B(KG)

Fig.1 T -vs-B and T.-vs-B

U

: ,frequency of wave x : wave length

0 0 0

y=0%

I

^o ^:Experiments

1.01 -

^:Theory (for y-0)

Fig. 3 Theoretical and Experimental values of EoZ-vs-B

References:

/1/ B.Lehnert: 2nd UN Geneva Con.

32,

349(1958) /2/ B.B.Kadomtsev and A.V.Nedospasov: J. Nucl.

Pt,Cl, 230(1960)

/3/ S.Imazu,K.Miura,T.Takamatsu,H.Shindo and T.Maruyama: Proc. X I 1 1 th. Int. Con. Phenom.

in Ion. Gas.

,

Berlin (1977)295