Making relativistic shocks with a spectral 1D PIC code UZEIN I. - PowerPoint PPT Presentation

Making relativistic ⊥ shocks with a spectral 1D PIC code UZEIN I. Plotnikov (IPAG), B. Lembège (LATMOS) october 3th 2012 1/1 PIC IAP 2012

Particles-In-Cells (PIC) codes : Principles Birdsall&Langdon 1985 “Plasma Physics via Computer Simulation” , Dawson 1983 Scheme of 1 iteration Finite size macroparticles : shape Eq. mvt. (x, v) Interpolation factors Interpolation grille -> part. part. -> grille Time step : ∆ t < 0 . 2 ω − 1 pe ; Force Lorentz Densité Grid step : ∆ x ∼ λ D ; sur grille et courant CFL stability criterion : c ∆ t < ∆ x ; Relativistic bias : grid-Cerenkov Eq. Maxwell. radiation Champs Elec. et Magn. 2/1 PIC IAP 2012

UZEIN 1D code : Lembège & Dawson 1986 Normalisation : Move particles : d � p i � p i q i ( � γ i m i c × � x = E + B ) x ˜ = d t ∆ ˜ = ω pe t t Solved by time-centered finite difference method. v ˜ v = ω pe ∆ Update fields on grid ( x → k transformed) : p α p α ˜ = m α ω pe ∆ E x ( k x ) = − i 4 πρ ( k x ) / k x q E ˜ ∂ B y E = = ick x E x ( k x ) m e ω 2 pe ∆ ∂ t q B ∂ B z ˜ B = = − ick x E y ( k x ) m e ω 2 pe ∆ ∂ t ∂ E y J ˜ = − ick x B z ( k x ) − 4 π J y ( k x ) = J ∂ t q ω pe ∆ ∂ E z J � � = ick x B y ( k x ) − 4 π J z ( k x ) and not ∂ t n 0 q ω pe ∆ 3/1 PIC IAP 2012

Input parameters and physical quantities Example set of input parameters for Hada et al. 03 simulation : Param. Valeur Quant. Valeur ∆ x 1 ˜ ρ L , e 0.4 0.1 ω − 1 ∆ t ρ L , i ˜ 26.74 pe ˜ c 3 ∆ x · ω pe ω pe 1 n 0 50 part/ ∆ /spec ω pi 0.1 J 0 -110 λ De = V th , e 0.2 B 0 1.5 ω ce 0.5 Θ 0 90 ω ci 0.006 m i / m e 84 V A 0.149 T e / T i 1.58 β e 0.0356 4/1 PIC IAP 2012

A realistic simulation ? Some numbers : For an electron-proton shock in the ISM ω pe ≃ 10 4 s − 1 we need : ∆ t inf 10 − 4 ω − 1 pe 1 m i / m e = 1835 ρ I = 10 5 ∆ 2 c / v A ≃ c / c s ≃ 10 4 1 Ω CI ∼ 310 9 ∆ t 3 T simulation ≥ Ω − 1 ci . Box length = 10 6 − 10 8 ∆ , and N p > 10 8 In the code fiducial setup we have : ∆ t = 0 . 1 ω − 1 pe 1 m i / m e ∼ 100 ρ I ∼ 10 2 ∆ 2 c / v A ≃ c / c s ≃ 10 1 Ω CI = 10 2 − 10 3 ∆ t 3 T simulation ≥ Ω − 1 ci . Box length = 10 4 ∆ , and N p < 10 7 → Need to focus on relevant physics. 5/1 PIC IAP 2012

1D PIC Shock structure Non-relativistic... Kinetic structure 1D PIC shock Downstream Upstream Vs Reflected Ions Thermalized ions and elecs Rankine-Hugeniot conditions ± ok. Px Shock front reformation (non-rel). Blue: ions Black: elecs Electron heating. X 6/1 PIC IAP 2012

3 different methods tested 1 Magnetic Piston 2 Reflexion on wall 3 Beam injection in the plasma at rest 7/1 PIC IAP 2012

Piston method (1) Electron-ion simulation (phase space X-Px) Electron-positron simulation ( V s = 0 . 97 c ) (e.g. Hada et al. 03) 8/1 PIC IAP 2012

Piston method (2) Relativistic electron-ion shock ? Need J 0 >> 0, non-linear behavior V s = f ( J 0 ) . Difficult to control the shock speed and go up to the relativistic regime. Other methods tested. 9/1 PIC IAP 2012

Reflexion of bulk plasma on a wall (1) Most popular method, but not suited to the spectral code... k = 0 problem. v in � v × � Difficult to handle � E 0 + � B 0 = 0 in all plasma components. Moderate injection speed + low Moderate injection speed + high magnetisation : magnetisation : 10/1 PIC IAP 2012

Beam injection in a plasma at rest (1) Blue: ions x-px Plasma at rest Vacuum Vacuum Black: elecs x-px n0=40 part/cell/species t=0 Vacuum Vacuum Plasma at rest 0<t<T(form) Beam inj. (n_b) Down SF Vacuum CD Vacuum Up t>>T(form) 11/1 PIC IAP 2012

Beam injection (2) : non-relativistic "Benchmarks" Biskamp & Welter 72 (Whistlers) Hoshino & Shimada 2002 Hada et al. 03 Scholer & Matsukiyo 04 12/1 PIC IAP 2012

Beam injection (3) : Variyng the γ beam Parameters : n b = n 0 , ˜ c = 3, γ b ∈ [ 1 , 30 ] . Example : γ b = 3 . 12 V S / c = 0 . 76, γ S = 1 . 54 Buneman instability in the shock foot. Shock reformation at τ ref ∼ τ ci / 3. 13/1 PIC IAP 2012

Beam injection (4) : Variyng the γ beam Parameters : n b = n 0 , ˜ c = 3, γ b ∈ [ 1 , 30 ] . Shock formation time and shock speed as function of the beam speed. m i / m e .vs. T form γ b .vs. T form γ b .vs. γ S γ s .vs. n d / n 0 14/1 PIC IAP 2012

Beam injection (5) : Highly magnetised plasma Animation : red line : B z / 4, magenta : � E x / 4. Black dots u x , elec / m i / m e . Distributions downstream V S ≃ 0 . 9 c . 15/1 PIC IAP 2012

Conclusions and Perspectives 1 Difficult to deal with initial inhomogeneous plasma drifts in a spectral code. 2 Time formation for a relativistic shock... 3 Electron acceleration ? 16/1 PIC IAP 2012

Thank You ! 17/1 PIC IAP 2012