Stability of an MHD shear flow with a piecewise linear velocity profile

In this paper we present the results of the stability analysis of a simple shear flow of an incompressible fluid with a piecewise linear velocity profile in the presence of a magnetic field. In the flow, a finite transitional magnetic-free layer with a linear velocity profile is sandwiched by two semi-infinite regions. One of these regions is magnetic-free and the flow velocity in the region is constant. The other region is magnetic and the fluid in it is quiescent. The magnetic field is constant and parallel to the flow in the transitional layer. The fluid density is constant both in the magnetic as well as the magnetic-free regions, while it has a jump-type discontinuity at the boundary between the transitional layer and the magnetic region. The effect of gravity is included in the model, and it is assumed that the lighter fluid is overlaying the heavier one, thus no Rayleigh-Taylor instability is present. The dispersion equation governing the normal-mode stability of the flow is derived and its properties are analysed. We study stability of two cases: (i) magnetic-free flow in the presence of gravity, and (ii) magnetic flow without gravity. In the first case, the flow stability is controlled by the Rayleigh number, R. In the second case, the control parameter is the inverse squared Alfvénic Mach number, H . Stability of a particular monochromatic perturbation also depends on its dimensionless wavenumber α. We combine the analytical and numerical approaches to obtain the neutral stability curves in the (α,R)-plane in the case of the magnetic-free flow, and in the (α,H)-plane in the case of the magnetic flow. The dependence of the instability increment on R in the first case, and on H in the second case is treated. We apply the results of the analysis to the stability of a strongly subsonic portion of the heliopause. Our main conclusion is as follows: The inclusion of a transitional layer near the heliopause into the model increases by an order of magnitude the strength of the interstellar magnetic field required to stabilize this portion of the heliopause in comparison with the corresponding stabilizing strength of the magnetic field required when modelling the heliopause as a tangential discontinuity

Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion

Many flexible parameterizations exist to represent data on the sphere. In addition to the venerable spherical harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper we focus on the latter two: spherical wavelets developed for geophysical applications on the cubed sphere, and the Slepian "tree", a new construction that combines a quadratic concentration measure with wavelet-like multiresolution. We discuss the basic features of these mathematical tools, and illustrate their applicability in parameterizing large-scale global geophysical (inverse) problems.Comment: 15 pages, 11 figures, submitted to the Proceedings of the SPIE 2011 conference Wavelets and Sparsity XI

The linear impulse response for disturbances in an oscillatory Stokes layer

A brief review is given of numerical simulation results for the evolution of disturbances in a flat oscillatory Stokes layer. A spatially localised form of impulsive forcing is applied to trigger the disturbances. For the linearized case, the disturbance development displays an intriguing family tree-like structure, which involves the birth of successive generations of wavepackets. Although some features of the wavepacket behaviour may be accounted for using a Floquet linear stability analysis, for Fourier modes with prescribed wavenumbers, the discovery of the tree-like structure was completely unexpected. Simulation results were also obtained for nonlinear disturbances, where highly localised spikes were found to develop

Dynamic Control Flow in Large-Scale Machine Learning

Many recent machine learning models rely on fine-grained dynamic control flow for training and inference. In particular, models based on recurrent neural networks and on reinforcement learning depend on recurrence relations, data-dependent conditional execution, and other features that call for dynamic control flow. These applications benefit from the ability to make rapid control-flow decisions across a set of computing devices in a distributed system. For performance, scalability, and expressiveness, a machine learning system must support dynamic control flow in distributed and heterogeneous environments. This paper presents a programming model for distributed machine learning that supports dynamic control flow. We describe the design of the programming model, and its implementation in TensorFlow, a distributed machine learning system. Our approach extends the use of dataflow graphs to represent machine learning models, offering several distinctive features. First, the branches of conditionals and bodies of loops can be partitioned across many machines to run on a set of heterogeneous devices, including CPUs, GPUs, and custom ASICs. Second, programs written in our model support automatic differentiation and distributed gradient computations, which are necessary for training machine learning models that use control flow. Third, our choice of non-strict semantics enables multiple loop iterations to execute in parallel across machines, and to overlap compute and I/O operations. We have done our work in the context of TensorFlow, and it has been used extensively in research and production. We evaluate it using several real-world applications, and demonstrate its performance and scalability.Comment: Appeared in EuroSys 2018. 14 pages, 16 figure

Absolute and convective instabilities in an inviscid compressible mixing layer

We consider the stability of a compressible shear flow separating two streams of different speeds and temperatures. The velocity and temperature profiles in this mixing layer are hyperbolic tangents. The normal mode analysis of the flow stability reduces to an eigenvalue problem for the pressure perturbation. We briefly describe the numerical method that we used to solve this problem. Then, we introduce the notions of the absolute and convective instabilities and examine the effects of Mach number, and the velocity and temperature ratios of each stream on the transition between convective and absolute instabilities. Finally, we discuss the implication of the results presented in this paper for the heliopause stability.Comment: 5 pages, 6 figures, accepted by Astronomical Notes (Astron. Nachrichten