Physics Letters A 374 (2010) 2461–2463

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Physics Letters A

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Generation of zonal magnetic fields by drift waves in a current carryingnonuniform magnetoplasma

Nitin Shukla a,∗,1, P.K. Shukla b,2

a Department of Physics, Umeå University, SE-90187 Umeå, Swedenb Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 March 2010Accepted 2 April 2010Available online 10 April 2010Communicated by V.M. Agranovich

It is shown that zonal magnetic fields (ZMFs) can be nonlinearly excited by incoherent drift waves (DWs)in a current carrying nonuniform magnetoplasma. The dynamics of incoherent DWs in the presenceof ZMFs is governed by a wave-kinetic equation. The governing equation for ZMFs in the presence ofnonlinear advection force of the DWs is obtained from the parallel component of the electron momentumequation and the Faraday law. Standard techniques are used to derive a nonlinear dispersion relation,which depicts instability via which ZMFs are excited in plasmas. ZMFs may inhibit the turbulent cross-field particle and energy transport in a nonuniform magnetoplasma.

© 2010 Elsevier B.V. All rights reserved.

Multiscale fluctuations are ubiquitous in magnetized space [1,2]and laboratory [3–6] plasmas. In magnetized plasmas, both elec-trostatic and electromagnetic fluctuations of different scale-sizescan coexist. Different scale-size nonthermal fluctuations play avery important role in regulating cross-field turbulent trans-port in nonuniform space and fusion plasmas. Recently, therehave been growing interests [7–10] in identifying mechanismsthat are responsible for generating zonal magnetic fields (ZMFs)[3,11,12], current filaments [5,6] and associated electromagneticstructures [10] in magnetized plasmas.

In this Letter, we consider nonlinear interactions between in-coherent DWs and ZMFs/magnetostatic (MS) modes in a currentcarrying nonuniform magnetoplasma. The DWs [13–15] are thelow-frequency (in comparison with the ion gyrofrequency) pseudothree-dimensional electrostatic waves (with the external magneticfield-aligned phase speed much smaller than the electron thermalspeed) supported by the Boltzmann distributed adiabatic electronsand two-dimensional magnetized ions, while ZMFs/MS modes areradially inhomogeneous (qx �= 0, where qx is the component of thewave vector along the x-axis in a Cartesian co-ordinate system) or-dinary mode electromagnetic fluctuations [16–21] with zero den-sity fluctuations. Our coupled DW–ZMF turbulence model, as pro-posed here, can be regarded as a characteristics of a “predator–prey” system in which the population of incoherent DWs (prey),

* Corresponding author.E-mail address: ps@tp4.rub.de (P.K. Shukla).

1 Also at the Grupo de Lasers e Plasmas, Instituto de Plasmas e Fusão Nuclear, In-stituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisbon, Portugal.

2 Also at Scottish Universities of Physics Alliance (SUPA), Department of Physics,University of Strathclyde, Glasgow, Scotland G4 ONG, United Kingdom.

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.04.016

growing via a linear instability, would generate ZMFs (predator)through the modulational instability [22] in a current carryingnonuniform magnetoplasma. Subsequently, the growth of ZMFswould reduce the strength of the prey; a scenario similar tothe DW–electrostatic zonal flows (ZFs) turbulence system [22–27]in a nonuniform magnetoplasma, where the DW eddies becomeof smaller amplitudes and longer sizes due to random shearingcaused by large-scale ZMFs. Accordingly, the turbulent cross-fieldparticle transport is significantly reduced in a current carryingnonuniform magnetoplasma.

Let us consider a nonuniform plasma in the presence of thepre-existing incoherent electrostatic DWs in an inhomogeneousmagnetic field B(x)z, where B(x) is the strength of the magneticfield and z is the unit vector along the z-axis. The equilibriumdensity gradient ∂n0/∂x is along the x-axis, where n0 is the unper-turbed plasma number density. The frequency of obliquely (to z)propagating DWs is

ωk = ω∗(1 + k2⊥ρ2

s ), (1)

which is deduced by setting zero the dielectric constant

ε(ω,k) = 1 + 1

k2λ2De

+ ω2pik

2⊥ω2

cik2

+ ω2piκiky

ωωcik2, (2)

and by assuming that k2λ2De � 1. Here ω∗ = ky V∗ , V∗ = Cs|κi |ρs ,

Cs = (kB Te/mi)1/2 is the ion sound speed, ρs = Cs/ωci is the ion

sound gyroradius, ωci = eB/mic is the ion gyrofrequency, κi =(∂ ln(n0/B0)/∂x) < 0, λDe = (kB Te/4πn0e2)1/2 is the electron De-bye radius, ωpi = (4πn0e2/mi)

1/2 is the ion plasma frequency, kB

is the Boltzmann constant, Te is the electron temperature, e is

2462 N. Shukla, P.K. Shukla / Physics Letters A 374 (2010) 2461–2463

the magnitude of the electron charge, mi is the ion mass, and cis the speed of light in vacuum. The wave vector k = k⊥ + zkz ,and k2⊥ = k2

x + k2y , where the subscripts x, y, and z stand for the

radial, azimuthal, and axial components of the wave vector, re-spectively. Furthermore, in deducing Eq. (2), we have assumed thatkz V T i � ω � ωci,kz V T e,k2

zωce/ky |κi |, where V T i and V T e are theion and electron thermal speeds, respectively, ωce = eB/mec is theelectron gyrofrequency, and me is the electron mass. The motionof ions along z has been ignored, so that the ion sound wave [28]is decoupled from our system.

The energy density of the DWs is [14,15]

Ek =[

∂

∂ω(ωε)

]|Ek|2, (3)

where Ek = −ikφk is the DW electric field and φk is the DW po-tential. By using (2) we have from (3)

Ek = (1 + k2⊥ρ2

s

)∣∣∣∣ eφk

kB Te

∣∣∣∣2

Eth, (4)

where Eth = 4πn0kB Te is the kinetic energy density of the plasmaparticles.

The DW action is defined as

Nk = Ek

ωk= (1 + k2⊥ρ2

s )2

ω∗

∣∣∣∣ eφk

kB Te

∣∣∣∣2

Eth. (5)

The nonlinear interaction between the random phase DWs andZFs is governed by a wave-kinetic equation [13,24,25]

∂Nk

∂t+ V gx

∂Nk

∂x− ∂ωn

k

∂x

∂Nk

∂kx= S, (6)

where S = γk Nk − �ωk N2k represents the source term due to the

wave growth and damping due to linear and nonlinear mecha-nisms, where γk is the linear growth rate, and �ωk N2

k is thedamping caused by nonlinear resonance broadening effects [24].We note that ωn

k arises due to the nonlinear electron Lorentz forceu0B⊥/B , where B = ∇ψ × z ≡ −y(∂ψ/∂x) is the sheared magneticfield of ZMFs and y is the unit vector along the y-axis. Assumingthat small scale DW turbulence is close to a stationary state, onecan set S = 0. The radial group velocity of the DWs is

V gx = −∂ωk

∂kx≡ − 2kxρ

2s ωk

(1 + k2⊥ρ2s )

. (7)

Furthermore, the nonlinear frequency ωnk involving the ZMF paral-

lel (to z) vector potential ψz is

ωnk = kyuzm ≡ −kyu0

B

∂ψz

∂x, (8)

where the ZMF speed in the azimuthal direction is uzm =−(u0/B)∂ψz/∂x, and u0 is the equilibrium magnetic field-alignedelectron speed in a current carrying magnetoplasma.

The evolution of the ZMF vector potential in the presence ofnonlinear advection force of the DWs is governed by(

1 − λ2e

∂2

∂x2

)∂ψ

∂t− c

ωce

∑k

⟨(z × ∇φ∗

k · ∇)vek

⟩ + c.c. = 0, (9)

where λe = c/ωpe is the electron skin depth, ωpe = (4πn0e2/me)1/2

is the electron plasma frequency, vek is the parallel componentof the electron fluid velocity in the DW potential, and the angu-lar bracket denotes an ensemble average over the period 2π/ωk .The asterisk and c.c. denote the complex conjugate. We note thatEq. (9) has been deduced from the parallel component of theelectron momentum equation by using the ZMF parallel electron

fluid velocity (c/4πn0e)∂2ψ/∂x2 and the ZMF parallel electric field−c−1∂ψ/∂t . The nonlinear term in the left-hand side of (9) rep-resents nonlinear advection force and arises from the couplingbetween the parallel electron fluid velocity and the (c/B)z × ∇φkelectron flow in the DW electric fields. For simplicity, we haveneglected the effects of electron–ion/neutral collisions on the dy-namics of ZMFs. We note that the ZMFs have zero electron numberdensity fluctuations, and that ions do not participate in the dynam-ics of ZMFs.

We now study instability of incoherent DWs against the ampli-tude modulation caused by ZMFs. For this purpose, we express [29]

Nk = Nk0 + Nk1 exp(iϕ) and ψz = ψ exp(iϕ), (10)

where Nk1(� Nk0) is a small perturbation of the DW action aroundthe equilibrium value Nk0, and ϕ = qx − Ωt is phasor. The fre-quency and radial wave number of ZFs are denoted by Ω and q,respectively.

Inserting (10) into (6) and (9) and Fourier analyzing the resul-tant equations, we have

Nk1 = − iq2kyu0

B(Ω − qV gx)

∂Nk0

∂kxψ (11)

and

Ωψ = − 2icTeρ2s

eωce(1 + q2λ2e )

∫ωk

kzk2⊥kykx|Φk|2 d2k, (12)

where Φk = eφk/kB Te . In deriving (12) we have used

vek = ωk

kzk2⊥ρ2

s Φk. (13)

By using (5) we can express |φk|2 in terms of Nk1, and insertingthat expression into (12) and using (11) we have the dispersionrelation

Ω = βq2λ2eρ

2s u0

(1 + q2λ2e )Eth

∫ kxk2yω

2k

(Ω − qV gx)

k2⊥(1 + k2⊥ρ2

s )

∂Nk0

∂kxd2k, (14)

where β = 8πkBn0Te/B2.We analyze (13) in two limiting cases. First, for the resonant-

type instability, we replace the resonance function R = (Ω −qV gx)

−1 by −iπδ(Ω − qV gx), where δ is the Dirac-delta function.Letting Ω = iγq in the resultant equation, we obtain the growthrate

γq = − βπq2λ2eρ

2s u0

(1 + q2λ2e )Eth

∫ kxk2yk2⊥ω2

k

kz(1 + k2⊥ρ2s )

∂Nk0

∂kxδ(Ω − qV gx)d2k,

(15)

which ensures instability if ∂Nk0/∂kx < 0.Second, we consider a nonresonant hydrodynamic type insta-

bility. We assume that Nk0 = N0δ(k − k0), with k0 = (kx0,ky0).Carrying out integration by parts in (14), we have

1 = βq2λ2eρ

2s u0

(1 + q2λ2e )Eth

∫ k2yk2⊥ωk V ′

g Nk0

kz(1 + k2⊥ρ2s )(Ω − qV gr)2

d2k, (16)

where the DW group dispersion is

V ′g = ∂V gx

∂kx= −2ωkρ

2s (1 + k2

yρ2s − 3k2

xρ2s )

(1 + k2⊥ρ2s )2

. (17)

Eq. (16) yields[Ω − qV gx(k = k0)

]2

= − βq2λ2ek2⊥0k2

y0ρ2s u0 I0

k (1 + q2λ2)(1 + k2 ρ2)

(1 + k2

y0ρ2s − 3k2

x0ρ2s

), (18)

z0 e ⊥0 s

N. Shukla, P.K. Shukla / Physics Letters A 374 (2010) 2461–2463 2463

where I0 = |eφk0/kB Te|2. Eq. (18) predicts an oscillatory instabilitywith a real frequency qV gx(k = k0), and the growth rate

γr =√

βky0 V∗(u0 I0/kz0)k⊥0ky0qλeρs

(1 + q2λ2e )

1/2(1 + k2⊥0ρ2s )

(1 + k2

y0ρ2s − 3k2

x0ρ2s

)1/2,

(19)

provided that 1 + k2y0ρ

2s > 3k2

x0ρ2s .

To summarize, we have considered nonlinear interactions be-tween finite amplitude DWs and ZMFs in a nonuniform currentcarrying magnetoplasma. In our model, the DWs are treated asa beam of quasi-particles, and their dynamics in the presenceof ZMFs/MS modes is governed by a wave-kinetic equation. Fur-thermore, the equation for ZMFs in the presence of a nonlinearelectron advection force of the DWs is deduced from the parallelcomponent of the electron momentum equation. The nonlinearlycoupled DW–ZMF system is found to be modulationally unsta-ble. Consequently, ZMFs (magnetic filaments) would grow at theexpense of the free energy of DW quasi-particles. Since electronflows in ZMFs are along the azimuthal direction, ZMFs will tearapart DW eddies and would keep their amplitudes low. Accord-ingly, the cross-field turbulent electron energy transport will bedrastically reduced due to longer sizes and smaller amplitudesDWs in a turbulent current carrying magnetically confined fusionplasma [3].

Acknowledgement

This work was partially supported by the Deutsche Forschungs-gemeinschaft through project SH 21/3-1 of the Research Unit 1048.

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