Super-Fermi Acceleration in Multiscale MHD Reconnection Stephen Majeski1aand Hantao Ji1b Department of Astrophysical Sciences Peyton Hall 4 Ivy Lane Princeton University Princeton

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Super-Fermi Acceleration in Multiscale MHD Reconnection

Stephen Majeski1, a) and Hantao Ji1, b)

Department of Astrophysical Sciences, Peyton Hall, 4 Ivy Lane, Princeton University, Princeton,

NJ 08544

(Dated: 3 April 2023)

We investigate the Fermi acceleration of charged particles in 2D MHD anti-parallel plasmoid reconnection,

ﬁnding a drastic enhancement in energization rate ˙εover a standard Fermi model of ˙ε∼ε. The shrinking

particle orbit width around a magnetic island due to ~

E×~

Bdrift produces a ˙εk∼ε1+1/2χ

kpower law with

χ∼0.75. The increase in the maximum possible energy gain of a particle within a plasmoid due to the

enhanced eﬃciency increases with the plasmoid size, and is by multiple factors of 10 in the case of solar ﬂares

and much more for larger plasmas. Including eﬀects of the non-constant ~

E×~

Bdrift rates leads to further

variation of power law indices from >

∼2 to <

∼1, decreasing with plasmoid size at the time of injection. The

implications for energetic particle spectra are discussed alongside applications to 3D plasmoid reconnection

and the eﬀects of a guide ﬁeld.

I. INTRODUCTION

Energy conversion in magnetic reconnection is pivotal

to understanding reconnection’s role throughout the Uni-

verse1–3. In solar ﬂares, estimates have found as much

as half of electrons being energized to non-thermal ener-

gies4,5. Moreover, within the solar wind and the earth’s

magnetotail, electron acceleration and power law energy

spectra are often found associated with plasmoids and

compressing or merging ﬂux ropes6–10. Recent years have

seen considerable eﬀort to explain these observations, fo-

cusing on three leading mechanisms during reconnection:

direct acceleration by reconnection electric ﬁeld11–13 or

by localized instances of magnetic ﬁeld-aligned electric

ﬁelds14, betatron acceleration due to ﬁeld compression

while conserving particle magnetic moments15–17, and

Fermi acceleration by “kicks” from the motional electric

ﬁeld within islands18–21. Fermi acceleration operates pri-

marily in multiscale, or plasmoid, reconnection which is

thought to be pervasive from solar ﬂares to magneto-

spheric substorms to accretion disks22–25. In these en-

vironments, it takes place within the large volume of

magnetic islands which pervade plasmoid-unstable cur-

rent sheets26. A unique characteristic of Fermi accelera-

tion which makes it particularly promising for explaining

power law distributions, is that the acceleration rate is it-

self a power law in energy18. This has led to simulations

ﬁnding Fermi-generated power law distributions over a

range of Lundquist numbers, Lorentz factors, guide ﬁelds,

and more27,28.

Analytical estimates of ﬁrst-order Fermi acceleration

are frequently based oﬀ of the seminal work of Drake

et al, which found that the particle acceleration rate

is linear in the particle energy, ˙ε∼ε(in what follows

we will refer to acceleration rate power law indices with

p,i.e. ˙ε∼εp)18. Note that we are concerned here in

a)Electronic mail: smajeski@princeton.edu

b)Also at Princeton Plasma Physics Laboratory

this work only with ﬁrst-order Fermi acceleration which

should not be confused with less eﬃcient, second-order,

or stochastic, Fermi acceleration. Other approaches have

described Fermi acceleration in more MHD-like plasmoid

mergers via conservation of the bounce invariant Jk17,21.

Building oﬀ of these concepts, energetic particle spectral

indices over a range of values larger than 1 have been

explained through a combination of Fermi acceleration,

various drifts, and particle-loss processes20,26. Eﬀorts

have also been made to implement the kinetic physics of

Fermi acceleration without resolving small scales29. Un-

fortunately, most analytical particle acceleration studies

are developed to explain the results of kinetic simulations

which are computationally limited in the scale separation

between large MHD magnetic islands and the Larmor ra-

dius (ρL) of accelerating particles. Yet many astrophys-

ical systems showing promise as a source for energetic

particles are deep within the MHD regime2,30. Such lack

of scale separation leads to diﬃculty in capturing eﬀects

like the conservation of adiabatic invariants, increasing

loss rates from magnetic islands through pitch-angle scat-

tering31,32. Additionally, for lower energy but still weakly

collisional particles, their bounce motion may not be fast

enough to assume conservation of Jk. We therefore pro-

pose a new model of Fermi-like acceleration in 2D MHD

anti-parallel reconnection, which focuses on systems with

large scale separation between thermal particle Larmor

radii and plasmoid sizes. With the aid of guiding-center

test particle simulations, we ﬁnd that enhanced particle

conﬁnement to compressing magnetic ﬁeld lines yields

an O(1) correction to the linear Fermi power law index

p= 1.

A. Linear Fermi acceleration

Consider a plasmoid embedded in a current sheet un-

dergoing 2D anti-parallel MHD reconnection with elec-

tric and magnetic ﬁelds ~

Eand ~

B, respectively. Away

from the x-point, the dominant electric ﬁeld component

is the motional ﬁeld which drives the “E cross B” drift

arXiv:2210.06533v4 [physics.plasm-ph] 30 Mar 2023

~uE=c~

E×~

B/B2, which, along with all other electric ﬁeld

components, is out-of-plane in this setup33. If a magne-

tized particle within a plasmoid is to gain energy, it must

experience net motion along this electric ﬁeld, in this case

via guiding center drift. The only drift in this circum-

stance satisfying this constraint is the curvature drift ~vC.

Note that we’ve assumed drifts arising from explicit time

dependence can be neglected, unlike those resulting from

particle motion along gradients in ˆ

b. This is due to the

slow nature of the MHD background compared to the

relatively fast motional time derivatives experienced by

high energy (and importantly super-Alfv´enic) particles.

Figure 1 shows the process of Fermi acceleration in such

a setup.

FIG. 1: Diagram of Fermi acceleration process. Blue

lines represent the magnetic ﬁeld (separatrix dashed),

and the curvature drift is given for a positively charged

particle.

As a magnetized, µ=mv2

⊥/2Bconserving particle

travels along a ﬁeld line within the plasmoid (with m

the particle mass and v⊥the particle velocity perpen-

dicular to ~

B), it enters a narrow region (with respect to

the orbit’s vertical height h) near the neighboring x-point

of thickness ∆ which is deﬁned by a large value of the

curvature of the magnetic ﬁeld. This region is generally

somewhat larger than the current sheet thickness δ, but

approaches that value with increasing proximity to the x-

point. The magnetic tension in this high curvature region

drives the magnetic ﬁeld to rapidly straighten out, there-

fore within ∆ the ~

E×~

Bdrift velocity is also large. In 2D

anti-parallel reconnection, the ~

E×~

Bassociated electric

ﬁeld and the curvature drift are aligned, therefore the

parallel energy of the particle is increased according to

˙εk= 2q~

E·~vC/m, where εk.

=v2

k, and

~vC=mεk

qB ˆ

b×(ˆ

b· ∇ˆ

b)≈ −2mεk

∆qB ˆz. (1)

Note we have assumed here that the gradient scale of

bis approximately ∆/2. The increase in εkgained by

the particle during its transit of ∆ is then estimated

as ˙εk∆/√εk≈4huEi∆√εk, with hi∆representing the

average over the narrow layer ∆. We have also used

|uE|=|E/B|, and assumed that vk |uE|in keeping

with Drake et al18. This process occurs each time the

particle transits the island width w, which takes a time

dtw≈w/√εk, yielding the linear Fermi acceleration rate:

dεk

dtF≈4huEi∆

εk

w.(2)

This expression is identical in appearance to that of

Drake et al, with key diﬀerences in meaning18. The

assumptions under which this equation was derived are

MHD without a guide ﬁeld, not kinetic, meaning no Ek

or Hall magnetic ﬁeld component is present. Equation

(2) has the appearance of being linear in energy, however

we will show that during a particle’s acceleration huEi∆

and ware not constant, leading to deviation from the

linear dependence.

II. TEST PARTICLE SIMULATIONS

To investigate possible variation of huEi∆and win

Eq.(2), we performed guiding center simulations of test

particles in a plasmoid reconnection scenario. To be

precise, we solved the following set of simpliﬁed non-

relativistic guiding-center equations 24,34

dt=vkˆ

b+~uE(3a)

dvk

dt=q

mEk+~uE·hvkˆ

b+~uE· ∇ˆ

bi−µ

mˆ

b· ∇B,

(3b)

by an adaptive time step 2nd order-accurate midpoint

method35, using time-evolving background data from a

2D MHD simulation36,37. The code which provided the

background ﬁelds solves the fully-compressible resistive

MHD equations via ﬁnite diﬀerences with a ﬁve point

spatial stencil and second-order trapezoidal leapfrog time

stepping38. These guiding center equations have been

simpliﬁed assuming that the time dependent drifts are

weak due to the slow nature of the MHD background

compared to the motional time dependence of super-

Alfv´enic particles. Out-of-plane motion of the guiding

center (but not out-of-plane acceleration) is ignored given

the 2D symmetry, and in the MHD simulation data used,

Ek= 0. An example snapshot of uEfrom the simu-

lation is shown in ﬁgure 2. Note that when interpret-

ing the magnitude of uE, the density and magnetic ﬁeld

away from the current sheet in this simulation approach

ρ0=B0= 1 in dimensionless numerical units. The spa-

tial grid size is 2000 (x) by 4000 (y) and time outputs

are available at intervals of one-tenth of the primary cur-

rent sheet Alfv´en time (for context, the snapshot ﬁg. 2

shows a zoomed-in portion of the grid which is 1000 by

100 cells). As a result, linear interpolation from the MHD

grid to the particle’s time and position is used. The back-

ground plasma beta is β= 1, with a uniform Lundquist

number of S= 105. The adaptive particle time step is

calculated as a fraction (CFL number) of the simulation

grid cell-crossing time for the particle’s velocity, including

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Super-FermiAccelerationinMultiscaleMHDReconnectionStephenMajeski1,a)andHantaoJi1,b)DepartmentofAstrophysicalSciences,PeytonHall,4IvyLane,PrincetonUniversity,Princeton,NJ08544(Dated:3April2023)WeinvestigatetheFermiaccelerationofchargedparticlesin2DMHDanti-parallelplasmoidreconnection,ndingadrasticen...

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Super-Fermi Acceleration in Multiscale MHD Reconnection Stephen Majeski1aand Hantao Ji1b Department of Astrophysical Sciences Peyton Hall 4 Ivy Lane Princeton University Princeton

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