Non-linear spiral waves in accretion discs

TL;DR

This paper derives a non-linear equation for spiral density waves in accretion discs, revealing that non-linear spirals can avoid shocking and are often more loosely wound than linear predictions, explaining observed spiral structures.

Contribution

It introduces a simple, accurate non-linear global equation for spiral waves in accretion discs, including solutions that show spirals can be non-shocking and more loosely wound.

Findings

Non-linear spirals can avoid shocking at low amplitudes due to dispersion.

Highly non-linear spirals are more loosely wound than linear ones.

Numerical simulations support the theoretical predictions.

Abstract

We derive a simple, accurate, non-linear, global equation governing spiral density waves in thin, non-self-gravitating, inviscid accretion discs. These discs may have any slowly varying surface density or temperature profile. For specific 'self-similar' disc profiles, solutions to our equation match (novel) smooth non-linear exact spiral solutions derived via a separate method, which highlight that non-linear spiral waves need not shock. Indeed, at low amplitudes, we find that dispersion can overcome wave steepening, and may prevent the inner spiral wakes excited by low mass planets (below roughly 1% of a thermal mass) embedded in protoplanetary discs from shocking. At high amplitudes, we find a simple universal description of non-linear spiral waves with shocks, as well as caps on the possible amplitude and wave action flux of non-linear spirals both with and without shocks, depending on how many arms they have. We further find that highly non-linear spirals are far more loosely wound than their linear counterparts. These developments shed light on why two-armed spirals are prevalent across a range of astrophysical systems which don't necessarily possess an intrinsic twofold symmetry, and why they appear surprisingly loosely wound in observations. These results are supported by very high-resolution numerical simulations.