# Linear and nonlinear evolution of the vertical shear instability in accretion discs

**Authors:** Richard P. Nelson, Oliver Gressel, Orkan M. Umurhan  
**arXiv:** [1209.2753](https://arxiv.org/abs/1209.2753)  
**Date:** 2012-09-12  
**Summary page:** https://www.summarizepaper.com/en/paper/1209.2753v1/

## Key points

- The study investigates stability and evolution of power-law accretion disc models
- Two different hydrodynamic codes were used to perform 2D-axisymmetric and 3D simulations
- A stability analysis was presented to interpret the simulation results
- Discs whose temperature or entropy profiles cause their equilibrium angular velocity to vary with height are unstable to modes with wavenumber ratios |k_R/k_Z| >> 1; this instability is known as Goldreich-Schubert-Fricke (GSF) or 'vertical shear' linear instability
- This instability excites vertical breathing and corrugation modes in the disc which become a feature in its nonlinear saturated state
- It operates when dimensionless disc kinematic viscosity nu < 10^{-6} (Reynolds numbers Re > 2500)
- In 3D simulations it generates a quasi-turbulent flow with Reynolds stress producing an effective viscosity coefficient whose mean value reaches alpha ~ 6 x 10^{-4} by end-of-simulation
- The implications for realistic thermal physics have yet to be examined; however, these results suggest that it will have significant consequences for their internal dynamics, transport properties, and observational appearance.
- Overall, this study provides valuable insights into how power-law accretion discs evolve over time under different conditions and highlights potential implications for their behavior and appearance.

## Summary

This study investigates the stability and evolution of power-law accretion disc models. These models have midplane densities that follow radial power-laws and temperature or entropy distributions that are power-law functions of cylindrical radius. To examine the long-term evolution of these disc models as a function of the power-law indices of the temperature or entropy, thermal relaxation time of the fluid, and viscosity, two different hydrodynamic codes were used to perform 2D-axisymmetric and 3D simulations. A stability analysis was presented to interpret the simulation results. The authors found that when either the thermodynamic response of the fluid is isothermal or when the thermal evolution time is comparable to or shorter than the local dynamical time scale, discs whose temperature or entropy profiles cause their equilibrium angular velocity to vary with height are unstable to modes with wavenumber ratios |k_R/k_Z| >> 1; this instability is known as Goldreich-Schubert-Fricke (GSF) or 'vertical shear' linear instability. This instability excites vertical breathing and corrugation modes in the disc which become a feature in its nonlinear saturated state. It operates when dimensionless disc kinematic viscosity nu < 10^{-6} (Reynolds numbers Re > 2500). In 3D simulations it generates a quasi-turbulent flow with Reynolds stress producing an effective viscosity coefficient whose mean value reaches alpha ~ 6 x 10^{-4} by end-of-simulation. The implications for realistic thermal physics have yet to be examined; however, these results suggest that it will have significant consequences for their internal dynamics, transport properties, and observational appearance. Overall, this study provides valuable insights into how power-law accretion discs evolve over time under different conditions and highlights potential implications for their behavior and appearance.