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Equation(3.47), on the other hand, was derived without any explicit statement of the collisionrate. Since the two equations are identical except for the E and B terms, canEq. (3.47) really describe a plasma species? The answer is a guarded yes, and thereasons for this will tell us the limitations of the fluid theory.In the derivation of Eq. (3.47), we did actually assume implicitly that there weremany collisions. This assumption came in Eq. (3.39) when we took the velocitydistribution to be Maxwellian. Such a distribution generally comes about as theresult of frequent collisions.
However, this assumption was used only to take theaverage of v2xr . Any other distribution with the same average would give us the sameanswer. The fluid theory, therefore, is not very sensitive to deviations from theMaxwellian distribution, but there are instances in which these deviations areimportant. Kinetic theory must then be used.There is also an empirical observation by Irving Langmuir which helps the fluidtheory.
In working with the electrostatic probes which bear his name, Langmuirdiscovered that the electron distribution function was far more nearly Maxwellianthan could be accounted for by the collision rate. This phenomenon, calledLangmuir’s paradox, has been attributed at times to high-frequency oscillations.There has been no satisfactory resolution of the paradox, but this seems to be one ofthe few instances in plasma physics where nature works in our favor.Another reason the fluid model works for plasmas is that the magnetic field,when there is one, can play the role of collisions in a certain sense. When a particleis accelerated, say by an E field, it would continuously increase in velocity if it wereallowed to free-stream.
When there are frequent collisions, the particle comes to alimiting velocity proportional to E. The electrons in a copper wire, for instance,drift together with a velocity v ¼ μE, where μ is the mobility. A magnetic field alsolimits free-streaming by forcing particles to gyrate in Larmor orbits. The electronsin a plasma also drift together with a velocity proportional to E, namely,vE ¼ E B=B2 . In this sense, a collisionless plasma behaves like a collisionalfluid. Of course, particles do free-stream along the magnetic field, and the fluidpicture is not particularly suitable for motions in that direction. For motionsperpendicular to B, the fluid theory is a good approximation.643.3.53 Plasmas as FluidsEquation of ContinuityThe conservation of matter requires that the total number of particles N in a volumeV can change only if there is a net flux of particles across the surface S bounding thatvolume.
Since the particle flux density is nu, we have, by the divergence theorem,∂N¼∂tð∂ndV ¼V ∂tþnu dS ¼ð∇ ðnuÞdVð3:49ÞVSince this must hold for any volume V, the integrands must be equal:∂nþ ∇ ðnuÞ ¼ 0∂tð3:50ÞThere is one such equation of continuity for each species. Any sources or sinks ofparticles are to be added to the right-hand side.3.3.6Equation of StateOne more relation is needed to close the system of equations. For this, we can usethe thermodynamic equation of state relating p to n:p ¼ Cργð3:51Þwhere C is a constant and γ is the ratio of specific heats Cp/Cv.
The term ∇p istherefore given by∇p∇n¼γpnð3:52ÞFor isothermal compression, we have∇ p ¼ ∇ ðnKT Þ ¼ KT∇nso that, clearly, γ ¼ 1. For adiabatic compression, KT will also change, giving γ avalue larger than one. If N is the number of degrees of freedom, γ is given byγ ¼ ð2 þ N Þ=Nð3:53ÞThe validity of the equation of state requires that heat flow be negligible; that is, thatthermal conductivity be low. Again, this is more likely to be true in directionsperpendicular to B than parallel to it. Fortunately, most basic phenomena can bedescribed adequately by the crude assumption of Eq. (3.51).3.4 Fluid Drifts Perpendicular to B3.3.765The Complete Set of Fluid EquationsFor simplicity, let the plasma have only two species: ions and electrons; extensionto more species is trivial.
The charge and current densities are then given byσ ¼ ni qi þ ne qeð3:54Þj ¼ ni qi vi þ ne qe veSince single-particle motions will no longer be considered, we may now usev instead of u for the fluid velocity. We shall neglect collisions and viscosity.Equations (3.1)–(3.4), (3.44), (3.50), and (3.51) form the following set:ε 0 ∇ E ¼ ni qi þ ne qe∇E¼ð3:55ÞB_ð3:56Þ∇B¼0ð3:57Þμ0 1 ∇ B ¼ ni qi vi þ ne qe ve þ ε0 E_ð3:58Þ∂v jþ v j ∇ v j ¼ q jn j E þ v j Bm jn j∂t∂n jþ ∇ n jv j ¼ 0∂tγp j ¼ C jn j j∇pjj ¼ i, ej ¼ i, ej ¼ i, eð3:59Þð3:60Þð3:61ÞThere are 16 scalar unknowns: ni, ne, pi, pe, vi, ve, E, and B. There are apparently18 scalar equations if we count each vector equation as three scalar equations.However, two of Maxwell’s equations are superfluous, since Eqs.
(3.55) and (3.57)can be recovered from the divergences of Eqs. (3.58) and (3.56) (Problem 3.3). Thesimultaneous solution of this set of 16 equations in 16 unknowns gives a selfconsistent set of fields and motions in the fluid approximation.3.4Fluid Drifts Perpendicular to BSince a fluid element is composed of many individual particles, one would expectthe fluid to have drifts perpendicular to B if the individual guiding centers have suchdrifts.
However, since the ∇p term appears only in the fluid equations, there is a driftassociated with it which the fluid elements have but the particles do not have. Foreach species, we have an equation of motion663 Plasmas as Fluidsð3:62ÞConsider the ratio of term ① to term ③:Here we have taken ∂/∂t ¼ iω and are concerned only with v⊥.
For drifts slowcompared with the time scale of ωc, we may neglect term ①. We shall also neglectthe ðv ∇ Þv term and show a posteriori that this is all right. Let E and B be uniform,but let n and p have a gradient. This is the usual situation in a magnetically confinedplasma column (Fig. 3.4). Taking the cross product of Eq. (3.62) with B, we have(neglecting the left-hand side)Therefore,v⊥ ¼EBB2∇pB vE þ vDqnB2ð3:63ÞwherevE vD EBB2∇pBqnB2driftð3:64ÞDiamagnetic driftð3:65ÞEBThe drift vE is the same as for guiding centers, but there is now a new drift vD, calledthe diamagnetic drift. Since vD is perpendicular to the direction of the gradient,our neglect of ðv ∇ Þv is justified if E ¼ 0.
If E ¼ ∇ϕ 6¼ 0, ðv ∇ Þv is still zero if∇ϕ and ∇p are in the same direction; otherwise, there could be a more complicatedsolution involving ðv ∇ Þv.With the help of Eq. (3.52), we can write the diamagnetic drift asvD ¼ γKT ^z ∇neBnð3:66Þ3.4 Fluid Drifts Perpendicular to B67Fig. 3.4 Diamagnetic driftsin a cylindrical plasmaIn particular, for an isothermal plasma in the geometry of Fig. 3.4, in which0∇n ¼ n ^r , we have the following formulas familiar to experimentalists who haveworked with Q-machines2:0vDi∂n0<0n ∂rKT i n ^¼θeB nð3:67Þ0vDe ¼KT e n ^θeB nThe magnitude of vD is easily computed from the formulavD ¼KT ðeV Þ 1 mBðT Þ Λ secð3:68Þwhere Λ is the density scale length jn0 /nj in m.The physical reason for this drift can be seen from Fig.
3.5. Here we have drawnthe orbits of ions gyrating in a magnetic field. There is a density gradient toward theleft, as indicated by the density of orbits. Through any fixed volume element thereare more ions moving downward than upward, since the downward-moving ionscome from a region of higher density. There is, therefore, a fluid drift perpendicularto ∇n and B, even though the guiding centers are stationary. The diamagnetic driftreverses sign with q because the direction of gyration reverses.
The magnitude of vD2A Q-machine produces a quiescent plasma by thermal ionization of Cs or K atoms impinging onhot tungsten plates. Diamagnetic drifts were first measured in Q-machines.683 Plasmas as FluidsFig. 3.5 Origin of thediamagnetic driftFig. 3.6 Particle drifts in abounded plasma,illustrating the relation tofluid driftsdoes not depend on mass because the m 1/2 dependence of the velocity is cancelledby the ml/2 dependence of the Larmor radius—less of the density gradient issampled during a gyration if the mass is small.Since ions and electrons drift in opposite directions, there is a diamagneticcurrent. For γ ¼ Z ¼ 1, this is given byjD ¼ neðvDivDe Þ ¼ ðKT i þ KT e ÞB ∇nB2ð3:69ÞIn the particle picture, one would not expect to measure a current if the guidingcenters do not drift. In the fluid picture, the current jD flows wherever there is apressure gradient. These two viewpoints can be reconciled if one considers that allexperiments must be carried out in a finite-sized plasma.
Suppose the plasma werein a rigid box (Fig. 3.6). If one were to calculate the current from the single-particlepicture, one would have to take into account the particles at the edges which have3.4 Fluid Drifts Perpendicular to B69Fig. 3.7 Measuring thediamagnetic current in aninhomogeneous plasmacycloidal paths. Since there are more particles on the left than on the right, there is anet current downward, in agreement with the fluid picture.The reader may not be satisfied with this explanation because it was necessary tospecify reflecting walls.
If the walls were absorbing or if they were removed, onewould find that electric fields would develop because more of one species—the onewith larger Larmor radius—would collected than the other. Then the guidingcenters would drift, and the simplicity of the model would be lost.
Alternatively,one could imagine trying to measure the diamagnetic current with a current probe(Fig. 3.7). This is just a transformer with a core of magnetic material. The primarywinding is the plasma current threading the core, and the secondary is a multiturnwinding all around the core. Let the whole thing be infinitesimally thin, so it doesnot intercept any particles. It is clear from Fig.
3.7 that a net upward current wouldbe measured, there being higher density on the left than on the right, so that thediamagnetic current is a real current. From this example, one can see that it can bequite tricky to work with the single-particle picture. The fluid theory usually givesthe right results when applied straightforwardly, even though it contains “fictitious”drifts like the diamagnetic drift.What about the grad-B and curvature drifts which appeared in the single-particlepicture? The curvature drift also exists in the fluid picture, since the centrifugalforce is felt by all the particles in a fluid element as they move around a bend in themagnetic field.
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