Breakdown of chiral symmetry during saturation of the Tayler instability

We study spontaneous breakdown of chiral symmetry during the nonlinear evolution of the Tayler instability. We start with an initial steady state of zero helicity. Within linearized perturbation calculations, helical perturbations of this initial state have the same growth rate for either sign of helicity. Direct numerical simulations (DNS) of the fully nonlinear equations, however, show that an infinitesimal excess of one sign of helicity in the initial perturbation gives rise to a saturated helical state. We further show that this symmetry breaking can be described by weakly nonlinear finite-amplitude equations with undetermined coefficients which can be deduced solely from symmetry consideration. By fitting solutions of the amplitude equations to data from DNS, we further determine the coefficients of the amplitude equations.

Figure 1

The phase portrait for $\mu <{\mu }_{*}$. This is the typical situation in which $S_2$ and $S_3$ are attractive and $S_4$ is a saddle point.

Figure 2

The dispersion relation for the dimensionless growth rate $\Gamma$ for the $m=\pm 1$ mode (solid line) and for the $m=\pm 2$ mode (dashed). Higher values of $\left|m\right|$ have even smaller growth rates. This curve is obtained for a linear model with physical parameters corresponding to the nonlinear model $\mathtt{\text{Held}}$, for which we indicate, with a rhombus, the growth rate for its faster growing mode.

Figure 3

Eigenfunction $v_{1s}$ for the $m=1$ mode for $q=16$. The result of the simulation, model $\mathtt{\text{Held}}$ in Table (solid line), is overplotted on the eigenfunction obtained solving (20) (dashed line); see [13] for more details. This is observed at $t/t_A=9$, that is, during the linear growth of the instability.

Figure 4

Kinetic, current, and magnetic helicities for two different runs (models $\mathtt{\text{Hel}}$ and $\mathtt{\text{Helm1}}$ in Table ) with helical perturbation and $m=\pm 1$. $t$ is in units of the Alfvén travel time $t_A$. The viscous time is $t_{\nu }\approx {10}^2*t_A$ and the magnetic diffusion is $t_{\eta }\approx {10}^9*t_A$. The difference in the evolutions of the kinetic, current, and magnetic helicities can be clearly seen in the first, second, and third panel. These plots show how these quantities grow with the same rate, but different sign, depending on the sign of the initial perturbation, that is, the sign of $m$. Note that for each model, the magnetic helicity has an opposite sign to that of the kinetic and current helicities.

Figure 5

Time evolution for the logarithmic derivative of kinetic energy (solid line) $E$ and kinetic helicity $H$ (dashed) as measured in DNS for models $\mathtt{\text{Hel}}$ and $\mathtt{\text{Helm1}}$ (see Table ). $t$ is in units of the Alfvén travel time $t_A$. We overplot a fit of the model with Eqs. (13). The best fit is obtained for $\gamma =2.71/t_A$, $\mu =7.5t_A/s_{\mathrm{out}}^2$, and ${\mu }_{*}=18t_A/s_{\mathrm{out}}^2$ and the solutions are overplotted on the DNS results.