---
abstract: 'We describe a new mechanism that leads to the destabilisation of non-axisymmetric waves in astrophysical discs with an imposed radial temperature gradient. This might apply, for example, to the outer parts of protoplanetary discs. We use linear density wave theory to show that non-axisymmetric perturbations generally do not conserve their angular momentum in the presence of a forced temperature gradient. This implies an exchange of angular momentum between linear perturbations and the background disc. In particular, when the disturbance is a low-frequency trailing wave and the disc temperature decreases outwards, this interaction is unstable and leads to the growth of the wave. We demonstrate this phenomenon through numerical hydrodynamic simulations of locally isothermal discs in 2D using the FARGO code and in 3D with the ZEUS-MP and PLUTO codes. We consider radially structured discs with a self-gravitating region which remains stable in the absence of a temperature gradient. However, when a temperature gradient is imposed we observe exponential growth of a one-armed spiral mode (azimuthal wavenumber $m=1$) with co-rotation radius outside the bulk of the spiral arm, resulting in a nearly-stationary one-armed spiral pattern. The development of this one-armed spiral does not require the movement of the central star, as found in previous studies. Because destabilisation by a forced temperature gradient does not explicitly require disc self-gravity, we suggest this mechanism may also affect low-frequency one-armed oscillations in non-self-gravitating discs.'
author:
- |
    Min-Kai Lin$^{1}$ [^1]\
    $^1$Department of Astronomy and Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
bibliography:
- 'ref.bib'
title: 'One-armed spirals in locally isothermal, radially structured self-gravitating discs'
---

accretion, accretion discs, hydrodynamics, instabilities, methods: numerical, protoplanetary discs

Introduction {#intro}
============

An exciting development in the study of circumstellar discs is the direct observation of large-scale, non-axisymmetric structures within them. These include lopsided dust distributions [@marel13; @fukagawa13; @casassus13; @isella13; @perez14; @follette14; @plas14] and spiral arms [@hashimoto11; @muto12; @boccaletti14; @grady13; @christiaens14; @avenhaus14].

The attractive explanation for asymmetries in circumstellar discs is disc-planet interaction. In particular, spiral structures naturally arise from the gravitational interaction between a planet and the gaseous protoplanetary disc it is embedded in [see, e.g. @baruteau13b for a recent review]. Thus, the presence of spiral arms in circumstellar discs could be signposts of planet formation [but see @juhasz14].

Spiral arms are also characteristic of global gravitational instability (GI) in differentially rotating discs [@goldreich65; @laughlin96b; @laughlin98; @nelson98; @lodato05; @forgan11]. Large-scale spiral arms can provide significant angular momentum transport necessary for mass accretion [@lynden-bell72; @papaloizou91; @balbus99; @lodato04], and spiral structures due to GI are potentially observable with the Atacama Large Millimeter/sub-millimeter Array [@cossins10; @dipierror14]. GI can be expected in the earliest stages of circumstellar disc formation [@kratter10b; @inutsuka10; @tsukamoto13], and may be possible in the outer parts of the disc [@rafikov05; @matzner05; @kimura12].

Single-arm spirals, or eccentric modes, corresponding to perturbations with azimuthal wavenumber $m=1$, have received interest in the context of protoplanetary discs because of their global nature [@adams89; @heemskerk92; @laughlin96; @tremaine01; @papaloizou02; @hopkins10]. In the ‘SLING’ mechanism proposed by [@shu90], an $m=1$ gravitational instability arises from the motion of the central star induced by the one-armed perturbation, and requires a massive disc [the former may have observable consequences, @michael10].

In this work we identify a new mechanism that leads to the growth of one-armed spirals in astrophysical discs. We show that when the disc temperature is prescribed (called locally isothermal discs), the usual statement of the conservation of angular momentum for linear perturbations acquires a source term proportional to the temperature gradient. This permits angular momentum exchange between linear perturbations and the background disc. This ‘background torque’ can destabilise low-frequency non-axisymmetric trailing waves when the disc temperature decreases outwards.

We employ direct hydrodynamic simulations using three different grid-based codes to demonstrate how this ‘background torque’ can lead to the growth of one-armed spirals in radially structured, self-gravitating discs. This is despite the fact that our disc models do not meet the requirements for the ‘SLING’ mechanism. Although our numerical simulations consider self-gravitating discs, this ‘background torque’ is generic for locally isothermal discs and its existence does not require disc self-gravity. Thus, the destabilisation effect we describe should also be applicable to non-self-gravitating discs.

This paper is organised as follows. In §\[model\] we describe the system of interest and list the governing equations. In §\[wkb\] we use linear theory to show how a fixed temperature profile can destabilise non-axisymmetric waves in discs. §\[methods\] describes the numerical setup and hydrodynamic codes we use to demonstrate the growth of one-armed spirals due to an imposed temperature gradient. Our simulation results are presented in §\[results2d\] for two-dimensional (2D) discs and in §\[results3d\] for three-dimensional (3D) discs, and we further discuss them in §\[discussions\]. We summarise in §\[summary\] with some speculations for future work.

Acknowledgments {#acknowledgments .unnumbered}
===============

I thank K. Kratter, Y. Wu and A. Youdin for valuable discussions, and the anonymous referee for comments that significantly improved this paper. All computations were performed on the El Gato cluster at the University of Arizona. This material is based upon work supported by the National Science Foundation under Grant No. 1228509.

[^1]: minkailin@email.arizona.edu
