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Saturation of the Neoclassical Tearing Mode Islands

Saturation of the Neoclassical Tearing Mode Islands. F. Militello 1 , M. Ottaviani 2 , F. Porcelli 1 , J. Hastie 1. 1 Burning Plasma Research Group Politecnico di Torino Italy. 2 CEA Cadarache France. Outline. NTMs and the generalizations of the Rutherford Equation.

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Saturation of the Neoclassical Tearing Mode Islands

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  1. Saturation of the Neoclassical Tearing Mode Islands F. Militello1, M. Ottaviani2, F. Porcelli1, J. Hastie1 1 Burning Plasma Research Group Politecnico di Torino Italy 2 CEA Cadarache France Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  2. Outline • NTMs and the generalizations of the Rutherford Equation. • The asymmetric saturation, mathematical technique, nonlinear solution. • Resistivity models, self-consistent solution, saturated width relations. • The Symmetric Model. • The Code and our Results. • Theory and Numerics, do they agree? • Summary and conclusions. Theoretical model for Asymmetric saturation (simplified model) Numerical analysis of the Symmetric NTM Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  3. The NTMs • NTMs may decrease tokamak performance. • It is important to have a reliable prediction of the size of the saturated NTMs islands. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  4. The nonlinear solution of the model for the island width, w, can be obtained by using an asymptotic matching procedure. The Generalized Rutherford Equation THE MODEL After Rutherford (PoP ’73) x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  5. The Generalized Rutherford Equation • Bootstrap current term, drive for the nonlinear instability when D’<0 Hegna & Callen (‘92) Fitzpatrick (‘95) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  6. The Generalized Rutherford Equation • Polarization current term, proportional to the magnetic island poloidal rotation frequency, w Smolyakov (’89) Waelbroeck et al (’01,’05) x x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  7. The Generalized Rutherford Equation • Term related to the shape of the equilibrium current density: Militello & Porcelli (’04) Escande & Ottaviani (’04) x x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  8. The Generalized Rutherford Equation • Term related to the shape of the equilibrium current density: Militello & Porcelli (’04) Escande & Ottaviani (’04) x x Valid only for symmetric equilibria. Corrections are required in cylindrical geometry !!!! Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  9. The Limit of the Asymmetric Saturation • Previous investigations on classical asymmetric saturation had two major flaws: 1) limiting model for resistivity, 2) no self-consistency (Ansatz required). We can do better !! Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  10. Simplified Model Equations • Vorticity equation: • Ohm’s law: • Energy equation Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  11. Mathematical technique • Following Rutherford, we employ an Asymptotic Matching procedure, justified by the smallness of the island width w compared to the macroscopic length, L: w<<L. Mout(c) Min(c) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  12. Inner Nonlinear Solution • The matching function depends on Jin: • From the averaged Ohm’s law: • The flux surface average is: Resistivity Model ! Metric term ! A.Thyagaraja, Phys. Fluids 24, 1716 (1981) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  13. Resistivity Models • Parallel heat transport is very efficient: • then, the perpendicular transport acts on scale length of order: • Below this threshold the perturbation of the temperature are smoothed by perpendicular transport. • Where the perp. transport is negligible T=T(y) • Cf. R. Fitzpatrick, Phys. Plasmas 2, 825(1995) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  14. Resistivity Models II 1) Small Island case: 2) Non-relaxed Large Island case: 3) Relaxed Large Island case: ←Core Edge→ Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  15. Small Island Case • The shape of the flux surfaces is defined by Ampere’s law, that can be solved by employing a perturbative technique: x x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  16. Large Island Cases • Now, the Ampere’s law is: • where T(y) is given by the constrain condition: Metric term again ! Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  17. Small Island Relation • SI Saturation Relation: Hastie, Militello, Porcelli PRL (2005) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  18. Small Island Relation • SI Saturation Relation: • Thyagaraja: • Pletzer et al.: Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  19. Non-Relaxed Large Island Relation • NRLI Saturation Relation: • No log(w) and A contributions! • The thermal boundary layer around the separatrix (where h≠h(y) ) brings them back (but multiplied by wc). Hastie, Militello, Porcelli PRL (2005) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  20. Relaxed Large Island Relation • RLI Saturation Relation: • Similar to the NRLI relation • log(w) contribution from the outer solution, not from the inner solution as in the SI case! • The thermal boundary layer around the separatrix (where h≠h(y) ) would introduce additional log(w) and A terms. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  21. The Complete Model • The full model contains many additional effects. • The Generalized Rutherford Equation is obtained by using strong physical assumptions. • The effect of rotation is not completely clarified. • With numerical investigations it is possible to check the assumptions and shed some light on the relevant physics. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  22. The 4-Field Model -The model evolves the 4 fields: j: Stream function y: Magnetic flux n: Perturbed density v: Parallel ion velocity -2D, slab geometry -Symmetric equilibrium -Constant h x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  23. Symmetric NTM • Complete model (Small Island Case): • And symmetric case: Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  24. Simplifying the Model… • Averaging Ohm’s law: • J is “almost” a function of y Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  25. Simplifying the Model… • Averaging Ohm’s law: • The density equation gives: From Ohm’s Law Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  26. Simplifying the Model… • Averaging Ohm’s law: • The density equation gives: Transport Equation Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  27. Numerical Matching • All the terms can be substituted in Min and evaluated numerically. • The islands rotates at and the Polarization term is small. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  28. Numerical Matching • All the terms can be substituted in Min and evaluated numerically. • The islands rotates at and the Polarization term is small. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  29. The bootstrap current term must be corrected. Boundary Conditions Magnetic field – Contour plot Numerical solution: -spectral code, -double periodicity, x x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  30. Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  31. Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  32. Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

  33. Conclusions • A systematic investigation of the saturation of the NTMs has been carried out with theoretical and numerical tools. • New terms describing the asymmetric saturation have been added to the Generalized Rutherford Equation. • Theoretical models have been compared to the numerical data obtained with solving the complete symmetric model. Three new saturation relations describing different physical scenarios Good agreement but the position of the tangent bifurcation is not well predicted Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

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