Linear and Weakly Nonlinear Analysis of the Magneto-Rotational-Instability in Thin Keplerian Discs

The linear instability of thin, vertically-isothermal Keplerian discs, under the influence of axial magnetic field is investigated. Solutions of the stability problem are found explicitly by asymptotic expansions in the small aspect ratio of the disc. It is shown that the perturbations are decoupled into in-plane and vertical modes. Exact expressions for the growth rates as well as the number of unstable modes are derived. Those are the discrete counterpart of the continuous infinite homogeneous cylinder magnetorotational (MRI) spectrum. In addition, a weakly nonlinear analysis of the MRI is performed. It is shown that near the instability threshold the latter is saturated by the stable magnetoacoustic modes.Comment: 18 pages, 6 figures, to appear in the proceedings of the workshop "Waves and Instabilities in Space and Astrophysical plasmas" (WISAP-2011), June 201

About disposition of energy levels

The unique properties of central potential of the form $-\beta e^{-r}r^{\gamma}$ were studied using the recently developed critical parameter technique. The particular cases of $\gamma=0$ and $\gamma=-1$ yield, respectively, the exponential and Yukawa potentials widely used in the atomic, molecular and nuclear physics. We found different behavior of the energy levels of this potential for three different ranges of the value of $\gamma$. For $\gamma\geq0$ it was found that the energy of bound states with the same principal quantum number $N$ decreases with increasing angular momentum $\ell$. The Gaussian and Woods-Saxon potentials also show this behavior. On the contrary, for $-2\leq\gamma\leq-1$ increasing $\ell$ gives a higher energy, resembling the Hulthen potential. However, a potential with $-1<\gamma<0$ possesses mixed properties, which give rise to several interesting results. For one, the order of energy levels with different quantum numbers is not preserved when varying the parameter $\beta$. This leads to a quantum degeneracy of the states, and in fact, for a given value of $\gamma$ we can find the values $\beta_{thr}$ for which two energy levels with different quantum numbers coincide. Another interesting phenomena is the possibility, for some values of $\gamma$ in this range, for two new energy levels with different quantum numbers to appear simultaneously when $\beta$ reaches their common critical value.Comment: 10 pages, 3 table