10-11-Mirror,Pinch.pdf

Introduction to Nuclear Fusion. Prof. Dr. Yong-Su Na. To build a sun on earth. 2. To build a sun on earth. 3. Magnetic Confinement • Bring the Sun on the Earth Quantity. ITER. Sun. Ratio. Diameter. 16.4 m. 140x104 km. ~ 1/108. Central temp.. 200 Mdeg. 15 Mdeg. 10. Central density. ~1020/m3. ~1032/m3. ~1/1012. Central press.. ~5 atm. ~1012 atm. ~1/1011. Power density. ~0.6 MW/m3. ~0.3 W/m3. ~2x106. Reaction. DT. pp. Plasma mass. 0.35 g. 2x1030 kg. ~1/1034. Burn time const.. 200 s. 1010 years. ~1/1015. M. Kikuch, “steady state tokamak reactor and its physics issues”, Talk at SNU, Korea, September 30, 2011. 4. To build a sun on earth. - Open magnetic confinement - Closed magnetic confinement. 5. What is open magnetic confinement?. 6. Open Magnetic Confinement • Equilibrium – Radial Force Balance   ∇p = J × B   ∇ × B = µ0 J  ∇⋅B = 0.   J ×B. ∇p. Bθ. X Jz. Jz x Bθ. Jθ. Jθ x Bz. X Bz. http://lasp.colorado.edu/education/outerplanets/solsys_star.php. 7. Open Magnetic Confinement. Z pinch. θ pinch. screw pinch. Magnetic mirror. http://phys.strath.ac.uk/information/history/photos.php http://www.frascati.enea.it/ProtoSphera/ProtoSphera%202001/6.%20Electrode%20experiment.htm. 8. What is magnetic mirror?. 9. Magnetic Mirror. - Suffering from end losses. 10. Magnetic Mirror. Gamma 10 and Hanbit. - To reduce end-leakage by establishing an increasing magnetic field at the two ends - Many of charged particles are trapped due to the imposed constraints on particle motion with regards to conservation of energy and the magnetic moment. - First proposed by Enrico Fermi as a mechanism for the acceleration of cosmic rays. 11. Magnetic Mirror. What is the motion of particles?. 12. Magnetic Mirror • Single particle picture - As the ion approaches the ends, it is increasingly subjected to drifts due to the inhomogeneity of the mirror field. - Drifts occur in azimuthal directions and the particles are still bound to their magnetic surfaces. mv||2 R 0 × B 0. v D , R = v gc , x =. rL =. v⊥ m q Bz. qB02. R2. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 13. 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) 14. Magnetic Mirror • Conservation of kinetic energy and magnetic moment. d d 1 2 1 2 E0 =  mv⊥ + mv||  = 0 dt dt  2 2 . d (µ ) = 0 dt. mv⊥2 / 2 µ= B. 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. Magnetic Mirror Enrico Fermi (1901-1954) Nobel Laureate in physics in 1938 Cf. Marshall Rosenbluth (Doctoral student). CP-1 (Chicago Pile-1, the world's first human-made nuclear reactor) and Drawings from the Fermi–Szilárd "neutronic reactor" patent. 17. Magnetic Mirror. 18. Magnetic Mirror. 1 2 1 2 1 2 mv = mv⊥ + mv|| = const. 2 2 2 1 2 1 2 2 mv⊥ mv sin θ 2 2 = = const. μ= B B. E0 =. 19. Magnetic Mirror • Fermi as a genuine scientist. 20. Magnetic Mirror • Condition for trapping of particles v|| B ≤B r. 1 mv⊥2 F|| = − ∇|| B = − µ∇|| B 2 B. =0. max. 1 2 1 2 mv⊥ min mv⊥ r 2 2 µ= = Bmin Br. 2 ⊥ min 2 ⊥r. v v. =. Bmin Br. E=. v. 1 2 1 1 1 1 mv||min + mv⊥2 min = mv||2r + mv⊥2 r = mv⊥2 r 2 2 2 2 2. 2 ⊥ min. v||2min + v⊥2 min. =. Bmin Br. Bmin. Br Bmax. 21. Magnetic Mirror • Condition for trapping of particles v|| B ≤B r. 1 mv⊥2 F|| = − ∇|| B = − µ∇|| B 2 B. =0. max. 1 2 1 2 mv⊥ min mv⊥ r 2 2 µ= = Bmin Br. E=. 1 2 1 1 1 1 mv||min + mv⊥2 min = mv||2r + mv⊥2 r = mv⊥2 r 2 2 2 2 2 v||. 2 ⊥ min 2 ⊥r. v v. =. Bmin Br. v. 2 ⊥ min. v||2min + v⊥2 min. =. 2 ⊥ min 2 min. Bmin v = Br v. loss cone. = sin 2 θ. Condition for trapping of particles. Bmin B = sin 2 θ ≥ min Br Bmax. loss cone 22. Magnetic Mirror. Bmin Bmin 2 = sin θ ≥ Br Bmax. • Mirror ratio θ0 π ∞   f (v ) 2  3 φ θ θ θ θ v dv + sin sin d d d   f (v ) d v ∫ 2 ∫ ∫ ∫ ∫  0 4πv  0 0 π −θ 0 double cone = ∞ = 2π π ∞  3 f (v ) 2 φ θ θ sin d d ( ) f v d v ∫0 ∫0 ∫0 ∫0 4πv 2 v dv. θ 0 = arcsin. 2π. f loss. v||. Bmin Bmax. = 1 - cosθ 0. f trap = 1 − f loss. B = cosθ 0 = 1 − min Bmax. f trap = 1−. mirror ratio: Bmax ≡ Rm Determining the effectiveness Bmin of confinement. 1 Rm A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 23. Stability in mirror?. 24. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. http://www.threes.com/index.php?view=article&id=20:greek-columns&option=com_content&Itemid=39 25. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. +. due to curvature of B in Z. v D , ∇B. E. -. B × ∇B 1 = ± v⊥ rL 2 B2. - Particle picture: The curvature drift leading to azimuthal polarisation to create the E-field resulting in the ExB drift displacing the plasma particles radially outward. v D,R. mv||2 R 0 × B 0 = qB02 R 2 26. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. Magnetic well. minimum-B field configuration: field lines are (almost) everywhere concave into the plasma. 27. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. x x F. F. Chen, “An Indispensable Truth”, Springer (2011) http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=&currentPage=1&listtype=0 http://en.wikipedia.org/wiki/File:St_Louis_Gateway_Arch.jpg. 28. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. x x F. F. Chen, “An Indispensable Truth”, Springer (2011) http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=&currentPage=1&listtype=0 http://en.wikipedia.org/wiki/File:St_Louis_Gateway_Arch.jpg. 29. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. Yin-Yang coil Ioffe bars. Magnetic field profiles in a Ioffe-Pritchard trap A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 30. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. Baseball coil http://www.wangnmr.com/Copper_magnet_catolog.htm. 31. Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field. Flux surfaces. Plug/Barrier (potential plugging) ECRH. Baseball coils (MHD stabilising) ICRF. Gamma 10 Tandem mirror (Univ. of Tsukuba, Japan). 32. Magnetic Mirror • Instabilities - Velocity-space instability: driven by the non-Maxwellian velocity distribution due to the preferred loss of particles with large v||/v⊥. - Enhancing the velocity-space diffusion into the loss cone - Observed that such it is less harmful to plasma confinement when the mirror device is short in dimension. v||. loss cone. loss cone. 33. Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time.. 34. Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time.. ion. ion loss cone. loss cone. ion electron. 35. Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time..  Bmax   A T ln Bmin   Bmax     : mirror confinement time ≈ τ ii ln 4 ≈C B n Z i i  min  3/ 2 Ai1/ 2 (kTi ) 1 : ion-ion collision time ∝ τ = 1/ 2 3 / 2 i i. τM. ii. ni σ s vr. ni qi4 σ s vr. - Mirror confinement time does not depend on the actual magnitude of B or the plasma size but on the size of the loss cone. - Higher density enhances the scattering into the loss cone.. 36. What is Z-pinch?. 37. Magnetic Pinch. Bθ. • The Z Pinch. X Jz. Jz x Bθ. - The Lorenz force created by a lightning strike crushed this hollow lightning rod and led to the discovery of the pinch.. B J. G. Thomson and P. Thonemann http://joiimg.tistory.com/416 http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?p=731&sid=824cd77d679b252e92355fcbf212ed7a. http://www.plasma-universe.com/Pinch_(plasma_physics)?title=Special:Booksources&isbn=0387975756 http://ask.nate.com/qna/view.html?n=6318216. 38. Magnetic Pinch. 39. Magnetic Pinch • The Z Pinch - The plasma carries an electric current and is confined by the magnetic field induced by this current. - As the current is increased, the larger magnetic field compresses the plasma and also raises its temperature by Joule-heating. - Confinement and heating is simultaneously provided. - The Lorenz force created by a lightning strike crushed this hollow lightning rod and led to the discovery of the pinch.. B J. G. Thomson and P. Thonemann http://joiimg.tistory.com/416 http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?p=731&sid=824cd77d679b252e92355fcbf212ed7a. http://www.plasma-universe.com/Pinch_(plasma_physics)?title=Special:Booksources&isbn=0387975756 http://ask.nate.com/qna/view.html?n=6318216. 40. Magnetic Pinch. Bθ. • The Z Pinch. X Jz. - The plasma carries an electric current and is confined by the magnetic field induced by this current. - As the current is increased, the larger magnetic field compresses the plasma and also raises its temperature by Joule-heating. - Confinement and heating is simultaneously provided.. P. C. Thonemann et al, Nature 181 217 (1958). ZETA (1954-58, UK). Jz x Bθ. Where did the neutrons come from in ZETA?. 41. Magnetic Pinch • The Z Pinch. self field induced by JZ. How about the pinch effect in a electric cable/wire?. longitudinal plasma current. Large current produced by capacitor. - Equilibrium Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0 2. Ampere’s law: μ0J = ▽xB 3. The momentum equation: JxB = ▽p.   ∇p = J × B   ∇ × B = µ0 J  ∇⋅B = 0. J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note. 42. Magnetic Pinch • The Z Pinch Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0 1 ∂B θ. r ∂θ. =0. 2. Ampere’s law: µ0J = ▽xB. Jz =. 1 d (rBθ ) µ 0 r dr. 3. The momentum equation: JxB = ▽p. dp Bθ d + (rBθ ) = 0 dr µ 0 r dr. J z Bθ = −. Bθ2  Bθ2 d  p+ + =0   dr  2µ0  µ0 r. dp dr. particle pressure + magnetic pressure force tension force by the curvature of the magnetic field lines. 43. Magnetic Pinch. Bθ2  Bθ2 d  p+ + =0   dr  2µ0  µ0 r. • The Z Pinch. - It is the tension force and not the magnetic pressure gradient that provides radial confinement of the plasma.. Bθ = Jz =. µ0 I 0 r 2π r 2 + r02 I0. 2 0. r π (r 2 + r02 ) 2. µ 0 I 02 r02 p= 8π 2 (r 2 + r02 ) 2. Bθ. Jz x Bθ. X Jz. Bennett profiles (Bennett, 1934). - The magnetic pressure and plasma pressure each produce positive (outward forces) near the outside of the plasma. - The plasma is confined by magnetic tension. J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note http://scienceray.com/technology/industry/where-do-rubber-bands-come-from/. 44. Magnetic Pinch • The Z Pinch - A number of linear Z-pinch experiments were constructed during the early years of the fusion program. - Large currents of ~0.1 MA are needed thus rendering the Z pinch to operate only in short pulses. - Exhibiting disastrous instabilities, often leading to a complete quenching of the plasma after ~1-2 μs. Sausage or pinch instability. Helical kink instability. A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000). 45. Magnetic Pinch • The Z Pinch - A number of linear Z-pinch experiments were constructed during the early years of the fusion program. - Large currents of ~0.1 MA are needed thus rendering the Z pinch to operate only in short pulses. - Exhibiting disastrous instabilities, often leading to a complete quenching of the plasma after ~1-2 μs.. "kink instability" in one of the earliest Z-pinch devices, a pyrex tube used by the AEI team at Aldermaston, or earlier while still at Imperial College 46. Magnetic Pinch • The Z Pinch - A number of linear Z-pinch experiments were constructed during the early years of the fusion program. - Large currents of ~0.1 MA are needed thus rendering the Z pinch to operate only in short pulses. - Exhibiting disastrous instabilities, often leading to a complete quenching of the plasma after ~1-2 μs.. θ-direction tension for equilibrium, z-direction tension for stability. 47. What is θ-pinch?. 48. Magnetic Pinch. Jθ. • The θ Pinch. X Bz. induced diamagnetic current. Jθ x Bz. externally driven. 49. Magnetic and Kinetic Pressure • Plasma Equilibrium   → Force balance kinetic pressure ∇p = J × B balanced by JxB (Lorentz) force   → Ampere‘s law ∇ × B = µ0 J  → Closed magnetic field lines ∇⋅B = 0  B ⋅ ∇p = 0.  J ⋅ ∇p = 0. induced by the pressure gradient: causing a decrease in B → diamagnetism Diamagnetic current .  ∇p × B v D , ∇p = − nqB 2     B × ∇p J = ni qi vD ,i + ne qe vD ,e = B2 - If B is applied, plasma equilibrium can be built  by itself due to induction of diamagnetic current. ∇p = J × B. ion. ←∇p xB. 50. Magnetic Pinch. Jθ. • The θ Pinch. X Bz. induced diamagnetic current. Jθ x Bz. externally driven. - Equilibrium Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0 2. Ampere’s law: μ0J = ▽xB 3. The momentum equation: JxB = ▽p.   ∇p = J × B   ∇ × B = µ0 J  ∇⋅B = 0. J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note. 51. Magnetic Pinch • The θ Pinch Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0. ∂Bz =0 ∂z. 2. Ampere’s law: µ0J = ▽xB. Jθ = −. 1 ∂Bz µ 0 ∂r. 3. The momentum equation: JxB = ▽p. d  Bz2   p +  = 0 dr  2µ0 . dp J θ Bz = dr. B02 Bz2 = p+ 2µ0 2µ0. - At any local value of r, the sum of the particle pressure and the magnetic pressure is a constant, equal to the externally applied magnetic pressure. - The plasma is confined radially by the pressure of the externally applied magnetic field.. 52. Magnetic Pinch • The θ Pinch - Typical example. Plot the forces.. p = p0 exp(−r 2 / a 2 ) Bz = B0 [1 − β 0 exp(−r 2 / a 2 )]1/ 2. β 0 ≡ 2 µ 0 p / B02 B02 Bz2 = p+ 2µ0 2µ0 - The plasma is confined radially by the pressure of the externally applied magnetic field. 53 J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note. Magnetic Pinch • The θ Pinch - One of the early successes of the fusion program in terms of the performance. - Ti ~ 1-4 keV, n ~ 1-2x1022 m-3, β(0) ~ 0.7-0.9 - High temperature using the implosion heating method: rapidly rising magnetic field acting like a piston - No indication of macroscopic instability (neutrally stable) - Severe end loss (τ ~ L/VTi, e.g. 10 μs for a 5 m device). 54. What is screw pinch?. 55. Magnetic Pinch • The General Screw Pinch - A hybrid combination of Z pinch and θ pinch - This combination of fields allows the flexibility to optimise configurations w.r.t. force balance and stability. Bθ. Jz x Bθ. X Jz. - Equilibrium Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0 2. Ampere’s law: μ0J = ▽xB 3. The momentum equation: JxB = ▽p. Jθ. ·. Jθ x Bz Bz.   ∇p = J × B   ∇ × B = µ0 J  ∇⋅B = 0 56. Magnetic Pinch • The General Screw Pinch - Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0 1 ∂B ∂B θ. r ∂θ. +. 2. Ampere’s law: µ0J = ▽xB. z. ∂z. =0. Jθ = −. 1 ∂Bz µ 0 ∂r. 3. The momentum equation: JxB = ▽p. Jz =. 1 d (rBθ ) µ 0 r dr. dp J θ Bz − J z Bθ = dr. Bθ2 + Bz2  Bθ2 d   p +  + =0 dr  2µ0  µ0 r - Even though the equations are nonlinear, the forces superpose because of symmetry. 57. Magnetic Pinch • The General Screw Pinch - Because of its flexibility the general screw-pinch relation describes a wide variety of configurations.. 58.