Author name code: spiegel ADS astronomy entries on 2022-09-14 author:"Spiegel, Edward A." ------------------------------------------------------------------------ Title: Rapidly rotating Rayleigh-Bénard convection with a tilted axis Authors: Novi, Lyuba; von Hardenberg, Jost; Hughes, David W.; Provenzale, Antonello; Spiegel, Edward A. Bibcode: 2019EGUGA..21.5155N Altcode: We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector (supposed vertical). The angle φ between the rotation axis and the horizontal plane measures the tilting of the rotation axis and it can be taken as a proxy for latitude, if we think of a local Cartesian representation of the convective dynamics in a rotating fluid shell. The results of the simulations indicate the existence of three different convective regimes, depending on the value of φ: (1) sheared, intermittent large-scale winds in the direction perpendicular to the plane defined by the gravity and rotation vectors, when rotation is "horizontal" (φ = 0); (2) a large-scale cyclonic vortex tilted along the rotation axis, when the angle between the rotation axis and the gravity vector is relatively small (φ between about 45∘ and 90∘); (3) a new intermediate regime characterized by vertically sheared large-scale winds perpendicular to both gravity and rotation. In this regime, the winds are organized in bands that are tilted along the rotation axis, with unit horizontal wavenumber in the plane defined by gravity and rotation, at values of φ less than about 60∘. This intermediate solution, studied for the first time in this work, is characterized by weaker vertical heat transport than the cases with large-scale vortices. For intermediate values of φ (between about 45∘ and 60∘) the banded, sheared solution coexists with the large-scale vortex solution, and different initial conditions lead to one or the other dynamical behavior. A discussion of the possible implications of these results for the dynamics of rapidly-rotating planetary atmospheres is provided. Title: Intermittency and large-scale winds in horizontally anisotropic convection Authors: von Hardenberg, Jost; Goluskin, David; Provenzale, Antonello; Spiegel, Edward Bibcode: 2016EGUGA..18.2177V Altcode: We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection, confined between free-slip horizontal plates and rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the vertical turbulent heat flux is much lower on average than when there is no wind. Convection takes place in irregular, strongly intermittent bursts and the flow alternates between wind-dominated longer stages and convection-dominated events. Our findings support the conjecture that the upscale cascade of energy in anisotropic turbulent convection, which here drives sheared winds, drives differential rotation in the equatorial regions of planetary atmospheres and stellar convective zones, with interesting consequences associated with the strong intermittency of the convective events. Title: Generation of Large-Scale Winds in Horizontally Anisotropic Convection Authors: von Hardenberg, J.; Goluskin, D.; Provenzale, A.; Spiegel, E. A. Bibcode: 2015PhRvL.115m4501V Altcode: 2015arXiv150107308V We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection, confined between free-slip horizontal plates and rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the vertical turbulent heat flux is much lower on average than when there is no wind. Title: Rotating relativistic stars Authors: Spiegel, Edward Bibcode: 2015GApFD.109..462S Altcode: No abstract at ADS Title: Convectively driven shear and decreased heat flux Authors: Goluskin, David; Johnston, Hans; Flierl, Glenn R.; Spiegel, Edward A. Bibcode: 2014JFM...759..360G Altcode: 2014arXiv1408.4802G We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh-Bénard convection, focusing on its ability to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ($Pr$) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ($Ra$) sufficiently, and we explore the resulting convection for $Ra$ up to $10^{10}$. When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $Ra\to\infty$. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $Ra$. When the large-scale shear is present with $Pr\lesssim2$, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $Ra$ for $Pr=1$. When the shear is present with $Pr\gtrsim3$, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $Ra$, but the growth rates are slower than any previously reported for Rayleigh-Bénard convection without large-scale shear. We find the Nusselt numbers grow proportionally to $Ra^{0.077}$ when $Pr=3$ and to $Ra^{0.19}$ when $Pr=10$. Analogies with tokamak plasmas are described. Title: Uhlenbeck's Complaint Authors: Spiegel, E. A. Bibcode: 2014arXiv1411.4934S Altcode: The passage from kinetic theory to fluid dynamics as discussed by Hilbert has perplexed Uhlenbeck. Herein, I try to smooth over the discord. Title: The Right Side of Einstein's Equation Authors: Spiegel, Edward A Bibcode: 2014arXiv1401.5513S Altcode: Recent developments in observational cosmology have led to attempts to make modifications on both sides of the Einstein equation to explain some of the puzzling new findings. What follows is an examination of the source of gravity that we usually find on the right hand side of Einstein's equation. The outcome is a modified version of the stress-energy tensor that is the source of the gravitational field. The derivation is based on the kinetic theory of a gas of identical particles with no internal structure. The presentation here is in two parts. In Part I, I describe the stress tensor that Xinzhong Chen and I have proposed for the matter tensor for a nonrelativistic gas with input from Hongling Rao and Jean-Luc Thiffeault. Our derivation of the equations of fluid dynamics is based on kinetic theory without recourse to the standard Chapman-Enskog approximation. In Part II, I present the analogous derivation of our form for the stress-energy tensor in the relativistic case. Then I exhibit its application to the usual isotropic cosmological model. The result of that, in addition to the Friedmann solution, is a second solution that arises from terms discarded in the usual Chapman-Enskog approximation. The new solution is a temporal analogue of a spatial shock wave. Just as the usual shock waves make transitions in properties within a mean free path, the new solution can change its properties appreciably in a mean flight time. Whereas the Friedmann solution is not dissipative, the new solution produces entropy at a rate that may be of cosmological interest. For the calculation of cosmic entropy production I use a formula derived in the ultrarelativistic limit in which particle masses are negligible. Independently of the cosmological aspects, the fluid dynamical equations that we derive are causal, even for the heat equation (or Fourier equation). Title: Causal relativistic fluid dynamics Authors: Chen, X.; Spiegel, E. A. Bibcode: 2011CQGra..28q5003C Altcode: 2011arXiv1107.0319C We derive causal relativistic fluid dynamical equations from the relaxation model of kinetic theory as in a procedure previously applied in the case of non-relativistic rarefied gases in Chen et al (2000 Phys. Lett. A 271 87). By treating space and time on an equal footing and avoiding the iterative steps of the conventional Chapman-Enskog (CE) method, we are able to derive causal equations in the first order of the expansion in terms of the mean flight time of the particles. This is in contrast to what is found using the CE approach. We illustrate the general results with the example of a gas of identical ultrarelativistic particles such as photons under the assumptions of homogeneity and isotropy. When we couple the fluid dynamical equations to Einstein's equation we find, in addition to the geometry-driven expanding solution of the FRW model, a second, matter-driven nonequilibrium solution to the equations. In only the second solution, entropy is produced at a significant rate. Title: Patterns of convection in solidifying binary solutions Authors: Keating, Shane; Spiegel, E. A.; Worster, M. G. Bibcode: 2011GApFD.105..304K Altcode: During the solidification of two-component solutions a two-phase mushy layer often forms consisting of solid dendritic crystals and solution in thermal equilibrium. Here, we extend previous weakly nonlinear analyses of convection in mushy layers to the derivation and study of a pattern equation by including a continuous spectrum of horizontal wave vectors in the development. The resulting equation is of the Swift-Hohenberg form with an additional quadratic term that destroys the up-down symmetry of the pattern as in other studies of non-Boussinesq convective pattern formation. In this case, the loss of symmetry is rooted in a non-Boussinesq dependence of the permeability on the solid-fraction of the mushy layer. We also study the motion of localized chimney structures that results from their interactions in a simplified one-dimensional approximation of the full pattern equation. Title: The Theory of Turbulence Authors: Spiegel, Edward A. Bibcode: 2011LNP...810.....S Altcode: No abstract at ADS Title: Spotted discs Authors: Bracco, A.; Provenzale, A.; Spiegel, E. A.; Yecko, P. Bibcode: 2010tbha.book..254B Altcode: No abstract at ADS Title: Destabilizing Taylor-Couette flow with suction Authors: Gallet, Basile; Doering, Charles R.; Spiegel, Edward A. Bibcode: 2010PhFl...22c4105G Altcode: 2009arXiv0911.2403G We consider the effect of radial fluid injection and suction on Taylor-Couette flow. Injection at the outer cylinder and suction at the inner cylinder generally result in a linearly unstable, steady, spiraling flow, even for cylindrical shears that are linearly stable in the absence of a radial flux. We study nonlinear aspects of the unstable motions with the energy stability method. Our results, though specialized, may have implications for drag reduction by suction, accretion in astrophysical disks, and perhaps even in the flow in Earth's polar vortex. Title: Cosmic vortices in hot stars and cool disks Authors: Spiegel, Edward A. Bibcode: 2010ThCFD..24...77S Altcode: The radiation that permits us to observe cosmic bodies also plays a role in their structure and evolution. While the thermal aspects of the radiation are familiar to fluid dynamicists, at least qualitatively, the dynamical effects of the radiation are perhaps less so, though these effects are becoming quite important in current astrophysical studies. This subject, which I have provisionally been calling photofluiddynamics after some discussion with the late James Lighthill, has a number of applications to cosmic objects. The most massive stars known are very hot and are the sites of vigorous fluid dynamical activity. The processes involved are of interest, not only in themselves, but also in the way they affect the observed features of the hottest stars by forming coherent vortices and magnetic flux tubes. Similar structures in accretion disks, particularly in protoplanetary systems, arise and play important roles in the evolution of those objects. Here, we shall consider only disks that, like the primitive solar nebula, are relatively cool and in which vortices may participate in the formation of planets. Title: Chaos and Intermittency in the Solar Cycle Authors: Spiegel, Edward A. Bibcode: 2009SSRv..144...25S Altcode: 2008SSRv..tmp..189S Where a magnetic flux tube of sufficient strength and cross section protrudes from the sun, convection is locally inhibited and a sunspot appears. The number of spots on the sun at any time varies in a cyclic, but aperiodic, manner. Models with chaos and intermittency can capture the main qualitative aspects of this temporal variability, especially if they display the mechanism of on-off intermittency. Capturing the spatio-temporal aspects of the sunspot cycle requires a more complicated model but a description in terms of waves of excitation seems promising. To clarify these possibilities, qualitative introductory remarks about chaos theory itself are included in this narrative. Title: Chaos and Intermittency in the Solar Cycle Authors: Spiegel, Edward A. Bibcode: 2009odsm.book...25S Altcode: Where a magnetic flux tube of sufficient strength and cross section protrudes from the sun, convection is locally inhibited and a sunspot appears. The number of spots on the sun at any time varies in a cyclic, but aperiodic, manner. Models with chaos and intermittency can capture the main qualitative aspects of this temporal variability, especially if they display the mechanism of on-off intermittency. Capturing the spatio-temporal aspects of the sunspot cycle requires a more complicated model but a description in terms of waves of excitation seems promising. To clarify these possibilities, qualitative introductory remarks about chaos theory itself are included in this narrative. Title: Modeling a Maunder minimum Authors: Brandenburg, A.; Spiegel, E. A. Bibcode: 2008AN....329..351B Altcode: 2008arXiv0801.2156B We introduce on/off intermittency into a mean field dynamo model by imposing stochastic fluctuations in either the alpha effect or through the inclusion of a fluctuating electromotive force. Sufficiently strong small scale fluctuations with time scales of the order of 0.3-3 years can produce long term variations in the system on time scales of the order of hundreds of years. However, global suppression of magnetic activity in both hemispheres at once was not observed. The variation of the magnetic field does not resemble that of the sunspot number, but is more reminiscent of the 10Be record. The interpretation of our results focuses attention on the connection between the level of magnetic activity and the sunspot number, an issue that must be elucidated if long term solar effects are to be well understood. Title: Reflections on the solar tachocline Authors: Spiegel, Edward A. Bibcode: 2007sota.conf...31S Altcode: No abstract at ADS Title: Phenomenological photofluiddynamics Authors: Spiegel, E. A. Bibcode: 2006EAS....21..127S Altcode: Using a variety of laboratory analogues and some simple models, I try to portray the fluid dynamics of very hot stars and disks. By analogy to fluidized beds, we may expect the formation of photon bubbles that stir things up into a sort of turbulence. Because of rotation, the bubbles merge to make vortices whose presence causes inhomogeneous radiative outflows and produces spots. These phenomena will allow the Eddington limit to be exceeded and, where the radiative outflows are locally large, mass loss will be encouraged. Title: Obituary: Kevin H. Prendergast, 1929-2004 Authors: Spiegel, Edward A. Bibcode: 2005BAAS...37.1555S Altcode: Kevin H. Prendergast, Emeritus Professor of Astronomy at Columbia University, died 8 September 2004 at the age of 75 from complications of lung cancer. He had been at Columbia for more than fifty years.

I first met Kevin in the summer of 1955, during a brief visit to the Yerkes Observatory. I had gotten into a heated discussion about double stars with a fellow graduate student, who suggested that we seek arbitration from a postdoc who was just then passing by. That postdoc was Kevin Prendergast. Kevin went straight to the blackboard, unleashed a learned and insightful lecture on binary stars, and then continued on his way. He wasted no motion, then nor ever, in our long association. Kevin was not at the time particularly concerned with double stars, though he made two significant contributions to their study somewhat after our meeting. The first of these was an early discussion (1960) of the dynamics of gaseous streams in binary systems that made use of theory gleaned from a book on the gulf stream by Henry Stommel (himself a former astronomer). The second was the important suggestion, made with G.R. Burbidge, that X-rays from binary stars are produced when gas from one star falls onto a compact companion (1968).

Kevin was a native of Brooklyn and, after a stint at Brooklyn Technical High School, he attended Columbia University for his undergraduate and graduate studies. He received the PhD in 1954 for an astrometry thesis under Jan Schilt. While attending Columbia, Kevin also studied at the Julliard School of Music, and he became a very accomplished musician. As a pianist, he was about as good as one can get and still be called an amateur, according to my musically knowledgeable friends.

From Columbia, he went to the Yerkes Observatory for postdoctoral work with S. Chandrasekhar and developed an interest in MHD. His model of a magnetic star with a global force-free field holds an important place in the subject of stellar magnetism. The relativistic solution for a magnetized expanding sphere that he later developed has recently been published posthumously through the efforts of Donald Lynden-Bell (MNRAS 359, 725).

By 1956, Kevin was an assistant professor at the University of Chicago and began teaching at the Yerkes Observatory. Norman Lebovitz, who was in one of his classes, has told me that often when the time came for Kevin's afternoon class, the students had to go and roust him out of bed so that he could give his lecture. Around then (1958) he produced another memorable paper, this one on the role of dissipation in the elastic tumbling of asteroids which led to a better understanding of their interesting light curves. This was one of seven papers that he published in the 1954-58 period, of which three were with Chandrasekhar. The productivity increased in 1959 when Kevin began a collaboration with the Burbidges on the determination and interpretation of rotation curves of galaxies. They produced well over twenty papers in the next eight years on this topic.

Kevin spent 1961-62 at the Institute for Advanced Studies in Princeton and 1962-63 at the Goddard Institute for Space Studies on a National Academy Fellowship. He returned to Columbia in 1963 as an associate professor. He was made full professor in 1966 and, when Lo Woltjer left to direct ESO in 1976, Kevin became Chairman of the Department of Astronomy, a position he held on two occasions for a total of seven years.

In 1968, with R.H. Miller, Kevin began developing numerical schemes to study dynamics in disk galaxies. One of their main ideas was to discretize the phase space so as to remove the irreversibility found in many simulations of stellar dynamics. They also developed a gas dynamical procedure (``the beam scheme'') which made clever use of the moments of the discretized kinetic equation. With Kevin's student W.J. Quirk, they put together a simulation with gas and stars, and even introduced a star formation algorthim. They produced films of galactic evolution that were shown quite widely in colloquia and symposia. The films revealed phenomena of qualitative interest such as mergers, bridges, and tails, and the formation of bars. Similar results were also being obtained by Hohl around that time and both pieces of work were no doubt influential in shaping the thinking of people working in this field. One striking feature of the calculations was that spiral arms formed initially but were transient. To keep the spiral patterns from collapsing it seemed necessary to artificially heat the disks. Only later, when the existence of massive halos was recognized (by Ostriker and Peebles), could the true cause of stability be surmised.

From the mid-seventies on, Kevin worked on topics in astrophysical fluid dynamics and applied mathematics, largely with students. Some of this work was published, but it has to be said that much of his best work was not. A good example of the latter is his three-part handwritten manuscript on the dynamics of barred spirals that he distributed to several people over thirty years ago. Many of his other unpublished calculations have been deposited in the Columbia Library, and there are no doubt several things of interest to be found among his papers.

While one can only speculate on why so much of his work went unpublished, I find a remark by de Kooning quite helpful in thinking about it. In a review of book about the painter, Peter Schjeldahl reported that "He [de Kooning] made ...paintings...and destroyed nearly all of them, to his subsequent regret....He explained `I was so modest then that I was vain.'" When I accused Kevin of a similar mindset, he chuckled and said "You are right, but don't tell anyone."

Kevin was widely read and he had a remarkable awareness ofliterature. He was especially devoted to the work of P.G. Wodehouse. He also loved the Marx Brothers and late in life discovered Zero Mostel of whom he became an instant fan. He was a sailor and a snorkler, and enjoyed trading quips with anyone who was worthy of his steel. He was, in short, a person worth knowing.

Kevin is survived by his wife Jane, two daughters, Laura and Cathy, and a younger brother, Robert, an emeritus professor of medicine from Johns Hopkins who rowed too much. Title: Continuum equations for stellar dynamics Authors: Spiegel, Edward A.; Thiffeault, Jean-Luc Bibcode: 2003safd.book..377S Altcode: 2002astro.ph.10185S The description of a stellar system as a continuous fluid represents a convenient first approximation to stellar dynamics, and its derivation from the kinetic theory is standard. The challenge lies in providing adequate closure approximations for the higher-order moments of the phase-space density function that appear in the fluid dynamical equations. Such closure approximations may be found using representations of the phase-space density as embodied in the kinetic theory. In the classic approach of Chapman and Enskog, one is led to the Navier-Stokes equations, which are known to be inaccurate when the mean free paths of particles are long, as they are in many stellar systems. To improve on the fluid description, we derive here a modified closure relation using a Fokker-Planck collision operator. To illustrate the nature of our approximation, we apply it to the study of gravitational instability. The instability proceeds in a qualitative manner as given by the Navier-Stokes equations but, in our description, the damped modes are considerably closer to marginality, especially at small scales. Title: Luminosity effects in projected fractals Authors: Thieberger, R.; Spiegel, E. A. Bibcode: 2002BASI...30..461T Altcode: 2002astro.ph..4485T The use of two-dimensional catalogues in unraveling the largescale distribution of extra-galactic objects can reveal more information than has been supposed if the objects have approximate scaling properties such as observations suggest. After a brief general discussion of this issue, we turn to specific examples of projected fractals for the case where the objects studied have a Schechter luminosity function. We analyze the effects of projection on the characteristics of such a fractal distribution. Our results indicate that two-dimensional catalogues of sources could be of value in detecting the effects of luminosity functions as well as of large-scale structure. Title: Shear and Mixing in Oscillatory Doubly Diffusive Convection Authors: Paparella, Francesco; Spiegel, Edward A.; Talon, Suzanne Bibcode: 2002GApFD..96..271P Altcode: 2002astro.ph..3282P To investigate the mechanism of mixing in oscillatory doubly diffusive (ODD) convection, we truncate the horizontal modal expansion of the Boussinesq equations to obtain a simplified model of the process. In the astrophysically interesting case with low Prandtl number (traditionally called semiconvection), large-scale shears are generated as in ordinary thermal convection. The interplay between the shear and the oscillatory convection produces intermittent overturning of the fluid with significant mixing. By contrast, in the parameter regime appropriate to sea water, large-scale flows are not generated by the convection. However, if such flows are imposed externally, intermittent overturning with enhanced mixing is observed. Title: Continuum description of rarefied gas dynamics. III. The structures of shock waves Authors: Chen, Xinzhong; Spiegel, Edward A.; Rao, Hongling Bibcode: 2002PhRvE..65c6304C Altcode: 2001astro.ph..5348C We use the one-dimensional steady version of the equations derived in paper I to compute the structure of shock waves and find good agreement with experiment. Title: Continuum description of rarefied gas dynamics. I. Derivation from kinetic theory Authors: Chen, Xinzhong; Rao, Hongling; Spiegel, Edward A. Bibcode: 2001PhRvE..64d6308C Altcode: 2001astro.ph..5346C We describe an asymptotic procedure for deriving continuum equations from the kinetic theory of a simple gas. As in the works of Hilbert, of Chapman, and of Enskog, we expand in the mean flight time of the constituent particles of the gas, but we do not adopt the Chapman-Enskog device of simplifying the formulas at each order by using results from previous orders. In this way, we are able to derive a new set of fluid dynamical equations from kinetic theory, as we illustrate here for the relaxation model for monatomic gases. We obtain a stress tensor that contains a dynamical pressure term (or bulk viscosity) that is process dependent and our heat current depends on the gradients of both temperature and density. On account of these features, the equations apply to a greater range of Knudsen number (the ratio of mean free path to macroscopic scale) than do the Navier-Stokes equations, as we see in the accompanying paper. In the limit of vanishing Knudsen number, our equations reduce to the usual Navier-Stokes equations with no bulk viscosity. Title: Continuum description of rarefied gas dynamics. II. The propagation of ultrasound Authors: Chen, Xinzhong; Rao, Hongling; Spiegel, Edward A. Bibcode: 2001PhRvE..64d6309C Altcode: 2001astro.ph..5347C The equations of fluid dynamics developed in Paper I [X. Chen, H. Rao, and E. A. Spiegel, Phys. Rev. E 64, 46308 (2001)] are applied to the study of the propagation of ultrasound waves. There is good agreement between the predicted propagation speed and experimental results for a wide range of Knudsen numbers. Title: Radiative bulk viscosity Authors: Chen, Xinzhong; Spiegel, Edward A. Bibcode: 2001MNRAS.323..865C Altcode: 2001astro.ph..2022C Viscous resistance to changes in the volume of a gas arises when different degrees of freedom have different relaxation times. Collisions tend to oppose the resulting departures from equilibrium and, in so doing, generate entropy. Even for a classical gas of hard spheres, when the mean free paths or mean flight times of constituent particles are long we find a non-vanishing bulk viscosity. Here we apply a method recently used to uncover this result for a classical rarefied gas to radiative transfer theory, and derive an expression for the radiative stress tensor for a grey medium with absorption and Thomson scattering. We determine the transport coefficients through the calculation of the comoving entropy generation. When scattering dominates absorption, the bulk viscosity may be much larger than either the shear viscosity or the thermal conductivity. Title: Resonant Vibrational Instabilities in Magnetized Stellar Atmospheres Authors: Birch, A. C.; Kosovichev, A. G.; Spiegel, E. A.; Tao, L. Bibcode: 2001SoPh..199..291B Altcode: We perform linear stability analysis on stratified, plane-parallel atmospheres in uniform vertical magnetic fields. We assume perfect electrical conductivity and we model non-adiabatic effects with Newton's law of radiative cooling. Numerical computations of the dispersion diagrams in all cases result in patterns of avoided crossings and mergers in the real part of the frequency. We focus on the case of a polytrope with a prevalent, relatively weak, magnetic field with overstable modes. The growth rates reveal prominent features near avoided crossings in the diagnostic diagram, as has been seen in related problems (Banerjee, Hasan, and Christensen-Dalsgaard, 1997). These features arise in the presence of resonant oscillatory bifurcations in non-self adjoint eigenvalue problems. The onset of such bifurcations is signaled by the appearance of avoided crossings and mode mergers. We discuss the possible role of the linear stability results in understanding solar spicules. Title: The Radiative Stress Tensor Authors: Chen, Xinzhong; Spiegel, Edward A. Bibcode: 2000ApJ...540.1069C Altcode: 2000astro.ph..3469C We use the transfer equation in relativistic form to develop an expansion of the one-photon distribution for a medium with constant photon mean free path, ɛ. On carrying out appropriate integrations and manipulations, we convert this expansion into one for the frequency-integrated intensity. We regroup the terms of the intensity expansion according to both the power of ɛ and the angular structure of the various terms and then carry out angle integrations to obtain the expansions for the components of the stress energy tensor: the radiative energy density, the radiative flux, and the pressure tensor. In leading order, we recover Thomas' results for the viscosity tensor and his expression for the viscosity coefficient, which are correct for short mean free paths. As had been done earlier for the radiative heat equation, we keep at each order in the expansion a dominant portion, but this time one with a richer angular structure. Then, after some rearrangement of the various summations in the expressions for the moments, we replace the sum of the calculated higher order terms by a Padé approximant, or rational approximation, to provide an improved closure approximation for the radiative stress tensor. The resulting radiative viscosity tensor may be expressed either as a simple integral operator acting on the Thomas stress tensor or as the solution of an inhomogenous, linear partial differential equation. The expression obtained for the radiative viscosity tensor applies for media with long, as well as short, photon mean free paths. We also develop results applicable for relatively smooth flows by using the form of the Thomas stress tensor with generalized transport coefficients derived by the application of a suitable operator to the bare Thomas coefficients. Title: Particle aggregation in a turbulent Keplerian flow Authors: Bracco, A.; Chavanis, P. H.; Provenzale, A.; Spiegel, E. A. Bibcode: 1999PhFl...11.2280B Altcode: 1998astro.ph.10336B For the problem of planetary formation one seeks a mechanism to gather small dust particles together into larger solid objects. Here we describe a scenario in which turbulence mediates this process by aggregating particles into anticyclonic regions. If, as our simulations suggest, anticyclonic vortices form as long-lived coherent structures, the process becomes more powerful because such vortices trap particles effectively. Even if the turbulence is decaying, following the upheaval that formed the disk, there is enough time to make the dust distribution quite lumpy. Title: Instability of Nonadiabatic Oscillations in a Magnetized Polytrope Authors: Birch, A. C.; Kosovichev, A. G.; Spiegel, E. A.; Tao, L. Bibcode: 1999AAS...194.9307B Altcode: 1999BAAS...31S.989B We apply linear stability analysis to a stratified plane parallel perfectly conducting atmosphere with a vertical magnetic field and Newton's law of cooling. For the case of a polytropically stratified atmosphere we find that there are overstable modes. The instability is due to the background temperature gradient combined with the cooling. We show perturbation results for the case of the cooling time much longer or much shorter than the oscillation period. Numerical calculations of the dispersion relations are shown for the weak and strong magnetic field cases, for long and short cooling times, and for polytrope and isothermal atmospheres. Generically the k-omega diagrams show a complicated pattern of avoided crossings. The damping or growth rates in general show features where branches undergo avoided crossings. The results, in particular the existence of overstable modes in the polytrope atmosphere, may be significant to the investigation of MHD phenomena in solar and stellar atmospheres, for example solar spicules. Title: A Resummed Radiative Stress Tensor Authors: Chen, X.; Spiegel, E. A. Bibcode: 1999AAS...194.3208C Altcode: 1999BAAS...31R.871C We extend Thomas' (1930) development of the solution of the equation of radiative transfer to include terms of third order in photon mean free path. The sum of the calculated higher order terms is then replaced by a Pade approximant (or rational approximation; see Baker, 1975), to provide an improved closure approximation for the radiative stress tensor, as has been done for the kinetic theory of gases by Rosenau (1993) and Slemrod (1997), and for the radiative heat equation in a grey medium (Unno and Spiegel, 1966). The results are uniformly valid in mean free path and so work well for both the optically thick and thin limits. Using simple examples, we contrast the results obtained for the transmission of shear stresses with those found with the conventional form for the radiative viscosity. These illustrate how the usual form of viscous tensor fails for transparent media and is rectified by the resummation procedure. Similarly, we exhibit the formula obtained for the radiative bulk viscosity and use it to extend the calculation of the cosmological entropy to cases with long photon mean free paths. Title: Photofluid instabilities of hot stellar envelopes. Authors: Spiegel, E. A.; Tao, L. Bibcode: 1999PhR...311..163S Altcode: 1998astro.ph.10225S Beginning from a relatively simple set of dynamical equations for a fluid permeated by a radiative field strong enough to produce significant forces, the authors find the structure of plane-parallel equilibria and study their stability to small acoustic disturbances. In doing this, they neglect viscous effects and complications of nongrayness. They find that acoustic instabilities occur over a wide range of conditions below the Eddington limit. This result is in line with findings reported 20 years ago but it contradicts a more recent report of the absence of instabilities. The authors briefly attempt to identify the causes of the discrepancies and then close with allusions to the possible astrophysical interest of such instabilities. Title: Gravitational Screening Authors: Spiegel, E. A. Bibcode: 1999oep..conf..465S Altcode: 1998astro.ph..1014S Gravitational Stopping Power A Drag Crisis Saved by Self-Gravity The Message Is The Medium Title: Negative Energy Modes and Gravitational Instability of Interpenetrating Fluids. Authors: Casti, A. R. R.; Morrison, P. J.; Spiegel, E. A. Bibcode: 1998NYASA.867...93C Altcode: 1998astro.ph..7310C; 1998ndca.conf...93C We study the longitudinal instabilities of two interpenetrating fluids interacting only through gravity. When one of the constituents is of relatively low density, it is possible to have a band of unstable wave numbers well separated from those involved in the usual Jeans instability. If the initial streaming is large enough, and there is no linear instability, the indefinite sign of the free energy has the possible consequence of explosive interactions between positive and negative energy modes in the nonlinear regime. The effect of dissipation on the negative energy modes is also examined. Title: Stellar Oscillons. Authors: Umurhan, O. M.; Tao, L.; Spiegel, E. A. Bibcode: 1998NYASA.867..298U Altcode: 1998ndca.conf..298U; 1998astro.ph..6209U We study the weakly nonlinear evolution of acoustic instability of a plane- parallel polytrope with thermal dissipation in the form of Newton's law of cooling. The most unstable horizontal wavenumbers form a band around zero and this permits the development of a nonlinear pattern theory leading to a complex Ginzburg-Landau equation (CGLE). Numerical solutions for a subcritical, quintic CGLE produce vertically oscillating, localized structures that resemble the oscillons observed in recent experiments of vibrated granular material. Title: A Nonlinear Model for Solar Spicules Authors: Birch, A. C.; Spiegel, E. A.; Tao, L.; Kosovichev, A. G. Bibcode: 1998AAS...19310001B Altcode: 1998BAAS...30.1397B We develop a weakly nonlinear theory for the growth of optically thin perturbations to a plane-parallel chromosphere-like layer described by the MHD equations in an attempt to explain the dynamics of solar spicules. Damping of the perturbations is by Newton's law of cooling and the magnetic field is assumed to be vertical. The theory leads to a nonlinear PDE that describes the time evolution of perturbations to the layer. Numerical results are presented. Title: Scaling Regimes in the Distribution of Galaxies. Authors: Murante, G.; Provenzale, A.; Spiegel, E. A.; Thieberger, R. Bibcode: 1998NYASA.867..258M Altcode: 1998astro.ph..9116M; 1998ndca.conf..258M If we treat the galaxies in published redshift catalogues as point sets, we may determine the generalized dimensions of such sets by standard means, outlined here. For galaxy separations up to about 5 Mpc, we find the dimensions of the galaxy set to be about 1.2, with not a strong indication of multifractality. For larger scales, out to about 30 Mpc, there is also good scaling with a dimension of about 1.8. For even larger scales, the data seem too sparse to be conclusive, but we find that the dimension is climbing as the scales increase. We report simulations that suggest a rationalization of such measurements, namely that in the intermediate range the scaling behavior is dominated by flat structures (pancakes) and that the results on the smallest scales are a reflection of the formation of density singularities. Title: Spotted disks. Authors: Bracco, A.; Provenzale, A.; Spiegel, E. A.; Yecko, P. Bibcode: 1998tbha.conf..254B Altcode: 1998astro.ph..2298B; 1998npad.conf..254B Rotating, turbulent cosmic fluids are generally pervaded by coherent structures such as vortices and magnetic flux tubes. The formation of such structures is a robust property of rotating turbulence as has been confirmed in computer simulations and laboratory experiments. The authors defend here the notion that accretion disks share this feature of rotating cosmic bodies. In particular, they show that the intense shears of Keplerian flows do not inhibit the formation of vortices. Given suitable initial disturbances and high enough Reynolds numbers, long-lived vortices form in Keplerian shear flows and analogous magnetic structures form in magnetized disks. The formation of the structures reported here should have significant consequences for the transport properties of disks and for the observed properties of hot disks. Title: Gravitational Instability of Interpenetrating Fluids Authors: Casti, A.; Morrison, P.; Spiegel, E. Bibcode: 1997AAS...191.8601C Altcode: 1997BAAS...29.1352C We examine the gravitational instability of two interpenetrating, barotropic fluids interacting only through gravity. Previous work on the problem (Fridman and Polyachenko, 1984) indicates that two fluids with initial relative motion are unstable to small perturbations at all wavenumbers in some range of Mach numbers. We expand upon these results by exploiting the analogy between this problem and the two-stream instability of plasma physics, and demonstrate the existence of negative energy modes from the indefiniteness of the energy functional. This allows for explosive nonlinear growth even in situations for which the linear theory predicts absolute stability. We further investigate the effects of negative energy modes in a non-conservative situation where one of the fluids is radiatively damped. reference: "Physics of Gravitating Systems I and II," by A.M. Fridman & V.L. Polyachenko, Springer-Verlag, 1984. Title: Acoustic Overstability of Plane Parallel Polytropes: an Astrophysical Thermoacoustic Engine Authors: Umurhan, O. M.; Spiegel, E. A. Bibcode: 1997AAS...191.1208U Altcode: 1997BAAS...29.1228U We examine the stability of nonadiabatic acoustic modes in a plane parallel atmosphere in a constant gravitational field where the source of nonadiabaticity is thermal conduction with constant coefficient. This prescription filters out the effects of the kappa -mechanism. The static state is described by polytrope solutions where the density is related to the linear temperature profile via a power law relation described by the usual polytropic index m. We solve the resulting fourth order boundary-value/eigenvalue problem describing perturbations by utilizing a variety of asymptotic and numerical methods. Disturbances are shown to be unstable under a variety of parameter regimes including conditions where the background polytrope is either super- or subadiabatic. Two main instabilities emerge and in one of them its resemblance to the thermoacoustic engine is drawn. We also find that it is most clearly heard when the conduction parameter is strong. For this extreme circumstance marginal stability conditions are derived and are used to motivate and derive weakly nonlinear amplitude equations for this problem. Title: Density singularities and cosmic structures Authors: Murante, G.; Provenzale, A.; Spiegel, E. A.; Thieberger, R. Bibcode: 1997MNRAS.291..585M Altcode: 1997astro.ph..4188M We show that a random superposition of singular density concentrations accounts for the scaling exponents observed in the luminous matter distribution in the Universe for scales below 10 Mpc. This model provides a good description of the matter distribution on those smaller scales. Title: Cosmic lacunarity Authors: Provenzale, A.; Spiegel, E. A.; Thieberger, R. Bibcode: 1997Chaos...7...82P Altcode: The present distribution of galaxies in space is a remnant of their formation and interaction. On a large enough scale, we may represent the galaxies as a set of points and quantify the structures in this set by its generalized dimensions [Beck and Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1986); Paladin and Vulpiani, Phys. Rep. 156, 147 (1987)]. The results of such evaluation are often taken to be evidence of a fractal (or multifractal) distribution of galaxies. However, those results, for some scales, may also reveal the presence of singularities formed in the gravitational processes that produce structure in the galaxy distribution. To try to make some decision about this issue, we look for the more subtle galactic lacunarity. We believe that this quantity is discernible in the currently available data and that it provides important evidence on the galaxy formation process. Title: Catastrophes, chaos and cycles Authors: Spiegel, E. A. Bibcode: 1997ppvs.conf..311S Altcode: Astromathematics Sensitive systems Gradient systems Excitability Chaotic systems Intermittency The macculate Sun Variable solar maccularity The solar tachocline Solar solitoids Cycles A model solar oscillation Solar activity waves Title: Through a Glass Darkly: Distinguishing Chaotic from Stochastic Resonance. Authors: Graf von Hardenberg, J.; Paparella, F.; Provenzale, A.; Spiegel, E. A. Bibcode: 1997NYASA.808...79G Altcode: 1997nsia.conf...79G No abstract at ADS Title: Nonlinear ringing of polytropic disks. Authors: Balmforth, N. J.; Spiegel, E. A. Bibcode: 1996PhyD...97....1B Altcode: The authors derive equilibria and axisymmetric normal modes for a rotating polytropic fluid under the influence of both its own gravity and that of a massive halo. By tuning the ratio of the self-gravity to imposed gravity, they find sequences of models that pass through the point of instability. They then construct amplitude equations in Hamiltonian form for the weakly unstable modes. When there are two unstable modes, one recovers the Hénon-Heiles system as a special case. The authors close with a brief exploration of the effects of the many neutrally stable modes in such systems. Title: Chaotic Dynamics of the Solar Cycle Authors: Spiegel, Edward A.; Zahn, Jean-Paul Bibcode: 1996cuny.rept.....S Altcode: The cyclic variation of solar activity is both irregular and intermittent. We have sought to isolate and illuminate the physical mechanisms of this behavior and to provide a mathematical description of it. Our work has brought out three ingredients of the solar cycle that we believe to be central to its operation. (1) The seat of the solar cycle is in a shear layer just below the solar convection zone. We have investigated the structure of this layer (which we call the solar tachocline) in some detail. (2) The spatio-temporal development of the solar cycle is represented by the propagation of robust solitary waves which are affected by dissipation and instability. We have studied the structure and interactions of such waves, which we call solitoids. (3) On top of the simple propagative behavior of the solar solitoids there are intermissions during which the number of sunspots remains quite small. We attribute these intermissions (such as the Maunder minimum) to a form of interaction between the convection zone and the tachocline which is characteristic of a process that we have developed and that we call on/off intermittency. These three ingredients make up some of the key features of the solar cycle and may be expected to play a role in future simulations of the solar cycle. Title: Sinuous oscillations and steady WARPS of polytropic disks Authors: Balmforth, N. J.; Spiegel, E. A. Bibcode: 1995STIN...9534216B Altcode: In an asymptotic development of the equations governing the equilibria and linear stability of rapidly rotating polytropes we employed the slender aspect of these objects to reduce the three-dimensional partial differential equations to a somewhat simpler, ordinary integro-differential form. The earlier calculations dealt with isolated objects that were in centrifugal balance, that is the centrifugal acceleration of the configuration was balanced largely by self gravity with small contributions from the pressure gradient. Another interesting situation is that in which the polytrope rotates subject to externally imposed gravitational fields. In astrophysics, this is common in the theory of galactic dynamics because disks are unlikely to be isolated objects. The dark halos associated with disks also provide one possible explanation of the apparent warping of many galaxies. If the axis of the highly flattened disk is not aligned with that of the much less flattened halo, then the resultant torque of the halo gravity on the disk might provide a nonaxisymmetric distortion or disk warp. Motivated by these possibilities we shall here build models of polytropic disks of small but finite thickness which are subjected to prescribed, external gravitational fields. First we estimate how a symmetrical potential distorts the structure of the disk, then we examine its sinuous oscillations to confirm that they freely decay, hence suggesting that a warp must be externally forced. Finally, we consider steady warps of the disk plane when the axis of the disk does not coincide with that of the halo. Title: Instability of rapidly rotating polytropes Authors: Balmforth, N. J.; Howard, L. N.; Spiegel, E. A. Bibcode: 1995SJAM...55..298B Altcode: The linear stability theory of rapidly rotating, self-gravitating polytropes is developed by an asymptotic, shallow-layer method. This reduces the general three-dimensional stability problem to an integro-differential eigenvalue problem (a Fredholm integral equation of the second kind) for normal modes. At leading order, the asymptotic analysis produces familiar, zero-thickness disk equations. In subsequent orders, stabilizing effects due to compressibility enter. We solve the stability equations numerically and construct approximate solutions using short-wavelength arguments. The eigenspectrum of a disk can have various forms; instabilities of a pressure-less configuration form a continuous piece of the spectrum, but polytropic disks can have discrete, unstable eigenvalues. A further example is provided by the rigidly rotating disk, which, at leading order, can be solved exactly. Title: Sinuous modes and steady warps of polytropic disks. Authors: Balmforth, N. J.; Spiegel, E. A. Bibcode: 1995NYASA.773...55B Altcode: The authors build models of polytropic galactic disks of small but finite thickness that are subjected to prescribed, external gravitational fields. First they estimate how a symmetrical potential distorts the structure of the disk, then they examine its sinuous modes to confirm that they are stable, hence suggesting that a warp must be externally forced. Finally the authors consider steady warps of the disk plane when the axis of the disk does not coincide with that of the halo. Title: A prelude to stellar convection theory Authors: Spiegel, Edward A. Bibcode: 1995LNP...458..129S Altcode: No abstract at ADS Title: Autogravity waves in a polytropic layer Authors: Qian, Z. S.; Spiegel, E. A. Bibcode: 1994GApFD..74..225Q Altcode: We study nonlinear waves in polytropic slabs under the joint influence of both self-gravitational and externally imposed gravitational fields. In the long wavelength limit, these obey nonlinear evolution equations akin to the Boussinesq and the KdV equations, with modifications resulting from the effects of self-gravity, or autogravity. We present pole solutions of these equations, including, in particular, a travelling solitary wave. These are surface gravity waves that we call autogravity waves because they are excited by selfgravity. Title: The Chaotic Solar Cycle Authors: Spiegel, E. A. Bibcode: 1994lspd.conf..245S Altcode: No abstract at ADS Title: Patterns of Aperiodic Pulsation Authors: Spiegel, E. A. Bibcode: 1993Ap&SS.210...33S Altcode: 1993IAUCo.134...33S Techniques for deriving amplitude equations for stellar pulsation are outlined. For the simplest such equations with multiple instabilities, the derivation of a map for the patterns of pulsation phases is described. This map gives the time between two successive maxima of pulsation in terms of the time between the previous pair, under suitable conditions. The phase differences can be regular, chaotic or hyperchaotic. Title: Chaotic dynamics of the solar cycle Authors: Spiegel, Edward A. Bibcode: 1993cuny.reptR....S Altcode: A dynamical model of the solar activity cycle has been developed that incorporates the key chaotic mechanism of on/off intermittence. It has been proposed that the solar butterfly diagram represents the motion of activity waves which are very stable nonlinear solitary waves. The waves are driven by an instability under the convection zone in a layer called the tachocline. The result are for one wave in one solar hemisphere but show the correct spatio-temporal dynamics. A new approach to dynamo theory is being developed based on theoretical techniques developed in chaos theory called the thermodynamic formalism. Possible sources of activity in other turbulent, rotating stars has been studied. Title: Equilibria of rapidly rotating polytropes Authors: Balmforth, N. J.; Howard, L. N.; Spiegel, E. A. Bibcode: 1993MNRAS.260..253B Altcode: The equilibrium structures of self-gravitating polytropes are found in the limit of rapid rotation by the method of matched asymptotic expansions. For physical values of the polytropic index, the discs are thin throughout most of their radii, but they flare out at their edges where the local Jeans length of the configuration becomes large. This signals a breakdown in the asymptotic expansion that we correct by including a boundary layer at the rim of the disc. A variety of results are illustrated for differing rotation curves and polytropic indices. Title: On-off intermittency: A mechanism for bursting Authors: Platt, N.; Spiegel, E. A.; Tresser, C. Bibcode: 1993PhRvL..70..279P Altcode: On-off intermittency is an aperiodic switching between static, or laminar, behavior and chaotic bursts of oscillation. It can be generated by systems having an unstable invariant (or quasi-invariant) manifold, within which is found a suitable attractor. We clarify the roles of such attractors in producing intermittency, provide examples, and relate them to previous work. Title: Advection of Vector Fields by Chaotic Flows Authors: Balmforth, N. J.; Cvitanović, P.; Ierley, G. R.; Spiegel, E. A.; Vattay, G. Bibcode: 1993NYASA.706..148B Altcode: 1993chao.dyn..7011B We have introduced a new transfer operator for chaotic flows whose leading eigenvalue yields the dynamo rate of the fast kinematic dynamo and applied cycle expansion of the Fredholm determinant of the new operator to evaluation of its spectrum. The theory hs been tested on a normal form model of the vector advecting dynamical flow. If the model is a simple map with constant time between two iterations, the dynamo rate is the same as the escape rate of scalar quantties. However, a spread in Poincaré section return times lifts the degeneracy of the vector and scalar advection rates, and leads to dynamo rates that dominate over the scalar advection rates. For sufficiently large time spreads we have even found repellers for which the magnetic field grows exponentially, even though the scalar densities are decaying exponentially. Title: The intermittent solar cycle Authors: Platt, N.; Spiegel, E. A.; Tresser, C. Bibcode: 1993GApFD..73..147P Altcode: A prominent feature of the solar cycle is the rise and fall of the number of sunspots on the surface with a timescale of approximately eleven years. The mathematical description of this behavior is complicated by the interruption of the cycle for 75 years starting around 1650. Similar previous intermissions of this kind are implied by the available data. We explore the possibility of modeling such temporal variations of the sunspot number with a deterministic dynamical system of relatively low order. The system we propose manifests on/off intermittency in which the cyclic variations of the solar activity switch off almost completely for extended periods. We also offer an explanation of the variation of the fluctuating part of the sunspot number over the cycle. Title: Astrophysical fluid dynamics. Authors: Spiegel, E. A. Bibcode: 1993afd..conf....1S Altcode: Contents: 1. Kinematics of the continuum. 2. Fluid dynamics. 3. Atmospheric waves. 4. Cosmology's fictitious forces. 5. Conclusion. Title: The solar tachocline. Authors: Spiegel, E. A.; Zahn, J. -P. Bibcode: 1992A&A...265..106S Altcode: Acoustic sounding of the Sun reveals that the variation of angular velocity with latitude is independent of depth in the convection zone. By contrast, deep within the radiative zone, the rotation appears to be rigid. The transition between the two rotation laws occurs in a thin, unresolved layer that the authors here call the tachocline. This paper is an examination of the structure and previous evolution of this layer. It is assumed that the stress exerted by the convection zone is prescribed, much as oceanographers take the wind stress on the sea surface as given. It is concluded that the helioseismic observations are best rationalized by a scenario in which, after an initial adjustment or spindown period, the subconvective rotation settles into a quasisteady state with a turbulent boundary layer. In the tachocline, the advection of angular momentum is controlled by horizontal turbulence. If this turbulence is intense enough, the tachocline is thin and is unresolved. Title: Chaotic dynamics of the solar cycle Authors: Spiegel, Edward A. Bibcode: 1992cuny.reptR....S Altcode: In modeling the solar cycle, we proceed on the assumption that the processes driving the solar cycle are deterministic. In that case, a chaotic model is a good choice for a description of its complexity. In our modeling, we suppose that the solar activity variation is composed of two distinct, coupled processes: one a conventionally chaotic system, and the other a nonlinear oscillator. This idea comes directly from our analysis of the observations. Since the sun's rotation period is one month, we do not use the daily sunspot number, but work with its monthly average. This quantity shows both the cyclic variation on the eleven year time scale and additional strong fluctuations. If we smooth the data to remove periods less than a year to two, we see the solar cycle clearly exposed. When we subtract this smoothed sunspot number from the monthly average, we obtain the fluctuations in the sunspot number. We show a comparison between the monthly averaged number and the fluctuations for a few cycles. There is a clear correlation between the level of solar activity, as measured by the sunspot number, and the amplitude of the fluctuations in this number. We suggest that the fluctuations and the cyclic behavior correspond to two distinct but interacting processes. Title: Vortices on accretion disks Authors: Abramowicz, M. A.; Lanza, A.; Spiegel, E. A.; Szuszkiewicz, E. Bibcode: 1992Natur.356...41A Altcode: 1992Nat...356...41A EVERY rotating cosmic fluid that can be observed sufficiently closely displays either vortices or magnetic flux tubes on its surface; examples are tornadoes in the Earth's atmosphere1, the Great Red Spot and other vortices in Jupiter's atmosphere, and sunspots. We suggest here that hot accretion disks also produce coherent objects, and that these vortices and magnetic flux tubes will cause significant dissipation and other observable physical effects. They will facilitate the escape of collimated radiation from deep within hot disks, producing spectral changes and time variability in the radiation from the disk. In the case of active galactic nuclei, modification of X-ray spectra due to the presence of vortices on accretion disks permits us to explain several observational puzzles, including short-term variability and the low degree of linear polarization. Title: A Model for the X-Ray Variability of the Seyfert Galaxy NGC6814 Authors: Abramowicz, M. A.; Bao, G.; Fiore, F.; Lanza, A.; Massaro, E.; Perola, G. C.; Spiegel, E. A.; Szuszkiewicz, E. Bibcode: 1992pagn.conf...61A Altcode: No abstract at ADS Title: The Turbulent Tachycline Authors: Spiegel, E. A.; Zahn, J. -P. Bibcode: 1992ASPC...26..188S Altcode: 1992csss....7..188S No abstract at ADS Title: Global equilibria of turbulent accretion disks. Authors: Balmforth, N. J.; Meacham, S. P.; Spiegel, E. A.; Young, W. R. Bibcode: 1992NYASA.675...53B Altcode: 1992naad.work...53B In the study of accretion disks around central objects with specified gravitational potentials, it is commonly assumed that the distribution of the mean azimuthal velocity is Keplerian. A similar assumption of centrifugal balance underlies the usual determinations of galactic mass distributions from rotation curves. The authors offer a justification for this assumption for the case of fully turbulent disk, starting from ideas about the general influence of turbulence on the mean state of a fluid. Title: Complexity from thermal instability Authors: Elphick, Christian; Regev, Oded; Spiegel, E. A. Bibcode: 1991MNRAS.250..617E Altcode: The consequences of the basic nonlinear equation for thermal instability, with the cooling function represented analytically, are explored in some depth. Ths equation leads to a dissipative analog of the virial theorem that has import for the unfolding of the instability as a phase separation process. This is modeled with a one-dimensional cloudy medium and equations of motion are derived for the fronts separating the hot and cold (locally) stable phases. It is found that fronts come together and annihilate, snuffing out the 'cloud' they enclosed. The lifetime of an isolated cloud is an exponentially increasing function of the cloud size. Hence, the medium exhibits an inverse cascade, with increasingly larger clouds predominating, and arrives finally at a state with one or no surviving fronts. To sustain structure in the medium, a spatially periodic excitation source is introduced, leading, for appropriate choices of parameters, to a state of spatial chaos. Title: On thermonuclear convection: I shellular instability Authors: Ghosal, Sandip; Spiegel, Edward A. Bibcode: 1991GApFD..61..161G Altcode: As the sun evolves, a sharp compositional peak of He3 builds up in the core. Nuclear reactions involving He3 are very temperature sensitive, as a result, this He3 layer is susceptible to thermal instability. The small horizontal wavenumber g-modes have large time scales, comparable to the thermal time scale. Using a two-layer model, we find that such "shellular modes" are the most unstable. As a result of nuclear heating, these modes may be excited in the solar core in a shallow layer confined to the He3 zone. A possible effect of such shellular convection on the solar neutrino problem is discussed. In this paper we discuss the linear theory; the nonlinear effects will be treated in a subsequent paper. Title: Waves of Solar Activity Authors: Proctor, M. R. E.; Spiegel, E. A. Bibcode: 1991LNP...380..117P Altcode: 1991IAUCo.130..117P; 1991sacs.coll..117P We develop a theory of the sunspot cycle predicated on the assumption that the observed bands of activity are packets of dynamo waves. An approximate equation is proposed to describe the dynamics of these packets, using standard ideas from bifurcation theory. We show that in a certain limit the system can be described in terms of a slowlyevolving solitary wave, and that periodic behavior, like that of the observed butterfly diagram, can easily be found. Generalizations of the theory are discussed. Title: Stellar and Jovian vortices. Authors: Dowling, T. E.; Spiegel, E. A. Bibcode: 1990NYASA.617..190D Altcode: 1990nafd.conf..190D The authors begin their discussion with the basic problem of vortex maintenance. With this as background, they then examine what is known observationally about Jupiter's Great Red Spot, by far the most extensively studied Jovian vortex. This will point up the importance of zonal wind profiles in the Jovian vortex problem. Further, the authors enter briefly into observational and theoretical aspects of solar vortices. It seems likely that solar vortices arise by straightforward convective driving, a process that is also important on Jovian planets, with possibly a bit of help from solar rotation. By contrast, the violent activity on hot young stars, which are rapid rotators, may well arise in a large and active vorticity field driven by intense zonal winds. The authors speculate on this area of vortex dynamics, whose essential novelty lies in its two-fluid character, and whose basic conservation laws have yet to be exploited. Title: Chaos: A Mixed Metaphor for Turbulence Authors: Spiegel, E. A. Bibcode: 1987RSPSA.413...87S Altcode: There are special circumstances when the equations of fluid mechanics can be asymptotically reduced to third- or higher-order differential equations that admit chaotic solutions. For physically extended systems, similar reductions lead to simplified partial differential equations whose solutions contain coherent structures that interact in complicated and erratic ways. It is suggested here that analogous reductions of the fluid equations are possible even when the fluid is in a turbulent state. From this we conclude that, more than being a metaphor for turbulence, chaos is a basic property of turbulent fluids. Title: Strange Accumulators Authors: Smith, L. A.; Spiegel, E. A. Bibcode: 1987NYASA.497...61S Altcode: No abstract at ADS Title: Chaos and the Solar Cycle Authors: Spiegel, E. A.; Wolf, Alan Bibcode: 1987NYASA.497...55S Altcode: No abstract at ADS Title: Book-Review - Chaos in Astrophysics Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A.; Tavakol, R. Bibcode: 1987Obs...107...89B Altcode: No abstract at ADS Title: Book-Review - Chaos in Astrophysics Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A. Bibcode: 1987ApL....25S.266B Altcode: No abstract at ADS Title: Strange Accumulators Authors: Smith, L. A.; Spiegel, E. A. Bibcode: 1987cpa..work...61S Altcode: No abstract at ADS Title: Hydrostatic Adjustment Time of the Solar Subconvective Layer Authors: Spiegel, E. A. Bibcode: 1987ASSL..137..321S Altcode: 1987isav.symp..321S Support is given for the suggestion that the solar cycle is driven by magnetohydromagnetic activity below the convection zone. The solar magnetocline is modeled as possessing amplitude-modulated waves of instability travelling across it in latitude, with the envelope taking 11 years to go across. A thin vertical sheet of magnetic disturbance is postulated which produces pertubations in magnetic field, pressure, convective activity, and temperature. Although the Kelvin-Helmoltz time for the subconvective layer is one million years, it is argued that the layer adjusts its hydrostatic structure in response to highly nonradial disturbances in time scales of the order of a few years. Title: Chaos and the Solar Cycle Authors: Spiegel, E. A.; Wolf, A. Bibcode: 1987cpa..work...55S Altcode: No abstract at ADS Title: Book-Review - Chaos in Astrophysics Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A. Bibcode: 1986Sci...232.1045B Altcode: No abstract at ADS Title: Chaos in astrophysics Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A. Bibcode: 1985ASIC..161.....B Altcode: 1985cia..conf.....B Topics addressed include stellar variability, a perturbative approach to stellar pulsations, chaos and noise, the information aspects of strange attractors, the rapid generation of magnetic field, and ordered and chaotic motions in Hamiltonian systems. Consideration is also given to the transition to chaos in galactic models of two and three degrees of freedom; many-mode couplings in connection with nonlinear nonradial adiabatic stellar oscillations; chaotic oscillations in a simple stellar model; X-ray bursters; and compressible MHD turbulence as a mechanism for the heating of stellar coronas. Title: Chaos in astrophysics. Proceedings of a NATO Advanced Research Workshop, held at Palm Coast, Florida, USA, 9 - 11 April 1984. Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A. Bibcode: 1985capn.book.....B Altcode: No abstract at ADS Title: Cosmic arrhythmias. Authors: Spiegel, E. A. Bibcode: 1985chas.conf...91S Altcode: The author examines models that are inspired, not so much by the detection of specific physical processes, but by the observation of qualitative kinds of spatial or temporal variability. The models discussed are mathematical in spirit. Often they are not physically derived, but are equations that can be written down without detailed knowledge of the physical process that causes the observed phenomenon. For example, in studying the temporal behavior of the solar oscillations, one can find an interesting model system once one concludes that the sun is overstable. It is not neccessary for further exploration to know the explicit physical mechanism. Title: Chaos in astrophysics. Proceedings of a NATO Advanced Research Workshop, held at Palm Coast, Florida, USA, 9 - 11 April 1984. Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A. Bibcode: 1985chas.conf.....B Altcode: No abstract at ADS Title: The dynamics of triple convection Authors: Arneodo, A.; Coullet, P. H.; Spiegel, E. A. Bibcode: 1985GApFD..31....1A Altcode: A numerical analysis of the dynamics of triple convection is presented. It is shown that in the parameter space of a fluid subject to triple convection, there is a critical hypersurface on which three linear growth rates vanish, and all the remaining rates are negative. Parameter values chosen to place a triply unstable system near the critical condition in the hypersurface may lead to complicated temporal behavior, and in some cases, chaotic behavior. The problem is illustrated using the example of Arenodo (1982) from geophysical fluid dynamics: a two-dimensional, Boussinesq thermohaline convection in a plane parallel layer. In the example, it is assumed that the parallel layer is in rotation around a vertical axis, and is subject to convenient boundary conditions. The theoretical calculations from the example are applied to other types of triply unstable systems, and the possibility of chaotic temporal behavior is exmined. Title: Cosmic arrhythmias Authors: Spiegel, E. A. Bibcode: 1985ASIC..161...91S Altcode: 1985cia..conf...91S An introductory discussion concerning chaos is presented. This is followed by a consideration of normal form theory, involving the construction of amplitude equations for mildly unstable situations. Examples of astrophysical chaos are then discussed: qualitative approaches are taken to stellar pulsation and the solar cycle; and modulational chaos and solar waves are discussed. Title: Magnetic buoyancy and the Boussinesq approximation Authors: Spiegel, E. A.; Weiss, N. O. Bibcode: 1982GApFD..22..219S Altcode: The full Boussinesq equations for hydromagnetic convection are derived and shown to include the effects of magnetic buoyancy. Instabilities caused by magnetic buoyancy are analyzed and their roles in double convection are brought out. Title: Time-dependent solutions of multimode convection equations Authors: Toomre, J.; Gough, D. O.; Spiegel, E. A. Bibcode: 1982JFM...125...99T Altcode: Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Numerical studies dealing with two or three modes with triad interactions are discussed. Rich time dependence was found in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favorably. Except at the highest Rayleigh number considered (10,000,000), the theoretical Nusselt numbers agree well with experiment. Title: Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties Authors: Depassier, M. C.; Spiegel, E. A. Bibcode: 1982GApFD..21..167D Altcode: We study the bifurcation to steady two-dimensional convection with the heat flux prescribed on the fluid boundaries. The fluid is weakly non-Boussinesq on account of a slight temperature dependence of its material properties. Using expansions in the spirit of shallow water theory based on the preference for large horizontal scales in fixed flux convection, we derive an evolution equation for the horizontal structure of convective cells. In the steady state, this reduces to a simple nonlinear ordinary differential equation. When the horizontal scales of the cells exceed a certain critical size, the bifurcation to steady convection is subcritical and the degree of subcriticality increases with increasing cell size. Title: A prospectus for a theory of variable variability Authors: Childress, S.; Spiegel, E. A. Bibcode: 1981NASCP2191..273C Altcode: 1981vsc..conf..273C It is proposed that the kind of stellar variability exhibited by the Sun in its magnetic activity cycle should be considered as a prototype of a class of stellar variability. The signature includes long 'periods' (compared to that of the radial fundamental model), erratic behavior, and intermittency. As other phenomena in the same variability class we nominate the liminosity fluctuations of ZZ Ceti stars and the solar 160 m oscillation. We discuss the possibility that analogous physical mechanisms are at work in all these cases, namely instabilities driven in a thin layer. These instabilities should be favorable to grave modes (in angle) and should arise in conditions that may allow more than one kind of instability to occur at once. The interaction of these competing instabilities produces complicated temporal variations. Given suitable idealizations, it is shown how to begin to compute solutions of small, but finite, amplitude. Title: The large-scale structure of compressible convection Authors: Depassier, M. C.; Spiegel, E. A. Bibcode: 1981AJ.....86..496D Altcode: We study convection of small but finite amplitude in plane-parallel layer of perfect gas. The conductivity is assumed constant, hence the static state is polytropic. The heat flux on the boundaries is held fixed. When the polytropic index is not too large, the critical horizontal wave number for the onset of convection is zero and there is finite-amplitude instability. The finite-amplitude instability persists into the thin-layer limit provided that there are no geometrical limits to the horizontal scale of the convection. This result contradicts conclusions drawn from the strict Boussinesq approximation and it suggests that results based on that approximation are not generic for convection with flux prescribed on the boundaries. For all the Rayleigh numbers and layer thicknesses accessible to our amplitude expansions, convection with very large horizontal extent is expected to be prevalent at significant amplitudes. We suggest ways in which the nonlinear solutions found here may be useful in the interpretation of large-scale solar convection. Title: Magnetic activity and variations in solar luminosity Authors: Spiegel, E. A.; Weiss, N. O. Bibcode: 1980Natur.287..616S Altcode: Attempts to detect changes in the solar luminosity suggest that the solar constant has been misnamed1-4. Although the Nimbus satellite data4 show no significant fractional variations above 5 × 10-3 during the period 1975-78, results from recent balloon5 and rocket6 flights show changes of 4 × 10-3. Intense magnetic fields in sunspots hamper convection locally7 but active regions are believed not to be directly responsible for long term variations in luminosity of the Sun8 or of RS CVn and BY Dra stars9. The cause of luminosity variations over spot cycles should be sought in more deep-seated global features. Here we indicate how strong magnetic fields at the base of the convective zone can alter the local convection. The resulting changes in thermal energy are large enough to produce variations of order 0.1% in the solar luminosity over the 11-yr sunspot cycle. Title: The onset of convection in a radially pulsating star Authors: Poyet, J. P.; Spiegel, E. A. Bibcode: 1979AJ.....84.1918P Altcode: An attempt to find the analogue of the Schwarzschild criterion in radially pulsating stars is described. The procedure is to transform to pulsating coordinates to make the problem resemble that of ordinary convective stability theory, but with fictitious forces acting. Attention is confined to layers in a star's envelope that are so thin that the pulsation is homologous within them. This also permits the use of the Boussinesq and the plane-parallel approximations. It is found that pulsation is destabilizing and that it promotes monotonic convective instability for large horizontal scales and overstability for small horizontal scales. Title: Penetration of the molecular-weight barrier. Authors: Huppert, H. E.; Spiegel, E. A. Bibcode: 1977ApJ...213..157H Altcode: The penetration of meridional currents from the radiative envelope of a rotating star into a core with a molecular-weight barrier (mu-barrier) is investigated using a plane-parallel model constrained by numerous simplifying assumptions. The circulation time in the presence of mu-gradients is calculated along with the penetration depth of the motions into the region with stabilizing mu-gradients, and the response of the core is analyzed. The results suggest that a mu-barrier can be penetrated and that allowance must be made for some rotational mixing. A scheme is developed for including the effects of such mixing in calculations of stellar evolution. Title: Ring galaxies. II. Authors: Theys, J. C.; Spiegel, E. A. Bibcode: 1977ApJ...212..616T Altcode: Dynamical problems concerning the formation and evolution of ring galaxies are discussed with the help of a simulation code for disk systems. The dynamical age of ring galaxies is found to be of the order of 100 million yr, in agreement with other estimates of age. On a longer time scale, the ring breaks up into about six smaller objects which subsequently collide and merge. It is thought that this system forms a galaxy with a gas-enriched active nucleus. The simulations are used to study the effect on a normal disk galaxy of a passage through its center by a second galaxy. The gas in the disk forms a ring, and the stellar core is distended. The galaxy-collision model for ring galaxies, therefore, has difficulty in rationalizing the RE galaxies, in which the original core is not in evidence. The difficulty is alleviated if the target core loses some mass to the intruding galaxy so that its remnant disperses. This process is simulated numerically for a pure population I target galaxy with the help of a simple model. The model is axisymmetric but has dispersion normal to the disk plane. It gives an impression of the three-dimensional aspects of the collision as seen in projection. Title: Erratum: the Equations of Photohydrodynamics Authors: Hsieh, S. -H.; Spiegel, E. A. Bibcode: 1977ApJ...212..593H Altcode: No abstract at ADS Title: Problems of stellar convection Authors: Spiegel, E. A.; Zahn, J. -P. Bibcode: 1977LNP....71.....S Altcode: 1977IAUCo..38.....S; 1977stco.coll.....S The papers deal with various aspects of stellar convection theory ranging from mixing-length formalisms to computations based on full equations. The major subjects encompass mixing-length theory, linear theory, observational aspects of convection, numerical solutions to stellar-convection problems, effects of rotation and magnetic fields, penetration of convection, special types of convection, waves, and turbulence. Specific topics include the current state of mixing-length theory, dynamical instabilities in stars, observations bearing on convection, the evolutionary pattern of exploding granules, numerical methods in convection theory, convection in rotating stars, and convective dynamos. Other contributions examine penetrative convection in stars, convective overshooting in the solar photosphere, thermosolutal convection, Urca convection, photoconvection, convection in the helium flash, and the roles of determinism and chaos in turbulence. Title: Photoconvection Authors: Spiegel, E. A. Bibcode: 1977LNP....71..267S Altcode: 1977stco.coll..267S; 1977IAUCo..38..267S; 1977psc..conf..267S Convection under the influence of dynamically significant radiation fields (photoconvection) is considered. A set of approximate equations is given for plane-parallel photoconvection, hydrostatic solutions are examined, and the onset of convective instability is treated schematically. Some arguments are outlined for believing that photon bubbles occur in the nonlinear regime. It is tentatively concluded that the two-fluid nature of the photoconvection process may make for some qualitative differences from basic Boussinesq convection. Title: On taking mixing-length theory seriously Authors: Gough, D. O.; Spiegel, E. A. Bibcode: 1977LNP....71...57G Altcode: 1977IAUCo..38...57G; 1977psc..conf...57G; 1977stco.coll...57G A clarification of mixing-length theory is recommended, and the determination of the equations of motion of quasi-particles is discussed. The treatment makes use of a specific model which considers the star to be composed of a background fluid through which discrete well-defined parcels of fluid move. These parcels may be thought of as quasi-particles whose number density is sufficiently high so that they constitute a second fluid permeating the background fluid. The convective model is a two-fluid model analogous to a composite model of radiation and matter except that the quasi-particle fluid is more complicated than the photon gas. Two approaches to the equations of motion are examined - one in which the quasi-particles are treated as idealizations of buoyant thermals, and the other in which the solution of the differential equations is sought and then applied in conjunction with hypotheses about the distribution of initial conditions of quasi-particles to compute heat flow. Title: Numerical solutions of single-mode convection equations Authors: Toomre, J.; Gough, D. O.; Spiegel, E. A. Bibcode: 1977JFM....79....1T Altcode: In the Boussinesq approximation, single-mode equations describing thermal convection are constructed by expanding the fluctuating velocity and temperature fields in a complete set of functions (or planforms) of the horizontal coordinates and retaining just one term. Numerical solutions of the single-mode equations are investigated, chief consideration being given to hexagonal planforms. Extensive surveys of steady solutions are presented for various Rayleigh numbers, Prandtl numbers, and horizontal wavenumbers. The dependences on Rayleigh number and Prandtl number at very large Rayleigh number are in satisfactory agreement with the results of asymptotic expansions. Title: Seiches in supergranules Authors: Gough, D. O.; Pringle, J. E.; Spiegel, E. A. Bibcode: 1976Natur.264..424G Altcode: THE recent report by Hill, Stebbins and Brown1 of oscillations of an apparent solar radius has stirred a mild controversy2. In particular, the failure of Grec and Fossat3 to detect Doppler variations with similar periods in solar spectral lines provides striking contrast with the results of Hill et al. Both sets of observations have been carefully performed and thoughtfully analysed and, if one accepts both sets of results at face value, the problem of reconciling the two must be confronted. Here we consider seiches in supergranules as a possible cause of the discrepancy. Title: Ring galaxies. I. Authors: Theys, J. C.; Spiegel, E. A. Bibcode: 1976ApJ...208..650T Altcode: A classification scheme is proposed for ring galaxies, or R galaxies, with three subclasses. RE galaxies have crisp empty rings; RN galaxies are like REs but for an off-center nucleus; RK galaxies have single dominant knots or condensations in their rings. From the projected ellipticities of the rings and the relative positions of their companions, simple three-dimensional models are deduced. With the help of these models, photometric and spectroscopic observations we employed to estimate global properties of R galaxies. They have galactic sizes, masses of the order of 100 billion solar masses, and kinematical time scales of the order of 100 million yr. The present interpretation of these data suggests that rings are formed when an intruding galaxy passes nearly through the center of a normal disk galaxy. Title: Stellar convection theory. II. Single-mode study of the second convection zone in an A-type star. Authors: Toomre, J.; Zahn, J. -P.; Latour, J.; Spiegel, E. A. Bibcode: 1976ApJ...207..545T Altcode: The anelastic modal equations presented in Paper I are considered in their simplest version: only one mode is retained in the representation of the fluctuating dynamic and thermodynamic variables of convection theory. These single-mode equations are used to examine the structure of the second convection zone of an A-type star. Two- and three-dimensional numerical solutions are obtained for a range of parameters in the theory, and a simple analysis is provided for their interpretation. The principal results are for three-dimensional motions, since these are most likely to be relevant to stellar convection. Such motions produce a convective heat flux several orders of magnitude greater than predicted by standard mixing-length theory for the same situation; we find that convection carries up to 6 percent of the total flux. The most significant astrophysical implication of our results is that they suggest strong overshooting into the adjacent radiative zones. We anticipate that mixing will extend to the overlying hydrogen convection zone. This would rule out some interpretations of metallic-line stars which invoke diffusive element separation between the two convection zones. Subject headings: convection - stars: interiors - stars: metallic-line Title: Stellar convection theory. I. The anelastic modal equations. Authors: Latour, J.; Spiegel, E. A.; Toomre, J.; Zahn, J. -P. Bibcode: 1976ApJ...207..233L Altcode: Methods are developed for dealing with the various dynamical problems that arise because of convective zones in stars. A system of equations for stellar convection is derived from the full equations of compressible fluid dynamics with the aid of two major approximations. The first of these is the anelastic approximation, which involves both the filtering out of acoustic waves and a suitable linearization of the fluctuating thermodynamic variables. The second one approximates the horizontal structure of convection by expanding the motion in a set of horizontal cellular platforms and severely truncating the expansion. The resulting system of partial differential equations, referred to as the anelastic modal equations, is outlined along with suggested boundary conditions and techniques for solving the equations. Ways of assessing the overall validity of the present treatment are discussed. Title: The equations of photohydrodynamics. Authors: Hsieh, S. -H.; Spiegel, E. A. Bibcode: 1976ApJ...207..244H Altcode: A system of equations governing the dynamics of a fluid under the influence of radiative forces is presented. The starting point is Thomas's form of the equation of transfer, but the final equations are good only to order v/c, where v is a typical fluid speed and c is the speed of light. Continuous absorption and Thomson scattering are treated, and they influence the energy balance differently. The radiation field is described by the first two moments of the transfer equation, and the resulting system may be thought of as a two-fluid system Title: Convective Instability in a Compressible Atmosphere. II Authors: Gough, D. O.; Moore, D. R.; Spiegel, E. A.; Weiss, N. O. Bibcode: 1976ApJ...206..536G Altcode: The onset of steady convection in polytropic atmosphere with constant viscosity is studied numencally. Subject headings: convection : atmospheres Title: Photohydrodynamic instabilities of hot stellar atmospheres Authors: Spiegel, E. A. Bibcode: 1976pmas.conf...19S Altcode: No abstract at ADS Title: Radiative transfer through a flowing refractive medium. Authors: Anderson, J. L.; Spiegel, E. A. Bibcode: 1975ApJ...202..454A Altcode: Propagation of photons in a nondispersive refractive medium with arbitrary velocity and gravity fields is discussed. Photons move along the geodesics associated with the optical metric given by Gordon (1923) in the limit where geometrical optics applies. The known properties of geodesics then simplify the task of finding the world lines of photons in a variety of problems. The formalism for describing photon propagation, summarized here, makes it possible to derive a transfer equation for a flowing refractive medium which is formally identical to the relativistic equation of radiative transfer. However, the theory is phenomenological since it presumes that the index of refraction is specified. Title: Modal analysis of convection in a rotating fluid Authors: Baker, L.; Spiegel, E. A. Bibcode: 1975JAtS...32.1909B Altcode: The Boussinesq modal equations for convection in a horizontal fluid layer rotating about a vertical axis are expanded in the planform functions of linear theory. A finite difference technique is used to solve the one-mode equations at arbitrary Rayleigh number. For large Rayleigh numbers, moderate Prandtl numbers and rigid boundaries, steady solutions are found which display nonmonotonic dependence of heat flux on rotation rate even when the horizontal wavenumber is fixed. It is concluded that rotation does not necessarily suppress convection and reduce heat flux. It is shown that the one-mode approximation permits simulation of time-dependent rotating convection with an entirely modest computing effort. Title: The Second Convection Zone in an A-type Star Authors: Latour, J.; Spiegel, E. A.; Toomre, J.; Zahn, J. P. Bibcode: 1975BAAS....7..526L Altcode: No abstract at ADS Title: Modal equations for cellular convection Authors: Gough, D. O.; Spiegel, E. A.; Toomre, J. Bibcode: 1975JFM....68..695G Altcode: We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possible useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of 'modal equations' is analyzed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion. Title: Highly stretched meshes as functionals of solutions Authors: Gough, D. O.; Spiegel, E. A.; Toomre, J. Bibcode: 1975LNP....35..191G Altcode: No abstract at ADS Title: Photon Bubbles Authors: Prendergast, K. H.; Spiegel, E. A. Bibcode: 1973CoASP...5...43P Altcode: 1973ComAp...5...43P No abstract at ADS Title: Radial Velocities in the Tail of NGC 4676A Authors: Theys, J. C.; Spiegel, E. A.; Toomre, Juri Bibcode: 1972PASP...84..851T Altcode: A spectrum of ffie long tail of NGC 4676A shows [011] x3727 in emission. The radial velocity measured from one plate varies by about 400 km sec-1 along the length of the tail. Key words: peculiar galaxy - radial velocities Title: Spectra of Extragalactic Rings. Authors: Theys, J. C.; Spiegel, E. A.; Toomre, J. Bibcode: 1972BAAS....4..213T Altcode: No abstract at ADS Title: Effect of a Magnetic Field on Finite Amplitude Convection Authors: Murphy, J. O.; Spiegel, E. A.; Van Der Borght, R. Bibcode: 1972PASA....2...93M Altcode: 1972PASAu...2...93M It is well known that a magnetic field has an inhibiting effect on thermal convection. It is also likely that a magnetic field might have a marked effect on the preferred cell-size and it is therefore of some importance to investigate the effect of such a field on finite amplitude convection. We restrict ourselves to fluids in which the Boussinesq approximation holds. For astrophysical application, especially in a study of the convective layer of the Sun, one should consider the compressible case but unfortunately the full compressible equations are much more difficult to handle numerically. In any case, the integration of the equations for the incompressible case will yield at least an indication of the effect of a magnetic field. Title: The Moment Method in Relativistic Radiative Transfer Authors: Anderson, J. L.; Spiegel, E. A. Bibcode: 1972ApJ...171..127A Altcode: The moment method of Grad is applied to the problem of radiative transfer in a medium with relativistic differential motions. If a mean absorption coefficient is used, the method readily leads to a closed system of equations. The first approximation gives the relativistic analog of the classical Eddington approximation. In the limit of small photon mean free path, the Eddington approximation does not reproduce Thomas's radiative-viscosity terms which were derived from the exact transfer equation. To recover Thomas's results it is necessary to go to the second approximation. It is suggested that this second approximation will be of use when scales of interest are not optically thick, although in general, such problems may be fairly complicated. Title: Convection in Stars. II. Special Effects Authors: Spiegel, E. A. Bibcode: 1972ARA&A..10..261S Altcode: No abstract at ADS Title: A History of Solar Rotation Authors: Spiegel, E. A. Bibcode: 1972NASSP.300...61S Altcode: 1972pss..conf...61S; 1972poss.conf...61S No abstract at ADS Title: Some Fluid Dynamical Problems in Cosmogony Authors: Spiegel, E. Bibcode: 1972oss..conf..165S Altcode: No abstract at ADS Title: Transport Mechanisms in Stars Authors: Spiegel, E. A. Bibcode: 1972stev.conf..493S Altcode: No abstract at ADS Title: Galactic Wakes Authors: Ruderman, M. A.; Spiegel, E. A. Bibcode: 1971ApJ...165....1R Altcode: The hypersonic flow of intergalactic gas past a spherical galaxy is discussed. Isentropic motion near the downstream symmetry axis is described by a similarity solution to the fluid-dynamical equations. The upstream flow is represented by orbit theory and matched to the fluid flow across a conical shock. The downstream flow exhibits high density and low temperature near the axis. These results are used to estimate the X-radiation from galactic wakes. Title: Turbulence in Stellar Convection Zones Authors: Spiegel, E. A. Bibcode: 1971CoASP...3...53S Altcode: 1971ComAp...3...53S No abstract at ADS Title: Book reviews Authors: Švestka, Z.; Spiegel, E. A.; van Herk, G. Bibcode: 1971SSRv...11..867S Altcode: No abstract at ADS Title: Convection in Stars: I. Basic Boussinesq Convection Authors: Spiegel, E. A. Bibcode: 1971ARA&A...9..323S Altcode: No abstract at ADS Title: Temperature Variation and the Solar Oblateness Authors: Ingersoll, Andrew P.; Spiegel, Edward A. Bibcode: 1971ApJ...163..375I Altcode: Dicke and Golden berg's oblateness measurement may be explained by an equatorial temperature excess of 30 K, smoothly distributed in optical depths <0.01 The resulting brightness variation with solar latitude is concentrated close to the limb, and it is not possible, with data presently available, to distinguish such variation from true oblateness. Title: Aperiodic behaviour of a non-linear oscillator Authors: Baker, N. H.; Moore, D. W.; Spiegel, E. A. Bibcode: 1971QJMAM..24..391B Altcode: No abstract at ADS Title: Instabilities of Differential Rotation Authors: Spiegel, E. A.; Zahn, J. -P. Bibcode: 1970CoASP...2..178S Altcode: 1970ComAp...2..178S No abstract at ADS Title: A History of Solar Rotation Authors: Spiegel, E. A. Bibcode: 1970pss..conf...63S Altcode: No abstract at ADS Title: Thermodynamics and Cosmology Authors: Schucking, E. L.; Spiegel, E. A. Bibcode: 1970CoASP...2..121S Altcode: 1970ComAp...2..121S No abstract at ADS Title: Cortical Projection of Labyrinthine Impulses: Study of Averaged Evoked Responses Authors: Spiegel, E. A.; Szekely, E. G.; Moffet, H.; Egyed, J. Bibcode: 1970NASSP.187..259S Altcode: No abstract at ADS Title: The Gas Dynamics of Accretion Authors: Spiegel, E. A. Bibcode: 1970IAUS...39..201S Altcode: No abstract at ADS Title: On the Circulation of Gas Near the Galactic Center Authors: Spiegel, E. A. Bibcode: 1970IAUS...38..441S Altcode: 1970ssg..conf..441S No abstract at ADS Title: Fluidization on the Moon (?) Authors: Murray, J. D.; Spiegel, E. A.; Theys, J. Bibcode: 1969CoASP...1..165M Altcode: 1969ComAp...1..165M No abstract at ADS Title: Semiconvection Authors: Spiegel, E. A. Bibcode: 1969CoASP...1...57S Altcode: 1969ComAp...1...57S No abstract at ADS Title: On the Flow of Gas from the Galactic Center Authors: Moore, D. W.; Spiegel, E. A. Bibcode: 1968ApJ...154..863M Altcode: We consider a steady axisymmetric flow out of the galactic center, neglecting magnetic and turbulent stresses and the self-gravitation of the gas. The flow is in a disk and possesses angular momentum, but otherwise the problem is like solar-wind theory, and we assume that initial conditions are specified at the edge of the nuclear disk. The equations of motion are then quite simple, and their solution indicates that the flow must contain a nearly discontinuous transition from supersonic to subsonic conditions. We argue that this transition must be a compressible hydraulic jump, or bore, which generates intense turbulence and gives rise to H non the downstream edge of the bore. The bore causes radial deceleration of the gas and hence density enhancement, and we suggest that this phenomenon is associated with the expanding arm Title: The Mixing of Lithium Authors: Spiegel, E. A. Bibcode: 1968HiA.....1..261S Altcode: No abstract at ADS Title: An overstability of gravity waves Authors: Souffrin, P.; Spiegel, E. A. Bibcode: 1967AnAp...30..985S Altcode: No abstract at ADS Title: The Solar Atmospheric Convection Zone -- The Theory of Turbulent Convection Authors: Spiegel, E. A. Bibcode: 1967IAUS...28..347S Altcode: No abstract at ADS Title: Radiative damping of sound waves. Authors: Stein, R. F.; Spiegel, E. A. Bibcode: 1967ASAJ...42..866S Altcode: No abstract at ADS Title: A Thermally Excited Non-Linear Oscillator Authors: Moore, Derek W.; Spiegel, Edward A. Bibcode: 1966ApJ...143..871M Altcode: Instability in the form of growing oscillations (overstahility) can occur iii convectively unstable fluids which rotate, have magnetic fields, or are compressible, so long as thermal dissipation operates. To clarify the manner in which thermal dissipation causes instability, a model oscillator which exhibits overstability is constructed. The governing equations are derived and the linear stability is discussed. The non4inear behavior of the oscillator is then investigated. The governing non4inear equation is third order in time, and it therefore is a simple representative example of the type of equation which describes non-linear stellar pulsations. The equation contains two parameters, and a great variety of solutions is found, depending on the values taken. One kind of solution shows relaxation oscillations with superposed variations, while, in a particular range of the governing parameters, the numerical solutions of the governing equation are aperiodic or irregular. A mathematical explanation of this irregularity is suggested, and the possibility that it might be relevant to irregular variability in stars is raised. The general conclusion is suggested that a great variety of oscillatory phenomena, analogous to several of those observed in variable stars, can be generated from a single instability mechanism, provided the essential non4inearities are retained and the law of dissipation is appropriately chosen. Title: The Eddington Approximation in the Radiative Heat Equation Authors: Unno, W.; Spiegel, E. A. Bibcode: 1966PASJ...18...85U Altcode: The Eddington approximation in the theory of radiative transfer is generalized to the uon-planeparalld case and is applied to the radiative heat equation. The resulting equation reduces to the exact equation for the problem of radiative relaxation of small temperature fluctuations in the limits of large and small optical thickness of perturbations, and its solutions are reasonably accurate over the whole range of optical thicknesses. A derivation of the Eddington approximation is exhibited which brings out the reason for its wide range of applicability and which suggests a methed of developing higher approximations. Title: Energy Transport by Turbulent Convection Authors: Spiegel, Edward A. Bibcode: 1966stev.conf..143S Altcode: No abstract at ADS Title: Nonlinear oscillations in the one-zone model for stellar pulsation Authors: Baker, N. H.; Moore, D. W.; Spiegel, E. A. Bibcode: 1966AJ.....71S.844B Altcode: No abstract at ADS Title: The Solar Hydrogen Convection Zone and its Direct Influence on the Photosphere Authors: Spiegel, E. A. Bibcode: 1966IAUTB..12..539S Altcode: No abstract at ADS Title: Convective Instability in a Compressible Atmosphere. I. Authors: Spiegel, Edward A. Bibcode: 1965ApJ...141.1068S Altcode: The linear equations for time4ndependent convection in a plane-parallel layer of perfect gas are studied for the case of constant viscosity and conductivity. These equations determine the condition for the onset of steady convection. The equations are treated from three points of view. First a perturbation expansion in terms of layer thickness is carried out to first order. The zeroth-order terms give the Boussinesq equations studied by Lord Rayleigh. The first-order terms show that if the Rayleigh number is evaluated at the mid-height of the layer, its eigenvalues are stationary with respect to variations in layer thickness. The first-order eigenfunctions are numerically calculated for a particular polytropic index. Next, two variational statements are written and some sample numerical results are presented. It is found that variational techniques are not effective in the present problem. The WKB approach is then explored and extended to the determination of uniformly valid asymptotic solutions. These provide not only stability criteria (from which explicit numerical results are derived), but also produce analytic approximations for all the eigenmodes. This method appears to be generally useful for problems of this kind. Title: On Convective Overstability. Authors: Moore, D. W.; Spiegel, E. A. Bibcode: 1965AJ.....70S.327M Altcode: Chandrasekhar has shown that the application of a uniform rotation or magnetic field to a convectively unstable layer of fluid can give rise to oscillations which grow exponentially in time. This phenomenon, called overstability, does not occur unless thermal dissipation (i.e., conduction or radiative transfer) acts, as Cowling has discussed. To clarify the physical mechanism by which then~al dissipation leads to overstability, we have studied a simple mechanical oscillator which is likewise destabilized by thermal dissipation. The physical meaning of this is briefly explained. The dynamics of this overstable oscillator is of astrophysical interest for two reasons. First, it suggests that other oscillatory motions in a convectively unstable layer may be destabilized in this way. In- deed, it has now been verified that sound waves in a convectively unstable layer are unstable. We suggest that this mechanism may be relevent to th~ problem of coronal heating. The second astrophysical problem for which the overstable oscillator may be of interest is nonlinear oscillations of stars. Mathematically, there are similarities between the model oscillator and the more complicated stellar problem features which previously discussed model oscillators, such as van der Pol's, do not possess. In particular, the model equations are third order in time and the nonlinear behavior is quite interesting. For example, in certain cases the model has relaxation oscillations on which are superposed additional, more regular oscillations. This behavior is illustrated and a possible astrophysical analogue indicated. Title: The Effect of Radiative Transfer on Convective Growth Rates. Authors: Spiegel, Edward A. Bibcode: 1964ApJ...139..959S Altcode: Calculations of the convective growth rate, , are carried out for an inviscid, radiating, polytropic ?. The effect of a radiative damping is studied in two special cases. First we assume a layer thickness ch less than the scale height, but with arbitrary optical thickness of disturbances. In this case the wth rate is a monotonically fncreasing function of the horizontal wavenumber, k. Second, the case layer of arbitrary vertical extent is considered, but only for disturbances with large k. In this case, ton's law of cooling is valid for the perturbation equations. When the polytropic index, m, is 6, it is onstrated that for the convective modes, `iapproaches the non-radiative value for large k, while the acoustic modes overstability can occur. Arbitrary values of m are then treated in the boundaryr approximation. For m <6 the effect of radiation is vanishingly small for large k; for m > 6 radiasuppresses convective modes and does not promote overstability. t is concluded that radiative transfer does not qualitatively alter the dependence of on k, for contively unstable modes in most cases of interest. Thus `i is expected to be generally a monotonically .easing function of k with no radiative cutoff. This points up an apparent discrepancy between theory the granulation observations. Title: The Generation and Propagation of Waves in a Compressible Atmosphere. Authors: Moore, Derek W.; Spiegel, Edward A. Bibcode: 1964ApJ...139...48M Altcode: The equations governing the aerodynamic generation and the propagation of waves in a compressible atmosphere are exhibited. The fluctuating terms which are the turbulent sources for aerodynamic noise are approximated by an externally applied, time-harmonic, point force. Lighthill's results for the asymptotic radiation field in an anisotropic medium are then applied to an isothermal atmosphere. In this way, the surfaces of constant phase, group velocity, and intensity of the far field are computed. For finite frequencies above the critical frequency for vertical propagation, a monopole component is produced in the field by gravitational effects. The propagation problem is also studied for arbitrary temperature profiles, and it is found that in certain regions in the solar atmosphere there exist finite bands of non-propagating frequencies It is suggested that the oscillations in the solar atmosphere result from forced excitations of these non-propagating frequencies and that these are excited by turbulence arising from shear instability and penetrative convection Title: A Generalization of the Mixing-Length Theory of Turbulent Convection. Authors: Spiegel, Edward A. Bibcode: 1963ApJ...138..216S Altcode: The mixing4ength theory as currently employed is valid only when the mixing length is sufficiently small. The present work attempts to remove this limitation by writing a heat-transfer integral for convecting fluid elements. There then follows an integrodifferential equation for the mean temperature in a convecting medium. it is indicated how this equation may be used to include the effects of penetration into convectively stable regions from adjacent stable regions. Title: Thermal Turbulence at Very Small Prandtl Number Authors: Spiegel, Edward A. Bibcode: 1962JGR....67.3063S Altcode: The equations of thermal turbulence are derived for the case of small Prandtl number, a case of a relatively simple realizable turbulent flow. The power spectrum of velocity is described using the transfer functions of Heisenberg and of Kovasznay. It is seen that these do not give uniformly good approximations, since they force spectral energy to flow from low to high wave numbers, though the Kovasznay approximation may be useful for large Rayleigh numbers. A general comparison of the present study with the Malkus theory of convection indicates disagreement which probably results from the dominance of nonlinear terms in the low-Prandtl-number limit. Title: On Convective Growth-rates in a Polytropic Atmosphere Authors: Spiegel, E. A.; Unno, W. Bibcode: 1962PASJ...14...28S Altcode: An exact asymptotic form of convective growth-rates for large horizontal wave numhers is derived analytically for a polytropic atmosphere without viscosity and conductivity. Title: On the Spectrum of Turbulent Convection. Authors: Ledoux, P.; Schwarzschild, M.; Spiegel, E. A. Bibcode: 1961ApJ...133..184L Altcode: A procedure is described by which, under certain assumptions, the turbulence spectrum can be derived for the motions in a convectively unstable layer. The energy input from the buoyancy forces is assessed in this procedure by deriving the growth rates of the laminar modes obtained from the relevant linearized equations On the other hand, the exchange of energy between modes is assumed to follow Heisenberg's elementary theory of turbulence. The procedure is carried through for an exceptionally simple case for which a closed solution was found for the spectrum. The results, though not strictly applicable, are applied to the convection in the solar photosphere for purposes of orientation. Title: The Convective Instability of a Radiating Fluid Layer. Authors: Spiegel, Edward A. Bibcode: 1960ApJ...132..716S Altcode: The thermal stability of a gray, radiating fluid layer with an adverse temperature gradient is studied. It is assumed that the vertical dimension of the layer is much less than the scale height of density or pressure. By essentially dimensional reasoning it is shown that convection can occur when a non-dimensional parameter, analogous to the Rayleigh number, exceeds a certain critical value. The formal study begins from the same equations as the classical study of Rayleigh, with the addition of a radiative term to the heat equation. It is shown that if the difference between the temperature gradient and the adiabatic gradient is constant, overstability cannot occur. An equation of marginal stability is then derived. By means of a variational principle, critical values of the non-dimensional stability parameter and of the scale of the most unstable disturbance are computed for the case of rigid bounding surfaces. The critical value for instability is given as a function of the optical thickness of the layer. Finally, the convective stability of the atmosphere of a B0 star is discussed in terms of these results. Title: On The Trumpler Shift in Early Stars. Authors: Spiegel, E. A. Bibcode: 1960AJ.....65..353S Altcode: star atmospheres, about 20 km/sec. Indeed, Trumpler's shifts correspond to about 7 km/sec and this is consistent with what might be expected. It is not difficult to check the foregoing suggestion observationally by measuring radial velocities for lines of different excitation potential. A preliminary study has been made by Struve (unpublished) on two high dispersion plates of 10 Lac (09.5). The observations seem to corroborate the foregoing explanation to within the experimental accuracy. Title: On the Boussinesq Approximation for a Compressible Fluid. Authors: Spiegel, E. A.; Veronis, G. Bibcode: 1960ApJ...131..442S Altcode: The full, non4inear equations governing thermal convection in a compressible fluid have been re-examined in order to determine the conditions under which the Boussinesq approximation is applicable These conditions are (a) the vertical dimension of the fluid is much less than any scale height, and (b) the motion-induced fluctuations in density and pressure do not exceed, in order of magnitude, the total static variations of these quantities. Under these conditions the equations are formally equivalent to those for an incompressible system when the temperature gradient is replaced by its excess over the adiabatic and Cp replaces C . Title: The Princeton balloon observations Authors: Spiegel, E. Bibcode: 1960IAUS...12..319S Altcode: No abstract at ADS Title: The Onset of Thermal Convection in a Radiating Atmosphere. Authors: Spiegel, Edward Alexander Bibcode: 1959PhDT.........2S Altcode: No abstract at ADS Title: The Smoothing of Temperature Fluctuations by Radiative Transfer . Authors: Spiegel, Edward A. Bibcode: 1957ApJ...126..202S Altcode: A time-dependent equation for the temperature field of a medium with deviations from radiative equilibrium is derived. It is assumed that the medium is graylike and changes quasi-statically. Perturbations of small amplitude imposed on a homogeneous medium are shown to decay exponentially in time. The dependence of decay time on a characteristic length of the perturbation is found. Title: Problems of radiative hydrodynamics. Authors: Spiegel, Edward A. Bibcode: 1957AJ.....62R.146S Altcode: The equation of heat transfer for a radiating medium (Spiegel 1957) has been applied to three hydrodynamical theories with a view toward adapting these to problems of stellar atmospheres. The hydrodynamical theories so treated are: I.The study of the thermal instability of a plane layer heated from below; 2.The behavior of isotropic temperature fluctuations in a turbulent medium; 3.The propagation of sound. In all cases only the effects of integrated radiation have been considered, and only mean absorption coefficients figure in the theory. The stability investigation begins with the Rayleigh-Boussinesq equations in combination with the radiative heat transfer equation. The stability of the layer in a stellar atmosphere where the gradient is superadiabatic is found to depend on a dimensionless parameter analogous to the Rayleigh number. The scale of the most unstable perturbation is of the order of the layer thickness, and its exact value is given as the solution of a transcendental equation involving the parameters of the medium. In the study of turbulent fluctuations it is assumed that the amplitude of the fluctuations is small and that their behavior does not appreciably affect the velocity field. The amplitude of velocity fluctuations need not be small. The development follows Corrsin (1950, 1951) for the non-radiative case. It is possible to obtain a number of results about the temperature fluctuations, particularly a universal spectrum for the scales in which there is local isotropy. The effects of conductivity and radiative transfer on sound propagation are a space- damping and a reduction of propagation speed. The former effect is the more important and the space-damping of sound waves by radiative transfer has been found in the general case where the mean free path of photons may be comparable with the acoustic wave length. Corrsin, S. 1950, J. AppI. Phys. 22, 469. 1951, J. Aer. Sci. iS, 417. Spiegel, E. A. 1957, Ap. J. in press. University of Michigan Observatory, Ann Arbor, Mich. Title: Turbulence in the atmosphere of 31 Cygni. Authors: Spiegel, Edward A.; Aller, Lawrence H. Bibcode: 1953AJ.....58Q.229S Altcode: We have been attempting to utilize data gathered in the 1951 atmospheric eclipse of 31 Cygni to investigate the hydrodynamics in the atmosphere of the K star. From radial velocities measured during the egress of the 1951 eclipse1 we have calculated correlation coefficients as defined below. The observati9ns here used are velocities at separate points in space-time; hence the correlation coefficients depend on space-time intervals. Moreover, the measured velocities are not velocities at single points in the atmosphere, but are mean velocities along the line of sight, the means being weighted with respect to the contribution of each element of atmosphere to the line intensities. For our purposes, the correlation coefficient is T - ~ v5(t) Vz(t + n) R5 (n) = [~))%52~t) T 52(t + n)i-s' t=n where n is a time interval in days, P is the number of days over which observations are made, t is the index (from I to P), the z direction is the line of sight direction, and v is the observed velocity corrected for orbital motion. The correlation Co- efficient is expressed as a function of n alone, since the transverse velocity of the B star is very nearly constant during the observation period. In this case, n completely specifies the space-time interval on which the correlation depends. If there exists turbulence in the hydrodynamical sense in the giant's atmosphere, moderately large eddies, i.e., those whose characteristic lengths are comparable to half the thickness of the atmosphere, will dominate the observed velocities. Smaller eddies than these will have effects analogous to those of thermal broadening, and larger ones will, if they are much larger than the transverse distance spanned by the observations, alter the observed mean velocities. In terms of this turbulence model, the length scale of the energy-containing eddies can be obtained, if the eddies lie within the detectable range discussed in the preceding paragraph. This is accomplished by replacing n by a corresponding space interval. The space interval is linearly related to n through the transverse velocity of the B star. We have calculated the above correlation for velocities derived from the calcium K line.1 Twenty-two observations, taken over a 3~-day period, were used. The calculated correlation Coefficients imply that, on the basis of this provisional turbulence model, the length scale of the energy-containing eddies is of the order of 2 X I0~ km. I.D. B. McLaughlin, Ap. J. ~~6, 550, 1952. University of Michigan Observatory, Ann Arbor, Mich.