In this section we show that the growth of the m = 1 mode in the disc results in the formation of a spiral density wave with a constant global pattern speed ΩP. As the disc is differentially rotating this means that there is a location in the disc, the so-called corotation radius rco, where the spiral density wave has the same orbital velocity as the fluid in the disc. Inside rco the wave travels slower than the fluid, which means that it has negative angular momentum with respect to the fluid. While the wave amplitude is linear, it does not interact with the fluid, but once non-linear amplitude effects come into play, the spiral density wave can couple to the fluid via dissipation (Papaloizou & Lin 1995; Goodman & Rafikov 2001; Heinemann & Papaloizou 2012). When this happens, the wave begins to transport angular momentum outwards as the fluid loses angular momentum to the wave inside the corotation radius. The development and persistence of the spiral density wave are shown in Fig. 3. The figure shows the fractional change in density between two successive snapshots of the tracer particles Δρ/ρ at nine different times of the disc evolution, as well as the tracer particles that are located at rco (grey circles). The location of the corotation radius is defined as the radius where the m = 1 pattern speed and the fluid orbital velocity are equal. As usual, the non-axisymmetric modes in the disc are analysed by means of an azimuthal Fourier transform of the rest-mass density ρ (Zurek & Benz 1986; Heemskerk et al. 1992):

(5)

\begin{equation}
D_m = \int \, \alpha \, \sqrt{\gamma } \,\rho \, {\rm e}^{-i\,m\,\phi } \, d^3x \,,
\end{equation}

with α and γ being the space–time lapse function and the determinant of the three-dimensional metric, respectively. Similarly, the mode analysis performed using the tracers is accomplished by means of the following sum

(6)

\begin{equation}
D_m = \sum _j^N \, m_j \, {\rm e}^{-i\,m\,\phi },
\end{equation}

where mj is the mass of each tracer particle. From the mode amplitudes, the pattern speed of an azimuthal mode with mode number m is defined as (see e.g. Heemskerk et al. 1992)

(7)

\begin{equation}
\Omega _{\rm P} = \frac{1}{m}\frac{d \phi _m}{dt},
\end{equation}

where the phase angle ϕm is given by

(8)

\begin{equation}
\phi _m = \mathrm{tan}^{-1} \left(\frac{\mathrm{Im}(-D_m)}{\mathrm{Re}(D_m)} \right).
\end{equation}

For our simulation, we obtain an orbital period of the m = 1 pattern of PP ∼ 1.96 ms, which is slightly shorter than twice the initial orbital period of the disc at the location of the density maximum (see Table 1). We note that we have chosen the same range of the fractional change in the rest-mass density in all snapshots shown in Fig. 3, which corresponds to the interval [−0.1, 0.1]. At these wave amplitudes, non-linear effects are negligible compared to the linear effects (Masset & Tagger 1997). This plot range has, however, been chosen for visualization purposes and the wave amplitudes are actually much larger than 0.1 after the saturation of the m = 1 growth, reaching almost unity. At these wave amplitudes non-linear effects are important and the spiral density wave can couple to the fluid via dissipation. To show the development of weak shocks, we plot the fractional change in the entropy S = p/ρΓ for tracers with Δρ/ρ > 0.1 in Fig. 4. From this plot, we clearly see an increase in the fluid entropy in the inner and outer regions of the spiral density wave. In some snapshots of Fig. 3, we can see fractional changes in ρ in the form of two spiral arms in the inner regions of the disc. The dominant m = 1 mode developing in the disc should not produce these, which means there could be a different mechanism at play. Standing shocks in the inner regions of tilted accretion discs in the form of two spiral arms have been observed and analysed in Henisey, Blaes & Fragile (2012) and Generozov et al. (2014) and are a distinctive feature of tilted accretion discs. Finally, we note that the mesh refinement boundaries of our computational grid also produce entropy changes due to numerical dissipation. However, these changes are about an order of magnitude smaller than the physical increase of entropy due to the development of shocks in the spiral density wave.
Citation: Zurek & Benz 1986
Citation function: Uses; Uses