The Coefficient of Bohm Diffusion in Fully Ionized Plasma and its Theoretical Proof

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1 Utah State University Graduate Student Presentations Browse all Graduate Research The Coefficient of Bohm Diffusion in Fully Ionized Plasma and its Theoretical Proof Ahmad Talaei Utah State University Reza Amrollahi Follow this and additional works at: Part of the Physics Commons Recommended Citation Ahmad Talaei and R. Amrollahi Coefficient of Bohm diffusion in fully ionized plasma and its theoretical proof Proceeding of 2th IPM international conference on plasma dynamics, Tehran, Iran, 2006 This Presentation is brought to you for free and open access by the Browse all Graduate Research at It has been accepted for inclusion in Graduate Student Presentations by an authorized administrator of For more information, please contact

2 The Coe cient of Bohm Di usion in Fully Ionized Plasma and its Theoretical Proof Ahmad Talaei a,, Reza Amrollahi a a Department of Nuclear Engineering and Physics, Amirkabir University of Technology, No. 424, Hafez Ave., P.O.Box , Tehran, Iran 2 th IPM International Conference on Plasma Dynamics, Tehran, Iran, Abstract As we know, the classical di usion can not be able to respond the experimental results of the diffusion in the fully ionized and isothermal plasma. In this article, we purpose to extract the empirical coe cients of the Bohm di usion and its confinement time, theoretically, in the fully ionized plasma, using the single fluid MHD equations. Keywords: Classical di usion, Bohm di usion, Bohm time, Single fluid MHD equations, Fick s low. 1 Introduction As we know, in the weakly ionized plasma in the absence of magnetic fields, charged particles collide primarily with neutral atoms rather that with one another. But considering the weakly ionized plasma in a existence of the magnetic field, charged particles will move along by di usion and mobility. In this case, we can write the perpendicular coe cient of the di usion to the magnetic field as [1]: D? / r2? (1.1) Where r? is the Larmor radius and the mean time between collisions for particles. In fully ionized plasmas (composed of ions and electrons alone), all collisions are Coulomb collisions between charged particles. In the absence of gravity, for a steady state plasma, the perpendicular coe cient of the di usion to the magnetic field using the single fluid MHD equations, becomes[1]: D? /?n P KT B 2 (1.2) Corresponding author. addresses: ahmad.talaei@gmail.com (Ahmad Talaei) 1

3 D? is so called the classical di usion coe cient for a fully ionized gases where n is the plasma density and? the perpendicular resistance to the magnetic field. As we see Eq.(1.2), D? is proportional to B 2, just like in the case of weakly ionized plasma. But the laboratory verification of the B 2 dependence of the D? in a fully ionized plasma elude from the all experiments. In almost previous experiments, D? scaled as B 1, rather than B 2, and the decay of plasma was found to be exponential, rather than reciprocal with time. Furthermore, the absolute value of D? was far larger than that given by Eq.(1.2). Bohm, gave the coe cient of this poor magnetic confinement di usion empirical as [1]: D? = 1 KT e 16 eb D B (1.3) Di usion following this law is called Bohm di usion. This formula was obeyed in a surprising number of di erent experiments. As we see, D B is independent of density and proportional to B 1. The confinement time in a cylindrical plasma column of radius R and length L can be estimated from Eq.(1.3) as follows [1]: The B is often called the Bohm time. = R2 2D B B (1.4) 2 Therotical Equation of Bohm Di usion in Fully Ionized Plasma We use the single fluid MHD equations for obtaining Eqs.(1.3) and = j B rp + g (2.1) E + V B = + r ( V ) = + r ( j ) = 0 (2.4) Where is the mass density of plasma, V the velocity of one phase plasma, j the current density of plasma, B the magnetic field, p the Maxwellian pressure, g gravitational field,? specific resistivity and charge density in one phase plasma. We write the perpendicular component to B in Eq.(2.1), as V? = j? B rp + g In the absence of gravity, for a steady state plasma, we have: Taking the cross product with B, we have: j? B rp = 0 (2.6) 2

4 ( j? B ) B rp B = 0 (2.7) The scalar product of current density perpendicular to magnetic field with B is zero, thus: j? = As we know, j? = ne V?. Thus, we have: rp B B 2 (2.8) rp B V? = neb 2 ) V? = ( ekt e + i KT i ) rn B neb 2 (2.9) For simplicity, we assumed e KT e = i KT i (for an isothermal plasma) so we have: V? = 2 e KT e rn B neb 2 (2.10) The perpendicular component of the flux associated with di usion to magnetic field,? is as:? = n V? = 2 e KT e rn B eb 2 (2.11) On the other hand, Fick s low of di usion is equal: = D rn (2.12) Where D is di usion coe cient. We compare Eq.(2.11), with the perpendicular component of the flux in Eq.(2.12) so we can write the perpendicular di usion coe cient to B as: D? =2 KT e eb (2.13) As we obtained, D? is proportional to B 1, and it is not proportional to n. Also, the absolute value of D? in Eq.(2.13) is nearly equal to value of coe cient of the Bohm di usion. For obtaining the Bohm time, we use the equation of continuity + r = 0 (2.14) For perpendicular component of Eq.(2.14) to B, and using Fick s law (that we assume D? is independent of geometry), we D? r 2 n = 0 Eq.(2.15) can be solved by using the method of separation of variables. Therefore, we consider: Combining Eqs.(2.15) and (2.16), we have: n(r, t) =T (t)s(r) (2.16) 1 dt (t) = D? T (t) dt S(r) r2 S(r) (2.17) 3

5 Since the left side is a function of time alone and the right side a function of space alone, they must both be equal to the same constant, which we shall call 1. The function T(t) obeys the equation: 1 dt (t) = D? T (t) dt S(r) r2 S(r) = 1 (2.18) 1 dt (t) = 1 T (t) dt ) T (t) =T e t (2.19) Therefore, density decays exponentially with time, as one would expect. Also, the spatial part of Eq.(2.18), S(r), in cylindrical geometry obeys the equation: D? S(r) r2 S(r) = 1 ) d2 S(r) dr r ds(r) dr + S(r) D? = 0 (2.20) As we know, S(r) is called a Bessel function of order of zero, and Eq.(2.20) is called Bessel s equation of order zero. With the solution we have: r S(r) =AJ ( p D? )+BY ( r p D? ) (2.21) We would expect the density to be finite at the center of cylinder (r=0). condition {S(r 0) = finite} requires: This boundary r S(r) =AJ ( p D? ) (2.22) Also, we would expect the density to be nearly zero at the walls. Since once ions and electrons reach the wall, they recombine there. The density near the wall, therefore, is essentially zero. Thus: R S(r R) =0) AJ ( p D? )=0) R p D? = n (2.23) n is points that the Bessel function of order of zero is zero in these points, such as 2.408,5.87,etc.. R For satisfying the boundary condition {n(r R, t) =0}, we must set p D? equal to the first zero of J, namely, Therefore, for the decay time, we have: R p D? =2.408 ) = R 2 (2.24) 5.798D? The valve of confinement time in Eq.(2.24) is nearly equal to value of the Bohm time. As one sees, we extract the empirical coe cients of the Bohm di usion and Bohm time, theoretically, using the single fluid MHD equations. Consequently, by combining Eqs.(2.16),(2.19),(2.22) and (2.24), the density of plasma as function of time and space in the cylindrical geometry is equal: n(r, t) =T (t)s(r) =n J ( 2.408r R )e t (2.25) 4

6 3 Conclusion In this work, we extract Bohm di usion and Bohm time in fully ionized plasma for a steady state (that Bohm gave the empirical formulas for them) using the single fluid MHD equations. Consequently, we suggested a plasma density with time and space components in a cylindrical column of radius R and length L. References [1] Francis F.Chen, Introduction to Plasma Physics and Controlled Fusion, Plennum Press, New York, Second Edition, Vol. 1, Plasma Physics,

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