Abstract
When the Hall effect is included in the magnetohydrodynamics equations (Hall-MHD model) the wave propagation modes become coupled, but for propagation parallel to the ambient magnetic field the Alfvén mode decouples from the magnetosonic ones, resulting in circularly polarized waves that are described by the derivative nonlinear Schrödinger (DNLS) equation. In this paper, the DNLS equation is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for time integration. Firstly, the nondiffusive DNLS equation is considered to test the validity of the method by verifying the analytical condition of modulational stability. Later, diffusive and excitatory effects are incorporated to compare the numerical results with those obtained by a three-wave truncation model. The results show that different types of attractors can exist depending on the diffusion level: for relatively large damping, there are fixed points for which the truncation model is a good approximation; for low damping, chaotic solutions appear and the three-wave truncation model fails due to the emergence of new nonnegligible modes.
Highlights
Alfven waves are one of the most characteristic features of magnetized laboratory and space plasmas
Nonlinear Alfven waves were extensively detected in the solar wind [3] and they are believed to be responsible for the turbulent heating of stellar coronas [4]
The truncation model is obtained looking for solutions of the derivative nonlinear Schrodinger (DNLS) in the form b (z, t) = ∑aj exp [i], (23)
Summary
Alfven waves are one of the most characteristic features of magnetized laboratory and space plasmas. The DNLS equation is reduced to a set of three ordinary differential equations where the free variables are the two components of the transverse magnetic field and the phase wave [20] In this way, a continuous three-dimensional dynamical system is obtained, which would allow retaining the nonlinear evolution of driven conservative and dissipative Alfven waves that is registered in more complicated high-dimensional models [21]. The numerical results are compared with those obtained by a three-wave truncation model which was carried out to represent Alfven wave fronts generated by orbiting conductive tethers interacting with the ambient magnetic field in the ionosphere [5, 26, 27] This analysis allows establishing the application range of the truncation method by inspecting the power spectrum, in a similar way as the work by Ghosh and Papadopoulos [21].
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