Magnetic Reconnection during Turbulence and the Role it

Plays in Dissipation and HeatingMichael Shay

Bartol Research InstituteUniversity of Delaware

C. Haggerty, T. Parashar, W. Matthaeus, R. BandyopadyayT. D. Phan, J. F. Drake, Y. Yang, M. Wan, S. Servidio,

P. WuM. A. Shay, C. C. Haggerty, W. H. Matthaeus, T. N. Parashar, M. Wan, and P. Wu, Turbulent heating due to magnetic reconnection, Physics of Plasmas, 25, 012304 (2018);

Haggerty, C. C., T. N. Parashar, W. H. Matthaeus, M. A. Shay, Y. Yang, M. Wan, P. Wu, and S. Servidio, Exploring the statistics of magnetic reconnection X-points in kinetic particle-in-cell turbulence, Physics of Plasmas, 24, 102308, 2017, doi:10.1063/1.5001722.

Motivation• Focus: Turbulence

– Not reconnection generated turbulence• Significant advances recently on the nature of plasma heating

during magnetic reconnection– e.g., Phan et al., 2013, 2104; Yamada et al., 2014, Shay et al., 2014;

Haggerty et al., 2015; Wang et al., and many more• Kinetic simulations of turbulence: some inertial range to electron

scales– e.g., Howes et al., Tenbarge et al., Karimabadi et al., Parashar et al,

Gary et al., and many more. • Can we apply our understanding of “simple” magnetic

reconnection to turbulent heating?– Short Answer: Yes …. But …..

Overview• Background• Laminar Reconnection Studies

– Heating in Reconnection Exhausts• Framework: Apply Heating Predictions to

Turbulence• Kinetic PIC Turbulence Simulations

– Test Framework• Statistics of Reconnection: Kinetic PIC

Simulations

Overview• Background• Laminar Reconnection Studies

– Heating in Reconnection Exhausts• Framework: Apply Heating Predictions to

Turbulence• Kinetic PIC Turbulence Simulations

– Test Framework• Statistics of Reconnection: Kinetic PIC

Simulations

Solar Wind Turbulence

Notice the Comet!

“Powerlaws everywhere”!

• Solar wind

• Corona

• Diffuse ISM

• Geophysical flows

Interstellar medium: Armstrong et al

SW at 2.8 AU: Matthaeus and Goldstein

Broadband self-similar spectra are a signature of cascade

Coronal scintillation results (Harmon and Coles)Tidal channel: Grant, Stewart and Moilliet

Heating by turbulence cascade

energy containing

energy input

inertial dissipation

heati

ng

cascade

En

er

gy s

pectr

um

E(k

)

Log(wavenumber)

• heat the plasma, • increasing the pressure gradient • adding momentum ! producing the solar wind

…in corona, turbulence can:

Where the heating occurs is important!

Plasma Heating - Magnetic Dissipation?• Something is heating the

solar wind

!8

• Something is heating the solar corona

Wang et al., JGR, 106, 29401, 2001

Model of Photosphere/Corona Transition “Physics of the Solar Corona,”

Aschwanden, 2005.

Physical Dissipation Mechanisms for Kinetic Turbulence

Three mechanisms have been proposed:

(1) Collisionless Wave-Particle Interactions (Landau damping) (Leamon et al., 1998, 1999, 2000; Quateart & Gruzinov, 1999; Howes et al.

2008; Schekochihin et al. 2009; TenBarge & Howes 2012)

(3) Dissipation in Current Sheets (Dmitruk et al. 2004; Markovskii & Vasquez 2011; Matthaeus & Velli 2011;

Osman et al. 2012; Servidio 2011)

(2) Stochastic Heating(Johnson & Cheng, 2001; Chen et al. 2001; White et al., 2002; Voitenko &

Goosens, 2004; Bourouaine et al., 2008; Chandran et al. 2010; Chandran 2010)

Key Goal:-To compare and contrast the different turbulence theories -To identify observational and numerical tests to distinguish these

distinct models

From SHINE conference, 2014

Howes/Shay Session Description

Phan et al., Nature, 2018• Reconnection in Turbulent Magnetosheath

– “Electron-Only” Reconnection

BLMN

Vi LMN

ExB/B2

ELMN

(mV/m)

j.(E+VexB) (nW/m3)

E||

Ve LMN

jLMN(µA/m2)

MMS 1 and MMS 3 on opposite sides of the X-line: smoking gun evidence for reconnection

veL

MMS 1

veM

jL jM

EN EM

45 ms

ΔVeL < 0

EN > 0

Δ(ExB/B2)L

veL veM ΔVeL > 0Δ(ExB/B2)L

EN < 0

MMS 3 45 ms

electron jet

MMS 1

MMS 3EN

EN

Magnetic-to-electron energy conversion

Overview• Background• Laminar Reconnection Studies

– Heating in Reconnection Exhausts• Framework: Apply Heating Predictions to

Turbulence• Kinetic PIC Turbulence Simulations

– Test Framework• Statistics of Reconnection: Kinetic PIC

Simulations

Vx

Character of Reconnection Heating• Heating Tends to Occur as

beams– Especially: Ion Heating

Gosling et al., 2005

f

v||

Lottermoser et al., 1998

Simulations

Solar Wind Observations

“Simple” Problem: Heating in Reconnection Exhausts

• Focus on Understanding Heating in Reconnection Exhausts

• Heating Definition:

– Counter-streaming beams are considered “heating”.

• Systematic Kinetic PIC Simulations

T =1

n

Zd3vf(x, v)(v � u)(v � u)

T =1

3Tr[T]

Energy Budget

δ

D

Vin

Vin

B

B

VoutVout

• Magnetic Energy In: • Flow Energy, Thermal Energy, Heat Flux out

B2

4⇡VinD =

1

2minV

2out +

�

� � 1�Ten+

�

� � 1�Tin

�Vout� +Qix +Qex

Energy Budget: Heating• Energy Conservation

Divide eqn by left hand side

• 1 = αflow + αTe + αTi + αQi + αQe

• αT = % of released energy that heats a species.

• Important Questions – Is αT a constant? What does it depend on? – What is the value of αT?

– What is 𝛾/(𝛾-1)? (5/2 for adiabatic) Review in: Shay et al., 2014

↵T ⇡ �

� � 1

�T

B2/(4⇡n)

B2

4⇡VinD =

1

2minoutV

3out +

�

� � 1�Tenout Vout +

�

� � 1�Tinout Vout

�� + Qix + Qex

<latexit sha1_base64="GE3DpxYVR9QPh7+Mxpw1V2PUdUc=">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</latexit><latexit sha1_base64="GE3DpxYVR9QPh7+Mxpw1V2PUdUc=">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</latexit><latexit sha1_base64="GE3DpxYVR9QPh7+Mxpw1V2PUdUc=">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</latexit><latexit sha1_base64="GE3DpxYVR9QPh7+Mxpw1V2PUdUc=">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</latexit>

Laminar Reconnection Simulations• Reconnection heating depends

strongly on parameters upstream of x-line.

– cAr: Alfven speed based on reconnection field

– Br: reconnection field– B: Total Field

• MTi, MTe are constants• ΔTi consistent with

Drake et al., 2009• Relative heating is:

mi c2Ar

B2r

B2

ΔTi

mi c2Ar

Br

B

ΔTe

Dataset limited to:

Bguide > 0.2Te/Ti < 1.25βi < 2

�Ti = MTi c2Ar

B2r

B2

�Te = MTe c2Ar

Br

B

�Ti

�Te/ Br

B

Question• Can we take this knowledge about laminar

reconnection and apply it to reconnection during turbulence?

Overview• Background• Laminar Reconnection Studies

– Heating in Reconnection Exhausts• Framework: Apply Heating Predictions to

Turbulence• Kinetic PIC Turbulence Simulations

– Test Framework• Statistics of Reconnection: Kinetic PIC

Simulations

Heating in Turbulence Due to Reconnection?

• Multifaceted problem – Magnitude and Character of Heating

due to Reconnection?• What parameters does the heating

depend on?– Properties of Magnetic Reconnection

in Turbulence?• How many x-lines? How many

reconnecting x-lines?– Relative role of heating due to

magnetic reconnection versus other sources of heating

• Kinetic PIC Simulation Study– Can we apply our understanding of

“simple” magnetic reconnection to turbulent heating?

– What are the statistics of X-lines in kinetic turbulence simulations?

Haggerty el al., 2017Kinetic PIC Turbulence Simulation

Magnetic Flux Contours with X-llines

Flux Bundles Reconnecting• Energy released into ions:

Single Flux Bundle:

• Sum Over All Bundles:

CAr

`>

`<

X

Y

Servidio, 2010

Magnetic Flux Contours

" =MT

2

X

x-line i

`<i `>i B2ir

B2ir

B2

" =

MT

2

X

x-line i

`<i `>i

!B2

irB2

ir

B2

"⇥ = 2⇡�T `< `>nr

"⇥ =MT

2`< `> B2

rB2

r

B2

Determine Bir?• Assume mean field is constant and B0 >> Bir • First estimate: Width of reconnection sites typically ~ di

– What are magnetic fluctuations at scale size di?• Assume δZ at di scales like reconnection Alfven speed: δZdi ∝ Car

• Where cA is Alfven speed based on global RMS B• 𝜏nl(di) is nonlinear time at ion inertial scale

• 𝜏c is cyclotron time based on global RMS B ≈ B0

Protons:

Electrons: "e =�Z3

di

c3A

MT

2B2

X

x-line i

`<i `>i

!= ↵3

nl ( Flux Bundle Details )

"p =�Z4

di

c4A

MT

2B2

X

x-line i

`<i `>i

!= ↵4

nl ( Flux Bundle Details )

cAr

cA=

Br

B=

�Zdi

cA

didi

=⌧ci

⌧nl(di)⌘ ↵nl

Key Points• Turbulent heating due to reconnection

estimated to be:

• Will ( Flux Bundle Details ) remain invariant as we change turbulence properties?

• Proton to electron heating scaling expected to be more accurate.

"p = ↵4nl ( Flux Bundle Details )

"e = ↵3nl ( Flux Bundle Details )

"p"e

/ ↵nl

Overview• Background• Laminar Reconnection Studies

– Heating in Reconnection Exhausts• Framework: Apply Heating Predictions to

Turbulence• Kinetic PIC Turbulence Simulations

– Test Framework• Statistics of Reconnection: Kinetic PIC

Simulations

Kinetic PIC Turbulence Simulations

• Wu et al, 2013, Parashar et al., 2016.

• 2 1/2 Dimensions• Initial uniform Bz = 5• Initial Perturbation:

Z20 ⌘ hv2i+ h B2

4⇡mini

NASA Phan et al Publicity Movie, 2018

Turbulence Simulations• Constant Mean field B0 = 5, B >> Br

5

run family βi βe Z0 B0 λc L Nx = Ny Qi/Qe τc/τnl ppg krun805.1 PAPJ15 0.08 0.08 1.1 5.0 0.445 1.28 64 0.584 0.534 200 {1,2}run805.2 PAPJ15 0.08 0.08 0.825 5.0 0.445 2.56 128 0.887 0.340 200 {1,2}run805.3 PAPJ15 0.08 0.08 0.737 5.0 0.445 5.12 256 0.435 0.260 200 {1,2}run805.4 PAPJ15 0.08 0.08 0.713 5.0 0.445 10.24 512 0.331 0.209 200 {1,2}run805.5 PAPJ15 0.08 0.08 0.707 5.0 0.445 20.48 1024 0.343 0.169 200 {1,2}run809.1 OTVdbB 0.08 0.08 0.718 5.0 3.820 20.48 1024 0.551 0.169 200 {1,2}run810.1 OTVdbB 0.08 0.08 1.075 5.0 3.928 20.48 1024 0.588 0.251 200 {1,2}run811.1 OTVdbB 0.08 0.08 1.433 5.0 4.286 20.48 1024 0.754 0.325 200 {1,2}run812.1 OTVdbB 0.08 0.08 1.790 5.0 4.608 20.48 1024 0.945 0.397 200 {1,2}Turb808 k13 0.25 0.25 1.0 5.0 0.349 25.6 2048 0.404 0.173 400 {1,3}Turb812 k13 0.25 0.25 2.5 5.0 0.871 25.6 2048 0.542 0.404 400 {1,3}Turb813 k13 0.25 0.25 4.0 5.0 1.394 25.6 2048 0.857 0.606 400 {1,3}Turb814 k24lb 0.25 0.25 2.5 5.0 0.868 25.6 2048 0.697 0.419 400 {2,4}Turb815 k24lb 0.25 0.25 4.0 5.0 1.389 25.6 2048 1.159 0.646 400 {2,4}Turb816 k24sb 0.10 0.10 2.5 5.0 0.868 25.6 2048 0.546 0.427 400 {2,4}Turb817 k24sb 0.10 0.10 4.0 5.0 1.389 25.6 2048 1.010 0.664 400 {2,4}

TABLE 1Properties of simulation runs. All simulations employ P3D particle-in-cell code in 2.5D (3 components for vectors; 2 dimensional grid)periodic boxes of lengths described in the L column and grid described in the Nx = Ny column. The table does not list the runs from Wu

et al. (2013). Listed are proton beta βi, electron beta βe, turbulence amplitude Z0, out of plane unfirm magnetic field B0, correlationscale λc, box size L, grid points in plane Nx = Ny , ratio of average ion/electron heating rates Qi/Qe, ratio of proton cyclotron time to

nonlinear time at di ion inertial scale τc/τnl, number of particles per grid point ppg and wavenumber band of initial conditions. Alllengths are in di. The b and v fluctuations in the initial conditions were excited for wave-numbers having the values given in the k column

and with a given initial spectrum. The time step in all the simulations was chosen to satisfy the CFL condition for the speed of light.

wave-numbers. The energy then cascades to fill in the complete wave-number range available and produce turbulence.Typical parameters for these runs (except for Wu et al. (2013) runs) are given in Table 1.

REFERENCES

W. H. Matthaeus and M. Velli, Space Sci. Rev. pp. 273–+ (2011).A. Lazarian, G. L. Eyink, and E. T. Vishniac, Phys. Plasmas 19,

012105 (pages 8) (2012).M. Mac Low, Astrophys. J. 524, 169 (1999).B. D. G. Chandran and S. C. Cowley, Phys. Rev. Lett. 80, 3077

(1998).N. Banerjee and P. Sharma, Mon. Not. Roy. Astron. Soc. 443, 687

(2014).J. R. Jokipii, Astrophys. J. 146, 480 (1966).H. Karimabadi, V. Roytershteyn, M. Wan, W. H. Matthaeus,

W. Daughton, P. Wu, M. Shay, B. Loring, J. Borovsky,E. Leonardis, et al., Physics of Plasmas 20, 012303 (2013).

P. Wu, M. Wan, W. H. Matthaeus, M. A. Shay, and M. Swisdak,Phys. Rev. Lett. 111, 121105 (2013).

T. de Karman and L. Howarth, Proc. Roy. Soc. London Ser. A164, 192 (1938).

M. Hossain, P. C. Gray, D. H. Pontius Jr., W. H. Matthaeus, andS. Oughton, Phys. Fluids 7, 2886 (1995).

M. Wan, S. Oughton, S. Servidio, and W. H. Matthaeus, J. FluidMech. 697, 296 (2012), ISSN 0022-1120.

C. W. Smith, K. Hamilton, B. J. Vasquez, and R. J. Leamon,Astrophys. J. 645, L85 (2006).

W. H. Matthaeus, J. M. Weygand, P. Chuychai, S. Dasso, C. W.Smith, and M. G. Kivelson, Astrophys. J. 678, L141 (2008).

P. Dmitruk, W. H. Matthaeus, and N. Seenu, Astrophys. J. 617,667 (2004).

J. F. Drake, M. Swisdak, H. Che, and M. A. Shay, Nature 443,553 (2006).

B. D. G. Chandran, B. Li, B. N. Rogers, E. Quataert, andK. Germaschewski, Astrophys. J. 720, 503 (2010).

R. J. Leamon, C. W. Smith, N. F. Ness, W. H. Matthaeus, andH. K. Wong, J. Geophys. Res. 103, 4775 (1998).

M. A. Shay, J. F. Drake, R. E. Denton, and D. Biskamp, J.Geophys. Res. 103, 9165 (1998).

M. R. Brown, C. D. Cothran, and J. Fung, Phys. Plasmas 13,056503 (2006).

W. H. Matthaeus, S. Oughton, K. T. Osman, S. Servidio,M. Wan, S. P. Gary, M. A. Shay, F. Valentini, V. Roytershteyn,H. Karimabadi, et al., Astrophys. J. 790, 155 (2014), 1404.6569.

S. A. Markovskii, B. J. Vasquez, C. W. Smith, and J. V. Hollweg,Astrophys. J. 639, 1177 (2006).

T. N. Parashar, S. Servidio, M. A. Shay, B. Breech, and W. H.Matthaeus, Phys. Plasmas 18, 092302 (2011).

S. Dalena, A. F. Rappazzo, P. Dmitruk, A. Greco, and W. H.Matthaeus, Astrophys. J. 783, 143 (2014).

M. Wan, W. H. Matthaeus, V. Roytershteyn, H. Karimabadi,T. Parashar, P. Wu, and M. Shay, Physical Review Letters 114,175002 (2015).

A. Retino, D. Sundkvist, A. Vaivads, F. Mozer, M. Andre, andC. J. Owen, Nature Physics 3, 236 (2007).

K. T. Osman, W. H. Matthaeus, A. Greco, and S. Servidio,Astrophys. J. 727, L11 (2011).

J. Ambrosiano, W. H. Matthaeus, M. L. Goldstein, and D. Plante,J. Geophys. Res. 93, 14 383 (1988).

J. F. Drake, M. Swisdak, P. A. Cassak, and T. D. Phan, Geophys.Res. Lett. 41 (2014).

S. Dalena, A. Greco, A. F. Rappazzo, R. L. Mace, and W. H.Matthaeus, Phys. Rev. E 86, 016402 (pages 13) (2012).

P. Dmitruk and W. H. Matthaeus, Physics of Plasmas 13, 2307(2006).

S. A. Markovskii and B. J. Vasquez, Astrophys. J. 739, 22 (2011).J. Wang, Y. Yang, Y. Shi, Z. Xiao, X. T. He, and S. Chen, Phys.

Rev. Lett. 110, 214505 (2013).K. Makwana, V. Zhdankin, H. Li, W. Daughton, and F. Cattaneo,

Phys. Plasmas 22, 042902 (2015).J. A. le Roux, G. P. Zank, G. M. Webb, and O. Khabarova,

Astrophys. J. 801, 112 (2015).M. A. Shay, C. C. Haggerty, T. D. Phan, J. F. Drake, P. A. Cassak,

P. Wu, M. Oieroset, M. Swisdak, and K. Malakit, Physics ofPlasmas 21, 122902 (2014), 1410.1206.

S. Servidio, F. Valentini, F. Califano, and P. Veltri, Phys. Rev.Lett. 108, 045001 (2012).

M. Wan, W. H. Matthaeus, V. Roytershteyn, T. N. Parashar,P. Wu, and H. Karimabadi, Physics of Plasmas 23, 042307(2016).

A. Zeiler, D. Biskamp, J. F. Drake, B. N. Rogers, M. A. Shay,and M. Scholer, Journal of Geophysical Research (Space Physics)107, 1230 (2002).

T. N. Parashar, W. H. Matthaeus, M. A. Shay, and M. Wan,Astrophys. J. 811, 112 (2015).

P. Wu, S. Perri, K. Osman, M. Wan, W. H. Matthaeus, M. A.Shay, M. L. Goldstein, H. Karimabadi, and S. Chapman, ApJ763, L30 (2013).

Determining Nonlinear Time• Estimate of 𝜏nl(di) based on von Karman-

Kolmogorov phenomenology

– λ is correlation or energy containing scale• Is this estimate applicable to our system?

⌧nl(`) =`

�Z`= ⌧nl

✓`

�

◆2/3

�Z` = Z

✓`

�

◆1/3

Turbulence Simulations: Scaling of Heating

"p"e

/ ↵nl

Approximate Best Fit Not Consistent"p / ↵3

nl "e / ↵2nl

Ratio of Heating Matches Well

Conclusions: Heating• Turbulence Simulations:

• Theory Predictions:

• Why the difference?• Is ΔB at di scale best measure of magnetic fields upstream of

current sheets?• 𝜏nl(di) determined ignoring intermittency. Justified?• Are the number of x-lines changing with changing

parameters?• Filling factor of reconnection exhausts?

• εp/εe matches reconnection prediction quite well. – Even if some details of scaling of reconnection parameters wrong,

taking the ratio removes the discrepancy.

"p"e

/ ↵nl "p / ↵3nl "e / ↵2

nl

"p"e

/ ↵nl "p / ↵4nl "e / ↵3

nl