Unbalanced instabilities of rapidly rotating stratified shear flows

The linear stability of a rotating, stratified, inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. An energy argument is used to show that unstable perturbations must have large wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia-gravity waves with or without turning points. Although, the wave phase speeds are found to be real to all algebraic orders in the Rossby number, we establish that the flow, whether cyclonic or anticyclonic, is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. We derive asymptotic estimates for the instability growth rates, which are exponentially small in the Rossby number, and confirm them by numerical computations. Our results, which extend those of Kushner et al (1998) and Yavneh et al (2001), highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia-gravity waves and hence predict the stability of the Couette flow. They are also relevant to the stability of Taylor-Couette flows and of astrophysical accretion discs.Comment: 6 figure

Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows

The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two- and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit Runge-Kutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p- and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated

Multilevel Turbulence Simulations

A novel multigrid method for the accurate and efficient simulation of turbulent flows is described and demonstrated. The method's efficiency relative to direct simulations is of the order of the ratio of required integration time to the smallest-eddy turnover time, potentially resulting in orders-of-magnitude improvement for a large class of turbulence problems.Earth and Planetary Science

Non-Axisymmetric g-Mode and p-Mode Instability in a Hydrodynamic Thin Accretion Disk

It has been suggested that quasi-periodic oscillations of accreting X-ray sources may relate to the modes named in the title. We consider non-axisymmetric linear perturbations to an isentropic, isothermal, unmagnetized thin accretion disk. The radial wave equation, in which the number of vertical nodes (n) appears as a separation constant, admits a wave-action current that is conserved except, in some cases, at corotation. Waves without vertical nodes amplify when reflected by a barrier near corotation. Their action is conserved. As was previously known, this amplification allows the n=0 modes to be unstable under appropriate boundary conditions. In contrast, we find that waves with n >0 are strongly absorbed at corotation rather than amplified; their action is not conserved. Therefore, non-axisymmetric p-modes and g-modes with n>0 are damped and stable even in an inviscid disk. This eliminates a promising explanation for quasi-periodic oscillations in neutron-star and black-hole X-ray binaries.Comment: A new version of the paper. The technical error in version 1 has been correcte

On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient

Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

Algebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented

Sparsity-based single-shot sub-wavelength coherent diffractive imaging

We present the experimental reconstruction of sub-wavelength features from the far-field intensity of sparse optical objects: sparsity-based sub-wavelength imaging combined with phase-retrieval. As examples, we demonstrate the recovery of random and ordered arrangements of 100 nm features with the resolution of 30 nm, with an illuminating wavelength of 532 nm. Our algorithmic technique relies on minimizing the number of degrees of freedom; it works in real-time, requires no scanning, and can be implemented in all existing microscopes - optical and non-optical