Proc. 5th IAEA Tech. Com. Meeting Alpha-particles, JET (1997)

Global Alfvén Eigenmodes Stability in ITER

A. Jaun , J. Vaclavik , L. Villard
Alfvén Laboratory, EURATOM-NFR Association, KTH, 100 44 Stockholm, Sweden
CRPP-EPFL, Association Euratom-Confédération Suisse, 1015 Lausanne, Switzerland

Abstract. The stability of Alfvén eigenmodes with low to intermediate toroidal mode numbers n=1-12 is examined for a reference equilibrium using two different models for the plasma response which have previously been thoroughly validated against measurements from JET and DIII-D. The fluid LION code generally predicts instability thresholds for the volume averaged alpha-particle pressure around 0.5% for global TAEs and 0.1% for core-localized TAEs, while the calculations performed with the gyrokinetic PENN code show that Alfvén eigenmodes are stable for a large variety of burn conditions.

INTRODUCTION

  The stability of Alfvén eigenmodes (AE) is one of the key issues which needs to be addressed to build tokamak reactors [1], but cannot really be tested experimentally in the smaller, near to break-even plasmas achievable today. Theoretical models are therefore required for the prediction of the instability and have beforehand to be validated against measurements carried out under approaching conditions. Different types of eigenmodes have been identified and result from the propagation of the shear- (BAE, TAE, EAE, GAE), the kinetic- (KAE) and drift-kinetic- (DKAE) Alfvén waves. Resonant wave-particle interactions provide the main source of drive / damping and depend sensitively both on the spatial structure of the eigenmode and the velocity distribution of the particles. In DIII-D, the coupling to a drift-kinetic Alfvén wave in the plasma core has been shown to be responsible for the destabilization of AE by neutral beams [2, 3] and the wavelength modifications due to the magnetic shear in the plasma core/edge to change dramatically the damping rates measured in JET [4, 5]. In this paper, two models are used to predict the stability of AE in the finite single-null 21 MA reference equilibrium obtained using the PRETOR code [6] by the ITER-JCT in San Diego.

FLUID MODEL

  The LION code [7] computes global shear-Alfvén wavefields using a regularization of the fluid wave-equations and yields therefore a ``continuum damping'' which is questionable when more than one resonance is present inside the plasma [8]. Wave-particle power transfers due to resonant Landau interactions are evaluated perturbatively within the drift-kinetic approximation [9], assuming Maxwellian distributions for the bulk species and a slowing-down distribution for the particles with density profiles of the form and , the latter with a half-width varying in . The marginal stability of TAE modes is determined globally as a function of the critical volume averaged pressure , the half-width and the ratio of the birth to Alfvén velocity on axis , and is achieved when the particle drive (directly proportional to ) balances exactly the total absorption .

Fig.(a,b) illustrate that the n=-2 TAE wavefield at 42 kHz is very global, peaking at the radial locations of the shear-Alfvén gaps and the position of the Alfvén resonances. The corresponding local power transfers per species at marginal stability , where , show in fig.(d) that the main source of the particle drive stems from the core s<0.6 and in particular from the neighborhood of the Alfvén resonance at s=0.23, whereas the Landau damping is shared between the core ions s<0.7 and the edge electrons s>0.7. It is not surprising to find in fig.(e) that this mode is destabilized only for relatively peaked profiles and , when the particle pressure exceeds %.

Two n=-8 TAE modes illustrate calculations performed for intermediate toroidal mode numbers: fig.(g,j) show that the first mode at 52 kHz has a relatively global extension out to the plasma edge while the second at 55 kHz is core-localized [10] in the m=8,9 gap. The instability threshold % computed in fig.(k) is then lowest for the core-localized mode, when the pressure gradient is maximal around the m=8,9 gap.

Complete studies have been carried out for different toroidal mode numbers n= -1, -2, -3, -8,-12 varying the density profile so as to align the gaps and increasing the value of the magnetic shear in the plasma core. While both affect somewhat the wavefield structures and the species power transfers , the marginal stability threshold of the eigenmodes does not vary by more than a factor two.

To summarize, the fluid predictions for ITER show that global TAE modes from |n|=2 to 12 can become unstable for pressures exceeding %, while only core localized modes can have smaller threhsolds. Fluid models are however questionable in presence of Alfvén resonances (in particular for ``continuum damped'' modes such as the one in fig.(a,d,e)), so that it is useful to repeat the analysis with the gyrokinetic model which does not suffer from these short-comings.

GYROKINETIC MODEL

  The PENN code [11, 12] computes global wavefields including the propagation, damping and mode-conversion of the fast-, the kinetic- and the drift-kinetic-Alfvén waves, assuming an approximative functional dependence of the parallel wavevector to calculate the response of the plasma. The code has been thoroughly tested against measurements from JET and DIII-D and yields good agreement for the wavefield structures [13], frequencies [14, 5], damping rates [5] and instability thresholds [3].

Fig.(c,f) show that the n=-2 wavefield structure of an eigenmode at 45 kHz is somewhat similar to the fluid TAE illustrated in fig.(a,d). Because the TAE wavelength at the m=2 resonance meets the characteristic scale of the kinetic-Alfvén wave (KAW) around s=0.2, mode-conversion takes place and the associated gives rise to Landau interaction with all three species. The local power transfer oscillates radially around the resonance and because of the enhanced Landau damping yields a total power transfer showing that the mode is stable for all values of .

To illustrate the calculations carried out with intermediate mode numbers, an n=-12 AE at 77 kHz is displayed in fig.(i,l) with a core-localized wavefield in the m=11,12 gap extending somewhat to the adjacent neighbors. Mode coupling takes place throughout the interval where rapid wavefield variations induced by the toroidicity gaps meet the characteristic scale length of the KAW, while the fluid resonances do not contribute at all. Because of the higher frequency, the electron Landau damping exceeds the bulk-ion damping and with an oscillating power transfer which is again globally positive, the eigenmode is also stable for all values of .

Complete studies have been performed for different toroidal mode numbers n=-1, -2, -3, -6, -12, varying the density profile so as to align the gaps and changing the value of the magnetic shear in the plasma core. Both affect the wavefield and the power transfers rather strongly, but in all the cases analyzed so far the mode-conversion has always been strong enough to completely stabilize the eigenmode.

In summary, the gyrokinetic predictions for mode numbers from |n|=1 to 12 show that the mode-conversion to the kinetic-Alfvén wave has a strong stabilizing effect and that AE in ITER should generally be stable.

DISCUSSION

  The uncertainties in the prediction of the instability thresholds are of three different origins. Most obvious is the pertinance of the theoretical model. Fig.(d,f) show that the wavefields and the resulting power transfers obtained with LION and PENN are very different. The assumption made for the gyrokinetic PENN calculations seems to be rather well verified in the computed kinetic-Alfvén wavefields, but is certainly a bad estimate below 20 kHz when couplings occur to drift-waves. A second source of uncertainty is the sensitivity to the equilibrium profiles which seems here to be relatively weak for both models and may be comparable with the 30% which have been estimated from the damping comparisons with JET [5]. Third, the errors induced by the discretization with 3-4 cubic finite elements per wavelength are negligible compared with the pertinance of the model and the uncertainties of the equilibrium profiles.

To conclude this analysis, one can say that Alfvén eigenmodes n=1-12 above 20 kHz should be stable for a relatively large variety of burning ITER plasmas. Because different sources of ``strong damping'' are intrinsically built into the plasma configuration (high magnetic field, X-point, weak central shear), the degree of confidence in the prediction is relatively high.

The ITER reference equilibrium has been provided by O.Sauter from the ITER-JCT in San Diego. This work was supported in part by the Swedish, the Swiss National Science Foundations and the computer centers in Linköping and Manno.

Figure: Examples of global AE stability calculations performed with the ITER reference equilibrium using the fluid LION (left, a,d,e,g,j,k) and the gyrokinetic PENN codes (right, c,f,i,l), for low (top, a-f) to intermediate toroidal mode numbers (bottom, g-l). The fluid model generally predicts instability thresholds above % for global modes and above % for core-localized modes, while the AE computed using the more elaborate gyrokinetic model are all found to be stable.  

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