Microscopic structure of electromagnetic whistler wave damping by kinetic mechanisms in hot magnetized Vlasov plasmas Anjan Paul1 2and Devendra Sharma1 2

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Microscopic structure of electromagnetic whistler wave damping by kinetic

mechanisms in hot magnetized Vlasov plasmas

Anjan Paul1, 2 and Devendra Sharma1, 2

1Institute for Plasma Research, Bhat, Gandhinagar, India, 382428

2Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India

(Dated: October 26, 2022)

The kinetic damping mechanism of low frequency transverse perturbations propagating parallel

to the magnetic ﬁeld in a magnetized warm electron plasma is simulated by means of electromag-

netic (EM) Vlasov simulations. The short-time-scale damping of the electron magnetohydrodynamic

whistler perturbations and underlying physics of ﬁnite electron temperature eﬀect on its real fre-

quency are recovered rather deterministically, and analyzed. The damping arises from an interplay

between a global (prevailing over entire phase-space) and the more familiar resonant-electron-speciﬁc

kinetic damping mechanisms, both of which preserve entropy but operate distinctly by leaving their

characteristic signatures on an initially coherent ﬁnite amplitude modiﬁcation of the warm electron

equilibrium distribution. The net damping results from a deterministic thermalization, or phase-

mixing process, largely supplementing the resonant acceleration of electrons at shorter time scales,

relevant to short-lived turbulent EM ﬂuctuations. A kinetic model for the evolving initial transverse

EM perturbation is presented and applied to signatures of the whistler wave phase-mixing process

in simulations.

I. INTRODUCTION

The electromagnetic turbulence prevails in collective

excitations of charged matter interacting with, and by

means of, the electromagnetic ﬁeld over a vast range of

spatiotemporal scales, usually terminated by dissipation

at the ﬁner scales. The solar-wind spectrum, for exam-

ple, shows that beyond a frequency breakpoint a deviation

exists from the inertial rage characterized by exponent -

5/3 of power law [1, 2]. In one of the plousible scenarios,

the whistler ﬂuctuations can be the fundamental mode

and central means of dissipation in this weak turbulence

regime [3]. A steepening present in the spectrum lead-

ing to considerably high spectral exponents (∼2-3) sug-

gests presence of considerable damping alongside to the

intra-spectral energy transfer [3]. Besides by conversion

to electrostatic modes [4], damping by kinetic transverse

wave-particle interaction must operate on the short lived

excitations [5–8] initiated by sponteneous ﬁeld ﬂuctua-

tions. Fresh perturbations, so triggered, excite warm

plasma eigenmodes by leaving long lasting remnants of

their initially enforced phase-space structure in the mem-

ory of nonthermal kinetic distributions [9]. The asymp-

totic long-time solutions of the collisionless kinetic for-

mulation [10, 11] applied to them therefore have large

scope of sophistication by admitting a strong determin-

istic thermalization, or phase-mixing [9], alongside the

damping evaluated in usual time asymptotic, t→0 limit.

The general kinetic evolution produced collisionless

damping of electromagnetic ﬂuctuations [12] involves a

rather complicated phase-space dynamics and is most

accessible by deterministic Vlasov simulations [13, 14].

Only a limited number of studies have rather determin-

istically simulated the dynamics of the transverse elec-

tromagnetic excitations and their damping/stability in a

hot collisionless magnetized plasma [15, 16], even as the

process remains critical for determining the operational

state of turbulence and the transport associated with it

both in space plasmas [3, 17] as well as in modern mag-

netic conﬁnement fusion experiments [18].

In the collisionless limit, the modiﬁcations made to

temperature, or width, of an initially equilibrium warm

electron velocity distribution produce a higher order cor-

rection to the resonant wave particle interaction process.

The analytical model predicts a related downward shift

in the whistler wave frequency in collisionless plasmas

with hotter electrons [8, 19, 20]. The recovered strength

of damping due to wave particle interaction however re-

mains underestimated in comparison with that produced

by the computer simulations implemented with reason-

ably low collisionality.

This paper addresses above aspects of kinetic whistler

damping mechanism, subsequent to the recovery of gen-

eral electromagnetic modes of a magnetized plasma

and their dispersive characterization in our ﬂux-balance

based [13, 21] Vlasov simulations. This is accompanied

by illustration of its detailed phase-spatiotemporal evolu-

tion. The interaction of electromagnetic modes, propa-

gating parallel to the magnetic ﬁeld with the resonant

particles is studied, particularly recovering the damp-

ing of the whistler waves via full kinetic mechanism

and comparison of the simulation results with those an-

alytically prescribed in the linear Landau theory limit

[9, 10]. Presented simulations and analysis enter the

ﬁner regime of kinetic phase-mixing of the electromag-

netic mode uniquely achievable by Vlasov simulations.

We qualitatively recover the phase-mixing eﬀects show-

ing the dominance of frequency v·kof the ballistic term

(∝exp(ikvt)) [9] accounting for short time response, in

addition to time asymptotic Landau damping results that

are routinely applied to turbulent electromagnetic ﬂuc-

tuations of suﬃciently short life time. First quantitative

analysis of the phase-spatiotemporal evolution of a 4D

electron phase-space distribution perturbation associated

arXiv:2210.13764v1 [physics.plasm-ph] 25 Oct 2022

with the transverse electromagnetic whistler mode simu-

lated in a hot magnetized Vlasov plasma is presented.

The present paper is organized as follows. In Sec. II

the electromagnetic Vlasov-Maxwell model considered

for the simulation is presented. Vlasov simulation im-

plementation and its characterization by dispersion of

transverse EM perturbation is presented in Sec. III. In

Sec. IV the simulations of Landau damping of transverse

whistler wave perturbations in a warm electron plasma

is presented followed by their comparison with analyt-

ical results. Based on the evolution of simulated per-

turbations, the signatures of phase-mixing supplemented

landau damping of the initial perturbation are presented

in Sec. V. The kinetic model for initial transverse EM

perturbations is presented and solved in Sec. V B. Con-

clusions are presented in Sec. VI.

II. THE ELECTROMAGNETIC VLASOV

PLASMA MODEL

The electromagnetic magnetized plasma modes simu-

lated in this paper follow pure kinetic formulation and

are well represented by the solutions of collisionless, fully

nonlinear Vlasov equation for species α,

∂fα

∂t +v·∂fα

∂x+qα

mαE+v×B

c∂fα

∂v= 0.(1)

The evolution of electric ﬁeld Eand magnetic ﬁeld B

for the electromagnetic processes follows the Maxwell’s

equations,

∇ × B=1

∂E

∂t +4π

cJ,(2)

∇ × E=−1

∂B

∂t ,(3)

∇ · B= 0 (4)

where the current density Jis described by

J=X

αZ∞

−∞

dvqαvfα(5)

Consistent with the electron magnetohydrodynamic

regime excitations, we use an externally applied constant

magnetic ﬁeld B0and inﬁnitely massive ions. The sub-

script for species αtherefore only assumes value, e, repre-

senting electrons (and henceforth omitted), which are the

only mobile species and contributing to the perturbation.

The full nonlinear kinetic model (1)-(5) is implemented

for the case of waves propagating parallel to an applied

magnetic ﬁeld (B0kk) where principle modes are high

and low frequency right and left handed circularly po-

larized modes. This set up includes the whistler waves

that are right handed polarized and propagate in the fre-

quency range Ωci < ω < Ωce, where Ωcα is the gyrofre-

quency of the species α.

III. KINETIC SIMULATIONS OF

ELECTROMAGNETIC WAVES IN

MAGNETIZED WARM VLASOV PLASMA

The simulations presented in this analysis progress

by numerically evolving the magnetized plasma (elec-

tron) velocity distribution function faccording to the

dynamics of the phase-ﬂuid ﬂow [9] which is governed

by the collisionless Vlasov equation (1), and associated

Maxwell’s equations (2)-(4). The 4-dimensional (4D ≡

1x-3v) phase-space simulations are performed using an

advanced ﬂux balance technique [13, 21, 22] generalized

to simulate the electromagnetic plasma modes in a large

range of magnetization of plasma species. The results of

simulations are characterized ﬁrst against the analytical

dispersion relation of left and right handed circularly po-

larized low and high frequency modes and then against

both analytic and numerical evaluation of the linear Lan-

dau damping descriptions [8, 9, 23].

A. The simulation set-up

In order to simulate the waves propagating parallel to

an ambient magnetic ﬁeld B0in a wide range of frequency

ωand wave vector k, we have assumed a setup where

both the B0and kare aligned to z-axis and the periodic

boundary condition is used at both the boundaries of the

one-dimensional simulation zone located between z= 0

and L. The setup therefore assumes symmetry along

both ˆxand ˆydirections with ﬁnite spread of electron

velocities along these dimensions, besides the direction ˆz.

The three dimensional equilibrium velocity distribution

for the electron species is considered to be a Maxwellian,

f(z, v) = 1

2πv2

th 3/2

j=x,y,z

exp −(vj− hvji)2

2v2

th ,(6)

where vth = (Te/me)1/2is electron thermal velocity, Teis

electron temperature in energy units and meis electron

mass.

At time t= 0, the equilibrium distribution is per-

turbed with sinusoidal perturbations (having variation

along ˆz) with initial amplitudes of hvi1,E1and B1

(∝exp(ikz)) consistent with transverse electromagnetic

(right or left handed circularly polarized) linear magne-

tized plasma modes propagating with a desired k[5, 24].

B. Dispersion characterization of electromagnetic

plasma modes

We present the dispersion relations recovered for the

case of very small electron thermal velocity (vth = 0.001

c) simulated and its comparison with the corresponding

analytical cold plasma dispersion relation [24, 25]. Spe-

ciﬁc to simulation cases presented in this section, we have

FIG. 1. Dispersion of simulated frequency (dots) compared

to the Right and Left handed polarized branches of the ana-

lytical dispersion relation (solid line).

used a 3-dimensional velocity space grid of rather mod-

erate size having 32 ×32 ×32 grid points, in combination

with spatial grid size also of 32 grid points. In Fig. 1,

the comparison is presented of the simulated values of

frequency ωof the perturbation plotted as function of

k, with the right and left handed circularly polarized

(RHCP and LHCP) branches of the analytical dispersion

relation in the limit of inﬁnitely massive ions (ωpe ωpi),

R,L =ω2

c2"1−ω2

ω(ω∓Ωce)#.(7)

Considering the parameters, vth = 0.001 c, the ratio

of electron cyclotron frequency and plasma frequency

Ωce/ωpe = 0.1 and suﬃciently small initial perturba-

tion exclusively in electron average velocity amplitude

∆V=hvi1max − hvi1min = 2 ×10−3c, the high fre-

quency (ω > Ωce) RHCP and LHCP mode phase ve-

locities are suﬃciently high for these waves to stay out

of resonance with the cold electrons chosen for this case.

Moreover, for the low frequency whistler waves excited

(dots on the red curve in the region 0 < ω < Ωce in

Fig. 1), the electron thermal velocity vth = 0.001ccho-

sen is still suﬃciently low for no signiﬁcant resonant elec-

tron population to be available at the resonant velocity,

vz=vres = (ω−Ωce)/k. In rest of this analysis we exclu-

sively characterize this low frequency whistler branch of

the perturbation for relatively warmer electrons in order

to analyze its resonant damping.

IV. LANDAU DAMPING OF THE

TRANSVERSE WHISTLER MODE

An advanced simulation set up, with grid size of

64 ×64 ×64 ×64 is implemented in the following sets

of simulations, by adopting the electron velocity range

[-0.26, 0.26] c. The velocity distribution of electrons as

function of the parallel velocity vk≡vzis plotted in

Fig. 2 at vx=vy= 0 for a range of electron thermal ve-

locity vth = 0.001c(inner most proﬁle) to 0.026c(outer

FIG. 2. The electron distribution function (normalized to its

maximum value) plotted as function of parallel velocity vk≡

vzat vx=vy= 0. The thermal velocity of electrons for the

curves with increasing width ranges from vth = 0.001c(inner

most proﬁle) to 0.026c(outer most proﬁle) , respectively. The

vertical dotted lines indicate resonant velocity vres and phase

velocity vpof the low frequency whistler mode at k= 1.0ωp/c.

most proﬁle) explored in the cases presented in this sec-

tion. The resonant velocity vres = (ω−Ωce)/k and phase

velocity vpof the low frequency whistler mode for pa-

rameters k= 1.0ωp/c and Ωce/ωp= 0.1, are indicated by

vertical dotted line. The proﬁles for smaller values of vth

show the population of resonant electrons drops to neg-

ligible values such that no resonant damping is present

for these cases.

In Fig. 3(a), the time evolution of the amplitude of

the velocity perturbation ∆Vis plotted in the simu-

lations done for the range of electron thermal velocity

vth = 0.001cto 0.026cfor the value of k= 0.8ωpe/c.

Two additional set of simulations done for the values of

k= 0.897 and k= 1.0ωpe/c are presented in Fig. 3(b)

and Fig. 3(c), respectively. The time evolution of ∆Vin

all the cases above cover the initial evolution of the wave

amplitude only for the time duration ∆t∼3ω−1

p(i.e.,

about a fraction of one complete cycle of the whistler

wave with frequency ω < Ωce while Ωce/ωp= 0.1) which

is the time duration in which the linear Landau damp-

ing rate remains reasonable estimate for the wave damp-

ing. The short time evolution in Fig. 3 has suﬃciently

low numerical widening of ffor resolving the eﬀect of

Tevariation, which is varied with relatively much larger

increments in this study.

A. Comparison of simulations with analytic

whistler damping rates

The damping rates of whistler velocity perturbation

amplitude can be compared with the available analytic

approximations of the Landau damping rate of whistlers.

For the purpose of comparison we have used the analytic

results from kinetic formulation in certain limiting cases.

The numerical evaluation of analytical expression is done

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MicroscopicstructureofelectromagneticwhistlerwavedampingbykineticmechanismsinhotmagnetizedVlasovplasmasAnjanPaul1,2andDevendraSharma1,21InstituteforPlasmaResearch,Bhat,Gandhinagar,India,3824282HomiBhabhaNationalInstitute,TrainingSchoolComplex,Anushaktinagar,Mumbai400094,India(Dated:October26,2022)Th...

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Microscopic structure of electromagnetic whistler wave damping by kinetic mechanisms in hot magnetized Vlasov plasmas Anjan Paul1 2and Devendra Sharma1 2

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