Magnetic effects on the low-$T/|W|$ instability in differentially rotating neutron stars

Dynamical instabilities in protoneutron stars may produce gravitational waves whose observation could shed light on the physics of core-collapse supernovae. When born with sufficient differential rotation, these stars are susceptible to a shear instability (the “low-$T/|W|$ instability”), but such rotation can also amplify magnetic fields to strengths where they have a considerable impact on the dynamics of the stellar matter. Using a new magnetohydrodynamics module for the Spectral Einstein Code, we have simulated a differentially-rotating neutron star in full 3D to study the effects of magnetic fields on this instability. Though strong toroidal fields were predicted to suppress the low-$T/|W|$ instability, we find that they do so only in a small range of field strengths. Below $4\times 10^{13}\mathrm{G}$, poloidal seed fields do not wind up fast enough to have an effect before the instability saturates, while above $5\times 10^{14}\mathrm{G}$, magnetic instabilities can actually amplify a global quadrupole mode (this threshold may be even lower in reality, as small-scale magnetic instabilities remain difficult to resolve numerically). Thus, the prospects for observing gravitational waves from such systems are not in fact diminished over most of the magnetic parameter space. Additionally, we report that the detailed development of the low-$T/|W|$ instability, including its growth rate, depends strongly on the particular numerical methods used. The high-order methods we employ suggest that growth might be considerably slower than found in some previous simulations.

Figure 1

Illustrations of magnetic field lines at early ($t=0$, above) and intermediate ($t=2160$, below) times. Contours represent regions of similar rest-mass density. Magnetic field lines are seeded at coordinate radii of $2{\mathrm{M}}_{\odot }$ (large vertical extent; yellow) and $4{\mathrm{M}}_{\odot }$ (small vertical extent; pink).

Figure 2

Illustration of $x\text{−}z$ slice of domain decomposition. The shaded region with a bold outline represents the initial star. The dashed rectangle represents the finite-difference domain, which has a coordinate width of $25{\mathrm{M}}_{\odot }$ and a coordinate height of $14.5{\mathrm{M}}_{\odot }$. For spectral subdomains, the actual reference grid has twice as many collocation points in each direction as are shown in the figure.

Figure 3

Consistency of the growth rate of the low-$T/|W|$ instability when using WENO5 reconstruction at various resolutions (no magnetic field is present). The black dashed line represents the approximate growth rate found by Corvino et al. for m.1.200. Results from resolutions of $\mathrm{\Delta }x\lesssim 0.2{\mathrm{M}}_{\odot }$, while not formally convergent, are in good agreement and are clearly distinct from those of Corvino et al. “SLev” indicates the spectral resolution level, with higher levels corresponding to finer resolution (the “reference” grid uses SLev 4), and grid spacings are measured in solar masses.

Figure 4

Growth and saturation of the unmagnetized low-$T/|W|$ instability as expressed in the “plus” polarization of the distortion parameter $\eta$. The “cross” polarization exhibits the same behavior with a phase shift. Compare to Corvino et al. Fig. 3.

Figure 5

Relative power of $\rho$ in azimuthal modes for $m=1–4$. Note that measurements of $m=4$ power have a noise floor of $10^{-3}$ due to the Cartesian nature of the grid.

Figure 6

Range of behavior of distortion parameter $\eta$ at different magnetic field strengths for $n_s=1$. Curves that terminate at early times developed significant outflows, making further evolution impractical on our grid.

Figure 7

Range of behavior of distortion parameter $\eta$ at different magnetic field strengths for $n_s=2$, showing same classes of behavior as when $n_s=1$ (see Fig. 6).

Figure 8

Energy exchange for three magnetic field strengths ($n_s=1$ for each case). The change in gravitational energy is inferred from the sum of the changes in the other energies.

Figure 9

Lagrangian displacement of tracer particles seeded at various cylindrical radii for an unmagnetized star. For each initial radius, 12 tracers were distributed uniformly in azimuth. The corotation radius for this system is at $\varpi \approx 4.25{\mathrm{M}}_{\odot }$.

Figure 10

Spectrograms of the quadrupole moment $I_{xy}$ for six cases. Power spectral density (PSD) estimated via FFT periodogram using Welch’s method with a Hann window.

Figure 11

Magnitude of radial component of $B$-field in the $y\text{−}z$ plane at $t=3760{\mathrm{M}}_{\odot }$ for $B_0=5\times 10^{14}\mathrm{G}$, $n_s=2$.

Figure 12

Magnitude of radial component of $B$-field vs radius vs time in the $z=1$ plane for three configurations ($n_s=1$ in all cases), illustrating the onset of turbulence. Color bars are scaled relative to the initial $B$-field strength. Plot inspired by the analysis of Franci et al. [22].

Figure 13

Power of $b^2$ in azimuthal modes for $m=1–4$. Except in the most strongly magnetized systems, the $m=4$ power does not rise above that of the ambient grid mode. The growing strength of the magnetic field is factored out by normalizing by the $m=0$ power; thus, trends shown here represent growth of the proportional power of nonaxisymmetric modes.

Figure 14

Growth of the maximum of the cylindrical components of the $B$-field for three cases: $B_0=5\times 10^{14}\mathrm{G}$, $n_s=1$ (top), $B_0=5\times 10^{14}\mathrm{G}$, $n_s=2$ (middle), and $B_0=2\times 10^{15}\mathrm{G}$, $n_s=1$ (bottom). The temporal resolution during the period of rapid growth for the last case is $10\times \text{finer}$ than our default.

Figure 15

Rest-mass density at $t=t_{\text{final}}$ for the shock tests described in Table 3, shown for two resolutions ($N=400$ and $N=4000$ points).

Figure 16

Velocity at $t=t_{\text{final}}$ for the shock tests described in Table 3, shown for two resolutions ($N=400$ and $N=4000$ points).

Figure 17

Error in the final value of $u^y$ for the wave test at 3 resolutions ($N=50$, $N=100$, $N=200$), rescaled for the expected second-order convergence.

Figure 18

Error norm for the Bondi test at three resolutions, rescaled for second-order convergence.