Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. When the Hall effect is added to the minimum magnetohydrodynamics (MHD), the MHD waves become dispersive and different nonlinear interactions are expected. The emergent turbulent state will thus be expected to be different. For incompressible Hall MHD we develop a reduced model for wave-wave interactions concentrating on those processes that will lead to phase coherent modifications to the linear dispersion of a given wave. We show that these special interactions provide an amplitude-dependent contribution to the linear dispersion relation, which yields nonlinear frequency shifts. The resonance-driven frequency shifts are dominant and add damping or growth to the linear dispersion. The damping/growth rates represent the nonlinear time scales for energy redistribution and can be used in conjunction with a conjecture like the "critical balance" to estimate the energy spectral content.
The ability to support waves is one of the primary features that distinguishes magneto plasmas from neutral fluids; it readily translates into different nature of turbulence manifested in the respective models-magnetohydrodynamics (MHD) and hydrodynamics-used to describe them. In MHD, the Alfvén wave connects the oscillating fluctuations in the velocity and magnetic field perturbations. 1 Alfvén waves are a generic name for a host of relatively low frequency (lower than ion cyclotron frequency) wave motions accessible to a magneto-plasma. Due to their essential role in dictating plasma properties, Alfvén waves constitute a highly investigated research area. It is not our intention to survey this vast field but we do give here a list of representative references that the reader may consult. Alfvén waves play a fundamental role in determining the stability of and turbulence in magneto plasmas; they are also an efficient way of heating them. 2,3 In the solar wind, durable observed correlations between the magnetic and velocity field perturbations reflect Alfvénic activity. 4,5 A discrete spectrum has been found for Alfvén waves on the inclusion of the parallel electric field in a cylindrical plasma. 6 The particular manifestations of Alfvén waves in toroidal geometry have also received much attention, 7,8 and of particular importance has been the destabilization of Alfvén eienmodes by energetic particles. [9][10][11][12] Many observations of Alfvén eigenmodes in tokamak plasmas and their impact on fast ion confinement and achieving fusion gain have been documented, 13 and their appearance in optimized stellarators remains an active area of research. 14,15 Most of the aforementioned studies deal with the linear wave dynamics of a fluid plasma system. However, the eventual impact of these waves on the plasma dynamics must be understood in a nonlinear, possibly, turbulent state. The latter, of course, originates in the interactions of these waves. [16][17][18][19][20][21][22][23][24] As an example, turbulence in the solar wind exhibits power law spectra according to the relationship of the magnetic and velocity field for the interacting waves. [25][26][27] The goal of many of the cited turbulence theories is the derivation of energy spectrum (often a power law) for an inertial range in which driving mechanisms balance the effects of dissipation. In incompressible magnetohydrodynamics (MHD), for example, Goldreich and Sridhar introduced the critical balance conjecture, which states that the wave spectral content is obtained by balancing the linear oscillation timescale with the nonlinear time scale of wave energy transfer. 19 This assumption, owing to phenomenological arguments, exploits the eddy-damped quasinormal Markovian (EDQNM) closure to calculate a k -5/3 ⊥ spectrum. Another approach to calculation of the spectrum is the Zakharov transformation, which through an assumption of scale invariance in the energy transfer function leads to power law exponents for a stationary spectrum. 28 This has been used by Galtier et al. to calculate a k -2 ⊥ spectrum for incompressible MHD, 22 and a knee in the incompressible Hall MHD spectrum corresponding to where the Hall effect becomes most important. 23 The spatiotemporal chaos which characterizes turbulence also leads to energy content at a range of frequencies. This may be inferred from the spatial distribution of energy from interactions of eddies of similar and disparate scales, straining and sweeping motions in phase space. In each case, this leads to a power law frequency spectrum. [29][30][31] Specifically focusing on the linear waves of the system, simulations of compressible Alfvénic turbulence have found a Lorentzian frequency spectrum centered about the Alfvén, slow, and fast modes, with broadening caused by the nonlinear wave interactions. 32 Where low order resonances are difficult to find in a discrete wavenumber system, the phenomenon of nonlinear broadening allows simulations of resonant interactions because resonances exist within the broadening width. 33 The existence of long time scales of energy transfer is highlighted in the work of Mahajan 2021, which, inter alia, constructs a reduced model of Hall MHD turbulence by considering a three-wave subset of the total energy spectrum. 24 This is an exactly integrable system for which a timescale of nonlinear energy transfer may be constructed from the Jacobi elliptic function timescale.
The turbulent correction to linear frequencies has been described in generality by Galtier. 28 This approach uses the method of two-timing, in which the high frequency of a linear system is corrected by longer period behavior. 34 Such an approach eliminates secular perturbative solutions to the van der Pol and Duffing oscillators, 35 and is also used by Whitham to analyze nonlinear corrections to a linear dispersion relation for the Klein-Gordon equation. 36 This paper explores a different approach to estimate the nonlinear
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