Abstract
We consider here the evolution of a 1D spectrum in the expanding (radial) wind, that is, with wavevectors all aligned in a given direction. We consider two cases: wavevectors in the radial direction, and wavevectors in a transverse direction (perpendicular to the radial). Based on these assumptions, we present a model of turbulent heating that is then compared to direct numerical simulations of Hydrodynamic turbulence in 1D.
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Notes
- 1.
Wave-action is defined as the energy of a wave over the frequency of the wave, \(E/\omega \). In the case of sound waves, \(E=\rho \delta u^2\) and \(\omega =k c_s=\omega _0 c_s/ (U_0+c_s)\). Note that in the absolute frame of reference, the sound waves transported in a fluid with mean velocity \(U_0\) have a frequency \(\omega _0=k(U_0+c_s)\).
The conservation of wave-action states
$$\begin{aligned} \partial _t(E/\omega )+\nabla \cdot ((U_0+c_s)(E/\omega ))=0. \end{aligned}$$(9.8) - 2.
The opposite happens for the transverse propagating waves, as the WKB theory predicts a decrease of Mach number as \(M=\sqrt{\delta u/c_s}\propto R^{-1/12}\). In the non-expanding simulations compressibility also decreases in time, as velocity fluctuations are dissipated by the viscous terms.
References
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Grappin R, Velli M, Mangeney A (1993) Nonlinear wave evolution in the expanding solar wind. Phys Rev Lett 70(14):2190–2193
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Montagud-Camps, V. (2019). 1D Turbulent Heating. In: Turbulent Heating and Anisotropy in the Solar Wind. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-30383-9_9
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DOI: https://doi.org/10.1007/978-3-030-30383-9_9
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