Magnetic phase transition and confinement regimes

Size: px

Start display at page:

Download "Magnetic phase transition and confinement regimes"

Andrew Rose
5 years ago
Views:

1 Magnetic phase tansition and confinement egimes Emilia R. Solano 1,2, Richad D. Hazeltine 3 1 Laboatoio Nacional de Fusión, CIEMAT, Madid, Spain 2 JET EFDA CSU 3 Institute fo Fusion Studies, Univ. of Texas at Austin, Austin, USA Acknowledgements: E. Fable (IPP Gaching), P. McCathy (U.C. Cok, Ieland) 15 th Euopean Fusion Theoy Confeence Oxfod, United Kingdom 23 d -26 th Septembe 2013 Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 1/29

2 Inspiation and motivation Since the discovey of the H-mode, many models of the L-H tansition have been poposed The now conventional model is based on electostatic tubulent votices shedded by otation shea. It is highly developed, can include sophisticated pedato-pey model, it is now beginning to move towads electomagnetic consequences of electostatic fluctuations It is a vey attactive model to many, but then I saw this movie of an L-H tansition in MAST and I stated thinking about phase tansitions. (L-mode s, H-mode late) One of the bette studied phase tansitions in physics is the magnetic phase tansition. So that got me thinking some moe Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 2/29

3 Usual set-ups Micotubulence γ > 0 electostatic dift velocity Sophisticated, non-linea models λ~ (1-10) ρ s MHD γ > 0 electo-magnetic V themal o V Alfvén Linea models Enegy pinciple λ~ p/p o λ~ a Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 3/29

4 Usual set-ups Micotubulence γ > 0 electostatic dift velocity Sophisticated, non-linea models λ ~ (1-10) ρ s MHD γ > 0 electo-magnetic V themal o V Alfvén Linea models Enegy pinciple λ~ p/p o λ~ a Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 4/29

5 Index Plasma oveall magnetisation (cylindical tokamak appoximation) Magnetisation of tubes (field aligned pessue petubations) Motion of magnetised tubes in magnetised plasma ( z gadient) Effect on pofiles Connections to intechange stability theoy Expeimentally testable citeion Data! Refeences: E. R. Solano, Plasma Phys. Contol. Fusion 46 L7 (2004) E. R. Solano & R. D. Hazeltine Nucl. Fusion (2012) Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 5/29

6 Plasma Equilibium Plasma foce balance: p= j = j + j dp p' d Ψ j ζ 0 ζ θ θ ζ ( Rp' FF'/ ( R) ) = + µ µ j = F', F( Ψ ) = R θ θ ζ 0 In cylindical appoximation : d + p d + = 2 µ µ z θ θ 0 0 j ( ) ( R p' FF'/ R ) = + µ z d µ 0jθ= d z Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 6/29

7 Plasma magnetization of a cylindical tokamak Integating cylindical foce balance: a pds 2 2 za 0 z 2za za z θa µ 0 θa θa βθ = + /2 ( ) d d z = µ 0 j θ (β θ -1) is elated to nomalised aveage plasma magnetisation z β θ <1 p< j β θ > 1 ζ θ p> j ζ θ z inceased by j θ paamagnetism, low pessue z educed by j θ diamagnetism, high pessue z z z0 z0 Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 7/29

8 MAGNETS The tokamak plasma is a magnet. /2 pds a vac 2 vac z z µ 0 θa µ 0 z 0 the diffeence between poloidal magnetic and kinetic pessue detemines if it is a paa-magnet o a dia-magnet Paamagnets incease the backgound magnetic field move towads high field egions Diamagnets decease the backgound magnetic field move towads low field egions so fa I have just eviewed well known facts Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 8/29

Diamagnetic levitation A fog (diamagnetic) dopped in a

M V ey and A K Geimz Eu. J. Phys. 18 (1997) 307 313.

9 Diamagnetic levitation A fog (diamagnetic) dopped in a stong magnetic field levitates because it ties to get away fom the high field. It moves towads the lowe field, aanged to be upwads. M V ey and A K Geimz Eu. J. Phys. 18 (1997) Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/2013 9/29

Magnetised plasma element The poloidal cuent density aound a tokamak plasma is esponsible fo plasma magnetisation: the diffeence between the

filaments, with paallel cuent density Next conside a fieldaligned plasma element with a pessue petubation elative to the backgound plasma

10 Magnetised plasma element The poloidal cuent density aound a tokamak plasma is esponsible fo plasma magnetisation: the diffeence between the extenally applied tooidal field and the local tooidal field inside the plasma Y Kinetic Pessue Contous paa dia X Not conventional cuent filaments, with paallel cuent density Next conside a fieldaligned plasma element with a pessue petubation elative to the backgound plasma pessue. What is its magnetisation? Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

11 Magnetism in cylindical tube with pessue hill/hole ɶM ρ pɶ F ρ dvρ = mn = ɶ ρp+ dt ( ɶj ) ρ ɶ b ɶp j = Diamagnetic cuent: if inside the tube thee is a pessue hill (moe pessue than in the backgound plasma), the associated pependicula cuent educes z : diamagnetism Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

12 Magnetism in cylindical tube with pessue hill/hole ɶM ɶM ρ pɶ pɶ F ρ dvρ = mn = ɶ ρp+ dt ( ɶj ) ɶ b ɶp j = Diamagnetic cuent: if inside the tube thee is a pessue hill, the associated pependicula cuent educes z : diamagnetism Paamagnetic cuent: if inside the tube thee is a pessue hole (less pessue than in the backgound plasma), the associated pependicula cuent inceases z : paamagnetism ρ Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

13 Magnetism in cylindical tube with pessue hill/hole ɶM ɶM ρ pɶ F ρ dvρ = mn = ɶ ρp+ dt ( ɶj ) ɶ b ɶp j = Diamagnetic cuent: if inside the tube thee is a pessue hill, the associated pependicula cuent educes z : diamagnetism Paamagnetic cuent: if inside the tube thee is a pessue hole, the associated pependicula cuent inceases z : paamagnetism ρ pɶ Magnetization of the blob: b p ɶ Mɶ ɶ =µ dmˆ 0 = d 1 ρb p( ρ') p Μɶ ɶ ɶ = λ d ρ ' λ b 0 ρ' < 0, dia > 0, paa Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

14 Movement of magnetized object in field gadient (see Jackson) z0 dv dt m V = ( ) z0 M ɶ M ɶ z V (. ) dv ( ɶj) ( ) = dv dv 0z M ɶ z blob magnetization dv mnv Mɶ z z0= µ 0M ɶ zj dt θ the cold tube (paamagnetic) seeks high field the hot tube (diamagnetic) seeks low field aveaged d z /d contols motion of magnetised plasma tubes: Anti-potential leads to magnetic phase sepaation Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

15 Paamagnetic plasma: L-mode p() p() Motion of pessue blobs depends on d z /d mn V dv dt Mɶ ζ d d ζ0 paamagnetic cold blobs move inwad, diamagnetic hot blobs move outwad outwad themal enegy convection at the expense of inwad magnetic enegy convection z () pɶ blobs gow, instability Paamagnetism L-mode Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

16 Diamagnetic plasma: H-mode p() p() Motion of pessue blobs depends on d z /d mn V dv dt Mɶ ζ d d ζ0 diamagnetic hot blobs move inwad, paamagnetic cold blobs move outwad inwad themal enegy convection at the expense of outwad magnetic enegy convection z () diamagnetism pɶ blobs decease, satuation H-mode Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

17 Magnetic ounday: phase tansition p() p inceases somewhee, ceating diamagnetic egion at plasma edge. z () Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

18 Magnetic ounday: phase tansition p() At a magnetic phase bounday blobs of the same type accumulate diamagnetic blobs (heat) seek wells z () paamagnetic blobs seek hilltops With multiple blobs moving, p and z pofiles evolve Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

19 Magnetic ounday: phase tansition p() Pessue gadient inceases in diamagnetic egion Deceases in paamagnetic egion z () Magnetization, of both signs, inceases. Phase tansition is self-einfocing. Pessue pedestal foms, gows. Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

20 Evolution equations Ideal 1 fluid MHD evolution magnetization foce tems nmi dt dv = F 3 p =. Q + H 2 t = ( ) η t n m V i dv dt v ( j j ni) z = ( vθ) M M M 0z 3 p d = (pv) ɶ 2 t d t d = Mɶ ζ d z Fo now, conside what the magnetisation foce does, disegad othe tanspot mechanisms Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

Pedestal fomation at magnetisation bounday Assume dashed z (), p() initial pofiles Ideal MHD

d Integating one tempoal step t pessue steepens in diamagnetic egions, inceases diamagnetism

Initial pofiles - - - Final pofiles Magnetic phase sepaation dives pedestal fomation Emilia R.

21 Pedestal fomation at magnetisation bounday Assume dashed z (), p() initial pofiles Ideal MHD with magnetization foce n m V i dv dt M M d = Mɶ ζ d 0z 3 p d = (pv) ɶ 2 t d z d = (v ɶ 0z ) t M d Integating one tempoal step t pessue steepens in diamagnetic egions, inceases diamagnetism flattens in paamagnetic egions, inceases paamagnetism ~ v ~ p() ζ - ζ0 d ζ d ~ M ζ dia paa dia Initial pofiles Final pofiles Magnetic phase sepaation dives pedestal fomation Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

22 Recap and applicability conditions plasma tubes with an excess o defect of pessue ae convected adially, depending on the plasma magnetisation. phase sepaation occus at the flux suface in which j q changes sign. Unde what conditions? The seed pessue petubation is stong enough to potude above o below the backgound pessue pofile. plasma elements must be long enough to aveage out the 1/R vaiation of the vacuum field: λ > qr. Othewise conventional, ballooning-like tanspot would dive shot diamagnets towads low R. Collisionality/esistivity: the tempeatue must be high enough fo paticles to sample LFS and HFS befoe being scatteed out of the tube. Edge pessue must be high enough to allow negative petubations as well as positive. Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

23 Caveats So fa we have teated the plasma tubes as test paticles with a magnetic moment ignoed geometical magnetisation fom j No evolution equation fo tube magnetisation. the magnetic inteaction between plasma elements and bulk plasma is quite complex, and ou model vey simple (too simple?) We hope that a moe detailed calculation, up to second ode on the spatial vaiation of F=R to, can be caied out. Kind of neoclassical magnetisation, instead of classical Much hade to do Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

24 Intechange instability 1 pesent when a adial foce acts equally on electons and ions equivalent to the Rayleigh-Taylo instability in a fluid. magnetization gadient acting on magnetized plasma blobs eplace gavitational field o cuvatue. Magnetization intechange g heavy fluid light fluid γ = g/ λ 1 ɶp dζ 1 γ = ρ d λρ m,blob Magnetization intechange gowth faste fo high magnetisation, stong seed, low field & mass 1 M.N. Rosenbluth and C.L. Longmie, Annals of Physics, Volume 1, Issue 2, May 1957,120 Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

25 Suydam citeion fo intechange instability. R. Suydam, Poc. 2nd UN Conf. on Peaceful Uses of Atomic Enegy, Geneva, Rq κ q' 2 s µ 0 4q β ' > magnetic shea opposes intechange of tubes diven by cylindical cuvatue and β Genealization: add magnetization foce to cylindical cuvatue κ d 0z q' Mɶ z 2 s 0 d 4q Rq β ' + > µ d In magnetically mixed states 0z Mɶ z < 0 d magnetisation foce adds to cuvatue, instability, until the magnetic shea q o the vaiation of d z /d changes. Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

26 Evolution towads magnetic phase tansition As heating is applied, low pessue paamagnetic plasmas have degaded confinement, diven by low p When sufficient heating is applied, p gows until zeo magnetization is obtained somewhee inside the plasma: j = θ 0 p= j + j = 0 ζ θ θ ζ Estimate citical pessue gadient as dp d = j = E η ζ θ loop Spitze θ Need database of typical p, loop voltage, esistivity and θ to test pedictions o measuements of j θ Explaining T e theshold fo L-H tansition via η Spitze? and associated pessue gadient (β θ ) theshold Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

27 Expeimental evidence? AUG (10 ) j θ zeo cossing at L H tansition P. J. McCathy, P4.115, 40th EPS Confeence on Plasma Physics, Espoo, 2013 Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

28 Expeimental evidence? AUG II In a slow L-H tansition followed by type III ELMs j θ emains diamagnetic afte tansition ut at least one counte example has been found in a slow tansition (still unpublished). Moe analysis needed, as well as moe efined model. Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

29 Summay and comments Fist-pinciples model of plasma magnetization and magnetic phase tansition as the basis fo tiggeing confinement tansitions The magnetic state of the plasma detemines convective motion of high and low pessue tubes. Paamagnetic plasma egions attact cold tubes, become moe paamagnetic. Diamagnetic plasma egions attact hot tubes, becoming moe diamagnetic. A pedestal stuctue builds up in the magnetic phase bounday. Magnetic bounday defines citical magnetization: Magnetization foce dives the magnetic intechange mechanism in closed field line egion, simila to cuvatue intechange in SOL. Emilia R. Solano Magnetic Phase Tansitions & Confinement Regimes 15 th Eu. Fus. Th. Conf., Oxfod, 24/09/ /29

Magnetohydrodynamics (MHD) I

Magnetohydrodynamics (MHD) I Magnetohydodynamics (MHD) I Yong-Su Na National Fusion Reseach Cente POSTECH, Koea, 8-10 May, 006 Contents 1. Review Confinement & Single Paticle Motion. Plasmas as Fluids Fluid Equations 3. MHD Equations