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An erratum to this chapter can be found athttps://doi.org/10.1007/978-3-319-22309-4_11© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_61871886 Equilibrium and StabilityFig. 6.1 Mechanical analogy of various types of equilibriumOf the two problems, equilibrium and stability, the latter is easier to treat. One canlinearize the equations of motion for small deviations from an equilibrium state. Wethen have linear equations, just as in the case of plasma waves.
The equilibriumproblem, on the other hand, is a nonlinear problem like that of diffusion. In complexmagnetic geometries, the calculation of equilibria is a tedious process.6.2Hydromagnetic EquilibriumAlthough the general problem of equilibrium is complicated, several physicalconcepts are easily gleaned from the MHD equations. For a steady state with∂/∂t ¼ 0 and g ¼ 0, the plasma must satisfy (cf. Eq. (5.85))6.2 Hydromagnetic Equilibrium189Fig. 6.2 The j B force ofthe diamagnetic currentbalances the pressuregradient force in steadystate∇p ¼ j Bð6:1Þ∇ B ¼ μ0 jð6:2ÞandFrom the simple equation (6.1), we can already make several observations.(a) Equation (6.1) states that there is a balance of forces between the pressuregradient force and the Lorentz force. How does this come about? Consider acylindrical plasma with ∇p directed toward the axis (Fig.
6.2). To counteractthe outward force of expansion, there must be an azimuthal current in thedirection shown. The magnitude of the required current can be found by takingthe cross product of Eq. (6.1) with B:j⊥ ¼B ∇pB ∇n¼ ðKT i þ KT e ÞB2B2ð6:3ÞThis is just the diamagnetic current found previously in Eq. (3.69)! From asingle-particle viewpoint, the diamagnetic current arises from the Larmorgyration velocities of the particles, which do not average to zero when thereis a density gradient. From an MHD fluid viewpoint, the diamagnetic current isgenerated by the ∇p force across B; the resulting current is just sufficient tobalance the forces on each element of fluid and stop the motion.(b) Equation (6.1) obviously tells us that j and B are each perpendicular to ∇p.This is not a trivial statement when one considers that the geometry may bevery complicated.
Imagine a toroidal plasma in which there is a smooth radialdensity gradient so that the surfaces of constant density (actually, constant p)are nested tori (Fig. 6.3). Since j and B are perpendicular to ∇p, they must lieon the surfaces of constant p. In general, the lines of force and of current maybe twisted this way and that, but they must not cross the constant-p surfaces.1906 Equilibrium and StabilityFig. 6.3 Both the j and B vectors lie on constant-pressure surfacesFig. 6.4 Expansionof a plasma streaminginto a mirror(c) Consider the component of Eq. (6.1) along B. It says that∂ p=∂s ¼ 0ð6:4Þwhere s is the coordinate along a line of force.
For constant KT, this means that inhydromagnetic equilibrium the density is constant along a line of force. At firstsight, it seems that this conclusion must be in error. For, consider a plasmainjected into a magnetic mirror (Fig. 6.4). As the plasma streams through,following the lines of force, it expands and then contracts; and the density isclearly not constant along a line of force.
However, this situation does not satisfythe conditions of a static equilibrium. The (v · ∇)v term, which we neglectedalong the way, does not vanish here. We must consider a static plasma withv ¼ 0. In that case, particles are trapped in the mirror, and there are moreparticles trapped near the midplane than near the ends because the mirror ratiois larger there. This effect just compensates for the larger cross section at themidplane, and the net result is that the density is constant along a line of force.6.3The Concept of βWe now substitute Eq. (6.2) into Eq.
(6.1) to obtainhi11ðB ∇ ÞB ∇B2∇ p ¼ μ10 ð∇ BÞ B ¼ μ02ð6:5Þ6.3 The Concept of β191Fig. 6.5 In a finite-βplasma, the diamagneticcurrent significantlydecreases the magneticfield, keeping the sum of themagnetic and particlepressures a constantorB21∇ pþ¼ ðB ∇ ÞBμ02μ0ð6:6ÞIn many interesting cases, such as a straight cylinder with axial field, the right-handside vanishes; B does not vary along B. In many other cases, the right-hand side issmall. Equation (6.6) then says thatpþB2¼ constant2μ0ð6:7ÞSince B2/2 μ0 is the magnetic field pressure, the sum of the particle pressure and themagnetic field pressure is a constant.
In a plasma with a density gradient (Fig. 6.5),the magnetic field must be low where the density is high, and vice versa.The decrease of the magnetic field inside the plasma is caused, of course, by thediamagnetic current. The size of the diamagnetic effect is indicated by the ratio ofthe two terms in Eq. (6.7). This ratio is usually denoted by β:PParticle pressurenkTð6:8Þβ 2¼B =2μ0 Magnetic field pressureUp to now we have implicitly considered low-β plasmas, in which β is between103 and 106. The diamagnetic effect, therefore, is very small.
This is the reasonwe could assume a uniform field B0 in the treatment of plasma waves. If β is low, itdoes not matter whether the denominator of Eq. (6.8) is evaluated with the vacuumfield or the field in the presence of plasma. If β is high, the local value of B can begreatly reduced by the plasma.
In that case, it is customary to use the vacuum valueof B in the definition of β. High-β plasmas are common in space and MHD energyconversion research. Fusion reactors will have to have β well in excess of 1 % in1926 Equilibrium and Stabilityorder to be economical, since the energy produced is proportional to n2, while thecost of the magnetic container increases with some power of B.In principle, one can have a β ¼ 1 plasma in which the diamagnetic currentgenerates a field exactly equal and opposite to an externally generated uniformfield. There are then two regions: a region of plasma without field, and a region offield without plasma.
If the external field lines are straight, this equilibrium wouldlikely be unstable, since it is like a blob of jelly held together with stretched rubberbands. It remains to be seen whether a β ¼ 1 plasma of this type can ever beachieved. In some magnetic configurations, the vacuum field has a null inside theplasma; the local value of β would then be infinite there. This happens, for instance,when fields are applied only near the surface of a large plasma.
It is then customaryto define β as the ratio of maximum particle pressure to maximum magneticpressure; in this sense, it is not possible for a magnetically confined plasma tohave β > 1.6.4Diffusion of Magnetic Field into a PlasmaA problem which often arises in astrophysics is the diffusion of a magnetic field intoa plasma. If there is a boundary between a region with plasma but no field and aregion with field but no plasma (Fig.
6.6), the regions will stay separated if theplasma has no resistivity, for the same reason that flux cannot penetrate a superconductor. Any emf that the moving lines of force generate will create an infinitecurrent, and this is not possible. As the plasma moves around, therefore, it pushesthe lines of force and can bend and twist them. This may be the reason for thefilamentary structure of the gas in the Crab nebula. If the resistivity is finite,however, the plasma can move through the field and vice versa. This diffusiontakes a certain amount of time, and if the motions are slow enough, the lines of forceFig.
6.6 In a perfectly conducting plasma, regions of plasma and magnetic field can be separatedby a sharp boundary. Currents on the surface exclude the field from the plasma6.4 Diffusion of Magnetic Field into a Plasma193need not be distorted by the gas motions. The diffusion time is easily calculatedfrom these equations (cf.
Eq. (5.91)):∇ E ¼ B_ð6:9ÞE þ v B ¼ ηjð6:10ÞFor simplicity, let us assume that the plasma is at rest and the field lines are movinginto it. Then v ¼ 0, and we have∂B=∂t ¼ ∇ η jð6:11ÞSince j is given by Eq. (6.2), this becomes∂Bηη¼ ∇ ð∇ BÞ ¼ ∇ ð∇ BÞ ∇ 2 B∂tμ0μ0ð6:12ÞSince ∇ · B ¼ 0, we obtain a diffusion equation of the type encountered in Chap. 5:∂Bη¼ ∇2 B∂tμ0ð6:13ÞThis can be solved by the separation of variables, as usual.
To get a rough estimate,let us take L to be the scale length of the spatial variation of B. Then we have∂Bη¼B∂tμ0 L2ð6:14ÞB ¼ B0 et=τð6:15Þτ ¼ μ0 L2 =ηð6:16ÞwhereThis is the characteristic time for magnetic field penetration into a plasma.The time τ can also be interpreted as the time for annihilation of the magneticfield.
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