Tayler-Spruit Dynamo

• The Tayler-Spruit Dynamo predicts magnetic field generation in differentially rotating, stratified astrophysical fluids.
• Research demonstrates its significance in forming proto-neutron star magnetars and aiding angular momentum transport.
• Recent simulations provide scaling laws for magnetic amplitudes, crucial for stellar evolution models.

The Tayler–Spruit dynamo is a theoretical and numerically validated mechanism for magnetic field generation and angular momentum transport in differentially rotating, stably stratified astrophysical fluids—most notably in stellar radiative zones and proto-neutron stars. Its key feature is the closure of the dynamo loop by the Tayler (kink) instability, which arises in predominantly toroidal magnetic fields wound up from poloidal seeds by differential rotation. The Tayler instability is inherently non-axisymmetric, and its emergence in regions of strong stratification and rotation yields large-scale, often toroidal-dominated, magnetic fields whose amplitudes, topology, and transport effects are governed by precise scaling relations derived in recent direct numerical simulations (Barrère et al., 2024). These properties make the Tayler–Spruit dynamo essential for understanding angular momentum coupling in stars, magnetar formation, and for calibrating magnetic prescriptions in stellar evolution simulations.

1. Physical Mechanism: Differential Rotation, Stratification, and Tayler Instability

The Taylor–Spruit dynamo operates where differential rotation (shear $q \equiv r\,\partial_r\ln\Omega$) winds up weak poloidal magnetic fields into toroidal fields. As the toroidal field ($B_\phi$) strengthens, it becomes susceptible to the Tayler instability—a current-driven, $m=1$ non-axisymmetric “kink” mode that rapidly induces fluctuations and can regenerate poloidal field components, thus closing the dynamo loop. The stabilizing influence of stratification is quantified by the Brunt–Väisälä frequency $N$, which restricts vertical displacements and the effective radial length scale to $\ell_{\mathrm{TI}} \lesssim r\omega_A/N$, with $\omega_A = B_\phi/\sqrt{4\pi\rho}\,r$ as the local Alfvén frequency. Thermal and magnetic diffusion further reduce the effective stabilizing frequency to $N_{\mathrm{eff}} = N \sqrt{\eta/\kappa}$.

This dynamo operates under Boussinesq MHD, with dimensionless numbers: shell aspect ratio $\chi = r_i/r_o=0.25$, Ekman $E=\nu/(d^2\Omega_o)=10^{-5}$, magnetic Prandtl $Pm=\nu/\eta = 1$--$4$, thermal Prandtl $Pr=\nu/\kappa=0.1$, and the stratification ratio $N/\Omega_o$ varied from $0.1$ to $10$ (Barrère et al., 2024).

2. Saturated Field Strengths and Scaling Laws

Rigorous 3D-MHD simulations (MagIC code; resolution $[257,256,512]$) have confirmed the scaling laws of Fuller et al. (2019) for magnetic field amplitudes in the saturated dynamo state. For local shear $q$, rotation $\Omega_o$, and effective stratification $N_{\mathrm{eff}}$, the axisymmetric toroidal and poloidal fields follow:

• $$B_\phi^{m=0} \simeq \alpha\, (4\pi\rho)^{1/2} r\, \Omega_o\, [ q\Omega_o / N_{\mathrm{eff}} ]^{1/3}$$
• $$B_\mathrm{pol}^{m=0} \sim B_\mathrm{dip} \simeq \alpha^2\, (4\pi\rho)^{1/2} r\, \Omega_o\, [ q^2\Omega_o^5 / N_{\mathrm{eff}}^5 ]^{1/3}$$

The best-fit dimensionless normalization in the simulations is $\alpha \simeq 1.7 \times 10^{-2}$, with $\alpha^2 \sim 10^{-3}$--$10^{-4}$ for poloidal/dipole components. These values imply significantly reduced dynamo efficiency compared to earlier analytic models (Barrère et al., 2024).

For a proto-neutron star ($r_o=12~\mathrm{km}$, $\rho=4\times10^{14}~\mathrm{g~cm}^{-3}$, $N=10^3~\mathrm{s}^{-1}$) and surface periods $P=1$--$10~\mathrm{ms}$, the dynamo generates:

• Toroidal fields $B_\phi^{m=0} \simeq 1.2$--$2\times10^{15}~\mathrm{G}$ for $P=1~\mathrm{ms}$,
• Dipolar fields $B_\mathrm{dip} \simeq 1.4$--$3.0\times10^{13}~\mathrm{G}$ for $P=1~\mathrm{ms}$,
• For $P \lesssim 6~\mathrm{ms}$, classical magnetar-like dipoles $B_\mathrm{dip} \gtrsim 4\times10^{13}~\mathrm{G}$ are produced (Barrère et al., 2024).

3. Dynamo Regimes: Stationary, Intermittent, and Subcriticality

The Tayler–Spruit dynamo exhibits various magnetic regimes:

• Self-sustained stationary dynamos exist up to $N/\Omega_o \simeq 1$ at $Pm=1$, pushing to $N/\Omega_o \simeq 4$ at higher $Pm$, and transient solutions up to $N/\Omega_o \simeq 10$ at $Pm=4$. The mechanism remains active even where linear hydrodynamic stability would suppress field generation (“subcriticality”).
• Intermittent dynamos occur at $N/\Omega_o \gtrsim 2$ ($Pm=2,4$), where non-axisymmetric energy and axisymmetric poloidal field oscillate by factors $\gtrsim 10^2$ with period $3$--$30~\mathrm{s}$ and duty cycles $\alpha_{\mathrm{cyc}} \sim 0.4$--$0.7$. Spectral analysis shows energy in large-scale $m\neq 0$ modes is suppressed by $\sim 100$ compared to axisymmetric ($m=0$) $B_\phi$ (Barrère et al., 2024).

4. Angular Momentum Transport and Prescription Calibration

Angular momentum transport is mediated by Maxwell and Reynolds torques:

• Maxwell torque: $$T_M \sim r^2\,\Omega_o^2\,q\,(\Omega_o/N_{\mathrm{eff}})^2$$,
• Reynolds torque: $$T_R \sim r^2\,\Omega_o^2\,q^{5/3}\,(\Omega_o/N_{\mathrm{eff}})^{10/3}$$.

Simulations reveal the normalization factor for the Fuller et al. angular momentum transport (AMT) prescription is $\alpha^3 \sim 10^{-6}$, leading to AMT rates that are six orders of magnitude smaller than unity. Implications are that Tayler–Spruit induced coupling is weak for moderate field strengths, and additional processes may be needed to match observed stellar core rotation rates (Barrère et al., 2024).

5. Key Equations, Stability Criteria, and Implementation

The Boussinesq MHD system (in frame rotating at $\Omega_o$): $\nabla \cdot \mathbf{v} = 0,$

$$D_t\mathbf{v} = -\frac{1}{\rho}\nabla p' - 2\Omega_o\times\mathbf{v} - N^2\,\Theta\,\mathbf{e}_r + \frac{1}{4\pi\rho}(\nabla\times\mathbf{B})\times\mathbf{B} + \nu\nabla^2\mathbf{v},$$

$$D_t\Theta = \kappa \nabla^2 \Theta, \quad \partial_t\mathbf{B} = \nabla\times(\mathbf{v}\times\mathbf{B}) + \eta\nabla^2\mathbf{B}, \quad \nabla\cdot\mathbf{B}=0,$$

where the buoyancy variable obeys $\Theta \sim -(g/N^2)\,\rho'/\rho$.

The Tayler $m=1$ kink-mode stability criterion (Goossens & Tayler 1980): $$B_\phi^2\,(1-2\cos^2\theta) - \sin\theta\cos\theta\,\frac{\partial B_\phi^2}{\partial \theta} > 0$$ identifies magnetostatically stable regions (Barrère et al., 2024).

6. Astrophysical Implications: Magnetar Formation and Stellar Interiors

Numerical evidence confirms that the Tayler–Spruit dynamo can amplify poloidal and toroidal fields to magnetar-strength in proto-neutron stars spun up by fallback, spanning $B_\mathrm{dip}$ fields in the $10^{13}$--$10^{15}$ G range and $B_\phi$ up to $2\times10^{16}$ G for $N=10^3~\mathrm{s}^{-1}$, $P=1$--$10~\mathrm{ms}$ (Barrère et al., 2024). This supports fallback-driven magnetar formation scenarios and provides constraints for magnetic field evolution in compact objects.

In stellar radiative zones, the dynamo governs angular momentum transport and enforces rotational coupling, but the reduced normalization implies that transport alone may be insufficient to match asteroseismic observations without recalibration or additional processes.

7. Research Frontiers and Calibration Directions

The reduction in dynamo efficiency from earlier analytic estimates ($\alpha \ll 1$) is now well established via direct numerical simulation. The existence of intermittent dynamo behavior and subcriticality, as well as the precise scaling exponents for field amplitudes and torque, are critical for refining 1D stellar evolution models and population synthesis. Future directions involve exploring the interplay between compositional gradients, multi-mode symmetry breaking, and non-stationary regimes, as revealed in recent high-resolution simulations.

The Tayler–Spruit dynamo provides a physically constrained, numerically validated pathway for magnetic field amplification and angular-momentum transport in stratified, differentially rotating astrophysical plasmas, underpinning modern models of stellar magnetism and magnetar birth (Barrère et al., 2024).