Tayler Instability: Stars & Liquid Metals

• Tayler instability is a current-driven, non-axisymmetric kink instability in toroidal magnetic fields, observed in both stellar and laboratory contexts.
• Its linear growth occurs on the Alfvén timescale, while rotation, stratification, and diffusivity significantly modify its dynamic evolution.
• The instability plays a key role in angular momentum transport in stars and poses design challenges in liquid-metal battery technology.

The Tayler instability is a current-driven, non-axisymmetric instability of predominantly toroidal magnetic fields, usually manifesting as an $m=1$ kink of azimuthal field lines. It arises in systems as diverse as stellar radiative zones, where differential rotation winds weak poloidal fields into toroidal configurations, and liquid-metal columns carrying strong axial currents. In stellar work it is closely associated with the Tayler–Spruit dynamo paradigm and with angular-momentum transport in radiative interiors; in laboratory and technological settings it appears in low-${\rm Pm}$ liquid metals and constrains the design of liquid-metal batteries (Asatiani et al., 6 Jul 2025, Weber et al., 2012, Weber et al., 2014).

1. Definition, field geometry, and ideal criteria

In stellar applications the instability is usually formulated for a toroidal field $B_\phi(r,\theta)$ embedded in a stably stratified fluid. A representative toroidal background used in Ap-star calculations is

$$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$

where $\Omega_{\rm A}$ sets the field amplitude through an Alfvén angular frequency (Kitchatinov et al., 2020). In cylindrical liquid-metal configurations the corresponding basic state is the azimuthal field generated by a homogeneous axial current, often approximated by $B_\phi \propto R$ (Weber et al., 2012).

The dominant unstable disturbance is typically the non-axisymmetric $m=1$ kink mode. In stellar language this is a sideways displacement of toroidal loops; in pinch language it is the familiar kink of a current channel. The instability extracts energy from the electric current or, equivalently, from the free magnetic energy stored in the toroidal field (Kitchatinov et al., 2020, Ruediger et al., 2012).

For ideal cylindrical configurations the classical Tayler criterion is commonly written as an instability condition for non-axisymmetric modes,

$$\frac{\partial}{\partial r}\bigl(r B_\varphi^2(r)\bigr) > 0,$$

which is satisfied by the $B_\varphi \propto r$ profile generated by a homogeneous axial current (Seilmayer et al., 2011, Weber et al., 2012). In more general stellar energy-principle formulations the criterion is expressed through the sign structure of quadratic forms in the displacement field. For a purely toroidal field, Tayler’s conditions can be written in terms of coefficients $a_m,b_m,c_m$, with instability present when one of

${\rm Pm}$0

is satisfied somewhere in the star (Gusakov et al., 23 Jun 2026).

The linear growth rate is of order the Alfvén frequency associated with the toroidal field. In stellar-evolution applications this is often summarized schematically as

${\rm Pm}$1

with rotation and stratification reducing the effective rate (Asatiani et al., 6 Jul 2025). In non-rotating laboratory columns two asymptotic regimes are distinguished: a weak-field, high-diffusivity regime with growth quadratic in field strength, and a strong-field regime with ${\rm Pm}$2 (Ruediger et al., 2012).

2. Rotation, stratification, and admissible mode structure

Stable stratification is central to the stellar form of the instability. In radiative zones the Brunt–Väisälä frequency ${\rm Pm}$3 suppresses vertical motion and drives the instability toward short radial or vertical length scales, so that perturbations are predominantly horizontal (Ji et al., 2022, Guerrero et al., 2019). In one spherical linear treatment the control parameter is

${\rm Pm}$4

and the measured nonlinear growth rates obeyed ${\rm Pm}$5 in the strong-stratification regime, consistent with the linear prediction ${\rm Pm}$6 (Guerrero et al., 2019). This strong suppression at large ${\rm Pm}$7 has been proposed as one ingredient in explaining why weak toroidal fields can be only marginally unstable in strongly stratified stars (Guerrero et al., 2019).

Rotation introduces Coriolis stabilization and, in many stellar analyses, modifies the mode frequency as much as the growth rate. For sub-equipartition fields in Ap-star models, the drift frequency in the co-rotating frame is counter-rotational and the growth rate scales as ${\rm Pm}$8; for stronger fields the growth approaches ${\rm Pm}$9 (Kitchatinov et al., 2020). In the inertial frame this can make the magnetic pattern rotate much more slowly than the stellar matter itself when $B_\phi(r,\theta)$0, a feature used to explain super-slowly rotating Ap stars (Kitchatinov et al., 2020).

An important formal issue is that the instability criterion depends on the dynamical approximation. In fully compressible MHD, Tayler’s original energy principle gives criteria in terms of coefficients $B_\phi(r,\theta)$1. In the Lantz–Braginsky–Roberts anelastic approximation, however, the admissible displacements must satisfy the anelastic constraint, and the stability conditions become

$B_\phi(r,\theta)$2

Some equilibria unstable in the fully compressible case are stable in the anelastic case (Goldstein et al., 2018). This is not a minor technicality: it changes the linearly unstable parameter space used in many stellar simulations.

3. Nonlinear saturation, helicity, and dynamo relevance

The nonlinear saturation of the Tayler instability remains one of its central unresolved problems. Low-$B_\phi(r,\theta)$3 liquid-metal simulations show that the primary $B_\phi(r,\theta)$4 mode nonlinearly excites $B_\phi(r,\theta)$5 and $B_\phi(r,\theta)$6 components; in that regime the quadratic combination of the $B_\phi(r,\theta)$7 perturbations produces velocity components that suppress further growth, while the turbulent $B_\phi(r,\theta)$8 effect remains nearly constant (Weber et al., 2015, Stefani et al., 2016). At $B_\phi(r,\theta)$9 and $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$0, the instability saturates after the $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$1 and $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$2 mode energies become comparable to the $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$3 energy (Stefani et al., 2016).

The same low-$$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$4 work shows spontaneous chiral symmetry breaking during the exponential phase: the system selects one helicity sign, producing a nonzero $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$5 effect. In the saturated state, however, two regimes appear. For very low Lundquist number the final state tends to restore zero net helicity, whereas at higher supercriticality the saturated state exhibits helicity oscillations around zero; for Lundquist numbers of order unity, chiral symmetry breaking can persist in the saturated magnetic field (Weber et al., 2015). This distinction matters because the Tayler–Spruit dynamo requires a finite $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$6 effect to regenerate poloidal field.

Three-dimensional rotating, stratified simulations in cylindrical geometry have clarified a different regime, closer to stellar ordering. There the linear phase is again dominated by $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$7, but the simulated nonlinear saturation appears to be caused by secondary shear instabilities. For the parameter range reached numerically, the instability generated enough horizontal shear that a criterion of the form $$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$8 was met near saturation (Ji et al., 2022). The same simulations also showed amplification of an axisymmetric poloidal magnetic field, which suggests that Tayler instability can contribute to dynamo action in radiative regions (Ji et al., 2022).

Whether this shear-mediated saturation is the relevant stellar mechanism is disputed. One line of argument is that real stars operate at much larger scale separation than present simulations, so saturation should instead occur through Alfvénic turbulence or weak magnetic-wave damping rather than secondary shear instabilities (Ji et al., 2022). This suggests that transport coefficients calibrated in one numerical regime should not be transferred uncritically to stellar-evolution calculations.

4. Stellar radiative zones, angular-momentum transport, and chemical mixing

In one-dimensional stellar-evolution work the Tayler instability is generally not evolved as an explicit magnetic field but encoded through effective transport coefficients. In Geneva models using shellular rotation, angular momentum satisfies

$$\vec{B} = \vec{e}_\phi \sqrt{4\pi\rho}\,\sin\theta\cos\theta\, r\,\Omega_{\rm A},$$9

with meridional circulation treated as advection and shear or magnetic transport entering through $\Omega_{\rm A}$0 (Asatiani et al., 6 Jul 2025). In magnetic models adopting the asteroseismically calibrated prescription of Eggenberger et al. (2022), Tayler transport strongly flattens $\Omega_{\rm A}$1, suppresses $\Omega_{\rm A}$2, and modifies meridional circulation; chemical transport remains

$\Omega_{\rm A}$3

so the magnetic influence on abundances is indirect (Asatiani et al., 6 Jul 2025).

This formulation has an observational consequence in massive B stars because boron is destroyed at relatively low internal temperatures, $\Omega_{\rm A}$4 K, not far below the surface. In models without magnetic transport, strong differential rotation leads to overly efficient shear mixing and excessive boron depletion. Including Tayler-driven angular-momentum transport produces much flatter rotation profiles and better agreement with observed boron abundances, projected rotational velocities, and evolutionary states of moderate and fast rotators (Asatiani et al., 6 Jul 2025). The same comparison also indicates a limitation: for slow rotators, current Tayler prescriptions may overestimate angular-momentum transport and predict too little boron depletion (Asatiani et al., 6 Jul 2025).

The instability has also been used to interpret surface magnetic and abundance patterns in Ap/Bp stars. For super-slowly rotating Ap stars, the non-axisymmetric eigenmodes of the Tayler instability drift in the counter-rotational direction in the co-rotating frame, and for $\Omega_{\rm A}$5 the inertial-frame pattern can rotate very slowly; this has been proposed as an explanation of magnetic periods far longer than the stellar spin period (Kitchatinov et al., 2020). In 56 Ari, numerical calculations with $\Omega_{\rm A}$6 yield $\Omega_{\rm A}$7 and a retrograde pattern drift consistent with the observed secular period increase of $\Omega_{\rm A}$8–$\Omega_{\rm A}$9 s per $B_\phi \propto R$0 yr, while the surface geometry of the unstable $B_\phi \propto R$1 mode reproduces a dipole with $B_\phi \propto R$2 (Potravnov et al., 11 Apr 2025). These applications use the drift of the non-axisymmetric pattern, not a change in the matter’s angular velocity, as the observable.

5. Laboratory realization and liquid-metal technology

The instability has been realized in liquid-metal columns, where an axial current produces the toroidal field. The standard control parameter is the Hartmann number,

$B_\phi \propto R$3

which measures the ratio of Lorentz to viscous forces (Weber et al., 2014). For non-rotating GaInSn columns in the wide-gap limit, the critical threshold is $B_\phi \propto R$4, corresponding to $B_\phi \propto R$5 and $B_\phi \propto R$6 G (Ruediger et al., 2012). A closely related linear study for a 10 cm diameter gallium column found a minimum current of $B_\phi \propto R$7 and a growth time of $B_\phi \propto R$8 s for $B_\phi \propto R$9 (Ruediger et al., 2010). Experiments using up to $m=1$0 and 14 fluxgate sensors have detected the expected non-axisymmetric $m=1$1 pattern and found growth rates in reasonable agreement with numerical predictions (Seilmayer et al., 2011).

In the accessible low-conductivity regime the non-rotating growth rate obeys

$m=1$2

with $m=1$3 for wide gaps and small $m=1$4 (Ruediger et al., 2012). This regime also shows an unusual independence of the growth rate from the overall system size (Ruediger et al., 2012). Rotation suppresses the instability, modifying the weak-field scaling to $m=1$5 and raising the critical field (Ruediger et al., 2010, Ruediger et al., 2012).

In liquid-metal batteries the same instability is technologically important because large axial currents generate the unstable toroidal field in the stratified metal layers. The current-driven kink can threaten the integrity of the electrolyte layer, which makes TI a size-limiting factor for large cells (Stefani et al., 2010, Weber et al., 2014). Two mitigation strategies emerge directly from Tayler’s criterion. One is to introduce a hollow central bore, which raises the critical current. The more effective one is to return the current through a central conductor, so that the field profile becomes

$m=1$6

with a choice of axial current satisfying $m=1$7; in the ideal analysis this removes the instability entirely (Stefani et al., 2010). Realistic current collectors and feeding lines also matter because they introduce electro-vortex flows that can coexist with, or even suppress, Tayler modes depending on collector height and conductivity ratio (Weber et al., 2014).

6. Open problems, competing formulations, and extensions

Several theoretical controversies remain. First, the correct transport efficiency in stars is unresolved. The original Spruit scaling, Fuller et al. (2019), and the asteroseismically calibrated Eggenberger et al. (2022) prescription imply different effective viscosities; in B-star boron studies, however, surface abundances distinguish mainly between efficient and inefficient angular-momentum transport rather than among the various magnetic scalings (Asatiani et al., 6 Jul 2025). This suggests that abundance data alone do not yet uniquely identify the correct Tayler prescription.

Second, the proper dynamical framework is unsettled. Fully compressible criteria and anelastic criteria are not equivalent, and some equilibria unstable in the former are stable in the latter (Goldstein et al., 2018). Likewise, nonlinear low-$m=1$8 simulations show hydrodynamic saturation routes and helicity oscillations (Weber et al., 2015), whereas rotating, stratified simulations at more stellar ordering suggest dynamo-relevant axisymmetric poloidal amplification and challenge the fast-dissipation assumption of older transport models (Ji et al., 2022). These are not mutually exclusive results; they pertain to different parameter regimes.

Third, the classical Tayler instability may not exhaust the spectrum of toroidal-field instabilities. A complementary current-driven instability has recently been proposed for non-rotating, stably stratified stars. In ideal MHD it grows on the Alfvén timescale, is intrinsically global in angle, and can manifest as shellular differential rotation about an axis perpendicular to the magnetic symmetry axis. Its radial wavenumbers satisfy

$m=1$9

or, more explicitly, $$\frac{\partial}{\partial r}\bigl(r B_\varphi^2(r)\bigr) > 0,$$0, whereas the classical Tayler instability occupies the complementary high-$$\frac{\partial}{\partial r}\bigl(r B_\varphi^2(r)\bigr) > 0,$$1 range $$\frac{\partial}{\partial r}\bigl(r B_\varphi^2(r)\bigr) > 0,$$2 (Gusakov et al., 23 Jun 2026). Because diffusive damping scales as $$\frac{\partial}{\partial r}\bigl(r B_\varphi^2(r)\bigr) > 0,$$3, this global mode may prevail over the Tayler instability in some dissipative regimes (Gusakov et al., 23 Jun 2026). This suggests that “Tayler-driven” transport in stellar evolution may, in part, be a shorthand for a broader family of toroidal-field instabilities.

Finally, observational inference remains indirect. Asteroseismic rotation profiles, boron depletion, projected rotation rates, photometric drift periods in Ap/Bp stars, and liquid-metal experiments all probe different aspects of the same instability class, but no single observable yet fixes the nonlinear saturation law, the relevant mean-field coefficients, or the relative importance of local kink modes versus global toroidal-field instabilities (Asatiani et al., 6 Jul 2025, Kitchatinov et al., 2020, Ji et al., 2022, Gusakov et al., 23 Jun 2026). The present state of the subject therefore combines robust linear physics with active debate over saturation, transport coefficients, and the correct reduced description in stellar interiors.