1. SWIM Meeting Madison, WI 12/4/07 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. SWIM Meeting Madison, WI 12/4/07 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. SWIM Meeting Madison, WI 12/4/07 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 F rf = -  J rf (x,t)B/B with the scalar J rf a function to be chosen. Crudely, J rf 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. SWIM Meeting Madison, WI 12/4/07 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. SWIM Meeting Madison, WI 12/4/07 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 LaHaye, PoP ‘06 Unmodulated ECCD predicted to not completely stabilize NTM Modulated ECCD may not completely stabilize NTM

6. SWIM Meeting Madison, WI 12/4/07 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. SWIM Meeting Madison, WI 12/4/07 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. SWIM Meeting Madison, WI 12/4/07 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/E D << 1 (E D = Dreicer field) is a small parameter neE ~ F rf, E/E D ~ F rf /neE D << 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. SWIM Meeting Madison, WI 12/4/07 By assuming a lowest order Maxwellian, the addition RF sources in the fluid equations can be written as fluid variables With f s ~f Ms –F rf and S rf are now expressed as functions of low order fluid moments. –Once the quasilinear diffusion tensor D is specified, the fluid equation sources F rf and S rf are determined. –However, we’re not done yet ---- RF contributions also modify the closure moments

10. SWIM Meeting Madison, WI 12/4/07 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 df M /dt, we have Additional source terms for the kinetic distortion due to RF

11. SWIM Meeting Madison, WI 12/4/07 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    = n e e 2 /m e e (not the Spitzer resistivity)

12. SWIM Meeting Madison, WI 12/4/07 In the collisional limit, the kinetic equation can be solved via expanding in Laguerre polynomials Expanding in Laguerre polynomials (x = v 2 /v T 2) Taking L i 3/2 moments of the kinetic equation yields the matrix equations RF modified closure

13. SWIM Meeting Madison, WI 12/4/07 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 L s /w Solve by expanding in Cordey eigenfunctions, etc.

14. SWIM Meeting Madison, WI 12/4/07 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.