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In the region of large v, there is classical diffusion, but it is modifiedby currents along B. The theoretical rate for neoclassical diffusion hasbeen observed experimentally by Ohkawa at La Jolla, CA, and by Chen atPrinceton, NJ.Fig. 5.21 A banana orbit of a particle confined in the twisted magnetic field of a toroidalconfinement device. The “orbit” is really the locus of points at which the particle crosses theplane of the paper1825 Diffusion and ResistivityFig. 5.22 Behavior of theneoclassical diffusioncoefficient with collisionfrequency νProblems5.7.
Show that the mean free path λei for electron–ion collisions is proportionalto T2e .5.8. A tokamak is a toroidal plasma container in which a current is driven in thefully ionized plasma by an electric field applied along B (Fig.
P5.8).How many V/m must be applied to drive a total current of 200 kA in aplasma with KTe ¼ 500 eV and a cross-sectional area of 75 cm2?5.9. Suppose the plasma in a fusion reactor is in the shape of a cylinder 1.2 m indiameter and 100 m long. The 5-T magnetic field is uniform except for shortmirror regions at the ends, which we may neglect. Other parameters areKTi ¼ 20 keV, KTe ¼ 10 keV, and n ¼ 1021 m3 (at r ¼ 0). The density profileis found experimentally to be approximately as sketched in Fig. P5.9.(a) Assuming classical diffusion, calculate D⊥ at r ¼ 0.5 m.(b) Calculate dN/dt, the total number of ion-electron pairs leaving thecentral region radially per second.Fig.
P5.85.10Bohm Diffusion and Neoclassical Diffusion183Fig. P5.9(c) Estimate the confinement time τ by τ N/(dN/dt). Note: a roughestimate is all that can be expected in this type of problem. The profilehas been simplified and is not realistic.5.10. Estimate the classical diffusion time of a long plasma cylinder 10 cm inradius, with n ¼ 1021 m3, KTe ¼ KTi ¼ 10 keV, B ¼ 5 T.5.11. A cylindrical plasma column has a density distributionn ¼ n0 1 r 2 =a2where a ¼ 10 cm and n0 ¼ 1019 m3. If KTe ¼ 100 eV, KTi ¼ 0, and the axialmagnetic field B0 is 1 T, what is the ratio between the Bohm and the classicaldiffusion coefficients perpendicular to B?5.12.
A weakly ionized plasma can still be governed by Spitzer resistivity ifνei νe0 where νe0 is the electron–neutral collision frequency. Here aresome data for the electron–neutral momentum transfer cross section σ e0 insquare angstroms (Å2):HeliumArgonE ¼ 2 eV6.32.5E ¼ 10 eV4.113.8For singly ionized He and A plasmas with KTe ¼ 2 and 10 eV (four cases),estimate the fractional ionization f ni/(n0 + ni) at which νei ¼ νe0, assumingthat the value of σv ðT e Þ can be crudely approximated by σ ðEÞjvj ðEÞ; whereE ¼ KTe. (Hint: For νe0, use Eq.
(7.11); for νei, use Eqs. (5.62) and (5.76).5.13. The plasma in a toroidal stellarator is ohmically heated by a current alongB of 105 A/m2. The density is uniform at n ¼ 1019 m3 and does not change.The Joule heat ηj2 goes to the electrons. Calculate the rate of increase of KTein eV/μsec at the time when KTe ¼ 10 eV.5.14.
In a θ-pinch, a large current is discharged through a one-turn coil. The risingmagnetic field inside the coil induces a surface current in the highlyconducting plasma. The surface current is opposite in direction to the coilcurrent and hence keeps the magnetic field out of the plasma.
The magneticfield pressure between the coil and the plasma then compresses the plasma.1845 Diffusion and ResistivityThis can work only if the magnetic field does not penetrate into the plasmaduring the pulse. Using the Spitzer resistivity, estimate the maximum pulselength for a hydrogen θ-pinch whose initial conditions are KTe ¼ 10 eV,n ¼ 1022 m3, r ¼ 2 cm, if the field is to penetrate only 1/10 of the way tothe axis.z ; and5.15. Consider an axisymmetric cylindrical plasma with E ¼ Er r^ , B ¼ B^∇ pi ¼ ∇ pe ¼ r^ ∂ p=∂r.
If we neglect the (v · ∇)v term, which is tantamountto neglecting the centrifugal force, the steady-state two-fluid equations canbe written in the form (Fig. P5.14)enðE þ vi BÞ ∇ pi e2 n2 ηðvi ve Þ ¼ 0enðE þ ve BÞ ∇ pe e2 n2 ηðve vi Þ ¼ 0Fig. P5.14(a) From the θ components of these equations, show that vir ¼ ver.(b) From the r components, show that vjθ ¼ vE + vDj ( j ¼ i, e).(c) Find an expression for vir showing that it does not depend on Er.5.16. Use the single-fluid MHD equation of motion and the mass continuityequation to calculate the phase velocity of an ion acoustic wave in anunmagnetized, uniform plasma with Te Ti.5.17.
Calculate the resistive damping of Alfvén waves by deriving the dispersionrelation from the single-fluid equations (5.85) and (5.91) and Maxwell’sequations (4.72) and (4.77). Linearize and neglect gravity, displacementcurrent, and ∇p.(a) Show that 2ω2B02 iωη¼ c ε0ρ0k2(b) Find an explicit expression for Im (k) when ω is real and η is small.5.10Bohm Diffusion and Neoclassical Diffusion1855.18. If a cylindrical plasma diffuses at the Bohm rate, calculate the steady stateradial density profile n(r), ignoring the fact that it may be unstable. Assumethat the density is zero at r ¼ 1 and has a value n0 at r ¼ r0.5.19.
A cylindrical column of plasma in a uniform magnetic field B ¼ Bz z^ carries auniform current density j ¼ jz z^ ; where z^ is a unit vector parallel to the axis ofthe cylinder.(a) Calculate the magnetic field B(r) produced by this plasma current.(b) Write an expression for the grad-B drift of a charged particle with vjj ¼ 0in terms of Bz, jz. r, v⊥, q, and m. You may assume that the fieldcalculated in (a) is small compared to Bz (but not zero).(c) If the plasma has electrical resistivity, there is also an electric field E¼ Ez z^ : Calculate the azimuthal electron drift due to this field, taking intoaccount the helicity of the B field.(d) Draw a diagram showing the direction of the drifts in (b) and (c) for bothions and electrons in the (r, θ) plane.Chapter 6Equilibrium and Stability6.1IntroductionIf we look only at the motions of individual particles, it would be easy to design amagnetic field which will confine a collisionless plasma.
We need only make surethat the lines of force do not hit the vacuum wall and arrange the symmetry of thesystem in such a way that all the particle drifts vE, v∇B, and so forth are parallel tothe walls. From a macroscopic fluid viewpoint, however, it is not easy to seewhether a plasma will be confined in a magnetic field designed to contain individualparticles.
No matter how the external fields are arranged, the plasma can generateinternal fields which affect its motion. For instance, charge bunching can createE fields which can cause E B drifts to the wall. Currents in the plasma cangenerate B fields which cause grad-B drifts outward.We can arbitrarily divide the problem of confinement into two parts: the problemof equilibrium and the problem of stability. A tight-rope walker can easily find anequilibrium, but it is not stable unless he holds a drooping rod.
The differencebetween equilibrium and stability can also be illustrated by a mechanical analogy.Figure 6.1 shows various cases of a marble resting on a hard surface. An equilibriumis a state in which all the forces are balanced, so that a time-independent solution ispossible. The equilibrium is stable or unstable according to whether small perturbations are damped or amplified. In case (F), the marble is in a stable equilibrium aslong as it is not pushed too far.
Once it is moved beyond a threshold, it is in anunstable state. This is called an “explosive instability.” In case (G), the marble is in anunstable state, but it cannot make very large excursions. Such an in stability is notvery dangerous if the nonlinear limit to the amplitude of the motion is small. Thesituation with a plasma is, of course, much more complicated than what is seen inFig. 6.1; to achieve equilibrium requires balancing the forces on each fluid element.The original version of this chapter was revised.
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