Alfvénic fluctuations in the solar wind exhibit a high degree of velocities and magnetic field correlations consistent with Alfvén waves propagating away and toward the Sun. Two remarkable properties of these fluctuations are the tendencies to have either positive or negative magnetic helicity (–1 ≤ σ m ≤?+1) associated with either left- or right- topological handedness of the fluctuations and to have a constant magnetic field magnitude. This paper provides, for the first time, a theoretical framework for reconstructing both the magnetic and velocity field fluctuations with a divergence-free magnetic field, with any specified power spectral index and normalized magnetic- and cross-helicity spectrum field fluctuations for any plasma species. The spectrum is constructed in the Fourier domain by imposing two conditions—a divergence-free magnetic field and the preservation of the sense of magnetic helicity in both spaces—as well as using Parseval's theorem for the conservation of energy between configuration and Fourier spaces. Applications to the one-dimensional spatial Alfvénic propagation are presented. The theoretical construction is in agreement with typical time series and power spectra properties observed in the solar wind. The theoretical ideas presented in this spectral reconstruction provide a foundation for more realistic simulations of plasma waves, solar wind turbulence, and the propagation of energetic particles in such fluctuating fields.
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