Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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

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

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

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

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

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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 Simplified Model Equations Vorticity equation: Ohms law: Energy equation

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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. M out ( )M in ( )

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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( ) Cf. R. Fitzpatrick, Phys. Plasmas 2, 825(1995)

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Large Island Cases Now, the Amperes law is: where T( ) is given by the constrain condition: Metric term again !

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

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

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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 ) would introduce additional log(w) and A terms.

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

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… Averaging Ohms law: J is almost a function of

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… Averaging Ohms law: The density equation gives: From Ohms Law

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… Averaging Ohms law: The density equation gives: Transport Equation

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

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

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

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 C b =1 C b =2 C b =1.4

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 C b =1 C b =2 C b =1.4

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 C b =1 C b =2 C b =1.4

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 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