Tutorial: Incorporating kinetic aspects of RF current drive in MHD simulation
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- Joshua Mathews
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1 kinetic aspects of RF current with a focus on ECCD stabilization of tearing modes RF current Lorentz Workshop: Modeling Kinetic Aspects of Global MHD Modes 4 Dec 2013, Leiden, Netherlands
2 Outline radio frequency (RF) heating electron cyclotron current (ECCD) suppression of tearing modes Review of models for ECCD flux function model of n current (Yu, Günter) anisotropic diffusive model (Giruzzi, Yu, Gianakon) RF force model (Kruger, Jenkins) convective model (Pratt, ) basics of EC current two equation convective model single equation convective model results from the reduced MHD JOREK Discussion of the physics RF current
3 Tearing Modes Magnetic reconnection breaks up the nested flux surfaces in a tokamak. This creates regions of closed magnetic field lines, called magnetic islands. The tearing instability produces magnetic islands that grow in size. Large islands increase radial transport, cause loss of confinement. A neoclassical tearing mode, in contrast to a classical tearing mode, is n by reduction of bootstrap current. RF current
4 Tearing Mode Suppression Maraschek 2012 Nuc. Fus. Suppression can be accomplished by replacing current inside the islands using: electron cyclotron resonance heating/current, lower hybrid resonance heating/current. Microwave power is injected at the electron cyclotron resonance frequency. ECCD targets the island center, at a surface of rational safety factor q around which the island forms. RF current
5 Electron cyclotron current A small group of electrons resonates with the RF waves. The current produced is localized tight control over current profile. ECCD produces a steady-state, non-maxwellian distribution of electrons. RF current Figure from: Pletzer and Perkins. Phys. Plasmas, Stabilization of neoclassical tearing modes using a continuous localized current. Nice short review: La Haye. Phys. Plasmas Neoclassical tearing modes and their control.
6 Flux Function Model Yu and Günter (PPCF 1998) model the contribution to the helical magnetic flux ψ Dψ Dt = E 0 + η(j j BS j EC ), η and j EC : functions of ψ, centered at O-point. RF current (b), (c), and (d) curves use different width and intensity of EC current Result: width of magnetic island abruptly drops when simple ECCD is applied, partial suppression of NTM!
7 Anisotropic Diffusive Model Sometimes referred to as the Giruzzi model, this model derives from the bounce-averaged kinetic equation for electrons: collisions EC waves radial diffusion f e = C(f e ) + Q EC (f e )+ 1 t ρ ρ ρd f e r ρ + ee Expressed as an evolution equation for the electron cyclotron current, this model includes parallel and perpendicular diffusive terms: Dψ Dt j EC t = E + η(j j BS j EC ) f e p = ν(j j EC ) + (χ j EC ) + (χ j EC ) RF current Giruzzi, G. PPCF Modelling of RF current in the presence of radial diffusion. Giruzzi, G., et al. Nuc. Fus Dynamical modelling of tearing mode stabilization by RF current.
8 Anisotropic Diffusive Current Drive Results Yu, et al. Phys Plasmas, 2000 & (TM code) Gianakon. Phys Plasmas (NIMROD code) RF current Even with the complicated equations mentioned above, the time evolution of RF current has not been completely described by our model...the major purpose of the present paper is on the stabilization of the NTM s by the RF current rather than on the RF current physics itself. dotted source at O-point dashed rotating source
9 RF force model ρ Du = p + j B Π t RF force E + u B = ηj + F rf e /n q e 3 2 ndt Dt = p v q + Π : u + Q + S rf RF energy (1) ignore the electron stress tensor (2) use Braginskii closure for parallel heat flux (3) ignore heat flux contributions to resistivity. Hybrid code: NIMROD (extended MHD) + ray-tracing for RF force: RF current w 4 B(n = 1) 2 results show reduction of magnetic island size. Jenkins et al. Phys. Plasmas Kruger et al. in Proceedings of the 5th IAEA Technical Meeting on the Theory of Plasma Instabilities, Austin, Texas, USA, 2011
10 Outline radio frequency (RF) heating electron cyclotron current (ECCD) suppression of tearing modes Review of models for ECCD flux function model of n current (Yu, Günter) anisotropic diffusive model (Giruzzi, Yu, Gianakon) RF force model (Kruger, Jenkins) convective model (Pratt, ) basics of EC current two equation convective model single equation convective model results from the reduced MHD JOREK Discussion of the physics RF current
11 Description of EC current Gyrophase-averaged kinetic equation for electrons: f e t collisions = C(f e ) + Q EC (f e ) v f e averaged EC wave-effect Q EC : quasi-linear diffusion model electron distribution is permitted to convect along the magnetic field. Hegna and Callen. Phys. Plasmas Two-fluid MHD equations are produced by taking moments: ( ) m s n s t + u s u s = p + n s q s (E s + u s B) RF current ( ) 3 2 n s t + u s Π s + R s + F rf stress s collisional RF force friction heat flux T s = n s T s u s q s RF energy Π s : u s + Q s + S rf s collisional energy exchange
12 Choice of operators For the collision operator we use a simple Krook operator: C(f e ) = ν(v)(f e f M ). We assume that the quasi-linear diffusion is non-relativistic and dominantly in the perpendicular direction, reasonable for EC resonance. Thus Q EC (f e ) = v D EC v f e, D EC Dδ(v v res )ˆv ˆv, RF current where D is a constant, and v res = (ω nω e )/k is the parallel velocity of the resonant electrons.
13 Effect of ECCD on electrons ECCD produces a perturbation δf e in the gyrophase-averaged electron distribution function. This perturbation: creates zero net parallel momentum. is localized at the electron cyclotron resonant parallel velocity. in perpendicular velocity takes the form of a velocity space hole for v < 2v th and a bump for v > 2v th. is convected along the magnetic field lines out of the deposition region. RF current δf e (v par =v EC,v ) bump 0 hole v 1 2v th v 2
14 Asymmetric Resistivity Collisions would eventually return the distribution function to the equilibrium Maxwellian state: δf e 0. But the asymmetric energy exchange between waves and electrons (heating electrons moving in one toroidal direction) produces an asymmetric collision rate, and thus an asymmetric resistivity. The hole is filled in more quickly than the bump is eroded, because of the velocity dependence of the collision frequency. RF current This is the Fisch-Boozer current mechanism, the dominant mechanism for ECCD. Net current decays at the slower collision rate of the high velocity electrons in the bump.
15 Developing a fluid model that uses asymmetric resistivity δf e can be reasonably represented by two delta functions at perpendicular velocities v 1 and v 2 (representing the hole and bump respectively) with different collision rates. We take a moment of the kinetic equation to get the EC current evolution: j EC = e R δfe e v t m j EC, e j EC (x, t) = e d 3 vv δf e, R δfe e = d 3 vm e vν(v)δf e. RF current Here R δfe e is the transient electron-ion friction associated with the EC-n quasi-linear modification of the distribution function.
16 Two equation convective model We model R δfe e in the standard way as the sum of a current generation and collisional decay. This produces a convective model with two equations for the EC current: Dψ = E 0 + η(j j Dt BS (j EC2 + j EC1 )) j EC1 t j EC2 t 4 parameters to be tuned: = S EC ν 1 j EC1 + v res j EC1 = +S EC ν 2 j EC2 + v res j EC2 RF current current source: S EC parallel velocity of the resonant electrons : v res collision frequency of the resonant electrons that contribute to the EC n current: ν 1 and ν 2
17 Single equation convective model Dψ Dt j EC t = E 0 + η(j j BS j EC ) = S ν 2 j EC + v res j EC j EC is the sum of j EC1 and j EC2. current source S is the flux surface average of S EC in the limit where the collision time of the slower electrons is sufficiently long that they travel around the entire flux surface. RF current parallel velocity of the resonant electrons : v res A more detailed discussion of the single equation model is presented in: E. and J. Pratt. Expression of electron cyclotron current in plasma fluid models. Proceedings of the 40th EPS Conference on Plasma Physics. Espoo, Finland, July 1st 5th 2013.
18 jec Evolution of the EC n current density along a field line current 2 equation model 1 equation model Time is normalized to the collision time and length is normalized to a electron-thermalmean-free-path. Parameters: v res = 2 v 1 = 0 ν 1 = 1/8 v 2 = 2.7 ν 2 = 1/38 RF current x/mean free path The EC n current is generated as the perturbation δf e flows out of the EC power deposition region, 0 x 10 3.
19 About JOREK Over the last decade, the 3D nonlinear JOREK has been developed by an international developers group centered at ITER/Cadarache. work performed with JOREK: edge localized modes, resonant magnetic perturbations, pellet pacing, resistive wall modes, disruptions poloidal plane treated with 2D Bezier finite elements (based on bicubic Bezier surfaces a generalization of cubic Hermite elements, elements are aligned with magnetic flux surfaces) toroidal direction treated with Fourier decomposition fully-implicit time-stepping (choice of Crank-Nicholson, BDF1 Implicit Euler, or BDF2 Gears method) RF current
20 Results: classical tearing mode suppression in JOREK RF current w/a without ECCD with ECCD ECCD application JOREK reduced MHD : high resistivity, low viscosity, low collision frequency, 8 toroidal harmonics t(s)
21 Results: RF current physics RF current
22 Results: RF current physics RF current
23 Results: RF current physics RF current
24 Results: RF current physics RF current
25 Results: RF current physics RF current
26 Results: RF current physics RF current
27 Results: RF current physics RF current
28 Results: RF current physics RF current
29 Results: RF current physics RF current
30 Results: RF current physics RF current
31 Results: RF current physics RF current
32 Results: RF current physics RF current
33 Results: RF current physics RF current
34 Results: RF current physics RF current
35 Results: RF current physics RF current
36 Results: RF current physics RF current
37 Results: RF current physics RF current
38 Results: RF current physics RF current
39 Results: RF current physics RF current
40 Results: RF current physics RF current
41 Results: RF current physics RF current
42 Results: RF current physics RF current
43 How do these models compare and what physics is important for practical results? j EC convection model = S ν 2 j t EC + v res j EC j EC = ν(j j t EC ) + (χ j EC ) + (χ j EC ) anisotropic diffusion model [ ( ) ] ψ ψ(r o ) 2 j EC = C exp 2 ψ(r o ) ψ(r o d/2) flux function model RF current Du s ρ s Dt ρ Du t RF force model H & C two-fluid eqs = p + n s q s (E s + u s B) Π s + R s + F rf s = p + j B Π E + u B = ηj + F rf e /n q e
44 Discussion What are the limits of validity for the given approximations/assumptions? How accurate does the physics of the RF current need to be to capture relevant features of NTM supression? What is the role of self-induction of the current produced by RF current? RF current How do we decide that the time evolution of the RF current is sufficiently described?... to predict power required in a tokamak, time required to reduce the island, minimum island width possible with a realistic RF source?
45 RF current Thanks! Many thanks to G.T.A. Huysmans, Marina Bécoulet, Matthias Hölzl, Wolf-Chrisian Müller and the participants of the ASTER project and the JOREK collaboration. This work was performed on the Helios system at the system at Computational Situational Centre, International Fusion Energy Research Centre (IFERC-CSC), Rokasho-Japan and the Cartesius system, the Dutch national supercomputer, at SURFsara, Amsterdam, Netherlands. The work in this tutorial talk has been performed in the framework of the NWO-RFBR Centre of Excellence (grant ) on Fusion Physics and Technology. This work, supported by the European Communities under the contract of Association between EURATOM/FOM, was carried out within the framework of the European Fusion Programme. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
46 References La Haye. Phys. Plasmas Neoclassical tearing modes and their control. Giruzzi, G. Modelling of RF current in the presence of radial diffusion. PPCF Giruzzi, G., et al. Nuc. Fus Dynamical modelling of tearing mode stabilization by RF current. Gianakon, T. A. Limitations on the stabilization of resistive tearing modes. Physics of plasmas 8 (2001): Hegna and Callen. Phys. Plasmas A closure scheme for modeling RF modifications to the fluid equations. Yu, Günter, Giruzzi, et al. Phys Plasmas, Modeling of the stabilization of neoclassical tearing modes by localized radio frequency current. Yu, Zhang, and Günter. Phys Plasmas Numerical studies on the stabilization of neoclassical tearing modes by radio frequency current. E. and J. Pratt. Expression of electron cyclotron current in plasma fluid models. Proceedings of the 40th EPS Conference on Plasma Physics. July 1st 5th RF current
47 Reduced MHD Formulation in JOREK Vector fields are represented in terms of u (velocity stream function) and ψ (poloidal magnetic flux): RF current B = ê φ 1 R ψ + F 0/Rê φ v = ê φ R u + v B JOREK solves for 6 scalar variables: also toroidal current density j, toroidal vorticity ω, density ρ, temperature T.
48 Reduced MHD Equations Poisson bracket [u, ψ] = ru zψ zu rψ ψ t + R [u, ψ] = η(j j BS,0 E 0 /η j EC ) F 0 u φ. e φ [ρ v t = ρ(v )v p + j B + µ v] j φ = R 2 (R 2 ψ) ρ t ρ T t ω = 2 polu = (ρv) + (D ρ) + S ρ = ρv T (κ 1)p v + (K T + K T ) + S T RF current K, are the parallel and perpendicular heat diffusivity, and κ = 5/3 is the ratio of specific heats. S ρ is a particle sources and S T is a heat source.
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