THE COUPLING OF ELECTROMAGNETIC POWER TO PLASMAS. R. Koch

Transcription

1 THE COUPLING OF ELECTROMAGNETIC POWER TO PLASMAS R. Koch Laborator for Plasma Phsics Ecole Roale Militaire - Koninklijke Militaire School B- Brussels, Belgium ABSTRACT This lecture complements the three previous lectures,2,3 on waves b addressing, on the basis of elementar and intuitive treatment, the process of coupling of electromagnetic power to plasma. Coupling is here meant in a broad sense. It consists of four different steps. (i) The first one is the coupling of vacuum electromagnetic power to plasma waves. An elementar antenna coupling theor is given. The state of the art in coupling models and status of comparisons with experiments are briefl discussed. (ii) The second is the transfer of plasma wave energ to particle energ. The resonant processes leading to this transfer are described in a heuristic wa. (iii) The third one is the build-up of fast particle populations. It will be outlined through a sketch of quasilinear diffusion for the simple case of Landau damping. (iv) The last step is the conversion of power through the resonant particle population to bulk plasma heating b collisions, which will be briefl addressed. I. INTRODUCTION Dielectric tensor notations are taken over from the lecture b Westerhof 2, but like in the previous lecture 4 we stick to the SI sstem, except for temperatures which will sometimes be expressed in ev, where explicitl stated in the text. The principle of wave heating is similar for all schemes and is sketched in Fig.. The electromagnetic energ is produced b a generator and sent to the machine area via transmission lines constituted of coaxial lines at low frequenc and waveguides at higher frequenc. At ver high frequenc optical transmission is also possible. Some matching circuitr has to be incorporated in the transmission sstem in order to prevent the reflected power to come back to the generator. The transmission line is connected to some launching structure (antenna, waveguide, ) that will couple the power inside the machine s vacuum chamber. The vacuum wave that exists Generator Matching circuit and transmission lines Vacuum Vessel Vacuum Wave Launcher Plasma Waves PLASMA Fig.. Principle of heating b plasma waves Absorption Zone inside the launching structure and at the ver edge of the plasma is then converted to a plasma wave that transports electromagnetic energ to some region inside the plasma where it will be absorbed. This is the region where the resonant process occurs. This process accelerates the population of particles that is in resonance with the wave, usuall a small fraction of the plasma particles. A slightl or strongl non-maxwellian resonant population builds up against the restoring force of collisions between this population and the remainder of the plasma. It is through the latter collisional process that the bulk of the plasma is heated up. To fix the ideas, it is useful to have in mind the tpical order of magnitude of the plasma parameters. Let us consider tpical parameters of a JET-tpe plasma:. The basic frequencies and velocities are given in Table. Note that electron frequencies (f ce, f pe, f UH ) are much higher than all other frequencies due to the smallness of m e /m i. In this introduction, we shall not describe the TABLE. Plasma parameters (Deuterium) T=5keV f ce =8GHz f LH =.8GHz V A =7. 6 m/s n=5 9 m -3 f pe =6GHz f ci =23MHz V te =3. 7 m/s B =3T f pi = GHz k // V A =MHz V ti =5. 5 m/s

2 technical parts of the launching sstems, i.e. generators, transmission lines, matching sstems, but shall focus on the phsics of power coupling. Let us first start with the launcher theor. II. II.A. ELEMENTARY WAVE COUPLING THEORY Tpes of launcher and coupling The simplest case is that of electron cclotron waves in large machines. In this case, the wavelength of the vacuum wave (k =ω/c is the vacuum wavevector) λ = 2π / k () is ver small as compared to the plasma cross-section. The wave is launched as a propagating wave pencil that will progressivel convert to a plasma wave. Because of the smallness of the wavelength, the boundar conditions at the conducting wall of the machine, as well as on the launching structure, pla no explicit role. The wave can be accuratel described in the geometric optics limit and the onl boundar conditions that matter are the initial launching angle and reflections at the wall, if an. If the vacuum wavelength becomes comparable to the antenna structure, the scale length of variations of edge plasma parameters or the plasma radius, the launcher environment and the plasma will affect the coupling process and a full boundar-value problem has to be solved to describe it. Such is usuall the case of Alfvén wave, ion cclotron, or lower hbrid wave launchers in medium or large-size machines. But this ma also be the case of electron cclotron launchers in low-field, small machines. One can still distinguish two different cases in the latter categor. Either the vacuum wave propagates inside the torus chamber, or it is evanescent. This depends first on the size of the machine relative to the wavelength. Indeed in a torus, the poloidal and toroidal wavenumbers will be quantified. To a first approximation, if we call R the major radius and r the minor radius of the machine, the fundamental toroidal wavenumber of the toroidal cavit will be / R and the poloidal one /r. Let us call m and n the poloidal and toroidal wavenumbers respectivel, then the dispersion relation for vacuum waves can be written (with k r the radial component of the wavevector) : k r 2 = k 2 m 2 / R 2 n 2 / r 2 (2) The wave can propagate towards the plasma inside onl if k r is real because the fields behave as E, B exp(±ik r r) (3) The existence of the m=n= mode is prohibited b the metallic boundar conditions. Hence, a necessar condition for the vacuum wave to propagate inside the torus is k > m / R (4) If k <r, then onl the m= mode can propagate and the rest of the poloidal spectrum is evanescent. It is usuall the case for medium size or large machines that part of the antenna spectrum is propagating, part evanescent. One can globall estimate in which categor a launching structure will fall onl from its dimensions and phasing between elements. Let us call k the poloidal wavenumber and k // the toroidal one [we shall associate to (x,,z) the radial, poloidal and toroidal directions respectivel]. If the launching structure has width w the support of the launcher field or current is (up to a translation) proportional to the function S // (z) = γ (z) γ (z w) (5) which power spectrum is proportional to S // 2 (k // ) sin 2 (k // w /2) (6) Therefore, the tpical parallel wavenumber of the structure can be taken as k //a =π/w. Similarl, if the structure has height h, the tpical poloidal wavenumber is k a =π/h. Therefore, one can estimate the tpical radial wavevector for the launcher from a formula similar to Eq.(2): k 2 r = k 2 k 2 2 a k // a (7) Assume, as is usuall the case in the lower frequenc range (f f ci ) that the antenna width is much smaller than the wavelength. Then the poloidal average (k =) of the field will deca as exp(ik r x) = exp( k //a x) = exp( πx / w) (8) This provides a simple rule to estimate the evanescence of the field launched b the antenna in vacuum. If the launcher is made of an arra of identical elements spaced b L z <w in the toroidal direction and excited with a phase difference φ, then the tpical parallel wavevector must be taken as k //a = φ/l z. (if >π/w, and similarl for the direction). Evanescence will then be stronger than in the previous case of an unphased structure. We shall now introduce coupling theor, on the basis of the simplest model. More sophisticated theories rest on similar principles but include more of a realistic geometr. II.B. Coupling model We consider the simplest case of an antenna facing a large plasma such that the plasma looks nearl uniform in the toroidal (z) and poloidal () directions (Fig.2). In the radial direction x the plasma is usuall non-uniform, and this is taken into account in standard coupling models, but here, in order to simplif the algebra, we consider a step model. The densit is zero for x< and constant densit

3 Fig. 3 wall antenna I j II z plasma III ik x B z e 2-D slab coupling model. The plasma is uniform in the (poloidal) and z (toroidal) directions. The width of the antenna is 2w z, the distance between the antenna and the plasma is a and the distance between the antenna and the wall is d. for x positive. We assume that the absorption is good and hence, there is no reflected wave. This is the so-called single-pass approximation. In addition, we shall also assume that the sstem is invariant in the -direction (which implies in particular that the antenna is infinite) and neglect propagation in the -direction (k =). Next, we have to choose a model for the plasma waves. To be specific, we choose the case of coupling in the ion cclotron frequenc range (ICRF). This case was treated b Westerhof 2, section III.A. The slow wave being evanescent in the plasma bulk, we shall consider onl coupling to the fast magnetosonic wave (FW). As the FW equations will be needed later on, we first derive them, starting from the cold wave theor 2. II.C. The fast magnetosonic wave equation We write the full cold dispersion relation 2 in terms of the parallel (N // =Nsinθ) and perpendicular (N =Ncosθ) components of the refractive index N // = k // k ; N = i k d dx We keep to N its operator meaning because x is the direction of inhomogeneit. In the parallel direction, the plasma is homogeneous and we use the Fourier transformed form. The dispersion equation can then be rewritten 2 S N // id N N // E x id S N 2 2 // N E 2 = () N N // P N E z It has been shown 2 that the uncoupling of the FW equations follows from the ver large value of P. Mathematicall, this amounts to taking the limit P () x (9) This is equivalent to assuming that the plasma conductivit is infinite in the parallel direction. It is also the same as taking the zero electron mass limit. This limit implies that the parallel electric field cannot penetrate the plasma, i.e. that the evanescence length of the slow wave is zero. Taking the limit Eq.() in Eq.(), we obtain k 2 2 S k // ik 2 D ik 2 D k 2 S k 2 // + d E 2 x E = (2) dx 2 and E z =. The other components of the fast wave field follow from Maxwell s equation iωb = E : H x = H = ;H z = de (3) iωµ dx Finall, eliminating E x from Eq.(2), we obtain the Fast wave equation d 2 E 2 + k dx 2 FW E = (4.) 2 k FW II.D. = k 2 2 N FW = k 2 2 ( S k // ) ( k 2 D) 2 k 2 2 S k // ( ) The plasma surface impedance (4.2) For a uniform plasma, the wave equation is Eq.(4.) has constant coefficient and the solutions are simpl exponential. The single-pass approximation allows us to impose at z a radiating boundar condition and the wave solution in the plasma can be written: E = C III exp(ik x) (5) where C III is a constant (relative to region III in Fig.2) to be determined. Eq.(3) then gives the H z field component: H z = ik E (6) iωµ The field in the plasma is thus known up to a multiplicative constant. Tangential filed components being continuous at the plasma-vacuum interface (II-III), their ratio is also continuous. This quantit is known as the surface impedance of the plasma Z S Z S = E H z = ωµ k (7) We express the continuit of this quantit at x= as [[Z S ]] Z S ( + ) Z S ( ) = (8) In the general case where all field components are to be considered, the equivalent of Eq.(7) is a vector relation and Z S is the surface impedance matrix: E H = Z S (9) E z H z

4 II.E. Fields in the vacuum region I-II Equipped with this boundar condition, the vacuum problem can be solved on its own. The plasma properties will enter its solution onl via the quantit Z S and the vacuum solution is therefore formall independent of the particular plasma model considered. The general electromagnetic field in vacuum can be decomposed into its TE (transverse electric) and TM (transverse magnetic) parts with respect to a given direction, here z. Maxwell s equations then appear in the form: B x iω ik ik E z E = k 2 2 k c 2 // db // ik // iω z (2.) dx E x iω ik // ik B z B = iω de k 2 2 k // ik // z (2.2) c 2 dx d 2 E z = (k 2 dx 2 B + k 2 // k 2 ) E z (2) z B z From this it can be seen that the problem can be solved independentl for E z and B z. The TM part of the field, which has a longitudinal (along z) E component does not couple to the plasma waves because E z = in the plasma ( II.C). Therefore, for the simplified problem considered here, we can retain the TE mode alone and ignore the field components deriving from E z.the solution for the two vacuum regions is elementar: Region I: B z = A I cosh(k // x) + B I sinh(k // x) (22.) Region I: B z = A II cosh(k // x) + B II sinh(k // x) (22.2) where A s and B s are constants. to be determined b appling the following boundar conditions. At the metallic wall x=-(a+d): E = (23.) The antenna is represented b an infinitel thin current sheet of finite width w and infinite length. This gives rise to a jump condition on the tangential magnetic field: [[B z ]] a = µ j (23.2) and a continuit condition [[ E ]] a = (23.3) At the plasma surface, x= : [[Z S ]] = (23.4) These conditions are sufficient to determine the 4 constants in Eq.(22). In particular, this gives the relation between all field components and their source, the current densit at the antenna j. In the general case one has a vector relation E = R (k, k // ) j (24) There is one additional condition, the continuit of E (or B z ) at x= that was not necessar to solve the vacuum problem. It can be used to determine C III in Eq.(5) as all field quantities in the vacuum region are now known. II.F. Ponting s theorem and antenna radiation Let us consider in the vacuum region I-II an arbitrar volume containing the antenna. Starting from Maxwell s equations, one can easil write down Ponting s theorem 2 E.j A dv = V 2 (E H* ) ds iω ( µ S 2 H 2 ε 2 E 2 )dv V (25) On the LHS appears the work done b the electric field on the antenna current. Strictl speaking, it should be zero because the antenna is a metallic conductor on which the tangential electric field should vanish. It is non-zero because the current distribution on the antenna has been assumed rather than self-consistentl computed. This is known as the induced e.m.f. method. Though it ma appear rough, this method usuall gives good results if the assumed current is a reasonable guess of the exact one. In more sophisticated computations 5, the current distribution on the antenna is self-consistentl determined. A theor completel similar to the above one can be done for waveguide 6 or aperture launchers. In these cases, the incoming wave field distribution on the aperture is given and the field reflected b the plasma and surrounding structures is the result of the computation. Alternativel, the above formalism can be applied without changes if the aperture boundar condition is expressed as an equivalent current densit 7. It is to be observed that the quantit on the LHS of Eq.(25) has both a real and an imaginar part, as explicitl apparent in the RHS. The real part is the power radiated b the antenna, while the imaginar part is related to the reactive properties of the antenna, as we will see. II.G. Antenna coupling properties The structure of the antenna modelled in the present exercise is basicall that of a strip-line, i.e. a conductor running above an infinite conducting plane. The field in such a strip line is known to have a TEM (transverse electro-magnetic) structure, like a coaxial transmission line. A TEM field has the propert that the electric and magnetic field structure in the transmission line crosssection is the same as that respectivel of the electrostatic and magnetostatic field it can sustain. Therefore, the antenna properties can be computed in the electrostatic and magnetostatic limits and used as in transmission line theor. We shall recast the results obtained above in terms of strip line characteristics. This formalism is often used in practice to represent properties of real ICRF antennas, which structure is indeed close to that of strip lines. First, we rewrite Ponting s theorem in terms of the k // field spectrum using Parseval s relation:

5 P = 2 E.j A = V 4π E A (k // ) j A (k // )dk // (26) Then, we equate the computed power to the the expression of the same quantit for a strip line: P = 2 (R iωl)i2 = E A (k // )j A (k // )dk 4π // (27) This relation assimilates the antenna radiated power to the equivalent power of a loss transmission line element with R and L the specific (i.e. per unit length) resistance and inductance of the line. The second equalit constitutes the definition of these two quantities in the present antenna model. I is the total current flowing on the antenna. No equivalent capacitance C appears in Eq.(27) because we have dropped the TM part of the field. It can however easil be obtained b solving the TM vacuum field equations with the boundar condition E = at the plasma x=. The three constants R,L,C completel determine the properties of the transmission-line equivalent to the antenna. In particular, Fig. 4. Distribution of the Ponting vector in the farβ = ω LC (28.) field region for three different phase is the propagation constant along the line. If the plasma is replaced b a metallic wall, this is simpl the vacuum differences φ between successive straps of a 4- strap antenna arra. From 8. wave vector ω/c, as in a coaxial line. In the presence of Z plasma, because the magnetic field leaks out of the E (x,k // ) = S sinh(k // d) j A (k // ) exp(ik x) vacuum region into the plasma, L (proportional to the sinh[k // (a + d)] + ik Z (29) // S cosh[k // (a + d)] ωµ magnetic energ) is larger than the vacuum value and the wavelength of the field along the antenna is shorter. One For a simple uniform current distribution on the antenna, therefore sas that the antenna is a slow-wave structure the current spectrum is: because the propagation velocit along it ω/β is lower j A (k // ) = sinh(k // w /2) (3) than c. The slow-wave structure propert is due to the k // w /2 anisotrop of the boundar x= which behaves as a perfect conductor along z and as a dielectric in the The real space distribution of the fields is obtained b inverse Fourier transform: direction. E (x, z) = 2π (x, k // )e ik // z dk // Z c = L /C (28.2) (3) = G(k // ) j A (k // )e i( k x+k // is the characteristic impedance of the antenna and z) dk // R iωl Z A = iωc tanh[ iωc(r iωl)l A ] (28.3) where the integrand has been recast in a form suitable for using the stationar phase approximation. In the far-field k is the antenna input impedance with l region x and z are large and the exponential is varing fast A the antenna length. This quantit is of while G and j A are slowl varing functions of k //. The primar interest for the design of the transmission and matching sstem. main contribution to the integral thus comes from the region where the phase is stationar: dk k II.H. Radiated field x + z = // dk // (32) Fig. 3 Fast wave dispersion Using the additional boundar condition at the plasma-vacuum interface, i.e. continuit of E, the field in the plasma can also be computed and used to determine the properties of the radiated far field. We start from the expression of E solution of the simplified boundar-value problem above: The relation between k and k // is determined b the dispersion relation [see Eq.(4)] k 2 2 = k FW (k // ) (33) and is shown schematicall in Fig.4. Equation (32) is the equation of the ras launched from the antenna. Each ra is a straight line with angular coefficient x/z uniquel determined b the value of k //. For k // =, the ra is launched perpendicularl to the wall, for nonzero k // it is launched at an angle and for k // corresponding to the cut-

6 off of the fast wave (k =), it is launched in the parallel direction. The ra direction can be shown to be identical to the Ponting vector direction as P x = 2 Re(E H* ).e x = k E 2ωµ dk dk // The electric field amplitude follows from the stationar 2 phase approximation of the integral Eq.(3): E (x, z) = (34) 2π ix d 2 k / dk // 2 G(k // )J A (k // )e i(k x+k //z) (35) where all quantities are evaluated using Eq.(32). In Fig.4, the far field Ponting flux distribution over constant phase surfaces is shown for a phased antenna arra in an ITERlike plasma. As the Ponting flux is the RF power flux, this shows how phasing, b sending the power at different parallel wave numbers sends it in different spatial directions as well. II.I State of the art Because the vacuum problem can be solved for itself, independentl of the complexit of plasma properties, sophisticated antenna models have been developed 9. The focus either on a precise determination of the antenna currents 5, on the recesses in which antennas are located, or on more complete description of plasma waves. Antenna models have been compared with success to experimental results in a number of cases, both in the ICRF,2,3 and Lower Hbrid range 4 and are used to predict the performance of ITER antennas 5. III. POWER COUPLING FROM WAVES TO PARTICLES III.A. Absorption mechanisms Once the wave has been launched in the plasma, one could think that it can be damped simpl because the accelerated particles experience a drag due to collisions. This is in general not the case. In the bulk of a hot plasma, e.g. T e T i 5keV, n=5 9 m -3 the collision frequenc is ν = nln Λ T 3/ 2 2kHz (36) In electromagnetic theor the ratio ν/ω, (ω=2πf) is characteristic of the importance of dissipative effects due to collisions with respect to reactive, i.e. wave oscillation, energ. For small values of ν/ω, the motion is almost dissipation-less and huge fields and large perturbations in the particle motion are necessar if an significant amount of energ is to be damped in the plasma. Equation (36) implies that, at frequencies in the MHz range or higher, direct dissipation of the wave b collisions will be negligible. In order to magnif collisional absorption one has recourse to resonances. Under resonance conditions, a small excitation will create either a huge response in the particle s motion (wave-particle resonance) or large wave-field build-up (wave resonance). In Fig. the absorption zone is the region where such a resonance takes place (the shape is of course onl smbolic). III.B. Wave resonances.. Wave resonances have been discussed in a previous lecture b Westerhof 2. Assuming that k // is for the essential determined b the antenna sstem, and thus fixed, and we shall characterise the resonance b k. An example of wave resonance is that of the fast wave, Eq.(4) when k 2 S k // 2 = (37) We shall examine in more detail how S varies across the plasma and gives rise to the ion-ion hbrid resonances in the ICRF subsequentl 6. Here, we are interested onl in understanding how a wave resonance can give rise to absorption. It is sufficient to notice that, as we proceed from the plasma edge to the plasma inside, e.g. along the major radius direction (x), S will var because both the magnetic field and the plasma densit var. As we shall see later in more detail, the simplest kind of resonance (assumed located at x=) can be schematicall represented b the wavevector variation 2 k FW = k 2 α x K 2 (x) (38) with α a positive constant. This variation is schematicall represented on Fig.5. Except in the vicinit of x=, the wave is propagating with constant k ±k. If the wave approaches the resonance from the left, it will slow-down more and more (its group velocit goes to zero) and come to rest at x=. In this simple WKB picture, all wave fronts coming from the left pile-up at the resonance leading to a large increase in wave amplitude. In addition, as the wave field Fig. 5 The simplest example of a oscillates a large wave resonance. number of times before propagating an significant amount of distance toward the resonance, it is obvious

7 that the effects of an damping mechanism will be considerabl magnified in the vicinit of the resonance. This simple picture has to be corrected for wavespecific effects. Indeed, part of the power will tunnel 2 through the evanescent region (where k <) and continue propagating on the other side (x ). The non-tunnelled fraction is absorbed at the resonance x=. This can be seen from Eq.(4.) with the wavevector from Eq.(38): d 2 E + K 2 (x) E dx 2 = (39) B multipling Eq.(39) b the complex conjugate of the electric field E * and taking the imaginar part, we get a conservation law for the electromagnetic power flux: d dx Im(E de * dx ) + 2 Im(K2 ) E = (4) Indeed, from Eqs.(3), we see that the electromagnetic power flux in the x-direction, i.e. the x-component of the Ponting vector is: P x = 2 Re(E H* ) x = 2 Re(E H * z ) * de dx ) = * (4) de Im(E 2ωµ dx ) = 2 Re(E iωµ which means that Eq.(4) is indeed a power conservation law that we can rewrite dp 2 x dx = Im(K ) 2 E 2ωµ (42) When K 2 is real, there is no loss anwhere and this equation implies that the power, is conserved. In the case of Eq.(38), integration across the singularit on the real x axis would be meaningless. One must remind here, that causalit requires Im(ω) to have a small positive imaginar part 2 (ν) such that the pole in Eq.(38) is slightl off the x-axis, and the singularit can be represented according to the Plemelj formula x = P( x ) ± iπδ ( x) (43) where the smbol P reminds that the principal part has to be taken when the integration is performed. The imaginar part represents half the residue at the pole and, in an practical case, the sign will follow from ν>. Accordingl, upon integration over x Eq.(4) implies that the power is constant for x and jumps abruptl at the crossing of the resonance. In the case of a cold or maxwellian plasma the jump correspond to a decrease in power, i.e. to wave energ absorption (b the plasma). This will be shown explicitl for the case of the ion-ion hbrid in a subsequent lecture 6. Power is thus absorbed at the resonance notwithstanding the fact that no absorption mechanism was explicitl considered in the original equation (4). Introducing some small dissipation in the dielectric tensor, which for the essential is similar to having a positive imaginar part iν in ω, would lead to the same effect. For small values of ν, the absorption at the resonance is nearl independent of ν and keeps a non-zero value as ν. III.C. Wave-particle resonances Landau damping. Wave-particle resonances appear as resonant denominators in the integrand of the expression of the kinetic dielectric tensor 2. Phsicall, the result from the fact that, in their reference frame, particles see a constant electric field and are therefore uniforml accelerated. Such a singular phenomenon can appear onl in an approximate treatment of the problem, namel in the linearised approximation. For example let us consider, as in Fig.6 a particle moving at the constant velocit v in an electric field directed along v and propagating at the phase velocit ω/k. The equations of motion of this particle v ω k Fig. 6 Wave-particle resonance. m dv dx = ZeE cos(kx ωt); dt can be solved exactl and ield two tpes of solutions 7 : (i) passing particles having a negative velocit or a too large positive velocit travel at the average speed v with an oscillation superposed on it, (ii) particles which travel at a velocit close to the phase velocit are trapped in the potential well of the wave and travel at the average velocit ω/k with a superposed oscillation which is the bouncing inside the well. These trajectories are integrable and there is no such thing as a resonant motion in the wave. (However a closer look reveals that there is a separatrix in phase space, corresponding to the trapped/passing boundar, which is extremel sensitive to perturbations, but this is another stor). However, when we linearise the problem we arrive at the equation of motion: m dv = ZeE cos(kx ωt) = ZeEcos[(kv ω)t] (45) dt where we have used in the expression for the field the unperturbed particle motion x=vt. The perturbed velocit v is oscillator and there is no energ exchange between the particles and the wave as long as the have a different zero-order velocit v ω/k. But if the have the same velocit, the particle sees a constant electric field and is uniforml accelerated: v = ZeE m t (46.) An alternative wa of looking at the same phenomenon is the Fourier transform approach, which here reduces to looking onl at the periodic solutions of the problem: dt = v

8 v = ZeE i( kv ω )t e (46.2) m i(kv ω) It can be seen that the singularit in the spectrum appearing at the resonance corresponds to the secularit ( t) in the time-representation. The corresponding absorption is known as Landau damping. III.D. Wave-particle resonances Transit time magnetic pumping Another frequentl quoted non-collisional absorption mechanism is transit time magnetic pumping or TTMP. It is similar to Landau damping, except that it is due to a propagating magnetic rather than electric- wave modulation: B = δbcos(kx ωt) (47) superposed on the static field B. This modulation gives rise to the µ B force acting on the particle s magnetic moment µ, according to the equation for the motion along the magnetic field lines: m dv = µ B dt (48) which is similar to Eq.(43). III.E. E Wave-particle resonances cclotron damping. v θ Fig.7. Cclotron resonance. linearl with time. Another tpe of wave-particle resonance is the cclotron resonance (Fig.7). Assume that the particle is rotating at the cclotron frequenc θ=ω c t and that we appl a rotating electric field at the same frequenc ω=ω c with a component along the particle velocit v: the particle s perpendicular energ will increase This ver sketch analsis puts into light a weakness of the linearised approach. Indeed what will happen in realit is not that uniform acceleration will take place indefinitel but rather that the accelerated particle will escape the resonant condition and thence terminate the resonant process. Resonant absorption can onl continue if, once this particle has left the interaction area, it is replaced b a fresh one that can continue the resonant interaction. In terms of distribution functions, this means that the wave will produce velocit diffusion such as to empt the interaction region from particles, which in practice means a flattening of the distribution function in the interaction zone. Once the distribution is flat the interaction has stopped. Collisions or other processes like stochasticit counteract this tendenc b restoring the gradients. If the latter processes are strong, the distribution function can remain maxwellian, however, generall speaking, the distribution function of a particle population heated b a resonance process is not maxwellian. IV. QUASILINEAR DIFFUSION AND TAILS In the previous sections, we have seen that the RF power could be absorbed b the plasma via wave-particle or wave resonances. In both cases, the wave equation tells us that the wave will be damped while travelling in the plasma but leaves open the question: where is this power going to? A wave resonance corresponds to infinit in phsical variables. At resonance, not onl the wave vector goes to infinit but field components as well, as is clear from the solution of the wave Eq.(4), which can be expressed analticall in terms of Bessel functions 7. Such infinities are the sign that some smallness hpothesis is violated and that additional terms should have been retained in the wave equation. Retaining these terms changes the resonance into a mode conversion 8 whereb the initial low-k wave is converted into high-k branch. Ultimatel, the latter can onl be absorbed through collisional or noncollisional (i.e. wave-particle resonance) processes. The existence of absorption through wave-particle resonance manifests itself b the presence of an antihermitian part of the magnetised hot-plasma dielectric tensor 2. The classical expression of the dielectric tensor is obtained b assuming that the unperturbed distribution function is a maxwellian. We shall see that this is onl an approximation and that the heating process onl takes place with a sub-class of plasma particles and necessaril leads to some deformation of the distribution function of the heated population. The corresponding theor is called quasilinear theor. The derivation of the quasilinear theor in the general case has been addressed in previous lectures,3. Here, we onl want to outline in which wa absorption and heating i.e. temperature increase, are inter-related. We do this for the simplest case, that of Landau damping in unmagnetised plasma. We start with Vlasov s equation in one dimension f t + v f z + ZeE f m v = (49) where E=E is the perturbed electric field (there is no equilibrium electric field). We decompose the distribution function into a slowl varing part (f ), both in time and space, and a perturbed part (f ), f=f +f, and insert this expression in Eq.(49). In order to isolate the slowl

9 P p = Z 2 e 2 E 2 varing part of the distribution function, we average this v f Im equation over time (man wave periods) and space (man 2m ω kv v dv (47) wavelengths). Denoting b <.> this averaging operation, On the other hand, we can also compute the power we obtain: f t + v < f z >+Ze m < E f dissipated b the waves, which follows from Maxwell s >= (5) equations: v P p = where we have used the fact that <E>=<f 2 Re [ E * j ] (58) >=. The second term in this equation is zero for uniform plasma and we can compute the perturbed particle current from and we are left with an equation that determines the the perturbed distribution function Eq.(53): evolution of the equilibrium distribution function under izee f the action of the first-order perturbations: j = Ze vf dv = Ze v f = Ze t m < E f m(ω kv) v dv v > (5) (59) = iz2 e 2 E v f m ω kv v dv The second term in this equation is the quasilinear term. In order to write it down explicitl, we must solve the Accordingl, equation for the perturbation of the distribution function f P w = Z2 e 2 E 2 iv f Re which we obtain b subtracting Eq.(5) from Eq.(49): 2m ω kv v dv f t + v f z + ZeE (6) f m v = (52) = Z 2 e 2 E 2 v f Im 2m ω kv v dv in which we recognise a simplified version of the and we have the important identit linearised Vlasov equation 2. Fourier-transforming in space P w = P p (6) and Laplace-transforming in time (f exp[i(kz-ωt)]), we easil obtain the solution of this equation: izee f f = (53) m(ω kv) v Inserting this expression into Eq.(5), we obtain, noting that the average of two oscillating quantities u(t) and v(t) is <u(t)v(t)> = (/2)Re(uv*): f = Ze t m 2 Re E * izee f v m(ω kv) v (54) = Z 2 e 2 E 2 2m 2 v Im f ω kv v This equation can be given a Fokker-Planck tpe form: f = t v D f ; D = Z 2 e 2 E 2 Im v 2m 2 ω kv (55) where D is the quasilinear diffusion coefficient. Note that b a formula similar to Eq.(43), the imaginar part appearing in the expression for D is nothing else than δ(ω-kv). This implies that onl the particles which are in resonance will be pushed in velocit space b the heating proccess. This evolution equation for the distribution function allows to estimate the heating power going to the particles: P p = t 2 mv2 f dv = 2 mv 2 v D f dv v (56) or, explicitl = m vd f v dv which expresses the fact that all the power that is lost b the waves is gained b the particle population. Although this might appear trivial in this present simple case, it is usuall not granted in more complex theories where wave propagation and absorption on one side, and quasilinear diffusion on the other, are treated distinctl, using different approximations. The lack of consistenc between the latter can break the equalit Eq.(6) 9. V. THERMALISATION If Eq.(55) is solved as it is, it can lead to a timeasmptotic stationar solution onl if the energ transferred b the wave to the particles is zero. Indeed, its r.h.s. corresponds to power input to the plasma and there is no loss term to balance it. Therefore the stationar solution of Eq.(55) must exhibit a quasilinear plateau, i.e. a zones around the resonant velocit where f is flat ( f / v=). This ensures that the heating power vanishes, see Eq.(57). In a situation where there is stationar power transfer to the plasma, the evolution equation for the particle s distribution functions, Eq.(55) is thus lacking a loss term. This is the collision term, which we denote b C(f ). It includes collisions on all particle species, including the heated ones. It also implies that the distribution function of the heated species will tend to relax to a maxwellian. From the different contributions to C(f ), one can then compute the power transfer to the different plasma components and the resulting temperature increases. The

10 complete equation for the evolution of f should thus be written f = t v D f + C( f ) (62) v This collisional thermalisation process is in close relation with the slowing down of fast NBI ions discussed earlier 4 7. T.H. STIX, Waves in plasmas, AIP New York (992). 8. D.G. SWANSON, Plasmas Waves, American Institute of Phsics (989). REFERENCES. P. LAMALLE, Kinetic theor of plasma waves: Part I Introduction These Proc. 2. E. WESTERHOF, Kinetic theor of plasma waves: Part II Homogeneous plasma These Proc. 3. P. LAMALLE, Kinetic theor of plasma waves: Part III Inhomogeneous plasma These Proc. 4. R. KOCH, Plasma heating b neutral beam injection These Proc. 5. PECOUL S., et al., Self-consistent computations of currents on ICRH antennae with the ICANT code, 26th EPS Conf. on Contr. Fus. and Plasma Phsics, Maastricht (999) 6. X. LITAUDON & D. MOREAU, Coupling of slow waves near the lower hbrid frequenc in JET, Nucl. Fus. 3, 47, (99) 7. R. KOCH, A global solution of the ICRH problem based on the combined use of a planar coupling model and hot-plasma ra-tracing in tokamak geometr, Computer Phs. Comm., 4,, (986) 8. R. KOCH et al., An asmptotic analsis of the radiation pattern of ion cclotron resonance heating antennas, J. Plasma Phs. 32, 29 (984). 9. R. KOCH et al., Progress in RF Theor: a sketch of recent evolution in selected areas, J. Plasma Phs. 4, A4, (998).. M.D. CARTER, et al., Three Dimensional Modelling of ICRF Launchers for Fusion Devices, Nucl. Fus. 36, 29, (996). M. BRAMBILLA, Evaluation of the Surface Admittance Matrix of a plasma in the Finite Larmor Radius Approximation, Nucl. Fus. 35, 265, (995) 2. R. KOCH et al., A Comparison Between ICRF Theor and Experiment, Plasma Phs. Contr. Fus., 3, 559, (988). 3. DESCAMPS P. et al., Global Modes Excitation in TEXTOR and Comparison with Theor, Plasma Phs. Contr. Fus., 33, 9, (99). 4. C. GORMEZANO, High power lower hbrid current drive on JET: results and prospects, Report JET-P(93)65 (993) 5. ITER team, et al., ITER Phsics basis, to be published in Nucl. Fus. (999 or 2) 6. R. KOCH, Ion cclotron, lower hbrid and Alfvén wave heating methods These Proc.

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