Nonadiabatic analysis of rosette modes of oscillations in rotating stars and their contribution to angular momentum transport

Rosette modes compose a peculiar group of oscillation modes in rotating stars. Although they have been investigated in some theoretical works, it has not been clear yet whether these modes are excited in real stars or not. In order to study this problem, a stability analysis of the modes is performed. It turns out that some rosette modes are unstable due to the kappa mechanism in a model of slowly pulsating B stars. It is also examined how unstable rosette modes can transport angular momentum in stars.


Introduction
Rosette modes are a class of eigenmodes of oscillations that are found in rotating stars. Their frequencies are in the range of gravity modes, but not below twice the rotation frequency, the domain in which the Coriolis force plays a dominant role in the oscillation motion. The most remarkable characteristic of the modes appears in the structure of the eigenfunctions. The kinetic energy of the oscillation distributes along rosette patterns on the meridional plane. This is the reason why the modes have their name. The modes have been discovered recently by accident [1] in numerical computations of oscillation modes of uniformly rotating polytropic models, when physical properties of gravito-inertial modes in rapidly rotating stars are studied [2].
The mechanism to form the rosette modes has been analysed based on a perturbation theory [3], assuming that rotation is slow (and uniform). The key point is considerable interaction caused by the second-order effect of the Coriolis force among multiple eigenmodes that have almost the same frequencies (close degeneracy) in the non-rotating limit. The possibility of such (close) degeneracy has already been recognised before in the perturbative treatment of the effect of rotation on stellar oscillations [e.g. 4]. However, a primary stress has usually been put on the case of two degenerate modes, whereas it is essential for the rosette modes to consider close degeneracy among many eigenmodes.
The properties of the rosette modes have been further studied. Although the modes have originally been found among axisymmetric modes, it has been shown that non-axisymmetric rosette modes can also be formed [5]. A detailed analysis has clarified that retrograde modes tend to have clearer rosette patterns than prograde modes. Moreover, the rosette patterns of the modes have been successfully described by simple relations that are derived based on an asymptotic analysis [6].
This paper presents two more steps of our analysis of the rosette modes. We first examine in section 2 whether the modes can be excited in real stars or not. We perform a linear stability analysis of the rosette modes of an early-type zero-age main sequence (ZAMS) stellar model based on the nonadiabatic calculation. Secondly, we evaluate in section 3 the contribution of the modes to the transport of angular momentum inside the model. A summary is finally given in section 4.

Stability analysis of rosette modes of an early-type ZAMS stellar model
In order to discuss whether the rosette modes can be excited in real stars or not, we examine the stability of oscillation modes of a 5 M ZAMS model, which can be regarded as a representative model of slowly pulsating B stars [7,8].
It is first confirmed in the case of no rotation that many high-degree gravity modes are destabilised by the iron opacity bump that is located at the temperature of about 2 × 10 5 K. Note that we have used the OP opacity [9] in this calculation. The left panel of figure 1 shows the plot of the real part of the eigenfrequency of the unstable modes versus the spherical degree, l. Among those unstable modes, we can identify a family of close degeneracy that is designated by the horizontal dashed line. This family is specified by a relation of 2n + 3l = 1, where n is the radial order. Although l of the unstable modes is limited to all the odd numbers between 7 and 17, we may regard that some stable modes with l ≤ 5 and l ≥ 19 also belong to the family. We concentrate on rosette modes that originate from this particular family. Examples of the (adiabatic) rosette modes are shown in figure 2 for the rotation rate, Ω, of 20 % of the Keplerian rotation rate, Ω K , at the equator.
We then use the program developed by [10] to calculate nonadiabatic rosette modes. It is found that some of the rosette modes are actually unstable if the rotation rate satisfies Ω 0.25Ω K . (The upper limit corresponds to the equatorial rotation velocity of about 150 km s −1 .) The right panel of figure 1 shows the growth rate of rosette modes as functions of the frequency. Corresponding to the fact that the eigenfunctions of rosette modes can generally be expressed as linear combinations of eigenfunctions in the case of no rotation, we can understand the stability property of the rosette modes based on that of their constituent modes in the non-rotating limit. In fact, only those rosette modes that have significant contribution from unstable modes (shown in the left panel of figure 1) tend to be excited. For example, the highly-stable prograde rosette mode with the normalised frequency of about 1.1425 in the right panel of figure 1 suffers from severe radiative damping caused by some exceptionally highdegree components. Similarly, as the rotation rate is higher, more modes, particularly those with higher degrees, are involved in the rosette modes because of stronger interaction caused by the Coriolis force. This results in the larger damping effect and eventually stabilises all of the rosette modes.

Angular momentum transport
An interesting topic related to unstable oscillation modes is their transport of angular momentum in stars. We examine how the rosette modes can contribute to the transport.
Following [11], we adopt the Lagrangian description of wave-mean flow interactions, which has been developed in meteorology [e.g. 12]. We particularly use equation (A28) of [11], which is given by with F defined by The meanings of the symbols in equations (1) and (2) are as follows: ρ is the density of the equilibrium structure; d/dt the Lagrangian time derivative (for a fixed mass element); the Lagrangian mean over the azimuthal angle; h the specific angular momentum around the rotation axis; φ the azimuthal angle; ξ, p and Φ are the (complex) eigenfunctions of nonadiabatic stellar oscillations, which represent the displacement vector, the Eulerian perturbation to the pressure and the Eulerian perturbation to the gravitational potential, respectively. As discussed in [11], the integral of the right-hand side of equation (1) over the whole solid angle is closely related to the derivative of the work function of the stability analysis, which is positive (negative) in the region of excitation (damping). In fact, equation (1) essentially describes the physical picture that angular momentum is given to (taken from) the equilibrium structure by waves that compose the oscillation modes at the places where prograde modes are damped (excited). (The situation is opposite for retrograde modes, because F is proportional to the azimuthal order, m.) We evaluate the right-hand side of equation (1) for a retrograde rosette mode with m = −1 that has been computed in section 2. While the distribution of the kinetic energy density is close to that shown in the rightmost panel of figure 2, the normalised distribution of −∇ · F is presented in figure 3. It is clear that the distribution preserves the rosette structure that changes significantly as a function of the latitude. This suggests that unstable rosette modes might uniquely contribute to the angular momentum transport in the latitudinal direction. On the other hand, we have also calculated the integral of −∇ · F over the whole solid angle. The comparison of the result to that of an ordinary (non-rosette) mode with a similar frequency shows no considerable difference. It is generally observed in the angle-integrated 05008-p.3 The Space Photometry Revolution distribution that large contribution is found only near the ionisation region of iron-peak elements, which is close to the surface.

Summary
A nonadiabatic analysis of rosette modes of a 5 M ZAMS stellar model has been performed. It is found that some rosette modes are destabilised by the standard kappa mechanism of iron-peak elements. The contribution of those modes to the transport of angular momentum has been examined. It is argued that rosette modes might provide a unique channel of angular momentum transport particularly in the latitudinal direction.
This work was partially supported by JSPS KAKENHI Grant Number 26400219.