Anisotropic linear waves and breakdown of the momentum expansion in spin magnetohydrodynamics

Abstract

We formulate spin magnetohydrodynamics (MHD) by including the magnetic-flux and total angular momentum conservation laws in the hydrodynamic framework. To specify the local angular momentum conservation, we choose the totally antisymmetric spin current. The entropy-current analysis allows for ten dissipative first-order transport coefficients including anisotropic spin relaxation rates and the conversion rate between a vorticity (shear) to a symmetric stress (antisymmetric torque), as well as anisotropic viscosities and resistivities. By employing the linear-mode analysis, we solve the first-order spin MHD equations to determine the dispersion relations with the complete information of anisotropy retained. Our analytic solutions indicate that the small-momentum expansion is spoiled by blow up of the higher-order terms when the angle between the momentum and the magnetic field approaches the right angle. This also reveals the existence of another expansion parameter, and, in light of it, we provide solutions in an alternative series expression beyond the critical angle. We confirm that these two series expansions work well in the appropriate angle ranges as compared with numerical results. Building on our findings regarding the breakdown of the small-momentum expansion in first-order theory, we proceed to discussing how these first-order solutions are modified when we include the relaxation dynamics for dissipative modes with the Israel-Stewart framework. We find that, due to the presence of the critical behavior in the first-order solutions, there remains a diffusive window even after the relaxation dynamics is introduced.

Figures (4)

• Figure 1: The dispersion relations for the Alfven modes. Blue curves show the "exact solution" without any expansion. Dotted curves show the small-$k$ expansion in Eq. (\ref{['NLO11']}). Red curves show the small-cosine expansion in Eq. (\ref{['quartic solution simple']}).
• Figure 2: The dispersion relations for the slow magneto-sonic modes. The blue curves represent the numerical solutions of the quartic equation. Dotted curves show the small-$k$ expansion in Eq. (\ref{['quartic NLO solution2']}). Red curves show the small-cosine expansion in Eq. (\ref{['quantic solution simple-sonic']}).
• Figure 3: Dispersion relations of the IS-modified Alfven modes (blue curves) and the first-order Alfven modes (orange curves) at fixed angles. Solid and dotted curves show the real and imaginary parts of the dispersion relations. See also Fig. \ref{['fig:IS-phase']} for general angles.
• Figure 4: Unshaded and shaded regions show diffusive and propagating regimes, respectively, in the IS-modified Alfven modes (blue shade) and the first-order Alfven modes (orange shade). The IS-modified Alfven modes retain a diffusive window in the unshaded "shark-fin" region.