Marginally unstable Holmboe modes

Marginally unstable Holmboe modes for smooth density and velocity profiles are studied. For a large family of flows and stratification that exhibit Holmboe instability, we show that the modes with phase velocity equal to the maximum or the minimum velocity of the shear are marginally unstable. This allows us to determine the critical value of the control parameter R (expressing the ratio of the velocity variation length scale to the density variation length scale) that Holmboe instability appears R=2. We then examine systems for which the parameter R is very close to this critical value. For this case we derive an analytical expression for the dispersion relation of the complex phase speed c(k) in the unstable region. The growth rate and the width of the region of unstable wave numbers has a very strong (exponential) dependence on the deviation of R from the critical value. Two specific examples are examined and the implications of the results are discussed.Comment: Submitted to Physics of Fluid

Stratified shear flow instabilities at large Richardson numbers

Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin-Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri =0.2, 2, 20. The flows for the three examined values are all unstable due to different modes namely: the Kelvin-Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and it is the first time that it is examined in the non-linear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the non-linear stage is smaller but comparable to first Holmboe mode. The increase of the potential energy however due to the second Holmboe modes is greater than that of the first mode. The Kelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments can not be excluded.Comment: Submitted to Physics of Fluid

Shell to shell energy transfer in MHD, Part I: steady state turbulence

We investigate the transfer of energy from large scales to small scales in fully developed forced three-dimensional MHD-turbulence by analyzing the results of direct numerical simulations in the absence of an externally imposed uniform magnetic field. Our results show that the transfer of kinetic energy from the large scales to kinetic energy at smaller scales, and the transfer of magnetic energy from the large scales to magnetic energy at smaller scales, are local, as is also found in the case of neutral fluids, and in a way that is compatible with Kolmogorov (1941) theory of turbulence. However, the transfer of energy from the velocity field to the magnetic field is a highly non-local process in Fourier space. Energy from the velocity field at large scales can be transfered directly into small scale magnetic fields without the participation of intermediate scales. Some implications of our results to MHD turbulence modeling are also discussed.Comment: Submitted to PR

The imprint of large-scale flows on turbulence

We investigate the locality of interactions in hydrodynamic turbulence using data from a direct numerical simulation on a grid of 1024^3 points; the flow is forced with the Taylor-Green vortex. An inertial range for the energy is obtained in which the flux is constant and the spectrum follows an approximate Kolmogorov law. Nonlinear triadic interactions are dominated by their non-local components, involving widely separated scales. The resulting nonlinear transfer itself is local at each scale but the step in the energy cascade is independent of that scale and directly related to the integral scale of the flow. Interactions with large scales represent 20% of the total energy flux. Possible explanations for the deviation from self-similar models, the link between these findings and intermittency, and their consequences for modeling of turbulent flows are briefly discussed

Non-local interactions in hydrodynamic turbulence at high Reynolds numbers: the slow emergence of scaling laws

We analyze the data stemming from a forced incompressible hydrodynamic simulation on a grid of 2048^3 regularly spaced points, with a Taylor Reynolds number of Re~1300. The forcing is given by the Taylor-Green flow, which shares similarities with the flow in several laboratory experiments, and the computation is run for ten turnover times in the turbulent steady state. At this Reynolds number the anisotropic large scale flow pattern, the inertial range, the bottleneck, and the dissipative range are clearly visible, thus providing a good test case for the study of turbulence as it appears in nature. Triadic interactions, the locality of energy fluxes, and structure functions of the velocity increments are computed. A comparison with runs at lower Reynolds numbers is performed, and shows the emergence of scaling laws for the relative amplitude of local and non-local interactions in spectral space. The scalings of the Kolmogorov constant, and of skewness and flatness of velocity increments, performed as well and are consistent with previous experimental results. Furthermore, the accumulation of energy in the small-scales associated with the bottleneck seems to occur on a span of wavenumbers that is independent of the Reynolds number, possibly ruling out an inertial range explanation for it. Finally, intermittency exponents seem to depart from standard models at high Re, leaving the interpretation of intermittency an open problem.Comment: 8 pages, 8 figure

Non-existence of stationary two-black-hole configurations: The degenerate case

In a preceding paper we examined the question whether the spin-spin repulsion and the gravitational attraction of two aligned sub-extremal black holes can balance each other. Based on the solution of a boundary value problem for two separate (Killing-) horizons and a novel black hole criterion we were able to prove the non-existence of the equilibrium configuration in question. In this paper we extend the non-existence proof to extremal black holes.Comment: 18 pages, 2 figure

Large scale flow effects, energy transfer, and self-similarity on turbulence

The effect of large scales on the statistics and dynamics of turbulent fluctuations is studied using data from high resolution direct numerical simulations. Three different kinds of forcing, and spatial resolutions ranging from 256^3 to 1024^3, are being used. The study is carried out by investigating the nonlinear triadic interactions in Fourier space, transfer functions, structure functions, and probability density functions. Our results show that the large scale flow plays an important role in the development and the statistical properties of the small scale turbulence. The role of helicity is also investigated. We discuss the link between these findings and intermittency, deviations from universality, and possible origins of the bottleneck effect. Finally, we briefly describe the consequences of our results for the subgrid modeling of turbulent flows

Critical exponents in zero dimensions

In the vicinity of the onset of an instability, we investigate the effect of colored multiplicative noise on the scaling of the moments of the unstable mode amplitude. We introduce a family of zero dimensional models for which we can calculate the exact value of the critical exponents $\beta_m$ for all the moments. The results are obtained through asymptotic expansions that use the distance to onset as a small parameter. The examined family displays a variety of behaviors of the critical exponents that includes anomalous exponents: exponents that differ from the deterministic (mean-field) prediction, and multiscaling: non-linear dependence of the exponents on the order of the moment

Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers

The effect of rotation is considered to become important when the Rossby number is sufficiently small, as is the case in many geophysical and astrophysical flows. Here we present direct numerical simulations to study the effect of rotation in flows with moderate Rossby numbers (down to Ro~0.1) but at Reynolds numbers large enough to observe the beginning of a turbulent scaling at scales smaller than the energy injection scale. We use coherent forcing at intermediate scales, leaving enough room in the spectral space for an inverse cascade of energy to also develop. We analyze the spectral behavior of the simulations, the shell-to-shell energy transfer, scaling laws, and intermittency, as well as the geometry of the structures in the flow. At late times, the direct transfer of energy at small scales is mediated by interactions with the largest scale in the system, the energy containing eddies with k_perp~1, where "perp" refers to wavevectors perpendicular the axis of rotation. The transfer between modes with wavevector parallel to the rotation is strongly quenched. The inverse cascade of energy at scales larger than the energy injection scale is non-local, and energy is transferred directly from small scales to the largest available scale. Also, as time evolves and the energy piles up at the large scales, the intermittency of the direct cascade of energy is preserved while corrections due to intermittency are found to be the same (within error bars) as in homogeneous turbulence.Comment: 12 pages, 16 figure