Progress on the nonlinear r-mode problem

Keith Lockitch, Penn State

Since the recent discovery that the r-modes of rotating stars are unstable to the emission of gravitational waves [1], much effort has been directed towards improving the physical models of the r-mode instability. In the last issue of Matters of Gravity, Nils Andersson gave an update on some of this work [2] - reviewing such effects as neutron star superfluidity, the nonlinear evolution of the r-modes, the damping associated with the formation of the crust and the effects of general relativity on the spectrum and growth timescales of the modes. (More detailed reviews may be found in [3].) My purpose here is to report further on very recent progress that has been made specifically on the nonlinear r-mode problem.

Early work suggested that the r-mode instability may limit the spin rate of newly formed, rapidly rotating neutron stars and that the radiation emitted while the star sheds its angular momentum may be detectable by LIGO II [4]. The spin-down model on which these tantalizing estimates were based assumed that the most unstable r-mode (with multipole indices $l=m=2$) would be able to grow to an amplitude of order unity before being saturated by some sort of nonlinear process. It was also assumed that the star would spin down along a sequence of stellar models each consisting of a uniformly rotating equilibrium star perturbed by the dominant r-mode.

The central issue is whether the instability found in idealized models survives the physics that governs a young neutron star: Will nonlinear coupling to other modes allow an unstable r-mode to grow to unit amplitude? Does the background star retain a uniform rotation law as it spins down or does a growing r-mode generate significant differential rotation? The importance of this last question was emphasized by Spruit [5] and by Rezzolla, Lamb and Shapiro [6] who argued that differential rotation would wind up a toroidal magnetic field and drain the oscillation energy of the r-mode. A number of different approaches have since been applied to the nonlinear r-mode problem in an attempt to address these questions.

One notable approach is the direct numerical evolution of the nonlinear equations describing a self-gravitating fluid. Stergioulas and Font [7] have performed 3-D general relativistic hydrodynamic evolutions in the Cowling approximation, and Lindblom, Tohline and Vallisneri [8] have performed 3-D Newtonian hydrodynamic evolutions with an added driving force representing gravitational radiation-reaction.

Stergioulas and Font [7] construct an equilibrium model of a rapidly rotating relativistic star and add to it an initial perturbation that roughly approximates its $l=m=2$ r-mode. They then evolve the perturbed star using the nonlinear hydrodynamic equations with the spacetime metric held fixed to its equilibrium value (the relativistic Cowling approximation). They find no evidence for suppression of the mode on a dynamical timescale, even when the mode amplitude, $\alpha$, is initially taken to be of order unity. Because of the approximate nature of the initial perturbation, other oscillation modes are excited in the initial data. For a star with a barotropic equation of state, the generic rotationally restored mode is not a pure axial-parity r-mode, but an r-g ``hybrid'' mode with a mixture of axial and polar parity components [8]. Stergioulas and Font [7] find that a number of these hybrid modes are excited in their initial data with good agreement between the inferred frequencies and earlier results from linear perturbation theory [8]. In their published work, they find no evidence that the dominant mode is leaking its oscillation energy to other modes on a dynamical timescale. Instead, a nonlinear version of an r-mode appears to persist over the time of the run, about 25 rotations of the star. In additional runs with amplitudes substantially larger than unity, however, one no longer sees a coherent r-mode. This may be evidence of nonlinear saturation, but further runs with more accurate initial data will be necessary to conclude this definitively [10].

These conclusions are consistent with preliminary results from studies of nonlinear mode-mode couplings at higher order in perturbation theory [11,12]. Other r-modes of a nonbarotropic star seem to give no indication of a strong coupling to the $l=m=2$ r-mode unless its amplitude is unphysically large ( $\alpha
{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}} 30$!) [12]. Work is still in progress on the nonlinear coupling of the dominant r-mode to the g-modes of nonbarotropic stars [12] and to the hybrid modes of barotropic stars [11].

The results of Stergioulas and Font [7] have also been confirmed and significantly extended by the calculation of Lindblom, Tohline and Vallisneri [8]. In Stergioulas and Font's calculation the growth of the unstable r-mode does not occur because the spacetime dynamics have been turned off. However, it would be impossible to model this growth anyway even in a fully general relativistic hydrodynamic evolution, because the timescale on which the mode grows due to the emission of gravitational waves far exceeds the dynamical timescale of a rapidly rotating neutron star.

To simulate the growth of the dominant r-mode in a calculation accessible to current supercomputers, Lindblom, Tohline and Vallisneri [8] take a different approach. They begin by constructing an equilibrium model of a rapidly rotating Newtonian star and add to it a small initial perturbation corresponding to its $l=m=2$ r-mode. They then evolve the perturbed star by the equations of Newtonian hydrodynamics with a post-Newtonian radiation-reaction force that drives the current quadrupole associated with the $l=m=2$ r-mode.

By artificially scaling up the strength of the driving force, they are able to shorten the growth time of the unstable r-mode by a factor of $4500$. In the resulting simulation the mode grows exponentially from an amplitude $\alpha=0.1$ to $\alpha=2.0$ in only about 20 rotations of the star.

With this magnified radiation-reaction force, Lindblom, Tohline and Vallisneri [8] are able to confirm the general features of the simplified r-mode spin-down models [4]. In their simulation, the star begins to spin down noticeably when the amplitude of the dominant mode is of order unity, and ultimately about $40\%$ of the star's angular momentum is radiated away. The evolution of the star's angular momentum as computed numerically agrees well with the predicted angular momentum loss to gravitational radiation. If their model is accurate, however, gravitational radiation would not be emitted steadily at a saturation amplitude, but would die out after saturation and then reappear as the mode regenerates.

Again, there is no evidence of nonlinear saturation for mode amplitudes $\alpha
1$. The growth of the mode is eventually suppressed at an amplitude $\alpha\simeq 3.4$, and the amplitude drops off sharply thereafter. Lindblom, Tohline and Vallisneri argue that the mechanism suppressing the mode is the formation of shocks associated with the breaking of surface waves on the star. They find no evidence of mass-shedding, nor of coupling of the dominant mode to the other r-modes or hybrid modes of their Newtonian barotropic model.

These various studies all provide evidence pointing to the same conclusion: the most unstable r-mode appears likely to grow to an amplitude of order unity before being suppressed by nonlinear hydrodynamic processes. It is important to emphasize, however, that the 3-D numerical simulations have probed nonlinear processes occurring only on dynamical timescales and that the actual growth timescale for the r-mode instability is longer by a factor of order $10^4$. It is possible that the instability may be suppressed by hydrodynamic couplings occurring on timescales that are longer than the dynamical timescale but shorter than the r-mode growth timescale. Further work clearly needs to be done before definitive conclusions can be drawn. Particularly relevant will be the results from the ongoing mode-mode coupling studies [11,12].

Turning to the question of differential rotation, deviations from a uniform rotation law are observed in both of the 3-D numerical simulations [7,8] It has been proposed that differential rotation will be driven by gravitational radiation-reaction [5] as well as being associated with the second order motion of the r-mode, itself [6]. In a useful toy model, Levin and Ushomirsky [13] calculated an exact r-mode solution in a thin fluid shell and found both sources of differential rotation to be present.

To address in more detail the issue of whether or not the r-mode instability would generate significant differential rotation, Friedman, Lockitch and Sá [14] have calculated the axisymmetric part of the second order r-mode. We work to second order in perturbation theory with the equilibrium solution taken to be either a slowly rotating polytrope (with index $n=1$) or an arbitrarily rotating uniform density star (a Maclaurin spheroid). The first order solution, which appears in the source term of the second order equations, is taken to be a pure $l=m$ r-mode with amplitude $\alpha$.

We find that differential rotation is indeed generated both by gravitational radiation-reaction and by the quadratic source terms in Euler's equation; however, the latter dominate a post-Newtonian expansion. The functional form of the differential rotation is independent of the equation of state - the axisymmetric, second order change in $v^\varphi$ being proportional to $z^2$ (in cylindrical coordinates) for both the polytrope and Maclaurin.

Our result extends that of Rezzolla, Lamb and Shapiro [6] who computed the order $\alpha^2$ differential drift resulting from the linear r-mode velocity field. These authors neglect the nonlinear terms in the fluid equations and argue (based on an analogy with shallow water waves) that the contribution from the neglected terms might be irrelevant. Indeed, for sound waves and shallow water waves, the fluid drift computed using the linear velocity field turns out to be exact to second order [15]; thus, one may safely ignore the nonlinear terms. However, for the motion of a fluid element associated with the r-modes, we find that there is in fact a non-negligible contribution from the second-order change in $v^\varphi$. Interestingly, the resulting second order differential rotation is stratified on cylinders. It remains to be seen whether the coupling of this differential rotation to the star's magnetic field does indeed imply suppression of the r-mode instability.


[1] Andersson, N., Astrophys. J., 502, 708, (1998);
Friedman, J. L. and Morsink, S. M., Astrophys. J., 502, 714, (1998)

[2] Andersson, N., An update on the r-mode instability, MOG No. 16, (2000)

[3] Friedman, J. L. and Lockitch, K. H., Prog. Theor. Phys. Supp., 136, 121 (1999); Andersson, N. and Kokkotas, K. D., The r-mode instability in rotating neutron stars, preprint gr-qc/0010102; Lindblom, L., Neutron star pulsations and instabilities, preprint astro-ph/0101136

[4] Lindblom, L., Owen, B. J. and Morsink, S. M., Phys. Rev. Lett., 80, 4843, (1998); Andersson, N., Kokkotas, K. and Schutz B. F., Astrophys. J., 510, 846, (1999); Owen, B. J., Lindblom, L., Cutler, C., Schutz, B. F., Vecchio, A. and Andersson, N., Phys. Rev. D, 58, 084020, (1998)

[5] Spruit, H. C., Astron. and Astrophys., 341, L1, (1999)

[6] Rezzolla, L., Lamb, F.K. and Shapiro, S.L., Astrophys. J. Lett., 531, L139, (2000)

[7] Stergioulas N. and Font, J.A., Nonlinear r-modes in rapidly rotating relativistic stars, Phys. Rev. Lett., in press (2001); preprint gr-qc/0007086

[8] Lindblom, L., Tohline, J. E. and Vallisneri, M., Non-linear evolution of the r-modes in neutron stars, Phys. Rev. Lett., in press (2001); preprint astro-ph/0010653

[9] Lockitch, K. H. and Friedman, J. L., Astrophys. J., 521, 764, (1999); Lockitch, K. H., Andersson, N. and Friedman, J. L., Phys. Rev. D, 63, 024019, (2000)

[10] Stergioulas, N., private communication (2001).

[11] Schenk, A. K., Arras, P., Flanagan, É. É., Teukolsky, S. A. and Wasserman, I., Nonlinear mode coupling in rotating stars and the r-mode instability in neutron stars, preprint gr-qc/0101092

[12] Morsink, S. M., private communication, (2000)

[13] Levin, Y. and Ushomirsky, G., preprint astro-ph/0006028, (2000)

[14] Friedman, J. L., Lockitch, K. H. and Sá, P. M., in preparation (2001)

[15] Lamb, F. K., Markovic, D., Rezzolla, L. and Shapiro, S. L., private communication (1999)

Jorge Pullin