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A term Fc f ¼ nmv2k =Rc ¼ nKT k =Rc has to be added to the righthand side of the fluid equation of motion. This is equivalent to a gravitational forceMng, with g ¼ KT k =MRc , and leads to a drift vg ¼ ðm=qÞðg BÞ=B2 , as in theparticle picture (Eq. (2.18)).703 Plasmas as FluidsFig. 3.8 In a nonuniform B field the guiding centers drift but the fluid elements do notThe grad-B drift, however, does not exist for fluids.
It can be shown on thermodynamic grounds that a magnetic field does not affect a Maxwellian distribution.This is because the Lorentz force is perpendicular to v and cannot change theenergy of any particle. The most probable distribution f(v) in the absence of B isalso the most probable distribution in the presence of B. If f(v) remains Maxwellianin a nonuniform B field, and there is no density gradient, then the net momentumcarried into any fixed fluid element is zero. There is no fluid drift even though theindividual guiding centers have drifts; the particle drifts in any fixed fluid elementcancel out. To see this pictorially, consider the orbits of two particles movingthrough a fluid element in a nonuniform B field (Fig.
3.8). Since there is noE field, the Larmor radius changes only because of the gradient in B; there is noacceleration, and the particle energy remains constant during the motion. If the twoparticles have the same energy, they will have the same velocity and Larmor radiuswhile inside the fluid element. There is thus a perfect cancellation between particlepairs when their velocities are added to give the fluid velocity.When there is a nonuniform E field, it is not easy to reconcile the fluid andparticle pictures. Then the finite-Larmor-radius effect of Sect. 2.4 causes both aguiding center drift and a fluid drift, but these are not the same; in fact, they haveopposite signs! The particle drift was calculated in Chap.
2, and the fluid drift can becalculated from the off-diagonal elements of P. It is extremely difficult to explainhow the finite-Larmor-radius effects differ. A simple picture like Fig. 3.6 will notwork because one has to take into account subtle points like the following: In thepresence of a density gradient, the density of guiding centers is not the same as thedensity of particles!Problems3.3 Show that Eqs.
(3.55) and (3.57) are redundant in the set of Maxwell’sequations.3.4 Show that the expression for jD on the right-hand side of Eq. (3.69) has thedimensions of a current density.3.5 Fluid Drifts Parallel to B713.5 Show that if the current calculated from the particle picture (Fig. 3.6) agreeswith that calculated from the diamagnetic drift for one width of the box, then itwill agree for all widths.3.6 An isothermal plasma is confined between the planes x ¼ a in a magneticfield B ¼ B0^z . The density distribution isn ¼ n0 1x2 =a2(a) Derive an expression for the electron diamagnetic drift velocity vDe as afunction of x.(b) Draw a diagram showing the density profile and the direction of vDe onboth sides of the midplane if B is out of the paper.(c) Evaluate vDe at x ¼ a/2 if B ¼ 0.2 T, KTe ¼ 2 eV, and a ¼ 4 cm.3.7 A cylindrically symmetric plasma column in a uniform B field hasnðr Þ ¼ n0 expr 2 =r 20andni ¼ ne ¼ n0 expðeϕ=KT e Þ:(The latter is the Boltzmann relation, Eq.
(3.73).)(a) Show that vE and vDe are equal and opposite.(b) Show that the plasma rotates as a solid body.(c) In the frame which rotates with velocity vE, some plasma waves (driftwaves) propagate with a phase velocity vϕ ¼ 0:5vDe . What is vϕ in thelab frame? On a diagram of the r θ plane, draw arrows indicating therelative magnitudes and directions of vE, vDe, and vϕ in the lab frame.3.8 (a) For the plasma of Problem 3.7, find the diamagnetic current density jD asa function of radius.(b) Evaluate jD in A/m2 for B ¼ 0.4 T, n0 ¼ 1016 m 3, KTe ¼ KTi ¼ 0.25 eV,r ¼ r0 ¼ 1 cm.(c) In the lab frame, is this current carried by ions or by electrons or by both?3.9 In the preceding problem, by how much does the diamagnetic current reduceB on the axis? Hint: You may use Ampere’s circuital law over anappropriate path.3.10 In 2013, the Voyager 1 spacecraft left the heliosphere, the region dominatedby solar winds, and entered outer space.
The plasma frequency jumped from2.2 to 2.6 kHz. What was the change in plasma density?3.5Fluid Drifts Parallel to BThe z component of the fluid equation of motion is723 Plasmas as Fluids∂vzþ ðv ∇ Þvz ¼ qnEzmn∂t∂p∂zð3:70ÞThe convective term can often be neglected because it is much smaller than the ∂υz/∂t term. We shall avoid complicated arguments here and simply consider cases inwhich vz is spatially uniform. Using Eq. (3.52), we have∂vz q¼ Ezm∂tγKT ∂nmn ∂zð3:71ÞThis shows that the fluid is accelerated along B under the combined electrostaticand pressure gradient forces. A particularly important result is obtained by applyingEq.
(3.71) to massless electrons. Taking the limit m ! 0 and specifying q ¼ e andE¼∇ϕ, we have3qEz ¼ e∂ϕ γKT e ∂n¼∂zn ∂zð3:72ÞElectrons are so mobile that their heat conductivity is almost infinite. We may thenassume isothermal electrons and take γ ¼ 1. Integrating, we haveeϕ ¼ KT e ln n þ Corn ¼ n0 exp ðeϕ=KT e Þð3:73ÞThis is just the Boltzmann relation for electrons.What this means physically is that electrons, being light, are very mobile andwould be accelerated to high energies very quickly if there were a net force onthem.
Since electrons cannot leave a region en masse without leaving behind a largeion charge, the electrostatic and pressure gradient forces on the electrons must beclosely in balance. This condition leads to the Boltzmann relation. Note thatEq. (3.73) applies to each line of force separately. Different lines of force may becharged to different potentials arbitrarily unless a mechanism is provided for theelectrons to move across B.
The conductors on which lines of force terminate canprovide such a mechanism, and the experimentalist has to take these end effects intoaccount carefully.Figure 3.9 shows graphically what occurs when there is a local density clump inthe plasma. Let the density gradient be toward the center of the diagram, andsuppose KT is constant. There is then a pressure gradient toward the center.
Sincethe plasma is quasineutral, the gradient exists for both the electron and ion fluids.3Why can’t vz ! 1 keeping mvz constant? Consider the energy!3.6 The Plasma Approximation73Fig. 3.9 Physical reason for the Boltzmann relation between density and potentialConsider the pressure gradient force Fp on the electron fluid. It drives the mobileelectrons away from the center, leaving the ions behind.
The resulting positivecharge generates a field E whose force FE on the electrons opposes Fp. Only whenFE is equal and opposite to Fp is a steady state achieved. If B is constant, E is anelectrostatic field E ¼∇ϕ, and ϕ must be large at the center, where n is large.This is just what Eq. (3.73) tells us. The deviation from strict neutrality adjusts itselfso that there is just enough charge to set up the E field required to balance the forceson the electrons.3.6The Plasma ApproximationThe previous example reveals an important characteristic of plasmas that has wideapplication.
We are used to solving for E from Poisson’s equation when we aregiven the charge density σ. In a plasma, the opposite procedure is generally used.E is found from the equations of motion, and Poisson’s equation is used only to findσ. The reason is that a plasma has an overriding tendency to remain neutral. If theions move, the electrons will follow.
E must adjust itself so that the orbits of theelectrons and ions preserve neutrality. The charge density is of secondary importance; it will adjust itself so that Poisson’s equation is satisfied. This is true, ofcourse, only for low-frequency motions in which the electron inertia is not a factor.In a plasma, it is usually possible to assume ni ¼ ne and ∇ E 6¼ 0 at the sametime.
We shall call this the plasma approximation. It is a fundamental trait ofplasmas, one which is difficult for the novice to understand. Do not use Poisson’sequation to obtain E unless it is unavoidable! In the set of fluid equations (3.55)–(3.61), we may now eliminate Poisson’s equation and also eliminate one of theunknowns by setting ni ¼ ne ¼ n.743 Plasmas as FluidsThe plasma approximation is almost the same as the condition of quasineutralitydiscussed earlier but has a more exact meaning. Whereas quasineutrality refers to ageneral tendency for a plasma to be neutral in its state of rest, the plasma approximation is a mathematical shortcut that one can use even for wave motions.
As longas these motions are slow enough that both ions and electrons have time to move, itis a good approximation to replace Poisson’s equation by the equation ni ¼ ne. Ofcourse, if only one species can move and the other cannot follow, such as in highfrequency electron waves, then the plasma approximation is not valid, and E mustbe found from Maxwell’s equations rather than from the ion and electron equationsof motion. We shall return to the question of the validity of the plasma approximation when we come to the theory of ion waves.
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