Axial $w$-modes of anisotropic neutron stars

We investigate the axial $w$-mode oscillations of anisotropic neutron stars. Stellar configurations are constructed using two realistic equations of state, BSk21 and SLy4, with two prescriptions for pressure anisotropy, the Horvat ansatz and the Bowers-Liang ansatz. The axial $w$-mode frequencies are computed by solving the linearized perturbation equations using a continued-fraction method. For each fixed anisotropy strength, the axial $w$-mode frequency decreases monotonically with increasing stellar mass along the stable branch, with its magnitude depending on both the equation of state and the nature of the anisotropy. At low stellar masses, configurations with dominant radial pressure ($p_r>p_t$) exhibit higher frequencies than those with dominant tangential pressure, whereas toward the upper end of the stable branch this ordering is reversed, and configurations with $p_t>p_r$ attain higher frequencies at the same mass. The axial $w$-mode frequency displays an approximately linear dependence on compactness, with anisotropy modifying both the slope and the intercept. The Bowers-Liang ansatz produces a wider spread in the frequency values compared to the Horvat ansatz. We also analyze the damping times associated with the axial $w$-modes and find that they increase with stellar mass, with a rapid rise toward the upper end of the stable branch. At a fixed mass, increasing the tangential pressure relative to the radial pressure leads to shorter damping times, while configurations with dominant radial pressure exhibit longer damping times. The sensitivity of the damping time to anisotropy is more pronounced for more compact stars, and the Bowers-Liang ansatz yields systematically larger damping times than the Horvat ansatz. Finally, we provide empirical expressions for the axial $w$-mode frequency and damping time as functions of stellar compactness and anisotropy strength.