Kinetic Alfvén Turbulence: An Update

Daniel GrošeljMax Planck Institute for Plasma Physics, Garching, Germany

12th Plasma Kinetics Working Meeting, Vienna, August 7th, 2019

Acknowledgements

Recent collaborators:Alfred Mallet (Berkeley), Chris Chen (QMUL), Silvio Cerri (Princeton), Luca Franci (QMUL),Ravi Samtaney (KAUST), Kai Schneider (I2M-CNRS), Frank Jenko (IPP Garching),

OSIRIS code [developed and distributed by the OSIRIS Consortium(UCLA & IST, Portugal)]:

3D, fully kinetic, fully explicit, & relativistic PIC code

⇒ Here: Utilized for space/astro 3D kinetic plasma turbulencesimulations

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Outline

1 Polarization alignment in kinetic Alfvén wave(KAW) turbulence?(a) generalized spectral field ratios(b) some exact (localized) wave solutions of the electron

reduced MHD eqs.(c) intermittency + alignment (simulated & observed)

2 3D local anisotropy of KAW turbulence(a) the “statistical eddies” of sub-ion range turbulence(b) anisotropy scalings(c) comparison with other kinetic simulations

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3D kinetic turbulence data: overview

Driven 3D fully kinetic simulation:βi ≈ βe ≈ 0.5, mi/me = 100, L⊥ ≈ 19di

L⊥/L‖ ≈ 0.4, non-relativistic regimespatial resolution 9282 × 1920, about 0.5 trillionparticles in total

SW measurements:7 h interval from Cluster (B data) [Chen et al. (2015)](βi ≈ 0.3, βe ≈ 0.6)159 s interval from MMS (B & ne data) [Gershman etal. (2018)] (βi ≈ 0.3, βe ≈ 0.03)

3D hybrid-kinetic simulations (kindly provided by mainauthors):

Cerri, Servidio & Califano, ApJ (2017),Arzamasskiy et al., ApJ (2019).

Note: In all simulations considered here, sub-ion range islimited to kdi . 10!

100 101k⟂di

10−810−710−610−510−410−310−210−1100

P b⟂k

⟂)

∼ k −2∼ 8⟂

⟂b)

0∼0 0∼5 1∼0 1∼5 2∼0t /tA

0∼20∼30∼40∼5

brms ⟂a)

100 101kzdi

10−810−610−410−2

P b⟂k

z)

100 101

k⟂di

100

101

F(k ⟂

)

Cluster(c)

ρ−1i Δρ=7⟂h

100

101

F(k ⟂

)

MMS(b)

d−1eρ−1

i Δρ=159⟂s100

101

F(k ⟂

)

simΔ(a)

ρ−1i ρ−1

ed−1e

δδ 4∥ ⟩⟨ δδ ∥

∥ ⟩∥ δn4e ⟩⟨ δn∥

e ⟩∥ δδ 4⟂ ⟩⟨ δδ ∥

⟂ ⟩∥

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Generalized spectral field ratiosratios of spectral amplitudes of δb⊥, δb‖, δne can be used to detect KAW polarization in a turbulent plasmawe introduced “generalized ratios” to probe the statistical polarizations within the large-amplitude, localizedturbulent structures (see Groselj et al., PRX (accepted), arXiv:1806.05741)

⇒ Large-amplitude structures (often considered as non-wavelike) preserve a linear wave footprintOK, but why?...

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Why do large-amplitude (nonlinear) structures carry a wave signature?

Nonlinear time = inversely proportional to fluct. amplitude ⇒ Naively one might think that intensestructures evolve faster than linear wavesBut, linear time scale ∝ `‖ and max. `‖ is limited from above by causality (implying critical balance) solinear time scale keeps upThe critical balance argument is maybe somewhat vague. Are there any additional arguments?

Kinetic Alfvén turbulence may be approximately described with the reduced electron MHD eqs.[Schekochihin et al. (2009)]:

∂tψ = −∂zne − z · (∇⊥ψ ×∇⊥ne), (1)∂tne = ∂z∇2

⊥ψ + z · (∇⊥ψ ×∇⊥∇2⊥ψ). (2)

How “robust” are the linear KAW solutions?A combination of co-propagating KAWs (with ψk = ±k⊥ne,k) with a fixed magnitude of k⊥ is an exactsolution [Schekochihin et al. (2009)]Is that all? No!

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Why do large-amplitude (nonlinear) structures carry a wave signature?

Nonlinear time = inversely proportional to fluct. amplitude ⇒ Naively one might think that intensestructures evolve faster than linear wavesBut, linear time scale ∝ `‖ and max. `‖ is limited from above by causality (implying critical balance) solinear time scale keeps upThe critical balance argument is maybe somewhat vague. Are there any additional arguments?

Kinetic Alfvén turbulence may be approximately described with the reduced electron MHD eqs.[Schekochihin et al. (2009)]:

∂tψ = −∂zne − z · (∇⊥ψ ×∇⊥ne), (1)∂tne = ∂z∇2

⊥ψ + z · (∇⊥ψ ×∇⊥∇2⊥ψ). (2)

How “robust” are the linear KAW solutions?A combination of co-propagating KAWs (with ψk = ±k⊥ne,k) with a fixed magnitude of k⊥ is an exactsolution [Schekochihin et al. (2009)]Is that all? No!

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Some exact wave solutions of the ERMHD eqs. 1/2

To find exact wave solutions we require that the nonlinear Poisson brackets vanish:Satisfied whenever the contours of ψ, ne, & ∇2

⊥ψ are aligned in every ⊥ plane (nonlinearity cancelsgeometrically)The alignment between the ⊥ contours of ψ and ∇2

⊥ψ restricts the geometry of the solutions

2 types of such exact wave solutions exist. Their ⊥ profiles are either:1 circularly symmetric (ne = ne(r⊥, z), ψ = ψ(r⊥, z)), or2 one-dimensional (fixed orientation of k⊥)

⇒ ne & ψ here need not satisfy a fixed-phase relation so solutions may be composed ofcounter-propagating KAWs⇒ there is no fixed k⊥ constraint so KAW packets can have a localized ⊥ envelope (at some t = tref)

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Some exact wave solutions of the ERMHD eqs. 1/2

To find exact wave solutions we require that the nonlinear Poisson brackets vanish:Satisfied whenever the contours of ψ, ne, & ∇2

⊥ψ are aligned in every ⊥ plane (nonlinearity cancelsgeometrically)The alignment between the ⊥ contours of ψ and ∇2

⊥ψ restricts the geometry of the solutions

2 types of such exact wave solutions exist. Their ⊥ profiles are either:1 circularly symmetric (ne = ne(r⊥, z), ψ = ψ(r⊥, z)), or2 one-dimensional (fixed orientation of k⊥)

⇒ ne & ψ here need not satisfy a fixed-phase relation so solutions may be composed ofcounter-propagating KAWs⇒ there is no fixed k⊥ constraint so KAW packets can have a localized ⊥ envelope (at some t = tref)

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Some exact wave solutions of the ERMHD eqs. 2/2

Implications for sub-ion scale turbulence:Simulations find that sub-ion scale structures are either elongated sheets or circular tubes (e.g., Boldyrev &Perez, ApJL (2012); Meyrand & Galtier, PRL (2013); Kobayashi et al., ApJ (2017))The idealized geometric versions of these two (1D sheets or circularly symmetric, field-aligned tubes) areexact wave solutions (for k‖ 6= 0) of ERMHDDue to wandering of field lines in turbulent flows, exact alignment cannot be reached [Boldyrev, PRL(2006)] but even if structures resemble the ideal solutions the nonlinearity is locally depleted andnonlinear time slows down

Is this reasonable?Consider the scale-dependent alignment between the ⊥ electron fluid velocity (∝ z×∇⊥ne) & magneticfield (∝ z×∇⊥ψ):

sin θ ≡ |δu⊥e × δb⊥|/|δu⊥e||δb⊥| (3)

Compute sin θ conditionally averaged on the (normalized) local KAW spectral energy density:⟨sin θ(k⊥)

∣∣∣LIM = EKAW (k⊥, r)⟨EKAW (k⊥, r)

⟩r> ξ

⟩r

(4)

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Some exact wave solutions of the ERMHD eqs. 2/2

Implications for sub-ion scale turbulence:Simulations find that sub-ion scale structures are either elongated sheets or circular tubes (e.g., Boldyrev &Perez, ApJL (2012); Meyrand & Galtier, PRL (2013); Kobayashi et al., ApJ (2017))The idealized geometric versions of these two (1D sheets or circularly symmetric, field-aligned tubes) areexact wave solutions (for k‖ 6= 0) of ERMHDDue to wandering of field lines in turbulent flows, exact alignment cannot be reached [Boldyrev, PRL(2006)] but even if structures resemble the ideal solutions the nonlinearity is locally depleted andnonlinear time slows down

Is this reasonable?Consider the scale-dependent alignment between the ⊥ electron fluid velocity (∝ z×∇⊥ne) & magneticfield (∝ z×∇⊥ψ):

sin θ ≡ |δu⊥e × δb⊥|/|δu⊥e||δb⊥| (3)

Compute sin θ conditionally averaged on the (normalized) local KAW spectral energy density:⟨sin θ(k⊥)

∣∣∣LIM = EKAW (k⊥, r)⟨EKAW (k⊥, r)

⟩r> ξ

⟩r

(4)

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Intermittent polarization alignment in KAW turbulence

100 101k⟂ di

0⟂2

0⟂4

0⟂8

⟨|δu⟂δ×δ

b ⟂|

|δu⟂δ||δ

b ⟂||L

IM>ξ

⟩ρ−1i ρ−1δd−1

δ

sim⟂

ρ=0ρ=1

ρ=2ρ=4

ρ=8ρ=1ξ

100 101k⟂ di

ρ−1i d−1δ

MMS

high-amplitude structures are indeed more aligned, similar to what was found in MHD (e.g., Beresnyak &Lazarian, ApJ (2006); Mallet et al., MNRAS (2016))trend is seen in 3D fully kinetic simulation & in MMS data but is weaker in the latter case (note that MMSinterval is only weakly intermittent)

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Are KAW eddies 3D anisotropic?

Introduce 3D conditional structure function in the local frame:

Sm(r, θ, φ) =⟨|∆f(r0, r)|m

∣∣r, θ, φ ⟩r0, (5)

where cos θ = r · Bloc & cosφ = r⊥ · δb⊥3 natural directions: the parallel direction, `‖ (θ = 0), fluctuation direction, ξ (θ = 90◦, φ = 0),“perpendicular” direction, λ (θ = 90◦, φ = 90◦)∆f is the field increment. For steep spectra, increments with more than 2 points are needed to measure thetrue scaling.For 5-point increments: ∆f(r0, r) = [f(r0 + 2r)− 4f(r0 + r) + 6f(r0)− 4f(r0 − r) + f(r0 − 2r)]/

√35.

Alternatively, the spectral method of Cho & Lazarian (2009; CL09) may be used to estimate the k‖(k⊥)scaling:

k‖(k⊥) ≈ 〈|B0k⊥ · ∇δbk⊥ |2〉1/2

〈|δbk⊥ |2〉1/2〈|B0k⊥ |2〉1/2 (6)

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Consistency check: (driven) MHD-scale turbulence

100 101 102

λ

100

101

102

ℓ ∥,ξ

Anisotropy of MHD− cale turbulence from 5−point ℓ2(λ, ξ,ℓ∥ ∥

ℓ∥ ∼π∼ξ∥ (CL09 method∥ℓ∥ξ∼ λ 1∼2

∼ λ 3∼4

reasonable agreement with predictions for (dynamically aligned) MHD turbulence [Boldyrev, PRL (2006)]:`‖ ∼ λ1/2, ξ ∼ λ3/4

spectral method (CL09) scaling in agreement with scalings from 5-point structure function

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The 3D statistical eddies of KAW turbulence

0123456ξ /di

0

1

2

3

4

5

6

λ/d i

0 2 4 6 8 10 12 14ℓ∥ /di

δℓ⟂

−9.96−8.28−6.59−4.91−3.23−1.540.141.82

log∥λ 2⟂

0123456ξ /di

0

1

2

3

4

5

6

λ/d i

0 2 4 6 8 10 12 14ℓ∥ /di

δℓ∥

−10.12−8.68−7.23−5.79−4.34−2.90−1.45−0.01

log∥λ 2)

Eddies do become more elongated along `‖ (in this simulation!) with decreasing scale, but there is hardly anyanisotropy in ⊥ local plane

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Parallel anisotropy scalings

10−1 100 101λ/di

100

101

ℓ ∥/d

i

δδ⟂

Parallel∥anisotropy

ℓ∥ ∼π/π∥ ∥⟂CL09∥method∼ℓ∥ ∥⟂from∥ℓ2∼ℓ∥ ∥⟂from∥ℓ4∼∼ λ 2/3

∼ λ 1/3

10−1 100 101

λ/di

100

101

ℓ ∥/d

i

δδ∥

Parallel∥anisotropy

ℓ∥ ∥(from∥ℓ2)ℓ∥ ∥(from∥ℓ4)∼ λ 2/∼

Structure function method gives weaker (but still scale-dependent) anisotropy than spectral method(consistent with what was found in the original Cho & Lazarian (2009) paper)Scalings deduced from isocontours of S2 and S4 are very similar ⇒ structures of different intensity havealmost identical parallel aspect ratios

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Perpendicular anisotropy scalings

10−1 100 101

λ/di

10−1

100

101

ξ/di

δδ⟂

Perpendicular⟂anisotropy

ξ⟂(from⟂ξ2)ξ⟂(from⟂ξ4)∼ λ

10−1 100 101

λ/di

10−1

100

101

ξ/di

δδ∥

Perpendicular∥anisotropy

ξ∥(from∥ξ2)ξ∥(from∥ξ4)∼ λ

It seems that the sub-ion scale statistical eddies do not get more sheetlike with decreasing scaleStructures of different intensities have also similar ⊥ aspect ratios

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Comparison with other 3D kinetic simulations

10−1 100 101λ/di

100

101

102

ℓ ∥/d

i

δℓ

Parallel∥anisotropy

Groselj∥et∥al.∥(2019)Cerri∥et∥al.∥(2017)Arzamasskiy∥et∥al.∥(2019) λ 1/∼ λ 2/∼ λ

100 101λ/di

100

101

ξ/di

δξ

Perpendicular anisotropy

Groselj et al. (2019)Cerri et al. (2017)Arzamasskiy et al. (2019) λ

The eddies from Arzamasskiy et al., ApJ (2019) have fixed parallel aspect ratio at kinetic scales and becomeslightly less sheetlike with decreasing scale (both things happen at approximately the same λ)

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Summary

Kinetic-scale structures may be linked to exact wave solutions of ERMHD via localnonlinearity depletion (supported by some simulation and observation data)The sub-ion scale eddies do not get more sheetlike with decreasing scaleThere is work to be done regarding the l‖(λ) scaling. Most 3D kin. sims (I haveanalyzed also other data) indicate scale-dependent anisotropy, but data fromArzamasskiy et al., ApJ (2019) do not show a scale-dependent anisotropy (solution:kinetic-scale tearing of sheetlike eddies??)

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