[Paper Review] Magnetic instability in a differentially rotating star

H. C. Spruit|arXiv (Cornell University)|Jul 9, 2006

Astro and Planetary Science3 citations

TL;DR

This paper investigates magnetic instabilities in differentially rotating stars, particularly focusing on a newly identified instability in compositional gradients (μ-gradient) driven by azimuthal magnetic fields. It proposes that such instabilities may lead to layer formation—steep compositional and rotational steps—enhancing angular momentum transport and reducing final stellar rotation rates, though the instability requires strong horizontal field gradients and is unlikely near poles.

ABSTRACT

Instabilities in magnetic fields wound up by differential rotation as reviewed in Spruit (1999) are discussed with some detail and new developments added. In stellar models which include magnetic torques, the differential rotation tends to accumulate in the gradients in composition. In view of this, instability in a $μ$-gradient is studied in more detail here, resulting in the detection of a second instability. Its relevance for angular momentum transport is uncertain, however, since it requires high horizontal field gradients and would not operate near the pole. Finally, the possibility is discussed that magnetic instability in a $μ$-gradient will lead to {\em layer formation}: the gradients breaking up into small steps of uniform composition and rotation rate. This would enhance the angular momentum transport across inhomogeneous zones, and decrease the rotation rates of the end products of stellar evolution. A recent { t astro-ph} submission by Dennisenkov and Pinsonneault proposing a modification of the instability conditions is shown to contain a mathematical error.

Motivation & Objective

• To analyze the stability of azimuthal magnetic fields in stellar interiors with compositional gradients, particularly in differentially rotating stars.
• To investigate whether magnetic instabilities in μ-gradients can drive angular momentum transport more effectively than standard models.
• To assess the potential for nonlinear layer formation in compositional gradients due to magnetic instabilities.
• To evaluate the implications of such layering for stellar evolution, especially in reducing final rotation rates of evolved stars.
• To correct a recent claim by Denshchikov and Pinsonneault regarding instability conditions, identifying a mathematical error in their derivation.

Proposed method

• Uses a linear stability analysis based on the dispersion relation from Acheson (1978), adapted for differentially rotating stars with compositional and thermal stratification.
• Applies the framework from Spruit (1999) to derive instability conditions for azimuthal magnetic fields in μ-gradients, incorporating thermal, magnetic, and compositional diffusion.
• Derives a new instability criterion (Eq. 40) for oscillatory modes in μ-gradients, showing dependence on field gradient index and diffusivity ratios.
• Analyzes the absence of a minimum wavenumber or field strength for instability due to cancellation of h²/f² terms, though viscous effects may restore such a minimum.
• Proposes a nonlinear mechanism—layer formation—where μ-gradients break into steps of uniform composition and rotation, enabling enhanced magnetic diffusion and angular momentum transport.
• Compares the proposed mechanism to double-diffusive convection and semiconvection, suggesting analogies in subcritical, nonlinear behavior.

Experimental results

Research questions

• RQ1 Can magnetic instabilities in azimuthal fields drive angular momentum transport in stars with compositional gradients?
• RQ2 What are the stability conditions for such instabilities in μ-gradients, and how do they differ from those in thermally stratified regions?
• RQ3 Does the absence of a minimum field strength in the instability criterion imply a viable mechanism for sustained dynamo action in μ-gradients?
• RQ4 Can nonlinear evolution of the instability lead to the formation of layered structures in composition and rotation rate?
• RQ5 What are the implications of such layering for angular momentum transport and final rotation rates in evolved stars like white dwarfs and neutron stars?

Key findings

• A new oscillatory instability is identified in μ-gradients, requiring a steeper field gradient than in thermally stratified cases, with instability condition (40) depending on the ratio of magnetic to compositional diffusivity.
• The instability does not require a minimum field strength or wavenumber due to cancellation of h²/f² terms, though viscosity may impose a lower bound.
• The instability is unlikely to operate near the poles due to the field gradient index p=1, which is stable under the derived criterion.
• Layer formation in μ-gradients is proposed as a nonlinear outcome, where the gradient breaks into steps of uniform composition and rotation, reducing the stabilizing effect of composition gradients.
• Layered structures could enhance angular momentum transport by enabling magnetic diffusion across steep boundaries, potentially explaining observed slow rotation in white dwarfs and neutron stars.
• The paper corrects a recent claim by Denshchikov and Pinsonneault, showing their instability condition contains a mathematical error.