Low‐Frequency Nonradial Oscillations in Rotating Stars. I. Angular Dependence

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We obtain the θ-dependence of the displacement vector of rotationally modulated low-frequency nonradial oscillations by numerically integrating Laplace's tidal equation as an eigenvalue problem with a relaxation method. This method of calculation is more tractable than our previous method in which the θ-dependence was represented by a truncated series of associated Legendre functions. Laplace's tidal equation has two families of eigenvalues. In one of these families, an eigenvalue λ coincides with l(l + 1) when rotation is absent, where l is the latitudinal degree of the associated Legendre function, Pml(cos θ). The value of λ changes as a function of ν ≡ 2Ω/ω, where Ω and ω are the angular frequencies of rotation and of oscillation (seen in the corotating frame), respectively. These eigenvalues correspond to rotationally modulated g-mode oscillations. In the domain of | ν | > 1, another family of eigenvalues exists. Eigenvalues belonging to this family have negative values for prograde oscillations, while they change signs from negative to positive for retrograde oscillations as | ν | increases. Negative λ's correspond to oscillatory convective modes. The solution associated with a λ that has a small positive value after changing its sign is identified as an r-mode (global Rossby wave) oscillation. Amplitudes of g-mode oscillations tend to be confined to the equatorial region as | ν | increases. This tendency is stronger for larger λ. On the other hand, amplitudes of oscillatory convective modes are small near the equator.