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The reason for the difference is the velocity dependence of theCoulomb cross section. Third, diffusion is automatically ambipolar in a fullyionized gas (as long as like-particle collisions are neglected). D⊥ in Eq. (5.99) isthe coefficient for the entire fluid; no ambipolar electric field arises, because bothspecies diffuse at the same rate. This is a consequence of the conservation ofmomentum in ion-electron collisions. This point is somewhat clearer if one usesthe two-fluid equations (see Problem 5.15).Finally, we wish to point out that there is no transverse mobility in a fullyionized gas.
Equation (5.96) for v⊥ contains no component along E which dependson E. If a transverse E field is applied to a uniform plasma, both species drifttogether with the E B velocity. Since there is no relative drift between the twospecies, they do not collide, and there is no drift in the direction of E. Of course,there are collisions due to thermal motions, and this simple result is only anapproximate one. It comes from our neglect of (a) like-particle collisions, (b) theelectron mass, and (c) the last two terms in Ohm’s law, Eq. (5.90).1765.95 Diffusion and ResistivitySolutions of the Diffusion EquationSince D⊥ is not a constant in a fully ionized gas, let us define a quantity A whichis constant:A ηKT=B2ð5:100ÞWe have assumed that KT and B are uniform, and that the dependence of η onn through the ln Λ factor can be ignored.
For the case Ti ¼ Te, we then haveD⊥ ¼ 2nAð5:101ÞWith Eq. (5.98), the equation of continuity (5.92) can now be written∂n=∂t ¼ ∇ ðD⊥ ∇nÞ ¼ A∇ ð2n∇nÞ∂n=∂t ¼ A∇2 n2ð5:102ÞThis is a nonlinear equation for n, for which there are very few simple solutions.5.9.1Time DependenceIf we separate the variables by lettingn ¼ T ðtÞSðrÞwe can write Eq. (5.102) as1 dT A 2 21¼ ∇ S ¼2 dtSτTð5:103Þwhere 1/τ is the separation constant. The spatial part of this equation is difficult tosolve, but the temporal part is the same equation that we encountered in recombination, Eq.
(5.41). The solution, therefore, is11t¼þT T0 τð5:104ÞAt large times t, the density decays as 1/t, as in the case of recombination. Thisreciprocal decay is what would be expected of a fully ionized plasma diffusingclassically. The exponential decay of a weakly ionized gas is a distinctly differentbehavior.5.9 Solutions of the Diffusion Equation5.9.2177Time-Independent SolutionsThere is one case in which the diffusion equation can be solved simply. Imagine along plasma column (Fig. 5.19) with a source on the axis which maintains a steadystate as plasma is lost by radial diffusion and recombination. The density profileoutside the source region will be determined by the competition between diffusionand recombination. The density falloff distance will be short if diffusion is smalland recombination is large, and will be long in the opposite case.
In the regionoutside the source, the equation of continuity isA∇2 n2 ¼ αn2ð5:105ÞThis equation is linear in n2 and can easily be solved. In cylindrical geometry, thesolution is a Bessel function. In plane geometry, Eq. (5.105) readsd 2 n2 α 2¼ nAdx2ð5:106Þwith the solutionhin2 ¼ n20 exp ðα=AÞ1=2 xð5:107Þl ¼ ðA=αÞ1=2ð5:108ÞThe scale distance isSince A changes with magnetic field while α remains constant, the change of l withB constitutes a check of classical diffusion. This experiment was actually tried on aQ-machine, which provides a fully ionized plasma. Unfortunately, the presence ofasymmetric E B drifts leading to another type of loss—by convection—made theexperiment inconclusive.Fig. 5.19 Diffusion of a fully ionized cylindrical plasma across a magnetic field1785 Diffusion and ResistivityFinally, we wish to point out a scaling law which is applicable to any fullyionized steady state plasma maintained by a constant source Q in a uniform B field.The equation of continuity then readsA ∇2 n2 ¼ ηKT ∇2 n2 =B2 ¼ Qð5:109Þn/Bð5:110ÞSince n and B occur only in the combination n/B, the density profile will remainunchanged as B is changed, and the density itself will increase linearly with B:One might have expected the equilibrium density n to scale as B2, since D⊥ / B2;but one must remember that D⊥ is itself proportional to n.5.10Bohm Diffusion and Neoclassical DiffusionAlthough the theory of diffusion via Coulomb collisions had been known for a longtime, laboratory verification of the 1/B2 dependence of D⊥ in a fully ionized plasmaeluded all experimenters until the 1960s.
In almost all previous experiments, D⊥scaled as B1, rather than B2, and the decay of plasmas was found to be exponential, rather than reciprocal, with time. Furthermore, the absolute value of D⊥ was farlarger than that given by Eq. (5.99). This anomalously poor magnetic confinementwas first noted in 1946 by Bohm, Burhop, and Massey, who were developing amagnetic arc for use in uranium isotope separation. Bohm gave the semiempiricalformulaD⊥ ¼1 KT e DB16 eBð5:111ÞThis formula was obeyed in a surprising number of different experiments.
Diffusionfollowing this law is called Bohm diffusion. Since DB is independent of density, thedecay is exponential with time. The time constant in a cylindrical column of radiusR and length L can be estimated as follows:τNnπR2 LnR¼¼dN=dt Γr 2πRL 2Γrwhere N is the total number of ion-electron pairs in the plasma. With the flux Γrgiven by Fick’s law and Bohm’s formula, we have5.10Bohm Diffusion and Neoclassical Diffusion179Fig.
5.20 Summary of confinement time measurements taken on various types of discharges inthe Model C Stellarator, showing adherence to the Bohm diffusion law (Courtesy of D. J. Grove,Princeton University Plasma Physics Laboratory, sponsored by the U.S. Atomic EnergyCommission)τnRnRR2¼ τB2DB ∂n=∂r 2DB n=R 2DBð5:112ÞThe quantity τB is often called the Bohm time.Perhaps the most extensive series of experiments verifying the Bohm formulawas done on a half-dozen devices called stellarators at Princeton. A stellarator is atoroidal magnetic container with the lines of force twisted so as to average out thegrad-B and curvature drifts described in Sect.
2.2.3. Figure 5.20 shows a compilation of data taken over a decade on many different types of discharges in the ModelC stellarator. The measured values of τ lie near a line representing the Bohm timeτB. Close adherence to Bohm diffusion would have serious consequences for thecontrolled fusion program. Equation (5.111) shows that DB increases, rather thandecreases, with temperature, and though it decreases with B, it decreases moreslowly than expected. In absolute magnitude, DB is also much larger than D⊥. Forinstance, for a 100-eV plasma in 1-T field, we have1805 Diffusion and Resistivity1 102 1:6 1019DB ¼¼ 6:25m2 = sec16 1:6 1019 ð1ÞIf the density is 1019 m3, the classical diffusion coefficient isð2:0Þ 5:2 105 ð10Þ2nKTη⊥ ð2Þ 1019 102 1:6 1019¼D⊥ ¼B2ð1Þ2ð100Þ3=2¼ ð320Þ 1:04 106 ¼ 3:33 104 m2 = secThe disagreement is four orders of magnitude.Several explanations have been proposed for Bohm diffusion.
First, there is thepossibility of magnetic field errors. In the complicated geometries used in fusionresearch, it is not always clear that the lines of B either close upon themselves oreven stay within the chamber. Since the mean free paths are so long, only a slightasymmetry in the magnetic coil structure will enable electrons to wander out to thewalls without making collisions. The ambipolar electric field will then pull the ionsout. Second, there is the possibility of asymmetric electric fields.
These can arisefrom obstacles inserted into the plasma, from asymmetries in the vacuum chamber,or from asymmetries in the way the plasma is created or heated. The dc E B driftsthen need not be parallel to the walls, and ions and electrons can be carried togetherto the walls by E B convection. The drift patterns, called convective cells, havebeen observed. Finally, there is the possibility of oscillating electric fields arisingfrom unstable plasma waves. If these fluctuating fields are random, the E B driftsconstitute a collisionless random-walk process. Even if the oscillating field is a puresine wave, it can lead to enhanced losses because the phase of the E B drift can besuch that the drift is always outward whenever the fluctuation in density is positive.One may regard this situation as a moving convective cell pattern.
Fluctuatingelectric fields are often observed when there is anomalous diffusion, but in manycases, it can be shown that the fields are not responsible for all of the losses. Allthree anomalous loss mechanisms may be present at the same time in experimentson fully ionized plasmas.The scaling of DB with KTe and B can easily be shown to be the natural onewhenever the losses are caused by E B drifts, either stationary or oscillating. Letthe escape flux be proportional to the E B drift velocity:Γ⊥ ¼ nv⊥ / nE=Bð5:113ÞBecause of Debye shielding, the maximum potential in the plasma is given byeϕmax KT eð5:114ÞIf R is a characteristic scale length of the plasma (of the order of its radius), themaximum electric field is then5.10Bohm Diffusion and Neoclassical Diffusion181ϕmax KT eReRð5:115Þn KT eKT e γ∇n ¼ DB ∇nR eBeBð5:116ÞEmax This leads to a flux Γ⊥ given byΓ⊥ γwhere γ is some fraction less than unity.
Thus the fact that DB is proportionalto KTe/eB is no surprise. The value γ ¼ 161 has no theoretical justification butis an empirical number agreeing with most experiments to within a factor oftwo or three.Recent experiments on toroidal devices have achieved confinement times oforder 1000τB. This was accomplished by carefully eliminating oscillations andasymmetries. However, in toroidal devices, other effects occur which enhancecollisional diffusion. Figure 5.21 shows a torus with helical lines of force. Thetwist is needed to eliminate the unidirectional grad-B and curvature drifts. As aparticle follows a B-line, it sees a larger jBj near the inside wall of the torus and asmaller jBj near the outside wall. Some particles are trapped by the magneticmirror effect and do not circulate all the way around the torus.
The guiding centersof these trapped particles trace out banana-shaped orbits as they make successivepasses through a given cross section (Fig. 5.21). As a particle makes collisions, itbecomes trapped and untrapped successively and goes from one banana orbit toanother. The random-walk step length is therefore the width of the banana orbitrather than rL, and the “classical” diffusion coefficient is increased. This is calledneoclassical diffusion. The dependence of D⊥ on collision frequency v is shown inFig. 5.22. In the region of small v, banana diffusion is larger than classicaldiffusion.
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