MHD turbulence in the solar corona and solar wind

MHD turbulence in the solar corona and solar wind Pablo Dmitruk Departamento de Física, FCEN, Universidad de Buenos Aires

Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms that accelerate charged particles in turbulent plasmas.

Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms that accelerate charged particles in turbulent plasmas. Relation to magnetic reconnection.

Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms that accelerate charged particles in turbulent plasmas. Relation to magnetic reconnection. Clues for dissipation mechanisms.

Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms that accelerate charged particles in turbulent plasmas. Relation to magnetic reconnection. Clues for dissipation mechanisms. Differential energization: T e T e, T i T i

Turbulence, magnetic reconnection, particle acceleration Understand the mechanisms that accelerate charged particles in turbulent plasmas. Relation to magnetic reconnection. Clues for dissipation mechanisms. Differential energization: T e T e, T i T i Direct approach: Test particles trajectories are followed in the turbulent fields obtained from a direct numerical solution of the MHD equations. Dmitruk, Matthaeus, Seenu, ApJ 617, 667 (2004) Dmitruk, Matthaeus, Seenu, Brown, ApJ 597, L81 (2003) Ambrosiano, Matthaeus, Goldstein, Plante, JGR 93, 14383 (1988)

Advantages Turbulent fields are not modeled but a direct result of the MHD simulations: energy cascade, coherent structures

Advantages Turbulent fields are not modeled but a direct result of the MHD simulations: energy cascade, coherent structures Current sheets (reconnection) are a natural result of the (resistive) MHD evolution (i.e. not set up as an initial condition)

Advantages Turbulent fields are not modeled but a direct result of the MHD simulations: energy cascade, coherent structures Current sheets (reconnection) are a natural result of the (resistive) MHD evolution (i.e. not set up as an initial condition) Disadvantages Extremely computationally demanding if we want to fully resolve from turbulent (MHD) to particle scales

Advantages Turbulent fields are not modeled but a direct result of the MHD simulations: energy cascade, coherent structures Current sheets (reconnection) are a natural result of the (resistive) MHD evolution (i.e. not set up as an initial condition) Disadvantages Extremely computationally demanding if we want to fully resolve from turbulent (MHD) to particle scales Lack of self-consistency in the MHD-kinetic physics interplay at the dissipative scales

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations b(x, y, z, t), v(x, y, z, t) fluctuating fields, with a background magnetic field B 0 ẑ, such that B = B 0 ẑ + b and we set B 0 = 8 δb, δb = b 2 1/2.

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations b(x, y, z, t), v(x, y, z, t) fluctuating fields, with a background magnetic field B 0 ẑ, such that B = B 0 ẑ + b and we set B 0 = 8 δb, δb = b 2 1/2. Start with some initially large scale eddies: energy containing scale (turbulent correlation length) L = 1/2π box size. Periodic boundary conditions.

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations b(x, y, z, t), v(x, y, z, t) fluctuating fields, with a background magnetic field B 0 ẑ, such that B = B 0 ẑ + b and we set B 0 = 8 δb, δb = b 2 1/2. Start with some initially large scale eddies: energy containing scale (turbulent correlation length) L = 1/2π box size. Periodic boundary conditions. Set v 2 1/2 = v 0 = δb/ 4πρ the plasma alfvenic speed based on the fluctuations (distinct from v A = B 0 / 4πρ the alfvenic speed based on the DC field, v A = 10v 0 ). Plasma β = v 2 T /v2 A = 0.16 v T = 4v 0, Mach number M s = v 0 /c s = 0.25

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations b(x, y, z, t), v(x, y, z, t) fluctuating fields, with a background magnetic field B 0 ẑ, such that B = B 0 ẑ + b and we set B 0 = 8 δb, δb = b 2 1/2. Start with some initially large scale eddies: energy containing scale (turbulent correlation length) L = 1/2π box size. Periodic boundary conditions. Set v 2 1/2 = v 0 = δb/ 4πρ the plasma alfvenic speed based on the fluctuations (distinct from v A = B 0 / 4πρ the alfvenic speed based on the DC field, v A = 10v 0 ). Plasma β = v 2 T /v2 A = 0.16 v T = 4v 0, Mach number M s = v 0 /c s = 0.25 Reynolds numbers R = v 0 L/ν = R m = v 0 L/µ = 1000 (limited by numerical resolution at 256 3 )

MHD Simulations We use turbulent fields obtained from a Direct Numerical Solution (pseudospectral code) of the compressible 3D MHD equations b(x, y, z, t), v(x, y, z, t) fluctuating fields, with a background magnetic field B 0 ẑ, such that B = B 0 ẑ + b and we set B 0 = 8 δb, δb = b 2 1/2. Start with some initially large scale eddies: energy containing scale (turbulent correlation length) L = 1/2π box size. Periodic boundary conditions. Set v 2 1/2 = v 0 = δb/ 4πρ the plasma alfvenic speed based on the fluctuations (distinct from v A = B 0 / 4πρ the alfvenic speed based on the DC field, v A = 10v 0 ). Plasma β = v 2 T /v2 A = 0.16 v T = 4v 0, Mach number M s = v 0 /c s = 0.25 Reynolds numbers R = v 0 L/ν = R m = v 0 L/µ = 1000 (limited by numerical resolution at 256 3 ) Run the code for two t 0 = L/v 0 (eddy turnover times), fully turbulent state developed, take a snapshot for pushing the test particles.

MHD turbulence energy spectrum: obtained 10 1 10 2 E k 10 3 10 4 10 5 1 10 100 k

MHD spatial structure: parallel current density Jz

Cross-sections: magnetic field over current density xz plane xy plane

Equations of motion for charged particles: Particles du dt = q m (1 c u B + E), dx dt = u

Equations of motion for charged particles: Particles du dt = q m (1 c u B + E), dx dt = u with electric field E = 1 c v B + v 0L J = inductive + resistive R m c

Equations of motion for charged particles: Particles du dt = q m (1 c u B + E), dx dt = u with electric field E = 1 c v B + v 0L J = inductive + resistive R m c Particle gyrofrequency w g = qb/(mc) (small) lengthscale r g = u /w g = u mc/(qb). If u = v 0, B = B 0 gyroradius r 0 = v 0 mc/(qb 0 ).

Turbulent length scales vs particle scales Dissipation length l d ρ ii (ion skin depth) Supported by solar wind observations, linear Vlasov theory, kinetic physics reconnection studies. ρ ii = ion skin depth = c/w pi = c/ e 2 4πn/m p

Turbulent length scales vs particle scales Dissipation length l d ρ ii (ion skin depth) Supported by solar wind observations, linear Vlasov theory, kinetic physics reconnection studies. ρ ii = ion skin depth = c/w pi = c/ e 2 4πn/m p ρ ii /L 1 for astrophysical and space plasmas (ρ ii /L 10 5 for solar wind)

Turbulent length scales vs particle scales Dissipation length l d ρ ii (ion skin depth) Supported by solar wind observations, linear Vlasov theory, kinetic physics reconnection studies. ρ ii = ion skin depth = c/w pi = c/ e 2 4πn/m p ρ ii /L 1 for astrophysical and space plasmas (ρ ii /L 10 5 for solar wind) We consider ρ ii /L = 1/32, L box 200ρ ii

Turbulent length scales vs particle scales Dissipation length l d ρ ii (ion skin depth) Supported by solar wind observations, linear Vlasov theory, kinetic physics reconnection studies. ρ ii = ion skin depth = c/w pi = c/ e 2 4πn/m p ρ ii /L 1 for astrophysical and space plasmas (ρ ii /L 10 5 for solar wind) We consider ρ ii /L = 1/32, L box 200ρ ii Particle (nominal) gyroradius vs turbulent dissipative scale r 0 l d = Z m m p δb B 0 electrons r e 0/l d = 6 10 5 protons r p 0/l d = 0.1

displacement [ L ] Electrons (rms displacement) 1.5 1.0 < z 2 > 1/2 (parallel) 0.5 < x 2 > 1/2 (transverse) 0.0 0.00 0.02 0.04 0.06 0.08 0.10 time [ t 0 ] Run stops when < z 2 > 1/2 L

velocity [ v 0 ] Electrons (rms velocity) 20 15 <u z 2 > 1/2 (parallel) 10 5 <u x 2 > 1/2 (transverse) 0 0.00 0.02 0.04 0.06 0.08 0.10 time [ t 0 ]

<u z 2 > / <ux 2 > Electrons (parallel/transverse velocity square) 150 100 50 0 0.00 0.02 0.04 0.06 0.08 0.10 time [ t 0 ] T e T e

N 1 dn/du j [v 0 1 ] Electrons (velocity distribution) 1.0000 0.1000 u x (transverse) 0.0100 0.0010 u z (parallel) 0.0001 160 80 0 80 160 u j [v 0 ] In t = 10 4 τ e = 0.1t 0 with τ e = 2πm e c/(eb 0 )

u z [ v 0 ] parallel Electrons (velocity scatter plot) 160 80 0 80 160 160 80 0 80 160 u x [ v 0 ] transverse

N 1 dn/dr g e Electrons (gyroradii) 10 6 [L 1 ] 10 5 10 4 10 3 l d 10 2 10 7 10 6 10 5 10 4 10 3 10 2 10 1 r e g [L] r e g = u m e c/(eb 0 )

Electrons (trajectories)

Electron parallel velocity gain: scaling du dt = e E = e v 0 L J, m e m e c R m dz dt = u

Electron parallel velocity gain: scaling du dt = e E = e v 0 L J, m e m e c R m dz dt = u z L in t = t = (2m e cr m /eu 0 J ) 1/2 u = (2eu 0 J /m e cr m ) 1/2 L

Electron parallel velocity gain: scaling du dt = e E = e v 0 L J, m e m e c R m dz dt = u z L in t = t = (2m e cr m /eu 0 J ) 1/2 u = (2eu 0 J /m e cr m ) 1/2 L which can be rewritten using J 0 = δb/l and ρ ii as u = [ 2m p m e L ρ ii 1 R m J J 0 ] 1/2 v 0

Electron parallel velocity gain: scaling du dt = e E = e v 0 L J, m e m e c R m dz dt = u z L in t = t = (2m e cr m /eu 0 J ) 1/2 u = (2eu 0 J /m e cr m ) 1/2 L which can be rewritten using J 0 = δb/l and ρ ii as u = [ 2m p m e L 1 J ] 1/2 v 0 ρ ii R m J 0 Using J max /J 0 = 47 we get t 0.03t 0 and u 74v 0 for the maximum velocity

Electron parallel velocity gain: scaling du dt = e E = e v 0 L J, m e m e c R m dz dt = u z L in t = t = (2m e cr m /eu 0 J ) 1/2 u = (2eu 0 J /m e cr m ) 1/2 L which can be rewritten using J 0 = δb/l and ρ ii as u = [ 2m p m e L 1 J ] 1/2 v 0 ρ ii R m J 0 Using J max /J 0 = 47 we get t 0.03t 0 and u 74v 0 for the maximum velocity Using J /J 0 = 4.5 we get t 0.09t 0 and u 20v 0 for the average velocity

velocity [ v 0 ] Protons (mean square velocities) 20 15 <u x 2 > 1/2 (transverse) 10 5 <u z 2 > 1/2 (parallel) 0 0.0 0.5 1.0 1.5 2.0 time [ t 0 ] T i T i

N 1 dn/du j [v 0 1 ] 1.0000 Protons (velocity distribution function) 0.1000 u z (parallel) 0.0100 0.0010 u x (transverse) 0.0001 160 80 0 80 160 u j [v 0 ] At t = 90τ p = 1.8t 0 with τ p = 2πm p c/(eb 0 )

u z [ v 0 ] parallel Protons (velocity scatter plot) 160 80 0 80 160 160 80 0 80 160 u x [ v 0 ] transverse

N 1 dn/dr g p Protons: gyroradii at t = 2τ p = 0.04t 0 1000 [L 1 ] 100 10 l d 1 10 5 10 4 10 3 10 2 10 1 10 0 r g p [L] r p g = u m p c/(eb 0 )

N 1 dn/dr g p Protons: gyroradii at t = 20τ p = 0.4t 0 100.00 [L 1 ] 10.00 1.00 0.10 l d 0.01 10 5 10 4 10 3 10 2 10 1 10 0 r p g [L]

N 1 dn/dr g p 100.00 Protons: gyroradii at t = 90τ p = 1.8t 0 [L 1 ] 10.00 1.00 0.10 l d 0.01 10 5 10 4 10 3 10 2 10 1 10 0 r p g [L]

Protons: trajectories

Protons: trajectories top view

Proton transverse velocity gain: scaling In the gyromotion u = w p r, with w p = eb 0 /(m p c) or equivalently u /v 0 = (B 0 /δb)(r/ρ ii ).

Proton transverse velocity gain: scaling In the gyromotion u = w p r, with w p = eb 0 /(m p c) or equivalently u /v 0 = (B 0 /δb)(r/ρ ii ). Energy gain stops at a critical gyroradius r.

Proton transverse velocity gain: scaling In the gyromotion u = w p r, with w p = eb 0 /(m p c) or equivalently u /v 0 = (B 0 /δb)(r/ρ ii ). Energy gain stops at a critical gyroradius r. Estimating r = Lρ ii, we get u B 0 v 0 δb [ L ] 1/2 ρ ii

Proton transverse velocity gain: scaling In the gyromotion u = w p r, with w p = eb 0 /(m p c) or equivalently u /v 0 = (B 0 /δb)(r/ρ ii ). Energy gain stops at a critical gyroradius r. Estimating r = Lρ ii, we get u B 0 v 0 δb [ L ] 1/2 ρ ii Using the numerical values u 60v 0

Effect of time dependent fields: just started, but...

Effect of time dependent fields: just started, but... Electrons move fast and with very small gyroradius, so, remain completely unaffected by the slow evolution of the MHD fields

Effect of time dependent fields: just started, but... Electrons move fast and with very small gyroradius, so, remain completely unaffected by the slow evolution of the MHD fields Ions are more sensitive to the time evolution of the MHD fields, (because they move slower and with larger gyroradius) resulting in slower energy gain, but still, we get large perp velocities in times of the order of the Alfven time. When u p >> v A then the dynamics of the particles becomes unaffected by the slow time evolution of the fields.

Effect of time dependent fields: just started, but... Electrons move fast and with very small gyroradius, so, remain completely unaffected by the slow evolution of the MHD fields Ions are more sensitive to the time evolution of the MHD fields, (because they move slower and with larger gyroradius) resulting in slower energy gain, but still, we get large perp velocities in times of the order of the Alfven time. When u p >> v A then the dynamics of the particles becomes unaffected by the slow time evolution of the fields. Another issue: influence of the Hall effect, negligible for electrons, it affects more (but not much) the behavior of ions (see Dmitruk & Matthaeus, Jour. Geophys. Res. 111, A12110, 2006)

Hall MHD turbulence Electric field E = v c B + J B nec + ηj

S E (k) Hall MHD turbulence Electric field E = v c B + J B nec + ηj Spectrum of the electric field 10 2 10 3 10 4 10 5 10 6 1 10 100 k k H k d

Distribution functions 10 0 10 1 N 1 dn/db x 10 2 10 3 10 4 10 5 2 0 2 b x 10 1 10 0 10 1 N 1 dn/de x 10 2 10 3 10 4 10 5 2 0 2 E x

Effect on reconnection changes the reconnection rate Dmitruk and Matthaeus, Phys. Plasmas 2006; also Reduced Hall MHD in Gomez, Dmitruk, Mahajan, Phys. Plasmas 2009; Martin, Dmitruk, Gomez, Phys. Plasmas 2010

electric field ratio b x, v z, b 2 1 0 1 2π/k H 2 2π/k d 0 5 10 15 20 j y 150 100 50 0 50 0 5 10 15 20 (v x b) y, (εj x b) y, µ j y 1.50 0.75 0.00 0.75 2π/k H 1.50 2π/k d 0 5 10 15 20 1.0 0.8 0.6 0.4 0.2 0.0 0 5 10 15 20 gridpoint

N 1 dn/dp j [(m e v 0 ) 1 ] Electrons (momentum distribution function) 1.0000 0.1000 p x (transverse) 0.0100 0.0010 p z (parallel) 0.0001 160 80 0 80 160 p j [m e v 0 ] At t = 10 4 τ e = 0.1t 0, for Hall and non-hall (light lines)

N 1 dn/dp j [(m p v 0 ) 1 ] Protons (momentum distribution function) 1.0000 0.1000 p z (parallel) 0.0100 0.0010 p x (transverse) 0.0001 150 75 0 75 150 p j [m p v 0 ] At t = 90τ p = 1.8t 0, for Hall and non-hall (light lines)

N 1 dn/du j [v 0 1 ] Results with time dependent fields Electrons (velocity distribution function) 1.0000 0.1000 u x (transverse) 0.0100 0.0010 u z (parallel) 0.0001 160 80 0 80 160 u j [v 0 ] At t = 10 4 τ e = 0.1t 0

N 1 dn/du j [v 0 1 ] Protons (velocity distribution function) 1.0000 0.1000 u z (parallel) 0.0100 0.0010 u x (transverse) 0.0001 40 20 0 20 40 u j [v 0 ] At t = 200τ p = 4t 0