Modelling of a plasma break-down for the SMART tokamak

Mas e ’s Deg ee Thesis In e -Uni e si y Mas e ’s Deg ee in Nuclea Physics Facul y o Physics, Uni e si y o Se ille Modelling o a plasma b eak-down o he SMART okamak Au ho : Daniel L´opez Ai es1 Supe iso s: D . Manuel Ga c´ıa Mu˜noz2 D . Ca los So ia del Hoyo3 Sep embe 7, 2020 1E-mail add es: [email p o ec ed] 2Depa men o A omic, Molecula and Nuclea Physics, Facul y o Physics, Uni e si y o Se ille 3Depa men o Elec onics and Elec omagne ism, Facul y o Physics, Uni e si y o Se ille Abs ac Achie ing con olled nuclea usion on Ea h could be a decisi e s ep on he ques owa ds g een ene gy p oduc ion since i would b ing a i ually enewable ene gy sou ce wi hou CO2 emissions. The Plasma Science and Fusion Technology g oup o he Uni e si y o Se ille is cu en ly designing a magne ic usion de ice, a sphe ical okamak o con olled nuclea usion esea ch, called SMall Aspec Ra io Tokamak (SMART). This hesis will model he i s phase o he ini ia ion o a okamak ( okamak s a -up), he b eak-down phase, in which he uel will ansi ion om he gas s a e o he plasma s a e, making use o he Fies a oolbox. The undamen al o okamak physics and okamak s a - up will also be e iewed. The cu en wa e o ms o he SMART coilse ha e been op imized o achie e he desi ed plasma equilib ium and allow he b eak-down o he p e- ill gas by educing he s ay poloidal magne ic ield and maximizing he loop ol age induced by he induc o solenoid. Se e al c i e ia ha e also been applied o es he easibili y o he b eak- down phase, such as he Paschen’s b eak-down cu e, he es ima ion o he a alanche ime, and he calcula ion o he connec ion leng h. The elec ic po en ial gained by he elec ons as hey ollow he magne ic ield lines has been compu ed o es ima e whe e he gas will b eak-down. B eak-down o he gas wi hou he use o any auxilia y hea ing me hod has been achie ed, las ing ew milliseconds o gas p essu es abou 10−4To . i Acknowledgemen s I would like o hank J. Lis e , who e e ed me o Izaskun Ga ido, who ga e me he main e e ence o unde s and he RZIp code. I would also like o hank Geo Cunningham o his help and commen s which help me ocus on he igh issues ega ding b eak-down, and which also p o ide us wi h he Fies a eposi o y on gi and a b ie Fies a documen a ion. Special hanks o my u o s, his mon hs o wo k ha e been eally aluable, I ha e lea ned a lo , especially abou eam wo k in a mul idisciplina y en i onmen . I also ha e o apologize o Ca los o no elling him all he apid upg ades he SMART eac o was su e ing o ensu e p ope b eak-down condi ions, isola ing him a bi om he g oup and om my wo k. Thanks o Eli, o he help supe ising his documen . Special hanks o Sco , who, despi e a i ing o he g oup in ma ch, ha e lea ned e y apidly and ha e played a c i ical ole in he changes in he SMART eac o , and o Alessio and Manu oo, who also played a c i ical ole in he changes om he enginee ing poin o iew. I would also like o hank Jes´us Poley, ou alks abou usion and science in gene al we e eally in e es ing and help ul o bo h o us. Thanks o An onio o his help wi h Inkscape. Special men ion o he CS-GO eam o all he good F iday nigh s du ing he Co id-19 qua - an ine. ii Con en s Con en s iii 1 In oduc ion 1 1.1 Nuclea usion ..................................... 1 1.1.1 De ini ion o a plasma ............................ 5 1.2 Con inemen o cha ged pa icles .......................... 6 1.3 Tokamaks ....................................... 9 1.4 Mo i a ion: SMART ................................. 12 1.5 Objec i es and ou line o he hesis ......................... 13 2 Fundamen als o okamak physics 15 2.1 Magne ohyd odynamic model o a plasma ..................... 15 2.2 G-S equa ion ..................................... 16 2.3 Pa ame e s ...................................... 18 2.3.1 Plasma shape and con ol in okamaks ................... 18 2.3.2 Sa e y ac o and no malized p essu e .................... 20 3 Tokamak s a -up 22 3.1 Plasma b eak-down .................................. 24 3.1.1 A alanche o b eak-down ime ........................ 28 3.1.2 Vol age and elec ic ield induced by he induc o solenoid ........ 29 4 Fies a oolbox 31 4.1 G ad-Sha ano sol e , EFIT ............................ 31 4.2 RZIp model ...................................... 32 5 Simula ion p ocedu e 37 5.1 Field line ace .................................... 39 iii CONTENTS CONTENTS 5.1.1 Po en ial .................................... 40 6 Resul s 41 6.1 S a ic and dynamic beha iou o SMART ...................... 41 6.1.1 Ta ge equilib ium .............................. 41 6.1.2 Cu en wa e o ms .............................. 42 6.1.3 Dynamic simula ions ............................. 45 6.2 B eak-down esul s .................................. 46 6.2.1 Magne ic ields ................................ 46 6.2.2 Connec ion leng h .............................. 48 6.2.3 Paschen cu e and a alanche ime ...................... 53 6.3 Discussion ....................................... 57 7 Summa y and conclusions 62 8 Bibliog aphy 63 A Calcula ion o he ol age induced by a solenoid o ini e wid h 68 B S a e space ep esen a ion 70 i Chap e 1 In oduc ion 1.1 Nuclea usion as an ene gy sou ce Nowadays, he human kind is beginning o unde s and he damage i s ac i i y is causing on Ea h, ha could lead o he des uc ion o he plane we li e in and, as a consequence, o ou sel es. A adical change is needed in human’s li e be o e i is oo la e. One undamen al s ep is o s op using ossil uels as an ene gy sou ce, and use enewable sou ces ins ead, like wind ene gy, sola ene gy, o geo he mal ene gy. Howe e , he e is ano he ene gy sou ce, i ually unlimi ed ha could p o ide a huge s ep o wa d his ansi ion, nuclea usion. Nuclea usion is a ype o nuclea eac ion in which wo o mo e a omic nuclei ( eac an s) Xand Yin e ac and p oduce a hea ie nuclei, gene ally in an exci ed s a e (X+Y)∗. This compound nuclei could de-exci a e by emi ing elec omagne ic adia ion and, i he exci a ion ene gy is su icien ly high, by eleasing neu ons (e apo a ion). X+Y→(X+Y)∗(1.1) Applying he conse a ion o ene gy o he eac ion, using he labo a o y e e ence ame EX+EY=E(X+Y)∗=E(X+Y)+Eexc ⇒Ti+ (mX+mY)c2=T + (mX+mY)c2+Eexc ⇒T −Ti≡Q=−Eexc, (1.2) Since he exci a ion ene gy Eexc is always posi i e, he Q ac o o he eac ion is nega i e, meaning ha he e is a e e ence ame in which T = 0, bu Tican ne e be ze o. This means ha his eac ion is an endo he mic eac ion, i needs ene gy o ake place. 1 1.1. NUCLEAR FUSION CHAPTER 1. INTRODUCTION Figu e 1.1. Coulomb ba ie be ween wo nu- clei o mass numbe Aand a.e=e/(4πε0). Sou ce: [1]. The need o ene gy o p oduce a nuclea usion eac ion can be easily unde s ood. Due o he posi- i e cha ges o he nuclei, hei Coulomb in e ac ion is epulsi e. Howe e , no always he in e ac ion is epulsi e, i he nuclei a e close enough, he nuclea in e ac ion appea s, and since i s much mo e in ense han he elec omagne ic in e ac ion, he dominan in- e ac ion is nuclea , which will a ac s he nuclei, en- abling hem o app oach enough so hey can use in o a new nuclei. I we plo ed he po en ial be ween wo nuclei, i would look simila o he one on igu e 1.1. The e a e wo egions, he egion o >>, in which he nuclei a e a away om each o he so he e is no nuclea in e ac ion be ween hem and he po en ial is he Coulomb po en ial, and he << egion, in which he nuclea in e ac ion appea s, so he po en ial is he nuclea po- en ial and he nuclei a ac each o he . The poin whe e Coulomb in e ac ion is compensa ed by he nuclea in e ac ion is usually es ima ed as RN≃1.45(A1/3 1+A1/3 2) m, whe e A1,A2a e he mass numbe o he nuclei (Aand ain igu e 1.1). The main challenge o nuclea usion eac ions is ha he Coulomb ba ie ha e o be o e comed, allowing he nuclei o app oach enough so hey can use in o a new nuclei. Al hough nuclea usion eac ions need ene gy o ake place, he eac ions could elease mo e ene gy han he needed o s imula e i . This can be unde s ood aking in o accoun he concep o binding ene gy B(N, Z), which is he ene gy needed o spli a nuclei in o i s componen s, B(A, Z)≡Zmp+ (A−Z)mn−[M(A, Z)−Zme])c2,(1.3) whe e M(A, Z) is he mass o an a om o a omic numbe Zand mass numbe A, and mp,mn and mea e he p o on, neu on and elec on masses espec i ely. I B/A is plo ed, igu e 1.2 is ob ained. B/A inc eases wi h A up o 56Fe, and hen i s a s dec easing. This means ha i combining elemen s o he le o 56Fe, he compound nuclei is mo e s able han he ini ial nuclei, and he di e ence o binding ene gy om he s able compound nuclei and he less s able sepa a e nuclei is eleased in he o m o kine ic ene gy. Equi alen ly, i a nuclei hea ie han 56Fe spli s in o ligh e nuclei, since he inal nuclei will ha e highe binding ene gy han he o iginal nuclei, he di e ence o binding ene gy will be eleased. 2 1.1. NUCLEAR FUSION CHAPTER 1. INTRODUCTION Figu e 1.2. Binding ene gy pe nucleon. The maximum B/A co espond o 56Fe, he mos s able elemen . Sou ce: [2]. Figu e 1.2 essen ially explains why en- e gy can be ob ained om ission eac ions, which ha e lead o he c ea ion o nuclea powe plan s o ob ain ene gy by ission eac ions. Bu , in he same way i sug- ges ha we could ob ain ene gy as well by pu suing nuclea usion eac ion. In bo h cases, he undamen al condi ion o ob ain ene gy is ha he ene gy eleased is g ea e han he ene gy applied. In ene gy con- ex , i is de ined a a iable called Q ac- o which is he a io be ween he powe ob ained and he powe applied o he sys- em, Q=Pob Papp .(1.4) How can we achie e nuclea usion eac ions on Ea h so i can be used as an ene gy sou ce (con olled nuclea usion)? The i s challenge is ha o nuclea usion o happen, he Coulomb ba ie needs o be o e comed (ac ually, conside ing quan um unneling, ene gy lowe han he needed o su pass he Coulomb ba ie would be needed o allow usion eac ions). Nuclea usion eac ions conside ed o be pu sued on Ea h in ol e Deu e ium, 2 1H because i is an abundan elemen , i exis on Ea h’s oceans comp ising 0.015 a om pe cen o he hyd ogen in sea wa e wi h he olume o abou 1.35 ·109km3[3] (sec ion 1.3). The possible eac ions a e: 1. 2 1H + 2 1H→3 1H(1.01MeV) + p(3.03MeV), 2. 2 1H + 2 1H→3 2He(0.82MeV) + n(2.45MeV), 3. 2 1H + 3 1H→4 2He(3.52MeV) + n(14.06MeV), 4. 2 1H + 3 2He →4 2He(3.67MeV) + p(14.67MeV), whe e he kine ic ene gy each p oduc ca ies is also indica ed. Thei c oss-sec ions a e plo ed in igu e 1.3, whe e he c oss-sec ion o he wo 2 1H-2 1H eac ions ha e been added, and he X-axis is he p ojec ile ene gy, 2 1H, assuming he a ge nuclei a es . 2 1H + 3 1H eac ion is he bes op ion since i s c oss-sec ion is he highes , and i peaks a he lowes ene gy1. 1T i ium, 3 1H, do no exis na u ally on Ea h’s, bu i could be p oduced by ce ain nuclea eac ions wi h 3 1.3. TOKAMAKS CHAPTER 1. INTRODUCTION a∇Bd i , ha d i he ions downwa d and he elec ons upwa d, since (1.14) depends on he cha ge q. This cha ge sepa a ion will c ea e an elec ic ield pe pendicula o he magne ic ield, so he pa icles will expe ience a ~ E∧~ Bd i , ha would d i ou wa d bo h ions and elec ons, p o ided ha his d i does no depend on he cha ge. These d i s a e shown in igu e 1.7 (a). (a) D i s in a o us. Sou ce: [5]. (b) To oidal (blue) and poloidal ( ed) di ec ions o a o us. Sou ce: google images, 2019. Figu e 1.7. D i s in a o us and de ini ion o he poloidal and o oidal di ec ions on a o us. To o e come he d i s discussed abo e, Tokamaks con ine he pa icles by wis ing he magne ic ield lines. Fo doing ha , in addi ion o he o oidal ield o he o us, a poloidal ield is added ( ield in he poloidal di ec ion, see igu e 1.7 (b)). This poloidal magne ic ield is c ea ed by he plasma i sel . Tokamaks need addi ional coils o con olling he plasma. The plasma i sel end o mo e adially ou wa d due o poloidal ield c ea ed by he plasma, which is g ea e in he inboa d egion han in he ou wa d, and due o he o oidal shape o he plasma, he plasma p essu e also c ea es an ou wa d o ce. This ou wa d o ce is called hoop o ce (see [11] o an illus a i e explana ion, and [3] o a igu ous ea men ). This coils a e called poloidal magne ic ield coils (PF coils) since i s ole is o c ea e poloidal ield whose ~ J∧~ B o ce balance he hoop o ce. Fo doing ha , he cu en lowing in his PF coils need o low in he opposi e di ec ion o he plasma cu en . In addi ion, his coils also help c ea e he elonga ed shape o okamak plasmas. An addi ional se o coils called di e o coils a e o en used o c ea ed a di e ed shape in he plasma, which will be explained la e . Figu e 1.8 shows a ske ch o a okamak wi h i s basics elemen s. The plasma is con ained in he acuum essel (VV) (g ey colou ed in he igu e). The o oidal ield (g een a ows) is c ea ed by he o oidal magne ic ield coils, and he poloidal ield is c ea ed mainly by he plasma i sel , 10 1.3. TOKAMAKS CHAPTER 1. INTRODUCTION and by he PF coils (plasma shaping). In he cen e o he de ice he e is a ans o me coil ha induces a o oidal cu en in he plasma ( ed a ows) ha c ea es he poloidal magne ic ield. Figu e 1.8. Ske ch o a okamak, showing i s basic elemen s, he ield lines and he plasma cu en . Sou ce: google images, 2019. The esul ing ield lines a e heli- cal lines (yellow a ows), which con ine mos o he plasma pa - icles. Since he plasma cu - en is induc i e, okamaks op- e a e in a pulsed egime. One he challenges o his de ice is he con ol o he plasma, and he elec omagne ic ins abili ies ha could a ise in he plasma, which could lead o he loss o he plasma ene gy. This e en s a e called Dis up ions. Fo he ini ializa ion o a okamak discha ge, called okamak s a -up, he o oidal magne ic ield mus be p e iously s ablished and he VV is illed wi h a gas. A e he induc o coil induced he elec ic ield by changing i s cu en , he gas will be ionized c ea ing a plasma ( his is called b eak-down). This plasma will s a o c ea e he poloidal ield ha con ines he pa icles. In he mean ime, he plasma needs o be hea ed and he PF coils will be con olling i s shape. The e a e se e al me hods o hea ing he plasma; o begin wi h, he plasma cu en will hea he plasma due o Joule’s e ec , which is called ohmic hea ing. Ex e nal me hods o hea ing could be he use o elec omagne ic wa es ( he elec omagne ic wa es will c ea e oscilla ions o he plasma pa i- cles, inc easing hei ene gy), injec ion o neu al pa icles ( he injec ed pa icles will accele a e he plasma pa icles by collisions wi h hem, inc easing hei empe a u e), and many mo e (see chap e 5 o [4] o a desc ip ion o hea ing me hods6). Finally, he plasma will a i e a he desi ed con igu a ion wi h he desi ed plasma cu en and shape, ending he s a -up phase. 6The websi es o cu en ly ope a ing okamaks also p o ide in o ma ion abou he way hey hea hei plasmas. 11 1.4. MOTIVATION: SMART CHAPTER 1. INTRODUCTION 1.4 Mo i a ion: SMART, a SMall Aspec Ra io Toka- mak o he Uni e si y o Se ille The Plasma Physics and Fusion Technology G oup o he Uni e si y o Se ille is designing a okamak ha will be ope a ing he nex yea 7. I s name will be SMall Aspec Ra io Tokamak (SMART), which e eal he main cha ac e is ic o he de ice, i will be a sphe ical okamak a he han a s anda d okamak. This eac o will no make usion eac ions; ins ead, as well as all he exis ing small and medium size eac o s, i will ocus on doing okamak physics esea ch such as plasma con inemen , shape, ins abili ies, e c. This in o ma ion will be used by la ge in e na ional acili ies such as JET o ITER, which will make usion eac ions. The SMART missions a e  S udy plasma anspo and con inemen in posi i e and nega i e iangula i ies.  De elop no el diagnos ic and con ol schemes.  Examine elec omagne ic s abili y and con ol o ene ge ic pa icles.  T ain nex gene a ion o usion physicis s and enginee s. The main di e ence be ween a sphe ical okamak and a okamak is he aspec a io o he de ice, which is he a io o he majo adius and he mino adius o he de ice. I he aspec a io o he okamak is <2, he de ice is called sphe ical okamak. Figu e 1.9 shows a s anda d okamak and a sphe ical okamak. Figu e 1.9. Tokamaks and sphe ical okamaks. Sou ce: [12]. Sphe ical okamaks a e a desi able app oach o con olled nuclea usion because hey a e mo e compac han egula okamaks, which means lowe cos s, and sphe ical okamak’s plas- mas displays be e plasma p ope ies such as he so called sa e y ac o qand he β, which will be explained la e ( e iews o sphe ical oka- maks ea u es s egula okamaks ea u es can be ound on [12,13]). Th ee ope a ional phases ha e been designed o he SMART okamak: 7h p://www.ps .eu/. 12 1.5. OBJECTIVES AND OUTLINE OF THE THESIS CHAPTER 1. INTRODUCTION  Phase 1. Fi s plasma and p oo -o -concep .  Phase 2. Inclusion o Neu al Beam Injec ion (NBI) hea ing sys em. The goal o his phase is he demons a ion o plasma shaping.  Phase 3. This phase will explo e usion- ele an ope a ions. The main pa ame e s o his h ee phases a e shown in able 1.1. This pa ame e s will be explained in he ollowing sec ions. A 3D model o he SMART okamak is shown on igu e 1.10. The coilse con igu a ion will be he same o he h ee phases. (a) Reac o . (b) Coilse . Figu e 1.10. 3D plo s o he SMART eac o . The coilse is symme ic wi h espec o he Z= 0 plane, and se e al coils a e inside he acuum essel (VV). The po s in he VV a e o plasma diagnosis and o plasma hea ing me hods such as NBI and ECRH, which will be explained la e . Sou ce: [14]. 1.5 Objec i es and ou line o he hesis The goal o his wo k is o model he ini ial phase o he okamak s a -up, he b eak-down o he gas, o SMART, op imizing he de ice so ha he gas b eaks-down and u ns in o a plasma o he i s ope a ional phase and he i s upg ade, phase wo, wi hou any addi ional hea ing me hod. This hesis is o ganized as ollows: chap e 2 p esen s he basics o okamak physics, he simples model o he plasma, and he undamen al equa ion o desc ibe he okamak equi- lib ium, he G ad-Sha ano equa ion, as well as he cha ac e iza ion o a okamak plasma. Chap e 3 e iews he okamak s a -up, ocusing on he b eak-down phase, in oducing he 13 1.5. OBJECTIVES AND OUTLINE OF THE THESIS CHAPTER 1. INTRODUCTION SMART Phase 1 Phase 2 Phase 3 VV adius(m) 0.8 VV heigh (m) 1.6 Majo plasma adius(m) 0.42 Mino plasma adius(m) 0.24 Plasma elonga ion κ κ < 2.1 Plasma iangula i y δ−0.51 < δ < 0.44 Plasma cu en (kA) 35 100 500 To oidal ield(T) 0.1 0.3 1.0 Fla - op ime(ms) 20 100 500 Ex e nal hea ing (KW) ECRH 6 [2.4GHz] 6 [7.5GHz] 200 [- GHz] NBI - 600 600 Table 1.1. Pa ame e s o he phases o he SMART okamak. To oidal ield is he o oidal ield alue a he plasma magne ic axis. basic physics o his phase, as well as se e al c i e ia o ensu e a p ope b eak-down o he gas. Chap e 4 e iews he Fies a oolbox used o compu ing he plasma equilib ium and he dynamic beha iou o he plasma using he RZIp model. Chap e 5 summa izes he simula ion p ocedu e ollowed in his wo k. Chap e 6 con ains he simula ion esul s, as well as he dis- cussion o he esul s wi h o he ope a ing okamaks. Finally, chap e 7 summa ized he wo k ca ied ou in his hesis, and discuss u u e wo k. 14 Chap e 2 Fundamen als o okamak physics In his chap e he undamen als o okamak physics will be e iewed. The i s sec ion includes he plasma model used in okamak physics, he second sec ion includes he de i a ion o he equa ion o he okamak equilib ium con igu a ion, and he hi d sec ion e iews some basic okamak pa ame e s. 2.1 Magne ohyd odynamic model o a plasma The Magne ohyd odynamic (MHD) model is a single luid desc ip ion o a plasma, i.e., i is a model ha ea s a plasma like a con inuum ma e , a he han as a se o pa icles. This is one o he simples models o s udy a plasma, and i assumes se e al hypo hesis, like quasineu ali y o negligible elec on ine ia (see any book abou plasma physics o u he de ails, like [5]). The se o equa ions a e ∂ρ ∂ +∇ · (ρ~ ) = 0,[Mass conse a ion] (2.1) ρh∂~ ∂ + (~ · ∇)~ i=~ j∧~ B− ∇p, [Momen um conse a ion] (2.2) ~ E+~ ∧~ B=η~ j, [Ohm’s law] (2.3) ∂ ∂ (pρ−γ)+(~ · ∇)(pρ−γ) = 0,[Adiaba ic beha iou ] (2.4) ∇ ∧ ~ E=−∂~ B ∂ ,(2.5) ∇ ∧ ~ B=µ0~ j, (2.6) ∇ · ~ B= 0,(2.7) whe e ρis he plasma densi y, ~ i s eloci y, ηi s esis i i y, pi s p essu e (in gene al he p essu e is a enso , bu o his simpli ied model, i is conside ed an scala magni ude), ~ j 15 2.2. G-S EQUATION CHAPTER 2. FUNDAMENTALS OF TOKAMAK PHYSICS i s cu en densi y, γis he adiaba ic index, and ~ Eand ~ B he elec ic and magne ic ield he gene a ed by he plasma. No e ha he las h ee equa ions a e a quasi-s a ic limi o Maxwell’s equa ions. 2.2 G ad-Sha ano equa ion Figu e 2.1(a) displays he coo dina e sys em o o oidal de ices. In an equilib ium si ua ion, he magne ic ield o a Tokamak p oduces an in ini e se o nes ed o oidal magne ic lux su aces1 as shown in igu e 2.2, and he magne ic ield lines ollow an helical pa h on hem as hey wind ound he o us. The poloidal lux Ψ and he o oidal lux Φ be ween wo magne ic su aces a e de ined by dΨ≡~ B·~ dSθ, dΦ≡~ B·~ dSφ,(2.8) whe e dSθ,dSφa e he poloidal and o oidal su ace elemen s, whose magni ude a e de ined in igu e 2.1(b) (i s uni a y ec o is pe pendicula o he su ace, and he sign is a bi a y, as usually in he magne ic luxes), and ~ Bis he magne ic ield. The basic condi ion o he (a) Cylind ical coo dina e sys em used in de ices wi h o odial symme y, (R, φ, Z). R0is called he majo adius o he o us, is called he mino adius. The ci cum e ence R=R0de ines he o oidal o magne ic axis. Sou ce: h p:// usionwiki.ciema .es/wiki/ To oidal_coo dina es. (b) To oidal (T) and poloidal (P) su ace elemen s be ween wo magne ic lux su aces. Sou ce: [15]. Figu e 2.1. Cylind ical coo dina e sys em o o oidal de ices, and de ini ion o he poloidal lux in a o us. equilib ium is ha he o ce on he plasma be ze o a all poin s, so he momen um conse a ion 1A gi en su ace is a magne ic lux su ace i i sa is ies ~ B·~n = 0, whe e ~n is he no mal ec o o he su ace. Tha is, he magne ic ield do no c oss he su ace. This is only a isual way o unde s and he magne ic ield, since he e would be an in ini e numbe o magne ic lux su aces inside a okamak. 16 2.2. G-S EQUATION CHAPTER 2. FUNDAMENTALS OF TOKAMAK PHYSICS equa ion (2.2) leads o ~ j∧~ B=∇p. (2.9) This implies ~ B· ∇p= 0, so he e is no p esu e g adien along he magne ic ield lines, which means he magne ic su aces a e also p essu e su aces. (2.9) also implies ~ j· ∇p= 0, and as a consequence he cu en lie in he magne ic su aces. In wha ollows he G ad-Sha ano equa ion, one o he mos undamen al equa ions o MHD equilib ium, will be de i ed. The idea o his equa ions is o ew i e (2.9) o ha e a scala equa ion ins ead o a ec o equa ion. F om (2.7), aking in o accoun he axysymme y, and using he coo dina e sys em o igu e 2.1(a), se ing R0≡0, 1 R ∂(RBR) ∂R +∂BZ ∂Z = 0.(2.10) The unc ion o ha scala equa ion will be he unc ion ψ, called he s eam unc ion, which is de ined as ψ≡RAφ, whe e Aφis he o oidal componen o he ec o po en ial ~ A. Wi h his unc ion, he poloidal magne ic ield can be w i en as BR=−1 R ∂ψ ∂Z , BZ=1 R ∂ψ ∂R,            ⇔~ Bθ=1 R∇ψ∧∧ φ, (2.11) whe e ∧ φis he o oidal uni ec o ( he magne ic ield can be exp essed as ~ B=~ Bθ+~ Bφ). I can be shown ha Ψ = 2πψ [15] (sec ion 6.2). I is usual o label he magne ic su aces wi h ψ, also called he magne ic lux. This means ha p=p(ψ), since magne ic su aces a e also p essu e su aces. F om he symme y o ~ j, i can be in oduced a unc ion ha e i ies jR=−1 R ∂ ∂Z , jZ=1 R ∂ ∂R,            ⇔~ jθ=1 R∇ ∧∧ φ . (2.12) Compa ing (2.12) wi h (2.6) leads o =RBφ µ0 ,(2.13) whe e µ0is he acuum magne ic pe meabili y and he subsc ip φindica es he o oidal com- ponen . I can be shown ha is a unc ion o ψ[4] (sec ion 2.3). Equa ion (2.9) can be expanded as ~ jθ∧∧ φ Bφ+jφ ∧ φ∧~ Bθ=∇p, (2.14) 17 2.3. PARAMETERS CHAPTER 2. FUNDAMENTALS OF TOKAMAK PHYSICS whe e jφ,~ jθa e he magni ude o he o oidal cu en , and he poloidal cu en densi y ec o espec i ely. Subs i u ing (2.12) and (2.11) in o (2.14), we ge , using ha ∧ φ·∇ψ=∧ φ·∇p= 0 (consequence o he o oidal symme y) Bφ R∇ +jφ R∇ψ=∇p. (2.15) Now, applying he chain ule on ∇ and ∇p, ∇ =d dψ∇ψ, ∇p=dp dψ∇ψ,      (2.16) In oducing (2.13) and (2.16) in o (2.15) lead o: −µ0 R2 d dψ∇ψ+jφ R∇ψ=dp dψ∇ψ⇒jφ=µ0 R d dψ +Rdp dψ.(2.17) ∇ψcan be emo ed om (2.17) since ∇ψ= 0 co espond o he i ial solu ion. To ge jφas a unc ion o ψ, we subs i u e (2.11) on (2.6), ob aining µ0~ j=µ0jφ ∧ φ+1 R∇(RBφ)∧∧ φ⇒ −µ0Rjφ=R∂ ∂R1 R ∂ψ ∂R+∂2ψ ∂Z2,(2.18) Finally, i we subs i u e (2.18) on (2.17), we ge he G ad-Sha ano equa ion, R∂ ∂R1 R ∂ψ ∂R+∂2ψ ∂Z2=−µ0R2dp(ψ) dψ −µ2 0 (ψ)d (ψ) dψ .(2.19) (2.19) is one o he undamen al equa ions o MHD equilib ium. I is a second o de pa ial di e en ial equa ion ha calcula es he equilib ium in o oidal de ices, gi en he unc ions p(ψ) and (ψ). In igu e 2.2 (b), we can see a ypical solu ion o his equa ion, showing ha he su aces a e shi ed wi h espec o he magne ic axis, which is he majo adius o he inne mos su ace. 2.3 Tokamaks pa ame e s The mos impo an pa ame e s o a okamak equilib ium, as well as abou plasma shape and con ol will be in oduced in his sec ion. 2.3.1 Plasma shape and con ol in okamaks The i s concep ha mus be in oduced is he plasma bounda y. The bounda y o he plasma is he ou e mos closed magne ic su ace con ained in he VV, called Las Closed magne ic Flux 18 2.3. PARAMETERS CHAPTER 2. FUNDAMENTALS OF TOKAMAK PHYSICS (a) Magne ic lux su aces o a oka- mak equilib ium, o ming a se o nes ed cylind ical su aces. Sou ce: [4]. (b) Typical solu ion o he G ad- Sha ano equa ion. Sou ce: [4]. Figu e 2.2. Magne ic lux su aces o a okamak equilib ium, and ypical solu ion o he G ad-Sha ano equa ion. Su ace, LCFS. The pa icles inside his ou e mos su ace ollow he ield lines ha emain in he plasma, bu pa icles ha ollow he ex e nal ield lines will end up escaping om he plasma and colliding wi h he VV ( he pa icles ollow he magne ic ield lines, bu i he magne ic ield lines a e no closed inside he VV, pa icles will collide wi h he VV as hey ollow hem). The bounda y can be c ea ed by a se o coils o by a limi ing ma e ial ha will ouch he plasma (ei he he acuum essel o ano he elemen ). The i s me hod is o c ea e he LCFS displaying one o mo e X-poin s by using a se o coils. X-poin s a e saddle poin s whe e ∂ψ ∂Z =∂ψ ∂R = 0, so he poloidal magne ic ield is ze o (see (2.11)). The ou e mos closed su ace is called sepa a ix, and a plasma con ined his way is called a di e ed plasma, and he coils used o c ea e i a e called di e o coils. The second me hod is o limi he plasma by he VV o an speci ic ma e ial, so ha he plasma is ouching ha ma e ial. The plasma is hen called a limi ed plasma. In igu e 2.3 (a) a limi ed plasma and a di e ed plasma wi h one X-poin is showed. Fo shape con ol o he plasma, he ollowing pa ame e s a e in oduced o desc ibe he 19 3.1. PLASMA BREAK-DOWN CHAPTER 3. TOKAMAK START-UP wi h each o he . The connec ion leng h can be calcula ed nume ically by in eg a ing he magne ic ield lines equa ion, bu also an empi ical o mula is used o es ima e i in he poloidal ield null egion [25]: L≃0.25ae Bϕ(Rnull) < Bθ>,(3.5) whe e Bϕ(Rnull) is he o oidal magne ic ield a he cen e o his egion Rnull and < Bθ> is he a e age poloidal ield in he su ace o his egion. Rega ding ae , he e a e wo isions in he bibliog aphy: i) ae is he linea dis ance o he closes wall. Some imes his is es ima ed as he mino adius o he plasma a ge equilib ium con igu a ion [21,23]. ii) ae is he mino adius o he ield null egion [20,25,19]. The a ia ion in he elec on densi y conside ing bo h ioniza ion and losses is dne d =ne(νion −νloss).(3.6) The ioniza ion a e can be assumed cons an since elec ons will acqui e a cons an speed quickly a e he s a o he ioniza ion p ocess. Abou he loss a e, he connec ion leng h may a y since as he o oidal elec ic ield is induced inside he VV, eddy cu en s a e also induced in he VV, which will a ec he ield, and hence al e he connec ion leng h L. Howe e , as a i s app oxima ion, we could assume L o be cons an . In ha case he elec on densi y as a unc ion o ime will be ne( ) = ne(0) exp [(νion −νloss) ],(3.7) whe e ne(0) is he elec on densi y a = 0, he ime a which he o oidal elec ic ield is induced. Fo a p ope ioniza ion o he gas, he ioniza ion a e mus be g ea e han he loss a e, so ha he elec on densi y inc eases. Imposing his, and in oducing (3.1) and (3.4) in (3.6) gi es dne d =ne k(α−1/L)>0⇒(α−1/L)>0⇒αL > 1,(3.8) aking in o accoun kis he module o he speed, e go, posi i e. Fo Townsend a alanche o p oceed, αL > 1 is needed. Se ing αL = 1 will gi e he limi condi ion so ha he a alanche no inc ease no dec ease. In oducing his condi ion on (3.2) gi es Eϕmin =C2p ln(C1pL),(3.9) whe e Eϕmin is he elec ic ield needed o his condi ion. Since his condi ion do no ensu es a alanche, [25] s a es ha Eϕ>2Eϕmin o a eliable s a -up, so ha he a alanche inc eases. 26 3.1. PLASMA BREAK-DOWN CHAPTER 3. TOKAMAK START-UP 10 -5 10 -4 10 -3 p (To ) 10 -1 10 0 10 1 10 2 10 3 10 4 Emin (V/m) Paschen's cu e o Hyd ogen L=10m L=50m L=100m Figu e 3.2. Paschen’s cu e o o H2as p e- ill gas, showing di e en connec ion leng hs. The plo o Eϕmin as a unc ion o p o a gi en Lis called he Paschen’s cu e. An example is shown on ig- u e 3.2, o H2as p e- ill gas. The cons an a e C1= 510m−1To −1 and C2= 1.25 ·104Vm−1To −1. I can be seen om he igu e ha all he lines displays a minimum elec- ic ield needed o a ce ain p es- su e. This can be easily unde s ood: o high p essu es, he mean ee-pa h will be oo sho ( he mean ee pa h λis gi en by λ= 1/(C1p) [21]), so o he elec ons o gain enough ene gy o ionize, he elec ic ield needs o be high. Fo low p essu es, he mean ee-pa h will be oo long meaning ha he e would be oo li le collisions be o e he elec ons a e los , al hough all he collisions will be ionizing collisions because he elec on will ha e gained enough ene gy. This esul s in a e - ical asymp o e, so ha o he le o ha asymp o e he loss a e is g ea e han he ioniza ion a e, ega dless o he ield, because he e is no enough molecules o c ea e a alanches. A widely used empi ical c i e ia o a eliable s a up is he so-called Lloyd’s c i e ia [24] Eϕ Bϕ Bθ >1000Vm−1.(3.10) In he case o plasma b eak-down assis ed by Elec on Ciclo on Resonance Hea ing (ECRH), which is a hea ing me hod based on he i adia ion o elec omagne ic wa es o ce ain equen- cies o he ionized gas so ha i abso bs hem helping ionize he gas [19], he minimum alue is 100Vm−1ins ead o 1000Vm−1[26]. In he DIII-D okamak3i has been shown [27] ha he b eak-down o he p e- ill gas did no occu whe e he s ay poloidal ield was minimum; ins ead, i ook place whe e he po en ial gained by he elec ons along hei pa h R ield line ~ E·~ dl was g ea es . This po en ial is no he elec os a ic po en ial because he elec ic ield has been c ea ed by a changing magne ic ield. An elec os a ic ield ~ Ecan be w i en as ~ E=−∇V, whe e Vis he elec os a ic po en ial. The line in eg al be ween wo poin s Aand Bis RB A~ E·~ dl =−V(B) + V(A), so i does no depend on he pa h ollowed, only on he s a ing and ending poin s. In he p esence o ansien magne ic ields, he elec ic ield is gi en by ~ E=−∇V−∂~ A/∂ , whe e ~ Ais he ec o 3h ps://www.ga.com/magne ic- usion/diii-d. 27 3.1. PLASMA BREAK-DOWN CHAPTER 3. TOKAMAK START-UP po en ial. The line in eg al in his case would be RB A~ E·~ dl =RB A−∇V·~ dl −RB A∂~ A/∂ ·~ dl= −V(B) + V(A)−RB A∂~ A/∂ ·~ dl. To compu e he las in eg al, he pa h ollowed om A o B needs o be known, and hence, he inal esul will depend on he pa h ollowed. This means ha an elec on gains ene gy a e doing a e olu ion inside he okamak e u ning o he s a ing poin . The NSTX okamak4also con i ms his empi ical esul s [28]. This seems unde s andable since o c ea e a alanches, he elec ons ha e o gained enough ene gy o ionize, so i is no only ele an ha hey ollow a pa h wi h he minimum de ia ion o collide as much as possible, bu ha hey gain as much ene gy as possible in i s pa h so ha mos o he collisions a e ionizing collisions. 3.1.1 A alanche o b eak-down ime When Coulomb collisions domina e o e neu al-elec on collisions, he uel s a s o beha e like a plasma, en e ing he bu n- h ough phase. This phase is eached usually when he ioniza ion ac ion o he gas is abou 5% [19]. A his s age, he a alanche as desc ibed abo e, wi h a gas being ionized by a o oidal elec ic ield s ops being alid, and he u he ioniza ion is made by he plasma i sel , i he powe losses a e coun e balanced. The ime when he his phase s a s (o when he b eak-down phase ends) can be es ima ed using eq (3.7). In oducing he concep o ioniza ion ac ion i, which is he a io be ween he elec on densi y c ea ed by ioniza ion and he neu al p e- ill gas densi y np e, which is i≡ne/2 np e ,(3.11) whe e he 2 accoun s o he 2 elec ons ha H2(and also He) gi es. The p e- ill gas densi y is ela ed o he gas p essu e by he ideal gas law, p=np eKBTp e. Subs i u ing his in o (3.7) yields, assuming ne(0)=1 which is an s anda d assump ion [19,23], i( ) = 1 2np e exp [(νion −νloss) ] = 1 2 KBTp e pexp h k(α−1 L) i.(3.12) Se ing i= 5% on (3.12) will gi e an es ima ion o he ime needed o he a alanche o b eak-down phase o end, he a alanche ime a a: a a = ln 2·0.05 ·p KBTp e  k(α−1 L) = ln 2·0.05 ·p KBTp e  43Eϕ phC1pexp −C2p Eϕ−1 Li.(3.13) No e ha i is posi i e since α−1/L > 0 which is he condi ion o he a alanche o occu . 4h ps://www.pppl.go /ns x. 28 3.1. PLASMA BREAK-DOWN CHAPTER 3. TOKAMAK START-UP 3.1.2 Vol age and elec ic ield induced by he induc o solenoid To s a he b eak-down phase, he induc o solenoid (Sol) is p e-cha ged wi h a ce ain cu en , and hen i s cu en i s amped down apidly o induce he elec ic ield. In his subsec ion he elec ic ield and he ol age induced will be de i ed. The ol age induced by he amp down o he Sol, called loop ol age, can be easily compu ed using Fa aday’s law in i s in eg al o m: ε≡Vloop =−d d ZS ~ B·~ndS, (3.14) whe e Sis he su ace enclosed by he loop, which is a ci cle o an a bi a y adius, and ~n i s uni ec o , which will go in he Zdi ec ion. The magne ic ield ~ Bo a solenoid also goes in he Zdi ec ion, so he loop ol age is, choosing ~n =∧ Z: Vloop =−d d ZS BSoldS =−ZS dBSol d dS. (3.15) The de i a i e can be in oduced in o he in eg al since he in eg a ion a iables do no a y o e ime. The only hing ha a y wi h ime is he Sol cu en . I s de i a i e can be compu ed easily since he Sol cu en is dec eased linea ly, so i sa is ies ISol( ) = m +n. (3.16) A = 0, ISol ≡I0, and he amp goes down un il ISol( 1)≡I1(I0> I1), so he Sol cu en is ISol( ) = I1−I0 1 +I0.(3.17) Sol ing (3.15) o he magne ic ield o a solenoid o inne adius Rin and ou e adius Rou > Rin wi h N/l u ns pe uni leng h, igno ing bo de e ec s (i.e., conside ing in ini e leng h), gi es (see appendix A) Vloop =−µ0 N l I1−I0 hπR2 in +π(R2 ou −R2 in)−2π Rou −Rin 1 3(R3 ou −R3 in)−Rin 2(R2 ou −R2 in)i. (3.18) Since he slope o he Sol cu en is nega i e, he loop ol age induced is posi i e. Once he loop ol age has been compu ed, he elec ic ield can also be compu ed. Fo compu ing i , he in eg al o m o Fa aday’s law will be used: Vloop =ε≡IΓ ~ E·~ dl =−d d ZS ~ B·~ndS, (3.19) whe e Γ is he loop’s pe ime e . To compu e he elec ic ield ~ E om his equa ion we mus know i s symme y. To do so, le ’s ake in o accoun ha (3.19) is e y simila o Amp`e e’s law IΓ ~ B·~ dl =µ0ZS ~ J·~ndS =µ0Ienclosed.(3.20) 29 3.1. PLASMA BREAK-DOWN CHAPTER 3. TOKAMAK START-UP I is widely known ha he magne ic ield o a in ini e cilind al conduc o ca ying a cu en densi y ~ J=J∧ Zis ~ B=B∧ ϕ(see any book o classic elec omagne ism, like [29]). And, since he oles o ~ Band ~ Jin Amp`e e’s law a e he same as he oles o ~ Eand ~ B espec i ely in Fa aday’s law, p o ided ha ~ B=B∧ Zin Fa aday’s law, he elec ic ield mus be ~ E=E∧ ϕ. Fu he mo e, aking in o accoun he axysymme y and he assump ion o no bo de e ec s (in ini e leng h), he elec ic ield can only depend on he adial coo dina e R, so ~ E=E(R)∧ ϕ. Conside ing his, he ield can be compu ed, since ~ dl =Rdϕ ∧ ϕ: Vloop =IΓ ~ E·~ dl = 2πRE(R)⇒E(R) = Vloop 2πR .(3.21) The elec ic ield will be mo e in ense nea he inne wall o he VV. The same applies o he o oidal magne ic ield, which also goes as 1/R. The po en ial gained in ha egion will be he g ea es , enhancing he ioniza ion p ocesses, and he magne ic ield lines will be mos ly o oidal lines in ha egion (since he o oidal ield will be g ea e in ha egion, he e ec o he poloidal ield will be educed as a consequence), educing he elec on losses. This wo ac o s explain why he p e- ill gas b eaks-down nea he inne side o he VV in gene al. 30 Chap e 4 Fies a oolbox The Fies a oolbox is an objec -o ien ed oolbox p og ammed in MATLAB by G. Cunningham a he Culham Cen e o Fusion Ene gy [30,31]. I was c ea ed o equilib ium calcula ions such as sol ing G ad-Sha ano equa ion and plasma shape con ol. Howe e , dynamic calcula ions a e also included, using he RZIp model. The eac o geome y is in oduced using i s objec a chi ec u e. 4.1 G ad-Sha ano sol e , EFIT Fies a con ains wha is called ee bounda y equilib ium sol e s. The main cha ac e is ic o his ype o sol e s is ha he plasma bounda y is no known (on he con a y o ixed bounda y p oblems) and hence will also be pa o he solu ion. The poloidal magne ic lux ψis di ided in o wo componen s, he c ea ed by he plasma and he c ea ed by he ex e nal cu en ca ying elemen s, i.e., he coils and he VV, which will ca y eddy cu en s. This ex e nal elemen s will be called s uc u e. The G ad-Sha ano equa ion is sol ed by an i e a i e me hod, acco ding o a gi en ole - ance. In pa icula , we ha e used he EFIT sol e [32], which is a ee-bounda y equilib ium sol e ha can ob ain he ex e nal cu en s needed o achie e a plasma equilib ium wi h plasma pa ame e s close o he desi ed plasma pa ame e s. The plasma cu en p o ile used is de ined by he so-called Topeol2 model,      dp dψ =j0 0 βθ(1 −ψN), d dψ =µ0 0j0(1 −βθ)(1 −ψN), (4.1) whe e 0is he X-poin adial coo dina e, j0 he plasma cu en and ψN he no malized lux, 31 4.2. RZIP MODEL CHAPTER 4. FIESTA TOOLBOX de ined as ψN≡ψ−ψaxis ψbounda y −ψaxis ,(4.2) whe e ψaxis and ψbounda y a e he alues o he poloidal lux a he cen e o he magne ic lux su ace and a i s bounda y espec i ely. The cons an s 0and j0a e adjus ed a each s ep o he i e a i e p ocess. The ield and he lux a he compu a ional g id a e calcula ed using G een’s unc ions (see [33] o a e iew o he G een’s unc ion o malism applied o okamaks). 4.2 RZIp model The igid cu en displacemen model o RZIp (R,Zand Ip) [34,35] is a model used o desc ibe he dynamic beha iou o a okamak equilib ium. The plasma is modelled as a igid conduc o so ha he plasma can mo e adially and e ically, bu he plasma shape and cu en dis ibu ion emain cons an . Hence, he plasma can be iden i ied by i s adial and e ical posi ion, Rgeo and Zgeo, and i s cu en , Ip. The RZIp model is a linea ised model based on he assump ion ha small pe u ba ions leads o small changes o he a ge equilib ium. I s equa ions a e he ci cui equa ion and he adial and e ical o ce balance equa ions. This model conside s ha a okamak is made o ac i e and passi e elemen s. An elemen is conside ed an ac i e elemen i is subjec ed o a ex e nal exci a ion such as cu en o ol age supplies and passi e in any o he case. The ac i e s uc u e will hen be he coilse , and he passi e s uc u e he acuum essel. The e a e se e al ways o calcula e he plasma sel -induc ance. Fies a’s RZIp uses his app oxima e o mula [35] Lp=µ0Rgeo4πRgeo |< BZp>| µ0Ip−βθ −βθ−1 2,(4.3) whe e BZpis he BZc ea ed by he plasma (in ou case, he plasma cu en lows in he posi i e o oidal di ec ion, esul ing in a nega i e < BZp>). The ou equa ions o he RZIp model a e, indica ing ma ices wi h bold le e s [34] d(LpIp+MpsIs+RGeoEIp) d +Ipρp=Vp[Ipequa ion],(4.4) d(LsIs+MspIp) d +ΩsIs=Vs[Isequa ion],(4.5) d(mp˙ Rgeo) d =1 2I2 p ∂Lp ∂Rgeo +Is ∂Msp ∂Rgeo Ip+EI2 p 2[Rgeo equa ion],(4.6) d(mp˙ Zgeo) d =Is ∂Msp ∂Zgeo Ip[Zgeo equa ion],(4.7) 32 4.2. RZIP MODEL CHAPTER 4. FIESTA TOOLBOX whe e Eis a cons an ma ix, Mps is he induc ance be ween he plasma and he s uc u e (bo h ac i e and passi e), Is he cu en o he s uc u e, Vsand Vp he ol age o he s uc u e and he plasma espec i ely and Ωs he esis ance ma ix o he s uc u e. Is is common o neglec he plasma mass mp, which is also wha Fies a’s RZIp does. This se o equa ions a e linea ized and cas in he s a e space o m o sol e i , whose gene al s uc u e is (see appendix B)    dx d =Ax +Bu, y=Cx +Du. (4.8) The linea iza ions o eq (4.4), (4.5), (4.6) and (4.7) se ing mp= 0 esul in:                      Ls ∂Mps ∂Zgeo 0 ∂Mps ∂Zgeo 0 ∂2Mps ∂Z2 geo 0 Is0 I0 p ∂Mps ∂Rgeo 0 ∂2Mps ∂Zgeo∂Rgeo 0 Is0 I0 p Mps 0 ∂Msp ∂Rgeo 0Mps0 ∂2Mps ∂Rgeo∂Zgeo 0 Is0 I0 p 0 1 2 ∂2Lp ∂R2 geo 0+∂2Mps ∂R2 geo 0 Is0 I0 p ∂Lp ∂Rgeo 0+∂Mps ∂Rgeo 0 Is0 I0 p +µ0 2πS l2βpR0 geo ∂Lp ∂Rgeo 0+∂Mps ∂Rgeo 0 Is0 I0 p +µ0 2πS l2βpR0 geo L0 p+µ0 2πS l2βpR0 geo                       ∗dx d + +               Ω0 s0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ω0 p               x=         I0 00 00 01              δVs δVp      , (4.9) whe e 0indica es equilib ium alue, so R0 geo indica es he alue a he a ge equilib ium, Sis 33 4.2. RZIP MODEL CHAPTER 4. FIESTA TOOLBOX he plasma c oss-sec ion o he equilib ium con igu a ion, and li s pe ime e . The s a e ec o xis x=         Is−Is0 (Zgeo −Z0 geo)I0 p (Rgeo −R0 geo)I0 p Ip−I0 p         =         δ(Is) δ(Zgeo)I0 p δ(Rgeo)I0 p δ(Ip)         .(4.10) and he inpu ec o uis u=  δVs δVp .(4.11) Howe e , Fies a’s RZIp do no uses he ol ages as inpu s, i uses he coilse cu en s as inpu s, so (4.9) is in e ed o ha e cu en s as inpu s. The sys em (4.9) has he o m Mdx d +P x =Qu, and compa ing i wi h he s a e space sys em (4.8), we conclude A=−M−1P, B=M−1Q. (4.12) The sel and mu ual induc ances on (4.9) a e compu ed wi h he G een’s unc ions o he magne ic ield by ela ing hem o he ield c ea ed by he s uc u e and by he plasma. Using (2.11) and he di ision o he lux in o he c ea ed by he plasma and he c ea ed by he s uc u e: ψ=ψp+ψs, BR=BRp+BRs, BZ=BZp+BZs, BRp=−1 R ∂ψp ∂Z , BRs=−1 R ∂ψs ∂Z , BZp=1 R ∂ψp ∂R , BZs=1 R ∂ψs ∂R . (4.13) The ou pu ec o ycon ains he s a e ec o a iables and diagnos ic measu emen s such as he poloidal ield o lux measu emen s. In ou case, we only use poloidal ield measu es so 34 4.2. RZIP MODEL CHAPTER 4. FIESTA TOOLBOX he ou pu ec o is y=Cx ⇒            δ(Is) δ(Zgeo)I0 p δ(Rgeo)I0 p δ(Ip) Bθn            = =             I0 0 0 01 0 0 00 1 0 00 0 1 ∂(~ B·~n)n ∂Is ∂(~ B·~n)n ∂(δ(ZGeo)I0 p) ∂(~ B·~n)n ∂(δ(RGeo)I0 p) ∂(~ B·~n)n ∂Ip                     δ(Is) δ(Zgeo)I0 p δ(Rgeo)I0 p δ(Ip)         , (4.14) whe e he subsc ip nin he poloidal ield Bθindica es he measu e n, and ~n is he no mal ec o o he loop ha measu es he poloidal ield [34] (sec ion 3.3). The eed- o wa d ma ix Dis ze o p o ided ha he e is no di ec ela ion be ween he inpu s and he ou pu s. Using (4.14), he coil cu en s needed o ob ain poloidal ield null in he egion whe e he senso s a e placed (null cu en s) can be compu ed by simply se ing o ze o he elemen s o y ela ed o he ield, and compu ing he coil cu en s needed, using only he coil cu en s in x, and he co esponding elemen s o he Cma ix. This is c ucial o he b eak-down phase, since he poloidal ield has o be null ou as much as possible o ensu e he Townsend a alanche. The sys em (4.9) is sol ed by a Runge-Ku a adap i e me hod ( unc ion ode45 om MAT- LAB), ob aining he ime e olu ion o he plasma cu en , he eddy cu en s in he acuum essel and he coilse and plasma ol ages, gi en he cu en p o ile o he coilse . As s a ed p e iously, RZIp is a pe u ba i e model on an equilib ium s a e. This equilib ium con igu a- ion ha e been ob ained wi h he G ad-Sha ano sol e o Fies a, so RZIp is applied a e he equilib ium con igu a ion is ob ained. Fies a is a complex oolbox, o housands o code lines, and i is sca cely documen ed, so di ec code eading is impe a i e. Howe e , o his hesis is su icien o unde s and he basis o he Fies a oolbox as desc ibed abo e. Figu e 4.1 shows a low diag am o he Fies a oolbox as i has been used. 35 6.1. STATIC AND DYNAMIC BEHAVIOUR OF SMART CHAPTER 6. RESULTS displays posi i e iangula i ies, abou 0.2, al hough plasmas wi h nega i e iangula i ies can also be ob ained wi h he same coilse con igu a ion i he PF2 coils ac as di e o coils, so ha hey ake he ole o he Di 1 coils, and uppe and lowe bulks will be placed on he PF2 coils ins ead o on Di 1. The pa ame e s q0 and q95 a e he sa e y ac o a a su ace wi h ΨN=0 and 0.95 espec i ely (ΨNis ze o o he magne ic axis, and 1 o he plasma bounda y o sepa a ix). Di 2 cu en is ze o in he a ge equilib ium because Di 2 is only used in he b eak-down phase. (a) Phase 1. (b) Phase 2. Figu e 6.1. Ta ge equilib ium o bo h phases, showing he coil cu en s o he equilib ium con igu a ion and se e al plasma pa ame e s a he igh . 3 bulk egion a e showed on bo h plo s (uppe , lowe and cen al), being he plasma in he cen al bulk. The colo ba indica es he poloidal lux Ψ. Eddy cu en s in he VV a e no included he e. 6.1.2 Cu en wa e o ms As s a ed on chap e 5, he cu en wa e o ms we e gi en as an inpu lacking on he cu en s du ing he b eak-down, which we e calcula ed using RZIp. P e ious o he RZIp simula ion, he egion whe e he poloidal ield will be nulled ou (poloidal null egion) had o be chosen. The size and loca ion was op imized manually o maximize he empi ical connec ion leng h (3.5) and educing he cu en needed o c ea e he null egion. The bes shape and size a e he one showed on igu e 6.2, an squa e o 0.30x0.30m2 cen e ed a Rnull = 0.31m. The senso s, o magne ic ield p obes a e shown. The egion is placed close o he inne wall since i s nea he inne wall whe e he gas b eaks-down, as explained on 42 6.1. STATIC AND DYNAMIC BEHAVIOUR OF SMART CHAPTER 6. RESULTS Coil I(kA) Phase 1 Phase 2 Sol -0.15 -0.30 PF1 -0.47 -1.49 PF2 -0.07 -0.14 Di 1 0.30 1.00 Di 2 0.00 0.00 Table 6.1. Coilse cu en s o he a ge equilib- ium con igu a ion o bo h phases. Di 2 cu en is ze o because i is used only o he b eak-down phase. Pa ame e Phase 1 Phase 2 Rgeo(m) 0.42 0.42 A1.82 1.84 κ1.95 2.01 δ0.20 0.24 q0 1.14 1.03 q95 7.23 7.02 βϕ(%) 2.92 3.71 βθ0.73 0.91 βN(%) 1.98 2.56 Table 6.2. Plasma pa ame e s on he a ge equi- lib ium o bo h phases. Zgeo=0 due o he sym- me y wi h espec he Z= 0 plane. sec ion 3.1. The ull cu en wa e o ms, a e compu ing he cu en needed o null ou as much as possible he poloidal ield in he senso egion wi h RZIp, is showed on igu e 6.3. This igu e also includes he cu en g adien a each ime, which will be deno ed wi h ∆. Each cu en wa e o m ha e 7 do s indica ing he phases o he plasma discha ge, as indica ed in igu e 6.3 (b):  Poin 1, he s a ing poin : all he cu en s a e ze o.  Poin 2, p e-pulse: he Sol coil is cha ged, and he PF and Di coils a e amped on o he cu en s needed o null ou he poloidal ield c ea ed by he Sol (null cu en s).  Poin 3, 1 º amp-down: a his poin , =0ms, he Sol cu en is amped down, inducing he loop ol age ha will ionize he p e- ill gas. The PF and Di coils a e he null cu en s o null ou he poloidal ield so he a alanche can succeed.  Poin 4, 2 º amp-down: a his poin , he Sol coil change i s dec easing slope o make i less ab up . The PF and Di coil cu en s a e s ill he null cu en s. The b eak-down o he p e- ill gas will happen be ween poin 3 and 4. The ime in e al be ween hem was ixed o ma ch he a alanche ime.  Poin 5, a ge equilib ium: a his poin he a ge equilib ium is eached. The PF and Di coil cu en s a e he alues a he a ge equilib ium. 43 6.1. STATIC AND DYNAMIC BEHAVIOUR OF SMART CHAPTER 6. RESULTS Figu e 6.2. Senso s o he poloidal null egion. In ha egion he poloidal magne ic ield will be nulled ou (as much as possible) by he PF and Di coils.  Poin 6, end a ge equilib ium: he a ge equilib ium is sus ained om poin 5 o poin 6, being he ime in e al be ween hem he la - op ime, which is 20ms on phase 1 and 100ms on phase 2. The Sol coil cu en a y so ha he desi ed plasma cu en is main ained, while he PF and Di cu en s a e kep cons an  Poin 7, e mina ion o he plasma discha ge: all he coil cu en s a e u ned o , ending he plasma discha ge. To model he plasma cu en , Spi ze ’s esis i i y ha e been used, using Ze = 2 as e ec i e a omic numbe o accoun o Ca bon impu i ies because he plasma- acing ma e ial in SMART will be Ca bon. Due o he low esis i i y on phase 1, he Sol cu en has o inc ease om poin 5 o 6 o p e en u he inc ease o he plasma cu en . The Sol wa e o m ha e 2 dec easing slopes, he i s one wi h a highe slope o induce he elec ic ield and he ollowing wi h a lowe slope o con inue inc easing he plasma cu en bu a a lowe a e. The eason o his wo slopes is because o enginee ing equi emen s. Wi h he design o he powe supplies o phase 1, he i s ope a ional phase, he coilse cu en g adien s could no su pass 50A/ms. Bu his slope was no enough o allow he b eak-down o he p e- ill gas ( he elec ic ield induced was below he paschen’s cu e, so b eak-down was no possible, he loss a e we e g ea e han he ioniza ion a e), so inally i was decided ha he e would be a i s slope o induce he loop ol age ha would su pass he ope a ional limi s and would be sus ained un il he b eak-down phase ends, and ano he slope wi hin he ope a ional limi s once he b eak-down phase was inished. No e ha he es o he amp cu en s o 44 6.1. STATIC AND DYNAMIC BEHAVIOUR OF SMART CHAPTER 6. RESULTS phase 1 a e below 50A/ms. The ime in e al be ween poin 1 and 2 was chosen so ha he cu en amp is lowe han 50A/ms. The second slope was designed so ha he plasma cu en ises up a a e om 1 o 10MA/s, as was ound in sphe ical okamaks wi h ohmic s a up [27,38,39,40]. The abo e discussion applies o phase 1 only, bu phase 2 was also buil wi h wo slopes since his wo slopes Sol wa e o m is less demanding o he powe supplies. -40 -20 0 20 40 60 80 ime (ms) -0.5 0 0.5 1 1.5 Iinpu (kA) Inpu cu en s ph1 Sol PF1 PF2 Di 1 Di 2 (a) Inpu cu en s o phase 1. 0 50 100 150 ime (ms) -2 -1 0 1 2 3 4 Iinpu (kA) Inpu cu en s ph2 Sol PF1 PF2 Di 1 Di 2 7 6 45 1 2 3 (b) Inpu cu en s o phase 2, indica ing he dis- c e e poin s o each o he wa e o ms. -40 -20 0 20 40 60 80 (ms) -350 -300 -250 -200 -150 -100 -50 0 50 I (A/ms) (Inpu cu en s) ph1 Sol PF1 PF2 Di 1 Di 2 (c) Cu en s g adien s o phase 1. 0 50 100 150 (ms) -300 -200 -100 0 100 200 I (A/ms) (Inpu cu en s) ph2 Sol PF1 PF2 Di 1 Di 2 (d) Cu en s g adien s o phase 2. Figu e 6.3. Coilse cu en s gi en as inpu s and cu en g adien s o bo h phases. The desi ed plasma cu en is sus ained o 20ms in phase 1 and o 100ms in phase 2 ( la - op ime). 6.1.3 Dynamic simula ions The RZIp ou pu s, he plasma cu en and he ne eddy cu en induced in he VV as a unc ion o ime a e shown on igu e 6.4. The coil wa e o ms ha e also been included o a be e unde s anding o he ou pu s. Rega ding he plasma cu en , i can be seen he di e en g ow h a e o i due o he 2 slopes o he Sol cu en , being he i s one mo e ab up . The plasma is sus ained o he desi ed ime, 20ms and 100ms o phase 1 and 2 espec i ely. A 45 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS he end o he discha ge he plasma cu en is no ze o. To null i , cooling down he plasma by allowing ai o en e he VV b eaking he acuum would be a me hod o null ou he plasma cu en . Rega ding he eddy cu en s, hey ollow he expec ed beha iou acco ding o he inpu cu en s. The highes alues a e abou 30kA, which is abou he plasma cu en in phase 1. The eddy cu en s a =0ms a e 1.6kA o phase 1 and 2.5kA o phase 2. Figu e 6.5 includes he plasma cu en and i s g adien , showing ha he g ow h a es a e kep wi hin he desi ed ange, om 1 o 10MA/s (1kA/ms=1MA/s). 6.2 B eak-down esul s 6.2.1 Magne ic ields The magne ic ield a he s a o he Sol amp down (do 3 in igu e 6.3 (b) co esponding o =0ms) ha e been calcula ed using he G ad-Sha ano sol e . This calcula ion includes he eddy cu en s bu no he plasma cu en , no s a ed ye a his ime. The Ea h’s ield was also added o a oid ex emely low alues o he poloidal magne ic ield, as commen ed on chap e 5. Figu e 6.6 includes plo s o he o oidal ield Bϕ, he poloidal ield Bθand he LLoyd’s c i e a, EϕBϕ/Bθ. Due o he symme y o he ield wi h espec o he Z=0 plane, only he uppe po ion o he VV is showed1. Small gaps a e app eciable in he VV co ne s due o ounding issues, wi hou u he in luence in he simula ions. Bϕis he ield c ea ed by he o oidal ield coils, which decays wi h he adial posi ion as 1/R. The Bθplo s displays a mo e complica ed pa e n due o he addi ion o he eddy cu en s. Wi hou he eddy cu en s, he lowes Bθis con ained wi hin he polidal ield null egion (dashed in g een), bu he addi ion o he eddys c ea e a mo e complica ed pa e n. This is a esul o he ac ha RZIp can no un using a a ge equilib ium which con ains eddy cu en s, esul ing in ha he cu en s o he poloidal ield null con igu a ion (null cu en s) do no ake in o accoun he e ec o he eddy cu en s. Hence, he eddy cu en s ha e o be added manually, wo sening he ield o he poin ha he lowes poloidal ield is no in he poloidal ield null egion. In he cen al egion o he VV, Bθis abou 1.1·10−4T in phase 1 and 2.1·10−4T in phase 2. Rega ding Lloyd’s empi ical c i e ia, i can be seen ha he c i e ia o ohmic b eak-down is sa is ied (EϕBϕ/Bθ>1000V/m) nea he inne wall in bo h phases, wi h a la ge egion in 1Ac ually, due o he addi ion o he Ea h’s ield, he ield is no comple ely symme ic wi h espec o he Z=0 plane, bu he di e ences a e negligible, abou 10−7T. 46 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS -40 -20 0 20 40 60 80 ime (ms) -0.5 0 0.5 1 1.5 Iinpu (kA) Inpu cu en s ph1 Sol PF1 PF2 Di 1 Di 2 (a) Inpu cu en s o phase 1. 0 50 100 150 ime (ms) -2 -1 0 1 2 3 4 Iinpu (kA) Inpu cu en s ph2 Sol PF1 PF2 Di 1 Di 2 7 6 45 1 2 3 (b) Inpu cu en s o phase 2, indica ing he dis- c e e poin s o each o he wa e o ms. -40 -20 0 20 40 60 80 ime (ms) 0 5 10 15 20 25 30 35 40 Ip (kA) Plasma cu en ph1 (c) Plasma cu en o phase 1. 0 50 100 150 ime (ms) 0 20 40 60 80 100 Ip (kA) Plasma cu en ph2 (d) Plasma cu en o phase 2. -40 -20 0 20 40 60 80 ime (ms) -10 -5 0 5 10 15 20 25 30 IVV (kA) Ne eddy cu en in he VV ph1 (e) Ne eddy cu en in he VV o phase 1. 0 50 100 150 ime (ms) -40 -30 -20 -10 0 10 20 30 40 IVV (kA) Ne eddy cu en in he VV ph2 ( ) Ne eddy cu en in he VV o phase 2. Figu e 6.4. Coise cu en s gi en as inpu s and RZIp ou pu s o bo h phases, plasma cu en and he ne eddy cu en induced in he acuum essel. The desi ed plasma cu en is sus ained o 20ms in phase 1 and o 100ms in phase 2 ( la - op ime). 47 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS -40 -20 0 20 40 60 80 ime (ms) 0 5 10 15 20 25 30 35 40 Ip (kA) Plasma cu en ph1 (a) Plasma cu en o phase 1. 0 50 100 150 ime (ms) 0 20 40 60 80 100 Ip (kA) Plasma cu en ph2 (b) Plasma cu en o phase 2, indica ing he disc e e poin s o each o he wa e o ms. -40 -20 0 20 40 60 80 (ms) -1 0 1 2 3 4 5 I (kA/ms) (plasma cu en ) ph1 (c) Plasma cu en g adien o phase 1. 0 50 100 150 (ms) -2 -1 0 1 2 3 4 I (kA/ms) (plasma cu en ) ph2 (d) Plasma cu en g adien o phase 2. Figu e 6.5. Plasma cu en and plasma cu en g adien o bo h phases. phase 2 since his phase has la ge Bϕ. 6.2.2 Connec ion leng h In his sec ion, he connec ion leng h compu ed using bo h ield line acing and he empi ical o mula will be showed, as well as he po en ial gained by he elec ons along hei pa h. Figu e 6.7 shows he connec ion leng h ob ained by ield line acing, he elec ic po en ial gained by he elec ons and he g id used o he in eg a ion. The g id con ains a meshg id o 20x20 poin s, and he poin s loca ed whe e he coils a e ha e been emo ed. The esolu ion o he g id is 0.034m in he R axis and 0.0842 in he Z axis, gi ing a esolu ion ela i e o he VV size o 5.26% in bo h axis. The i s hing o no e conside ing he connec ion leng h Lplo is ha i is no symme ic wi h espec o he Z= 0 plane . This is a consequence o he poloidal ield, i s componen s, BRand BZ. Figu e 6.8 shows he e ical and adial ield a =0ms o phase 1 (i displays he 48 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS (a) To oidal ield o phase 1. BT(RGeo) = 0.1T. (b) To oidal ield o phase 2. BT(RGeo) = 0.3T. (c) Poloidal ield o phase 1. IVV( = 0) = 1.6kA. (d) Poloidal ield o phase 2. IVV( = 0) = 2.5kA. (e) Lloyd’s c i e ia o phase 1. ( ) Lloyd’s c i e ia o phase 2. Figu e 6.6. Magne ic ields and Lloyd’s c i e ia a =0ms. Due o he symme y wi h espec o Z=0, only he uppe po ion is shown. The poloidal ield null egion is dashed in g een. No e in his plo he small gaps on he VV due o ounding issues can be seen. 49 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS (a) Connec ion leng h o phase 1. (b) Connec ion leng h o phase 2. (c) U/Vloop o phase 1. (d) U/Vloop o phase 2. 0.2 0.4 0.6 0.8 1 R (m) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Z (m) C oss-sec ion (e) G id o he line acing. I is a 20x20 g id, and he poin s whe e he coils a e lo- ca ed in ha e been emo ed. Figu e 6.7. Connec ion leng h by line acing and elec ic po en ial o SMART a =0ms. The compu a ional g id is also shown. The poloidal ield null egion is dashed in g een. No e he connec ion leng h plo displays a egion nea he lowe PF2 coil wi h high alues, he lines wind up a ound he PF2 coil wi hou colliding wi h i . This egion do no appea s in he elec ic po en ial plo con i ming i is no ele an o he b eak-down. 50 6.2. BREAK-DOWN RESULTS CHAPTER 6. RESULTS same beha iou o phase 2). The Sol coil p oduces nega i e e ical ield in he VV, and he PF and Di coils y o compensa e i by c ea ing posi i e e ical ield. The esul an e ical ield is mos ly nega i e in he whole VV, bu i s magni ude has educed. This explain he ac ha he Lplo is no symme ic, because he lines s a ing in he uppe po ion o he VV ha e on a e age highe connec ion leng h. Hence, he highes alues o La e loca ed a he uppe po ion o he VV. Highe alues a e ob ained nea he inne wall since he e he o oidal ield has i s g ea es alues (Bϕ∝1/R), making Bθless ele an , esul ing in lines ha mos ly goes in he o oidal di ec ion, wi h li le de ia ion. (a) Radial ield. (b) Ve ical ield. Figu e 6.8. Radial and e ical magne ic ields a =0ms. The e ical componen o he Sol ield is nega i e, and he PF and Di coils y o null i by c ea ing posi i e e ical ield. The esul an e ical is mos ly nega i e in he whole VV, bu i s magni ude has been educed. The adial ield is an i-symme ic and he e ical ield symme ic wi h espec o he Z= 0 plane. Ne e heless, he highes alues appea in he su oundings o he lowe PF2 coil (Z= −0.6m), because o he inhomogenei y o he ield a he su oundings o a coil. Fu he om he coils, he ield is mo e homogeneous, bu a he su oundings on a coil, he ield is mos ly he ield o he coil. Plo s o se e al ield lines a e shown on igu e 6.9, showing ha some lines in he su oundings o he lowe PF2 coil wind a ound hem wi hou colliding wi h hem ( he magne ic lines a e wis ed a ound he lowe PF2 coil in a simila way he magne ic lines a e wis ed in a okamak plasma). This igu e also shows ha , al hough he uppe po ion o he VV in gene al ha e highe connec ion leng hs due o he ac ha he e ical ield is nega i e, i he lines s a oo high, hey end up colliding apidly wi h he uppe coils o he uppe VV wall. Conside ing he U/Vloop plo , we see ha he wei d egion su ounding he lowe PF2 coil has low alue o U/Vloop in compa ison o he alue o he uppe po ion o he VV nea he 51 6.3. DISCUSSION CHAPTER 6. RESULTS (10−3.6713T) in he null egion, which is e y simila o GlobusM alue. Howe e , al hough he ag eemen be ween ou alues and VEST and GlobuM alues seems good enough, he e is a sou ce o imp ecision in ou simula ions, and is he ac ha RZIp do no un in a sel - consis en manne wi h he eddy cu en s, as commen ed on sec ion 6.2.1, so he null cu en s do no ake in o accoun he eddy cu en s, and he addi ion o he eddy cu en s wo sen he ield, so ha he lowes poloidal ield is no in he ield null egion. Ano he sou ce o e o is ha ou simula ions use he ideal cu en wa e o ms, while mo e ealis ic simula ions, such as VEST simula ions, also simula e he powe supplys o he coils. As a conclusion o his magne ic ield compa ison, o he e ec s such as e o s in he coil windings o magne ic ma e ial ma e ials su ounding he VV could also al e he magne ic ield. Conside ing now he connec ion leng h Lby line acing, igu e 6.14 include a compa ison wi h NSTX and phase 2, because NSTX has Bϕ= 0.3T [42]. The alues o bo h phase 2 and NSTX a e abou km long, bu he appea ance o bo h plo s a e no e y simila . This di e ence is caused by he ac ha SMART has mos o i s coils inside he VV, while NSTX has i s coils ou side, as can be seen on (d). To p o e his poin , a Lplo o an olde SMART con igu a ion wi h all he coils ou side (phase 1) is included (c), showing simila i ies wi h he NSTX plo ; bo h plo s displays high La m-like egions, poin ing upwa d and ou wa d ( o he ou e wall). Howe e , SMART do no ha e an a m poin ing downwa d while NSTX do, and his a m seems o be caused by a lowe PF coil o NSTX, PF1B, which do no appea s in he uppe po ion o he NSTX c oss-sec ion (i.e., NSTX c oss-sec ion do no displays symme y wi h espec o he Z= 0 plane). The connec ion leng h compu ed by he empi ical o mula is oughly an o de o magni ude lowe han he connec ion leng h ob ained by ield line acing. Di e ences la ge han a ac o o wo we e p edic ed in [23] and di e ences abou an o de o magni ude, as ob ained in his wo k, we e ob ained in ITER simula ions [25]. Rega ding Paschen’s cu e and a alanche ime, SMART loop ol age seem a bi low in compa ison o GlobusM, bu since he loop ol age is s ongly dependen on he con igu a ion o he machine, which is di e en o each machine, is no such a de e minan ac o . Ins ead, s udying o he a iables such as he a alanche ime would gi e mo e in o ma ion, and in ou case he a alanche ime con i m ha he b eak-down phase could be inished in a sho enough ime in e al, going om 1 o 4ms o p e- ill p essu es abou 10−4To . Da a econs uc ion in a VEST discha ge using ECRH assis ance shows ha he plasma has al eady o med and mo ed o he ou e wall 2ms a e he amp-down o he Sol, so he a alanche ime mus be ≤2ms [26]. This ime in e al lies be ween he ange o alues ob ained o SMART, wi h he di e ence ha he SMART esul s ha e been ob ained wi h an ohmic s a up, wi hou any 58 6.3. DISCUSSION CHAPTER 6. RESULTS (a) Poloidal ield o SMART phase 1. (b) Poloidal ield o VEST. 1G= 10−4T. (c) Poloidal ield o SMART phase 2. (d) Poloidal ield o GlobusM. (e) Lloyd’s c i e ia o SMART phase 1. ( ) Lloyd’s c i e ia o VEST. Figu e 6.13. Compa ison o he poloidal ield and Lloyd’s c i e ia wi h VEST [26] and GlobusM [41] (bo h simula ed da a) a he exac ime when he loop ol age is induced. Phase 1 is used o compa e wi h VEST because i he same Bϕ= 0.1T. Eϕ= 1.12V/m o VEST and 1.06V/m o phase 1. Bϕ<0.62T o GlobusM, so i is compa ed wi h phase 2, which has 0.3T. 59 6.3. DISCUSSION CHAPTER 6. RESULTS assis ance. Howe e , SMART will also ha e ECRH assis ance o he b eak-down jus in case i is needed o ionize he p e- ill gas. Ne e heless, he loop ol age calcula ed he e is he loop ol age induced by only he Sol, excluding he e ec o he eddy cu en s, which will dec ease he loop ol age acco ding o Lenz’s law. A mo e p ecise simula ion should compu e he loop ol age in a sel -consis en manne wi h he eddy cu en s. Finally, he simula ions ca ied ou he e, and in mos o he bibliog aphy ega ding b eak- down s udies, a e done igno ing he ime dependence o all he a iables such as he connec ion leng h o he poloidal ield, so hey ha e o be ea ed as a i s app oxima ion o he p oblem o he okamak s a -up ( o deduce (3.7), he ime dependence o he loss a e was neglec ed). Also, he plasma cu en needs o be conside ed since sho ly a e he loop ol age is induced, he plasma cu en will be high enough o c ea e non-negligible ield as compa ed o he s uc- u e’s ield. Fo deepe s udies o he s a -up phase, speci ic codes ha e been de eloped o simula e all he s ages o he s a -up phase such as DYON [21] o DINA, which conside he ime dependence o he loss a e and he ole o he plasma cu en in he magne ic ields. 60 6.3. DISCUSSION CHAPTER 6. RESULTS (a) Connec ion leng h o SMART phase 2. (b) Connec ion leng h o NSTX. (c) Connec ion leng h o an olde SMART phase 1 wi h he coils ou side (S1-014). (d) C oss-sec ion o NSTX. Figu e 6.14. Compa ison o he connec ion leng h plo wi h NSTX [28,43] (simula ed da a). Bϕ= 0.3T o NSTX and o phase 2. 61 Chap e 7 Summa y and conclusions The b eak-down phase o he u u e SMART eac o o he Uni e si y o Se ille has been modelled nume ically using he Fies a oolbox. The undamen als o okamak physics and okamak s a -up ha e been e iewed o unde s and he ounda ion o his wo k. The wo k o his hesis summa izes mon hs o wo king in he design o he SMART eac o along wi h he SMART eam o he PSFT g oup, composed o physicis s and enginee s s uden s and esea che s, showing he mos upda ed scena ios o he ini ial ope a ional phase o SMART and a u u e upg ade. The coil cu en s ha e been op imized o achie e he same a ge equilib ium on bo h phases and o ensu e he b eak-down o he p e- ill gas, hyd ogen, wi hou any assis ance me hod. Fo p e- ill p essu es abou 10−4To , he gas will las be ween 2 and 4 ms o b eak-down. The eddy cu en s ha e been included in he calcula ions, since hey play an impo an ole in he b eak-down phase al e ing he poloidal magne ic ield inside he acuum essel (VV). Poloidal ield alues a ound 1.1 and 2.1·10−4T a e ob ained in he poloidal ield null egion. 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