January 2016

EPL, 113 (2016) 25001 www.epljournal.org

doi: 10.1209/0295-5075/113/25001

Linear kinetic Alfven waves in inhomogeneous plasma:

Effects of Landau damping

R. P. Sharma1, R. Goyal1(a), Nidhi Gaur1 and Earl E. Scime2

1 Centre for Energy Studies, Indian Institute of Technology - Delhi-110016, India2 Department of Physics and Astronomy, West Virginia University - Morgantown, WV 26506-6315, USA

received 9 November 2015; accepted in final form 18 January 2016published online 10 February 2016

PACS 52.35.Bj – Magnetohydrodynamics waves (e.g., Alfven waves)PACS 52.25.Xz – Magnetized plasmasPACS 52.38.Hb – Self-focusing, channelling, and filamentation in plasmas

Abstract – The turbulent spectrum of kinetic Alfven waves in inhomogeneous plasma is investi-gated in the presence of Landau damping. Inhomogeneities in transverse and parallel directionsto the ambient magnetic field are incorporated in the dynamics. Numerical solutions of the equa-tions governing kinetic Alfven waves in the linear regime are obtained while retaining the effectsof Landau damping, which have a significant impact on the frequency spectrum generated bypropagating kinetic Alfven waves. A semi-analytical model developed to elucidate the physics ofthis process is also described.

Copyright c© EPLA, 2016

Introduction. – Kinetic Alfven waves (KAWs) play amajor role in various space phenomena [1]. The disper-sive properties of KAWs make them able to participatein a variety of phenomena [2] in which classic, nondisper-sive, Alfven waves play no role. The CLUSTER data [3]shows that mode conversion from surface mode Alfvenwaves to KAWs across the magnetopause [4] transportslarge electromagnetic energy fluxes across geomagneticfield lines from magnetosheath flows into the magneto-sphere. KAW’s importance in particle heating and theenergy cascade of solar wind turbulence is suggested bytheoretical as well as by experimental studies [5,6]. KAWsare also known to play an important role in the stochasticion heating mechanism in magnetopause [7].

The comparison of the experimental results in heliconplasmas [8] with the theoretically predicted dispersion re-lation for KAW including full kinetic effects [9] providesstrong evidence for the excitation of KAW in inhomo-geneous plasma. The kinetic theory may deviate KAWnonlinear processes from an ideal behaviour [10] quanti-tatively as well as qualitatively. But before going to afully developed kinetic theory, it is better to study Lan-dau damping in magnetic fluctuations. Landau dampingis an important effect associated with KAWs [11,12]. Bor-gogno et al. [13] have used the Landau fluid model to studythe filamentation of Alfven waves due to density chan-nels. Passot and Sulem [14,15] have derived a Landau fluid

(a)E-mail: ravig.iitd@gmail.com

model to describe the magnetohydrodynamic waves in col-lisionless plasma where Landau damping is the main dis-sipation process. Hasegawa and Chen [16] have describedthe kinetic process of plasma heating by resonant excita-tion of shear Alfven waves.

Motivated by the Houshmandyar and Scime observa-tions [17], Sharma et al. [18] and Goyal et al. [19] examinedthe localization of KAWs in inhomogeneous plasma. Thesestudies were based on a fluid model and kinetic effects likeLandau damping which may play a vital role in the en-ergy decay process were not considered. Landau dampingmay also affect the localization process and the magnitudeof turbulence generated. Here we examine the effects ofLandau damping on the linear propagation of KAWs in in-homogeneous plasma. The model equations are solved nu-merically to study the localization of KAW and resultingturbulent spectrum. A semi-analytical model developedto provide an insight into the physics is also described.

In the next section the dynamical equations for linearlypropagating KAW with Landau damping effects are devel-oped and then solved numerically. In the third section theresults of the simulation of the dynamics of propagatingKAW are presented together with a semi-analytic model ofKAW dynamics based on a paraxial approximation. Con-clusions and additional results are summarized in the lastsection.

Dynamical equations. – Consider the ambient mag-netic field �B0 along the z-axis. An inhomogeneous plasma

25001-p1

R. P. Sharma et al.

with density inhomogeneity in transverse as well as paral-lel directions is describable by n0 exp(−x2/L2

x − z2/L2z),

where n0 is the background plasma density, Lx (Lz) isthe inhomogeneity scale length in transverse (parallel)

direction to the ambient magnetic field �B0. KAWs withlow frequency ω and finite amplitude are assumed to bepropagating in the x-z plane, i.e., �k0 = k0xx+ k0z z wherek0x (k0z) is the component of the wave vector perpendicu-lar (parallel) to the background magnetic field. Follow-ing the standard method [20], the dynamical equationfor finite-amplitude KAW propagating in the x-z planeis obtained:

i) the momentum balance equation:

∂�υj

∂t+2γL�υj =

qj

mj

E+qj

cmj

(

�υj × B)

− kB

mjnj

∇njTj ;

(1)

ii) the continuity equation:

∂nj

∂t+ ∇ · (nj�υj) = 0; (2)

iii) Faraday’s law

∇ × �E = −1

c

∂ �B

∂t; (3)

iv) Ampere’s law

∇ × �B =4π

c�J +

1

c

∂ �E

∂t, (4)

where mj , �υj , nj , Tj , respectively, define the mass, fluidvelocity, density and temperature of the species j (j =electrons e) or ions (i), kB is the Boltzmann constant, γL

is the phenomenologically incorporated Landau dampingfactor given by

γL

ω0≈ −

√π

VA

υte

k2⊥

ρ2i

4(5)

which may be obtained by using the dispersion relationobtained by Lysak and Lotko [11] and Hasegawa andChen [12]. J = n0e(υi − υe) is the current density. Hereω0 is the real part of the frequency ω. From eq. (4):

∂By

∂t= c

∂Ez

∂x− c

∂Ex

∂z. (6)

Using eq. (2), we get

υix = iωni

/(

ikxn0 +∂

∂xδn

)

, (7)

where δn is the density perturbation.Using electron dynamics of eq. (1), we obtain

Ez =

[

ikz

kBTe

e+

me

e

(

iω2 + 2γLω

kz

)] (

δn

n0

)

. (8)

Using the velocity components and eq. (8) in the currentdensity conservation equation (∇ · J = 0), we get

∂Ex

∂t= −

{(

ω2ci +

∂2

∂t2

)/

cω2ci

}

V 2A

(

1 − δn

n0

)

∂By

∂z.

(9)The ∂2/∂t2 in eq. (9) comes due to finite-frequency effectsand in the context of low frequencies, this equation (9)reduces to the usual equation [20]. Combining eqs. (6), (8)and (9), we obtain the dynamical equation for KAW:

∂2By

∂t2+ 2γL

∂By

∂t=

(

∂2

∂t2+ 2γL

∂

∂t

)

λ2e

∂2By

∂x2

− V 2Aρ2

s

∂4By

∂z2∂x2+

V 2A

ω2ci

(

ω2ci+

∂2

∂t2

)

exp

[

x2

L2x

+z2

L2z

]

∂2By

∂z2,

(10)

where λe =√

c2me/4πn0e2 is the inertial length of theelectrons, V 2

A = B20/4πnimi is the Alfven speed, ρs =

cs/ωci is the ion sound gyroradius at the electron tempera-ture, cs =

√

Te/mi is the ion sound speed, υte =√

Te/me

is the electron thermal speed, υti =√

Ti/mi is the ionthermal speed. ρi = υti/ωci is the ion gyroradius and ωci

is the ion cyclotron frequency. The inhmogeneities usedin eq. (10) are just like perturbations and are very smallso their spatial derivatives may be omitted in comparisonto other terms in the equation. To study the localizationprocess, we substitute the envelope solution

By = B0(x, z, t)ei(k0xx+k0zz−ω0t) (11)

into eq. (10) and obtain

− 2iω0

V 2Ak2

0z

(

1 +iγL

ω0

)

∂B0

∂t− 2iγLω0

V 2Ak2

0z

B0 =

[

−2ik0x

λ2eω

20

V 2Ak2

0z

+ 4γL

ω0k0x

λ2eω

20

V 2Ak2

0z

+ 2ik0xρ2s

]

∂B0

∂x

+2ik2

0xρ2

s

k0z

∂B0

∂z+

[

ρ2s − λ2

eω20

V 2Ak2

0z

(

1 +2iγL

ω0

)]

∂2B0

∂x2

+k2

0xρ2

s

k20z

∂2B0

∂z2− B0 exp

(

x2/L2z + z2/L2

z

)

+ B0. (12)

Equation (12) after normalization becomes

(

1 +iγL

ω0

)

∂B

∂t+ 2c4

ξ1

ξ2

[

1 − c5 − 2ic5c6∂2

∂x2

]

∂B

∂x

+ c24

ξ1

ξ2

∂B

∂z− i

ξ1

ξ22

[

1 − c5 + 2ic5c6

(

ξ22 − ∂2

∂x2

)]

∂2B

∂x2

− iξ1

ξ23

c24

4

∂2B

∂z2+ iξ1

(

exp

(

x2

ζ2x

+z2

ζ2z

)

− 1

)

B = 0, (13)

where c4, c5, c6, ξ1, ξ2, ξ3 are constants given byc4 = k0xρs, c5 = λ2

e(1 + k20xρ2

s)/ρ2s(1 + k2

0xλ2e), c6 =

√π VA

υte

ρ2

i

4 ; ξ1 ≈ 650, ξ2 = ξ3 ≈ 50 and the nor-malizing parameters are xn = ξ2ρs, zn = 2ξ3/k0z,

25001-p2

Linear KAWs in inhomogeneous plasma: Effects of Landau damping

(a) (b)

Fig. 1: (Colour online) Normalized magnetic-field intensity variation of KAW with direction of propagation (z) and transversedirection (x) at t = 0.04 s (a) and at t = 0.06 s (b) with and without Landau damping.

tn = 2ω0ξ1/V 2Ak2

0zζx = Lx/ρs and ζz = Lz/zn are the

normalized inhomogeneity scale lengths of the plasmachannel. As our model is motivated from the experimentconducted in helium plasma, we have used the followinglaboratory plasma parameters [17] in our numerical studyas well as in the semi-analytical analysis: B0 ≈ 560 G,plasma density, n0 ≈ 6 × 1012 cm−3, Te = 81200 K,Ti = 2900 K. These values yield characteristic plasmaparameters: VA ≈ 2.5 × 107 cm/s, cs = 1.3 × 106 cm/s,β(= c2

s/V 2A) ≈ 0.002, υte ≈ 1.1 × 108 cm/s, υti ≈ 2.445 ×

105 cm/s, ωci = 2.68 × 106 rad/s, λe = 0.1085 cm, ρi ≈0.09 cm, ρs ≈ 0.49 cm, k0x ≈ 0.85 cm−1, k0z ≈ 0.01 cm−1

for ω0/ωci = f0/fci = 0.1 and ω0 = 2.68 × 105 rad/s. Forthe parameters defined above, xn ≈ 24.5 cm for k0x ≈0.85 cm−1, zn ≈ 104 cm for k0z ≈ 0.01 cm−1, tn ≈ 0.005 sfor ω0 = 2.68 × 105 rad/s.

Equation (13) is mathematically similar to the nonlinearSchrodinger (NLS) equation if we neglect the second andfourth terms and replace the last term on the left-handside by (BB∗)B. To solve eq. (13) numerically, we haveadopted the algorithm developed for solving the standardNLS equation. Therefore, after testing the invariants ofNLS up to 10−5 accuracy; we modified the algorithm forsolving the NLS to accommodate eq. (13). For the initialhollow Gaussian profile:

B(x, z, 0) = a0(x2/r2

10 + z2/r220) exp(−x2/r2

10 − z2/r220).(14)

Equation (13) is solved numerically, where a0 is the ini-tial value of the amplitude of the KAW and r10 (r20)is the scale size of the initial wave field in the trans-verse (parallel) direction to the background magnetic field,where r10 = k−1

x and r20 = k−1z with kx and kz the

respective normalized wave numbers. The specific algo-rithm used to solve eq. (13) was developed by Sharma andSingh [21] and is based on a pseudo spectral method forspace integration with periodic lengths lx = r10, lz = r20

and 64 × 64 grid points. A finite-difference method withpredictor corrector scheme was employed for the evolu-tion in time. The time step was taken to be of the orderof Δt ≈ 10−4. To control the dynamics of KAW evolu-tion, we choose a0 = 0.02 and kx = kz = 0.2 as param-eters where the value of a0 is based on the experimentalconditions [17] and it defines the wave launching field nor-malized by the background magnetic field B0. r10 andr20 respectively define the system lengths that extend be-yond the size of a hollow Gaussian profile in transverseand parallel directions to B0.

Simulation results and simplified model. – Thenormalized magnetic-field intensities as a function of timefor fixed values of transverse and parallel inhomogeneityscale lengths are shown in figs. 1(a) and (b). The effect ofLandau damping is clearly observed. When Landau damp-ing effects are included, the corresponding peaks havethe same spatial structures but have lower values of peakmagnetic-field intensity (normalized). As expected, Lan-dau damping clearly affects the linear behavior of KAWs.The magnetic-field fluctuations vs. time are shown in fig. 2for a fixed spatial location (x, z) with and without Landaudamping effects. Looking carefully at the initial magneticfield fluctuations in fig. 2, the fluctuations with Landaudamping effects have the same initial behavior then astime progresses the Landau damping effects contributeand the magnetic-field amplitude decays.

The variation of |Bω|2 with ω, the power spectra,obtained from Fourier transforms of the total magnetic-field fluctuations, are shown in fig. 3. The values shownin fig. 3 were obtained from the magnetic-field fluctua-tion data generated by the simulation for a time window∼ 300 μs. The spectra show that the levels of turbulencevs. frequency are quite different with and without Landaudamping for fixed values of inhomogeneity scale lengths.The nature (spectral structure) of the turbulence in fig. 3 is

25001-p3

R. P. Sharma et al.

A0(η, z) =B00√

f0

(

(√2 + η

)2/

2

)

exp

(

−(√

2 + η)2/

2

)

exp (−kiz/2), (20)

Fig. 2: (Colour online) Time series of magnetic field B withand without Landau damping.

Fig. 3: (Colour online) Variation of |Bω|2 vs. ω for KAWshowing the Fourier spectrum for the data of fig. 2.

unchanged when the Landau damping effects are included,demonstrating the linear behavior of KAW propagation.

Because of the Landau damping factor introduced inthe KAW dynamics, the localization changes with time forfixed values of inhomogeneity scale lengths, as describedby eq. (13) and shown in figs. 1(a) and (b). As the timeprogresses, the spectral components of these spatially lo-calized structures may evolve at different rates. There-fore, the magnetic-field fluctuation amplitude is expectedto vary with time at a particular spatial location, as shownin fig. 2.

To better understand the results of the numerical sim-ulation, we developed a simplified model using a semi-analytical approach and the paraxial approximation. Theplasma is assumed to be homogeneous along the back-ground magnetic field and the only inhomogeneity is inthe transverse direction (Lz ≫ Lx). Then eq. (10) can be

approximated as

∂2By

∂t2+ 2γL

∂By

∂t=

(

∂2

∂t2+ 2γL

∂

∂t

)

λ2e

∂2By

∂x2

− V 2Aρ2

s

∂4By

∂z2∂x2+

V 2A

ω2ci

(

ω2ci +

∂2

∂t2

)

exp

(

x2

L2x

)

∂2By

∂z2. (15)

Equation (15) is then solved assuming the paraxialapproximation (x ≪ r0f0) [22], here r0 is the scale sizeof the KAW in the transverse direction and f0 is a di-mensionless parameter that defines the beam width of theKAW. Substituting the envelope solution

By = B0(x, z)ei(k0xx+k0zz−ω0t) (16)

into eq. (15), the steady-state KAW equation becomes

− 2ik0z

∂B0

∂z− k2

0z

V 2A

(

υ2te

λ2e − λ2

eω20

k20z

)

∂2B0

∂x2− i

2γLω0

V 2A

B0

+ k20z

(

x2

L2x

)

B0 = 0. (17)

An additional eikonal S0 is introduced to separate eq. (17)into real and imaginary parts by assuming a solution of theform

B0 = A0(x, z) exp {ik0zS0(x, z)} . (18)

The solution of eq. (17) is accomplished with a tech-nique [23] analogous to the paraxial approximation inwhich eqs. (17) and (18) are expressed in terms of theparameters η and z, i.e.,

η =(

x/r0f0 −√

2)

. (19)

r0f0 is the beam width and is the position of maximumirradiance.

The solution of the real part is then [24]

see eq. (20) above

S0(η, z) =(

(√

2 + η)2

/2)

ς(z) + φ0 (21)

ς(z) = ar20f0

df0

dz, (22)

where a = V 2A{(υ2

teλ2e − λ2

eω20/k2

0z)k20z}−1 and ki =

ω0γLRd/k0zV2A, Rd = k0zr

20 .

The governing differential equation for the KAW beamwidth parameter f0 is

d2f0

dz2=

7

12a2R2df

30

− f0

aL2x

. (23)

25001-p4

Linear KAWs in inhomogeneous plasma: Effects of Landau damping

(a)

(b)

Fig. 4: (Colour online) The variation of beam width parameter(f0) of KAW along the direction of propagation (z) for ω =0.1ωci (a) and for ω = 0.4ωci (b).

The right-hand side of eq. (23) plays a crucial role in theconvergence and divergence of the wave as it containsthe terms arising from diffraction and the inhomogeneousplasma profile. If these terms balance each other, self-trapping of the KAW occurs. In that case, the convergenceor divergence of the wave does not occur (i.e. f0 = 1) as itpropagates along z. This gives us a critical transverse in-homogeneity scale length (L2

x(Cr) ∼ (12/7)aR2d ∼), where

Rd ≈ 4 cm for r0 ∼ 20 cm.

To solve eq. (23) numerically, we assume boundary con-ditions corresponding to a plane-wave front, i.e. f0 = 1and df0/dz = 0 at z = 0. The results are presented infigs. 4(a) and (b) for a fixed value of the inhomogeneityscale length Lx but for ω0 = 0.1ωci and 0.4ωci, respec-tively. By increasing the wave frequency from 0.1ωci to0.4ωci the beam width parameter f0 attains its minimumvalue faster than that at lower frequency. Therefore, thelocalization process becomes faster. This may be under-stood from eq. (23): the first term on the right-hand sideviz. the diffraction term is inversely proportional to a2R2

d.So, the diffraction term as a whole is inversely propor-tional to ω6

0 . On the other hand, the second term on theright-hand side of eq. (23) viz. the inhomogeneous term isinversely proportional to a. It is obvious from the expres-sion of a that a is proportional to ω2

0 ; therefore the inho-mogeneous term is inversely proportional to ω2

0 . Hence, asthe value of ω0 increases, the converging effect dominatesover the diverging effect and f0 attains its minimum valuefaster than that at lower frequency (figs. 4(a) and (b)).

(a)

(b)

Fig. 5: (Colour online) The variation of the normalizedmagnetic-field amplitude with distance of propagation (z) atω = 0.1ωci (a) and at ω = 0.4ωci (b).

In addition to providing an insight into the localiza-tion process, the semi-analytical model also helps to ex-plain the effect of Landau damping on the propagation oflinear KAW in the steady state (eq. (20)). Figures 5(a)and (b) show the variation of wave magnetic-field ampli-tude vs. the distance of propagation (z) for wave fre-quencies ω0 = 0.1ωci and 0.4ωci, respectively. With theincrease in wave frequency, the magnetic-field amplitudedecreases faster with increasing distance of propagation.This effect can be understood from eq. (20) in which thewave magnetic field A0 depends on f0 and the Landaudamping factor ki. From the expression of ki, it is clearthat ki linearly depends upon ω0. Therefore, at higherfrequencies, the Landau damping factor ki increases. Asdiscussed above, at higher frequency, f0 attains its mini-mum values faster than at lower frequency. Also, from theexpression of A0 in eq. (20), it is clear that the wave mag-netic field A0 decays exponentially faster with increasingwave frequency. The net effect is also seen in figs. 5(a)and (b).

The semi-analytical model also shows the Landaudamping effect for steady-state systems also observed inthe numerical simulations. Just as figs. 1(a) and (b)clearly indicate the role of Landau damping in KAW prop-agation in the spatio-temporal domain (the wave inten-sity is attenuated and the localization of KAW occursdue to inhomogeneous plasma effects), figs. 5(a) and (b)

25001-p5

R. P. Sharma et al.

demonstrate the same Landau damping effect on wavepropagation in inhomogeneous plasma in the spatial do-main. Thus, the results obtained using the semi-analyticalmodel are consistent with those obtained using numericalsimulations.

Conclusion. – With Landau damping, the modelequations have been derived and solved using the nu-merical simulation technique as well as a semi-analyticalmethod. With the inclusion of the term arising due tolinear Landau damping, the wave dynamics is shown tofollow modified NLS equation.

Landau damping is the main dissipation process dueto electrons in collisionless plasma. Collisionless heatingdepends on the parallel electric field and owing to its smallperpendicular wavelength, KAW accompanies this field.It is identified as one of the dissipation mechanisms ofKAW which appears because of this parallel electric field.The inhomogeneities included in the dynamics help informing wave packets with varying wave numbers result-ing into the localization of KAW. As Landau damping isalso wave number dependent, so it affects the amplitude oflocalization, level of turbulence etc. Landau damping hasa profound effect on wave propagation and energy decayas indicated by the wave amplitudes and the resultantfrequency spectra. This analysis suggests that Landaudamping of KAWs in inhomogeneous plasmas couldplay an important role in particle heating in space andlaboratory plasmas. Further, it may also be the probablemechanism for the damping of interplanetry magneticfield.

∗ ∗ ∗

This work is partially supported by DST (India) andISRO (India) under RESPOND program. EES was sup-ported by the US National Science Foundation grantNo. PHY-1360278.

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