MHD WAVES AND GLOBAL ALFVÉN EIGENMODES

Transcription

1 MHD WVES ND GLOBL LFVÉN EIGENMODES S.E. Sharapov Euratom/CCFE Fusion ssociation, Culham Science Centre, bingdon, Oxfordshire OX14 3DB, UK S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

2 Linearised MHD equations MHD waves Compressional waves Shear lfvén waves Kinetic lfvén waves Global lfvén Eigenmodes Summary OUTLINE OF LECTURE 4 S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

3 LFVÉN INSTBILITIES DRIVEN BY LPH-PRTICLES lpha-particles (He 4 ions) are born in deuterium-tritium nuclear reactions with birth energy 3.5 MeV. These fusion-born ions are super-lfvénic, V Ti ~1 6 m/s << V ~ m/s < V α = m/s << V Te ~ m/s, where V =B /(µ n i m i ) 1/, and the estimate is for D:T=5:5 ITER plasmas During slowing-down of alpha-particles, they pass the resonance condition V = V α and may excite lfvén waves with ω = k II V dispersion relation Free energy source: radial gradient of alpha-particle pressure. The instability results in radial re-distribution of alpha-particles Re-distribution may also cause losses of highly energetic alphas damage to the first wall We have to assess possible wave-particle interaction in burning plasmas S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

4 STRTING MHD EQUTIONS For describing plasma particles, we take velocity moments of the kinetic equations for electron and thermal ion distribution functions and obtain ρ + ( ρv) = ; t V ρ d = p+ 1 J B ; dt c p + V p+ γ p V = ; t 1 E+ V B= ; c For describing electromagnetic fields in the plasma, Maxwell s equations are used 4π B= J c 1B E= c t B = Here, scale lengths larger than Debye length are considered with n e = Zi n i i S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

5 THE LINERISTION PROCEDURE ll the field and plasma variables are represented as sums of equilibrium (denoted by subscript ) and perturbed (denoted by δ ) quantities: J= J+ δ J, B= B+ δ B, V= δ V, p= p+ p, ρ = ρ + δρ, E= δe, (*) where all the perturbed quantities satisfy δ << 1, i.e. δ J / J << 1 etc. Substitute the expressions (*) in the starting set of equations and obtain equations with terms: a) not having δ at all; b) having δ ; c) having δ etc. The terms NOT having δ are balanced thanks to the plasma equilibrium 1 p = J B c The relation between equilibrium quantities J, p, B MUST be kept in all equations with δ δ ll the equations are linearised then, i.e. only linear terms in δ are kept and terms with δ etc. are dropped off as small (since δ << 1) S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

6 LINERISED MHD EQUTIONS The linearised ideal MHD equations take the form: δρ + ( ρ δv) = ; t dδv 1 ρ = δp+ δb B dt 4π δb = [ δv B ]; t p δp= γ δρ ρ ; [ ] Introduce plasma displacement from the equilibrium,, related to δ V via δ V = / From the first and third equations we find then δρ div ( ρ ) t = ; δ = [ B ] B = B div + B where we used [ a b] = ( b ) a ( a ) b+ a divb b diva, and B ez ; z S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

7 Substitute the expressions for Where γ p /ρ EQUTION FOR IDEL MHD WVES t = c δρ, δb in the remaining two equations and obtain S div+ V div + V c S = is the ion sound speed, B /( 4πρ ) z V = is the lfvén velocity, This equation describes linear MHD perturbations of homogeneous ideally conducting plasma. Single vector equation gives three scalar equations for three types of waves S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

8 PLSM DISPLCEMENT IN MHD WVES Compressional lfvén and slow magnetosonic waves: the returning force is the magnetic and the kinetic pressure Shear lfvén wave: the returning force is the tension of magnetic field lines C S SM JG cr S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

9 S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1 COMPRESSIONL WVES - 1 Coming back to the main equation z V div V div c t S + + = (*) Consider two compressible types of waves, in which z and div The parallel displacement z is described by the parallel projection of equation (*): + = div z c z c t S z S z We obtain equation for div by taking divergence of perpendicular projection of (*): z c div z V div c t div z S S = Here, = div We see, that two equations for z and div are coupled

10 COMPRESSIONL WVES - Consider limit β c S / V << 1. In this case, equation for div reduces to div = V div t which decouples from z and describes Compressional lfvén Wave. The magnetic pressure B / 8π determines the returning force that acts perpendicular to B The displacement z parallel to B is described by z z = c S t z and such wave corresponds to Ion Sound Wave existing in plasma even without B However, if β is not small, the decoupling of compressible waves does not work! The ion sound wave is modified then by the magnetic pressure and becomes Slow Magnetosonic Wave. The coupled equations give for z, div exp( iωt+ ik r) the following relation between wave frequency and wave vector (dispersion relation): 4 ω ( c + V ) k ω + V c k k = S S z S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

11 SHER LFVÉN WVES In contrast to the compressional waves, the third type of MHD waves, so-called Shear lfvén wave, is incompressible: z = and div = For such waves the main MHD equation becomes simply = V t z which coincides with equation for string oscillations. The returning force is the tension of magnetic field lines, which act similarly to the strings In shear lfvén wave the fluid displacement vector and E ~ are perpendicular to the magnetic field B. The wave propagates along B : ω = ±k V ; V = 4π B i n i M i k = k B /B ; mong all the waves in plasmas, the lfvén wave (H. lfvén, rkiv. Mat. stron. Fysik 9B() (194)) constitutes the most significant part of the MHD spectrum and is probably the best studied S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

12 HOW THIS WVE EVOLVES IN INHOMOGENEOUS PLSM? ω = k ( r) V ( r) The life-time of a wave-packet of shear lfvén waves is limited by the phase mixing 1 d τ ( k ( r) V( r) ) dr If S wave packet cannot live long, why to care about S interacting with fast ions? Radial gradients of plasma may modify such waves of comparable wavelength. S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

13 INHOMOGENEOUS PLSM - 1 Consider in detail the model: 1-D slab non-uniform cold plasma, n ( x) n =, P, = B e B = Z, n (x) ω x x JG cr The presence of plasma gradients modifies our MHD equation. n externally excited electromagnetic wave with perturbed φ φ( x) exp ( ikyy iωt) is described by d dφ ( ω ω ( x) ) ky( ω ω( x) ) φ = dx dx ω ( x) k V ( x) S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

14 INHOMOGENEOUS PLSM - This equation has zero coefficient at high order derivative at the point x = x of the local lfvén resonance layer, where ω = ω ( x ) Investigating the equation in the vicinity of this point: d dφ dφ const ( ω ω( x) ) = = dx dx dx ω ω( x) Expand the local lfvén frequency in the vicinity of the point x = x : and obtain ω ( x ) + dω ( x) / dx ( x x ) = ω x= x ( x ) const ln x φ, x> x ( x x ) φ const ln + iπ, x< x The wave energy peaks up at x x = and resonant absorption of the wave energy occurs at this point continuum damping S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

15 LINER MODE CONVERSION TO KINETIC LFVÉN WVE Finite ion Larmor radius & finite electron parallel conductivity incorporated in the layer x x ρi remove the wave singularity Wave equation takes the form ( m / β <<1) : ω ρ i 3 4 e M i e ( 1 δ ) + ( 1 iδ ) φ + i T T i e << e i ω ω ( x) dφ ( ) k ( ω ω ( x) ) = d + y φ dx dx Solution of this equation satisfies the dispersion relation in the form of Kinetic lfvén Wave: 3 T ω k V ( x) 1+ x i i 1 4 Ti e ( k ρ ) ( 1 iδ ) + ( iδ ) = e In contrast to the shear lfvén wave, KW propagates across ~ ω /, and it has E KW kx B, S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

16 LINER MODE CONVERSION TO KINETIC LFVÉN WVE E x E O / Kx KINETIC LFVEN WVE JG c x E O / Kx d S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

17 DO NY SHER LFVÉN WVES EXIST WHICH DO NOT STISFY LOCL S DISPERSION RELTION ND RE FREE OF THE CONTINUUM DMPING? S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

18 DISCOVERY OF GLOBL LFVÉN EIGENMODE In cylindrical geometry, in addition to the continuous lfvén spectrum, ω =ω( r) k ( r) V( r), a discrete Global lfvén Eigenmode with frequency ω GE< ω exists in plasma with current (D.W.Ross et al. Phys. Fluids 5, 65 (198);K.ppert et al. Plasma Phys. 4, 1147 (198)) π K z Z J θ J J z r p r c r w JG c PLSM COLUMN COIL WLL The ideal plasma-coil-wall system used in the numerical investigation S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

19 DISCOVERY OF GLOBL LFVÉN EIGENMODE new high-quality, Q ω/γ ~ 1 3, resonance was discovered during these lfvén antenna studies, in plasmas with current 1 8 With current No current 6 Zr (Ω) Ω Real part of the coil impedance vs normalized frequency JG c S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

20 GE with GE ω GLOBL LFVÉN EIGENMODE ω < exists in Ideal MHD if the current profile determining minimum in lfvén continuum dω ( r) dr r=r =, i.e. 1 k dk dr 1 = V dv dr db ϑ / dr provides a 5 4 m = -3 ω 3 m = - m = -1 1 JG c.5 1. r / a The local minimum of the lfvén continuum provides a maximum of the perpendicular refraction index N = / ω. Similarly to fiber optics, the electromagnetic wave has to r ck r propagate in a wave-guide surrounding the region of the extremum refraction index. S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

21 NO CONTINUUM DMPING FOR GLOBL LFVÉN EIGENMODE Re (E r ) Re (E θ ) E (a.u.) Im (E r ) Im (E θ ) JG cr.5 1. r/a Ideal MHD GE with m=- ωge ω (r The eigenfrequency of GE does not satisfy the local lfvén resonance condition, ), for < r / a< 1. Therefore, this S mode has no singularity and does not experience continuum damping S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1

22 SUMMRY Linearised ideal MHD equations give the equation for plasma displacement t = c S div + V which describes compressional lfvén, slow magnetosonic, and shear lfvén waves The compressional waves can be decoupled at low-β, and they are coupled otherwise div In shear lfvén wave the fluid displacement vector and E ~ are perpendicular to the ω = ±k V magnetic field B. The wave propagates along B : In inhomogeneous plasma, shear lfvén wave experiences strong continuum damping The continuum damping in ideal MHD corresponds to linear mode conversion to short waved-length Kinetic lfvén Wave, which has a radial group velocity In cylindrical plasma with current, a discrete eigenmode may exist, Global lfvén Eigenmode, which has NO continuum damping + V z S.E.Sharapov, Lecture 3, ustralian National University, Canberra, 7-9 July 1