Nonlinear decay of $r$ modes in rapidly rotating neutron stars

We investigate the dynamics of $r$ modes at amplitudes in the nonlinear regime for rapidly but uniformly rotating neutron stars with a polytropic equation of state. For this, we perform three-dimensional relativistic hydrodynamical simulations, making the simplifying assumption of a fixed spacetime. To excite $r$ modes, we linearly scale exact eigenfunctions to large amplitudes. We find that for initial dimensionless amplitudes around three, $r$ modes decay on time scales around ten oscillation periods, while at amplitudes of order unity, they are stable over the evolution time scale. Together with the decay, a strong differential rotation develops, conserving the total angular momentum, with angular velocities in the range $0.5\dots 1.2$ of the initial one. We evolved two models with different rotation rates and found slower decay for the more rapidly rotating one. We present $r$-mode eigenfunctions and frequencies, and compare them to known analytic results for slowly rotating Newtonian stars. As a diagnostic tool, we discuss conserved energy and angular momentum for the case of a fixed axisymmetric background metric and introduce a measure for the energy of nonaxisymmetric fluid oscillation modes.

Figure 1

Two-dimensional eigenfunctions $\widehat{ϵ}(\vartheta ,z)$ (left half) and ${\widehat{v}}^a(\vartheta ,z)$ (right half), belonging to the $r$ mode of model MB85.

Figure 2

Perturbations corresponding to the $r$ mode of model MB85, as longitude/latitude plot at fixed coordinate radius $r=R_e/2$, with $R_e$ being the equatorial coordinate radius. The velocity perturbations $\delta v^{\varphi }$, $\delta v^{\theta }$ are plotted on the left half, and the specific energy perturbation $\delta ϵ$ on the right half. Note the patterns have a 180° periodicity in $\varphi$.

Figure 3

Decomposition of the $r$-mode velocity perturbation of model MB85 into vector spherical harmonics. The vertical lines mark the polar and equatorial radius.

Figure 4

Development of differential rotation, for model MB85 perturbed with an $r$ mode of initial amplitude $A=3$. Shown is the profile during the $r$-mode decay (LEFT) and when the $r$ mode is almost completely gone (RIGHT); compare also Fig. 8. Color coded (with identical scales) is the $\varphi$-averaged angular velocity in units of the angular velocity of the unperturbed model. The surface of the unperturbed star is marked by a thin dotted line, while the surface at which the $\varphi$-averaged density falls below 0.005 times the central density is marked by a thin solid line. A small region near the axis has been excluded to avoid problems when transforming from the Cartesian grid.

Figure 5

Time evolution for the same setup as Fig. 4 of the $\varphi$-averaged angular velocity at different sample points in the star. Also shown is the global average angular momentum change defined by Eq. (30).

Figure 6

Snapshots at different times of the $\varphi$-averaged angular velocity in the equatorial plane versus coordinate radius, for the same setup as Fig. 4. The lines end where the $\varphi$-averaged density falls below 0.005 times the central one.

Figure 7

Convergence of differential rotation. Shown is the $\varphi$-averaged change of angular velocity after 5 ms for an $r$ mode of model MB85 with initial amplitude $A=3$, at different grid resolutions.

Figure 8

Decay of the $r$-mode amplitude $A$ defined by Eq. (8) for model MB85 perturbed with different initial amplitudes. The two curves with initial amplitudes of $A=2.48$ and $A=2.51$ highlight the steep dependence of the decay time on the initial amplitude.

Figure 9

Decay of the $r$-mode amplitude at different initial amplitudes, for model MB70.

Figure 10

Energy budget for a decaying $r$ mode of model MB85 with initial amplitude $A=3$. Shown is the loss of total conserved energy $E$ defined by Eq. (22), as well as the loss of energy in the $r$ mode, computed by estimating the amplitude from the current multipole moment $J_{22}$ and the mode energy from Eq. (26). For the points labeled “Phase ignored,” we took the absolute value of the complex integrand when computing $J_{22}$.

Figure 11

Convergence of $r$-mode amplitude (a) and conserved energy (b), for model MB85 perturbed with an $r$ mode of initial amplitude $A=3$.

Figure 12

Time development of the $\varphi$-integrated $l=m=2$ current multipole integrand for a nonlinearly decaying $r$ mode of model MB85, with initial amplitude $A=3$. (a)–(c) Absolute value at different times. (d) Complex phase difference to the equator, near the end of the decay. The phase was computed such that the usual phase jump from $-\pi$ to $\pi$ is avoided, keeping the phase continuous.

Figure 13

Fourier spectra of the time evolution of $v^{\theta }$, $v^r$ at a sample point $x=y=z=0.24R$ of model MB85, during the $r$-mode decay with initial amplitude $A=3$. Shown are two spectra corresponding to the first and second half of the evolution. For comparison, vertical lines mark the known frequencies of the $r$ mode, the quasiradial $F$ mode, and the axisymmetric $f$ mode. The shaded area marks the transitional band between inertial and pressure modes (see main text).