Abstract In the present paper we have studied the nonlin-ear dynamical equation of Landau damped kinetic Alfvénwave (KAW) to investigate the nonlinear evolution of KAWand the resulting turbulent spectra in solar wind plasmas. Wehave introduced a parameter g which governs the couplingbetween the amplitude of the pump KAW and the densityperturbation. The numerical solution has been carried out tosee the dependence on the parameter g in the nonlinear partof our equation. Our results reveal the formation of dampedlocalized structures of KAW as well as steepening of the tur-bulent spectra by increasing g when damping is taken intoaccount. The power spectra of magnetic field fluctuationsindicate the redistribution of energy among the higher wavenumbers. Each power spectrum with and without dampingsplits up into two different scaling ranges, Kolmogorov scal-ing followed by a steeper scaling. The steepening in thepower spectra with Landau damping is more than withoutLandau damping case (for the same value of g). This type ofsteeper spectra has also been observed in the solar wind andis attributed to the Landau damping effects.
Keywords Alfvén waves · Solar wind · Turbulence
1 Introduction
Alfvén waves (AWs) are of fundamental importancethroughout the solar system. In 1945, Hannes Alfvén specu-lated their presence in the astrophysical plasmas. They weredetected in 1971 by the Mariner 5 spacecraft in solar wind
N. Gaur (�) · N.K. Dwivedi · R.P. SharmaCentre for Energy Studies, Indian Institute of Technology, Delhi110016, Indiae-mail: [email protected]
(Belcher and Davis 1971). AWs are also usually observedin space plasmas particularly in magnetosheath (Sahraouiet al. 2003) and the auroral regions (Norqvist et al. 1996;Stasiewicz et al. 2000). AWs are low frequency electro-magnetic waves in a conducting medium with some back-ground magnetic field. These waves attain dispersive naturewhen the wavelength perpendicular to the magnetic field isof the order of ion gyroradius (Chen and Hasegawa 1974)where these are known as kinetic Alfvén waves (KAWs). Inthe low-beta plasmas the wave is still dispersive but wavedispersion mainly follows from the electron inertia effect(Stasiewicz et al. 2000). Here KAW is specifically calledas inertial Alfvèn wave (IAW). Observations from FASTspacecraft (Chaston et al. 1999), sounding rocket flights(Boehm et al. 1990), and Freja spacecraft (Louarn et al.1994; Wahlund et al. 1994) have identified the presenceof the dispersive AW in the auroral zone. They have beendemonstrated to exist in the high-altitude cusp with frequen-cies less than the ion gyro frequency by Cluster spacecraftobservations (Sundkvist et al., 2005). Polar spacecraft ob-servations have also detected their presence while crossingthe plasma sheet boundary layer at altitude 4–6 RE (Wygantet al. 2002).
Hasegawa first implicated the presence of parallel electricfield in KAW which could lead to efficient acceleration ofparticles (Hasegawa and Chen 1976; Thompson and Lysak1996; Chaston et al. 2004). Recent theoretical and observa-tional results claim the importance of KAWs in particle heat-ing and in the turbulent cascade of energy in the solar wind(Howes et al. 2008; Sahraoui et al. 2009). Nonlinear KAWsare responsible for many observed phenomena in the spaceplasmas. These waves play a remarkable role in the acceler-ation of solar wind (Cranmer et al. 1999), in coronal heat-ing and in the dissipation of solar wind turbulence (Sahraouiet al. 2010). Dispersive AWs are responsible for energization
of minor heavy ions observed in the solar corona, magneticreconnection (Chaston et al. 2009), and formation of densitycavities in auroral plasma based on ponderomotive force as-sociated with these waves (Shukla and Stenflo 1999).
Space plasmas are dynamically evolving turbulent sys-tems where wave particle interactions play important role.It is pivotal to understand the dissipation of magnetic fluc-tuations in space plasmas to understand the nature of turbu-lence as well as the processes by which plasma species areheated. A power density spectrum shows power-law in fre-quency and wavenumber which indicates energy cascade. Inthe inertial range the solar wind spectrum generally followsthe Kolmogorov scaling law in which fluctuating energy isinjected at larger scales and cascades through successivelyshorter-scale lengths to eventual dissipation where the fluc-tuations are converted to thermal energy. The physical pro-cesses involved in dissipation range are crucial for better un-derstanding of plasma turbulence because it is in this rangethat turbulent energy is converted to thermal energy. Thisdissipation can lead to significant heating of solar wind. Ithas been observed from in situ satellite measurements thatat 1 A.U. energy spectrum typically shows, for low frequen-cies, a power law spectrum with slope of −5/3, implicativeof Kolmogorov-like inertial range (Goldstein et al. 1994);break in the spectra is observed at around 0.4 Hz, with asteeper power law at higher frequencies with a slope thatvaries between −2 and −4 (Leamon et al. 1998). The dy-namics responsible for the break in the spectrum and steeperslope is yet to be investigated properly. Recently, Howeset al. (2011) are using gyrokinetic simulations of KAW tur-bulence and Rudakov et al. (2011) are using ideas from weakturbulence theory to understand the break in the slope. Manyfactors like proton cyclotron damping (Leamon et al. 1998),Landau damping of KAW (Leamon et al. 2000) and disper-sive nature of Whistler waves (Stawicki et al. 2001) can sig-nificantly contribute to the steepening of the slope. By exam-ining the spectrum of turbulence Sahraoui et al. (2010) havespeculated that the turbulence is carried by highly obliqueKAWs.
In the present paper we have studied the nonlinear dy-namical equation of Landau damped kinetic Alfvén wave(KAW) to investigate the dependence of the coupling param-eter g on their nonlinear evolution and the resulting turbu-lent spectra in solar wind plasmas. The paper is organized asfollows: In Sect. 2 we present the model equations to studynonlinear dynamics of KAW. In Sect. 3 we derive the nonlin-ear modulational dispersion relations which have been usedto study the growth rate. Section 4 comprises of numericalsimulation of modified nonlinear Schrödinger equation ap-plicable to solar wind and the discussion of results. Section 5comprises of summary and conclusions.
2 Kinetic Alfvén wave dynamics
We have obtained the dynamical equation for nonlin-ear KAW propagating in the x–z plane, having ambient–magnetic field B0 along the z axis. Starting equations are asfollows:
(a) Equation of motion:
∂−→vj
∂t+ 2γ ∗
L−→vj = qj
mj
−→E + qj
cmj
(−→vj ×−→B0)− γj kTJ
mj
−→∇ nj
n0.
(b) The continuity equation:
∂nj
∂t+ �∇.(nj �υj ) = 0.
(c) Faraday’s law:
�∇ × −→E = −1
c
∂−→B
∂t,
where �υj is the velocity of species j = i, e (i = ions, e =electrons), Tj is the temperature, qj is the charge andmj is the mass of the ions and electrons respectively,n0 is the unperturbed plasma number density, and c isthe speed of light in vacuum. Landau damping in colli-sionless magnetized plasma has been taken into accountby introducing the damping operator γ ∗
L(x) in the equa-tion of motion. γ ∗
L(x) is nonlocal in real space but localin wave number space. As we solve the linear part ofMNLS equation in the Fourier transform space so thereis no need to specify the damping operator in x space.We need to know only the functional form of this oper-ator in wave number space.
Using the drift approximation, Maxwell’s equations, andfollowing Sharma et al. (2011), the dynamical equation gov-erning the propagation of the KAW with Landau dampingcan be obtained as
∂2By
∂t2+ 2γ ∗
L
∂By
∂t=
(∂2
∂t2+ 2γ ∗
L
∂
∂t
)(λ2
e
∂2By
∂x2
)
− ρ2s V 2
A
∂4By
∂x2∂z2
+ V 2A
(1 − δns
n0
)∂2By
∂z2, (1)
where VA(=√
B20/4πn0mi) is the Alfvén speed, λe =
(√
c2me/4πn0e2) is the collisionless electron skin depth,VT e(= √
Te/mi) is the electron thermal speed, δns = ne −n0 is the number density change, ne is the modified elec-tron density, ρs = cs/ωci is the ion acoustic gyroradius, cs =({γeκTe +γiκTi}/mi)
1/2 is the acoustic speed, the electronsand ions are assumed to be isothermal, i.e. γe = γi = 1,
Astrophys Space Sci
and ωci(= eB0/cmi) is the ion gyrofrequency. The Landaudamping of KAW (Hasegawa and Chen 1976) is given as:
γL(kx)
ω≈ −
√π
4β
me
mi
Te
Ti
(k2xρ
2i
),
where ρi(= VT i/ωci) is the ion gyroradius and VTi(=√
Ti/mi) is the ion thermal speed.Consider a plane wave solution of Eq. (1)
By = By(x, z, t)ei(k0xx+k0zz−ωt). (2)
Using Eq. (2) in (1), one gets the dynamical equation ofKAW as
− 2iω
V 2Ak2
0z
(1 + i
γ ∗L
ω
)∂By
∂t− 2i
k0z
∂By
∂z− 2ik0xρ
2s
∂By
∂x
+ 2ik0x
λ2eω
2
V 2Ak2
0z
∂By
∂x+ λ2
eω2
V 2Ak2
0z
∂2By
∂x2− ρ2
s
∂2By
∂x2
+ 2iλ2
eω2
V 2Ak2
0z
(γ ∗L
ω
)∂2By
∂x2
− 4k0x
λ2eω
2
V 2Ak2
0z
(γ ∗L
ω
)∂By
∂x−
(δns
n0
)By = 0. (3)
Here k0x(k0z) is the component of the wave vector per-pendicular (parallel) to zB0 and ω is the frequency of theKAW. Here the density can be modified by the ponderomo-tive force of KAW in intermediate β plasma (Shukla andStenflo 2000) and is given as
δns
n0= exp
[ξ |By |2
] − 1, (4)
where
ξ ={
1 − (ω2/ω2
ci
)(1 + mek
20x
mik20z
)}V 2
Ak20z
16πn0Teω2.
Using Eq. (4), Eq. (3) can be written in dimensionless formas
−i
(1 + i
γ ∗L
ω
)∂By
∂t+ Γ1
∂By
∂x+ Γ2
∂2By
∂x2
− iΓ3∂By
∂z− 1
2g
(e2g|By |2 − 1
)By = 0. (5)
Here, we have introduced g parameter for the sake of gener-ality. For g = 1 corresponds to the present case and g = 0leads to the well known quadratic nonlinearity case. Thecoupling to the density perturbation is in some way con-trolled by g. Moreover, here the dimensionless parametersare
Γ1 = 2i(k0xρs)Γ2,
Γ2 ={(
1 + 2iγ ∗L
ω
)λ2
e
ρ2s
(1 + k2
0xρ2s
1 + k20xλ
2e
)− 1
}and Γ3 = 1.
The normalizing values are zn = 2/k0z, tn = (2ω/V 2Ak2
0z),xn = ρs and Bn = [ξ ]−1/2.
3 Stability analysis
In this section we have studied the growth rate of the mod-ulational instability of MNLS equation (Eq. (5)). Now con-sider the initial value problem with large amplitude pumpfield and small perturbing magnetic field. Following thestandard technique used by Shen and Nicolson (1987), weexpress By as
By(x, z, t) = By0 + B+ exp[i(αxx + αzz − Ωt
]+ B− exp
[−i(αxx + αzz − Ω∗t
)]. (6)
Here By0 denotes the amplitude of the homogenous pumpKAW, |B+| and |B−| are perturbation amplitudes muchsmaller than |By0| and Ω is the frequency of perturba-tion. Moreover αx and αz are perpendicular and parallelwavenumbers of perturbation. For MNLS equation, on sub-stituting Eq. (6) in Eq. (5), one gets the following dispersionrelation without Landau damping
Ω2 − (Γ3αx + Γ3αz)Ω + Γ 21 α2
x + 2iΓ1Γ2α3x − Γ2Γ3α
3x
+ iΓ1Γ3α2x − Γ 2
2 α4x + 2Γ2α
2xB
2y0
− iΓ1Γ3αxαz + Γ2Γ3αzα2x + Γ 2
3 αxαz
− Γ3αzB2y0 + Γ3αxB
2y0 = 0. (7)
We will study two cases for MNLS equation without damp-ing (γL = 0)
(1) when αz = 0(2) when αz = 0.02.
It is appropriate to present the case of NLS also so that acomparison of the present case of MNLS with NLS can bedone. The dispersion relation of NLS equation as derived byShen and Nicolson (1987) is as follows
Ω2 − α4x + 2α2
xB2y0 = 0.
If Ω = Ωr + iΩi and Ωi = γ , then the growth rate as afunction of wave number αx for the purely growing modefor NLS case is
γ = αx
(2B2
y0 − α2x
)1/2. (8)
Astrophys Space Sci
Fig. 1 Growth rate γ (αx) as afunction of perturbationwavenumber αx for (a) NLSequation and MNLS equationwith (b) αz = 0 and (c)αz = 0.02
The most unstable mode is 1.0 for NLS equation whenBy0 = 1. The growth rate (γ ) as a function of perturba-tion wave number (αx) from Eq. (8) has been illustrated inFig. 1(a). The growth rate increases with αx , attains a max-imum value (γ = 1) and then decreases after achieving themaxima. In the MNLS case when αz = 0 the most unstablemode lies at αx = 0.26 with growth rate (γ ) equal to 0.35as shown in Fig. 1(b). When αz = 0.02 Fig. 1(c) shows themost unstable mode lying at αx = 0.3 with the value of thegrowth rate equal to 0.32. NLS equation has a larger maxi-mum growth rate as compared to other two MNLS forms.
The parameters for solar wind at 1 A.U. are as follows:β ≈ 0.7,B0 ≈ 6 × 10−5 G, n0 ≈ 3 cm−3, Te = 1.4 × 105 K,Ti = 5.8 × 105 K. Making use of these values we ob-tain VA ≈ 6.5 × 107 cm/s, λe ≈ 3.07 × 106 cm, ωci ≈0.57 rad/sec, ω = 0.06 rad/sec, VT i ≈ 6.93 × 106 cm/sec,ρi ≈ 1.2 × 107 cm and ρs ≈ 1.34 × 107 cm. For k0xρs ≈0.02 we have k0x ≈ 1.49 × 10−9 cm−1 and k0z ≈ 9.23 ×10−10 cm−1. The normalizing values are xn ≈ 1.34 ×107 cm, zn ≈ 2.17 × 109 cm, tn ≈ 33.34 sec, nn ≈ 7 cm−3
and Bn ≈ 5.38 × 10−5 G.
4 Numerical simulation and results
We have performed the numerical simulation of Eq. (5) byusing a 2D pseudo-spectral method in a (2π/αx) × (2π/αz)
periodic spatial domain with αx,αz = 0.2 (where αx andαz are the wavenumbers of the perturbation). A finite dif-ference method with a predictor corrector scheme has beenused for the evolution in time with a time step of dt = 10−5.Simulations have been carried out in (x, z) domain with
(128 × 128) grid points. We have used the following initialcondition of simulation
By(x, z,0) = |By0|(1 + ε cos(αxx)
)(1 + ε cos(αzz)
). (9)
Here |By0| = 1 is the amplitude of the homogenouspump KAW. The parameter ε = 0.1 governs the magni-tude of the perturbation. Before solving MNLS equation(Eq. (5)), we developed the algorithm for well known cu-bic NLS equation. The linear evolution was exactly inte-grated which forms an important feature of the code to accu-rately reproduce the instability. The constancy of the num-ber N = ∑
k |Bk|2governed the accuracy in the case of NLSequation. A fixed step size in time (dt = 10−5) was used inorder to monitor the invariants of NLS equation and it wasobserved that the conserved quantity was preserved to theorder of 10−5 during computation. The results were com-pared with the well known results of the cubic NLS equa-tion and after testing this algorithm we modified it for solv-ing MNLS equation (Eq. (5)). We have studied the effectof parameter g on the nonlinear evolution of the modula-tional instability of Landau damped KAW and on its turbu-lent spectra applicable to solar wind.
We here present the simulation results of KAW (Eq. (5)),applicable to solar wind at 1 A.U. We have studied |Bk|2against kz at a fixed value of kx when g = 0.05. Figure 2(a)represents the spectrum at almost early time (t = 1). Herethe energy is confined to pump at kz = 0, and very smallenergy lies in the perturbation. As the time elapses the non-linear evolution takes place and several harmonics are gen-erated due to localization of the pump KAW, as indicated inFig. 2(b). With Landau damping, initially whole of energy is
Astrophys Space Sci
Fig. 2 Variation of |Bk |2against kz with g = 0.05without damping at (a) t = 1,(b) t = 8 and with damping at(c) t = 1 and (d) t = 8
with pump KAW (Fig. 2(c); t = 1) but with passage of timethe level of harmonics generated decreases in comparison tothe case when damping is zero (Fig. 2(d); t = 8).
Furthermore to have more insight about the energy ex-change between different Fourier modes, the value of |Bk|is obtained for kz = 0, kz = αz and kz = 2αz at fixed valueof kx at different times (Figs. 3(a)–(b)). In NLS case initiallyat t = 0 whole of the energy is in kz = 0 mode and as KAWpropagates, the energy is transferred to higher modes andthen it is regrouped to kz = 0 mode viz Fermi-Pasta-Ulam(FPU) recurrence takes place (Fig. 3(a)). In our MNLS casewhen g = 0.001, initially whole of the energy lies in kz = 0mode and with the advancement of time it is distributed tohigher modes as illustrated in Fig. 3(b). The value of |Bk| fort > 20 is very low and gets saturated with small amplitudefluctuations. Here the FPU recurrence is no longer presentand the higher order harmonics first grow and then decayshowing oscillatory evolution.
As compared to NLS case the dynamics of KAW local-ization is more complex in MNLS case. Therefore, to havea detailed understanding we construct the phase space di-agrams [|By(0,0, t)|, d|By(0,0, t)|/dt] for all three cases(first NLS equation and second MNLS equation with g =0.001 and g = 0.05), the results are illustrated in Figs. 4(a)–(c) respectively. For NLS case the phase space trajectoryshows the regular orbit as shown in Fig. 4(a) while it is ev-ident from Figs. 4(b) and 4(c) that in MNLS case the regu-larity is destroyed for both values of g.
The time evolution of the intensity of the transverse mag-netic field is illustrated in Figs. 5(a)–(b) when g = 0.001
without Landau damping (when γL = 0 in Eq. (5)) and Figs.5(c)–(d) with Landau damping by means of snapshots at twoinstants of time (t = 8 and 10). For given initial condition,the magnetic field (of KAW) gets localized and delocalized.Perturbation takes energy from main KAW by nonlinear in-teraction, grows, and finally leads to their own localizedstructures. Therefore, KAW breaks up into localized struc-tures where the intensity is very high as shown in Figs. 5(a)–5(d). At the early time (t = 8), both the low and high inten-sity localized structures are formed as shown in the Fig. 5(a).With the passage of time (Fig. 5(b); t = 10), the system hasmore number of structures of larger intensity scattered ran-domly at different x and z locations. It is evident from theFigs. 5(c)–(d) that when the Landau damping of KAW isincluded in the wave dynamics, the localized structures getdamped. With the advancement of time (t = 10) the systemhas more randomly distributed localized structures havingvery less intensity as illustrated in Fig. 5(d). Figures 6(a)–(d) depict the time evolution of the transverse magnetic fieldwith g = 0.05. Here at t = 8 (Fig. 6(a)) KAW breaks upinto magnetic localized structures of moderate intensity. Att = 10 Fig. 6(b) shows that the magnetic field intensity pro-files become complex and the system is highly chaotic. Withdamping as illustrated in Figs. 6(c)–6(d), the localized struc-tures get damped as shown at t = 8 and t = 10 respectively.Nonlinear interactions are responsible for the formation oflocalized structures thereby transferring the energy from theinjection scales to the smaller scales, until dissipative effectsdue to Landau damping become dominant and stop the en-ergy cascade.
Astrophys Space Sci
Fig. 3 Time evolution of |Bk |of three initial Fourier modes ofKAW for kz = 0 (blue curve),kz = αz (green curve), andkz = 2αz (red curve) for(a) NLS equation and(b) MNLS equation withg = 0.05
Fig. 4 Phase space plot for(a) NLS equation, (b) MNLSequation with g = 0.001,(c) MNLS equation withg = 0.05
Next, we study the power spectra by plotting |Bk|2against k. As shown in Fig. 7(a) at t = 8 and Fig. 7(b)at t = 10 (when g = 0.001), in the limit k < 1 the scal-ing law approaches k−5/3 scaling without damping in boththe cases. At around k ≈ 1 a spectral break appears fromwhere the spectra becomes steeper. When k > 1, the powerspectrum follows k−2 at t = 8 and k−2.3 scaling at t = 10
without damping. Figures 7(c) and Fig. 7(d) illustrate thepower spectrum with damping at t = 8 and t = 10 respec-tively. It follows the usual k−5/3 scaling (a typical inertialrange scaling) in the limit k < 1 as shown in Fig. 7(c) att = 8 and Fig. 7(d) at t = 10. At t = 8 in the limit k > 1,the scaling of power spectrum follows k−3.2 (Fig. 7(c)) andwith the advancement of time at t = 10, the scaling fol-
Astrophys Space Sci
Fig. 5 The magnetic fieldintensity profile of KAW withg = 0.001 without Landaudamping at (a) t = 8, (b) t = 10and with Landau damping at(c) t = 8 and (d) t = 10
Fig. 6 The magnetic fieldintensity profile of KAW withg = 0.05 without Landaudamping at (a) t = 8, (b) t = 10and with Landau damping at (c)t = 8 and (d) t = 10
lows k−3.6 (Fig. 7(d)). This type of steeper power spectrak−η(η = 3 ∼ 4) has also been observed in the solar windpower spectrum at the higher frequency side (Leamon et al.1998).
At g = 0.05 the spectral slope follows k−2.1 scaling att = 8 (Fig. 8(a)) and k−2.6 scaling at t = 10 (Fig. 8(b)) fork > 1 without damping. With damping the spectral slope in-creases to k−4.3 scaling at t = 8 (Fig. 8(c)) and k−4.6 scalingat t = 10 (Fig. 8(d)) for k > 1. It has been observed that thespectral slope increases on increasing the value of parame-ter g. This slope is within the observational range. It is evi-dent from the power spectrum of magnetic field fluctuations
that energy may be distributed among the large and inter-mediate wavenumbers at k > 1 due to the nonlinear inter-actions. The role of the electron Landau damping in steep-ening of the spectra has also been pointed out by Sahraouiet al. (2010).
5 Summary and conclusions
We have presented the numerical simulation of MNLS equa-tion comprising of the nonlinear dynamics of KAW. The ef-fect of coupling parameter g and Landau damping on the lo-
Astrophys Space Sci
Fig. 7 The power spectrum ofmagnetic field fluctuations withg = 0.001 in the absence ofLandau damping at (a) t = 8,(b) t = 10 and with damping at(c), t = 8 (d) t = 10
Fig. 8 The power spectrum ofmagnetic field fluctuations withg = 0.05 in the absence ofLandau damping at (a) t = 8,(b) t = 10 and with damping at(c) t = 8, (d) t = 10
calized structures and the turbulent spectra has been studied.Our results bring out an accountable role of Landau damp-ing on the localized structures and the turbulent spectra. Theintensity of localized structures has reduced with damping.At g = 0.001 when k > 1, the scaling of power spectrumfollows the k−3.6 at later time (t = 10) with damping. Onincreasing the value of g the slope increases to k−4.6. Theinferred power spectrum of magnetic field fluctuations indi-
cates that the nonlinear interactions may be distributing en-ergy among large and intermediate wavenumbers at k > 1.
It should be mentioned here that the repeated interactionof plasma particles with these localized structures (as stud-ied in the present paper) can lead to velocity space diffu-sion process. Using this velocity space diffusion coefficientin the Fokker-Planck equation one can calculate the new dis-tribution function of particles. This work will be done in fu-
Astrophys Space Sci
ture. However, recently Rudakov et al. (2011) and Crabtreeet al. (2012) studied the interaction of KAW with the plasmaparticles and demonstrated that the energy of the KAW canbe converted into the formation of plateau in the distribu-tion function during the time of flight of solar wind to theearth. This plateau will reduce the subsequent linear Landaudamping rate and allow for nonlinear scattering by particles.
Acknowledgements This work is partially supported by CSIR, DST(India) and ISRO (India) under RESPOND program.
References
Belcher, J.W., Davis, L. Jr.: J. Geophys. Res. 76(16), 3534 (1971)Boehm, M.H., Carlson, C.W., McFadden, J.P., Clemmons, J.H., Mozer,
