6-Plasma Single Particle, Fluid.pdf

Introduction to Nuclear Fusion. Prof. Dr. Yong-Su Na. How to describe a plasma?. 2. Description of a Plasma • Thermonuclear fusion with high energy particles of. 10-20 keV energy → plasma. • Three approaches to describe a plasma - Single particle approach - Kinetic theory - Fluid theory. http://http://dz-dev.net/blog/tag/eruption-solaire. 3. Individual Charge Trajectories • Equation of motion - Basic relation determining the motion of an individual charged particle of. mass m and charge q in a combined electric (E) and magnetic (B) field - Neglecting electromagnetic fields generated by movement of the charge itself. dv m = q(E + v × B ) dt. 4. Individual Charge Trajectories • Homogeneous electric field. dv m = qE = qE z k dt v x (0) = v x , 0 , v y (0) = v y , 0 , v z (0) = v z , 0 v x (t ) = v x , 0 , v y (t ) = v y , 0 , v z (t ) = v z , 0 +. q Ezt m. x(t ) = v x , 0t , y (t ) = v y , 0t , 1 q  z (t ) = v z , 0t +  E z t 2 2m  A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 5. Individual Charge Trajectories • Homogeneous magnetic field. dv m = q ( v × B ) = q ( v × Bz k ) dt v x (0) = v x , 0 , v y (0) = v y , 0 , v z (0) = v z , 0 = v||. v x (t ) = v⊥ cos(ωc t + φ ), v y (t ) =  v⊥ sin(ωc t + φ ), v z (t ) = v z , 0 = v|| v⊥ = v + v , tan(φ ) =  2 x. x(t ) = x0 +. v⊥. ωc. 2 y. vx,0. sin(ωc t + φ ), y (t ) = y0 ±. (x − x0 ) + ( y − y0 ) 2. v y ,0. 2. q Bz Cyclotron , ωc = m frequency. v⊥. ωc 2. cos(ωc t + φ ), z (t ) = z0 + v||t. 5 T, 10 keV rL = 0.05 mm for e rL = 2.9 mm for d rL = 3.5 mm for t.  v⊥   v⊥ m   = rL2 Larmor radius =   =    ω c   q Bz  A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 2. 6. Individual Charge Trajectories • Homogeneous magnetic field. Magnetic field. ∆z = 2π z (t ) = z0 + v||t. v||. ωc. Pitch of the helix. 2. (x − x0 )2 + ( y − y0 )2.  v⊥ m   = rL2 =    q Bz . ion A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 7. Individual Charge Trajectories • Combined homogeneous electric and magnetic field. dv m = q(E + v × B ) = q[(E x i + E z k ) + v × Bz k ] dt v x (0) = v x , 0 , v y (0) = v y , 0 , v z (0) = v z , 0 = v||. v x (t ) = v⊥* cos(ωc t + φ * ), v y (t ) =  v⊥* sin(ωc t + φ * ) −  qE v z (t ) = v|| +  z  m.  t  2.  Ex  v cos φ * v = v +  v y +  = ⊥ φ , tan( )= * Bz  cos φ  * ⊥. 2 x. Ex , Bz. v y ,0 +. Ex Bz. v x.0. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 8. Individual Charge Trajectories • Combined homogeneous electric and magnetic field x(t ) = x0 +. v⊥*. ωc. sin(ωc t + φ ), y (t ) = y0 ± *. v⊥*. ωc. cos(ωc t + φ * ) −. Ex t, Bz. 1  qE  z (t ) = z0 + v||t +  z t 2 2 m .  2 (x − x0 ) +  y − y0 + Ex Bz . 2.   v0*  t  =     ωc . 2. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 9. Individual Charge Trajectories • Combined homogeneous electric and magnetic field v = v gc + v g. : Guiding centre + Gyro motion. 10. Individual Charge Trajectories • Combined homogeneous electric and magnetic field v = v gc + v g. : Guiding centre + Gyro motion. Hannes Alfvén (1908-1995) “Nobel prize in Physics (1970)”. H. Alfvén, Cosmical Electrodynamics, Oxford (1950). 11. Individual Charge Trajectories • Combined homogeneous electric and magnetic field v = v gc + v g. : Guiding centre + Gyro motion. dv = F + q (v × B ) m dt dv gc dv g +m = F + q (v gc × B ) + q (v g × B ) m dt dt. v gc ,|| (t ) = v gc ,|| (0) + v gc ,⊥. 1 F|| dt ∫ m. F×B = = v DF 2 qB. F = Eq v D,E. E×B = B2. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 12. Individual Charge Trajectories • Spatially varying magnetic field. dv m = q[v × B(r )] dt. ∇B =. m. ∂Bz j, v|| = 0 ∂y. dv gc dt. +m. dv g dt. = F + q (v gc × B ) + q (v g × B ). v DF =.  dB  v⊥2  v⊥2  j = − q F ≈ − q   ∇B 2ωc  dy  2ωc  q mv⊥2 / 2 = − v⊥ rL ∇B = − ∇B 2 B. v D , ∇B =. F×B qB 2. F×B qB 2. B × ∇B v⊥2 B × ∇B 1 =± = ± v r ⊥ L 2ωc B 2 2 B2 A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 13. Individual Charge Trajectories • Curvature drift. dv z   m = q(v × B ) = q(v × B0k ) + q v × B0 j  dt R   F≈. mv||2 R. 2. R0 m. dv gc dt. +m. dv gc dt. = F + q (v gc × B ) + q (v g × B ). v DF =. v D , R = v gc , x. F×B qB 2. mv||2 R 0 × B 0 = qB02 R 2. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 14. Individual Charge Trajectories • Axial field variation. dv m = q(v × Bz e z ) + q(v × Br e r ) dt ∇ ⋅ B = 0 → Br ≈ −. r ∂Bz 2 ∂z. Bθ = 0, Br (r = 0) = 0. 1 mv⊥2 F|| = − ∇|| B = − µ∇|| B 2 B μ : magnetic moment of the gyrating particle. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 15. Individual Charge Trajectories • Invariant of motion 1 d d 1  E0 =  mv⊥2 + mv||2  = 0 dt dt  2 2 . F|| = m. dv || dt. = − µ∇||B = − µ. mv⊥2 / 2 µ= B. m. dv|| dt. =−. µ dB v|| dt. µ dB ∂B ∂B ds dt ∂B ds 1 = −µ ⋅ ⋅ = −µ ⋅ ⋅ =− ∂s ∂s dt ds ∂s dt v|| v|| dt. mv||. dv|| dt. = −µ. dB dt. dB d 1 2  mv||  = − µ dt  2 dt . d 1 2 1 2 d dB   =0  mv⊥ + mv||  = (µB ) +  − µ 2 dt  2 dt dt   . d (µB ) = µ dB + B dµ dt dt dt. dµ = 0 : adiabatic invariant dt. 16. Individual Charge Trajectories. http://www.physics.ucla.edu/icnsp/Html/spong/spong.htm. J. P. Graves et al, Nature Communications 3 624 (2012). 17. Individual Charge Trajectories. B × ∇B 1 v D ,∇B = ± v⊥ rL 2 B2. (x − x0 )2 + ( y − y0 )2 = rL2. v D,E. E×B = B2. v D,R. mv||2 R 0 × B 0 = qB02 R 2 18. Kinetic Approach of Plasmas • Boltzmann equation  qα    dfα ∂fα  ∂fα   ∂f   = + v ⋅ ∇f α + a ⋅ ∇ v f α = + v ⋅ ∇f α + ( E + v × B ) ⋅ ∇ v fα =  α  dt ∂t mα ∂t  ∂t  c. Ludwig Boltzmann (1844-1906) James Clerk Maxwell (1831-1879).   ∂B ∇× E = − ∂t  ∇⋅B = 0    1 ∂E ∇ × B = µ0 J + 2 c ∂t  σ ∇⋅E =. ε0. 19. Fluid Approach of Plasmas • Plasmas as fluids - The single particle approach gets to be complicated. - A more statistical approach can be used because we cannot follow each particle separately. - Introduce the concept of an electrically charged current-carrying fluid.. http://www.tower.com/music-label/gulan-music-studio Album art of music OST “Plasma Pong”. 20. Plasmas as Fluids. dfα ∂fα  qα     ∂f  = + v ⋅ ∇f α + ( E + v × B ) ⋅ ∇ v fα =  α  dt ∂t mα  ∂t  c. • Two fluid equations.  dfα  ∂fα    ∫ Qi  dt −  ∂t c  dv = 0. Q1 = 1.  Q2 = mα v. momentum. Q3 = mα v 2 / 2 energy.   dnα   + nα ∇ ⋅ uα = 0   dt α       d u  α nα mα   = qα nα ( E + uα × B ) − ∇ ⋅ Pα + Rα  dt α    3  dTα  nα   = − Pα : ∇uα − ∇ ⋅ hα + Qα 2  dt α. mass.  Pα ≡ nα mα w2    Rα ≡ ∫ mα wCαβ dw. hα ≡.  ∂fα    = ∑ Cαβ  ∂t  c β.      v = uα (r , t ) + w, w = 0.  1 nα mα w2 w 2.  1 Qα ≡ ∫ mα wα2 Cαβ dw 2 25. Plasmas as Fluids • Single-fluid magnetohydrodynamics (MHDs) A single-fluid model of a fully ionised plasma, in which the plasma is treated as a single hydrodynamic fluid acted upon by electric and magnetic forces.. • The magnetohydrodynamic (MHD) equation. ρ = ni M + ne m ≈ n( M + m) ≈ nM. σ = (ni − ne )e. charge density. mass density. Hydrogen plasma, charge neutrality assumed.        v = (ni Mui + ne mue ) / ρ ≈ ( Mui + mue ) /( M + m) ≈ ui + (m / M )ue mass velocity      J = e(ni ui − neue ) ≈ ne(ui − ue )     m J   J ui ≈ v + , ue ≈ v − M ne ne. electron inertia neglected: electrons have an infinitely fast response time because of their small mass 22. Plasmas as Fluids. F. F. Chen, Ch. 5.7. • Ideal MHD model  ∂ρ + ∇ ⋅ ρv = 0 ∂t  dv   ρ = J × B − ∇p dt. Mass continuity equation (continuously flowing fluid).   = 0     E+v×B =0   ∂B ∇× E = − ∂t   ∇ × B = µ0 J  ∇⋅B = 0. Energy equation (equation of state): adiabatic evolution. d  p  γ dt  ρ. Single-fluid equation of motion. Ohm’s law: perfect conductor → “ideal” MHD Maxwell equations ε0 → 0 assumued. (Full → low-frequency Maxwell’s equations) Displacement current, net charge neglected 23. Plasmas as Fluids • Magnetohydrodynamics (MHD): 1946 by H. Alfvén. Hannes Alfvén (1908-1995) “Nobel prize in Physics (1970)”. 24. Plasmas as Fluids • Magnetohydrodynamics (MHD): 1946 by H. Alfvén. 25. Plasmas as Fluids • Ideal MHD - Single-fluid model - Ideal:. Perfect conductor with zero resistivity - MHD:. Magnetohydrodynamic (magnetic fluid dynamic) - Assumptions:. Low-frequency, long-wavelength collision-dominated plasma - Applications:. Equilibrium and stability in fusion plasmas 26. What is Ideal MHD? • Ideal MHD: η = 0. • Resistive MHD: η ≠ 0. 27. What is Ideal MHD? • Ideal MHD: η = 0. • Resistive MHD: η ≠ 0. 28. Applications. PF coils. X. ●. X Force. X. X. ●. X Plasma Eqauilibrium and Stability. https://fusion.gat.com/theory/File:Theory1.gif. 29.