Coupled radius-energy turbulent transport of alpha particles George Wilkie, Matt Landreman, Ian Abel, William Dorland 24 July 2015 Plasma kinetics working group WPI, Vienna Wilkie (Maryland) Coupled transport 24 July 2015 1 / 28
Outline Wilkie (Maryland) Coupled transport 24 July 2015 2 / 28
Overview of alpha particle physics Motivation www.euro-fusion.org The fate of alpha particles is critical to a burning reactor Fast ion destabilize Alfvén eigenmodes Damage to wall if energetic particles escape Possible sources of alpha particle transport: Neoclassical ripple loss Coherent modes (TAEs, sawteeth, etc.) Anomalous transport from turbulence Wilkie (Maryland) Coupled transport 24 July 2015 3 / 28
Overview of alpha particle physics What has been done? Estrada-Mila, Candy, Waltz (2006): Performed transport calculations with an equivalent Maxwellian Angioni, Peeters (2008): Developed a quasilinear model for alpha transport Hauff, Pueschel, Dannert, Jenko (2009): Scalings of turbulent transport coefficients as a function of energy Albergante, et al. (2009): Efficient coefficient-based model for transport of Maxwellian alphas. Waltz, Bass, Staebler (2013): Quasilinear model allowing radial transport to compete with collisional slowing-down. Bass and Waltz (2014): Stiff transport model for TAEs along with Angioni model for anomalous transport Wilkie (Maryland) Coupled transport 24 July 2015 4 / 28
Overview of alpha particle physics There is an analytic approximation to the alpha particle slowing-down distribution If alpha particle transport is weak compared to collisions: F 0α t = C [F 0α (v)] + σ 4πvα 2 δ(v v α ) We can obtain the slowing-down distribution as a steady-state by approximating the collision operator valid for v ti v v te : n α F s (v) = A vc 3 + v 3 for v < v α This does not capture the Maxwellianization and buildup of He ash Gaffey, JPP (1976) Helander and Sigmar (2002) Wilkie (Maryland) Coupled transport 24 July 2015 5 / 28
Overview of alpha particle physics Anatomy of the alpha particle distribution Different physical processes will be dominant at different energy scales: Transport? Maxwellian Ash (Ions) Collisional drag (Electrons) Source Wilkie (Maryland) Coupled transport 24 July 2015 6 / 28
Overview of alpha particle physics The radial flux of alpha particles is a strong function of energy φ(r) φ R α 140 140 120 120 y / ρ i 100 80 60 40 20 ρ α ρ i Y α / ρ i 100 80 60 40 20 0 0 20 40 60 80 100 120 140 x / ρ i 10 1 Γ(E) 10 0 10-1 10-2 10-3 0 0 20 40 60 80 100 120 140 X α / ρ i Γ p = Γ(E) d 3 v 10-4 10-3 10-2 10-1 10 0 E / E α Wilkie (Maryland) Coupled transport 24 July 2015 7 / 28
Overview of alpha particle physics Transport can compete with collisional slowing-down at moderate energies (a) Characteristic time (seconds) (b) τ c / τ Γ 10 1 10 0 10-1 10-2 v c 10-3 0 2 4 6 8 10 12 14 v/v td 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 ash τ Γ τ c v α Define characteristic times such that: C[F 0 ] F 0 τ c (E) r Γ(E) F 0 τ Γ (E) When τ Γ < τ c, transport is dominant over collisions and the Gaffey slowing-down distribution is invalid 0 2 4 6 8 10 12 14 v/v td Wilkie, et al, JPP (2015) Wilkie (Maryland) Coupled transport 24 July 2015 8 / 28
Low-collisionality gyrokinetics Outline Wilkie (Maryland) Coupled transport 24 July 2015 9 / 28
Low-collisionality gyrokinetics Low-collisionality ordering of gyrokinetics Gyrokinetic equation (electrostatic): h s t +v b h s + v ds h s + c B b φ R h s + q h s φ R E t C[h s ] = q F 0s φ R c E t B (b φ R ψ) F 0s ψ Caveats: Ignoring the parallel nonlinearity out of convenience: would require a factor O [1/ρ ] more energy grid points For the moment, ignoring the collision operator because H-theorem is questionable for non-maxwellian equilibria. Trace approximation makes this okay? Wilkie (Maryland) Coupled transport 24 July 2015 10 / 28
Low-collisionality gyrokinetics Low-collisionality ordering of gyrokinetics Transport equation: 1 ( V ) 1 ( V F 0 + V ) 1 ( ) t V Γ ψ + EΓE = C [F 0 ] + S ψ E E ψ. Γ ψ and Γ E are the flux in radius and energy respectively. Both are functions of energy. Collision operator appears on the transport scale Solve for full F 0 instead of n 0, T 0 Wilkie (Maryland) Coupled transport 24 July 2015 11 / 28
Low-collisionality gyrokinetics Coupled GK-transport workflow Maxwellian, or other analytic distribution:, Q Turbulent dynamics (GS2, (GS2, GENE, GYRO, etc.) etc.) Equilibrium evolution (Trinity, TGYRO) n( ), u( ), T( ) Candy, et al. PoP (2009) Barnes, et al. PoP (2010) Parra, Barnes. PPCF (2015) Wilkie (Maryland) Coupled transport 24 July 2015 12 / 28
Low-collisionality gyrokinetics Coupled GK-transport workflow Velocity distribution not known a priori: Turbulent dynamics (GS2, (GS2, GENE, GYRO, etc.) etc.) Equilibrium evolution (?) (?) F (E, ) 0 Wilkie (Maryland) Coupled transport 24 July 2015 13 / 28
Trace transport Outline Wilkie (Maryland) Coupled transport 24 July 2015 14 / 28
Trace transport Alpha particles are trace and do not affect the linear or nonlinear electrostatic physics at realistic concentrations Cyclone base case, electrostatic. 0.20 0.15 (a) 10 1 γ R/v ti 0.10 10 0 0.05 0.00 Slowing-down Equiv. Maxw. Diluted ions 0.05 0.10 0.15 0.20 0.25 n α /n e Q tot / Q GB 10-1 0.08 0.07 Slowing-down Equiv. Maxw. 10-2 n α =0.002n e n α =0.01n e 0.06 Diluted ions n α =0.1n e γ R/v ti 0.05 0.04 0.03 n α =20% n e 0 50 100 150 200 250 300 350 400 t v ti /R 0.02 0.01 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 k y ρ i Tardini, et al. NF (2007) Holland, et al. PoP (2011) Wilkie, et al, JPP (2015) Wilkie (Maryland) Coupled transport 24 July 2015 15 / 28
Trace transport Method to calculate flux of an arbitrary equilibrium The (collisionless) GK equation can be written schematically as: F 0 L [h] = b v v + b F 0 ψ ψ where b v, b ψ, and L depend on the particular problem, but not on F 0. So we can write the radial flux as: F 0 Γ(E) = D v v D F 0 ψ ψ 1 Run a GK simulation with two additional species with the desired charge and mass, but at different temperatures or radial gradients 2 Use Γ(E) for each of these to calculate D v and D ψ as functions of energy 3 Plug in any desired F 0 (E, ψ), iterate with transport solver to find profile Wilkie (Maryland) Coupled transport 24 July 2015 16 / 28
Trace transport Two ways to obtain diffusion coefficients 1 Run with multiple species with different gradients Easier to implement, more computationally expensive 2 Obtain directly from φ (r, t) Requires solving the GK equation, one energy at a time, with φ given 3 Analytic estimate? Γ ψ = F 0 ψ σ [ ] L 1 φ R c 2 φ R πbdλ y B 2 y 1 λb φ R y b φ R ψ L t + (v b + v ds + c B b φ R ) t,ψ Wilkie (Maryland) Coupled transport 24 July 2015 17 / 28
Trace transport Proof of principle: Reconstruct non-maxwellian flux in ITG turbulence from Maxwellian Γ α (E) / φ 2 (arb. units) 10 0 10-1 10-2 10-3 10-4 10-5 10-6 Quasilinear: Slowing-down Maxwellian Corrected maxw. 0.0 0.2 0.4 0.6 0.8 1.0 E / E α Γ(E) 10 0 10-1 10-2 10-3 Fully turbulent Reconstructed Direct simulation 10-4 10-3 10-2 10-1 10 0 E / E α Wilkie (Maryland) Coupled transport 24 July 2015 18 / 28
Trace transport New tool in development to calculate coupled radius-energy transport of trace energetic particles T3CORE Written in Trace Turbulent Transport, COupled in Radius and Energy Solves for the steady-state or time-dependent F 0 (v, r) Utilizes the trace approximation to determine turbulent transport coefficients Runs on a laptop once the seed nonlinear simulations are performed Wilkie (Maryland) Coupled transport 24 July 2015 19 / 28