MMSc thesisAnalysis of fast ion inducedinstabilities in tokamak plasmas
L´aszl´o Horv´ath
Supervisors:
Dr. Gergely Papp postdoctoral researcher
Max-Planck/Princeton Center for Plasma Physics
Dr. Gerg˝o Pokol associate professor
BME Institute of Nuclear TechniquesBudapest, Hungary
Budapest University of Technology and Economics2015 a r X i v : . [ phy s i c s . p l a s m - ph ] J un nalysis of fast ion induced instabilities in tokamak plasmas L´aszl´o Horv´athInstitute of Nuclear TechniquesBudapest University of Technology and Economics
Abstract
Nuclear fusion is a promising energy source of the future. One possibleway to achieve reliable energy production from fusion reactions is toconfine the high temperature deuterium-tritium fuel by magnetic fields. Inmagnetic confinement fusion devices like tokamaks, it is crucial to confinethe high energy fusion-born helium nuclei ( α -particles) to maintain theenergy equilibrium of the plasma. However, super-thermal energetic ions canexcite various instabilities which can lead to their enhanced radial transport.Consequently, these instabilities may degrade the heating efficiency and theycan also cause harmful power loads on the plasma-facing components of thedevice. Therefore, the understanding of these modes is a key issue regardingfuture burning plasma experiments.One of the main open questions concerning energetic particle (EP) driveninstabilities is the non-linear evolution of the mode structure. In this thesis,I present my results on the investigation of β -induced Alfv´en eigenmodes(BAEs) and EP-driven geodesic acoustic modes (EGAMs) observed inthe ramp-up phase of off-axis neutral beam injection heated plasmas inthe ASDEX Upgrade tokamak. These modes were well visible on severalline-of-sights of the soft X-ray cameras which made it possible to analyse thetime evolution of their spatial structure.In order to investigate the radial structure, the mode amplitude has tobe determined on different line-of-sights. I developed an advanced ampli-tude reconstruction method which can handle the rapidly changing modefrequency and the low signal-to-noise ratio. This method is based on shorttime Fourier transform which is widely applied in the thesis, because it isideal to investigate the time evolution of transient wave-like phenomena.The radial structure analysis showed that in case of the observeddownward chirping BAEs the changes in the radial eigenfunction were smallerthan the uncertainty of the measurement, while in case of rapidly upwardchirping EGAMs the analysis shows shrinkage of the mode structure. Theseexperimental results are shown to be consistent with the correspondingtheory. yors ionok hajtotta hull´amok vizsg´alata tokamak plazm´akban Horv´ath L´aszl´oNukle´aris Technikai Int´ezetBudapesti M˝uszaki ´es Gazdas´agtudom´anyi Egyetem
Kivonat
J¨ov˝onk ´ıg´eretes energiaforr´asa a nukle´aris f´uzi´o, mely megval´os´ıt´as´anakegyik lehets´eges m´odja m´agneses ¨osszetart´as´u berendez´es alkalmaz´asa. Ilyenberendez´es a tokamak, melyben a deut´erium-tr´ıcium f´uzi´os reakci´o sor´ankeletkez˝o nagy energi´aj´u h´elium atommagok ( α -r´eszecsk´ek) ¨osszetart´asak¨ul¨on¨osen fontos a plazma energiaegyens´uly´anak fenntart´asa szempontj´ab´ol.Ezek a termikusn´al j´oval nagyobb energi´aval rendelkez˝o ionok azonban olyanplazmainstabilit´asokat kelthetnek, melyek felgyors´ıthatj´ak a gyors ionokradi´alis transzportj´at. Ennek k¨ovetkezm´enyek´ent cs¨okken az α -r´eszecsk´ekf˝ut´esi hat´ekonys´aga, tov´abb´a a kisz´or´od´o gyors ionok s´ulyosan k´aros´ıthatj´aka plazm´at k¨or¨ulvev˝o falelemeket. Emiatt a gyors ionok ´altal hajtott in-stabilit´asok meg´ert´ese l´etfontoss´ag´u a j¨ov˝o energiatermel˝o f´uzi´os reaktoraiszempontj´ab´ol.A kutat´asi ter¨ulet egyik m´aig nyitott k´erd´ese a gyors ionok ´altalhajtott m´odusok szerkezet´enek nem-line´aris id˝ofejl˝od´ese. Jelen dolgozatbanaz ASDEX Upgrade tokamak semleges atomnyal´ab ´altal f˝ut¨ott plazm´ainakkezdeti szakasz´aban megfigyelt β -hajtott Alfv´en saj´atm´odusokkal ( β -inducedAlfv´en eigenmode, BAE) ´es gyors r´eszecsk´ek ´altal hajtott geod´ezikusakusztikus m´odusokkal (energetic particle driven geodesic acoustic mode,EGAM) kapcsolatos k´ıs´erleti eredm´enyeimet mutatom be. Ezen m´odusokj´ol megfigyelhet˝oek a l´agy-r¨ontgen diagnosztika sz´amos csatorn´aj´an, melyseg´ıts´eg´evel a m´odusok radi´alis strukt´ur´aj´anak id˝ofejl˝od´ese vizsg´alhat´o.A radi´alis strukt´ura tanulm´anyoz´as´ahoz a m´odus amplit´ud´omeghat´aroz´as´ara van sz¨uks´eg a l´agy-r¨ontgen diagnosztika k¨ul¨onb¨oz˝ol´at´ovonalain. Ennek ´erdek´eben kifejlesztettem egy amplit´ud´o rekonstrukci´oselj´ar´ast, amely seg´ıts´eg´evel kezelhet˝o a m´odus gyorsan v´altoz´o frekvenci´aja ´esaz alacsony jel-zaj viszony. Ez a m´odszer r¨ovid idej˝u Fourier-transzform´aci´onalapul, melyet sz´eles k¨orben haszn´altam a munk´am sor´an, mivel kiv´al´otranziens jelek elemz´es´ere.Vizsg´alataim megmutatt´ak, hogy a megfigyelt BAE-k eset´eben a m´odusradi´alis strukt´ur´aj´anak v´altoz´asa kisebb a m´er´es bizonytalans´ag´an´al. Ezzelszemben az EGAM-ok eset´eben a m´odus strukt´ura ¨osszeh´uz´od´asa figyelhet˝omeg. Ezek a k´ıs´erleti eredm´enyek ¨osszhangban vannak a jelens´eget le´ır´oelm´elettel. szakdolgozat ki´ır´asa Azonos´ıt´o: DM-2014-44¨Onfenntart´oan m˝uk¨od˝o szab´alyozott magf´uzi´os reaktor m˝uk¨od´ese szem-pontj´ab´ol kulcsfontoss´ag´u a termikusn´al j´oval nagyobb energi´aj´u ionok¨osszetart´asa. Ezen ionok rezon´ans r´eszecske-hull´am k¨olcs¨onhat´ason kereszt¨ulk¨ul¨onb¨oz˝o hull´amokat tudnak hajtani, mely hull´amok a r´eszecsk´ek meg-n¨ovekedett vesztes´eg´et okozz´ak, ez´altal rontva a plazmaf˝uz´es hat´ekonys´ag´at.A hallgat´o els˝odleges feladata a n´emetorsz´agi ASDEX Upgrade tokamakonmegfigyelt, gyors r´eszecsk´ekhez k¨othet˝o k¨ul¨onb¨oz˝o Alfv´en-saj´atm´odusok,ezen bel¨ul a gyors frekvenciav´altoz´assal jellemezhet˝o ´un. ”cs¨orp¨ol˝o” (az angol”chirp” sz´ob´ol) m´odusok vizsg´alata. Nyitott k´erd´es, hogy a cs¨orp¨ol˝o m´odusokradi´alis saj´atf¨uggv´enye hogyan v´altozik a cs¨orp ideje alatt: a m´er´esi bizonyta-lans´agon bel¨ul ´alland´o marad, vagy szignifik´ansan v´altozik? Ez az inform´aci´okulcsfontoss´ag´u a m´odusok meg´ert´ese, ´es k´es˝obbi predikt´ıv modellez´ese szem-pontj´ab´ol. A hallgat´onak ki kell dolgoznia egy elj´ar´ast amellyel a cs¨orp¨ol˝om´odusok saj´atf¨uggv´eny´enek id˝obeli v´altoz´asa vizsg´alhat´o, ´es a v´altoz´as / nemv´altoz´as t´enye a m´er´esi bizonytalans´agon bel¨ul szignifik´ansan kimutathat´o.A vizsg´alat sor´an els˝odlegesen felhaszn´aland´o adatok az ASDEX Upgradetokamakon rendelkez´esre ´all´o m´agneses diagnosztik´ak jelei, a vonalintegr´altl´agy r¨ontgen sug´arz´as m´er´ese illetve az elektron-ciklotron kibocs´ajt´asonalapul´o lokaliz´alt h˝om´ers´ekletm´er´es. Sz¨uks´eg eset´en m´as diagnosztik´ak isbevonhat´ok a vizsg´alatba. A feladat sor´an meg kell ismerni a diagnosztik´akm˝uk¨od´es´et, ki kell dolgozni egy elj´ar´ast a radi´alis saj´atf¨uggv´eny id˝obeliv´altoz´as´anak m´er´es´ere ´es a m´er´esi bizonytalans´agok becsl´es´ere. A m´er´esieredm´enyek f¨uggv´eny´eben, azok m´elyebb meg´ert´es´ehez sz¨uks´egess´e v´alhata vizsg´alt m´odusok szimul´aci´oja ´es szintetikus diagnosztik´ak alkalmaz´asa.Ezen szimul´aci´os programok meg´ır´asa a hallgat´onak nem feladata, a c´elra j´olm˝uk¨od˝o programcsomagok ´allnak rendelkez´esre a t´emavezet˝o int´ezet´eben.Az elemz´esek v´egrehajt´as´ahoz seg´ıts´eget ny´ujt a BME NTI-ben fejlesztettNTI Wavelet Tools programcsomag. A programcsomag t¨obb k¨ul¨onb¨oz˝oelemz˝o algoritmus mellett j´ol m˝uk¨od˝o k´ıs´erleti adatbeolvas´o-, manipul´al´o-´es ´abr´azol´o k´epess´egekkel rendelkezik, ez´altal c´elszer˝u a vizsg´alathoz a m´armegl´ev˝o programokat felhaszn´alni, sz¨uks´eg eset´en azokat m´odos´ıtani. AzNTI Wavelet Tools nagy r´esze IDL programnyelven ´ır´odott, ez´ert ennek anyelvnek ismerete vagy sz¨uks´eges el˝ofelt´etel, vagy megtanuland´o, mivel ahallgat´o feladata lesz a sz¨uks´eges v´altoztat´asokat is elv´egezni. A szakdolgozatt´em´aj´ahoz k¨ot˝od˝o kutat´omunka nemzetk¨ozi egy¨uttm˝uk¨od´esben zajlik, esetik¨ulf¨oldi kiutaz´asokkal. Magas szint˝u angol nyelvtud´as, alapos programoz´oi´es matematikai ismeretek sz¨uks´egesek. atement of Originality
This document is written by L´aszl´o Horv´ath who declares to take fullresponsibility for the contents of this document. I declare that the text andthe work presented in this document is original and that no sources other thanthose mentioned in the text and its references have been used in creating it. ¨On´all´os´agi nyilatkozat
Alul´ırott Horv´ath L´aszl´o, a Budapesti M˝uszaki ´es Gazdas´agtudom´anyiEgyetem fizika MSc szakos hallgat´oja kijelentem, hogy ezt a szakdolgozatotmeg nem engedett seg´edeszk¨oz¨ok n´elk¨ul, ¨on´all´oan, a t´emavezet˝o ir´any´ıt´as´avalk´esz´ıtettem, ´es csak a megadott forr´asokat haszn´altam fel. Minden olyanr´eszt, melyet sz´o szerint, vagy azonos ´ertelemben, de ´atfogalmazva m´asforr´asb´ol vettem, a forr´as megad´as´aval jel¨oltem.. . . . . . . . . . . . . . . . . . . . . . . . . . .Horv´ath L´aszl´o ontents hapter 1Introduction
Our world provides plenty of options to meet the energy needs of mankind.One of them, which is already in the service of humanity, is the release ofthe binding energy of atomic nuclei. Energy is produced by nuclear fissionin nuclear power plants all over the world. However, sustainable energyproduction to the electricity grid by fusion reactions has not yet beenachieved [1].
An example of reliable fusion energy production is the Sun. Its hydrogen ( H)fuel is converted to helium ( He) in different astrophysical reaction chains [2].These fusion reactions provided energy for several billions of years to lifeon Earth. In stellar cores the rate of fusion reactions is very slow, becauseproton-neutron decays have to occur. Therefore, astrophysical reaction chainshave very low cross-sections, thus the released power density of the Sun isless than 1 W/m [3].Reproduction of stellar core conditions on Earth for nuclear fusion powerproduction would be inefficient. In order to achieve sufficiently high powerfrom fusion reactions for electricity production, light isotopes should bechosen [3] such as deuterium ( H ≡ D), tritium ( H ≡ T) or He. Someof the fusion reactions possible with the aforementioned isotopes are listedhere [4]: D + D ⇒ He (0 .
82 MeV) + n (2 .
45 MeV) , D + D ⇒ T (1 .
01 MeV) + p (3 .
02 MeV) , D +++ T ⇒⇒⇒ He (3 .
52 MeV) +++ n (14 . , D + He ⇒ He (3 .
66 MeV) + p (14 . . Temperature [keV] R e a c t i o n r a t e [ m / s ] D - TD - DD - He -27 -27 -23 -21 Figure 1.1:
The fusion reaction rates of different isotopes [6].
A huge amount of deuterium is available in nature, because there is 1 atomof deuterium for every 6700 atoms of hydrogen in natural water resources [4].In addition to the enormous resources, deuterium can be easily extracted ata low cost. However, gathering tritium is difficult. Tritium is a radioactiveisotope and it has a half-life of 12 . Li + n (thermal) = He + T Li + n (energetic) = He + T + nThe tritium breeding process is illustrated in figure 1.2. The fusion-bornneutrons, by hitting the wall containing lithium, produce tritium. In thisway the fuel of a fusion power plant is deuterium and lithium. Both arenon-radioactive elements and are available in large quantities on Earth. Thefinal product of the fusion reaction and the tritium breeding reaction arehelium, thus neither the fuel, nor the end-product of the fusion and breedingcycle is radioactive. 8 i t h i u m deuterium deuterium heliumtritium neutron heliumtritium Figure 1.2:
Operational scheme of tritium breeding. The fusion-born neutrons,by hitting the wall containing lithium, produce tritium.
Fusion experiments with particle accelerators provided useful informationto determine the cross sections of the aforementioned reactions. However,sustainable energy production with accelerators cannot be achieved [6], sincethe cross-section of Coulomb scattering is several orders of magnitude higherthan the cross-section of the fusion reaction. Therefore, the most promisingmethod is to heat the DT fuel to sufficiently high temperature that thethermal velocities are high enough to produce fusion reactions. In this casethe Coulomb collisions just redistribute the energy between particles. Thenecessary temperature is in the order of hundred million Kelvin for DT fuel.The conditions required for a self-sustaining process are estimated by theLawson criterion for DT fuel and optimum temperature ( ∼
25 keV) [6]: nτ E > s/m , (1.1)where n is the density and τ E is the energy confinement time which measuresthe rate at which a system loses energy to its environment: τ E = WP loss , (1.2)where W is the plasma energy content and P loss is the total power loss of theplasma. 9 .2 Magnetic confinement fusion One way of confining the fuel which has such high temperature is to employa magnetic field. At temperatures necessary for the fusion reaction, theparticles are ionized and the fuel is in the plasma state. The charged particles(ions and electrons) start their helical motion in the plane along the magneticfield. The circular part of this motion in the plane perpendicular to the fieldline is called gyromotion. The helical motion is suitable to confine the plasma,because it does not allow the particles to move perpendicular to the magneticfield, but they are free to move parallel to the magnetic field. The problem oflosses at the ends of a linear device can be solved by bending the magneticfield lines into a torus. However, in a toroidal geometry, the magnetic field isnot homogeneous. The strength of the magnetic field is inversely proportionalto the distance measured from the centre of the torus, because of the highercurrent density in the toroidal field coils (indicated with red in figure 1.3)in the inner side of the torus results in a stronger magnetic field there. Thisinhomogeneity leads to a charge dependent drift ( ∇ B drift) [6], which causesthe electrons and ions to move in vertically opposite directions. The resultingvertical electric field creates a charge independent E × B drift [6], which movesthe entire plasma towards the outside of the torus. This electric field betweenthe top and the bottom of the torus can be shorted out by helically twistingthe magnetic field lines.A type of magnetic confinement device is the tokamak [6], where themagnetic field lines are helically twisted by a plasma current as it is illustratedin figure 1.3. In tokamak geometry the toroidal direction is the long way, thepoloidal direction is the short way around the torus. The toroidal plasmacurrent is driven by a transformer coil and it twists the magnetic fieldlines because it generates a poloidal magnetic field indicated with purple infigure 1.3. The twist of magnetic field lines is described by the safety factor.The safety factor q is the ratio of toroidal transits per single poloidal transitof a magnetic field line [8]. Due to the axisymmetry of a tokamak device,the magnetic field lines are organized into magnetic flux surfaces. Along themagnetic field lines, the particles can move freely, thus the plasma parameterson the magnetic flux surfaces are balanced very quickly, while the transportperpendicular to the surfaces is several orders of magnitude slower.In order to realize a self-sustaining magnetically confined plasma, thereleased fusion energy has to heat the plasma. The energy generated ina fusion reaction is distributed as kinetic energy to the fusion products.According to their mass ratio, 80 % of the energy is carried by the neutronswhich do not interact with the plasma and hit the plasma facing components.The remaining 20 % energy is carried by the He nuclei. These He ions have10 oroidal field coils Stabilizing coils Plasma current
Toroidal magnetic field
Poloidal magnetic field Transformator
Figure 1.3:
Schematic drawing of a tokamak. The superposition of the toroidalmagnetic field and the poloidal magnetic field generated by thetoroidal plasma current leads to helically twisted magnetic field linesin the torus. . ions ( α -particles) is called α -heating. In a hot plasmathe typical mean free path of fusion-born α -particles is very long comparedto the size of the device [9]. Thus, it is crucial to well confine the high energy α -particles.Beside the α -heating, auxiliary heating systems are needed to reach fusionrelevant conditions. Furthermore, these external heating methods are plannedto be used to tailor the plasma profiles during the fusion burn. Two types ofauxiliary heating systems are used to generate fast ions, namely the neutralbeam injection (NBI) and the ion cyclotron resonance heating (ICRH). Thesetechniques are capable of accelerating hydrogen isotopes up to the MeVenergy range. Since plasma heating scenarios are predominantly based onfast ions, their transport properties are fundamental to the success of anyburning plasma experiment. Enhanced radial transport of energetic particles(EPs) degrades the heating efficiency and it can also cause harmful powerloads upon the first wall of the device. Since, in the plasma core, the mostimportant transport process of EPs is their interaction with global plasmawaves [9], the thorough understanding of the EP driven instabilities is crucial.11 .3 Fast ion transport due to plasma waves Marginally stable plasma modes can be destabilized by EPs in tokamaks. Inaddition to normal modes, there are also energetic particle modes (EPMs)which are forced oscillations characterized by strong dependence on the fastion distribution function. First, a basic introduction of the shear Alfv´en wavesis presented.Shear Alfv´en waves are electromagnetic waves that propagate along themagnetic field with the following dispersion relation [10]: ω = k (cid:107) v A , (1.3)where v A = B (cid:112) µ (cid:80) i n i m i (1.4)is the Alfv´en velocity, k (cid:107) is the wave number in the direction of the magneticfield, (cid:80) i n i m i is the mass density of the plasma, i denotes the different ionspecies, B is the magnitude of the magnetic field and µ is the vacuumpermeability. In a torus, the periodic boundary condition requires that thewave solutions are quantized in toroidal and poloidal directions: k (cid:107) = n − m/q ( r ) R , (1.5)where n and m are the toroidal and poloidal mode numbers respectively, q isthe safety factor and R is the major radius of the plasma. In a toroidalplasma both k (cid:107) and v A are functions of the minor radius r . This leadsto the so called shear Alfv´en continuum. The dispersion relation of twosuccessive modes with poloidal mode number m = 2 and 3 is illustrated onfigure 1.4. Waves at different radii have different phase velocities which leadsto phase-mixing and a hypothetical wave packet would rapidly disperse. Thisprocess is called continuum damping [11]. In a torus with a large aspect ratio( R (cid:29) a ) the magnetic field can be approximated by B ≈ B (1 − (cid:15) cos( θ )) [6],where (cid:15) = a/R is the inverse aspect ratio and θ is the poloidal coordinate.The parallel component of the wave vector now also depends on the poloidalcoordinate θ which can lead to intersection of the dispersion relationsbelonging to different poloidal mode numbers m . The crossing is expectedto be at the radius where the wave number of the two counter-propagatingwaves equals: k m ( r m ) = − k m ( r m ) . (1.6)The crossing of m = 2 curve (blue) and the m = 3 curve (green) is visiblein figure 1.4. However, toroidicity resolves this degeneracy at the points12 AE gap
Alfvén continuum F r e q u e n c y [ k H z ] n = 1; m = 2n = 1; m = 3n = 1 coupled Figure 1.4:
Shear Alfv´en continuum plotted (without coupling) with dashed blueline (n = 1, m = 2) and with dotted green line (n = 2, m = 3).In toroidal geometry, a frequency gap is formed due to the toroidalcoupling between m = 2 and m = 3 modes. The Alfv´en continuumwith coupling is plotted with solid red lines. In order to calculate thesedispersion relations, the background plasma parameters are adoptedfrom Lesur’s PhD thesis [11]. The major radius is 3.3 m, the minorradius is 0.96 m and the magnetic field is 1.2 T on axis. The q profilehas been modeled by q ( r ) = 1 . . r/a ) . and the electron densityby n e ( r ) = 0 .
11 + 1 . − r /a ) . [10 m − ]. of intersection and produces gaps in the frequency continuum as shownin figure 1.4 with red. For a particular toroidal mode number, and to thelowest order in the inverse aspect ratio ( (cid:15) = a/R ) expansion parameter,each poloidal harmonic couples to its nearest neighbouring sidebands [12]and the eq. (1.3) dispersion relations changes: ω , /v A = k (cid:107) m + k (cid:107) m +1 ± (cid:113) ( k (cid:107) m − k (cid:107) m +1 ) + 4 (cid:15) k (cid:107) m k (cid:107) m +1 − (cid:15) ) . (1.7)Within these gaps global modes can exist, because waves do not experienceheavy continuum damping. These modes are called toroidicity-induced Alfv´eneigenmodes (TAEs). Analogous to the TAE gap further gaps arise caused bythe coupling of the m , m + 2 and m , m + 3 modes such as elipticity inducedgap and non-up-down-symmetric gap [9]. In these gaps radially extended,weakly damped modes can exist, which can be driven unstable by the fast13on population. The properties of these eigenmodes such as frequency or modestructure are mostly determined by the background plasma. A similar gapmode is the β -induced Alfv´en eigenmode (BAE). The BAE gap is introducedby the coupling between compressible acoustic waves and the shear Alfv´encontinuum [13].These modes can be driven unstable by the free energy in the fast iondistribution function [9]. Energy transfer between an ion and a wave requiresthat the velocity vector is not perpendicular to the electric field of the wave.Thus, only the drift velocity of ions contributes to the energy transfer [14].The energy transfer associated with the gyromotion averages to zero, sincethe gyromotion is very rapid compared to the mode frequency.The growth rate of a mode is generally small compared to the wavefrequency, because significant net energy transfer is gained only after dozensof orbital cycles. To avoid phase averaging to zero, a harmonic of thedrift-orbit frequency must match the wave frequency, so the followingresonance condition has to be fulfilled for net energy gain [14]: ω + ( m + l ) ω θ − nω ξ (cid:39) , (1.8)where ω is the wave frequency, ω θ is the frequency of the poloidal motion ofthe ion, ω ξ is the frequency of the toroidal motion of the ion, m is the poloidalmode number and n is the toroidal mode number of the mode. l denotesthe order of the harmonic of the drift-orbit frequency. In strongly shapedplasmas and for large drift-orbit displacements, higher order harmonics arealso important.The net energy exchange between the wave and the particles is determinedby several factors. It depends on the amplitude of the particular poloidalharmonic and on the drift velocity. Furthermore, the magnitude of theenergy exchange also depends on the gradient associated with the particularresonance in the EP distribution function. Particles with speed slightly belowthe resonant velocity gain energy, while particles with speed slightly above theresonant velocity lose energy. Thus, EPs can drive a mode where the gradientin the EP distribution is positive along the coordinate defining the resonance.Mostly the gradient in the EP distribution respect to the energy is negative,so the energy distribution usually damps the wave. This is true even for theslowing-down distribution in a burning plasma. Modes are usually driven bythe free energy in the spatial gradient. However, the anisotropy in the velocityspace can also drive the mode. Finally, the alignment of the particle orbitand the eigenmode is also a governing factor of the energy exchange. Modesare excited if the drive can overcome the damping of the background plasma.Important damping mechanisms include ion Landau damping, continuum14amping and radiative damping [9]. The damping rates are very sensitive toplasma parameters such as the temperature and the q profile.Several types of non-linear behaviour of the mode amplitude and fre-quency are observed on present-day tokamaks. This behaviour ranges fromsteady-state – where the amplitude saturates at nearly constant frequency– to an explosive growth. The type of behaviour is expected to significantlyinfluence the impact of the instabilities on the fast particle transport. Thus,their thorough understanding from both experimental and theoretical side isessential. In this thesis modes with bursting amplitude and rapidly changingmode frequency are experimentally investigated in detail. These modes arealso called “chirping” modes.So far, the discussion was restricted to Alfv´en modes, but if the EPpressure is sufficiently high compared to the thermal pressure, the EPdrive can overcome the continuum damping and energetic particle modesare excited. Their mode structure is independent from the shear Alfv´encontinuum, and sensitively depends on the EP distribution. Their frequencyusually correspond to a characteristic frequency of the EP orbital motion.One of the main open questions concerning EP driven Alfv´en eigenmodesand EPMs is the non-linear evolution of the mode amplitude. These instabil-ities constitute a non-linear system where kinetic and MHD non-linearitiescan both be important making it difficult to describe the phenomenon. Inthis thesis I present my results on the experimental investigation of betainduced Alfv´en eigenmodes (BAEs) and EP-driven geodesic acoustic modes(EGAMs) observed in the ramp-up phase of off-axis NBI heated plasmas inASDEX Upgrade (AUG) [15].The plasma scenario in which these modes were observed needs lowdensity operation and off-axis NBI heating. Low density is important inorder to keep the ratio of fast ion pressure and thermal pressure as highas possible. The excitation of these modes is possible if the damping is lowenough. The damping of EGAMs is lower at lower temperatures. Thus, theyare easily driven unstable at lower temperatures e.g. during the ramp-upphase of a discharge. The main damping mechanism of BAE is the ion Landaudamping [16] which scales with the background ion temperature and density.This way in low density, low temperature plasmas BAEs are more easilyexcited. On AUG, even in low density operation, the damping of BAEs inthe flat-top – where the main plasma parameters are maintained at constantvalues – is not sufficiently small to excite them with NBI. However, in theramp-up phase BAEs are routinely driven unstable with off-axis NBI heating.The role of off-axis NBI is twofold. One is that, further out from the magneticaxis, the temperature and density is lower which again allows the excitation15f these modes. Second, the dominant auxiliary heating system in futureburning plasma experiments will be off-axis NBI.The confinement of fast ions is crucial in a burning plasma experimentsuch as ITER where the heating of the plasma is based on the dominantself-heating by fusion-born α -particles. ITER is a large-scale scientificexperiment under construction that aims to demonstrate the technologicaland scientific feasibility of fusion energy [17]. In general, EP driven modeshave a massive contribution to the radial transport of fast ions. However, inview of that bursting BAEs and EGAMs are only observed in the ramp-upphase of discharges where damping is low enough due to the low density andtemperature the question arises what are their implications regarding ITERor any other burning plasma experiment?BAEs are driven by the radial gradient in the EP distribution func-tion [16]. Thus, their presence enhances the radial transport of fast ions.However, their stability in ITER is not yet clear. Simulations have shownthat in the ITER inductive baseline scenario the radial gradients in theEP distribution will not be sufficiently high to destabilize BAEs, but thedifference is marginal and the uncertainty of these results is high [18]. EGAMsthemselves do not cause radial redistribution of the fast ions directly, becausethey are driven by the velocity phase space gradient in the EP distribution.Therefore, the isotropic distribution of fusion-born α particles cannot affectEGAMs. However, NBI driven EGAMs can influence fast ion losses indirectlyvia mode-mode coupling with TAEs [19]. The impact of BAEs and EGAMs onthe fast ion confinement in ITER is not completely clear. On the other hand,the investigation of these modes is essential to understand the non-linearbehaviour of EP driven modes.The main goal of this thesis is to experimentally investigate the rapidchanges in the radial structure of bursting EP-driven modes during thenon-linear chirping phase. Due to diagnostic and data analysis complexitiesthis task has never been accomplished before. Some modes are expected toretain their radial structure, while others would be expected to change. Evenqualitative results can provide important information about the underlyingphysics and strengthen (or challenge) our present theoretical understanding.Furthermore, an analysis based on the methods developed here can serveas the basis of comparison with numerical codes which simulate the timeevolution of EP-driven modes.Typically, plasma modes cause magnetic, density and temperature fluc-tuations in the plasma which makes it possible to extensively examinethem in a well-equipped tokamak such as AUG. Since the observed modescause weak oscillations in the measured quantities, the used diagnosticstools were carefully selected by taking into consideration their spatial and16emporal resolution and their signal-to-noise ratio. The argument of theselection and the description of the applied diagnostic tools are presented insection 2.1. The analysis in question requires sophisticated signal processingtools. Since this work is focused on the investigation of transient, wave-likephenomena, the signal processing methods developed and used are based onshort time Fourier transform (STFT) [20]. The applied methods are describedin section 2.2. The amplitude reconstruction method derived in this thesiswhich allowed to investigate the radial structure of the observed modes ispresented in chapter 3. The dedicated experiments analysed are shown insection 4 where the results are also discussed in view of the physical picture.17 hapter 2Measurement set-up andanalysis principles The frequency of the modes investigated in this thesis is in the order of50 - 100 kHz. Both the diagnostic tools and the data processing method werechosen to deal with such high frequency oscillations.From the diagnostic point of view, fluctuation measurements are requiredwhich can measure either the magnetic, the density or the temperaturefluctuations in the plasma with higher temporal resolution than 100 kHz. Thepossible candidates were the magnetic pick-up coil, the reflectrometry, theelectron cyclotron emission (ECE) and the soft X-ray (SXR) measurements.Magnetic measurements do not have spatial resolution, thus it is not suitablefor the analysis of the radial structure, but it was used to examine the modestructure. The reflectrometry diagnostic can measure density fluctuationwith high temporal resolution, however due to the low spatial resolutionI only used it to localize the observed modes. The ECE [21] and theECE imaging (ECEI) [22] diagnostics provide local electron temperaturemeasurements. These would be good candidates due to their high time-and spatial resolution, however BAEs and EGAMs were not visible by ECEand ECEI in the investigated shots. Thus, ECE and ECEI are ruled outfrom the analysis. The signal of SXR measurement is a function of electrontemperature and density. Its sampling frequency is 2 MHz and due to thehigh number of SXR channels it also has a spatial resolution. Therefore,this diagnostic was chosen to investigate the changes in the radial structureof the modes. The magnetic probes, the soft X-ray measurements and thereflectrometry diagnostics are introduced in section 2.1.In addition to the suitable diagnostic tools, an appropriate data processingmethod is required. For this purpose, continuous linear time-frequencytransforms were chosen, because these are ideal to investigate transient18ignals. The amplitude reconstruction method developed is presented laterin chapter 3 is based on STFT [20] which is a linear time-frequencytransform. The mathematical background of these transforms are introducedin section 2.2.
The experiments presented in this thesis were carried out on the ASDEXUpgrade tokamak (Axially Symmetric Divertor EXperiment Upgrade) [23],which is a middle sized tokamak with a major radius of 1 . . The magnetic pick-up coils are mounted on the inner side of the vacuumchamber. Depending on the alignment these probes pick up different com-ponents of the magnetic field fluctuation. On AUG, so-called Mirnov coilsare placed on the vacuum vessel to measure the poloidal component of themagnetic field fluctuations ˜ B pol and so-called ballooning coils are placed closerto the plasma on the low field side to measure the radial component ˜ B r . Dueto their relative simplicity and compact size, several coils can be distributedover different locations within the plasma vessel, making them suitable forthe analysis of mode structure.The toroidal position of the ballooning coils is illustrated in figure 2.1awhere a top-down view of the tokamak is shown. These probes are placed inalmost identical poloidal positions which allows to investigate toroidal modenumbers. Furthermore, these coils are placed near the last closed flux surfaceto more efficiently detect the perturbations coming from the plasma. Theballooning coils are calibrated by taking into account the transfer functions ofthe different probes [25]. This way the systematic errors in the mode numberfitting are reduced which allows a better identification of mode numbers withreduced number of coils even for transient modes.19 PSLPSL z [ m ][ m ] R [m][m]0.0 1.0 2.0-2.0 -1.00.01.02.0-2.0-1.0
AUG (a) (b)
B31-13 B31-12 B31-01B31-03B31-02B31-14
Figure 2.1:
The position of magnetic probes used for the analysis on AUG. (a)
The six ballooning coils of the toroidal array are indicated withred in the top-down view of the tokamak. (b)
The 30 Mirnov coilsindicated with green in a poloidal cross section of the device. Probesindicated with empty rectangles were ruled out from the analysis,because the investigated modes were not visible in their signal. Thecyan rectangles indicate the passive stabilizing loops (PSLs).
In order to determine the poloidal mode number, the poloidal array ofMirnov coils was used. Mirnov coils are placed on the vacuum vessel, a littlefurther from the plasma and these are not calibrated. In turn, the 30 Mirnov-coils of the poloidal array constitute a full poloidal ring around the plasma.The position of these probes is shown in figure 2.1b in a poloidal cross sectionof AUG. Probes indicated with empty green rectangles are ruled out fromthe mode number analysis because the observed modes were hardly seen intheir signals. Most of the ruled out coils have a weak signal because these areplaced behind the divertor elements or the passive stabilizing loops (PSLs).Despite the difficulties concerning the use of Mirnov-coils, due to the highnumber of available probes, the poloidal mode number analysis gives accurateresults for low mode numbers.
The magnetic probes are very useful in mode identification and mode numberanalysis, but they cannot provide information on the radial structure of20lasma modes. For this purpose, an other fluctuation measurement waschosen: the soft X-ray diagnostic system. The major part of the radiationcoming from a hot fusion plasma is in the soft X-ray range (100 eV to10 keV). The energy spectrum of Soft X-ray radiation consists of a continuumradiation resulting from Bremsstrahlung, a recombination radiation resultingfrom free-bound transition, and a line radiation resulting from bound-boundtransition. In a plasma the light intensity from Bremsstrahlung radiation canbe estimated by the following formula [26]: P Br = 1 . · − · n e (cid:112) T e (cid:88) (cid:0) Z N ( Z ) (cid:1) , (2.1)where n e is the electron density in 1 / cm − , T e is the electron temperaturein eV and Z is the charge number of the given ionization state. The sum isexecuted on all ionization states and the radiation power is given in W/cm − .Eq. (2.1) tells us that the Bremsstrahlung emitted by a deuterium plasmais proportional to the square of the electron density and the square rootof the electron temperature. However, the soft X-ray radiation also involvesrecombination and line radiation. On AUG, the SXR detectors have a 75 µ m beryllium filter foil in front to suppress low energy photons. This way,photon energies below 1 keV are blocked by the beryllium and line radiationfrom light impurities is negligible for the Soft X-ray diagnostic. Thus, theyaffect the Soft X-ray radiation mainly via effective charge number Z eff . Onthe other hand, due to the full tungsten wall coverage of AUG, line radiationcan even exceed the continuum radiation seen in Soft X-ray diodes. From theanalysis point of view, it is sufficient to know that the measured radiation isa function of electron temperature and density.The SXR system of AUG consists of planar photo diode arrays withpinhole optics. A diode collects light from a thin cone shaped volume whichis referred to as the line-of-sight (LOS). AUG is equipped with 8 camerasconsisting about 200 LOSs [27]. During the analysis of chirping modes Iexamined the phenomenon on the majority of the LOSs that are shown infigure 2.2. Reflectrometry can give information on the density of the plasma. When anelectromagnetic wave of a certain frequency propagates through a plasmawith density increasing in the direction of propagation, it may arrive at apoint where the electron density equals the cut-off density [8]. At this pointthe wave is reflected and by detecting the reflected wave it is possible to use itto diagnose the plasma density and its fluctuation [21]. The phase shift of the21 .0 1.5 2.0 2.5-1.0-0.50.00.51.0 z [ m ] R [m]
AUG
HIJLM F&G K
Soft X-ray Line-of-sights
Figure 2.2:
The line-of-sights (LOSs) of the used soft X-ray channels in thepoloidal cross-section of AUG. Different colours indicate the differentcameras. reflected wave – which can be measured by using a microwave interferometer– gives information on the position of the cut-off layer.One operation mode of the diagnostic system is the profile measurement.By sweeping the frequency of the incident wave and recording the phase shiftof the reflected wave as a function of the frequency, a radial density profilecan be reconstructed. An other operation mode is the density fluctuationmeasurement which is carried out at a fixed frequency. From the timevariation of the phase shift, density fluctuations near the cut-off layer canbe determined. If the plasma is probed by several incident waves, the radiallocalization of the mode can be estimated. On AUG there are 4 channels forfluctuation measurements both on the low field side and the high field side.
To handle the rapidly changing mode frequency and the low signal-to-noiseratio of the observed EP-driven modes, an advanced amplitude reconstructionmethod was developed. This is based on Short Time Fourier Transform22STFT) which is a linear continuous time-frequency transform. It is idealto investigate the time evolution of transient wave-like phenomena. Themathematical background of linear continuous time-frequency transformsis presented in section 2.2.1 based on the book of Stephane Mallat: AWavelet Tour of Signal Processing [28]. Then, in section 2.2.2, the modenumber analysis method which is used to identify EP-driven modes is brieflyexplained. The introduction of analysis principles continues in section 3,where the amplitude reconstruction method developed in this thesis ispresented.
Linear continuous time-frequency transforms are calculated by expanding thesignal f ( t ) on the basis of families of so-called time-frequency atoms: T f ( u, ξ ) = (cid:104) f, g u,ξ (cid:105) = + ∞ (cid:90) −∞ f ( t ) g ∗ u,ξ ( t )d t, (2.2)where g u,ξ is a time-frequency atom, whose energy is well localized in bothtime and frequency. Variables u and ξ are the time and frequency indicesof the atom identifying its position on the time-frequency plane, and the ∗ represents the complex conjugation. The energy density distribution can thenbe calculated by taking the absolute value squared [28]: Ef ( u, ξ ) = | T f ( u, ξ ) | . (2.3) Ef ( u, ξ ) can be interpreted as energy density distribution on the time-frequency plane when the signal energy is defined as P = T (cid:90) | f ( t ) | d t. (2.4)The mode number analysis is based on the phase of the cross-transform whichis defined in the following way: ϕ kl ( u, ξ ) = arg { T f k ( u, ξ ) T f ∗ l ( u, ξ ) } , (2.5)where f k and f l represent the signals of probes placed in different positions.STFT is a type of linear continuous time-frequency transform when thefamily of time-frequency atoms are generated by shifting a real and symmetricwindow g ( t ) in time ( u ) and frequency ( ξ ): g u,ξ ( t ) = exp { iξt } g ( t − u ) , (2.6)23hich gives a uniform time-frequency resolution on the time-frequency plane.Figure 2.3 illustrates two atoms shifted in frequency ( ξ ). Following from time time Re (g(t)e i ξ t ) Re (g(t)e i t ) Figure 2.3:
Two time-frequency atoms shifted in frequency ( ξ ). eq. (2.2) STFT can be written as Sf ( u, ξ ) = (cid:104) f, g u,ξ (cid:105) = + ∞ (cid:90) −∞ f ( t ) exp {− iξt } g ( t − u )d t. (2.7)I applied Gabor atom [28] in the present thesis which means that g ( t ) is aGaussian function with a σ t standard deviation: g ( t − u ) = 1 (cid:112) √ πσ t exp (cid:26) − ( t − u ) σ t (cid:27) . (2.8)The energy density distribution defined in eq. (2.3) calculated from STFTis called a spectrogram, which is ideal to investigate the time-frequencyevolution of non-stationary modes [29, 30]. In order to apply a continuoustransform on sampled time signals, the calculation has to be discretizedin a way to avoid the degradation of the time-shift invariance property ofthe transform [31]. This is achieved by replacing the integrals with sumsand discretizing all variables in equation (2.7) with the smallest possiblesteps. The width (2 σ t in equation (2.8)) of the Gabor atom determinesthe time-frequency resolution of the transform. According to the specificapplication, the discretization can be done on a more sparse grid, whichis determined by a parameter called time step, but it has to be at most thequarter of the width of the atom to give a continuous transform. Since theGabor atom has an infinite support it has to be truncated [31]. To preservethe appealing properties of the continuous transform, only the part whereits values are several orders of magnitude smaller than the maximum areneglected. The resulting length of the window determines the frequency stepof the transform. In practice the transform is calculated in each time stepby fast Fourier transform (FFT) using Gaussian window having very longsupport as it is illustrated in figure 2.4.24 ttt short time Fourier transformTime signal FFT ∆ t Window Shifting in time
Figure 2.4:
The process of calculating STFT on discrete time signals. Thetransform is calculated in each time step by fast Fourier transform(FFT) using Gaussian window.
The evaluation of toroidal and poloidal mode numbers of the observed EP-driven modes were carried out by using signals of the magnetic pick-up coilsystem which is described in section 2.1.1. A brief introduction of the modenumber analysis method I used is given in this section, but a more detaileddescription can be found in our recently submitted paper [25].The measurement of magnetic field fluctuations produced by magnetohy-drodynamic (MHD) modes allows the reconstruction of their mode structure.Harmonics of global MHD eigenmodes are generally assumed in the following25orm [32]: A m,n ( ρ,θ ∗ ,φ,t ) = A m,n ( ρ,θ ∗ ) exp { i ( mθ ∗ + nφ − ωt ) } , (2.9)where A is the observable quantity (in our case the magnetic field perturba-tion) as a function of time t and the so-called straight field line coordinates( ρ,θ ∗ ,φ ) [33], A ( ρ,θ ∗ ) is the radial eigenfuntion and ω is the mode frequency.The mode structure of the mode is characterized by the m poloidal and n toroidal mode numbers. As follows from eq. (2.9), in the case of a single, puresinusoidal, globally coherent mode at fixed ( ρ,θ ∗ ,t ), but at different toroidalangles, the relative phase between A ( φ k ) and A ( φ l ) is proportional to therelative toroidal angle φ k − φ l of the probe locations and the ratio is thetoroidal mode number n : ϕ tor kl (cid:12)(cid:12) θk = θl = arg { A ( φ k ) } − arg { A ( φ l ) } = n ( φ k − φ l ) . (2.10)Similar formula can be constructed for the poloidal mode number at fixedtoroidal position: ϕ pol kl (cid:12)(cid:12) φk = φl = arg { A ( θ k ) } − arg { A ( θ l ) } = m ( θ k − θ l ) . (2.11)Due to its complexity, the θ ∗ transformation of the poloidal coordinates wasnot performed. This causes negligible error, because the investigated modesare core localized, thus the effect of the θ ∗ transformation would be small.Considering eq. (2.10) and (2.11) the mode number determination can behandled as a linear fitting problem. In order to evaluate the toroidal modenumber, one has to determine the phase of the mode in different toroidallocations ( φ i ), but at the same poloidal position. Eq. (2.10) shows that theslope of the linear curve, fitted on relative phases between all pairs of signalsas a function of the relative probe position, gives the toroidal mode number n .A similar argument stands for the poloidal mode number by consideringeq. (2.11). The cross-phases are evaluated from the cross-STFT which isdefined in eq. (2.5). The method searches for the integer mode number valuewhere the residual Q of the fit is minimal: Q = (cid:88) P (cid:107) ϕ P − nφ P (cid:107) , (2.12)where ϕ P is the phase between pairs of signals, φ P is the relative probeposition, n is the toroidal mode number, the sum is executed on the P signalpairs and (cid:107) . (cid:107) is the norm obtained by taking the optimum shift of ϕ P by2 πz , where z is an integer number. Eq. (2.12) can also be defined for poloidal26ode numbers. The sign of the mode number determines the direction ofpropagation in the device frame. If the helicity (which is determined by thedirection of the toroidal magnetic field and the plasma current) is known, thesign of the mode number can be related to the ion or electron diamagneticdrift direction. By performing the linear fit in each time-frequency point, atime-frequency resolved map of the mode numbers can be generated [34].Such mode number maps were evaluated to determine the mode numbers ofthe observed EP-driven modes in section 4.27 hapter 3Amplitude reconstruction ofchirping waves For the analysis of EP-driven modes STFT was chosen because it is ideal toexamine their time-frequency evolution. In order to investigate the radialstructure, the mode amplitude has to be determined on different SXRLOSs. STFT provides a good basis for the amplitude reconstruction, becausethe time-frequency resolution allows to eliminate the major part of thebackground noise and only the noise component interfering with the modeat the mode frequency causes uncertainty in the reconstruction. However,to handle the rapidly changing mode frequency and the low signal-to-noiseratio, I developed an advanced reconstruction method which is described inthis section.The derivation of the method is explained in section 3.1. A new approachfor interpolating STFT which is necessary for accurate reconstruction isdescribed in section 3.2. In section 3.3, the effect of background noise onthe reconstruction is discussed. The validation of the method is presented insection 3.4, where results on synthetic signals are shown. The linear chirpapproximation – which is finally applied to investigate chirping EP-drivenmodes – is investigated in more detail in section 3.5.
In order to perform the reconstruction, a signal model is needed. Thefluctuation on the soft X-ray signal caused by the observed mode is modelledas a frequency and amplitude modulated harmonic wave. The noise of themeasured signal is dominated by the soft X-ray light coming from thebackground plasma. The noise of the amplifier chain is negligible. The plasma28ackground noise is modelled as a Gaussian, additive, white noise z ( t ). Sincethe application of STFT serves as a narrow band filter, the white noiseapproximation only assumes that the spectral power of the noise is uniformin a narrow frequency range ( ∼
10 kHz). Thus, this is applicable to modelany kind of broadband noise. The sum of the chirping wave and the additivenoise takes the following form: f ( t ) = a ( t ) cos[ φ ( t )] + z ( t ) , (3.1)where a ( t ) is the instantaneous amplitude of the wave and the derivative ofthe phase φ ( t ) gives the instantaneous frequency (d φ/ d t = φ ( t ) (cid:48) ≡ ω ( t )).During the derivation of the amplitude reconstruction method, the cosinefunction in eq. (3.1) is substituted by exp { iφ ( t ) } . The results can be easilyconverted considering thatcos (cid:0) φ ( t ) (cid:1) = exp { iφ ( t ) } + exp {− iφ ( t ) } . (3.2)The reconstruction method is derived in the following way. In eq. (3.1), a ( t )and φ ( t ) are expanded in Taylor series around u : a T ,u ( t ) = ∞ (cid:88) p =0 a ( p ) ( u ) p ! ( t − u ) p , (3.3) φ T ,u ( t ) = ∞ (cid:88) p =0 φ ( p ) ( u ) p ! ( t − u ) p , (3.4) f T ,u ( t ) = a T ,u ( t ) cos[ φ T ,u ( t )] , (3.5)where ( p ) denotes the order of the derivation. Then, eq. (3.5) is substitutedinto the definition of STFT (eq. (2.7)). At this point noise z ( t ) is neglected.The effect of the noise on the results will be discussed in section 3.3. Byevaluating Sf T , u ( u, ξ ), a ( t ) can be expressed as a function of the STFTtransform and the time derivatives of the amplitude and frequency. Thisevaluation was carried out by Oberlin et al. [35, 36] in the first orderof the frequency. In the present thesis, a more careful treatment of theproblem is presented. In addition to the expansion of the frequency, thetime evolution of the amplitude is also taken into account by including itsTaylor expansion in eq. (3.5). Furthermore, the effect of higher order termsis also investigated. Substituting eq. (3.1) into the equation defining STFT29eq. (2.7)) and separating the zeroth order of the amplitude gives: Sf T ,u ( u, ξ ) = + ∞ (cid:90) −∞ a ( u ) exp { iφ T ,u ( t ) } exp {− iξt } g ( t − u )d t + (3.6)+ + ∞ (cid:90) −∞ ∞ (cid:88) p =1 a ( p ) ( u ) p ! ( t − u ) p exp { iφ T ,u ( t ) } exp {− iξt } g ( t − u )d t. By rearranging this equation, a ( u ) can be expressed as a function of theSTFT transform and the time derivatives of the amplitude and frequency: a ( u ) = (3.7)= Sf T ,u ( u, ξ ) − + ∞ (cid:90) −∞ ∞ (cid:88) p =1 a ( p ) ( u ) p ! ( t − u ) p exp { iφ T ,u ( t ) } exp {− iξt } g ( t − u )d t + ∞ (cid:90) −∞ exp { iφ T ,u ( t ) } exp {− iξt } g ( t − u )d t . Eq. (3.7) gives an exact formula to evaluate the instantaneous amplitudeif the background noise power is zero. However, this formula requires theknowledge of the instantaneous frequency and its derivatives ( φ ( p ) ( u )) and thederivatives of the amplitude ( a ( p ) ( u ), p ≥ a ( u ), because the knowledge of the amplitude is required tocalculate its derivatives. An estimation for the frequency and the amplitudecan be given considering eq. (3.6) in the zeroth order for the amplitude andin the first order for the phase: Sf T ,u ( u, ξ ) = + ∞ (cid:90) −∞ a ( u ) exp { iφ ( u ) + iφ (cid:48) ( u )( t − u ) } exp {− iξt } g ( t − u )d t. (3.8)This formula gives exact instantaneous amplitude in the case of constantamplitude and constant frequency waves. Considering the formula of the g ( t )Gaussian window function from eq. (2.8) the integral in eq. (3.8) can beanalytically evaluated: Sf T ,u ( u, ξ ) = a ( u ) (cid:113) σ t √ π exp { iφ ( u ) − iξu } exp (cid:26) − σ t φ (cid:48) ( u ) − ξ ) (cid:27) . (3.9)It is clear from eq. (3.9) that | Sf T ,u ( u, ξ ) | has its maximum where φ (cid:48) ( u ) = ξ .Therefore, the frequency of the mode can be estimated by searching for30he maxima on the spectrogram. It follows from eq. (3.9) that the modeamplitude can be evaluated from the maximum value of | Sf T ,u ( u, ξ ) | : a ( u ) = | Sf T ,u ( u, ξ = φ (cid:48) ( u )) | (cid:112) σ t √ π . (3.10)By using this approximation of the amplitude and frequency, the derivativesin eq. (3.7) can be estimated and a ( u ) can be evaluated numerically.In principle, the determination of p in eq. (3.7) above which the higherorder terms can be neglected, depends on the behaviour of the chirpingwave. The Gaussian window (eq. (2.8)) applied for STFT has a well-localizedenergy in time which time scale is characterized by the σ t standard deviation.Eq. (3.7) can give a good approximation with a limited p = p max if theremaining terms p < p max accurately describe the chirping wave on the timescale of the window width σ t . For example, if a linear chirp model (first orderin amplitude and second order in phase) was a good approximation of theobserved wave on the time scale of σ t , then the higher order terms would givenegligible contribution to the result.In summary, the amplitude reconstruction method – if higher order termsare taken into consideration – comprises the following steps: First, the instan-taneous frequency of the mode is estimated by using a maximum-searchingalgorithm. The time evolution of the maxima of the mode frequency istraced, which is called the ridge of the STFT transform. Second, by usingthe zeroth order approximation (eq. (3.10)) the instantaneous amplitude isestimated from the STFT values of the ridge. Then, the derivatives of theamplitude ( a ( p ) ( u )) and frequency ( φ ( p ) ( u )) are evaluated from the zerothorder approximation. Finally, the instantaneous amplitude is approximatedby calculating eq. (3.7). In order to avoid the effect of quantization error inthe numerical derivatives of the amplitude and frequency, an interpolation ofSTFT is required. The interpolation method that I used in this thesis could have a broaderapplicability than a simple interpolation algorithm. With this technique thereconstruction of the continuous STFT of a signal at any time-frequencypoint is possible. This method is based on a theorem introduced in the book ofMallat [28]. Let Ψ be an element of square integrable functions (Ψ ∈ L ( R )).31here exists f ∈ L ( R ) such that Ψ( u,ξ ) = Sf ( u,ξ ), if and only if,Ψ( u , ξ ) = 12 π ∞ (cid:90) −∞ ∞ (cid:90) −∞ Ψ( u, ξ ) K ( u , u, ξ , ξ )d u d ξ, (3.11)where K ( u , u, ξ , ξ ) = (cid:104) g u,ξ , g u ,ξ (cid:105) (3.12)is the reproducing kernel. This theorem shows that not any two dimensionaldistribution can be interpreted as the STFT of some f function. If thediscretization of the STFT is handled carefully as it is described insection 2.2.1, then the integral in eq. (3.11) can be well approximated by thediscrete STFT. In this way the continuous transform can be reconstructedat any given time-frequency point.I searched for the local maxima of the absolute value of the STFT ateach time-point by interpolating the transform in frequency. The method isillustrated in figure 3.1. The red circles show the values of the discrete STFTand the grey solid line presents the continuous transform. The transform isinterpolated in the frequency points halfway between the red points. Thisgives the cyan points which are the results of the first iteration of theinterpolation. Then, the highest value from the cyan points is selected. Afterthat the transform is interpolated again halfway between the neighbouringcyan points. This gives the green points which determine a more accurateestimation for the STFT ridge. In practice, this iteration was repeated 10 − The derivation presented in section 3.1 is valid if there is no additional noisein the signal. Thus, the effect of the noise has to be investigated. As it wasmentioned before, my signal model (eq. (3.1)) contains a Gaussian, additive,white noise: f ( t ) = a ( t ) cos[ φ ( t )] + z ( t ) = f chirp ( t ) + z ( t ) . (3.13)Since STFT is a linear transform: S (cid:0) f chirp ( t ) + z ( t ) (cid:1) = Sf chirp ( t ) + Sz ( t ) . (3.14)To illustrate the problem, the transform of the terms in eq. (3.14) arecalculated separately, then the behaviour of their sum is investigated on32 requency | S T F T | Maximum
Discrete STFT1 st iteration2 nd iteration Figure 3.1:
Illustration of the interpolation method which was used to accuratelyevaluate the STFT ridge. the complex plane. At a given time-frequency point ( u, ξ ) the value of thetransform of f chirp ( t ) gives a complex number whose amplitude and phaseare deterministic. This complex number is plotted as a white vector on thecomplex plane in figure 3.2a. The STFT of the Gaussian, additive, whitenoise z ( t ) at a given time-frequency point ( u, ξ ) is a complex number whoseamplitude is proportional to the noise power, and its phase is an uniformlydistributed random number. The distribution of this vector is illustratedwith the pink spot on the complex plane in figure 3.2a centred around thedeterministic component.Since the STFT of the Gaussian white noise has a 2D Gaussiandistribution on the complex plane, the magnitude of the sum of the twovectors has a Rice distribution [37]. The Rice or Rician distribution isthe probability distribution of the magnitude of a circular bivariate normalrandom variable with potentially non-zero mean [38]. Its probability densityfunction is the following: f ( x | ν, s ) = xs exp (cid:26) − ( x + ν )2 s (cid:27) I (cid:18) xνs (cid:19) , (3.15)where ν is the distance of the centre of the 2D Gaussian distribution from theorigin, s is the standard deviation of the 2D Gaussian distribution and I ( z ) isthe modified Bessel function of the first kind with order zero. The probabilitydensity of the Rice distribution with parameters ν = 5 and s = 3 is shownin figure 3.2b. The mean of a random variable x with Rice distribution isE[ x ] = s (cid:112) π/ L / ( − ν / s ) (3.16)33 f - P r o b a b i l i t y d e n s i t y Im Re x f (x | ν, s)ν = 5s = 3 (b)(a) Figure 3.2: (a)
This figure illustrates the sum of the STFT transform of adeterministic wave (white vector with length of ν = 5) and aGaussian, additive, white noise (pink spot with standard deviation σ = 3) at a given time-frequency point ( u, ξ ). (b) The probabilitydensity of the Rice distribution with parameters ν = 5 and s = 3. and its variance is Var[ x ] = 2 s + ν − πs L / ( − ν / s ) , (3.17)where L q ( x ) denotes a Laguerre polynomial.The background noise level, i.e. the s of the 2D Gaussian distribution isdetermined in the following way. A time-frequency interval is defined on thespectrogram of the given signal where no coherent mode appears. Since it isassumed that only the Gaussian white noise gives significant contribution tothe values over the selected time-frequency interval, the real or imaginary partof these values has a Gaussian distribution with standard deviation s , where s is the standard deviation of the 2D Gaussian distribution of the noise. Thus,the background noise level is estimated by calculating the standard deviationof the real part of the STFT values over the selected time-frequency interval.As ν becomes large or s becomes small the mean of the Rice distributionbecomes ν and the variance becomes s [39]. In the investigated real cases,the ratio of ν and s is around 10 where the oscillation amplitude is maximal.Therefore, in the present thesis, I use the ν (cid:29) s limit which gives a goodapproximation to estimate the uncertainty of the reconstructed amplitude.In figures showing the reconstructed amplitude in section 4, the error barswere derived from the standard deviation s of the background noise by takinginto account the error propagation in the reconstruction formula.34 .4 Numerical tests I performed numerical tests to investigate the effect of the higher order termsand the background noise on the amplitude reconstruction. These tests werecarried out on synthetic signals which had similar frequency and amplitudetime evolution as the measured signals. The amplitude and frequency of thegenerated signals presented in this thesis were the following: ω ( t ) = 2 π · (cid:0) exp {− − t } + 7 (cid:1) [Hz] , (3.18) a ( t ) = − exp { · t } + 10 . (3.19)First, the tests were carried out without adding noise to the signal toinvestigate the effect of higher order terms in the amplitude reconstruction.In figure 3.3a the pre-defined amplitude of the synthetic chirp is shown withred solid line. The other lines show the reconstruction taking into accountincreasing number of higher order terms. Maximum order corresponds tothe highest derivative of the amplitude and frequency taken into account.Figure 3.3a shows that the first order approximation gives a better estimationof the pre-defined amplitude evolution. However, the second order terms donot give significant contribution to the results. Real amp.0 th order1 st order2 nd order W/o noise With noiseAmplitude
Time [s] A m p l i t u d e [ a . u . ] Amplitude
Time [s] A m p l i t u d e [ a . u . ] Figure 3.3: (a)
The result of amplitude reconstruction taking into account higherorder terms for a synthetic signal without noise. (b)
Amplitudereconstruction for synthetic signal with additive Gaussian whitenoise.
In figure 3.3b the effect of additive noise on the reconstruction isexamined. In this case an additive Gaussian white noise was added to thesynthetic signal. The signal-to-noise ratio had similar magnitude as in theinvestigated real measurements. For the synthetic signal, this means that35he amplitude of the wave was around 8 and the standard deviation of theadditive Gaussian noise was set to be 15. These tests had already shown thatthe STFT based method effectively filters out the noise and can deal withthis very poor signal-to-noise ratio.The noise causes higher uncertainty in the zeroth order estimation ofthe amplitude and frequency. This uncertainty leads to high errors whenthe derivatives are calculated. As it is visible in figure 3.3b, the first orderapproximation gives a better estimation for the pre-defined amplitude thanthe zeroth order approximation. However, the effect of the noise is moresignificant in the case of the second order approximation due to the higherorder derivatives.The results of tests on synthetic signals suggest that the first orderapproximation estimates well the amplitude of the synthetic chirp. This is atrade-off between the magnitude of the correction given by the higher orderterms and the uncertainty arising from the derivatives of the noisy zerothorder approximation of the amplitude and the frequency. Since the tests haveshown that the first order approximation is sufficient for my purposes, thisis investigated in more detail.
Eq. (3.6) in the first order for the amplitude and second order for the phasegives Sf T ,u ( u, ξ ) = (3.20)= + ∞ (cid:90) −∞ (cid:0) a ( u ) + a (cid:48) ( u )( t − u ) (cid:1) exp (cid:110) i (cid:16) φ ( u ) + φ (cid:48) ( u )( t − u ) + φ (cid:48)(cid:48) ( u )2 ( t − u ) (cid:17)(cid:111) exp {− iξt } (cid:112) √ πσ t exp (cid:26) − ( t − u ) σ t (cid:27) d t. Even in case of a linear chirp, the absolute value of the transform has itsmaximum where φ (cid:48) ( u ) = ξ [35]. Evaluating eq. (3.20) and taking into account36hat φ (cid:48) ( u ) = ξ yields Sf T ,u ( u, ξ ) = 1 (cid:112) √ πσ t exp (cid:110) i (cid:16) φ ( u ) − ξu (cid:17)(cid:111) (3.21) (cid:34) a ( u ) + ∞ (cid:90) −∞ exp (cid:26)(cid:18) iφ (cid:48)(cid:48) ( u )2 − σ t (cid:19) ( t − u ) (cid:27) d t ++ a (cid:48) ( u ) + ∞ (cid:90) −∞ ( t − u ) exp (cid:26)(cid:18) iφ (cid:48)(cid:48) ( u )2 − σ t (cid:19) ( t − u ) (cid:27) d t (cid:35) . By performing variable substitution in the integral of the second term for t − u , it is clearly visible that this integral contains the product of an oddand an even function. Thus, the second term equals to zero which shows thatthe a (cid:48) ( u ) does not give contribution to the result. The first integral can beevaluated analytically, thus a ( t ) can be expressed as a ( u ) = (cid:115) − iφ (cid:48)(cid:48) ( u ) σ t σ t √ π exp {− i ( φ ( u ) − ξu ) } Sf T ,u ( u, ξ ) . (3.22)Taking into consideration that a ( u ) is a real number, its value is given by theabsolute value of eq. (3.22): a ( u ) = | Sf T ,u ( u, ξ ) | (cid:112) σ t √ π (cid:113) φ (cid:48)(cid:48) ( u ) σ t . (3.23)With this formula the amplitude of the wave can be reconstructed whenthe linear chirp is a good approximation on the time scale of σ t . Thisformula differs from the zeroth order approximation (eq. (3.10)) in thefollowing correction factor: (cid:112) φ (cid:48)(cid:48) ( u ) σ t . This correction depends on thederivative of the frequency and the width of the applied Gaussian windowwhich determines the time-frequency resolution of STFT. It is clear thatthe correction factor is lower if the window width is smaller, i.e whenthe time resolution of the transform is better. However, the window widthcannot be arbitrary small, because the better the time-resolution the worsethe frequency resolution. Therefore, a trade-off between the time- andfrequency resolution is needed. In a previous work I used the zeroth orderapproximation, but the time-frequency resolution was set to minimize theeffect of the neglected correction [40]. In this thesis I used the first orderapproximation, because it allows to apply a wider window function which37rovides better frequency resolution, thus more accurate reconstruction ofthe STFT ridge.Figure 3.4 shows the effect of the time-frequency resolution on thereconstruction. To confirm that the linear chirp approximation gives thebest estimation for the chirping modes investigated in real signals, in thiscase I reconstructed the amplitude of a real oscillation (BAE) with differenttime-frequency resolution. In figure 3.4a, the zeroth order approximation Time-frequencyresolution(2 σ width of theGaussian in time)0,25 ms Amplitude
Time [s] A m p l i t u d e [ a . u . ] th orderapproximation 1 st orderapproximation Amplitude
Time [s] A m p l i t u d e [ a . u . ] AUG (a) (b)
Figure 3.4: (a)
The result of amplitude reconstruction by using the zerothorder approximation (eq. (3.10)) on a real signal. (b)
The resultof amplitude reconstruction by using linear chirp approximation(eq. (3.23)) on a real signal. (eq. (3.10)) is used. As it is visible, the reconstructed amplitude depends onthe time-frequency resolution. Figure 3.4b shows the case of reconstructionwith the linear chirp approximation (eq. (3.23)). In this case the time-frequency resolution has a much weaker effect on the reconstruction.As it was discussed in this section, with an accurate signal model, theamplitude of rapidly chirping waves can be reconstructed. The effect ofadditive noise is handled separately by taking advantage of the linearity ofthe STFT. Numerical tests had shown that the linear chirp approximationgives a good approximation for the amplitude of chirping waves which havesimilar parameters as the ones observed in the measurement. Therefore, inthe following section, where the experimental observations are discussed, thefirst order approximation (eq. (3.23)) is applied.38 hapter 4Experimental observation ofEP-driven modes
The importance of EP-driven modes regarding future burning plasmaexperiments was discussed in section 1. Present section shows a detailedanalysis to experimentally characterize the behaviour of EP-driven plasmamodes. The diagnostic tools presented in section 2 and data processingmethods developed in section 3 allow to examine the rapid changes in theradial structure of bursting EP-driven modes during the non-linear chirpingphase.The experiments in question were carried out on ASDEX Upgrade (AUG).The scenario in which BAEs and EGAMs were observed was characterizedby low density ( ∼ · m − ) in order to maximize the fast particle pressurefraction. The modes were observed in the current ramp-up phase of thedischarge when off-axis NBI with 93 keV beam energy was applied. Theramp-up phase in a typical AUG discharge lasts about 1 sec. A series ofdischarges with strong Alfv´en activity were investigated. The list of analyseddischarges with the approximate time and frequency intervals where BAEsand EGAMs were observed is presented in table 4.1. The mode type wasidentified mainly by considering the toroidal mode number. EGAMs havea toroidal mode number n = 0 and BAEs have a nonzero toroidal modenumber.The main goal of this thesis is to investigate the changes in the radialstructure of the mode in the non-linear chirping phase. The only fluctuationmeasurement which has spatial resolution and these modes were visible in itssignal was the SXR diagnostic. Since the SXR diagnostic is a line-integratedmeasurement, it is not straightforward to reconstruct the radial structureof the mode, but many line-of-sights (LOSs) are available which makes itpossible to qualitatively investigate the time evolution of the radial structure.39 AEs EGAMsTime [s] Frequency [kHz] Time [s] Frequency [kHz] × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
Table 4.1:
Investigated shots with strong Alfv´en activity. The approximate timeand frequency intervals where BAEs and EGAMs were observed arepresented.
The identification of the modes was carried out by using the magneticpick-up coils. Magnetic spectrograms are excellent to follow the timeevolution of the modes and determine mode numbers. Note that in principleEGAMs should only be detectable with density fluctuation measurementsbecause it is and electrostatic mode, however, due to sideband coupling it isclearly visible on magnetic fluctuation measurements as well [9].In general, the signal-to-noise ratio of the SXR measurements was poor.In many cases, modes which were clearly visible on the magnetic spectrogramwere only visible on one or two LOSs of SXR. The strategy was to find caseswhere the mode is observable on at least 3 adjacent LOSs of a particularSXR camera. LOSs of SXR cameras F, G, H, I and J (see section 2.1.2)were investigated in the discharges listed in table 4.1 by calculating thespectrogram of their signal in the time-frequency range where the mode wasvisible on the magnetic spectrogram. In total, about 1000 SXR spectrogramswere analysed in this process. Finally, 3 cases for BAEs and 5 cases forEGAMs were found where the oscillation amplitude on the SXR signals wassufficiently high to identify them visually on the SXR spectrograms.40 .1 Beta-induced Alfv´en eigenmodes
First, the results from the observation of beta-induced Alfv´en eigenmodes(BAEs) are presented. As it is shown in table 4.1, BAEs were observed inseven discharges. However, the signal-to-noise ratio of the SXR signals wasonly appropriate for further analysis in shot t = 0 .
35 s), then off-axis NBI was injectedat t = 0 . Magnetic spectrogram F r e q u e n c y [ k H z ] E n e r g y d e n s i t y [ a . u . ] AUG
Figure 4.1:
Chirping BAEs with decreasing frequency in the range of 70 − An important step in mode identification is determining the toroidal andpoloidal mode numbers. The toroidal mode number was determined fromthe signals with correction [25] of the ballooning coils as it was describedin section 2.2.2. The result of the toroidal mode number evaluation ispresented in figure 4.2. In figure 4.2a, the time-frequency resolved modenumber calculation is shown where it is visible that BAEs have n = 1 toroidal41ode number. The mode numbers are plotted only in time-frequency pointswhere the residual of the fit is lower than the 5 % of the maximum. Thisfilter ensures that only well-fitting mode numbers are taken into account. Infigure 4.2b, the relative phases between all pairs of signals are plotted as afunction of relative probe position in the time-frequency point marked witha red star in figure 4.2a. The best fitting line with n = 1 slope is plotted withblack solid line. π ]-1.0-0.50.00.51.0 C r o ss - p h a s e s [ π ] F r e q u e n c y [ k H z ] -6-4-20246 T o r o i d a l m o d e n u m b e r AUG
Toroidal mode numbers Toroidal mode number fit (a) (b)
Figure 4.2:
Toroidal mode number calculation for BAEs. (a)
The result of time-frequency resolved mode number calculation. The mode numbers areplotted only in time-frequency points where the residual of the fit islower than 5 % of the maximum. (b)
The relative phases between allpairs of signals are plotted as a function of relative probe position inthe time-frequency point marked with a red star in figure 4.2a. Thebest fitting line with n = 1 slope is plotted with light green solid line. In order to determine the poloidal mode number, the poloidal array ofMirnov coils was used. Despite the difficulties concerning the use of Mirnov-coils (see section 2.1.1), the results presented in figure 4.3 clearly show thatthe poloidal mode number of the investigated BAEs is m = −
3. The negativesign means that the mode propagates in the ion diamagnetic drift direction.Since the Mirnov coils are not calibrated, in figure 4.3a a different filteris applied than in the case of toroidal mode numbers: the mode numbersare plotted only in time-frequency points where the value of the minimumcoherence [41] is higher than 0.3 for number of averages of 3. Figure 4.3bshows the relative phases between all pairs of signals as a function of relativeprobe position. 42 .0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0 C r o ss - p h a s e s [ π ] Poloidal mode number fit -6-4-20246 T o r o i d a l m o d e n u m b e r F r e q u e n c y [ k H z ] AUG
Poloidal mode numbers
Relative probe positions [ π ]Time [s] (a) (b)m = -3 Figure 4.3:
Poloidal mode number calculation for BAEs. (a)
The result of time-frequency resolved mode number calculation. The mode numbers areplotted only in time-frequency points where the value of the minimumcoherence is higher than 0.3. (b)
The relative phases between all pairsof signals are plotted as a function of relative probe position in thetime-frequency point marked with a red star in figure 4.3a. The bestfitting line with n = − The radial structure analysis of BAEs was carried out by using the LOSs ofSXR camera J. These LOSs are shown in figure 4.4b with red lines. Basedon the signal-to-noise ratio of the SXR signals three chirps were selectedfor further analysis. The soft X-ray spectrogram of channel J54 is shownin figure 4.4a. The instantaneous amplitude of the mode on each LOS iscalculated by using the first order approximation defined in eq. (3.23). Sinceeq. (3.23) contains the derivative of the instantaneous frequency, the precisedetermination of the instantaneous amplitude requires an accurate frequencyridge. Therefore, the frequency ridge was evaluated from the magneticspectrogram, because the magnetic signals have a higher signal-to-noise ratio.The ridge was traced by a local maximum searching algorithm [40, 42],which follows the mode evolution until the mode amplitude falls below thebackground noise level.The time evolution of the instantaneous amplitude of chirp . .63 0.64 0.65 0.66Time [s]708090100 F r e q u e n c y [ k H z ] SXR spectrogram E n e r g y d e n s i t y [ a . u . ] AUG z [ m ] R [m]
AUG
Soft X-ray Line-of-sights (a) (b)
Figure 4.4: (a)
Spectrogram calculated from the signal of SXR LOS J-54. Thedownchirping BAEs are also visible here similarly to the magneticspectrogram (figure 4.1, note the different time axis.). (b)
Six LOSsof SXR camera J (red lines) where the signal-to-noise ratio wassufficiently high for further analysis.
Smoothingwindow: 0.5 msJ50J51J52J53J54J55
SXR LOSs
AUG (a) (b)
Reconstructed amplitudes A m p l i t u d e s [ a . u . ] Smoothed amplitudes A m p l i t u d e s [ a . u . ] Figure 4.5:
The time evolution of the oscillation amplitude of BAE chirp (a)
Reconstructed amplitudesby using linear chirp approximation (eq. (3.23)). (b)
Amplitudessmoothed by a moving average with boxcar kernel of 0 . mode amplitude is calculated for each LOS and it is plotted as the function ofthe radial coordinate. This way the radial mapping of the mode amplitudescan be evaluated at any time instant. Since this work focuses on the relativechanges in the radial eigenfunctions of the modes, each curve of the radial44ap was normalized with its integral. This radial map for chirp Radial mapping of mode amplitude -0.3 -0.2 -0.1 0 0.1 0.2 0.30.00.51.01.5 N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
BAE (a)
AUG
J55 J54 J53 J52 J51 J50
BAE (b)
AUG
Radial mapping of mode amplitude -0.3 -0.2 -0.1 0 0.1 0.2 0.3Normalized radial coordinate N o r m a l i z e d a m p l i t u d e [ a . u . ] Radial mapping of mode amplitude -0.3 -0.2 -0.1 0 0.1 0.2 0.3 N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J55 J54 J53 J52 J51 J50
BAE (c)
AUG
Figure 4.6:
The radial mapping of the oscillation amplitude of BAEs. The resultsdo not show significant changes in the radial distribution of the modeamplitudes. .2 Energetic particle driven GAMs EGAMs were observed in the same (or similar) discharges (see table 4.1),but in different time-frequency intervals as BAEs. I found 5 cases wherethe signal-to-noise ratio was appropriate for further analysis. The strongestEGAMs were observed in discharge . ◦ respect to the horizontal axis). In thisdischarge 3 consecutive EGAMs were found where the signal-to-noise ratioof the SXR signals was sufficient. The evaluated fast ion distribution had apeak at ρ pol ≈ .
45 according to the fast ion D-alpha (FIDA) spectroscopymeasurements [43, 44]. ρ pol = Ψ / Ψ is the normalized poloidal flux whichacts as a flux label in this case. For mode identification and mode number analysis again the magnetic pick-up coils were used. The time evolution of the mode frequency was traced usingthe same ridge-following algorithm as for BAEs. The magnetic spectrogramfrom discharge
Magnetic spectrogram F r e q u e n c y [ k H z ] E n e r g y d e n s i t y [ a . u . ] EGAMs
Figure 4.7:
Chirping EGAMs with increasing frequency in the range of 45 -60 kHz are visible on the magnetic spectrogram. Numbers denotethe chirps which were investigated in detail. n = 0toroidal and m = − Time [s] F r e q u e n c y [ k H z ] -6-4-20246 T o r o i d a l m o d e n u m b e r Toroidal mode numbers
AUG (a)
Time [s] F r e q u e n c y [ k H z ] -6-4-20246 P o l o i d a l m o d e n u m b e r Poloidal mode numbers
Mirnov coils (b)n = 0 m = -2
Figure 4.8:
The result of time-frequency resolved mode number calculation. (a)
The toroidal mode numbers are plotted only in time-frequencypoints where the residual of the fit is lower than the 3 % of themaximum. (b)
The poloidal mode numbers are plotted only intime-frequency points where the value of the minimum coherenceis higher than 0.3.
The next issue to be resolved is to determine the radial location of themode. One estimation of the radial location can be given from the SXRmeasurements. Starting from the plasma core, the first SXR LOS on whichthe mode is not visible gives an outer boundary of the mode location.Considering the SXR spectrograms the investigated EGAMs are locatedinside the ρ pol = 0 . .
71, 1 .
35, 1 . . · / m . The detected modes are visible on the spectrograms ofchannels with frequencies 33 and 36 GHz. According to the density profilethis means that the mode is located around ρ pol ∼ . − .
45. Both resultsare consistent with the FIDA measurement from which it is expected that thestrongest drive occurs closer to the core than where the fast ion distributionpeaks ( ρ peak ∼ . .2.2 Changes in the radial structure of EGAMs The time evolution of the radial structure of EGAMs was investigated inthe same way as it was done for BAEs. Again, the LOSs of SXR cameraJ were chosen. The selected 6 LOSs where the signal-to-noise ratio wasappropriate are shown in figure 4.9b in a poloidal cross-section of AUG. Thesoft X-ray spectrogram of channel J54 with 3 consecutive EGAMs is shownin figure 4.9a.
Time [s] F r e q u e n c y [ k H z ] SXR spectrogram E n e r g y d e n s i t y [ a . u . ] z [ m ] R [m]
AUG
Soft X-ray Line-of-sights (b)
Figure 4.9: (a)
Soft X-ray spectrogram with 3 consecutive EGAMs. (b)
Theposition of the selected 6 SXR LOSs in a poloidal cross-section ofAUG. These channels were used for the radial structure analysis ofEGAMs.
The amplitude of the mode is evaluated for all LOSs by using theformula defined in eq. (3.23). The reconstructed amplitudes of chirp .
25 ms width. The uncertainty of theresult is indicated by the dashed lines.It is already visible in figure 4.10 that the time evolution of the amplitudeis different on the different channels. Since the SXR measurement is not welllocalized, the exact changes in the radial eigenfunction cannot be tracked,but its behaviour can be qualitatively described. The amplitudes shown infigure 4.10a are normalized to the amplitude at the beginning of the chirp forfurther analysis. The result is shown on figure 4.10b. In order to investigatethe changes of the radial structure, the radial mapping was evaluated, i.e.48 moothingwindow: 1.25 msJ49J50J51J54J55J56
SXR LOSs
Smoothed amplitudes
Time [s] A m p l i t u d e s [ a . u . ] Normalized amplitudes
Time [s] A m p l i t u d e s [ a . u . ] AUG (a) (b)
Figure 4.10: (a)
The time evolution of the oscillation amplitude of chirpingEGAM .
25 mswidth. (b)
The amplitudes shown in figure 4.10a are normalized tothe amplitude at the beginning of the chirp (indicated with dashedcyan line) for further analysis. the normalized amplitudes are plotted as a function of the radial position ofthe channels in different time points.The LOSs above and under the magnetic axis are handled separately andthe time evolution of the radial mappings are shown in figure 4.11. On LOSslocated above the magnetic axis (figure 4.11a-e), a shrinkage of the mode isvisible, since as time evolves the relative amplitude in the middle channel isrising compared to the outer channels. One example for the radial mappingcalculated from channels under the magnetic axis is presented in figure 4.11f.Regarding all 5 cases, radial mappings calculated from channels under themagnetic axis do not show any significant change. This is most probably dueto the longer distance from the observation point to the mode of these LOSsas it is visible in figure 4.9b. The LOSs located under the magnetic axisobserve a bigger volume, detecting more light from the background plasmaand relatively less from the flux surfaces where the mode is. However, theshrinkage of the mode is visible in radial mappings calculated from channelsabove the magnetic axis. As it is visible in figure 4.11a-e, this shrinkage wassignificant in all 5 investigated cases.49 .10 0.15 0.20 0.250.00.20.40.60.81.01.2
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J51 J50 J49
EGAM (c)
AUG +0.4 ms (54 kHz)+1.1 ms (56 kHz)+2.6 ms (58 kHz)+0.0 ms (52 kHz)0.8323 s
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J51 J50 J49
EGAM (a)
AUG +0.4 ms (54 kHz)+1.0 ms (56 kHz)+2.1 ms (58 kHz)+0.0 ms (52 kHz)0.8219 s
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J51 J50 J49
EGAM (b)
AUG +0.2 ms (62 kHz)+0.4 ms (64 kHz)+0.8 ms (66 kHz)+0.0 ms (60 kHz)0.5095 s
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J56 J55 J54
EGAM (f)
AUG +0.4 ms (54 kHz)+1.1 ms (56 kHz)+2.6 ms (58 kHz)+0.0 ms (52 kHz)0.8323 s -0.30 -0.25 -0.20 -0.15 -0.100.00.20.40.60.81.01.20.10 0.15 0.20 0.250.00.51.01.5
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J51 J50 J49
EGAM (e)
AUG +0.5 ms (54 kHz)+1.2 ms (56 kHz)+3.9 ms (58 kHz)+0.0 ms (52 kHz)0.8423 s
Radial mapping of mode amplitude N o r m a l i z e d a m p l i t u d e [ a . u . ] Normalized radial coordinate
J51 J50 J49
EGAM (d)
AUG +0.4 ms (64 kHz)+0.8 ms (68 kHz)+1.2 ms (72 kHz)+0.0 ms (60 kHz)0.8386 s
Above magnetic axis
Above magnetic axis
Above magnetic axis
Under magnetic axis
Above magnetic axis
Above magnetic axis
Figure 4.11:
The radial mappings of the oscillation amplitude of EGAMs. (a-e) above the magnetic axis. Shrinkage of the mode is visible in all cases. (f )
One case showing the radial mapping from SXR LOSs located under the magnetic. Radial mappings calculated from channelsunder the magnetic axis do not show any significant change. .2.3 Discussion In the following section the theoretical explanation of the experimental resultsare presented. The simulation results shown in this part are provided byPhilipp Lauber [19]. The observed shrinkage in the mode structure of EGAMsduring the non-linear chirping phase is consistent with our present theoreticalunderstanding. The mode coupled to the EGAM which was measured by themagnetic pick-up coils has a poloidal mode number m = −
2. However, theEGAM itself has a poloidal mode number m = 0. The resonance conditionto drive EGAMs simply follows from eq. (1.8) if n = 0, m = 0 and l = − ω − ω t = 0 , (4.1)where ω is the mode frequency and ω t is the transit frequency of theinteracting particles. This EGAM resonance condition shows that the modefrequency is equal to the transit frequency of the interacting ions.The exact phase space and real space coordinates of the wave-particleinteraction are determined by several factors. First, the EGAM drive isproportional to the velocity phase space gradient in the EP distributionfunction. Second, the strength of the drive is proportional to the EP density.In addition, the spatial dependence of the damping is an important factor.Since in case of EGAMs, the main damping mechanism is ion Landaudamping [16], the damping decreases with temperature. Thus, the dampingincreases towards the plasma center while the EP density – in the discharge ρ ≈ .
25, thus the EP distribution was investigated at this position.The EP distribution at ρ ≈ .
25 was calculated by the TRANSPsimulation code [19, 45] at the time instance when EGAMs appeared. The EPdensity ( F (Λ , E )) times its derivative ( ∂F/∂ Λ) is plotted in figure 4.12 as afunction of the pitch angle (Λ) and energy ( E ). The EP distribution can excitethe mode when the drive overcomes the damping. This can be reached wherethe drive is highest, i.e. at the peak of the F · ∂F/∂ Λ function which is around(Λ = 0 . , E = 40 keV) this case as it is shown in figure 4.12. The coordinatesof the peak of the F · ∂F/∂ Λ function determine the parameters of interactingparticles which are passing particles. The coloured curves in figure 4.12 showthe (Λ , E ) coordinates of particles with different transit frequencies (38, 50and 61 kHz). The curves corresponding to higher frequencies move from thegreen curve (38 kHz) towards the pink curve (61 kHz) as it is shown by thewhite arrow. 51
0 20 40 60 80 0.3 0.4 0.5 0.6 0.7 0.8 0.9Energy [keV] P i t c h a n g l e ( Λ ) Energetic particle distribution
Figure 4.12:
The contour plot shows the EP density ( F (Λ , E )) times its deriva-tive ( ∂F/∂ Λ) as a function of the pitch angle (Λ) and energy ( E )at ρ ≈ .
25 in discharge t = 0 .
83 s, where EGAMswere observed. The coloured curves show the (Λ , E ) coordinates ofparticles with different transit frequencies (38, 50 and 61 kHz). (Thisfigure was created from data provided by Philipp Lauber [19].)
The mode frequency of the observed EGAMs starts at around 45 kHz andincreases until 60 kHz as it is visible in figure 4.7. The curve correspondingto the initial 45 kHz frequency would intersect the region of the maximumpeak in figure 4.12. This means that - considering eq. (4.1) - the observedmode frequency is consistent with the simulated EP distribution functionand the velocity phase space coordinates of the interacting particles areapproximately (Λ = 0 . , E = 40 keV). The radial coordinate ρ ≈ . , E ) determine the orbit of theinteracting particles. The orbit of EPs is illustrated in figure 4.13 where theradial coordinate of the particle is plotted as the function of the normalizedcirculation time. The green curve corresponds to the particles having ( ρ =0 . , Λ = 0 . , E = 30 keV) coordinates and it shows that these interactingparticles have a ∆ ρ ≈ . − .
17 = 0 .
17 orbit width. The mode structure ofEGAMs is determined by the parameters of the drive, i.e. the orbit width ofthe interacting particles corresponds to the mode extent.The mode frequency increases until 60 kHz as it is visible in figure 4.7.This means that as time evolves, the resonance condition changes and52 .2 0.4 0.60.0 0.8 1.00.200.250.300.35 (Λ = 0.60, E = 30 keV)(Λ = 0.25, E = 50 keV)Normalized transit time
Orbit width of interacting particles N o r m a l i z e d r a d i a l c oo r d i n a t e Figure 4.13:
The orbit of EPs with different velocity phase space coordinates(Λ , E ). The radial coordinate of the particle is plotted as thefunction of the normalized circulation time. (This figure was createdfrom data provided by Philipp Lauber [19].) shifts to particles with higher transit frequency. Figure 4.12 shows only theequilibrium EP distribution, the evolution of the EP density due to thenon-linear interaction with fast ion cannot be followed. However, a qualitativedescription can be given. The pink curve in figure 4.12 shows the phase spacecoordinates of particles which have 61 kHz transit frequency. During the timeevolution of the mode the resonance condition moves from (Λ = 0 . , E = 40keV) toward the pink curve. This means that the resonance condition movesto the region of passing particles (particles with smaller Λ). The exacttime evolution of the resonance condition cannot be determined from theequilibrium EP distribution, but most probably the resonance follows theridge in the F · ∂F/∂ Λ function indicated with the white arrow in figure 4.12.More passing particles have a narrower orbit width which can explain theexperimentally observed shrinkage of the radial mode structure. The orbit ofEPs with coordinates ( ρ = 0 . , Λ = 0 . , E = 50 keV) is shown in figure 4.13with red. It is visible in this figure that the orbit width corresponding to theend of the EGAM time evolution (∆ ρ ≈ . − .
18 = 0 .
13) is narrowerwith approximately 25 % than the initial orbit width at the beginning of theinteraction. The consequence of the narrower orbit width is the shrinkage ofthe mode structure, which is consistent with the experimental observations.53 hapter 5Conclusions
The understanding of energetic particle (EP) driven plasma modes plays akey role regarding future burning plasma experiments. Super-thermal EPsin tokamak plasmas can excite various instabilities, and the most importanttransport process of EPs in the plasma core is their interaction with theseglobal plasma modes. The non-linear behaviour of the mode amplitudeand frequency may exhibit a wide range of different behaviours, whichsignificantly influences the impact of the instabilities on the fast particletransport. Therefore, in order to comprehensively understand the non-linearbehaviour of EP-driven instabilities, the investigation of these modes isessential. In this thesis the rapid changes in the radial structure of betainduced Alfv´en eigenmodes (BAEs) and EP-driven geodesic acoustic modes(EGAMs) were experimentally investigated during the non-linear chirpingphase.For the extensive characterization of the modes three diagnostic systemswere chosen, namely the magnetic pick-up coils, the soft X-ray (SXR) camerasand the reflectrometry measurements. In order to deal with the transientbehaviour of the phenomena, short time Fourier transform was chosen as abasis of the data processing tools developed in this thesis. The time evolutionof the radial structure of EP-driven modes was examined by using the SXRmeasurements. The SXR line-of-sights (LOSs) observe different parts of theplasma, which made it possible to analyse the spatial structure of the modes.To follow the amplitude of the oscillation caused by the mode on thedifferent LOSs, I developed an amplitude reconstruction method which canhandle the rapidly changing mode frequency. The derivation of the methodand its validation with synthetic signals was presented in detail. The effectof additive white noise on the reconstruction was also discussed in the thesis.The uncertainty of the measurements was estimated from the properties ofthe background noise. 54 investigated BAEs and EGAMs which were observed in the ramp-upphase of off-axis NBI heated plasmas in ASDEX Upgrade. In order to identifythe instabilities, a thorough mode number analysis was carried out. Then thetime evolution of their radial structure was examined. The radial structureanalysis showed that in case of the observed downward chirping BAEs thechanges in the radial eigenfunction were smaller than the uncertainty of themeasurement. This behaviour is consistent with that the radial structureof BAEs – as normal modes – strongly depends on the background plasmaparameters rather than on the EP distribution.In case of rapidly upward chirping EGAMs the analysis consistently showsshrinkage of the mode structure. This can be explained by the changingresonance condition in the velocity phase space of EPs. The mode structureof EGAMs is sensitive to the EP distribution. The rising frequency of themode indicates that, as time evolves, the EGAM is driven by more passingparticles which have narrower orbit width. This leads to the experimentallyobserved shrinkage of the mode structure.The results shown in the thesis will be presented at the 42nd EPSConference on Plasma Physics [46]. My proposal to carry out furtherexperiments in order to increase the event statistics of my investigationhas been accepted. Thus, dedicated discharges will be performed for furtheranalysis of the examined chirping EP-driven modes in the subsequentcampaign of the ASDEX Upgrade tokamak.55 cknowledgements
I would like to express my special thanks to my supervisors Gergely Papp andGerg˝o Pokol for their continuous support of my research, for their motivationand enthusiasm. I could not have imagined having better supervisors. Mysincere thanks also goes to Philipp Lauber. This thesis would not have beenpossible to write without his help.I would like to thank my collegues from BME-NTI, especially G´aborP´or and P´eter P¨ol¨oskei for the fruitful discussions and the collegues fromIPP Garching, especially Valentin Igochine and Marc Maraschek for theircontinuous help in diagnostic questions.A special thanks to my family. Words cannot express how grateful I amto my mother, and father as well as my siter and my brother for all of thesacrifices that they have made on my behalf. Finally I owe much to Eszti,without whose love I would not have completed this work.This work has been carried out with support of FuseNet – the EuropeanFusion Education Network – within the framework of the EUROfusionConsortium. This work has been carried out within the framework ofthe EUROfusion Consortium and has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programme under grantagreement number 633053. The views and opinions expressed herein do notnecessarily reflect those of the European Commission. I acknowledge thesupport of the Foundation for Nuclear Engineers (JNEA) and HungarianState grant NTP-TDK-14-0022. ibliography [1] European Physical Society. Energy for the future - EPS position paper onthe nuclear option . Press release of the European Physical Society, 2007. .[2] H. A. Bethe. Energy production in stars.
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