Resonant Decay of Kinetic Alfvén Waves and Implication on Spectral Cascading

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I. INTRODUCTION
Shear Alfvén waves (SAWs) are incompressible electromagnetic oscillations prevalent in both nature and laboratory plasmas [1].SAWs are often found to mode-convert into small-scale kinetic Alfvén waves (KAWs) [2] due to phase-mixing stemming from intrinsic plasma nonuniformities [3,4].KAWs are uniquely characterized by a finite parallel electric field component and perpendicular propagation across the mean magnetic field line, which contribute to early applications on laboratory plasma heating [5], geomagnetic pulsation [6,7], and solar corona heating [8].To quantitatively determine the effects of collisionless plasma transport and nonlocal wave energy transfer, one is supposed to acquire detailed knowledge of the nonlinear evolution of KAWs, with comprehensive consideration on both nonlinear wave-wave interactions and wave-particle interactions in typical weak turbulence theory [9].Resonant three-wave interaction [10], in this respect, is a basic process resulting in various nonlinear wave dynamic evolution.Previous studies on the nonlinear mode coupling process of SAW/KAW have focused on the resonant parametric decay into ion acoustic wave and/or ion-induced scattering, where qualitative difference between ideal-magnetohydrodynamic (MHD) results [9] and kinetic results [11,12] are addressed, which demonstrate the crucial importance of kinetic description in the study of KAW nonlinear processes.
Magnetized plasma turbulence in both space and fusion devices are very often, constituted by fluctuations with frequency ω much lower than the ion cyclotron frequency Ω i , and anisotropy in directions parallel and perpendicular to the magnetic field.The parallel wavelength can be up to the system size, while the perpendicular wavelength varying from system size to ion Larmor radius ρ i .One exam- ple is the drift wave (DW) type micro-turbulence excited by expansion free energy associated by plasma nonuniformities, and its nonlinear dynamics including spectral evolution in the strong turbulence limit can be described by the famous Charney-Hasegawa-Mima (CHM) equation [13,14].Charney-Hasegawa-Mima equation can also describe quasitwo-dimensional turbulence in atmospheric motion of a ro-tating planet [15], and it reveals essential features of conservation constraints like energy and enstrophy with respective cascading behaviors and self-organization processes [16] in analogy to two-dimensional neutral viscid fluid system described by the Navier-Stokes equation [17,18].From the dispersion relation of Alfvén waves, with predominantly ω ∝ k to the leading order and k being the wavenumber parallel to the equilibrium magnetic field, one can physically speculate that strong decays and coalescences could occur since the wavenumber and frequency matching conditions required for resonant three-wave interactions can be easily satisfied for Alfvén waves.Thus, this kind of self-interaction mechanism and strong coupling could result in high turbulent level and broad energy spectrum, which is universal in various systems and commonalities shall exist for Alfvénic turbulence with practical interest in solar wind [19][20][21], interstellar medium [22,23] and accretion disks [24,25].Generally speaking, the standard turbulence paradigm [26][27][28] involves long-wavelength energy-containing scales, inner dissipation scales, and disordered fluctuations filling up a broad range of intermediate scales (i.e., the inertial range), it is thus of necessity to adopt the nonlinear gyrokinetic theory [29,30] with the anisotropic assumption to fully describe Alfvénic turbulent cascading [31,32] both analytically and numerically with arbitrary spatial resolution.In the nonlinear gyrokinetic equation, we see that particle distribution functions in the gyrocenter phase space are nonlinearly phase-mixed by the gyroaveraged electric-field drift induced convective flows [28], which brings the energy injected at the outer scale down to collisional dissipations at particle gyroscales.The analytics shall further lead to concrete predictions for the spectra, conservative laws, statistical properties, ordered structures and selfconsistent states of Alfvénic turbulence.
Additionally, since SAW instabilities could be resonantly excited by energetic charged particles (EPs) [33,34] as discrete Alfvén eigenmodes (AEs) or energetic particle continuum modes [34] in fusion plasmas characterized by multiple short-wavelength |k ⊥ ρ i | ∼ O(10 −1 ) modes, with k ⊥ being the wavenumber perpendicular to the equilibrium magnetic field, the nonlinear interactions among SAW/KAW triplets can then transfer energy from a linearly unstable primary mode to stable modes, providing a fundamental nonlinear saturation mechanism for SAW instabilities in burning plasmas, which has already been observed in simulations [35][36][37] and analysed theoretically [38].The analysis on KAW/SAW nonlinear interactions, can thus be applied to study the spectral transfer among various AEs in fusion plasmas, with the ultimate goal of understanding the confinement of EPs and plasma performance in future fusion reactors.
In this work, we mainly address the potential spectral cascading behaviors, by deriving the general nonlinear equation describing resonant three-wave interactions among KAWs, and studying the parametric decay of a pump KAW, with emphasis on the condition for the process to spontaneously occur.The rest of the manuscript is organized as follows.In Sec.II, the gyrokinetic theoretical framework is introduced, which is used in Sec.III to derive the general nonlinear equation describing resonant three-KAW interactions.The parametric decay of a pump KAW is analyzed in Sec.IV, with emphasis on the condition for spontaneous decay.Finally the results are summarized in Sec.V, where future work along this line is also briefly discussed.

II. MODEL EQUATIONS
Considering a simple slab geometry and uniform plasma, we investigate the resonant nonlinear interaction among three KAWs in low-β magnetized plasmas using nonlinear gyrokinetic theory [29,30].The magnetic compression is systematically suppressed by the β O(10 −1 ) and k ≪ k ⊥ orderings of interest.Here, β = 8πP/B 2 0 is the ratio of thermal to magnetic pressure, P is the plasma thermal pressure, B 0 is the equilibrium magnetic field amplitude.Although the existence of KAW is related to intrinsic plasma nonuniformities [2,5] and/or wave-particle interactions [33,34,39], we, for the clarity of discussion, only focus on the essential physical mechanism of KAW resonant decay and the implication on spectrum cascading assuming uniform plasmas, while their linear stability, important for the spectrum cascading and final saturation, are not accounted for here.The effects of magnetic field geometry, plasma nonuniformities, and/or trapped particle effect, potentially impacting the KAW decay in magnetically confined high-temperature plasmas, are also neglected [40].Following the standard approach [30], the perturbed distribution function δ f j for species j = i, e is given by with q j being the particle's charge, F M j and T j being the local Maxwellian distribution function and the equilibrium temperature respectively, δ φ , and e −ρ j •∇ denoting the generator of coordinate transformation from guiding-center space to particle phase space.The nonadiabatic particle response δ g j can be derived from the nonlinear gyrokinetic equation with (...) α denoting gyro-averaging and b = B 0 /B 0 .The leading-order nonlinear E × B convection term is represented by the effective gyroaveraged potential δ L g, j α ≡ exp(ρ j • ∇)(δ φ − v δ A /c) α , which includes the contribution of both the perturbed electric-field drift term (c/B 0 ) b × ∇ ⊥ δ φ and the magnetic flutter term (v /B 0 )∇ ⊥ δ A × b.Assuming thermal ion species with unit electric charge e and particle density n 0 , the governing field equations are the quasineutrality condition and the nonlinear gyrokinetic vorticity equation [7,41] derived from the parallel Ampere's law δ J k = (c/4π)k 2 ⊥ δ A k , the quasi-neutrality condition, and the nonlinear gyrokinetic equation.Here, τ ≡ T e /T i is the temperature ratio, (...) v denotes the velocity-space integration and The two terms on the left-hand-side of Eq. (4) represent field line bending and plasma inertia, respectively; while the terms on the right-hand-side of Eq. ( 4) are the formally nonlinear terms from the Maxwell stress (MX) and generalized gyrokinetic Reynolds stress (RS) contribution.It is noteworthy that present nonlinear terms will be dominated by the polarization nonlinearity in the limit of |k ⊥ ρ i | 2 ≪ |ω/Ω i | [11,12,40], which has significant role in nonlinear MHD description [9,42].Therefore, the analyses of the present work are valid for |k ⊥ ρ i | 2 /|ω/Ω i | O(1) parameter regime, which should be kept in mind as we proceed to the investigation of cascading behaviors in future works.

III. NONLINEAR MODE EQUATION
Now we proceed to derive the governing equations describing resonant three-KAW interaction.For a KAW Ω k with frequency ω k and wavenumber k = k b + k ⊥ , strong scat- tering could occur for each pair of sidebands Without loss of generality, we assume that v te ≫ |ω k /k | ≫ v ti for all Fourier modes involved, and the linear particle responses of KAWs are derived as k,e ≃ − e T e F Me δ ψ k .
Here, the effective potential δ ψ k = ω k δ A k /(ck ) corresponding to the induced parallel electric field is introduced, and δ ψ k = δ φ k is equivalent to the ideal MHD condition (δ E = 0).Substituting linear particle responses into quasi-neutrality condition, one obtains which then yields, together with linear gyrokinetic vorticity equation, the linear dielectric function of KAW with indicating the deviation from ideal MHD constraint due to FLR effect and generation of finite parallel electric field that is crucial for plasma heating.Furthermore, v A = B 0 / √ 4πn 0 m i is the Alfvén speed.We note that, only the Hermitian part of KAW dielectric function is given in Eq. ( 8) for clarity of notation, while its anti-Hermitian part for linear stability crucial for later analysis of KAW parametric decay and spectrum cascading can be recovered straightforwardly.Eq. ( 8) yields, the familiar expression of KAW dispersion relation in the b k ≪ 1 limit with ρ 2 κ ≡ (3/4 + τ)ρ 2 i .The relation between field potentials simultaneously gives the polarization properties of KAW as and which suggest the difference between energy spectra of perpendicular electric and magnetic perturbations in both inertial and dissipation ranges of Alfvénic solar wind turbulence [43].
It has been demonstrated that even in a fully developed turbulence, the fluctuations could retain the bulk of linear physics of KAWs [44,45].
The nonlinear electron response to Ω k can be derived from the nonlinear gyrokinetic equation as while the nonlinear ion response is negligible due to Applying the quasi-neutrality condition, i.e.Eq.
(3), we obtain which indicates that the coupling of Ω k ′ and Ω k ′′ due to electron responses could nonlinearly contribute to additional par-allel electric field perturbation.The nonlinear gyrokinetic vorticity equation, Eq. ( 4), on the other hand, yields Substituting Eq. ( 13) into Eq.( 14), one then obtains the desired nonlinear equation for KAW resonant three-wave interactions with the nonlinear coupling coefficient expressed as Eq. ( 15) describes nonlinear dynamics of low-frequency KAW due to resonant three-KAW interactions in uniform systems, and is the most important result of the present work.The present form of the nonlinear coupling coefficient given by Eq. ( 16) allows us to analyze the contribution from different terms with clear physical meanings.The first two terms represent the contribution of RS and MX, both of which are negligible in the long-wavelength limit [11], and the last term originates from finite parallel electric field contribution via the field line bending term in vorticity equation.
We shall further defining s k as the direction of a given mode (ω k , k) propagation along the magnetic field line, with the value of either −1 or +1, and as the modulus of ω k /(k v A ).Meanwhile, α k increases with the perpendicular wavenumber monotoni- cally.The nonlinear coupling coefficient β k ′ ,k ′′ can, thus, be re-written as It is noteworthy that similar expression of the nonlinear coupling coefficient β k ′ ,k ′′ was also derived in Ref. 46, where the nonlinear mode equation is derived via integral along unperturbed particle orbit with the low-frequency (|ω/Ω i | ≪ 1) limit.The propagation direction of each mode, thus, crucially determines the magnitude of nonlinear coupling coefficient.Expanding β k ′ ,k ′′ up to the order of O(b k ) in the b k ≪ 1 limit, we obtain which indicates negligible coupling as two sidebands are co-propagating (s k ′ s k ′′ = 1), while retain finite coupling for counter-propagating sidebands (s k ′ s k ′′ = −1) due to additive effect of RS, MX and finite nonlinear parallel electric field contribution.Therefore, higher-order FLR correction O(b 2 k ) should be considered for co-propagating case and leads to Similarly, we can derive the simplified expression of β k ′ k ′′ in the short-wavelength limit b k ≫ 1 as These analytical expressions shall be used as verification for subsequent resultant cascading behaviors.As a brief remark, the nonlinear gyrokinetic equation and vorticity equation applied here are able to capture the crucial features of the low-frequency turbulent phenomena and depict the nonlinear behaviors of KAW turbulence in great detail.Similarity between the Charney-Hasegawa-Mima equation and the given Eq. ( 15) is evident since both electrostatic low-frequency drift wave turbulence and KAW turbulence could be characterized as strong turbulence with finiteamplitude fluctuations and broadband spectra in ω and k ⊥ .However, the more complicated form of nonlinear coupling coefficient indicates potentially richer turbulent phenomena for KAW turbulence.Furthermore, current analysis can be straightforwardly extended to account for plasma nonuniformities to describe general drift-Alfvén turbulence [41] in laboratory plasmas.At the present stage, to picture the universal spectra of KAW turbulence in both laboratory and nature plasmas, we shall investigate the potential turbulent cascading properties, using the paradigm model of a KAW resonantly decaying into two KAW sidebands.

IV. PARAMETRIC DECAY AND SPECTRAL CASCADES
Considering resonant three-KAW interaction process with frequency and wavenumber matching conditions Ω 0 = Ω 1 + Ω 2 satisfied, we look into the condition for spontaneous decay, with emphasis on the energy flow among these three waves.Without loss of generality, taking Ω 0 as the pump wave, the frequency and parallel wavenumber matching conditions are illustrated in Fig. (1), where cases for different propagation direction of Ω 2 with respect to Ω 0 are given.It also implies the perpendicular wavenumber matching condition and embodies possible cascading behaviors in k ⊥ , noting the dependence of the slope of ω k /k on b k , which will be elaborated in the following analysis.Taking δ φ k (t) = Φ k (t)e −iω k t with Φ k (t) being the slowly-varying amplitude, and noting for (a) dual-cascading and (b) inverse-cascading KAW triplets.Black vector arrows represent the values of frequency and parallel wavenumber for each modes Ω k = (ω k , k) and dashed lines constitute parallelograms for different propagation direction of Ω 2 .The modulus of slope of each dotted line is proportional to α k , which increases monotonically with k ⊥ . and ∂ t being the slowly temporal evolution and can be denoted as γ, and γ dk < 0 being the linear damping rate, we can rewrite the coupled equations as The resultant parametric dispersion relation is given as which then yields, the condition for spontaneous decay with the γ > 0 i.e., firstly, the sign of β k 2 ,k 0 β k 0 ,k 1 to be positive, and secondly, the nonlinear drive intensity γ 2 ND should exceed the threshold due to linear damping of sidebands, which determines the threshold on pump KAW amplitude.Note that, the threshold could also arise from the frequency mismatch, which is, however, not considered here.
, as well as the slopes of dotted lines in Fig. 1(a).While for the backward-scattering process with one sideband propagating in the direction opposite to that of the pump wave, as shown by the (Ω 0 , Ω ′ 1 , Ω ′ 2 ) combination in Fig. 1(a) or Fig. 1(b) (the difference is the slope of Ω 2 with respect to the pump KAW Ω 0 ), the counterpropagating KAW triplet could be manifested as either dualcascading or inverse-cascading within the circular region in Fig. 2(b), which is also confirmed by the analytical expression β k 2 ,k 0 β k 0 ,k 1 ∝ (α k 0 − α k 1 ).Above preliminary cascad- ing properties obtained from spontaneous decay condition can also be identified in Fig. (1), where the modulus of slope of each dotted line is proportional to α k = σ k b k /(1 − Γ k ) and thus increases with k ⊥ monotonically.Furthermore, from the contour of γ 2 ND /Ω 2 i , we conclude that the inverse cascade is dominant for counter-propagating case, since the interval with stronger nonlinear drive falls within the region of inverse cascading ((Ω 0 , Ω ′ 1 , Ω ′ 2 ) combination in Fig. 1(b)).These different cascading behaviors in perpendicular wavenumber spectrum would further determine the stationary energy spectrum of KAW turbulence and have significant implication on crossfield transport.For different value of temperature ratio τ, we can evaluate the maximized nonlinear drive γ ND /Ω i for different b k 0 , as shown in Fig. 3.The overall tendency is that the maximized nonlinear drive increases with b k 0 monotonically.For a specific τ, for relatively small b k 0 decay is more effective into counter-streaming KAWs, i.e. inverse cascade in k ⊥ .While in the short-wavelength region with relatively large b k 0 , the maximized nonlinear drive for co-propagating KAWs is slightly larger.Although the difference is not significant, the consequences on resultant saturated spectrum could be worth exploring, considering different types of cascadings.

V. CONCLUSION AND DISCUSSION
In this work, the generalized nonlinear equation governing the resonant interaction among three kinetic Alfvén waves (KAWs) in the kinetic regime with |k ⊥ ρ i | 2 /|ω/Ω i | O(1) is derived using nonlinear gyrokinetic equation, motivated to obtain a generalized nonlinear theoretic framework that can be applied to strongly turbulent system, with the potential application to the spectral cascading of KAWs in nature and laboratory plasmas.A generalized expression of the nonlinear coupling coefficient with concrete physics meaning is derived, which can be conveniently used for the cases with the two beating KAWs being co-or counter-propagating along the equilibrium magnetic field.The present analysis assumed uniform plasma with β ≪ 1 to simplify the analysis, however, extension to more realistic parameter regimes with plasma nonuniformity accounted for should be straightforward [47].
To reveal the crucial physics underlying the KAW spectral cascading, the condition for a pump KAW spontaneously decay into two sideband KAWs is analyzed.In the forwardscattering process with both sidebands being co-propagating with the pump KAW, it is found both analytically and numerically that, this corresponds to a dual cascading process in perpendicular wavenumber, similar to that of DW cascading described by the well-known Charney-Hasegawa-Mima equation.On the hand, in the back-scattering process with one of the sidebands being counter-propagating with respect to the pump KAW, both dual-and inverse-cascading are possible, and inverse-cascading may have a larger nonlinear cross-section.These aspects can be used in later analysis for the saturated KAW spectrum with linear drive/dissipation self-consistently accounted for.
The general equation (15) for resonant interactions among KAWs, with the nonlinear coupling coefficient given by Eq. ( 16), is the most important result of the present work, and can be used to study the potentially strongly turbulent KAW nonlinear spectral cascading with |δ ω/ω| ∼ O(1), noting that ω ∝ k to the leading order for KAWs, with clear analogy to drift waves with ω ∝ k ⊥ that is imbedded in the spectral evolution described by Charney-Hasegawa-Mima equation.For further analytical progress, in the long wavelength limit with b ≪ 1, Eq. ( 15) can be reduced to and for counter-and co-propagating KAWs, respectively, and symmetrization has been applied in the derivation of these equations [14].They are capable of describing the selfconsistent turbulent evolution of KAWs with multiple Fourier modes in long-wavelength regime.It is noteworthy that, the nonlinear term for counter-propagating case is similar to that of 2D Navier-Stokes/Euler equation and Charney-Hasegawa-Mima equation, while that for co-propagating case is quite different, suggesting both quantitatively and qualitatively different cascading dynamics.Therefore, the expected conservation laws, possible vortex solutions and direct numerical simulations of turbulent evolution will be given in future works.While applying the above Eqs.( 27) and ( 28), one needs to keep in mind that, they are derived assuming b ≪ 1, which defines the parameter regime for their application.As a final remark, while the interactions among KAWs are local in wavenumber space, it is expected that smallscale structures like convective cells [48] could be simultaneously excited, corresponding to zonal structures [49][50][51] prevalent in fusion and atmospheric plasmas, and these important physics are not included in the present model.Taking the zonal structure generation into account, one could study the nonlinear dynamics of KAWs, including the potential implications to fusion plasmas, could be analyzed in future projects.

FIG. 1 .
FIG. 1. Sketches of frequency and parallel wavenumber matching conditions (ω k0 = ω k 1 + ω k 2 , k 0 = k 1 + k 2 )for (a) dual-cascading and (b) inverse-cascading KAW triplets.Black vector arrows represent the values of frequency and parallel wavenumber for each modes Ω k = (ω k , k) and dashed lines constitute parallelograms for different propagation direction of Ω 2 .The modulus of slope of each dotted line is proportional to α k , which increases monotonically with k ⊥ .

FIG. 2 .
FIG. 2. Intensity profiles of γ 2 ND /Ω 2 i for the sideband Ω 2 being (a) co-propagating or (b) counter-propagating.Black dashed circles represent the separatrix between positive and negative values.The nonlinear drive γ 2 ND /Ω 2 i for co-and counterpropagating cases are visualized in two contour plots of Fig. 2, with the horizontal and vertical axes being k x1 ρ i and k y1 ρ i , respectively.The parameter used are β i = 0.01, τ = 1, pump KAW amplitude |δ B k 0 /B 0 | = 10 −3 , and b k 0 = 1.6 with k x,0 ρ i = 1.2 and k y,0 ρ i = 0.4.In Fig. 2, red and blue colours correspond to positive and negative values, and the boundary between stable and unstable regions are indicated by black dashed circles.For all three KAWs being co-propagating, i.e. a forward-scattering process as shown by the (Ω 0 , Ω 1 , Ω 2 ) combination in Fig. 1(a), the wavenumber spectrum exhibits a dual cascading character in k ⊥ , similar to that of the DW described by Charney-Hasegawa-Mima equation, which can be verified by the analytical expression

FIG. 3 .
FIG. 3. Parameter dependences of nonlinear drive γ ND /Ω i on b k 0 for different value of τ.