Generation of convective cells by kinetic Alfvén waves

Nonlinear interaction between kinetic Alfvén waves (KAWs) and the electrostatic and magnetostatic convective cells in plasmas is considered here. It is shown that the KAWs in the kinetic regime can excite only magnetostatic convective cells, but those in the inertial regime can excite only electrostatic convective cells. Moreover, there is a preferred spatial scale for the instability-generated electrostatic cells, but not for the magnetostatic cells. The significance of the present results for space plasmas is discussed.


Introduction
Convective cells are two-dimensional quasistationary spatially periodic plasma flow structures perpendicular to the external magnetic field [1,2]. They can be electrostatic or magnetostatic [3][4][5][6] and are associated with the zonal and other large-scale flows in systems with rotation and/or more complex magnetic fields [2,[7][8][9]. The electrostatic convective cells (ECCs) can cause macroscopic plasma convection and can play an important role in plasma transport and space weather [1,2,6,10]. The magnetostatic cells (MCs), on the other hand, are associated with quasistationary bending of the magnetic fields and can lead to the formation of magnetic islands and the enhancement of electron thermal transport [1,5,11]. The ECCs and MCs can lead to instabilities in the plasma by nonlinearly interacting with higher-frequency modes [12][13][14][15][16][17][18][19][20]. In the present study, we shall consider the nonlinear coupling of the ECC and MC modes with shear or kinetic Alfvén waves (KAWs).
KAWs are shear Alfvén waves having wavelengths (perpendicular to the ambient magnetic field, say B 0 = B 0ẑ ) comparable to the ion gyroradius or the electron inertial length [21][22][23][24][25][26]. The linear dispersion relation of the KAWs of interest here is ω = [1,26,27], where V A is the Alfvén speed, k z and k ⊥ are the parallel and perpendicular (to B 0 ) wave vectors, respectively, ρ = (ρ 2 i + ρ 2 s ) 1/2 is an effective gyroradius, ρ i is the ion gyroradius, ρ s is the ion-acoustic gyroradius and λ e is the electron inertial length.
Thus, we have ω = k z V A 1 + k 2 ⊥ ρ 2 in the limit k 2 ⊥ λ 2 e 1 and ω = k z V A / 1 + k 2 ⊥ λ 2 e in the limit k 2 ⊥ ρ 2 1 [28][29][30]. These two KAW regimes normally occur in m e /m i β 1 and β m e /m i plasmas, respectively. Here β is the ratio of thermal to magnetic pressure and m e /m i is the electron to ion mass ratio. KAWs in the two regimes have quite different perpendicular dispersion properties and both have been observed in laboratory experiments as well as space plasmas [26,31,32]. Sources of KAWs include mode conversion [21], instabilities due to parametric effects [33], particle beams and electric currents [34], etc. These waves are closely involved in particle acceleration and heating of space and tokamak plasmas [12,[35][36][37], and are believed to play important roles in the determination of solar and space weather [10]. Furthermore, the coupling of KAWs to the ECC and MC modes may be responsible for the nonlinear dynamics of the ionospheric Alfvén wave resonator and formation of the turbulent Alfvén boundary layer in the topside ionosphere and the auroral current system [38,39].
It has been shown [37] that KAWs with very long perpendicular wavelengths (k ⊥ → 0) cannot excite the ECC modes. When the perpendicular wavelength is finite, Sagdeev et al [40], 3 and others [41] showed that ECCs can be excited by KAWs through a modulational instability in m e /m i β 1 cold-ion plasmas. However, other studies [42][43][44] found that ECCs can be excited by KAWs only if the ion temperature T i satisfies T i > 4T e /3 > 0. Furthermore, Volokitin and Dubinin [41] found that in β m e /m i plasmas the corresponding inertial KAWs cannot excite ECCs, although Pokhotelov et al [38] and Onishchenko et al [39] found that such an excitation can take place. On the other hand, Yu et al [45] showed that modulation instability of the KAWs can excite MCs in m e /m i β 1 cold-ion plasmas, and Gruzinov et al [46] showed that MC-related (fast) dynamo action can occur in the presence of KAW turbulence.Thus, there is a controversy concerning the conditions of ECC excitation by KAWs, and the excitation of MCs by KAWs has not been investigated in detail. Accordingly, in this paper we shall reconsider the modulational excitation of the ECCs and MCs by KAWs, in particular the role played by the ion temperature. The results show that KAWs can excite ECCs in the inertial regime and can excite MCs in the kinetic regime.

The governing equations
To investigate the coupling of KAWs to ECCs and MCs in low-beta (β 1) plasmas, it is convenient to introduce the scalar potential ϕ and the parallel (to B 0 ) vector potential A zẑ : where E and B are the electric and magnetic field perturbations, respectively. For low-frequency motion (∂ t ω ci , where ω ci is the ion cyclotron frequency), one can obtain from the two-fluid equations after assuming quasineutrality [47][48][49][50] where where , v i⊥ is the perpendicular ion velocity, v e⊥ is the perpendicular electron velocity and v ez is the parallel electron velocity. The nonlinear terms in equation (2) are from the perpendicular ion convective flow and the magnetic field bending, and the nonlinear terms in equation (3) are from the parallel electron inertial force and Lorentz force.
We decompose ϕ and A z into their fast (KAW time scale) and slow (ECC or MC time scale) varying components: where ψ and A are for the KAWs and φ and a are for the ECCs and MCs, respectively. The particle velocities v i⊥ , v e⊥ and v ez , as well as the perturbed magnetic field B ⊥ , are also decomposed into their fast and slow varying components, The linear relations of the KAW quantities are and those of the convective cells are where µ 0 is the free-space permeability, n 0 is the background particle number, the perpendicular velocities v i⊥C and v e⊥C are of the ECCs, and the electron parallel velocity and perturbed magnetic field are of the MCs [1].
One can obtain from equations (2)-(7) the equations governing the nonlinear coupling of the ECCs and KAWs. For the ECCs, we have where the over bar denotes averaging over the fast, or Alfvén wave, time scale. The two nonlinear terms can be identified as the Reynolds stress tensor v i⊥K · ∇v i⊥K ×ẑ and the Maxwell stress tensor B ⊥K · ∇ B ⊥K ×ẑ, respectively. For the KAWs in the presence of ECCs, we have and which include the nonlinear effects arising from the perpendicular and the parallel (to B 0 ) particle motion. The coupling between the MCs and the KAWs is given by and Equation (11) shows that the nonlinear electron parallel convective inertia v e⊥K · ∇v ez K and the Lorentz force v e⊥K × B ⊥K ·ẑ drive the MCs. Equations (12) and (13) describe the nonlinear KAWs in the presence of the MCs. The nonlinear terms there are from the magnetic field bending and Lorentz force.

The generation of electrostatic convective cells (ECCs)
We now consider the modulation of a (pump) KAW by ECCs. The scalar potential of the ECCs can be written as where φ q , and q ⊥ are the amplitude, frequency and wave vector of the ECC potential. The scalar and vector potentials of the pump and sideband KAWs can be written as where the subscript 0 denotes the pump wave and ± denotes the upper (+) and lower (-) sidebands. In the interaction, the frequencies and wave vectors of the two sidebands approximately satisfy the matching conditions ω ± = ± ω 0 and k ± = q ⊥ ± k 0 . Substituting equations (14) and (15) into equation (8), we obtain where for simplicity and later use we have definedq 0 = (q ⊥ × k 0⊥ ) ·ẑ, R iq,0,± = 1 + k 2 q,0,±⊥ ρ 2 i and R 0,± = 1 + k 2 0,±⊥ ρ 2 . The expressions for ψ ± and A ± can be obtained from equations (9) and (10), where L 0,± = 1 + k 2 0,±⊥ λ 2 e , 6 and 1 = [26][27][28]. For weak modulational instability, one assumes that the pump KAW is little affected by the wave interaction, so that the relation between ψ k 0 and A k 0 can be obtained from equation (9) as Substituting expressions (17)- (19) into (16), we obtain The typical scale of the convective cells is usually much larger than that of the KAWs, or q ⊥ k ⊥ , and the perpendicular wavelength of the KAW is much larger than the ion gyroradius or the electron inertial length, i.e. ρ 2 k 2 ⊥ 1 or λ 2 e k 2 ⊥ 1. Accordingly, we obtain is the group velocity of the pump wave and δ = 1 and the solution of equation (22) is In the inertial regime β m e /m i , the electron inertia length is much larger than the ion gyroradius λ e ρ, and equation (23) becomes where δ I = 1 2 V A k 0z λ 2 e q 2 ⊥ . Equation (24) shows that the instability can occur when where θ is the angle between q ⊥ and k 0⊥ , and the relation ψ k 0 V A B k 0 /k 0⊥ has been used. The growth rate (24) is a function of the ECC wavenumber, and it is peaked at q 2 ⊥max = (4k 2 0⊥ sin 2 θ/λ 2 e k 2 0z )(B 2 k 0 /B 2 0 ). The maximum growth rate is For the kinetic regime m e /m i β 1, where ρ λ e , equation (23) reduces to where . Equation (28) shows that there is no ECC-modulated KAW instability in the kinetic regime.

The generation of magnetostatic convective cells
For MC modulation of KAWs, the vector potential of the MCs is given as a = a q e −i( t−q ⊥ ·r) + c.c. From equation (11), we obtain The expressions for ψ ± and A ± have the same forms as those in equation (17), except that here 1,2,2,4 are given by Inserting the expressions for ψ ± and A ± into equation (29) and using relations (19) and (21), we obtain 8 The corresponding solution is Thus, the solution for KAWs in the inertial regime is which shows that the MCs cannot be excited by modulation instability of KAWs in the inertial regime.
In the kinetic regime, equation (32) becomes so that modulation instability of the KAWs can occur if and the growth rate is

Discussion
We can see from equation (24) that the growth rate of ECC modulation of the KAWs in the inertial regime is a function of the cell scale q ⊥ and it peaks at q ⊥ = q ⊥max . There is therefore a characteristic or preferred spatial scale of the excited ECCs. Our result is qualitatively consistent with that of Pokhotelov et al [38]. In another work, Pokhotelov et al [42] considered the case T i = 0 and showed that ECCs cannot be excited by the KAWs in the kinetic regime. Here we found that this conclusion also holds when the ion temperature is finite. We also found that only KAWs in the kinetic regime can excite MCs. In contrast to the existence of a character scale for ECC excitation, the growth rate of the MCs increases with q ⊥ , so that within our assumptions there is no characteristic scale for MC excitation. Our results also agree qualitatively with those of Yu et al [45]. However, in the latter the nonlinear effects originating from the parallel electron motion, which as shown here can be comparable to that of magnetic field bending, have been ignored.
Our results show that energy can be transferred from the KAWs to the much larger scaled and much slower varying cell modes. The latter can play important roles in the study of phenomena observed in space plasmas. For example, plasma flow associated with large-scale ECCs can excite Kelvin-Stuart vortex streets in the ionosphere [51], and the electron flow of the MCs can be responsible for the field-aligned auroral currents [38,52]. The nonlinear coupling between the KAWs and the ECCs and MCs can also play a role in the ionospheric Alfvénic resonator [53,54]. Moreover, the unstable KAW-driven cells can develop into large-amplitude self-organized structures [1]. The excitation of large-scale convective flows studied here is also 9 closely related to the excitation of zonal and geodesic flows by KAWs in tokamak plasmas. However, for application to the latter, geometrical effects, as well as background density and temperature gradients, have to be taken into account [2,12,13,37,[55][56][57][58][59].
It should be mentioned that finite-amplitude KAWs can be involved also in other nonlinear interactions. In particular, they can participate in three-wave parametric decay among themselves ( [49] and references therein; [60]). In the long-wavelength regime the decay is mainly into two waves propagating in opposite directions. In the inertial regime, the nonlinear growth rate of the KAWs approximates and in the kinetic regime, it is given by Taking into account the electron Landau damping γ D = √ π/8v T e k z λ 2 e k 2 ⊥ , the existence condition for the decay is γ KAW > γ D , or (39) in the inertial regime and in the kinetic regime. Comparing equation (27) with (37) and equation (26) with (39), we obtain the following condition for the dominance of ECC generation over three-wave decay: where we have noted that the fastest growing ECC corresponds to θ = π/2, and the left inequality in the first relation comes from the condition ω 0 > γ max . We also found by comparing equation (36) with (38) and equation (35) with (40) that the three-KAW decay always dominates over the parametric generation of MCs. That is, we do not expect large-amplitude KAWs to generate effectively MCs if the three-KAW decay conditions can be met. There are many observations of kinetic and inertial AWs in space plasmas as well as the Earth's magnetosphere and ionosphere. For typical parameter values at the altitude ∼ 1000 km in the ionosphere [61,62], the perpendicular wavelength is ∼ 0.1-10 km, the frequency is ∼0.2-20 Hz, the Alfvén speed is ∼ 10 6 m s −1 , the electron inertial length is ∼80 m, the perturbed magnetic field is ∼50 nT and the background magnetic field is ∼0.1 G. The parallel wavenumber can be estimated to be k z ∼ 10 Hz (10 6 m s −1 ) −1 ∼ 10 −5 m −1 . For q ⊥ = 10 −2 k ⊥ , the condition for ECC generation obtained from equation (27) is B k 0 /B 0 > 10 −6 . This is much smaller than the observed value δ B/B 0 ∼ 5 × 10 −3 . The maximum growth rate can be estimated to be γ ∼ 0.9 s −1 for the KAW with perpendicular wavelength ∼10 km. The relative magnetic field perturbation δ B/B 0 ∼ 5 × 10 −3 also satisfies the relation (41). Thus, KAWs can indeed generate ECCs in the ionosphere, and it may be worth searching for them in the existing and future data.
It is also of interest to examine if MCs can also be excited in space plasmas. In the plasma sheet boundary layer [63], one finds that the perpendicular wave scale is ∼ 20-120 km, the wave frequency is ∼ 1 Hz, the Alfvén speed is ∼10 7 m s −1 , the ion gyroradius is ∼20 km, the disturbed magnetic field is ∼10 nT and the background magnetic field is ∼400 nT. The parallel wavenumber is then k z ∼ ω/V A ∼ 10 −7 m −1 , and the condition for the MC generation would be B k 0 /B 0 > 10 −3 . The latter is, however, smaller than the observed value δ B/B 0 ∼ 2 × 10 −2 , so that we do expect MC excitation by the KAWs. But here ρk ⊥ ∼ O(1), which would not be consistent with the long-wavelength assumption.
Finally, it should be mentioned that some of the assumptions made here, for example that the plasma is isothermal, collisionless and homogeneous and that the KAWs have relatively long perpendicular wavelengths (k ⊥ ρ 1 or k ⊥ λ e 1), may not be satisfied in very-smallscaled local structures in space plasmas or in situations where shorter-wavelength KAWs are dominant [61]. In this case, further quantitative analysis would be needed.