Nonlinear Evolution of Kinetic Alfvén Waves and Filament Formation

Solar Physics (2005) 229: 287–304 C© Springer 2005

NONLINEAR EVOLUTION OF KINETIC ALFVEN WAVESAND FILAMENT FORMATION

M. MALIK and R. P. SHARMACenter for Energy Studies, Indian Institute of Technology, Delhi 110016, India

(e-mail: [email protected])

(Received 18 March 2005; accepted 29 May 2005)

Abstract. The transfer of wave energy to plasma energy is a very crucial issue in coronal holes andhelmet streamer regions. Mixed mode Alfven waves, also known as kinetic Alfven wave (KAW) canplay an important role in the energization of the plasma particles because of their potential ability toheat and accelerate the plasma particles via Landau damping. This paper presents an investigationof the growth of a Gaussian perturbation on a non-uniform kinetic Alfven wave having Gaussianwave front. The effect of the nonlinear coupling between the main KAW and the perturbation hasbeen studied. The dynamical equations for the field of the main KAW and the perturbation havebeen established and their semi-analytical solution has been obtained in the low (β � me/mi � 1)and the high (β � me/mi � 1) β cases. The critical field of the main KAW and the perturbationhas been evaluated. Nonlinear evolution of the main KAW and the perturbation into the filamentarystructures and its dependence on various parameters of the solar wind and the solar corona have beeninvestigated in detail. These filamentary structures can act as a source for the particle accelerationby wave particle interaction because the KAWs are mixed modes and Landau damping is possible.Especially, in the solar corona, the low β and the high β cases could correspond to the coronal holesand the helmet streamer. The presence of the primary and the secondary filaments of the perturbationmay change the spectrum of the Alfvenic turbulence in the solar wind.

1. Introduction

Alfven waves are known to be the exact solution of the ideal magnetohydrodynamics(MHD) system, which assumes the concept of frozen-in field lines (Alfven, 1950).Alfven waves permeate the universe. The large amplitude Alfven waves have beenobserved in the solar wind and in other space and astrophysical plasmas (Tsurutani etal., 1994; Smith et al., 1997). Existence of the large amplitude Alfven waves accom-panying the fast solar wind have been shown by Helios 2 data (Hollweg et al., 1982)and recently by Ulysses observations (Smith et al., 1995). Alfven waves have beeninvestigated by number of authors to explain various observed phenomena in themagnetosphere–ionosphere coupling (Lysak, 1991; Trakhtengertz and Feldstein,1984; Streltsov and Lotko, 1999) and in space plasmas. Some of these examplesinclude turbulent heating of the stellar corona (Pettini, Nocera, and Vulpiani, 1985),coherent radio emission (Lakhina and Buti, 1988; Lakhina, Buti, and Tsinsadze,1990), interstellar scintillations of the radio sources (Spangler, 1991), generationof the stellar winds and extragalactic jets (Jatenco-Pereira, 1995), etc. Observations

288 M. MALIK AND R. P. SHARMA

of the Alfvenic intermittent turbulence are reported by Marsch and Liu (1993), Tuand Marsch (1995).

These waves have been subject of considerable interest for extensive studyof parametric instabilities (both analytically and through numerical simulations).Hoshino and Goldstein (1989) used both MHD theory and numerical simulation tostudy the nonlinear evolution of the parametric instability in waves fluctuating alongan unperturbed magnetic field. Vinas and Goldstein (1991) have studied the decayand the modulational instabilities along with many other parametric instabilities.Malara and Velli (1996) studied the parametric instability of large-amplitude non-monochromatic Alfven waves. An important parametric instability is the transversecollapse of a slightly perturbed small-amplitude Alfven wave; often referred as wavefilamentation (Champeaux, Passot, and Sulem, 1997, 1998).

The filamentation process has been studied in detail in the context of laser plasmainteraction (Kruer, 1988). For a non-uniform laser beam, because of the finite sizeof the beam, diffraction takes place. Also nonlinearity present in the plasma leadsto reduction of the beam size. When these two effects balance each other, thelaser beam will propagate without diverging or converging and is said to be in aself-trapped mode. The corresponding laser field (power) can be calculated and istermed critical field (power). If the laser field is more than the critical field then theintense region of intensity of the beam results, termed filaments or hot spots.

The surface temperature of the Sun is 6000 K, while the temperature of the solarcorona is 106 K. It has remained a mystery as to why the solar corona is hotterthan the Sun’s surface and how these high temperatures are maintained in spite ofthe radiative cooling. The physical mechanism responsible for the heating of thesolar corona and the generation of the fast solar wind is still not well established.A lot of work has been done on the wave heating of the solar corona and thesolar wind. The fluctuations in the solar wind are Alfvenic (the magnetic field andthe plasma velocity are often highly correlated) and move predominantly outwardfrom the Sun (Belcher and Davis, 1971), suggesting their solar origin. Alfvenwaves can provide the heat input and the wave ponderomotive force, also referredto as the wave pressure gradient, could provide necessary acceleration to the solarwind. Alfven wave can be dissipated in the solar corona by magnetic reconnection,current cascade, MHD turbulence, Alfven resonance, resonance absorption or phasemixing. Ulysses and Helios observations have shown that in the high speed streamsminor (heavy) ions flow faster than the protons by about local Alfven speed andthat they are hotter than the protons roughly in proportion of their masses. Remoteobservations by UVCS on SOHO indicate that in the vicinity of 3 Rs, the O5+

ions have a perpendicular temperature T⊥ ∼ 2 × 108 K, while the protons haveT⊥ ∼ 3×106 K. Also the bulk flow speeds of O5+ are faster than the proton speedsin r ≥ 2Rs. Furthermore, heavy ions are strongly anisotropic in temperature (seeHollweg and Isenberg, 2002 and references therein for a thorough review).

Hollweg and Isenberg (2002) proposed resonant interaction with cyclotronwaves as the possible mechanism responsible for heating and accelerating the

KINETIC ALFVEN WAVES AND FILAMENT FORMATION 289

coronal hole ions to generate the fast solar wind. Vinas, Wong, and Klimas (2000)showed that high-frequency electron plasma oscillations excited by low-frequencyobliquely propagating electromagnetic waves can heat the corona. Vointenko andGoossens (2004) have studied the possible heating produced by oblique kineticAlfven waves and found that the perpendicular heating of the heavy ions observed inthe corona may be explained by the superadiabatic acceleration of the ions across themagnetic filed induced by the kinetic Alfven waves. Some other alternative mecha-nisms for solar wind heating such as heating due to shocks and rotational discontinu-ities (RDs) have also been investigated (Lee and Wu, 2000; Vasquez and Hollweg,1999, 2001). Tsurutani et al. (2005) suggested that Alfven waves, discontinuities,proton perpendicular acceleration and magnetic decrease are interrelated. He arguedthat the discontinuities are the phase-steepened edges of the Alfven waves. The ionacceleration is associated with the dissipation of the phase-steepened Alfven waves.

Although a number of mechanisms for the coronal heating and acceleration ofthe solar wind have been investigated and are also able to explain many observations,problems still remain. The filamentation process (hot spots formation) may providea clue to the dissipation problem, as it is a fast (catastrophic) way to transport energyat small scales. Therefore, the filamentation of Alfven waves is important in thecontext of the solar wind and the solar corona.

When the thermal motion of the plasma particles is to be taken into account,the plasma has to be described by a distribution function (kinetic theory), whichdescribes the positions and velocities of the individual particles. Departure fromMHD allows shear or kinetic Alfven waves (Hasegawa and Mima, 1976; Lysak andCarlson, 1988 and recently summarized by Hollweg, 1999 and Gekelman, 1999)to exist as a normal mode of the plasma. Kinetic Alfven wave (KAW) is the shearAlfven wave modified by the short wavelength effects, and it can be created whenan obliquely propagating shear Alfven wave is affected by the electron temperaturesuch that a non-zero parallel electric field arises within the wave itself due to localcharge separation (Wu and Chao, 2004). It is because of the presence of the parallelelectric field, KAW can play an important role in accelerating and heating theplasma particles.

Most of the work related to filamentation is done in the framework of HallMHD (Champeaux, Passot, and Sulem, 1997, 1998). However, not much of theanalytical or numerical work has been done on the KAW filamentation. Shukla,Sharma, and Malik (2004) studied the KAW filamentation by taking a uniformplane Alfven wave with a Gaussian perturbation. But in more realistic cases, Alfvenwaves are assumed to be having localized structures in the form of the wave packets.Therefore, the filamentation should be studied when the perturbation is assumedto be superimposed on a wave packet rather than on a plane wave. The nonlinearcoupling between the main KAW and the perturbation is expected to form newfilaments (we call them secondary filaments here).

In this paper we have studied the filamentation process arising on account ofthe coupling between the main kinetic Alfven wave (having non-uniform intensity


distribution in a plane transverse to the direction of propagation) and the perturba-tion. The filamentation process could be a possible mechanism for the coronal heat-ing and high speeds of the solar wind. In order to investigate the nonlinear evolutionof the KAW, we derive the general analytical expressions for the wave amplitudesfor the low (β � me/m i � 1) and the high (β � me/m i � 1) β regimes. Espe-cially in the solar corona the low β and high β cases could correspond to the coronalholes and the helmet streamer. It should be mentioned here that for the arbitrary β

values (including intermediate β values) one has to use the Boltzmann equation.The second section provides the threshold (critical) field for filamentation (singlehot spot) taking into account the mutual nonlinear interaction between the mainKAW and the perturbation. We have studied the case of solar wind turbulence andthe coronal holes. The third section provides discussion and conclusion. Finally,the last section summarizes the paper/work.

2. Model Equations

We consider propagation of the linearly polarized kinetic Alfven wave in the mag-netized plasma having ambient magnetic field BB along the z axis. The initialdistribution is assumed to be of the form

B0 B∗0 = B2

00 exp

(−x2

r20

), (1)

where r0 is the transverse scale size of the main KAW. Let a perturbation be super-imposed on the main KAW such that its initial intensity distribution is given by

B1 B∗1 = B2

100 exp

(−x2

r210

), (2)

where r10 is the transverse scale size of the perturbation. The total magnetic fieldvector can be written as

B = B0 + B1. (3)

The dynamical equation governing the propagation of the KAW in the x–z-planecan be obtained by using the standard method (Bellan and Stasiewicz, 1998; Shuklaand Stenflo, 1999, 2000; Shukla and Sharma, 2002; Shukla, Sharma, and Malik,2004) and can be written as

∂2 B

∂t2= �1λ

2e

∂4 B

∂x2∂z2− �2V 2

Teλ2e

∂4 B

∂x2∂z2+ v2

A

(1 − δns

n0

)∂2 B

∂z2, (4)

where B is the total magnetic field of the KAW, with y component. Also �1 = 1and �2 = 0 for β � me/m i and �1 = 0 and �2 = 1 for β � me/m i. n0 is theelectron density of plasma in the absence of waves and δns is the number densitychange given by ne − n0. ne is the modified electron density. The density can be


modified due to both the nonlinearity driven by the nonlinear electron heating andthe ponderomotive force by the KAW. Also,

VTe

(=

√Te

me

)

is the electron thermal speed,

vA

(=

√B2

B

4πn0m i

)

is the Alfven speed and

λe

(=

√c2me

4πn0e2

)

is the electron inertia length.Consider a plane wave solution of Equation (4)

B = B(x, z)ei(kx x+kz z−ωt). (5)

Using Equations (4) and (5), one gets

−2ikz∂ B

∂z+ 1

υ2A

(�1λ

2eω

2 − �2V 2Teλ

2ek2

z

)∂2 B

∂x2− k2

z

∂ns

n0B = 0, (6)

where ω is the frequency of the KAW.Also,

∂ B

∂z� kz and

∂ B

∂z� kx B,

δns

n0= φ(B B∗), (7)

where φ(B B∗) for the low β can be written as (Bellan and Stasiewicz, 1998; Shuklaand Stenflo, 1999, 2000; Shukla and Sharma, 2002; Shukla, Sharma, and Malik,2004)

φ(B B∗) = exp(−α B B∗) − 1,

where α = 1+8k2x λ

2e

48πn0Te , and φ(B B∗) for the high β case can be written as

φ(B B∗) = exp(γ B B∗) − 1,

where γ = 1−α0(1+δ)16πn0T

υ2Akz

ω2 ,

α0 = ω2

ω2ci

,

δ = me

m i

(kx

kz

)2

.


Using Equation (6), the field vector of the main KAW (B0) and the perturbation(B1) satisfy the following equations

−2ikz∂ B0

∂z+ 1

υ2A

(�1λ

2eω

2 − �2V 2Teλ

2ek2

z

)∂2 B0

∂x2− k2

z φ(B0 B∗0 )B0 = 0, (8)

−2ikz∂ B1

∂z+ 1

υ2A

(�1λ

2eω

2 − �2V 2Teλ

2ek2

z

)∂2 B1

∂x2− k2

z {φ(B B∗) − φ(B0 B∗0 )}B0

− k2z φ(B B∗)B1 = 0. (9)

Following Akhmanov, Sukhorukov, and Khokhlov (1968) and Shukla andSharma (2002), the solution for B0 can be written as

B0 = A0 exp(ikz S0(x, z)),

A20 = B2

00

f0exp

( −x2

r20 f 2

0

),

(10)

S0 = β0x2

2+ φ0,

β0 = a1

f0

d f0

dz.

In Equation (10), f0 is the dimensionless beam width parameter governed by thedifferential equation in low β case

d2 f0

dz2= 1

a2 R2d f 3

0

+ αB200

ar20 f 2

0

exp

(−α B2

f 20

), (11)

where Rd = kzr20 .

Similarly for the high β case, the differential equation for f0 can be written as

d2 f0

dz2= 1

a2 R2d f 3

0

− γ B200

ar20 f 2

0

exp

(γ B2

f 20

). (12)

The initial conditions on f0 for the plain wave front ared f0

dz= 0, and f0 = 1 at z = 0.

To obtain the solution of Equation (9) in the low β case, we express

B1 = A1(x, z) exp(ιkz S1(x, z)), (13)

where S1 is the eikonal. Substituting for B1 from Equation (13) into Equation (9),we obtain the following equations after separating real and imaginary parts.

2∂S1

∂z+ 1

υ2A

(�1λ

2eω

2 − �2V 2Teλ

2ek2

z

)(∂S1

∂x

)2

,

= 1

k2z A1

1

υ2A

(�1λ

2eω

2 − �2V 2Teλ

2ek2

z

)∂2 A1

∂x2+ φeff, (14a)


where φeff = φ(B B∗) + cos(φp) A0

A1[φ(B B∗) − φ(B0 B∗

0 )] and

∂ A21

∂z+ 1

υ2A

(�1λ

2eω

2 − �2V 2T eλ

2ek2

z

)(∂ A21

∂x

∂S1

∂x+ A2

1∂2S1

∂x2

)−

− [αkz A2

0 sin(2φp) exp(−α B0 B∗0 )

]A2

1 = 0, (14b)

where φp = −kz(S1 − S0).φp is the measure of the change in the angle between the magnetic field of the

main wave and that of the perturbation. This angle changes with the distance ofpropagation because of the nonlinear effects. The value of φp depends on the phaseangle between the main KAW field and that of the perturbation field at z = 0 andthe phase shift introduced in the perturbation field due to the nonlinearity in themedium.

In writing Equation (14b) the nonlinear part has been expanded as

φ(B B∗) = φ(B0 B∗0 ) + ∂φ

∂ B B∗

∣∣∣∣B B∗=B0 B∗

0

(B B∗ − B0 B∗0 ). (15)

Following Akhmanov, Sukhorukov, and Khokhlov (1968), the solution ofEquations (14a) and (14b) can be written as

A1 = B100

f 1/21

e(−x2/2r210 f 2

1 )e−ki z, (16)

also

S1 = x2

2β1(z) + φ

;1(z), (17)

where

β1 = a11

f1

d f1

dz, (18)

where f1 is the dimensionless beam width parameter of the perturbation, inverse ofβ1 represents the radius of curvature of the wavefront and B100 is the field intensityof the perturbation at z = 0 and x = 0. The growth rate ki is given by

ki = −1

2kzα A2

0 sin(2φp)e−α A20, (19)

and

a1 =(

�1λ2e − �2V 2

Teλ2ek2

z

ω2

)−1υ2

A

ω2. (20)


Using Equations (15)–(20) in Equation (14a), the equation for f1 can be obtainedby equating coefficients of x2 in the resulting equation

d2 f1

dξ 2= R2

d

a21k2

z r410

1

f 31

+ f1

a1φ′, (21)

where

φ′ = αR2d B2

00

r20 f 3

0

e(−α B2/ f )

∣∣∣∣r=r10

− 3αR2d cos(φp)

2

(1

f1 f0

)1/2

×

× B00 B100

(2α B2

00

r20 f 3

0

− 1

r20 f 2

0

− 1

r210 f 2

1

)e(−ki z)e(−α B2

00/ f0) −

− 2αR2d cos2(φp)

f0

(α B2

00

r20 f 3

0

− 1

r20 f 2

0

)e(−α B2

00/ f0). (22)

Equation (21) determines the convergence/divergence of the perturbation. Theabove solution has been obtained in the paraxial approximation (x2 � r2

10 f 21 ). The

critical field of the perturbation for self-trapping is obtained by balancing two termsin the right side of Equation (21). The corresponding condition for β � me/m i and�1 = 1 and �2 = 0 for the critical magnetic field is given by

3αR2d cos(φp)

2B00 B100

(2α B2

00

r20

− 1

r20

− 1

r210

)e(−α B2

00) =

= αR2d B2

00

r20

e(−α B2)

∣∣∣∣r=r10

− 2αR2d cos2(φp) ×

×(

α B200

r20

− 1

r20

)e(−α B2

00) + R2d

a1k2z r4

10

. (23)

Similarly for the case when β � me/m i and �1 = 0 and �2 = 1, usingonly pondermotive density changes, the equation for f1 is given as shownbelow,

d2 f1

dξ 2= R2

d

a21k2

z r410

1

f 31

− f1

a1φ′, (24)

where

φ′ = γ R2d B2

00

r20 f 3

0

e(γ B2/ f0)

∣∣∣∣r=r10

− 3γ R2d cos(φp)

2

(1

f1 f0

)1/2

×

× B00 B100

(−2γ B200

r20 f 3

0

− 1

r20 f 2

0

− 1

r210 f 2

1

)e(−ki z)e(γ B2

00/ f0) −

− 2γ R2d cos2(φp)

f0

(−γ B2

00

r20 f 3

0

− 1

r20 f 2

0

)e(γ B2

00/ f0). (25)


The growth rate (ki ) for the high β case is

ki = 1

2kzγ A2

0 sin(2φp)eγ A20 . (26)

Just as in the low β case, the expression for the critical value of the field can beobtained as mentioned above

3αR2d cos(φp)

2B00 B100

(−2γ B2

00

r20

− 1

r20

− 1

r20

)e(γ B2

00) =

= −γ R2d B2

00

r20

e(γ B2)

∣∣∣∣r=r10

− γ 2R2d cos2(φp) ×

×(

−γ B200

r20

− 1

r20

)e(γ B2

00) + R2d

a1k2z r4

10

. (27)

1. The typical high β (β > m/m i) coronal parameters for β = 0.02 are as givenbelow

Te = 0.4 × 106 K; Ti = 0.2 × 106; BB = 3 Gauss;

fpe = 8.98 × 107 Hz; number of protons = 108 cm−3

Using these values one finds that

fci = 4.568 × 103 Hz; ρ = 245 cm; υA = 6.54 × 107 cm s−1;

VTe = 2.46 × 108 cm s−1.

Here, ρ = Cs/ωci, where Cs is the ion sound speed. For ω = 31.4×102 s−1 andkxρ = 0.01, we get kz = 4.78×10−5 cm−1; kx = 4.078×10−5 cm−1, where fci

is the ion cyclotron frequency and fpe the electron plasma frequency. For thesetypical parameters, the critical value of the magnetic field of the main KAW asgiven by Equation (12) comes out to be 2.43 × 10−4 G, for r0 = 6.54 km.

The critical value of the perturbation field as given by Equation (27) comes outto be 1.47×10−5 G for B00 = 7.29×10−4 G, r10 = 2.87 km and r0 = 6.54 km.

2. Next we consider the low β (β < me/m i) case of coronal holes. At the base of thesolar coronal holes the typical parameters for β = 2×10−4 are as given below:

Te = 0.4 × 106 K; Ti = 0.2 × 106 K; BB = 32 Gauss;

fpe = 8.98 × 107 Hz; n0 = 108 cm−3.

Using these values one finds that

fci = 4.8732 × 104 Hz; λe = 53.1993 cm;

υA = 6.9808 × 108 cm s−1; VTe = 2.4629 × 108 cm s−1.


for ω = 13.8 × 103 rad s−1 and kxλe = 0.001, we get kz = 1.9770 × 10−5 cm;kx = 1.8797 × 10−5 cm.

For these typical parameters, the critical value of the magnetic field ofthe main beam as given by Equation (12) for the low β comes out to be3.62 × 10−5 G for r0 = 15.8 km and the critical value of the perturbation asgiven by Equation (23) comes out to be B10 = 1.12 × 10−6 G for r1 = 6.9 km,B00 = 1.09 × 10−4 G, r0 = 15.8 km.

3. The typical high β (β > me/m i) solar wind parameters for β = 0.121 are asgiven below:

Te = 0.5 × 105 K; Ti = 0.2 × 105; BB = 10−4 G;

fpe = 20 × 103 Hz; number of protons = 5 cm−3.

Using theses values one finds that

fci = 0.152 Hz; ρ = 2.5 × 106 cm; υA = 9.8 × 106 cm s−1;

VTe = 8.7 × 107 cm s−1.

Here ρ = Cs/ωci, where Cs is the ion sound speed. For ω = 0.01 rad s−1 andkxρ = 0.01, we get kz = 6.44 × 10−9 cm−1; kx = 3.98 × 10−9 cm−1 .

For these typical parameters, the critical value of the magnetic field for themain kinetic Alfven wave as given by Equation (12) comes out to be 2.54 ×10−8 G for r0 = 48.8×103 km. The critical magnetic field value of the perturba-tion as given by Equation (27) comes out to be 1.28×10−9 G, for the main kineticAlfven wave field 6.1 × 10−8 G, r10 = 23.6 × 103 km and r0 = 48.8 × 103 km.

3. Discussion and Conclusion

In the present work we start with the propagation of kinetic Alfven waves withoutany perturbations. The initial wavefront of the main KAW is taken to be Gaussian.We have used the paraxial approximation (x � r10 f1) in our analysis. Further, thefield should be slowly converging/diverging which requires δns/n0 � 1. For thetypical parameters used here, for the solar wind and the coronal holes, this conditionis satisfied. We have studied the dependence of the filamentation of the main KAWon the initial magnetic field strength, for the typical parameters of the solar windand the solar corona in the low and the high β cases.

Figures 1(a) and (b) depict the intensity distribution of the main KAW in the solarcorona at different distances along the direction of propagation, but in a direction(plane) transverse to the direction of propagation, for the high β case. As observedin these figures, filamentary structures are seen at different locations. This can beexplained using Equation (12). When the initial magnetic field (maximum field atz = 0) of the main KAW is more than its critical magnetic field, the nonlinear term


Figure 1. The variation of the magnetic field intensity (normalized to background field intensity) ofthe main KAW, with x and normalized distance of propagation ξ ≡ z/Rd0 (≡ z, in the figures), for thehigh β (= 0.02) solar corona for different initial (viz. at z = 0) values of B00: (a) B00 = 4.1×10−4 G;(b) B00 = 6.3 × 10−4 G.

(second term in the right-hand side of Equation (12)) dominates and the value off0 decreases with the distance of propagation. But when f0 becomes very small,the diffraction term (first term in the right-hand side of Equation (12)) starts dom-inating. Therefore, f0 increases with the distance of the propagation till f0 becomeso large that the diffraction term becomes smaller in comparison to the nonlinearterm. f0 further decreases due to the nonlinear effects, till it becomes so small thatthe diffraction term again dominates and f0 starts diverging and this process re-peats. Hence, the main KAW attains certain minimum beam width parameter (f0),and the intensity of the main KAW in these small size structures becomes veryhigh.

Figures 1(a) and (b) display the variation of main KAW intensity for differentvalues of the initial magnetic field strength of the main KAW in the solar corona forthe high β case. It is seen from Figures 1(a) and (b) that the intensity of the filamentsof the main KAW depends on its initial magnetic field strength. If the initial magneticfield strength (maximum field value at z = 0) is less than the critical magnetic fieldstrength, the diffraction term dominates and the KAW will diverge. However, if


the initial field strength is more than the critical field value, KAW will split intothe filamentary structures, the intensity of which increases with the increase in theinitial magnetic field value. Also the number of filaments increases and their spacingdecreases. This can be explained as below. The increase in the initial magnetic fieldvalue of the KAW increases the nonlinearity in the wave. As the nature of f0 is alsogoverned by the nonlinear term, the increase in the nonlinear term results into morerapid changes in f0 with z and the balancing of the nonlinear term and the diffractionterm takes place at lower value of the f0. As a result the interspacing between thefilaments decreases and the intensity of the filaments increases. Similar behavior(as in Figures 1(a) and (b)) is observed in the low β solar corona (not shown here)and the high β solar wind as shown in Figures 2(a) and (b).

In the space plasmas it is not necessary that the KAW will propagate without anyperturbation. Perturbation (modulating agent) can be any perturbation for exampledue to fluctuation of the background field. Such perturbations take energy from themain KAW by nonlinear interactions, grow and finally can form their own filaments.So it is interesting to study how the filamentation process of the perturbationsevolves when the main KAW is already filamented.

Figure 2. The variation of the magnetic field intensity (normalized to background field intensity) ofthe main KAW, with x and normalized distance of propagation ξ ≡ z/Rd0 (≡ z, in the figures), for thehigh β (= 0.121) solar wind for different initial (viz. at z = 0) values of B00; (a) B00 = 4.07×10−8 G;and (b) B00 = 5.08 × 10−8 G.


Figure 3. The variation of the critical magnetic field of the perturbation with the magnetic fieldstrength of main KAW for different values of the transverse scale size of the perturbation for the highβ solar corona.

Therefore we have studied the nonlinear interaction between a Gaussian per-turbation present on the main KAW and the main KAW. As we have seen that theGaussian KAW splits into filamentary structures, when its initial intensity is higherthan the critical field intensity. In these filaments the magnetic field intensity be-comes very high. Because of the nonlinear interaction, the perturbation takes energyfrom the main KAW and can grow. If the intensity of the perturbation and that ofthe main KAW are higher than their respective critical field then it also results intofilamentation of the perturbation.

Figure 3 shows the variation of the critical magnetic field of the perturbationwith the main KAW magnetic field (at z = 0) for different values of the transversescale size of the perturbation in the solar corona for the high β case. As seen fromthe figure, the critical field value of the magnetic field of the perturbation decreaseswith the increases in the magnetic field of the main KAW. Also the critical magneticfield value of the perturbation increases with its transverse scale size correspondingto a particular initial magnetic field value of the main KAW. Similar behavior isobserved for the solar wind and the low β solar corona (not shown here).

Figures 4(a)–(d) illustrates the intensity pattern of the perturbation at differentdistances along the direction of propagation, in the solar corona for the high β case.It is seen in these figures that the spreading of the filaments are taking place alongthe direction of propagation. The more intense filaments (primary filaments) are inthe regions where the main KAW also has filaments. The number of the primaryfilaments is almost the same as that of the corresponding main KAW filaments.This clearly shows that the perturbation is taking energy from the main KAW bythe nonlinear interactions. Filaments with low intensity (secondary filaments) arealso observed along with the primary filaments in the figures. Secondary filamentsare formed even when the main KAW is propagating in self trapped mode and the


Figure 4. The variation of magnetic field intensity of the perturbation (normalized to backgroundmagnetic field intensity), with x and normalized distance of propagation, ξ ≡ z/Rd0 (≡ z, in thefigures) for high β (= 0.02) coronal holes for different initial (viz. at z = 0) of B00 and B100: (a)B00 = 4.1×10−4 G and B100 = 13.8×10−5 G; (b) B00 = 7.29×10−4 G and B100 = 28.1×10−5 G;(c) B00 = 4.1 × 10−4 G and B100 = 20.37 × 10−5 G; and (d) B00 = 7.29 × 10−4 G and B100 =28.1 × 10−5 G.

initial magnetic field strength of the perturbation is above its critical value as shownin Figure 4(d).

Figures 4(a) and (b) depict the intensity of the perturbation at different distancesalong the direction of propagation for different strength of the initial magnetic field(higher than the critical field strength) of the main KAW in the case of the high β

solar corona. Figures show that when the initial magnetic field strength of the mainKAW is increased above its critical field value, the intensity of the filaments of theperturbation increases. This implies that the energy gain by perturbation via nonlin-ear interaction increases when the main KAW has formed more intense filaments.Also the number of filaments increases. Similar behavior is observed in the case ofthe low β solar corona (not shown here) and the high β solar wind as displayed inFigures 5(a) and (b). Figures 4(a) and (c) display the intensity distribution of theperturbation at different distances along the direction of propagation for differentvalues of the initial magnetic field strength of the perturbation in the case of highβ solar corona. As observed in these figures, with increase in the initial magneticfield of the perturbation, the intensity of the secondary filaments increases. Similar


Figure 5. The variation of the magnetic field intensity of the perturbation (normalized to the back-ground magnetic field intensity), with x and normalized distance of propagation, ξ ≡ z/Rd0 (≡ z, inthe figures) in the case of high β solar wind for different initial values (viz. at z = 0) of B00 and B100: (a)B00 = 4.07×10−8 G and B100 = 1.36×10−8 G; (b) B00 = 5.08×10−8 G and B100 = 8.97×10−9 G;and (c) B00 = 4.1 × 10−4 G and B100 = 20.37 × 10−5 G.

behavior is observed in the case of the low β solar corona (not shown here) andalso for the solar wind as shown in Figures 5(b) and (c).

The above mentioned process of filamentation can have the following effectsin the solar coronal heating and the solar winds. These filamentary structures can


act as a source for particle acceleration by wave-particle interactions because theKAWs are mixed modes and Landau damping is also possible. The secondaryfilaments of the perturbation will provide additional kick to the plasma particles,modifying energy gain sequence. Hence the coronal heating is modified. Alsothe presence of the secondary filaments of the perturbation changes the spectrumof the Alfvenic turbulence in the solar wind. Due to this the spectral index isalso expected to change. These filaments may act as the source for parametricinstabilities of the Alfven waves and may act as a source of decay waves as wellas a source of further collapse of the KAW. The present mechanism may be usefulin the coronal heating, and may even have some interesting relevance about theinitiation of turbulence in the solar wind, specifically, at the small scales mayprovide the nonlinear means of dissipation and heating of the ambient solar windplasma. The heating rate in the coronal holes can be estimated by using the velocityspace diffusion coefficient and the Fokker Planck equation or from more rigorousmodels.

4. Summary

In this paper we have studied the nonlinear evolution of mixed mode Alfven wave(or known as kinetic Alfven wave) and filament formation. Filament formation onaccount of the main Alfven wave and the perturbation present on that have beenstudied for typical parameters in the low β regime of the solar corona and for thehigh β regime of the solar corona and the solar wind. The low β and high β casescould correspond to the coronal holes and the helmet streamer regions. Conditionsfor the filamentation are very well satisfied in these regions as mentioned below inTable I.

Localized region of the high magnetic field intensity separated by rarified re-gion are observed in steady state condition. This work finds application in particleacceleration and plasma heating in the solar wind and the solar corona.

TABLE I

The threshold magnetic field value of the main Alfven wave and the perturbation for the solar windand the solar corona parameters.

Main Alfven wave PerturbationFrequency

(rad/s) r0 (km) B00cr (G) r0 (km) r10 (km) B00 (G) B100cr (G)

Solar corona

High β (β = .02) 2.13 × 102 6.54 2.43 × 10−01 6.54 2.87 7.29×10−4 1.47×10−5

Low β (β = 2 × 10−04) 13.8 × 103 15.8 1.09 × 10−04 15.8 6.9 1.09 × 10−4 1.12×10−6

Solar wind

High β (β = .121) 0.01 48.8× 103 2.54 × 10−08 0.01 23.6× 103 6.1 × 10−8 1.28×10−9


Acknowledgements

This work is partially supported by DST (India). M. Malik is very grateful to CSIRfor providing financial assistance for the present work. The authors are also thankfulto Dr H. D. Pandey for the discussions in this work.

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Nonlinear Evolution of Kinetic Alfvén Waves and Filament Formation

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