Post on 27-Apr-2021
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