Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Secondly, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.Comment: 22 pages. To appear in JPA

Viscous and Resistive Effects on the MRI with a Net Toroidal Field

Resistivity and viscosity have a significant role in establishing the energy levels in turbulence driven by the magnetorotational instability (MRI) in local astrophysical disk models. This study uses the Athena code to characterize the effects of a constant shear viscosity \nu and Ohmic resistivity \eta in unstratified shearing box simulations with a net toroidal magnetic flux. A previous study of shearing boxes with zero net magnetic field performed with the ZEUS code found that turbulence dies out for values of the magnetic Prandtl number, P_m = \nu/\eta, below P_m \sim 1; for P_m \gtrsim 1, time- and volume-averaged stress levels increase with P_m. We repeat these experiments with Athena and obtain consistent results. Next, the influence of viscosity and resistivity on the toroidal field MRI is investigated both for linear growth and for fully-developed turbulence. In the linear regime, a sufficiently large \nu or \eta can prevent MRI growth; P_m itself has little direct influence on growth from linear perturbations. By applying a range of values for \nu and \eta to an initial state consisting of fully developed turbulence in the presence of a background toroidal field, we investigate their effects in the fully nonlinear system. Here, increased viscosity enhances the turbulence, and the turbulence decays only if the resistivity is above a critical value; turbulence can be sustained even when P_m < 1, in contrast to the zero net field model. While we find preliminary evidence that the stress converges to a small range of values when \nu and \eta become small enough, the influence of dissipation terms on MRI-driven turbulence for relatively large \eta and \nu is significant, independent of field geometry.Comment: Accepted to ApJ; version 2 - minor changes following review; 35 pages (preprint format), 10 figure