where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Since the observational detection of black holes is inherently difficult, it is important to begin with a clear idea of the general possibilities for such detection as they seem practical from observations of existing type. FIGURE 1 shows a schematic view of the paths leading to this objective. The primary attribute by which we hope to recognize a black hole is its gravitational mass, which is discernible through its effect on macroscopic bodies (orbital motions) or microscopic bodies (compressional heating with emission of x-rays). The combination of large mass, small radius, and small luminosity constitutes the unique signature of a massive black hole, and there do exist invisible components to numerous binaries that may fulfill these conditions. Since other small objects (white dwarfs and neutrons stars) cannot have masses exceeding about 1.5 Mo, large mass means M 2 2 Mo. The notions of small radius and small luminosity can be taken in general to mean that these quantities are small compared with the values expected for any conceivable stable astronomical body that could be present. Those binaries for which fairly serious black hole arguments have been made to date are e Aur,\"34,10 p Lyrae,3,\" Cygnus X1 = HDE 226868,',e and BM Ori.\" Space does not permit discussion of the merits of these cases, although counterarguments have been made regarding ,8 Lyrae',* and might well be made against Cyg X-1 and BM Ori. It has been predicted for most of a decade, if not longer, that black holes in binary systems might be x-ray sources. Indeed, Cygnus X-1 may be one such case. However, x-rays are not the only evidence that might uncover collapsed stars, and it is fortunate that this is so because of the intrinsic difficulty of unambiguous identification of a black hole. That is, we would like, if at all possible, to have several independent demonstrations that a black hole is present. A possible path is offered by spectroscopic and photometric observations of certain unusual binaries that have long histories of observational peculiarities and, in addition, have massive invisible secondary components. It would be quite convincing if we could travel both paths of FIGURE 1 for some particular binary system. For example, E Aur might turn on as an x-ray source, or Cyg X-1 might, on detailed examination, show properties similar to those of BM Ori. An appreciation for the left path of FIGURE 1 (optical observations) may be gained through consideration of the eclipsing binary E Aurigae. The most striking peculiarity of e Aur is that the eclipse is apparently total (flat bottom)-yet the spectrum of the eclipsed component remains visible at all times. This, of course, proves that the eclipse really is not total, so we must find another explanation. It can easily be shown' that the eclipse also is not annular and that the disk models by Huang,' by Kopal,5 and by Cameron' are not compatible with the observations.'\" Useful observations of E Aur have been made since 1848, and models for the eclipse mechanism have been offered over the past 40 years. To explain

Black holes in binary systems

Hydrodynamic and hydromagnetic stability

Taylor-Couette flow, the flow between two coaxial co- or counter-rotating cylinders, is one of the paradigmatic systems in the physics of fluids. The (dimensionless) control parameters are the Reynolds numbers of the inner and outer cylinders, the ratio of the cylinder radii, and the aspect ratio. One key response of the system is the torque required to retain constant angular velocities, which can be connected to the angular velocity transport through the gap. Whereas the low–Reynolds number regime was well explored in the 1980s and 1990s of the past century, in the fully turbulent regime major research activity developed only in the past decade. In this article, we review this recent progress in our understanding of fully developed Taylor-Couette turbulence from the experimental, numerical, and theoretical points of view. We focus on the parameter dependence of the global torque and on the local flow organization, including velocity profiles and boundary layers. Next, we discuss transitions between diff...

High-Reynolds number Taylor-Couette turbulence.

Taylor-Couette flow -- the flow between two coaxial co- or counter-rotating cylinders -- is one of the paradigmatic systems in physics of fluids. The (dimensionless) control parameters are the Reynolds numbers of the inner and outer cylinder, the ratio of the cylinder radii, and the aspect ratio. The response of the system is the torque required to retain constant angular velocities, which can be connected to the angular velocity transport through the gap. While the low Reynolds number regime has been very well explored in the '80s and '90s of the last century, in the fully turbulent regime major research activity only developed in the last decade. In this paper we review this recent progress in our understanding of fully developed Taylor-Couette turbulence, from the experimental, numerical, and theoretical point of view. We will focus on the parameter dependence of the global torque and on the local flow organisation, including velocity profiles and boundary layers. Next, we will discuss transitions between different (turbulent) flow states. We will also elaborate on the relevance of this system for astrophysical disks (Keplerian flows). The review ends with a list of challenges for future research on turbulent Taylor-Couette flow. 
The published version in ARFM: Annu. Rev. Fluid Mech. 48:53-80 (2016).

High Reynolds number Taylor-Couette turbulence

A pencil distributed finite difference code for strongly turbulent wall-bounded flows

Direct numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to 3×10 5 , corresponding to Taylor numbers of Ta=4.6×10 10 , were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/3E32E03826BF8F14869B66EC6F6B4C2A/S0022112014006181a.pdf/div-class-title-exploring-the-phase-diagram-of-fully-turbulent-taylor-couette-flow-div.pdf

Exploring the phase diagram of fully turbulent Taylor–Couette flow

We numerically simulate turbulent Taylor-Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh-B\'enard flow. Reynolds numbers of $Re_i=8\cdot10^3$ and $Re_o=\pm4\cdot10^3$ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers Ta up to $10^8$. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente turbulent Taylor-Couette ($T^3C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a = -\omega_o / \omega_i$ of about $a_{opt}=0.33-0.35$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of $a_{opt}=0.15$ for $Ta=2.5\cdot10^7$. An explanation for this shift is elucidated, consistent with the experimental result that $a_{opt}$ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities.

/pdf/optimal-taylor-couette-flow-direct-numerical-simulations-1yj58uivfr.pdf

Optimal Taylor-Couette flow: direct numerical simulations

Direct numerical simulations of turbulent Taylor-Couette flow are performed up to inner cylinder Reynolds numbers of {Re_i=10^5} for a radius ratio of {\eta=r_i/r_o=0.714} between the inner and outer cylinder. With increasing {Re_i}, the flow undergoes transitions between three different regimes: (i) a flow dominated by large coherent structures, (ii) an intermediate transitional regime, and (iii) a flow with developed turbulence. In the first regime the large--scale rolls completely drive the meridional flow while in the second one the coherent structures recover only on average. The presence of a mean flow allows for the coexistence of laminar and turbulent boundary layer dynamics. In the third regime the mean flow effects fade away and the flow becomes dominated by plumes. The effect of the local driving on the azimuthal and angular velocity profiles is quantified, in particular we show when and where those profiles develop.

Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor-Couette flow

Taylor-Couette flow with independently rotating inner (i) and outer (o) cylinders is explored numerically and experimentally to determine the effects of the radius ratio {\eta} on the system response. Numerical simulations reach Reynolds numbers of up to Re_i=9.5 x 10^3 and Re_o=5x10^3, corresponding to Taylor numbers of up to Ta=10^8 for four different radius ratios {\eta}=r_i/r_o between 0.5 and 0.909. The experiments, performed in the Twente Turbulent Taylor-Couette (T^3C) setup, reach Reynolds numbers of up to Re_i=2x10^6$ and Re_o=1.5x10^6, corresponding to Ta=5x10^{12} for {\eta}=0.714-0.909. Effective scaling laws for the torque J^{\omega}(Ta) are found, which for sufficiently large driving Ta are independent of the radius ratio {\eta}. As previously reported for {\eta}=0.714, optimum transport at a non-zero Rossby number Ro=r_i|{\omega}_i-{\omega}_o|/[2(r_o-r_i){\omega}_o] is found in both experiments and numerics. Ro_opt is found to depend on the radius ratio and the driving of the system. At a driving in the range between {Ta\sim3\cdot10^8} and {Ta\sim10^{10}}, Ro_opt saturates to an asymptotic {\eta}-dependent value. Theoretical predictions for the asymptotic value of Ro_{opt} are compared to the experimental results, and found to differ notably. Furthermore, the local angular velocity profiles from experiments and numerics are compared, and a link between a flat bulk profile and optimum transport for all radius ratios is reported.

Rodolfo Ostilla Monico

Papers

A pencil distributed finite difference code for strongly turbulent wall-bounded flows

Exploring the phase diagram of fully turbulent Taylor–Couette flow

Optimal Taylor-Couette flow: direct numerical simulations

Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor-Couette flow

Optimal Taylor-Couette flow: Radius ratio dependence