Stability analysis of the ideal internal kink mode in a toroidally rotating tokamak plasma with ultra flat q profile

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1 Stability analyi of the ideal internal kink mode in a toroidally rotating tokamak lama with ultra flat q rofile hriter Wahlberg To cite thi verion: hriter Wahlberg. Stability analyi of the ideal internal kink mode in a toroidally rotating tokamak lama with ultra flat q rofile. th International ongre on Plama Phyic, 5-9 October, Nice (France).. <hal-87> HL Id: hal-87 htt:hal.archive-ouverte.frhal-87 Submitted on 9 Oct HL i a multi-dicilinary oen acce archive for the deoit and diemination of cientific reearch document, whether they are ublihed or not. The document may come from teaching and reearch intitution in France or abroad, or from ublic or rivate reearch center. L archive ouverte luridicilinaire HL, et detinée au déôt et à la diffuion de document cientifique de niveau recherche, ublié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire ublic ou rivé.

2 Stability analyi of the ideal internal kink mode in a toroidally rotating tokamak lama with ultra flat q rofile. Wahlberg eartment of tronomy and Sace Phyic, EURTOMVR Fuion ociation, P.O. Box 55, Uala Univerity, SE-75 Uala, Sweden Introduction In ome tokamak exeriment the afety factor q i very cloe to unity in a wide area in the lama center, and increae u to q a ~ -5 (or larger) in the edge region. Thi i the cae for intance in tokamak dicharge in the o-called hybrid cenario [], and in the exeriment with the herical tokamak NSTX reorted in Ref. []. Static equilibria with thi tye of q-rofile are ucetible to an m n ideal, internal kink intability, with an eigenfunction of convective, or quai-interchange character [-5]. By the abence of normal awteeth it eem, however, that the ideal quai-interchange intability doe not develo for intance in the hybrid cenario dicharge in JET reorted in Ref. [6]. In NSTX, on the other hand, onet of m n MH activity i oberved, but ome ort of tabilization of thi activity by the toroidal rotation of the lama eem to take lace [7]. The rotation in NSTX i o trong (rotation frequencie u to % of the lfvén frequency) that the modification of the lama equilibrium due to the centrifugal force i detectable []. In the JET exeriment [6] the rotation, due to neutral beam injection (NBI), i robably much weaker, but can neverthele be exected to be of the order of a few ercent of the lfvén frequency []. Brunt-Väiälä effect otential mechanim for rotational tabilization of the quaiinterchange mode in the exeriment above, a well a in other, imilar exeriment, i the Brunt-Väiälä () effect, reviouly found to be able to tabilize both the uual (Buac), ideal internal kink mode [8] a well a the Mercier mode [9] in tokamak. In the reent work the effect on the quai-interchange mode i invetigated by extending the ideal MH theory of thi mode in toroidal lama, develoed in Ref. [-5], to equilibria with toroidal ma flow. The exreion for the frequency defined in Ref. [8-9] i given by ω q ( ) M M ( M ) Γ q Γ M q Γ Γ M.() It wa found in Ref. [8-9] that tabilization of the internal kink and Mercier intabilitie occur when the frequency above exceed the growth rate of the intability without

3 lama rotation. In Eq. (), the rotation frequency i aumed to be of order Ω ~ ω ~ εω, where aect ratio, ω ( ) B R ( µ ) ω Γ ρ R i the ound frequency, ε rr << the invere field. Furthermore, ρ M ( Ω R the lfvén frequency and B the toroidal magnetic ρ ) ~ i the onic Mach number, R the major radiu of the lama center and Γ ( 5) the adiabatic index. Stability analyi ytem of equation decribing the couling of the (m, n) (, ) and (m, n) (, ) radial comonent ξ and ξ of the Lagrangian lama dilacement ξ ~ ex( iωt) in a rotating, toroidal lama with large aect ratio and circular cro ection wa derived from the Frieman-Rotenberg equation in Ref. [8]. We aly thi ytem to a ituation where the magnetic hear i aumed mall and q q in the region r r, where q ~ ε, wherea in the edge region r r a, q ~ and the hear i of order unity. Furthermore, we aume that Ω i indeendent of r (rigid rotation) and that the reure rofile i arabolic, (r) ()( r a ). In thi cae one obtain a contant value of β in the low-hear region, given by β β (ε ), where a β µ () B and εa ar, and it turn out that the ideal MH tability roblem of the quai-interchange mode in the rotating lama i given by the following equation for ξ in the region r r, together with an integral condition on ξ coming from the boundary condition at r r : d q d ε a ( Ωˆ ˆ ω ) Ωˆ ˆ ω ˆ d ω ˆ ω ˆ ω d ˆ Ωˆ ξ ω M ( ˆ ˆ ˆ ˆ ω ω Ω Ω ) ˆ Γβ r ξ ˆ β r ( ˆ ω ˆ ω ), (a) ( ) ξ d. (b) 6β Here, r dx dr( r r ), ( ) ˆ r, ξ i the olution of the m ε a x ide-band equation for r r a, and x i normalized a x ( r ). The m amlitude ξ atifie, in addition, the boundary condition ξ (r ) and ξ (). Furthermore, ω ω Ω i the oler-hifted mode frequency a well a the eigenvalue of the roblem (a, b). Notice that ω in general i comlex, ω ω r iω i, with oitive ω i indicating intability. The normalized radiu and frequencie in Eq. (a) are defined a r ˆ r a, Ωˆ Ω ε a ω, ˆ ω ω ε aω and ˆ ω ω ε aω. detailed derivation of the Eq. (a, b) will be ublihed elewhere [].

4 In the cae Ω, Eq. (a) can be integrated analytically becaue of the roerty ˆ γ << ˆ ω of the eigenvalue [] (we ue the notation ω i γ, and ˆ γ γ ε a ω in the non-rotating cae). uming that q i a contant, the eigenvalue become γ ω β 8R r ( q). () Exreed in term of β ε a β thi reult i conitent with the tability criterion (6) derived by Hatie and Hender in Ref. []. When the lama rotate, we olve the eigenvalue roblem (a,b) numerically with a q-rofile given by: q q r r q( r) ( qa q )( r r ) () q ( a r ) r r a n examle of thi q-rofile, with q.5, r.5a and q a, i hown in Fig.. In Fig. the growth rate ω i of the quai-interchange intability i hown a a function of q. With increaing rotation, the growth rate a well a the range of untable q i een to decreae, and at ufficiently trong rotation the mode become table for all q. Thi behaviour i illutrated in more detail in Fig., where both the real and imaginary art of ω are lotted a function of the rotation frequency. It i een that the mode become fully tabilized for Ω > Ω crit, where Ω crit.8ω. In the untable regime, there i alo a mall, negative real art of the mode frequency. t the rotation frequency where the mode i tabilized, ω r change ign and aroache, a Ω increae, the frequency hown by the dahed curve in Fig.. The -frequency i calculated from Eq. () with r (and q ), which give the lowet -frequency in the low-hear region for the rotation frequency in the figure. Thu, the eigenfrequency of the quaiinterchange mode aroache the lowet -frequency in the low-hear region a the rotation frequency become large comared with Ω crit. The reaon for thi behaviour i exlained in Ref. []. Furthermore, it i hown in Ref. [] that Ω crit can be exreed in the form Ω ˆ r β H, (5) crit ˆ where the quantity H deend on ˆr and on the q-rofile in the edge region r r a, and ha to be calculated numerically in general. It turn out, however, that H for a large cla of q-rofile and radii r of the low-hear region, a hown in Fig. for the value q a, and 5 in Eq. (). Thu, Eq. (5) how a generic caling of the critical

5 rotation frequency with r and with β of the form Ωˆ ~ β. The critical rotation frequency i in Fig. 5 hown a a function of β for ε a. and q a., and for a few value of r a. The olid curve are calculated from the exact value of H, wherea the dahed curve are obtained from an analytical aroximation valid for mall value of r a given by Ω crit ω.57r R β crit []. It i een that, unle β and r a both are rather large, rotation frequencie of the order of a few ercent of the lfvén frequency are ufficient to tabilize the quai-interchange mode. By omitting the econd term in Eq. (a), the eigenvalue roblem can be formulated a the integral condition ( q i aumed here) [] 5 ( ˆ ω ˆ ω ) d ( ˆ ω ˆ ω )( ˆ ω ˆ ω ) β HF. (6) Here, i given by Eq. () and i the (high-frequency) root obtained if a lu ω HF ign i ued in front of the quare root in Eq. (). Furthermore, ωˆ and ωˆ HF are normalized in the ame way a the other frequencie in Eq. (a), and ˆ ω ˆ ω i a oitive quantity that become mall a the rotation frequency become large comared with Ωcrit, a een in Fig.. Furthermore, the -frequency i alway maller than, and for moderate Mach number (< ) much maller than, the ound frequency. Thi i hown in Fig. 6, which illutrate the radial deendence of and the ratio (r) () for everal Mach number M ( ) M () ( ). Obviouly, ω << ω hold u to relatively high Mach number, and we therefore alo have ˆ ω ˆ ω ˆ HF in Eq. (6). concern the frequency ω, thi quantity i given by ˆ ω when the Mach number i mall [8]. Then, for Mach number that are not too large, Eq. (6) i aroximated by ( ˆ ω ˆ ω ) β ˆ ˆ ˆ ω 5 d. (7) Thu, with increaing rotation frequency, and thereby increaing -frequency, the other arameter remaining fixed, ω ω ha to increae alo in order to fulfill the integral condition (7), and at ufficiently trong rotation the mode therefore become table. If we aume, in addition, that ra i mall, we can aroximate in the integral by it value for r. Thi lead, by uing Eq. (), to the eigenvalue ω ω ( ) γ. Thu, we obtain the ame tability condition for the quai-interchange mode, ω > γ, a wa found reviouly for the tabilization of the Buac and Mercier mode in Ref. [8-9]. oncluion The effect of toroidal rotation on the ideal MH tability of the quaiinterchange mode in tokamak with q in a wide area in the lama core ha been analyzed. Thi tability roblem can be formulated a Eq. (a) for the m amlitude

6 ξ in the region where q ( r r ), together with the integral condition in Eq. (b) and the boundary condition ξ (r ) and ξ () for ξ. For a tatic lama, the eigenvalue roblem (a,b) reroduce the reure-driven, quai-interchange, m n intability reviouly analyzed in Ref. [-5]. The numerical olution of Eq. (a,b) reented here how that thi mode i tabilized by rigid, toroidal rotation. The tabilization i caued by the modified lama equilibrium, and aociated Brunt-Väiälä () frequency, created by the centrifugal force. In the regime of lama rotation where the tabilization occur, the eigenvalue i given aroximately by ω ω γ, where γ i the growth rate of the mode in the abence of rotation and ω i the lowet frequency in the low-hear region of the lama. Thu, the tabilizing mechanim comete with the drive from the quai-interchange intability, and tability i achieved when exceed γ, which i the ame tability condition a wa found earlier for rotational tabilization of the uual ideal, internal kink mode and for the tabilization of Mercier mode that are untable in the abence of rotation in Ref. [8-9]. omarion with exeriment with low-hear q-rofile in JET [6] and in NSTX [, 7] indicate that the critical rotation for tabilization of the ideal m n mode found here aear to be of the ame order of magnitude a the actual lama rotation, driven by neutral beam injection, in thee exeriment (with ome uncertainty, though, for the actual rotation frequency in the JET exeriment). The Brunt-Väiälä mechanim could therefore be of interet for the interretation of the m n activity in thee, and in imilar tokamak dicharge. cknowledgement Thi work ha been uorted by the Euroean ommunitie under an aociation contract between EURTOM and the Swedih Reearch ouncil VR. Reference [] Litaudon X et al Plama Phy. ontrol. Fuion 6 9 [] Menard J E et al Nucl. Fuion [] Weon J 986 Plama Phy. ontrol. Fuion 8 [] Hatie R J and Hender T 988 Nucl. Fuion [5] Waelbroeck F L and Hazeltine R 988 Phy. Fluid 7 [6] Buratti P et al Proc. t EPS onf. on Plama Phy. (London, ) vol 8B, P-.65 [7] Menard J E et al Proc. th EPS onf. on ontr. Fuion and Plama Phy. (St Peterburg, ) vol 7, P-. [8] Wahlberg and Bondeon Phy. Plama 7 9 [9] Wahlberg and Bondeon Phy. Plama [] Wahlberg aer in rearation 5

7 . ε a. q. ω ω i.8.6 β. r.5a q. a Ω Ω.69ω. Ω.78ω r. Fig.. The growth rate ω i of the quai- interchange intability a a function of q for a lama with the arameter hown in the figure ra Fig.. Examle of the model q-rofile, defined in Eq. (), ued in the numerical calculation. The arameter of the q-rofile hown are q.5, r a.5 and q a q.. ω ω.. ε. a β. r.5a q. a q ω i ω r H.5. q a 5..5 ω r Ωω Fig.. The real (ω r ) and imaginary (ω i ) art of the eigenvalue ω a function of the rotation frequency Ω r a Fig.. The arameter H in Eq. (5) a a function of the radiu r of the hear-free region..5. Ω ω crit.5. ε a. q a. r a ( (r)ω ()) - - M () β Fig. 5. The critical rotation frequency a a function of β for a few value of the radiu r ra Fig. 6. The Brunt-Väiälä frequency in Eq. () a a function of r for a few value of the of the hear-free region. onic Mach number at r. 6