Coupled radius-energy turbulent transport of alpha particles

Transcription

1 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 / 28

2 Outline Wilkie (Maryland) Coupled transport 24 July / 28

3 Overview of alpha particle physics Motivation 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 / 28

4 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 / 28

5 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 / 28

6 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 / 28

7 Overview of alpha particle physics The radial flux of alpha particles is a strong function of energy φ(r) φ R α y / ρ i ρ α ρ i Y α / ρ i x / ρ i 10 1 Γ(E) X α / ρ i Γ p = Γ(E) d 3 v E / E α Wilkie (Maryland) Coupled transport 24 July / 28

8 Overview of alpha particle physics Transport can compete with collisional slowing-down at moderate energies (a) Characteristic time (seconds) (b) τ c / τ Γ v c v/v td 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 v/v td Wilkie, et al, JPP (2015) Wilkie (Maryland) Coupled transport 24 July / 28

9 Low-collisionality gyrokinetics Outline Wilkie (Maryland) Coupled transport 24 July / 28

10 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 / 28

11 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 / 28

12 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 / 28

13 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 / 28

14 Trace transport Outline Wilkie (Maryland) Coupled transport 24 July / 28

15 Trace transport Alpha particles are trace and do not affect the linear or nonlinear electrostatic physics at realistic concentrations Cyclone base case, electrostatic (a) 10 1 γ R/v ti Slowing-down Equiv. Maxw. Diluted ions n α /n e Q tot / Q GB Slowing-down Equiv. Maxw n α =0.002n e n α =0.01n e 0.06 Diluted ions n α =0.1n e γ R/v ti n α =20% n e t v ti /R 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 / 28

16 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 / 28

17 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 / 28

18 Trace transport Proof of principle: Reconstruct non-maxwellian flux in ITG turbulence from Maxwellian Γ α (E) / φ 2 (arb. units) Quasilinear: Slowing-down Maxwellian Corrected maxw E / E α Γ(E) Fully turbulent Reconstructed Direct simulation E / E α Wilkie (Maryland) Coupled transport 24 July / 28

19 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 / 28