Abstract
The process we will follow is similar to that of Chap. 2 for linear perturbations, but now we are also going to consider quadratic terms to define the perturbations (Sect. 4.1). We will show that, in the quadratic-perturbation approximation, modes couple in triplets, which satisfy a resonance condition (Sect. 4.2). Coupling of an unstable mode to other (stable) modes of the star can lead to the saturation of the unstable mode’s amplitude, through a mechanism known as parametric resonance instability (Sect. 4.3). For the saturation to be successful, some stability conditions, which determine the amplitude evolution of the coupled triplet, have to be satisfied (Sect. 4.4), with some interesting behaviours occurring throughout the parameter space, like limit cycles, chaotic orbits, and frequency synchronisation.
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- 1.
In fact, this was explicitly assumed in a previous step, namely during the derivation of Eq. (4.1.12) in Appendix D.2 [Eq. (D.2.6)]. Had we not made this assumption, the amplitude equation of motion would be given by the more general Eq. (D.2.5). However, since only resonant triplets contribute to the amplitude evolution, it is, in retrospect, valid.
- 2.
Equations (4.2.10) and (4.2.11) were derived for the coupling coefficient in the nonrotating limit, in which case each mode is described by a single spherical harmonic, thus reducing the angular part of the coupling coefficient to the simple integral (4.2.9). However, they should also be valid when rotation is included, as elegantly shown by Schenk et al. (2001), who also derived some additional, albeit less general, selection rules.
- 3.
Simple examples of parametric instability are pendula in which the length of the string is being varied periodically or the point of support oscillates vertically.
- 4.
The word “instability”, used to describe the phenomenon of parametric resonance, may cause some confusion, because, so far, we were only referring to stability or instability due to the presence of some damping or growth mechanism, like viscosity and/or gravitational radiation (see Sects. 3.5 and 3.6). The parametric resonance discussed here is a consequence of the resonant nonlinear coupling of an unstable mode to two stable modes, resulting in the growth of the latter and, thus, inducing an instability. Hence, to avoid any confusion, we will use phrases like “parametrically unstable”, when referring to modes undergoing the parametric resonance instability.
- 5.
- 6.
References
Aikawa, T. (1983). On the secular variation of amplitudes in double-mode Cepheids. Monthly Notices of the Royal Astronomical Society, 204, 1193–1202. http://adsabs.harvard.edu/abs/1983MNRAS.204.1193A.
Aikawa, T. (1984). Period shifts and synchronization in resonant mode interactions of non-linear stellar pulsation. Monthly Notices of the Royal Astronomical Society, 206, 833–842. http://adsabs.harvard.edu/abs/1984MNRAS.206..833A.
Anderson, D. (1976). Nonresonant wave-coupling and wave-particle interactions. Physica Scripta, 13, 117–121. https://doi.org/10.1088/0031-8949/13/2/010.
Arras, P., Flanagan, É. É., Morsink, S. M., Schenk, A. K., Teukolsky, S. A., & Wasserman, I. (2003). Saturation of the r-mode instability. The Astrophysical Journal, 591, 1129–1151. https://doi.org/10.1086/374657, arXiv:astro-ph/0202345.
Bondarescu, R., Teukolsky, S. A., & Wasserman, I. (2007). Spin evolution of accreting neutron stars: Nonlinear development of the r-mode instability. Physical Review D, 76, 064019. https://doi.org/10.1103/PhysRevD.76.064019, arXiv:0704.0799.
Bondarescu, R., Teukolsky, S. A., & Wasserman, I. (2009). Spinning down newborn neutron stars: Nonlinear development of the r-mode instability. Physical Review D, 79, 104003. https://doi.org/10.1103/PhysRevD.79.104003, arXiv:0809.3448.
Brink, J., Teukolsky, S. A., & Wasserman, I. (2004a). Nonlinear coupling network to simulate the development of the r-mode instability in neutron stars. I. Construction. Physical Review D, 70, 124017. https://doi.org/10.1103/PhysRevD.70.124017, arXiv:gr-qc/0409048.
Brink, J., Teukolsky, S. A., & Wasserman, I. (2004b). Nonlinear couplings of R-modes: Energy transfer and saturation amplitudes at realistic timescales. Physical Review D, 70, 121501. https://doi.org/10.1103/PhysRevD.70.121501, arXiv:gr-qc/0406085.
Brink, J., Teukolsky, S. A., & Wasserman, I. (2005). Nonlinear coupling network to simulate the development of the r mode instability in neutron stars. II. Dynamics. Physical Review D, 71, 064029. https://doi.org/10.1103/PhysRevD.71.064029, arXiv:gr-qc/0410072.
Buchler, J. R. (1983). Resonance effects in radial pulsators. Astronomy & Astrophysics, 118, 163–165. http://adsabs.harvard.edu/abs/1983A%26A...118..163B.
Buchler, J. R., & Regev, O. (1983). The effects of nonlinearities on radial and nonradial oscillations. Astronomy & Astrophysics, 123, 331–342. http://adsabs.harvard.edu/abs/1983A%26A...123..331B.
Buchler, J. R., & Goupil, M.-J. (1984). Amplitude equations for nonadiabatic nonlinear stellar pulsators I. The formalism. The Astrophysical Journal, 279, 394–400. https://doi.org/10.1086/161900.
Buchler, J. R., & Kovacs, G. (1986a). On the modal selection of radial stellar pulsators. The Astrophysical Journal, 308, 661–668. https://doi.org/10.1086/164537.
Buchler, J. R., & Kovacs, G. (1986b). The effects of a 2:1 resonance in nonlinear radial stellar pulsations. The Astrophysical Journal, 303, 749–765. https://doi.org/10.1086/164122.
Dappen, W., & Perdang, J. (1985). Non-linear stellar oscillations. Non-radial mode interactions. Astronomy & Astrophysics, 151, 174–188. http://adsabs.harvard.edu/abs/1985A%26A...151..174D.
Dimant, Y. S. (2000). Nonlinearly saturated dynamical state of a three-wave mode-coupled dissipative system with linear instability. Physical Review Letters, 84, 622. https://doi.org/10.1103/PhysRevLett.84.622.
Dziembowski, W. (1982). Nonlinear mode coupling in oscillating stars. I. Second order theory of the coherent mode coupling. Acta Astronomica, 32, 147–171. http://adsabs.harvard.edu/abs/1982AcA....32..147D.
Dziembowski, W. (1993). Mode selection and other nonlinear phenomena in stellar oscillations. In W. W. Weiss and A. Baglin, (Eds.), IAU Colloquia 137: Inside the Stars (vol. 40). Astronomical Society of the Pacific Conference Series. http://adsabs.harvard.edu/abs/1993ASPC...40..521D.
Dziembowski, W., & Kovács, G. (1984). On the role of resonances in double-mode pulsation. Monthly Notices of the Royal Astronomical Society, 206, 497–519. http://adsabs.harvard.edu/abs/1984MNRAS.206..497D.
Dziembowski, W., & Krolikowska, M. (1985). Nonlinear mode coupling in oscillating stars. II. Limiting amplitude effect of the parametric resonance in main sequence stars. Acta Astronomica, 35, 5–28. http://adsabs.harvard.edu/abs/1985AcA....35....5D.
Dziembowski, W., Krolikowska, M., & Kosovichev, A. (1988). Nonlinear mode coupling in oscillating stars. III. Amplitude limiting effect of the rotation in the Delta Scuti stars. Acta Astronomica, 38, 61–75. http://adsabs.harvard.edu/abs/1988AcA....38...61D.
Kumar, P., & Goldreich, P. (1989). Nonlinear interactions among solar acoustic modes. The Astrophysical Journal, 342, 558–575. https://doi.org/10.1086/167616.
Landau, L. D., & Lifshitz, E. M. (1969). Mechanics (vol. 1, 2nd ed.). Course of Theoretical Physics. New York: Pergamon Press. http://adsabs.harvard.edu/abs/1969mech.book.....L.
Morsink, S. M. (2002). Nonlinear Couplings between r-Modes of Rotating Neutron Stars. The Astrophysical Journal, 571, 435–446. https://doi.org/10.1086/339858, arXiv:astro-ph/0202051.
Moskalik, P. (1985). Modulation of amplitudes in oscillating stars due to resonant mode coupling. Acta Astronomica, 35, 229–254. http://adsabs.harvard.edu/abs/1985AcA....35..229M.
Moskalik, P. (1986). Amplitude modulation due to internal resonances as a possible explanation of the Blazhko effect in RR Lyrae stars. Acta Astronomica, 36, 333–353. http://adsabs.harvard.edu/abs/1986AcA....36..333M.
Nayfeh, A. H., & Mook, D. T. (1979). Nonlinear Oscillations. Pure & Applied Mathematics. New York: Wiley. http://adsabs.harvard.edu/abs/1979noos.book.....N.
Nowakowski, R. M. (2005). Multimode resonant coupling in pulsating stars. Acta Astronomica, 55, 1–41. http://adsabs.harvard.edu/abs/2005AcA....55....1N, arXiv:astro-ph/0501510.
Ott, E. (1981). Strange attractors and chaotic motions of dynamical systems. Reviews of Modern Physics, 53, 655–671. https://doi.org/10.1103/RevModPhys.53.655.
Passamonti, A., Stergioulas, N., & Nagar, A. (2007). Gravitational waves from nonlinear couplings of radial and polar nonradial modes in relativistic stars. Physical Review D, 75, 084038. https://doi.org/10.1103/PhysRevD.75.084038, arXiv:gr-qc/0702099.
Pnigouras, P., & Kokkotas, K. D. (2016). Saturation of the f-mode instability in neutron stars. II. Applications and results. Physical Review D, 94, 024053. https://doi.org/10.1103/PhysRevD.94.024053, arXiv:1607.03059.
Schenk, A. K., Arras, P., Flanagan, É. É., Teukolsky, S. A., & Wasserman, I. (2001). Nonlinear mode coupling in rotating stars and the r-mode instability in neutron stars. Physical Review D, 65, 024001. https://doi.org/10.1103/PhysRevD.65.024001, arXiv:gr-qc/0101092.
Smolec, R. (2014). Mode selection in pulsating stars. In J. A. Guzik, W. J. Chaplin, G. Handler and A. Pigulski (Eds.), Precision Asteroseismology (vol. 9). Proceedings of the IAU Symposium 301. Wroclaw, Poland. https://doi.org/10.1017/S1743921313014439, arXiv:1309.5959.
Stenflo, L., Weiland, J., & Wilhelmsson, H. (1970). A solution of equations describing explosive instabilities. Physica Scripta, 1, 46. https://doi.org/10.1088/0031-8949/1/1/008.
Takeuti, M., & Aikawa, T. (1981). Resonance phenomenon in classical cepheids. Science reports of the Tohoku University, Eighth Series, 2, 106–129. http://adsabs.harvard.edu/abs/1981SRToh...2..106T.
Van Hoolst, T. (1994a). Coupled-mode equations and amplitude equations for nonadiabatic, nonradial oscillations of stars. Astronomy & Astrophysics, 292, 471–480. http://adsabs.harvard.edu/abs/1994A%26A...292..471V.
Van Hoolst, T. (1994b). Nonlinear, nonradial, isentropic oscillations of stars: Hamiltonian formalism. Astronomy & Astrophysics, 286, 879–889. http://adsabs.harvard.edu/abs/1994A%26A...286..879V.
Van Hoolst, T., & Smeyers, P. (1993). Non-linear, non-radial, isentropic oscillations of stars: Third-order coupled-mode equations. Astronomy & Astrophysics, 279, 417–430. http://adsabs.harvard.edu/abs/1993A%26A...279..417V.
Vandakurov, Y. V. (1979). Nonlinear Coupling of Stellar Pulsations. Soviet Astronomy, 23, 421. http://adsabs.harvard.edu/abs/1979SvA....23..421V.
Verheest, F. (1976). Possible nonlinear wave-wave coupling between three or four waves in plasmas. Plasma Physics, 18, 225–234. https://doi.org/10.1088/0032-1028/18/3/008.
Verheest, F. (1990). Nonresonant three-mode coupling as a model for double-mode pulsators. Astrophysics and Space Science, 166, 77–91. https://doi.org/10.1007/BF00655609.
Verheest, F. (1993). Nonresonant mode coupling in double-mode pulsators. Astrophysics and Space Science, 200, 325–330. https://doi.org/10.1007/BF00627139.
Wersinger, J.-M., Finn, J. M., & Ott, E. (1980a). Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling. Physics of Fluids, 23, 1142–1154. https://doi.org/10.1063/1.863116.
Wersinger, J.-M., Finn, J. M., & Ott, E. (1980b). Bifurcations and strange behavior in instability saturation by nonlinear mode coupling. Physical Review Letters, 44, 453–456. https://doi.org/10.1103/PhysRevLett.44.453.
Wilhelmsson, H., Stenflo, L., & Engelmann, F. (1970). Explosive instabilities in the well-defined phase description. Journal of Mathematical Physics, 11, 1738–1742. https://doi.org/10.1063/1.1665320.
Wu, Y., & Goldreich, P. (2001). Gravity Modes in ZZ Ceti Stars. IV. Amplitude Saturation by Parametric Instability. The Astrophysical Journal, 546, 469–483. https://doi.org/10.1086/318234, arXiv:astro-ph/0003163.
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Pnigouras, P. (2018). Mode Coupling: Quadratic Perturbation Scheme. In: Saturation of the f-mode Instability in Neutron Stars. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-98258-8_4
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