Nuno Loureiro | XI Vienna meeting | July 2018

Turbulence in magnetized pair plasmas

Nuno Loureiro

Plasma Science and Fusion Center, MIT

Nuno Loureiro | XI Vienna meeting | July 2018 2

Magnetized pair plasmasFluid equations for a low beta, non-relativistic pair plasma

n±m

✓@v±

@t+ v± ·rv± � µ±r2v±

◆= ±n±e

✓E+

v± ⇥B

c

◆�rp± ⌥R

nm

✓@v

@t+ v ·rv +

1

n2e2j ·rj� µr2v

◆=

j⇥B

c�rp

m

ne2

✓@j

@t+ v ·rj+ j ·rv � µr2j

◆= E+

v ⇥B

c� ⌘j.

Start from two-fluid equations for electrons and positrons:

Sum and subtract:

v is center-of-mass velocity

Nuno Loureiro | XI Vienna meeting | July 2018 3

Magnetized pair plasmasFluid equations for a low beta, non-relativistic pair plasma

Impose reduced-MHD-like ordering: B = B0z +B?,with B?/B0 ⇠ ✏

v? = z ⇥r?�;B?p4⇡⇢

= z ⇥r? ,Introduce potentials:

@

@tr2

?�+ {�,r2?�} = { ,r2

? }+ VA@

@zr2

? + µr2?�,

@

@t

�1� d2er2

?� +

��, (1� d2er2

?) = VA

@�

@z+ ⌘r2

? � µd2er4?

momentum equation

Ohm’s law

These reduce to the familiar RMHD equations when electron inertia is neglected (unsurprisingly).

Nuno Loureiro | XI Vienna meeting | July 2018 4

Magnetized pair plasmasInvariants and waves

Alfvén wave modified at kinetic scales by electron inertia

energy

(generalized) cross helicity

These equations have two exact invariants at all scales:

Only one wave is supported:

Nuno Loureiro | XI Vienna meeting | July 2018 5

Turbulence in magnetized pair plasmas Turbulence at MHD scales

At MHD scales (kde << 1), the equations are the same as for ion-electron plasmas. So turbulence will be the same:

1. expect k-3/2 up until the reconnection scale,

2. followed by a transition to a k-3 (or -8/3) due to reconnection (?)

Vech et al., ApJ 2018

Nuno Loureiro | XI Vienna meeting | July 2018 6

Turbulence in magnetized pair plasmas Turbulence at kinetic scales

Let us work out the spectrum expected from a Kolmogorov-like energy cascade at kinetic scales (kde >> 1).

) 1

2

ZdV

�d2e(r2

? )2 + (r?�)

2 Expect equipartition between

these two terms (parallel and perpendicular kinetic energies) at these scales.

Nuno Loureiro | XI Vienna meeting | July 2018 7

Turbulence in magnetized pair plasmas Turbulence at kinetic scales (cont’d)

�� ⇠ "1/3k�4/3?

and same scaling for magnetic energy, by equipartition

This results in:

)

Finally, declare that the fluctuations are critically balanced at these scales: !l ⇠ !nl

Nuno Loureiro | XI Vienna meeting | July 2018 8

Turbulence in magnetized pair plasmas Dealing with the cross helicity

Estimate the flux of cross helicity as

Recall cross-helicity: It is not positive definite.

where R is a dimensionless cancelation factor at scale lambda. From the energy invariant:

so the cross helicity flux becomes

But energy flux (constant) is Thus:

Nuno Loureiro | XI Vienna meeting | July 2018 9

Conclusions

AcknowledgementsNSF-DOE Partnership in Basic Plasma Science and Engineering, Award No. DE-SC0016215 and NSF CAREER award no. 1654168.

Presented a tentative theory of turbulence in low beta, strongly magnetized, non-relativistic pair plasmas:

same as MHD at fluid scales

Energy cascade implies spectrum prediction at kinetic scales.

Critical balance then gives

Notice that reconnection is not allowed at (flux is no longer frozen) so no need to worry about that.

Loureiro & Boldyrev, arXiv:1805.09224

kk ⇠ k5/3?

k�11/3?

k?de � 1