HIGH-FREQUENCY KINETIC INSTABILITIES DRIVEN BY ANISOTROPIC ELECTRON BEAMS 20 March 2013 Anna Kómár 1, Gergő Pokol 1, Tünde Fülöp 2 1)Department of Nuclear Techniques, Budapest University of Technology and Economics, Association EURATOM 2)Department of Applied Physics, Chalmers University of Technology and Euratom-VR Association I. Chalmers Meeting on Runaway Electron Modelling

 Description of the instability  Runaway electron distributions  Wave dispersions  Growth rate of the waves  Critical runaway densities  Plans, current problems Outline 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 2/29

Runaway distribution 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling T. Fülöp, PoP 13(062506), 2006 p = γv e /c normalized momentum 3/29 Anisotropic → Instabilities

20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling response Imaginary part Growth rate Particle-wave interaction Dispersion of the plasma waves: (homogeneous plasma) Dielectric tensor Perturbative analysis k : wave number ω : wave frequency c : speed of light 4/29

General resonance condition Ultrarelativistic resonance condition Generalizations 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling Magnetosonic- whistler wave Electron-whistler, EXEL wave High electric field distribution function Near-critical field distribution function 5/29

Near-critical distributionAvalanche distribution Distribution function 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling T. Fülöp, PoP 13(062506), 2006P. Sandquist, PoP 13(072108), /29 p = γv e /c normalized momentum

Distribution function  Qualitatively similar  Lower electric field → less anisotropy  Growth rate of the waves:  Ensured by the anisotropy  Not affected by the details 20 March /29 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling

Electron plasma waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling  Relaxing the electromagnetic approximation:  Approximations for the dielectric tensor: 8/29 T.H. Stix, Waves in Plasmas (AIP, 1992)

Electron plasma waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling  Dielectric tensor:  Wave dispersion: 9/29

Electron plasma waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 10/29  Qualitatively different for B > 2.6 T

Electron plasma waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 11/29

Electron plasma waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 12/29  Cold plasma approximation is used (th. motion << gyro-motion)  Validity: 20 keV, 10 keV, 1 keV

Calculating the growth rate of the waves 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 13/29 Unperturbed dispersion: Perturbed dispersion: The wave frequency changes: → Calculating the runaway susceptibilities:

General resonance condition 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling  Resonance condition: implicit UltrarelativisticGeneral case 14/29 T.H. Stix, Waves in Plasmas (AIP, 1992)

Restrictions on the wave dispersion 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling Resonant momentum is physical if: (1) (2) and Whistler and high-k region of EXEL 15/29 and

Growth rate in near-critical field (Whistler) 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling Maximum out of the EW region of validity 16/29 E/E c = 1.3 B = 2 T n e = 5 ∙ m -3 n r = 3 ∙ m -3 Order of resonance:

Growth rate in near-critical field (EXEL) 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling No growth rate for k < 1300 m -1 17/29 E/E c = 1.3 B = 2 T n e = 5 ∙ m -3 n r = 3 ∙ m -3 Order of resonance: k || c = ω 0

Growth rate in near-critical field 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling Electron-whistler waveExtraordinary electron wave Near-critical case → max. energy (2.6 MeV, p = 5) 18/29 most unstable wave E/E c = 1.3, B = 2 T, n e = 5 ∙ m -3, n r = 3 ∙ m γ / ω ce

Damping rates of the wave, stability 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling  Collisional damping:  Convective damping:  The runaway beam has a finite radius, L r 19/29 M. Brambilla, PoP 2(1094), 1995 G. Pokol, PPCF 50(045003), 2008

Critical density (Whistler) 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling  Stability: Finding (Growth rate – Damping rates) = 0 (for the most unstable wave) Critical runaway density 20/29 T = 20 eV T = 1000 eV Unstable Stable E/E c = 1.3 n e = 5 ∙ m -3

Critical density (EXEL) 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 21/29  Orders of magnitude lower critical density  Break at B ~ 2.6 T (for high temperature) T = 20 eV T = 1000 eV E/E c = 1.3 n e = 5 ∙ m -3

What is different with EXEL? 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 22/29  Slightly higher growth rate  Orders of magnitude lower convective damping  Collisional damping smoothes this effect for low T E/E c = 1.3, n e = 5 ∙ m -3, n r = 3 ∙ m -3

What is different with EXEL? 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 23/29 E/E c = 1.3, n e = 5 ∙ m -3, n r = 3 ∙ m -3 Parameters of the most unstable wave change from θ ~ 0

What is different with EXEL? 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 24/29  For the parameters of the most unstable wave:

 Wave instability in near-critical electric field  Generalization  Relaxing the electromagnetic approximation  General resonance condition  Linear stability  The most unstable wave is dependent on the maximum runaway energy  Stability threshold is significantly lower for the EXEL 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling Conclusions 25/29

Plans, current problems 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 26/29  Growth rates for high electric field  Avalanche runaway distribution  No (or much higher) maximum energy  Whistler wave Electron-whistlerMagnetosonic-whistler T. Fülöp, PoP 13(062506), 2006 E/E c = 865 B = 2 T n e = 5 ∙ m -3 n r = 3 ∙ m -3

 Extraordinary electron wave  Most unstable wave would be in the region of no-growth rate (if there is a maximum runaway energy) Plans, current problems 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 27/29 Maximum runaway energy k || c = ω 0 This is not a problem!

Plans, current problems 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 28/29  For k || c > ω 0 :  n ≤ 0   For k || c < ω 0 :  n > 0  

Plans, current problems 20 March 2013 A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling 29/29 k || c > ω 0 k || c < ω 0 n = 0, -1 n = 1  Maximum at k || c = ω 0 : with n ≠ 0