An equilibrium state of the plasma confined by magnetic fields must exist in order to maintain the hot plasma away from the vessel walls. While the general solution of equilibrium states of a magnetically confined plasma constitutes a complicated problem, some useful physical relations follow readily from viewing the plasma as a single fluid (Sec. 6.3) subjected to the magnetohydrodynamic (MHD) description. The equilibrium exploration suggests that one consider the steady state MHD equations from which the plasma current density j and the

magnetic field В can be self-consistently derived, i. e., in agreement with Maxwell’s Equations. Consequently, an equilibrium plasma must satisfy the previously introduced force balance (Eq.(9.2))

Vp = j xB (10.22)

and the magnetostatic Maxwell equations

VxB = ^oj (10.23)

and

V-B = 0. (10.24)

As in Sec. 9.1, Eq. (10.22) implies that j and В lie on isobaric surfaces. Note that taking the divergence of Eq. (10.23) yields

V-j = 0 (10.25)

which corresponds with the quasi-neutrality assumption Ni ~ Ne.

For an axisymmetric tokamak plasma the isobaric surfaces will be nested toroidal surfaces, as previously mentioned. We assign now to each such surface a function ¥ = constant such that 2тгіЕ represents the poloidal magnetic flux through a toroidal plane extending from the minor axis (r = 0) to the considered isobaric surface at r represented by the shadowed area in Fig. 10.8, i. e.

2^: = jBe — dre. (10.26)

Since isobaric surfaces need not necessarily have circular cross sections, here r0 denotes the radial distance of an isobaric surface at a specified poloidal position

0. Then we may approximate the flux passing through the planar ribbon between the two isobaric surfaces labeled 4* and 4* + d^ and encircling the major axis with a radius R by

2л(‘¥ + cM>) — 2iW = JBe • dre — jBe • dre — Be2nRdr (10.27)

о о

with dr representing the radial distance between the considered surfaces. From Eq.( 10.27), the following relation between the poloidal magnetic field strength and the flux surface function ¥ is now apparent:

dfV

RBe=————————————————— (10.28)

dr

or, respectively, in vectorial form according to the orientation shown in Fig. 10.8,

Bfl=4*xv*- (io-29>

к

Similarly, due to the symmetry of j and В in Eq. (10.22), there exists a corresponding poloidal current function F^) such that the poloidal component of the current density can be expressed as

j e = 4e*xVF

which-via Eq. (10.23)-determines the toroidal field as

Вф = -^ЄФ — (Ю.31)

К

Since F = constant for ‘F = constant, it is therefore also seen that B^xR is constant on a magnetic surface.

Next, we decompose the poloidal magnetic field into its components along the R and z axes of Fig. 10.8 by multiplying Eq.( 10.29) with the respective unit vectors eR and ez. Noting that

we find

1 ЭЧ*

(,0.33a)

and

(10.33b)

Using these expressions the toroidal component of Eq. (10.23) is then

d_J_W 1 92vE dR R dR R dz2

which defines the operator

dR R dR dz2

used in the following. Upon introducing the flux and the current function in the force balance, Eq. (10.22),

Vp = (j, + je)x (в* + Be) = (i* "j^e* * VF j x (вф — x VV j, (10.36a)

and recognizing the pertinent orthogonalities, we obtain

Vp(T’) = ^-VT’-^-VF(T’). (10.36b)

R R

Since Vpf’E) = VT’ and VFf’F) = VT’, Eq.(10.36b) provides the relation

which is a nonlinear elliptic partial differential equation for the magnetic flux 4* in terms of the pressure distribution and the poloidal current function F. The corresponding boundary conditions are provided by the transformer-induced poloidal magnetic field outside the plasma. In evaluating Eq. (10.38) it is seen and has been discussed earlier that the total magnetic field must also possess a uniform vertical part Bv in order to prevent the current-carrying plasma from radial expansion; only then can a toroidal plasma be maintained in equilibrium. The topological magnetic field structure effected by the imposition of Bv is demonstrated in Fig. 10.9 where, in the resulting field, the domain of closed magnetic surfaces is separated by a so-called separatrix from the open surface contours intercepting the wall of the reactor chamber.

In practice, the Grad-Schlueter-Shafranov Equation, Eq.( 10.38), is solved numerically to find the geometrical location of the magnetic surfaces and the radial distribution of the axial current density in a way that is consistent with the

experimentally measured pressure profiles and the externally applied field. That is, the functions p(‘F) and F(vf/) are given for the numerical evaluation. For plasma pressures which are not too high, solutions of Eq.( 10.38) exist in the form of closed nested flux surfaces inside the separatrix indicated in Fig. 10.9.

which is the ratio of the volume-averaged plasma pressure to the energy density of the poloidal magnetic field averaged over the magnetic flux surface at r = a. Figure 10.10 exhibits some typical features of toroidal equilibria as they are affected by the plasma pressure. A more detailed analysis shows that the toroidal

field contributes to the pressure balance by the difference } ~ (5p2)

affected by the poloidal current components which may either increase or decrease the toroidal field in the plasma. A plasma is paramagnetic when the volume-averaged square of the toroidal field strength, (B^), exceeds the outer

flux surface average, (b^ ^ and it is diamagnetic for (b^ > (В,2). Asa

consequence, the diamagnetic plasma is associated with an average plasma pressure greater than ^Вe2^ J{2jio), that is with pp > 1, while pp < 1

corresponds to a paramagnetic plasma. Whereas the total beta of a tokamak plasma is constrained to values of a few percent for MHD stability reasons, the poloidal beta appears in the range 0.1 < pp < 2.5 for the equilibrium regimes displayed in Fig. 10.10.

It is observed that the magnetic axis of the torus column is displaced outwards with increasing plasma pressure until this displacement D is about half of the minor plasma radius, which can be shown to manifest the limit of

Surfaces on which p = constant

Surfaces on which j(j, = constant

diamagnetic behaviour of plasma decreases toroidal field; occurrence of reverse current

Fig. 10.10: Axisymmetric tokamak equilibrium characteristics varying with plasma
pressure. The symbol l; appearing in b) denotes the plasma internal inductance per unit

2 n

Byrdr

™2(Be)v(a)

equilibrium. The poloidal magnetic field must be capable of supporting at least the fraction a/R0 of the kinetic pressure, that is

2 Vo Ro

which obviously restricts (3P to values

Pp<R0/a. (10.40b)

Recalling the Kruskal-Shafranov stability limit in Sec. 10.1 which suggests that the safety factor at the plasma edge should be q(a) > 2.5, and further extracting from the geometric sketch in Fig. 10.11 the obvious relation

Be (а) аАв

by using the previous definitions l = Д0(Дф = 2%) and q = 2m.. Taking the square of Eq.(10.42) and assuming the approximate equivalence of

Be (a) with (я2 ) and of Я2 with [в] ) ^, we express the averages of

both Be2(a) and Вф2 by means of their (З-value to obtain <P> . о _ a PP ^ 1 a

Вф/2ц0 * Ro q2(<*)~ 6.25 Rl

which essentially is the toroidal beta, (3,, for large aspect ratios. Hence, for example, a tokamak featuring an aspect ratio Ro/a = 3 will be limited to (3 < 5% by principal MHD stability considerations, while the poloidal beta may amount to values in the order of Ro/a.

Another important consequence resulting from the poloidal-beta limitation

becomes evident when the poloidal magnetic induction is expressed-via Ampere’s Law-by the plasma current generating the Be-field. Doing so, we introduce Eq. (9.50) into Eq. (10.42) to obtain

JL->q(a)^- (10.47)

Vo1 a

2m

That is, the plasma current cannot be increased arbitrarily in order to achieve higher temperatures through Ohmic heating. Its magnitude appears to be bounded for stability reasons. We inject to note that the above derivation of plasma-beta constraints associated with simple MHD stability limits represents just a rough assessment nonetheless demonstrating the useful and important relations in an illustrative manner. Thorough analysis would provide more accurate factors in the several inequalities at the expense of extended expressions and more detailed concepts to be introduced.

The history of tritium accumulation in the breeding blanket and its eventual extraction for recycle to the fusion reactor core is of paramount importance. Consider then a unit volume in the blanket in which tritium gains and losses occur:

(14.23)

The gain-rate is clearly proportional to the rate of neutron production in the core so that, in view of the preceding section, we write

R+t = Rne., = Ct Rdt • (14.24)

Tritium removal in a unit volume of the breeding blanket occurs by radioactive decay and extraction:

• (14’25)

Here, the radioactive decay component is

and the extraction rate is taken to be a constant fraction of the tritium in the blanket:

Here xlb is the mean residence time of the tritium in the blanket. Hence, the dynamical equation for the tritium inventory in the blanket, Eq.( 14.23), is therefore

where an equivalent time constant for tritium in the blanket xb had been defined as

— = A,+ — . (14.29)

With Rdt, C, and xb as constant, Eq.( 14.28) is in a standard form amenable to solution given an initial tritium density in the blanket. Suppose that start-up of a fusion reactor is of interest so that N, ib(0) = 0, then the solution of Eq. (14.28) is

NtM(t)=CtRdtzb(l-eM. (14.30)

The substitution Rd, = Pdt/Q dt renders the tritium accumulation in the unit volume of the blanket particularly evident.

The mean residence time of the tritium bred in the blanket, x, b of Eq.( 14.27), now appears as an important operational parameter. For cases where T,,b is short relative to the inverse tritium decay constant, i. e., x, b « 1/A,, or, alternatively, very long, i. e., x, ib » 1/?ц, a most distinct tritium accumulation and equilibrium concentration in the blanket results, Fig. 14.5. The asymptotic quantity of accumulated tritium is thus directly proportional to the power and the time constant xb.

Some numerical experimentation with Eq.( 14.30) will show that it may take decades for the bred tritium to approach the equilibrium value Nt b(°°). It may also be desirable to introduce either continuous or periodic removal of the tritium while its accumulation is still small; we also illustrate this periodic batch removal of the tritium in Fig. 14.5.

Fast Tritium Extraction (xt b «1/XJ

1.1 Determine the energy released per mass-of-atoms initially involved for a chemical process, Eq.(1.2), for nuclear fission, Eq.(1.3), for nuclear fusion, Eq.(1.5) and for a water molecule falling through a 100 m elevation difference in a hydroelectric plant.

1.2 Calculate the reaction Q-values for each of the two branches of the d-d fusion reaction, Eq.(1.21).

1.3 What fraction of the original mass in d-t fusion is actually converted into energy? Compare this to the case of nuclear fission, Eq.(1.3).

1.4 Calculate the kinetic energies of the reaction products h and a resulting from p-6Li fusion, Eq.( 1.24a), ignoring any initial motion of the reactants.

1.5 Calculate the total fusion energy, in Joules, residing in a litre of water if all the deuterons were to fuse according to Eq.(1.21).

1.6 Redo problem 1.5 including the burning of the bred tritium according to Eq. (1.20).

1.7 Consult an astronomy text in order to estimate the mean fusion power density (Wm‘3) in the sun; compare this to a typical power density in a fission reactor.

1.8 The first artificial nuclear transmutation without the use of radioactive substances was successfully carried out in the Rutherford Laboratory by Cockcroft and Walton when they bombarded Lithium (at rest) with 100 keV proton canal rays (protons accelerated by a voltage of 100 kV and passing through a hole in the cathode). By scintillations in a Zincblende-screen, the appearance of a-particles with a kinetic energy of 8.6 keV was determined.

(a) Formulate the law of energy conservation valid for the above experiment referring to the nuclear reaction

]Li+H -» 2 "He

and find therefrom the reaction energy, Qp7u, via the involved rest masses (mp, m«= 6.64455 x 10’27 kg, m7Li = 11.64743 x 10’27 kg).

(b) In the Cockcroft-Walton experiment, conservation of momentum was proven by cloud chamber imaging whereby it was observed that the tracks of the two a-particles diverge at an angle of 175°. What angle follows from the law of momentum conservation by calculation?

(c) A further reaction induced by the protons in natural lithium is

з U+H-*lHe+ He

Provide an argument that shows the a-particles detected in the Cockcroft-Walton experiment cannot stem from this reaction.

A more comprehensive d-t isothermal bum analysis needs to include not only the destmction of deuterium and tritium by fusion reactions but also the injection and leakage rates of ions into and from the unit volume of interest. That is, the rate density equation, Eq.(7.3), referring to a fixed point in space is now given by the following:

The ion injection rate density, which is a reactor operations function, will be represented by the fueling rate density

where here, and in the following, we omit the + in front of the j. Ion leakage from a unit volume may be taken to be proportional to the ion density so that

where Xj is a characteristic time parameter representing the local mean residence time of the ions. The determination of this time quantity in terms of various reactor parameters is a continuing task of fusion research but for present purposes here, Xj is taken as a known parameter. Recalling our previous analysis leading to a global particle confinement time and noting that

L=E±=J_ r

x* Nj Nj{Tj(r, t)

suggests thereby that the global particle confinement time is the density-weighted volume average of the local mean residence time. Both these residence times
generally vary with time.

The more complete fuel balance equations for d-t bum are therefore given by the coupled set of equations dNd

dt

dN,

dt

or combining, as was done for Eq.(7.5), d

dt

(7.21)

(7.22)

(7.23)

where the subscript і refers to all fuel ions. This then leads to the dynamical equation for the fuel ion density at a fixed position in space and at some time t, for the three processes here identified, as

dNi

dt Т/ 2

This is a nonlinear first order differential equation, frequently called Ricatti’s equation, and like many nonlinear equations in general, possesses some subtle features. However, by combining several analytical, geometrical and physical notions, we can extract some useful qualitative information about the time variation of the fuel ion density N;(t) at a fixed position in space and hence about the power density Pdt(t), Eq.(7.2b).

For algebraic simplicity, we write Eq.(7.24) as

—— = ao + ai N + a.2 N2 (7.25)

dt

with ao = Fj, ai = -1/ti and a2= -<Ov>dt/2. For the special case when these parameters are constant, then the following information about the slope of an N vs. t plot at an arbitrary time can be specified:

ao > 0 for sufficiently small N

slope (=ao + aiN + a2N )=<

[~ a2 N <0 for sufficiently large N. ^ ^

Further, for the special case of as constant, any fixed points of Eq.(7.25)-to be represented by №-are the solutions of dN/dt = 0; that is

ao + ai № + a2( № f = 0

The above information about the slope of N(t) for small and large N’s, and the recognition that N(t) —» № for t —> °o-and the assumption that ao, aj and a2 can be adequately represented as suitable constants in time-allows us to sketch the solution of Nj(t) of Eq.(7.24) as depicted in Fig.7.3.

The unique design features of an ICF power system relate to the reaction chamber and ancillary components. Underlying these features-with critical consequences on design criteria-are the nuclear-atomic energetics processes leading to fusion with the evident requirement of economic and self-sustaining performance of the system.

Fuel pellet manufacture involves spherical coating technology at the micro­scale of composition and geometry. Entry of the pellets into the bum chamber will occur by gravity combined with pneumatic injection demanding, however, extreme trajectory precision. Then, an inordinate quantity of energy has to be deposited into this small pellet by laser or ion beam impingement within the short time of about 10’9 s. The expected multiplied quantity of energy over the bum time Хь = 10’8 s, now residing in the high kinetic energy of the fusion reaction products as well as in various electromagnetic flows and an assortment of debris, will spread out striking a liquid or metallic first wall surface. Both surface and internal radiation damage will occur as well as energy deposition-which needs to be recovered at the average rate that it is deposited. Simultaneously, tritium breeding by neutron capture will occur providing thereby eventual replacement of the scarce tritium fuel. Further aspects of the processes and reaction involved in the blanket surrounding the fusion chamber are addressed in Ch. 13.

Following each pulse, rapid purging of the reaction chamber needs to be undertaken in preparation for the next pulse. This operational cycling is expected to be at a frequency of about 1 Hz or greater, the associated fuel injection rate correspondingly being F+i* as previously introduced.

Extremes of power transport, energy conversion, material flow, radiation damage, and highly co-ordinated electro-mechanical functions will evidently characterize the eventual operation of an ICF power system. Considerable research, design, and testing will still need to be undertaken to arrive at the continuingly elusive goal of such a working power station.

Problems

11.1 Evaluate Ть, Eq.( 11.10) as a function of fb for kT = 20 keV; take рь = 500 Pt-

11.2 Discuss the averaging process for < OV >dt and explain the approximation for it made to evaluate Eq.(l 1.15).

11.3 Derive a relation between pR and the compression ratio, pb < pt, of a simple spherical target. Is a spherical target (e. g. microballoon) target advantageous compared to a disk or planar target?

11.4 Calculate the laser energy required to heat a spherical 50:50% D-T pellet, which attains pbRb = 3 g em’2, to an average kinetic temperature of kT = 20 keV as a function of the pellet density p assuming that only 5% of the laser light is absorbed in the pellet fusion plasma. Evaluate this expression for the cases of

(a) solid density (frozen state) ps = 0.22 g em’3

(b) pb=104ps

and ascertain the corresponding pellet radius in each case as well as the respective confinement times, XjC, the laser power requirements and the fusion energy release for a 10% bumup fraction.

11.5 Formulate the exact calculation of Efu* in Eq.(11.33). Can the suggested

Inertial Confinement Fusion proportionality be validated?

11.6 Undertake an analysis of the difference between the physical processes involved in energy deposition by laser beams and ion beams.

An important consequence of scattering in fusion plasmas is bremsstrahlung radiation, which refers to the process of radiation emission when a charged particle accelerates or decelerates. It involves the transformation of some particle kinetic energy into radiation energy which-due to its relatively high frequency (X-ray wavelength range of ~ 10’9 m)-may readily escape from a plasma; thus, the kinetic energy of plasma particles is reduced, plasma cooling occurs, and a compensating energy supply may be required in order to maintain the desired plasma temperature.

A rigorous derivation of bremsstrahlung power emission in a hot plasma involves considerations of quantum mechanics and relativistic effects and is both tedious and time consuming. Indeed, even advanced methods of analysis require the imposition of simplifying assumptions if the formulation is to be at all tractable. It is, however, less difficult to develop an approximation of the dominant effect of the bremsstrahlung processes by the following considerations. A particle of charge qe and moving with a time varying velocity v(t) will — according to classical electromagnetic theory in the nonrelativistic limit-emit radiation at a power

In Fig. 3.6, we suggest this process for an electron moving in the electrostatic field of a heavy ion. To estimate the energy radiated per encounter, we replace Idv/dtl in Eq.(3.35) by an average acceleration a to be calculated from Newton’s Law, a = Fc / m, with Fc representing the magnitude of the average Coulomb

force approximated here by the electrostatic force between the two interacting particles when separated by the impact parameter r0. That is, we take

7 Fc(r„) Zq a ~——- = ;

for the average acceleration experienced by a particle of mass m and charge qe in the field of a charge qi = — Zqe. Since an electron possesses a mass 1/1836 of that of a proton-with an even smaller ratio existing when compared to a deuteron or triton-the electrons in a plasma will therefore be the main contributors to bremsstrahlung radiation. We suggest this in Fig. 3.7 for a representative electron trajectory undergoing significant directional changes in a background of sluggish ions.

Imagining an electron (me, qe) to be in the vicinity of an ion (m„ |q;| = IZqel) for a time interval

(3.37)

with vr denoting the average relative speed between the electron and ion, we combine Eqs.(3.35) to (3.37) to assess the bremsstrahlung radiation energy per collision event as

.. ~^r. (3-38)

collision nie Го Vr

Note this expression refers to the collision of a single electron with an ion at the specific impact parameter r0. Extending these considerations to a bulk of electrons of density Ne all approaching the ion with r, then the number of electrons colliding per unit time with the same ion at the same impact parameter r0 is given by

— ~Nevr2Krodr0. (3.39)

ion

Multiplying this relation by the ion density Nj we obtain the differential electron — ion collision rate density

dR0~ NeNiVr2Krodr0 (3.40)

thereby accounting for all electron-ion interactions per unit volume and per unit time occurring about a specific impact parameter r0.

Next, in order to obtain the total bremsstrahlung radiation power density associated with the entire electron-ion force impact area, Eq. (3.38) needs to be multiplied by Eq.(3.40) and integrated over the range of the impact parameter Го. тіп to romax; that is we now obtain for the bremsstrahlung power the proportionality

While r0>max = 00 can be imposed, a finite minimum impact parameter needs to be specified due to the integral singularity for r0 —> 0. We choose to identify ro min with the DeBroglie wavelength of an electron, that is,

г_ = — Л — (3.42)

me ve

where h is Planck’s constant and ye represents the average thermal electron speed defined by Eq.(2.19b):

With the integration limits thus specified, the integral of Eq.(3.41) is readily evaluated so that the electron bremsstrahlung radiation power density is found to exhibit the following dominant dependencies:

Ры = Аы Ni Ne Z2 JkT (3.44)

where Abr is a constant of proportionality. For kT in units of eV and particle densities in units of m’3,

which yields Pbr in units of Wm. An important point to note here is that Pbr is proportional to (kT)1/2.

Problems

3.1 Examine the relationship between 0C and 0L for the limiting cases of

mb/ma = {0, 1, =»}.

3.2 For a typical case of d-t interaction at 5 keV, plot as(0c), 0 < 0C < n.

3.3 For the conditions in problem 3.2, calculate as assuming a background particle density of 1020 m’3.

3.4 Calculate the bremsstrahlung power increase if a d-t plasma were to contain totally stripped oxygen ions at a concentration of 1% of the electron density.

3.5 Calculate the ratio of bremsstrahlung power to fusion power for a d-t plasma with N; = Ne = 1020 rn3 at 2 keV and 20 keV.

3.6 Determine from a plot of power density versus kinetic temperature (use a double logarithmic scale) for a 50:50% D-T fusion plasma

(a) the so called ideal d-t ignition temperature T, i. e. the temperature at which the plasma fusion power density equals the loss power density due to bremsstrahlung.

(b) the temperature T jgn more relevant to fusion ignition since it refers to the plasma operational state where the bremsstrahlung loss power is balanced by the energy transfer from the fusion alphas to the background plasma per unit time by Coulomb collisions. Assuming that the charged particle fusion power is entirely transferred, T ign is found from

with fcdt representing the fraction of d-t fusion power allocated to charged particles (in this case a’s). What does this condition mean for the overall power balance of the fusion plasma if other heat losses were neglected?

3.7 A fusion reactor using two opposing accelerators is proposed, where a 30 keV tritium beam from one is aimed head on at a 30 keV deuterium beam from the other. Would this work, and can you suggest improvements? What would you estimate is the maximum energy gain possible with this system (see Ch. 8)?

Part II Confinement, transport, Burn

In Chs. 4 and 5, we have introduced some basic considerations of magnetic confinement. To begin this more detailed look, we consider the containment capacity of magnetic fields and subsequently discuss specific reductions.

The macroscopic fluid description of a plasma does not clearly suggest how individual charged particles could be confined by a magnetic field. Even if the external В-field is taken to consist of smooth straight field lines, the plasma could generate internal fields which might cause drift motion leading to particle escape from the confinement region. An effective containment will require the plasma particles to be confined to some region for a sufficient time period. We may therefore assert that, in general, this is associated with a state of the macroscopic fluid in which all identified forces are exactly in balance providing thereby a time-independent solution. We consider therefore Eq.(6.25) for steady state, i. e. 3Vj/5t = 0, and obtain for the isotropic case (Vj-V)V, = 0 in the absence of an electric field, the defining equation

Vpj = pCjjx B. (9.1)

Imposing this equation for plasma ions, j = i, and electrons, j = e, we write for the total plasma pressure gradient

Vp = W Pi + pe)

= (р-У; + pcee)x В (9.2)

= jxB.

Evidently, j is the electric current density which must also satisfy Maxwell’s equation

j = Vx— . (9.3)

V0

We note that Eq.(9.2) asserts that in equilibrium, there exists an equivalence between the pressure gradient force and the Lorentz force. Further, the electric current density j and the magnetic field В are perpendicular to Vp; that is, as suggested in Fig. 9.1, j and В lie on an isobaric surface because everywhere Vp is a normal to the surfaces p = constant.

Introducing Eq.(9.3) into (9.2) leads, upon some differential vector analysis, to the expression

>2

or, equivalently,

(9.5b)

This relationship is an important characterization of a hydrodynamic equilibrium state. It makes evident that the total sum of kinetic pressure and magnetic field energy density

Emag _ BH _ B2 V ~ 2 ~2/л0’ [2]

where H is the local field strength, will be a constant everywhere within the confined plasma. Hence, the pressure gradient Vp induces a current in the direction shown in Fig. 9.1 and which, by induction, causes an according decrease in the magnetic field in the plasma. From a particle perspective, we could as well have used Eq.(5.36) to derive a drift due to the pressure gradient force — Vp with the velocity

_ VP x B тал

VD. Vp——————— T

NqB

for a fluid element containing N particles. In view of the differences in ion and electron drifts, we define the electric current density by

В x Vp

В and label it the diamagnetic current. The above expression is essentially equivalent with the previous current density definition in Eq.(9.2) which, when cross-multiplied by B, results in Eq. (9.8), recognizing that j is perpendicular to B.

Characterizing the thermodynamic state of the plasma by the Ideal Gas Law

pV = N*kT (9.9)

with N* as the total number of particles in volume V, we obtain an expression for the total kinetic pressure for an ensemble of ions and electrons as

р = ЖкТ+^кТе

V V

= NikT + N±T

= (Ni + Ne )kT.

Both ions and electrons are, in this last expression, taken to be characterized by the same temperature Tj = Te = T for their Maxwellian distributions.

The extent of coupling between the magnetic pressure Eq.(9.6) and corresponding kinetic pressure Eq.(9.10) is a characteristic of a particular magnetic system and defines the dimensionless beta-parameter as the ratio of these two pressure terms:

a _ ( Ni + Ne )kT

B2/2p0

In a plasma with a pressure gradient the magnetic field is, as suggested by Eq.(9.5b), low where the particle pressure-or mass density-is high and vice — versa. Some magnetic configurations exist which produce a diamagnetic current such that the internal magnetic field may, in a limited region, be reduced to a minimum value or even to B=0; obviously, the local (З-value would then approach infinity. As a consequence, it is common to define (3 as the ratio of maximum particle pressure-e. g. at the centre of the plasma cylinder, if we refer to Fig. 9.1-to the maximum magnetic pressure-e. g. at the outer surface of the
plasma cylinder-thereby limiting the P-value to P < 1. In most magnetic configurations, fusion plasma confinement requires an imposed magnetic pressure which significantly exceeds the intrinsic particle kinetic pressure. A low beta facility is typically characterized by P < 0.1. In Ch. 4 it was shown that the fusion power density varies as p2 so that a high P facility is desirable recognizing however a limitation due to plasma instabilities.

The first wall surrounding a fusion plasma is bombarded by both electromagnetic radiation and escaping plasma particles. Numerous effects such as temperature changes, thermal stresses, and erosion must be considered in wall design. One important effect we will discuss here is that of material erosion by sputtering. In this process, the incident ions or neutrals from the plasma possess sufficient energy to cause "billiard ball"-type collisions in the wall material, possibly leading to the ejection of wall atoms into the plasma, as suggested in Fig. 13.4.

To gain some insight into this important plasma-wall interaction, consider the process suggested in Fig. 13.5. A plasma ion of mass mi and velocity Vj enters the
first-wall and collides elastically with a stationary atom of mass m2. Using the notation of Fig. 13.5, we write the momentum and energy balances for such elastic events as

/ /

mxvx=mxvx cos 6x+m2v2 cos 62 (component 11 to vt) (13.4a)

/ /

mxvx sin01=m2v2 sin62 (component _Lto v,) (13.4b)

and

where Ei„c is the kinetic energy of the incident ion.

The other important parameter is the average distance a plasma ion moves into the first wall before it collides with an atom. Since all two-body collisions can be characterized by a microscopic cross-section, we use

<Jcoi = nR I2 (13.8)

where Ri>2 is the distance of closest approach of an ion of mass m, and energy Einc to characterize the interaction with the wall atom of mass m2 in the elastic collision process. Taking these latter atoms to be of density N2 then gives the relevant macroscopic cross-section £coi as

1ы = о col N2 • (13.9)

Further, the mean-free-path of material penetration is

Ai,2

Hence, it follows that

(13.13)

F i>u oc EincGcol N2

where К is a proportionality constant and the latter expression emphasizes the particularly important dependencies.

Experimentally measured sputtering ratios are presented in Fig. 13.6 for various first wall materials bombarded by monoenergetic deuterons. The sputtering ratios generally increase with energy of the incident ion-as suggested in Eq.(13.13)-until a point is reached where most collisions occur at such a depth that the probability of escape of the knock-ons is substantially reduced; thereafter, the yield decreases with incident energy. Hence, Eq.(13.13) is only applicable to the lower energy region of these curves since it neglects the effect of subsequent collisions or knock-ons.

In addition to physically weakening the wall as material is sputtered away, sputtered material that enters the plasma can have serious effects on plasma energetics. This will be considered next.

Consider then charged particle motion in a magnetic field В whose field lines possess a radius of curvature R. In examining such a case we specify here the curved magnetic field lines by

В = B0k + B0~ri

К

when referred to in the vicinity of the у-axis, that is for Iz/RI « 1. A graphical depiction is provided in Fig. 5.10.

Introducing this field configuration, Eq. (5.52), into the Eorentz-force equation, Eq.(5.6), yields

m— = q( x В k) + q dt V ’

For small curvatures, i. e. Iz/RI « 1, we may, as in the previous cases, approximate the product z(vxj) by the undisturbed uniform solutions given in

(v, c x k) + —[- viiі + v± cos( ft> r)k] (5.55)

at m mR L J

where we specified z0 = 0 and the initial phase angle of Eq. (5.48) as ф = 0. Further substituting qBJm by Eq. (5.12), we separate Eq. (5.55) into its Cartesian components and write

Apparently, Eq. (5.56c) can be solved straightforwardly upon knowing that vgc z(0) = V||. Averaging the resulting velocity vgcz over a gyration period will suppress the small oscillations with the cyclotron frequency cog and indicate a steady drift anti-parallel to the В-field; this velocity drift, however, is seen to depend on the initial condition, v± = v0, and is therefore not classified as a drift.

To solve for the velocity components perpendicular to В we uncouple the

mutual dependencies of Eqs. (5.56a) and (5.56b) by differentiation and subsequent substitution to obtain

d Vf’X + C02g vgc, x = — sign(q)a>g • (5.57)

dt R

The solution of this inhomogeneous differential equation is accomplished by summing of the homogeneous solution of the form Acos(COgt + a) and a stationary solution to the inhomogeneous equation, which, by inspection of Eq.(5.57), we take as (sign(q)v12/(0gR). Referring to the initial conditions vgc, x(0)=vgc y(0)=0 associated with the undisturbed field at z(0) = 0, we determine

the constants A and a and finally find

2

Vll

vgc, x = sign(q)—’!—[cos(cog t) -1 ] (5.58a)

cogR

and

2

V|.

Vgc, y = —^ [C0gt — sin((Ogt)] . (5.58b)

cogR

The resulting drifts are, as preceedingly, revealed by averaging over a gyration period and are thus

where t represents an average length of time corresponding to one half of the interval of averaging. The latter equation clearly demonstrates that the guiding centre is accelerated in the direction of j at the magnitude V||2/R which, of course, is known to be the centripetal acceleration acp required to make the particle follow the curved В-field line. The only real steady drift is then accounted for by (Eq. 5.59a), in which we substitute according to the geometry of Fig. 5.10

i = jxk = -^x5i (5.60)

R Bo

and reintroduce cog = lqlB0/m to derive the curvature drift velocity as

which is thus again dependent upon the sign of the charge. Fig. 5.11 illustrates this drift of an ion spiraling around a curved magnetic field line.

A curved В-field is shown below to be accompanied by a gradient B-field, thus both specific drift velocity’s components add to yield the resultant drift. We remark that, although the expressions for the different drift velocities derived
here are correct, the assumptions and approximations used in the derivation are inconsistent in the sense that the suggested nonuniform В-fields do not satisfy Maxwell’s equations

V xB = 0 (5.62)

and V-B = 0. (5.63)

In order that these fundamental equations are met, it is seen that an inhomogeneous magnetic field has always to be associated with some curvature and vice versa. Hence grad-B drift and curvature drift will occur simultaneously.

Considering plasma states which are not in perfect thermodynamic equilibrium (no exact Maxwellian distribution), even though they represent equilibrium states in the sense that the force balance is equal to 0 and a stationary solution exists, means their entropy is not at the maximum possible and hence free energy appears available which can excite perturbations to grow. Such an equilibrium
state is unstable. The stability of a plasma confined by a toroidal and poloidal magnetic configuration is therefore seen to be determined by the free energies associated with eventual currents parallel to В (force-free currents) and with the plasma pressure. The ratio of these two free energies turns out to be (3P. Apparently, the gradients of plasma current magnitude and pressure, Vj and Vp, are the destabilizing forces in connection with the ‘bad’ magnetic field curvature discussed in Sec. 9.4 and necessarily inherent to a tokamak.

The occurring instabilities can be divided into three types: (a) Ideal MHD Modes:

Amongst all types, the ideal MHD instabilities are the most virulent due to their fast growth and the possible extension over the entire plasma. Thus, on the short time scale of relevance (microseconds) the resistivity of the plasma is negligible.

A toroidally confined plasma sees ‘bad’ convex curvature of the helical magnetic field lines on the outboard side of the torus. Consequently, such a configuration is subject to the onset of flute-type interchange instabilities of which the driving mechanisms have already been demonstrated in Sec. 9.4. However, because the magnetic field lines in a tokamak are more concentrated on the inboard side where there is ‘good’ curvature (concave as seen from the plasma), the average curvature of В-field lines over a full poloidal rotation is ‘good’ for windings with a rotational transform l < 2n, i. e., q > 1. Therefore, interchange perturbations do not grow in normal (q > 1) tokamaks. It is however observed that the perturbations can locally grow or ‘balloon’ in the outboard ‘bad’ curvature region. In that case a high local pressure gradient is responsible for driving the so-called ballooning instability. By establishing appropriate pressure profiles and appropriate magnetic field line windings, those modes can be suppressed almost everywhere in the plasma.

Another instability which represents the most dramatic one in ideal MHD is the kink instability already discussed in Sec. 9.6. It causes a contortion of the helical plasma column and consequently of the magnetic flux surfaces. Preferably it occurs in tokamak plasmas at low pressures and is driven by the radial gradient of the toroidal current. Fortunately these instabilities are bounded to small intervals of q(a) lying close below integer values. For the current profile distributions typical of tokamak fusion plasmas, such unstable modes arise mainly when q(a) is just a little less than 2. The associated kink distortion of the plasma column can be stabilized by an enhanced toroidal magnetic field strength such that the Kruskal-Shafranov condition

Be

is fulfilled, which we had used previously and extended to a more stringent criterion at the plasma edge, Eq. (10.42).

As seen in the preceding section, adjusting the safety factor q to the
appropriate value is associated with a limitation of the plasma beta. In order to avoid the major MHD unstable activities the overall P is limited by the maximum critical beta

ftri, [%] = CT^~ (10.50)

aB

with the so-called Troyon factor CT (dimensionless) ranging from 2.8 to 5 depending on the nature of the instabilities, if the plasma current I is in MA, the minor radius a in m and the confining field in Tesla. Since the plasma cross section need not necessarily be of circular shape (actually, most tokamak plasmas of today’s experiments feature an elliptic, bean or D-shaped cross section which permit optimization of energy confinement and plasma pressure profiles), parameters which account for the actual cross-sectional contour will also enter the Troyon factor. Note that the rough analysis in the preceding section, Eqs.(10.40) — (10.48), can also provide for the functional relation given in Eq. (10.50).

(b) Resistive MHD Modes

Additional types of macroscopic instabilities are attributable to the electrical resistivity of a tokamak plasma, which makes the instability grow more slowly. Characteristically, the growth times are of the order of 10’4 to 10’2 s which, however, is still short compared with the energy confinement time Tt (seconds). Resistive MHD modes result from the diffusion or tearing of magnetic field lines relative to the plasma fluid and can thereby destroy the nested topology of the magnetic flux surfaces. For helically resonant B-perturbations, magnetic field diffusion may preponderate the ideal MHD effects in thin boundary layers around surfaces having a rational q. Then the magnetic field lines can reconnect in these layers thereby producing nonaxisymmetric helical islands as suggested in Fig. 10.12. The tearing mode instability in a tokamak is driven by the radial gradient of the equilibrium current density. Upon formation of a magnetic island filament it grows until it acquires all the accessible free energy of the current. Outside these so-called resonant surfaces (q is rational) the plasma undergoes a sequence of MHD equilibria.

For low-(3 plasmas it is possible that these resistive modes couple nonlinearly amongst each other (different rational q), which leads to a disruption of the plasma current. Increasing the plasma current, which-according to Eq. (10.48)-is associated with lowering the safety factor q, is seen to diminish the existence of current distributions which are stable against tearing modes.

Another effect brought about by the nonlinear evolution of resistive MHD modes and observed in tokamak plasmas is the ‘sawtooth’ behaviour of some plasma parameters such as the electron temperature and the current density at the centre of the plasma column where q can drop below 1. Sawtooth oscillations can be delayed or prevented by modifying the current profile near the q = 1 magnetic surface. These sawtooth oscillations are not deemed catastrophic since their activity is constrained within a small region internal to the q = 1 surface, where it comes to a redistribution of the plasma energy.

(c) Microinstabilities

Microinstabilities are often associated with non-Maxwellian velocity distributions. The deviation from thermodynamic equilibrium means that there is free energy which can drive instabilities, often evolving into plasma turbulence. Also, nonuniformity and anisotropy of distributions can give rise to instabilities. Hence it is the particle kinetic effects that play an important role here, and the plasma cannot be expected to behave as a simple fluid and therefore cannot be treated as such anymore. It rather requires a kinetic description.

The electron velocity distribution becomes increasingly anisotropic as the plasma density decreases since, in order to carry a given plasma current, the individual electrons are further required to align their velocities with the current density direction. This drift velocity tends to make the velocity distribution function more and more asymmetric and hence unstable. On the other hand, the electrons will encounter collisions which randomize their velocities and thus reestablishes symmetry in the velocity distribution. However, as the density is further decreased, this stabilizing effect due to collisions is ultimately overcome by the destabilizing effect of the increasing drift velocity.

Further, anisotropy occurs in a plasma when it is confined by mirror fields, since particles having a large V||-component will escape and are therefore lost from the distribution. Hence, also the trapped particles in a tokamak, which bounce back and forth in the local mirror fields, can constitute a source of instabilities, preferably when the perturbation frequency is less than the bounce frequency. These then are classified as trapped particle instabilities and are still being investigated for effects on increased cross-field diffusion in tokamaks.

As a current is driven through the plasma or a beam of high energetic particles is injected, the different species will drift relative to one another. The drift energy can excite waves in the plasma. Since oscillation energy may be gained at the expense of the drift energy, the disturbance can grow. Such an instability is called a ‘two-stream’ or beam-plasma instability.

Where there is a steep density gradient in the plasma, an instability may appear due to the electron drift caused by the gradient and is called the drift instability. It can be stabilized by appropriate magnetic shear.

Microinstabilities are a large and complex field in plasma physics, which is still under investigation and many theoretical predictions are still to be proven by experiments.

Generally, there are three effective ways to prevent plasma instabilities: (i) magnetic shear, (ii) minimum-B configuration and (iii) dynamic stabilization by oscillating E — or В-fields or by proper-phase force feedback.