Temperature-Gradient-Driven Tearing Modes

Transcription

1 1 TH/S Temperature-Gradient-Driven Tearing Modes A. Botrugno 1), P. Buratti 1), B. Coppi ) 1) EURATOM-ENEA Fusion Assoiation, Frasati (RM), Italy ) Massahussets Institute of Tehnology, Cambridge (MA), USA ontat of main author: antonio.botrugno@frasati.enea.it Abstrat. The observation of spontaneous islands growth in tokamak experiments indiates that the tearing mode ould also be linearly unstable. However, in the framework of resistive magnetohydrodynamis, previous works have demonstrated that drift effets, eletron heat transport along magneti field lines and ion Larmor radius effets have stabilizing effets. Reently, numerial alulations and analytial theory in the ollisional regime have demonstrated that inlusion of perpendiular eletron heat transport arries out a new type of linear tearing mode driven by a large eletron temperature gradient. In this paper, we review the linear tearing mode equations in order to identify the relevant instability regimes. The relevane of some regimes to experiments is disussed and analytial results in different regimes are presented. In all ases, realisti value of the perpendiular eletron heat transport produes a strong destabilizing term proportional to the eletron temperature gradient in the expression of the island growth rate. 1. Introdution Tearing instabilities are related to a deformation of the magneti field topology leading to generation of magneti islands via magneti reonnetion due to finite plasma resistivity. The presene of magneti islands degrades onfinement and is presently a major limit to the ahievement of high beta plasmas. The physial mehanism of tearing modes onset remains an outstanding problem of magneti onfinement. In the simplest model of linear resistive magnetohydrodynamis, the stability of tearing modes is ruled by the Δ' parameter, whih represents the onvexity of the mode eigenfuntion at the resonant surfae k B 0. If Δ' 0 urrent filamentation is energetially favourable and the mode is unstable. Modes with poloidal number m typially have Δ 0. Drift-tearing theories involving the ombined effet of magneti reonnetion, eletron density and temperature gradients resulted in slower growth rate and instability ondition Δ' Δ' rit, Δ' rit being a large positive threshold [1-3]. All these theories inluded parallel eletron energy transport as a key ingredient but negleted the perpendiular one. These results were in marked ontrast with experiments, in whih tearing modes are often observed. In order to resolve this ontrast tearing modes were assumed to be metastable, i.e. growing from a seed island generated by other instabilities suh as sawtooth rashes, fishbones or edge loalized modes (ELMs). However, tearing modes an also grow without any trigger event. Spontaneous tearing modes have been observed in JET hybrid disharges [4] in FTU disharges and in various TFTR disharges [5]. These observations indiate that tearing mode ould be linearly instable. More reently, analyti theory in the ollisional regime [6] and numerial modelling in ollisional and weakly ollisional regimes [7, 8] have demonstrated that, with the inlusion of realisti values of the (anomalous) perpendiular eletron heat diffusivity, a new type of linear tearing mode appears, whih is driven unstable even at Δ 0 by a suffiiently large eletron temperature gradient.

2 TH/S In this paper the results of linear analytial theory are reviewed and extended to a weakly ollisional regime. The paper is organized as follows. In setion the experimental bakground is presented and realisti values of the key parameters are determined. In setion 3 the basi equations are presented and different instability regimes are highlighted. In setion 4 results relevant to the different regimes are summarized. Conluding remarks are given in setion 5.. Experimental bakground Tearing modes an grow without any trigger event in tokamak experiments. This has been learly observed in various disharges in the hybrid regime of JET, with high normalized kineti pressure (β N ). It is usually diffiult to distinguish tearing modes whih grow spontaneously from metastable tearing modes, beause of the MHD ativity normally present in the hybrid regime as fishbones, internal kink modes, Alfven eingemode, and espeially ELMs an disturb the interpretation of experimental signals. However, in some ases, there are no doubts about the spontaneous growth of tearing modes. As example, we show in FIG. 1 (lower frame) the spetrogram of a magneti oils signal in JET disharge Two tearing modes with toroidal number n=3 and n= grow in absene of any other MHD instabilities, the first one after 5.3 s. with frequenies between 0 and 5 khz, the seond one after 5.6 s. at about 15 khz. FIG. 1. Magneti oil signal (upper frame) and its spetrogram (lower frame) in JET disharge number Spikes in the signal are due to ELMs. Tearing modes with toroidal numbers n= and 3 an be seen as ontinuous red lines in the spetrogram. Both modes appear during ELM-free periods. Similar phenomenology an be observed in FTU disharges. As example, we show in FIG. (upper frame) the spetrogram of a magneti oil signal in the disharge number It is evident a tearing mode with toroidal and poloidal number n=1 m= at 6 khz starting at about 0.68 s. without any other evident MHD ativities.

3 3 TH/S FIG.. Spetrogram of magneti oil signal (upper frame), urrent (entral frame) and density (lower frame) in FTU disharge number Tearing modes with toroidal number n=1 an be seen as ontinuous dark lines in the spetrogram. TABLE I: CHARACTERISTIC FREQUENCIES AND LENGTHS FOR DISCUSSED DISCHARGES */khz) ei (khz) S (mm) D (mm) mm) mm) d e (mm) JET FTU Values are alulated from experimental data, assuming perpendiular heat diffusivity values of 1 m /s, lassial parallel heat ondution and Spitzer resistivity. JET Data are evaluated at onset of the n= mode shown in FIG. 1. FTU data are evaluated at mode shown in FIG. and t=0.7 s. 3. Basi equations of the linear model We desribe the tearing mode as a perturbation with frequeny and wave vetor k direted along y, in a sheared slab with shearing length L S in x diretion. The eletromagneti fields are given by A E = 1 z φ and B = Bz A z t where A is the z-omponent of the vetor eletromagneti potential and φ is the salar eletromagneti potential. We onsider a set of Braginskii-type transport equations in two fluids form. We assume linear expansions of the form F f0( x) f ( x)exp( iky it), and notation with 0 subsript for equilibrium quantities and without subsript for perturbations. The first equation of the linear system is the parallel eletron momentum balane equation, in whih pressure gradient and thermal fore effets are inluded:

4 4 TH/S 1 B j B E B p B T (1) en e Where n, p and T are eletron density, pressure and temperature, e is the unit harge and =0.71 is the thermal fore oeffiient. B Bx d The parallel gradient operator to first order of linear expansion beomes k, B B0 dx Bk kx where k k x and B x ika. The total time derivative is d kφ = iω i B L dt B 0 S The x-derivatives give rise to diamagneti frequenies of the form ω f = k T 0 df 0. The eb 0 f 0 dx parallel eletri field is E i A ik. Then, equation (1) an be rewritten as ηj = i ω A ik φ + T 0 ik en n i ω n A + 1+α ik 0 e T i 1+α ω T A d dx. or, defining δω = ω ω n αω T and α = 1 + α = 1.71, as ηj = i δω A ik T φ + ik 0 α n + ik en T () 0 e The seond equation is the parallel eletron ontinuity equation whih has the form dn = D n dt ( j) x e (3) Negleting perpendiular partile diffusion D = 0, the previous equation an be written as ωn = ω n n 0 eφ T 0 1 e k j (4) and using equation () to eliminate j we obtain x i n = x en 0 k T 0 δω A x αn 0 T 0 T + en 0 with = ω ηe n 0 k T 0 1 = ων ei k v e T 0 x + i ω n ω φ (5), where ν ei and v e are ollision frequeny an eletron thermal veloity respetively. The parameter gives the length sale of density response; values alulated for the experimental examples are given in Table I. The third equation is the parallel harge neutrality equation in the limit of strongly magnetized ions k j = ω ω di 4πV A d φ dx (6) where V A is the Alfvén veloity and ω di is the ion diamagneti frequeny. Substituting j from () and n from (3) we have

5 5 TH/S ρ s i Δ C 4 d φ x dx = 1 ± ω n ω φ δω k x A α e T where ρ s = T 0 m i Ω i is the ion sound Larmor radius and Δ 4 C = η ω ω di 4πV A k ρ s Δ D. The spatial sale for the eletrostati potential is then Δ C if Δ C ρ s and ρ s if ρ s Δ C. Equation (6) only holds if the ion larmor radius is smaller than the urrent layer width, otherwise a form aounting for non-loal effets has to be used [3]. The fourth equation is the eletron energy balane equation 3 iωt + iω d Teφ = χ T dx k χ T + k χ ω T e A + i α e T 0 n 0 k j (7) where χ and χ are perpendiular and parallel eletron heat diffusivities. Introduing the harateristi lengths Δ = χ 1 4 k and Δ χ = 3 1 ω k, equation (7) an be reast as χ 4 Δ d T dx x T + xω T e A i ω T Δ k ω eφ + iδ T + i α T 0 3 e k x n 0 ω Δ j = 0 (8) For lassial χ and η we have 3 χ = 1.07 T 0 ηe n 0 and the parallel transport sale Δ = Moreover we have the Ampère law and the boundary ondition 4π A j dx = 1 A A"dx = Δ (9) Where Δ is the jump in the logarithmi derivate of B x inside the slab x=+0 d dx ln A x= 0 We express the dispersion relation of the perturbation in the form ω = ω R + iγ, it an be obtained by solving equations () and (4) for j and integrating the result aording to (9). In the following we assume the onstant A approximation, whih an be applied if magneti diffusion aross the urrent layer during one osillation yle is negligible. Analytial solutions in different regimes will be onsidered in the following. Different regimes of ollisionality depend on the ordering of the harateristi lengths: Collisional regime ρ s. In this ase Δ C, so that the spatial variation of potential is more rapid than that of density and temperature. The indutive term δω A in equation () is then balaned by the eletrostati one k φ and the spatial width of j is Δ C. This regime learly applies to FTU data in Table I. Semiollisional regime ρ s. In this ase Δ C, so the indutive term is ompensated by density and temperature perturbations and the spatial width of j is. This regime applies for JET data as given in Table I, but it is important to notie that interation between miroturbulene and the tearing mode an inrease the

6 6 TH/S effetive eletron ollision frequeny and restore the ollisional regime even at high plasma temperature [6], (see also setion 4.). Collisionless regime. The eletron motion beomes insensitive to ollisions if the eletron mean free path λ mfp is larger than the parallel wavelength, i.e. if k λ mfp 1. Using k k, the ondition redues to ω ν ei. 4. Analytial results 4.1. Negligible perpendiular transport Δ = 0 This ase was treated in referenes [, 3] for different ollisionality regimes. The general finding was that the temperature gradient plays a stabilizing role. This is partiularly simple to illustrate in the semiollisional regime, for whih ω R ω n + 0.5αω T and γ η π ω R ω n 3 ω ρ s. The seond term in γ is stabilizing and it is proportional to ω T 3. 4π Δ 4. Collisional ase with strong perpendiular transport ρ s Δ C Δ This ase was treated in referene [1]. The dispersion relation yields ω R = ω n + αω T and γ = a η Δ 1 + a 4π Δ ω Δ 3 C R C Δ, where a1, a are numerial oeffiient of order one. The growth rate an be positive even with negative values of the Δ index, i.e. as a onsequene of perpendiular heat transport the temperature gradient an drive the instability even if the urrent profile is stabilizing. In the same paper the possibility of ollisionality enhanement by mirosopi reonneting modes was proposed. The enhanement fator is 1 + with this, the ondition Δ C > ρ s would apply in all experimental onditions. ω ν ei kd e ; 4.3. Semiollisional ase with strong perpendiular transport Δ ρ s In this ase the eletri potential varies on the largest length sale, so that it remains negligible aross the urrent hannel width. Use of equation 6 in this regime requires ρ i ρ S, whih does not apply to experiments like JET, in whih T i T e. Nevertheless, this ordering allows the most transparent derivation of the dispersion relation, so it is instrutive to onsider it. Equation () beomes iηj = +ix δω A + k x e αt Δ 4 D +ix 4 +x 4 (10) Temperature an be alulated expliitly sine Δ Δ, so that the energy balane equation redues to 4 d Δ T dx x T + xω T e k A = 0, (11) In the following we use the onstant A approximation, whih requires 4π ω approximation the solution of equation (11) is η 1. In this

7 7 TH/S T(x) = ω T k Δ e A z π 0 dθ sinθexp z osθ with z = x Δ. The dispersion relation gives ω R = ω n + αω T and γ = η 4π Δ + α ω T π π Δ I i Δ D I Δ r where Δ I r = Δ zy(z) z Δ dz 0 and I i = z 3 Y(z) z Δ dz =.78 with Y z = z π dθ sinθexp z os θ. 0 In the Δ = 0 ase the sale of temperature variation is instead of Δ (see FIG. 3) and the T-term in (10) gives no ontribution to the growth rate. The key effet of is to inrease the distane between the resonant surfae ( k B 0 ) and the isothermal region, in 1 whih B T 0. For 0 the distane is D / L S k []; the orresponding temperature perturbation is shown by the blue urve in figure 1. With realisti values of, the profile is smoother (red trae in figure 1) and the isothermal region is shifted to Δ Δ. The negative ontribution ρ s found in [] for the semiollisional regime (see setion 4.1) does not appear in the growth rate expression sine terms of first order in ρ s have been negleted in equations (10) and (11). FIG. 3. Profiles of the eletron temperature eigenfuntions with negligible (blue) and realisti (red) perpendiular transport versus normalized distane from the resonant surfae. 5. Disussion The possibility of tearing modes driven by the temperature gradient has been demonstrated in different asymptoti regimes (see setions 4. and 4.3). Similar results were obtained by inluding perpendiular heat transport in numerial simulations [7, 8]. These results point to a new interpretation of observations on spontaneous tearing modes. Comparison between

8 8 TH/S analyti results and JET data is not possible beause separation of length sales is not large enough, while results from setion 4. an be applied to the ollisional (FTU) experimental ase. With the parameters given in Table I, the mode turns out unstable unless /k = 14 (the typial result of alulations being /k = 1). This shows that the temperature gradient drive is a strong effet, and raises the question about what prevents tearing modes growth beoming ubiquitous). Referenes [1] B. COPPI et al., Phys. Rev. Lett. 4 (1979) 1058 [] J.F. DRAKE et al., Phys. Fluids 6 (1983) 509 [3] S.C. COWLEY, R.M. KULSRUD AND T.S. HAHM, Phys. Fluids 9 (1986) 330 [4] P. BURATTI et al., 34th EPS Conf. on Plasma Phys, Warsaw, July ECA 31F O4018 [5] E.D. FREDRICKSON, Phys. Plasmas 9 (00) 548 [6] B. COPPI, in Colletive phenomena in marosopi systems Eds. G.Bertin, Publ. Word Sientifi, 007 [7] Q. YU, S. GÜNTER, B.D. Sott, Phys. Plasmas 10 (003) 797 [8] S. NISHIMURA ET AL., J. Phys. So. Jpn 76 (007)