Linear and Nonlinear Evolution and Diffusion Layer Selection in Electrokinetic Instability

In the present work fournontrivial stages of electrokinetic instability are identified by direct numerical simulation (DNS) of the full Nernst-Planck-Poisson-Stokes (NPPS) system: i) The stage of the influence of the initial conditions (milliseconds); ii) 1D self-similar evolution (milliseconds-seconds); iii) The primary instability of the self-similar solution (seconds); iv) The nonlinear stage with secondary instabilities. The self-similar character of evolution at intermediately large times is confirmed. Rubinstein and Zaltzman instability and noise-driven nonlinear evolution to over-limiting regimes in ion-exchange membranes are numerically simulated and compared with theoretical and experimental predictions. The primary instability which happens during this stage is found to arrest self-similar growth of the diffusion layer and specifies its characteristic length as was first experimentally predicted by Yossifon and Chang (PRL 101, 254501 (2008)). A novel principle for the characteristic wave number selection from the broadbanded initial noise is established.Comment: 13 pages, 8 figure

On statistically stationary homogeneous shear turbulence

A statistically stationary turbulence with a mean shear gradient is realized in a flow driven by suitable body forces. The flow domain is periodic in downstream and spanwise directions and bounded by stress free surfaces in the normal direction. Except for small layers near the surfaces the flow is homogeneous. The fluctuations in turbulent energy are less violent than in the simulations using remeshing, but the anisotropy on small scales as measured by the skewness of derivatives is similar and decays weakly with increasing Reynolds number.Comment: 4 pages, 5 figures (Figs. 3 and 4 as external JPG-Files

A Comparison of Measured Crab and Vela Glitch Healing Parameters with Predictions of Neutron Star Models

There are currently two well-accepted models that explain how pulsars exhibit glitches, sudden changes in their regular rotational spin-down. According to the starquake model, the glitch healing parameter, Q, which is measurable in some cases from pulsar timing, should be equal to the ratio of the moment of inertia of the superfluid core of a neutron star (NS) to its total moment of inertia. Measured values of the healing parameter from pulsar glitches can therefore be used in combination with realistic NS structure models as one test of the feasibility of the starquake model as a glitch mechanism. We have constructed NS models using seven representative equations of state of superdense matter to test whether starquakes can account for glitches observed in the Crab and Vela pulsars, for which the most extensive and accurate glitch data are available. We also present a compilation of all measured values of Q for Crab and Vela glitches to date which have been separately published in the literature. We have computed the fractional core moment of inertia for stellar models covering a range of NS masses and find that for stable NSs in the realistic mass range 1.4 +/- 0.2 solar masses, the fraction is greater than 0.55 in all cases. This range is not consistent with the observational restriction Q < 0.2 for Vela if starquakes are the cause of its glitches. This confirms results of previous studies of the Vela pulsar which have suggested that starquakes are not a feasible mechanism for Vela glitches. The much larger values of Q observed for Crab glitches (Q > 0.7) are consistent with the starquake model predictions and support previous conclusions that starquakes can be the cause of Crab glitches.Comment: 8 pages, including 3 figures and 1 table. Accepted for publication in Ap

Low magnetic Prandtl number dynamos with helical forcing

We present direct numerical simulations of dynamo action in a forced Roberts flow. The behavior of the dynamo is followed as the mechanical Reynolds number is increased, starting from the laminar case until a turbulent regime is reached. The critical magnetic Reynolds for dynamo action is found, and in the turbulent flow it is observed to be nearly independent on the magnetic Prandtl number in the range from 0.3 to 0.1. Also the dependence of this threshold with the amount of mechanical helicity in the flow is studied. For the different regimes found, the configuration of the magnetic and velocity fields in the saturated steady state are discussed.Comment: 9 pages, 14 figure

A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ over $\Omega$ with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball $B$, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}(\overline{\Omega})$ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure

Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence

We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to $1024^3$ collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field. Our study concentrates on the dependence of various statistical properties of both decaying and statistically steady MHD turbulence on the magnetic Prandtl number ${\rm Pr_M}$ over a large range, namely, $0.01 \leq {\rm Pr_M} \leq 10$. We obtain data for a wide variety of statistical measures such as probability distribution functions (PDFs) of moduli of the vorticity and current density, the energy dissipation rates, and velocity and magnetic-field increments, energy and other spectra, velocity and magnetic-field structure functions, which we use to characterise intermittency, isosurfaces of quantities such as the moduli of the vorticity and current, and joint PDFs such as those of fluid and magnetic dissipation rates. Our systematic study uncovers interesting results that have not been noted hitherto. In particular, we find a crossover from larger intermittency in the magnetic field than in the velocity field, at large ${\rm Pr_M}$, to smaller intermittency in the magnetic field than in the velocity field, at low ${\rm Pr_M}$. Furthermore, a comparison of our results for decaying MHD turbulence and its forced, statistically steady analogue suggests that we have strong universality in the sense that, for a fixed value of ${\rm Pr_M}$, multiscaling exponent ratios agree, at least within our errorbars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.Comment: 49 pages,33 figure

A spectral method for elliptic equations: the Dirichlet problem

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure