A closure scheme for modeling RF modifications to the fluid equations

1. A closure scheme for modeling RF modifications to the fluid equations C. C. Hegna and J. D. Callen University of Wisconsin Madison, WI Acknowledge useful discussions with: J. Carlsson, E. D. Held, S. E. Kruger, D. Schnack and C. R. Sovinec and members of the SWIM team

2. Theses • Efforts to model the interaction of MHD modes with RF require a proper theoretical formulation. • RF sources modify the fluid equations via the addition of new terms to the fluid equations and through modification of the fluid closures. • A closure scheme for modeling RF effects in the fluid equations is presented based on a Chapman-Enskog-like approach. • The problem of ECCD stabilization of tearing modes (neoclassical and conventional) is emphasized as a starter problem • The extended Spitzer problem with RF current sources • Steps towards a more general CEL treatment are outlined

3. Motivation • One of the long term goals of the SWIM project is to model the interaction of RF sources with long time scale MHD activity • Principal application - ECCD stabilization of NTMs • Most prior theoretical treatments of ECCD stabilization of NTMs use an ad-hoc source term in parallel Ohm’s law (CCH and JDC, PoP ‘97; Zohm PoP ‘97; etc.) where Frf = - hJrf(x,t)B/B with the scalar Jrf a function to be chosen. Crudely, Jrf is characterized by its amplitude, localization width relative to island width and phase relative to the island phase. • Other quasi-ad-hoc models for ECCD stabilization have been used (Gianakon et al, ‘03) none of which come from a first principals formulation. • For more general problems, it is highly desirable to derive a more rigorous model for use in simulation. • This work --- an effort to develop a procedure on how to include RF effects in a fluid formulation

4. Experimentally, ECCD stabilization of NTMs work remarkably well • Stabilization of 3/2 NTM on DIII-D (from LaHaye et al ‘06) • NTM Stabilization demonstrated on DIII-D, AUG, JT60U, etc. • 2/1 and 3/2 modes can be stabilized. • The modified Rutherford theory largely models the island evolution properties and RF stabilization.

5. Attempts to model the effects of localized ECCD on NTMs use the modified Rutherford equation • The effect of localized RF on NTM is modeled using the modified Rutherford equation • Modeling efforts point to NTM control as a crucial issue for ITER • Unmodulated ECCD predicted • to not completely stabilize NTM • Modulated ECCD may not completely stabilize NTM LaHaye, PoP ‘06

6. A general kinetic equation is considered as a starting point in the calculation • A kinetic equation in the form with collision operator C(f) • For many applications (such as ECCD), a quasilinear diffusion operator describes Q(f) (after averaging over gyrophase) where the diffusion tensor D is needed from RF codes. • For ECCD, ray tracing is probably sufficient

7. The RF sources modify the fluid equations • Taking moments of the fluid equations yields: • The additional terms due to the RF are given by • The RF is assumed to produce no particles Additional RF contributions to the fluid equations

8. While the fluid equations are exact, a treatment of the closure problem is needed • In addition to providing terms in the fluid equations, the RF will also modify the closures; calculations for the heat fluxes and stress tensors are needed. • Since the problem of interest is principally a modification to the Ohm’s law (for localized current drive), the closest analogy is with the Spitzer problem. Solve via a perturbation theory • E/ED << 1 (ED = Dreicer field) is a small parameter • neE ~ Frf, E/ED ~ Frf/neED << 1 the RF terms are “small” • Reasonable approximation for ECCD, probably not a good assumption for other forms of RF heating. • Nonetheless, this approximation allows for analytic progress on the closure problem with RF.

9. By assuming a lowest order Maxwellian, the addition RF sources in the fluid equations can be written as fluid variables • With fs ~fMs • Frf and Srf are now expressed as functions of low order fluid moments. • Once the quasilinear diffusion tensor D is specified, the fluid equation sources Frf and Srf are determined. • However, we’re not done yet ---- RF contributions also modify the closure moments

10. Using a Chapman-Enskog-like approach, a kinetic equation for the kinetic distortion is derived with RF source terms • The CEL ansatz: the kinetic distortion has no density, temperature or momentum moments • The kinetic equation takes the form • Using the fluid equations to evaluate dfM/dt, we have Additional source terms for the kinetic distortion due to RF

11. As a simple application, we can revisit the Spitzer problem with RF sources • For simplicity, let’s assume a time-independent, homogeneous magnetic field • The kinetic equation reduces to where the … terms are even in v • In the collisional limit, the parallel component of this equation is equivalent to the Spitzer problem with 0= ne e2/mee (not the Spitzer resistivity)

12. In the collisional limit, the kinetic equation can be solved via expanding in Laguerre polynomials • Expanding in Laguerre polynomials (x = v2/vT2) • Taking Li3/2 moments of the kinetic equation yields the matrix equations RF modified closure

13. For more general problems, methods for solving the kinetic equation need to be developed • The more general problem entails solving the kinetic equation • A sequence of extensions to the modified Spitzer problem need to be developed ---- calculations for a bumpy cylinder, toroidal equilibrium, time-dependent processes, multiple length scales, magnetic island effects, etc. • Assuming parallel streaming is a dominant effect, an equation akin to that developed by Held et al for heat flux near an island emerges • Use multiple length scale expansion l ~ qR << L ~ qR Ls/w Solve by expanding in Cordey eigenfunctions, etc.

14. Summary • The beginning stages of an effort to address modeling efforts incorporating RF effects in fluid codes is underway • RF effects produce additional contributions to the fluid equations and modify fluid closure moments. • A kinetic theory is developed assuming the RF produces a “small” distortion away from a background Maxwellian. • A Chapman-Enskog-like framework is developed to outline a calculation procedure for the closure moments. • A simple application of the modified Spitzer problem is addressed. • Much further work is needed for modeling more general problems.