A summary of my research

The Rossby wave instability is a hydrodynamical instability in accretion disks, that can be excited if the potential vorticity as a function of the radius has a local maximum. Simplified one can say, that it exists in regions where the density or temperature strongly changes. Examples are gaps formed e.g. by protoplanets or also at the boundary of the so-called dead zone, in which the magneto-rotational instability is inefficient due to a low ionization rate. The Rossby wave instability leads to the formation of vortices that can merge at later times. The evolution depends strongly on the numerical viscosity, i.e. it can be an ideal test to analyze the diffusivity of a numerical scheme. In this study, we run global, two-dimensional simulations with the moving mesh code AREPO and use a similar setup as in Ono et al. (2018). To further analyze the evolution I implemented a Fourier filter, that is able at run time to filter out specified Fourier modes and allows me to measure the linear growth of individual modes. The results compare very well with the analytical growth rates and we concluded that AREPO on an arbitrary mesh shows similar accuracy as a static mesh code on a polar grid, that is optimized for this problem. We note that the higher-order flux integration discussed below is essential to reduce numerical noise in this setup. The results will be published in a future paper.

The molecular viscosity of diffuse gas is by several orders of magnitude too small to explain the required amount of angular momentum transported in accretion disks. A possible solution is an effective viscosity that can be created by turbulence in the disk, and which in turn can be generated by different fluid instabilities. One of the most promising candidates for the main culprit is the magnetorotational instability (MRI) in ionized regions since it requires only a weak initial field within the limit of ideal MHD. In this study, we employ the implementation of the shearing box approximation in AREPO to study the properties of the MRI. We perform simulations with different setups and compare the results with previous results in the literature. In general, our results compare well with those obtained with static grid codes, although the MRI turbulence seems to be a bit weaker. In unstratified simulations with a zero-net field, the strength of the MRI does not decrease with increasing resolution, which might give a hint that the numerical Prandtl number scales differently than in static grid codes. More information can be found here.

The moving mesh method tries to combine the advantages of particle-based Lagrangian methods with the high accuracy of Eulerian methods on a static mesh. But it is well-known from the literature that it can suffer from so-called grid noise, which means noise on the scale of individual cells. This noise is especially strong in shear flows, which we can find e.g. in rotating disks. During the implementation of the shearing box approximation, we found, that this noise is created by an approximation when we integrate the flux over the surface of a cell. This flux is typical a third-order polynomial but AREPO only used a first-order accurate integration rule. By using higher-order Gauss-Legendre rules this integral can be made exact and the noise disappears. While in two-dimensional simulations the overhead of the additional flux calculations is small, it becomes significant in three-dimensional simulations. To reduce this overhead I implemented an optimized version using explicit vector instruction based on the vectorclass library. This leads for pure ideal MHD simulations to an increased run time of around 30% compared to the original AREPO code. More information can be found in this paper.

AREPO allows to split and merge of cells to locally increase the resolution. I implemented a higher-order scheme that takes into account linear gradients in those processes which then can reduce noise. This is especially important in shearing box simulations that require high accuracy.

During my master thesis I performed simulations of Milky way like disk galaxies with the SPH code GADGET-3. The model included self-gravity, cooling, star formation, stellar feedback based on the work of Dobbs et al. (2011) and static potentials to described the gravitational effects of the dark matter halo, the stellar disk and a bulge. We used the model to analyze structure and especially turbulence in different phases of the ISM. Some results are summarized in an upcoming paper written by a student I cosupervised.

In my bachelor thesis, I analyzed the stability of an inclined layer of mercury to convection. Mercury is a typical liquid metal with a small Prandtl number (Pr = 0.025) and we used the standard setup of Rayleigh-Bénard convection (RBC), inclined by an angle gamma (see left side). The system allows for a heat conducting ground state with a linear temperature profile and a cubic flow profile, which disappear without inclination. Above a critical temperature difference, this state becomes linearly unstable and convection rolls form. For even higher temperature differences secondary instabilities form that can lead to different patterns. Using the standard Oberbeck–Boussinesq equations in combination with a Galerkin method I first analyzed the linear stability of the ground state as a function of gamma. Afterwards, I constructed static, nonlinear convection roll solutions and analyzed their stability towards secondary instabilities. In the end, I performed simulations with a pseudo-spectral code to verify the stability diagrams and analyze the nonlinear behaviour. The results were published here.