spiegel

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Author name code: spiegel
ADS astronomy entries on 2022-09-14
author:"Spiegel, Edward A."

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Title: Rapidly rotating Rayleigh-Bénard convection with a tilted axis
Authors: Novi, Lyuba; von Hardenberg, Jost; Hughes, David W.;
Provenzale, Antonello; Spiegel, Edward A.
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.

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Title: Intermittency and large-scale winds in horizontally anisotropic
convection
Authors: von Hardenberg, Jost; Goluskin, David; Provenzale, Antonello;
Spiegel, Edward
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.

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Title: Generation of Large-Scale Winds in Horizontally Anisotropic
Convection
Authors: von Hardenberg, J.; Goluskin, D.; Provenzale, A.; Spiegel,
E. A.
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.

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Title: Rotating relativistic stars
Authors: Spiegel, Edward
2015GApFD.109..462S Altcode:
No abstract at ADS

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Title: Convectively driven shear and decreased heat flux
Authors: Goluskin, David; Johnston, Hans; Flierl, Glenn R.; Spiegel,
Edward A.
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.

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Title: Uhlenbeck's Complaint
Authors: Spiegel, E. A.
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.

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Title: The Right Side of Einstein's Equation
Authors: Spiegel, Edward A
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).

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Title: Causal relativistic fluid dynamics
Authors: Chen, X.; Spiegel, E. A.
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.

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Title: Patterns of convection in solidifying binary solutions
Authors: Keating, Shane; Spiegel, E. A.; Worster, M. G.
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.

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Title: The Theory of Turbulence
Authors: Spiegel, Edward A.
2011LNP...810.....S Altcode:
No abstract at ADS

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Title: Spotted discs
Authors: Bracco, A.; Provenzale, A.; Spiegel, E. A.; Yecko, P.
2010tbha.book..254B Altcode:
No abstract at ADS

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Title: Destabilizing Taylor-Couette flow with suction
Authors: Gallet, Basile; Doering, Charles R.; Spiegel, Edward A.
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.

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Title: Cosmic vortices in hot stars and cool disks
Authors: Spiegel, Edward A.
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.

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Title: Chaos and Intermittency in the Solar Cycle
Authors: Spiegel, Edward A.
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.

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Title: Chaos and Intermittency in the Solar Cycle
Authors: Spiegel, Edward A.
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.

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Title: Modeling a Maunder minimum
Authors: Brandenburg, A.; Spiegel, E. A.
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.

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Title: Reflections on the solar tachocline
Authors: Spiegel, Edward A.
2007sota.conf...31S Altcode:
No abstract at ADS

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Title: Phenomenological photofluiddynamics
Authors: Spiegel, E. A.
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.

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Title: Obituary: Kevin H. Prendergast, 1929-2004
Authors: Spiegel, Edward A.
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.

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Title: Continuum equations for stellar dynamics
Authors: Spiegel, Edward A.; Thiffeault, Jean-Luc
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.

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Title: Luminosity effects in projected fractals
Authors: Thieberger, R.; Spiegel, E. A.
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.

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Title: Shear and Mixing in Oscillatory Doubly Diffusive Convection
Authors: Paparella, Francesco; Spiegel, Edward A.; Talon, Suzanne
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.

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Title: Continuum description of rarefied gas dynamics. III. The
structures of shock waves
Authors: Chen, Xinzhong; Spiegel, Edward A.; Rao, Hongling
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.

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Title: Continuum description of rarefied gas dynamics. I. Derivation
from kinetic theory
Authors: Chen, Xinzhong; Rao, Hongling; Spiegel, Edward A.
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.

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Title: Continuum description of rarefied gas dynamics. II. The
propagation of ultrasound
Authors: Chen, Xinzhong; Rao, Hongling; Spiegel, Edward A.
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.

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Title: Radiative bulk viscosity
Authors: Chen, Xinzhong; Spiegel, Edward A.
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.

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Title: Resonant Vibrational Instabilities in Magnetized Stellar
Atmospheres
Authors: Birch, A. C.; Kosovichev, A. G.; Spiegel, E. A.; Tao, L.
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.

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Title: The Radiative Stress Tensor
Authors: Chen, Xinzhong; Spiegel, Edward A.
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.

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Title: Particle aggregation in a turbulent Keplerian flow
Authors: Bracco, A.; Chavanis, P. H.; Provenzale, A.; Spiegel, E. A.
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.

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Title: Instability of Nonadiabatic Oscillations in a Magnetized
Polytrope
Authors: Birch, A. C.; Kosovichev, A. G.; Spiegel, E. A.; Tao, L.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 &amp; V.L. Polyachenko,
Springer-Verlag, 1984.

---------------------------------------------------------
Title: Acoustic Overstability of Plane Parallel Polytropes: an
Astrophysical Thermoacoustic Engine
Authors: Umurhan, O. M.; Spiegel, E. A.
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.
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.
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.
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.
1997NYASA.808...79G Altcode: 1997nsia.conf...79G
No abstract at ADS

---------------------------------------------------------
Title: Nonlinear ringing of polytropic disks.
Authors: Balmforth, N. J.; Spiegel, E. A.
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
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.
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.
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.
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.
1995LNP...458..129S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Autogravity waves in a polytropic layer
Authors: Qian, Z. S.; Spiegel, E. A.
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.
1994lspd.conf..245S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Patterns of Aperiodic Pulsation
Authors: Spiegel, E. A.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
1992pagn.conf...61A Altcode:
No abstract at ADS

---------------------------------------------------------
Title: The Turbulent Tachycline
Authors: Spiegel, E. A.; Zahn, J. -P.
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.
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.
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.
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.
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.
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.
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.
1987NYASA.497...61S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Chaos and the Solar Cycle
Authors: Spiegel, E. A.; Wolf, Alan
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.
1987Obs...107...89B Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Book-Review - Chaos in Astrophysics
Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A.
1987ApL....25S.266B Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Strange Accumulators
Authors: Smith, L. A.; Spiegel, E. A.
1987cpa..work...61S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Hydrostatic Adjustment Time of the Solar Subconvective Layer
Authors: Spiegel, E. A.
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.
1987cpa..work...55S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Book-Review - Chaos in Astrophysics
Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A.
1986Sci...232.1045B Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Chaos in astrophysics
Authors: Buchler, J. R.; Perdang, J. M.; Spiegel, E. A.
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.
1985capn.book.....B Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Cosmic arrhythmias.
Authors: Spiegel, E. A.
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: The dynamics of triple convection
Authors: Arneodo, A.; Coullet, P. H.; Spiegel, E. A.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
1977ApJ...212..593H Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Problems of stellar convection
Authors: Spiegel, E. A.; Zahn, J. -P.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
1976pmas.conf...19S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Radiative transfer through a flowing refractive medium.
Authors: Anderson, J. L.; Spiegel, E. A.
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.
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.
1975BAAS....7..526L Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Modal equations for cellular convection
Authors: Gough, D. O.; Spiegel, E. A.; Toomre, J.
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.
1975LNP....35..191G Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Photon Bubbles
Authors: Prendergast, K. H.; Spiegel, E. A.
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
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.
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.
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.
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.
1972ARA&A..10..261S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: A History of Solar Rotation
Authors: Spiegel, E. A.
1972NASSP.300...61S Altcode: 1972pss..conf...61S; 1972poss.conf...61S
No abstract at ADS

---------------------------------------------------------
Title: Some Fluid Dynamical Problems in Cosmogony
Authors: Spiegel, E.
1972oss..conf..165S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Transport Mechanisms in Stars
Authors: Spiegel, E. A.
1972stev.conf..493S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Galactic Wakes
Authors: Ruderman, M. A.; Spiegel, E. A.
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.
1971CoASP...3...53S Altcode: 1971ComAp...3...53S
No abstract at ADS

---------------------------------------------------------
Title: Book reviews
Authors: Švestka, Z.; Spiegel, E. A.; van Herk, G.
1971SSRv...11..867S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Convection in Stars: I. Basic Boussinesq Convection
Authors: Spiegel, E. A.
1971ARA&A...9..323S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Temperature Variation and the Solar Oblateness
Authors: Ingersoll, Andrew P.; Spiegel, Edward A.
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 &lt;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.
1971QJMAM..24..391B Altcode:
No abstract at ADS

---------------------------------------------------------
Title: Instabilities of Differential Rotation
Authors: Spiegel, E. A.; Zahn, J. -P.
1970CoASP...2..178S Altcode: 1970ComAp...2..178S
No abstract at ADS

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Title: A History of Solar Rotation
Authors: Spiegel, E. A.
1970pss..conf...63S Altcode:
No abstract at ADS

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Title: Thermodynamics and Cosmology
Authors: Schucking, E. L.; Spiegel, E. A.
1970CoASP...2..121S Altcode: 1970ComAp...2..121S
No abstract at ADS

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Title: Cortical Projection of Labyrinthine Impulses: Study of Averaged
Evoked Responses
Authors: Spiegel, E. A.; Szekely, E. G.; Moffet, H.; Egyed, J.
1970NASSP.187..259S Altcode:
No abstract at ADS

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Title: The Gas Dynamics of Accretion
Authors: Spiegel, E. A.
1970IAUS...39..201S Altcode:
No abstract at ADS

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Title: On the Circulation of Gas Near the Galactic Center
Authors: Spiegel, E. A.
1970IAUS...38..441S Altcode: 1970ssg..conf..441S
No abstract at ADS

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Title: Fluidization on the Moon (?)
Authors: Murray, J. D.; Spiegel, E. A.; Theys, J.
1969CoASP...1..165M Altcode: 1969ComAp...1..165M
No abstract at ADS

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Title: Semiconvection
Authors: Spiegel, E. A.
1969CoASP...1...57S Altcode: 1969ComAp...1...57S
No abstract at ADS

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Title: On the Flow of Gas from the Galactic Center
Authors: Moore, D. W.; Spiegel, E. A.
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

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Title: The Mixing of Lithium
Authors: Spiegel, E. A.
1968HiA.....1..261S Altcode:
No abstract at ADS

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Title: An overstability of gravity waves
Authors: Souffrin, P.; Spiegel, E. A.
1967AnAp...30..985S Altcode:
No abstract at ADS

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Title: The Solar Atmospheric Convection Zone -- The Theory of
Turbulent Convection
Authors: Spiegel, E. A.
1967IAUS...28..347S Altcode:
No abstract at ADS

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Title: Radiative damping of sound waves.
Authors: Stein, R. F.; Spiegel, E. A.
1967ASAJ...42..866S Altcode:
No abstract at ADS

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Title: A Thermally Excited Non-Linear Oscillator
Authors: Moore, Derek W.; Spiegel, Edward A.
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.

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Title: The Eddington Approximation in the Radiative Heat Equation
Authors: Unno, W.; Spiegel, E. A.
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.

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Title: Energy Transport by Turbulent Convection
Authors: Spiegel, Edward A.
1966stev.conf..143S Altcode:
No abstract at ADS

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Title: Nonlinear oscillations in the one-zone model for stellar
pulsation
Authors: Baker, N. H.; Moore, D. W.; Spiegel, E. A.
1966AJ.....71S.844B Altcode:
No abstract at ADS

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Title: The Solar Hydrogen Convection Zone and its Direct Influence
on the Photosphere
Authors: Spiegel, E. A.
1966IAUTB..12..539S Altcode:
No abstract at ADS

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Title: Convective Instability in a Compressible Atmosphere. I.
Authors: Spiegel, Edward A.
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.

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Title: On Convective Overstability.
Authors: Moore, D. W.; Spiegel, E. A.
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.

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Title: The Effect of Radiative Transfer on Convective Growth Rates.
Authors: Spiegel, Edward A.
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 &lt;6 the effect of radiation is vanishingly
small for large k; for m &gt; 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.

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Title: The Generation and Propagation of Waves in a Compressible
Atmosphere.
Authors: Moore, Derek W.; Spiegel, Edward A.
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.
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.

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Title: Thermal Turbulence at Very Small Prandtl Number
Authors: Spiegel, Edward A.
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.
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.

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Title: On the Spectrum of Turbulent Convection.
Authors: Ledoux, P.; Schwarzschild, M.; Spiegel, E. A.
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.

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Title: The Convective Instability of a Radiating Fluid Layer.
Authors: Spiegel, Edward A.
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.

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Title: On The Trumpler Shift in Early Stars.
Authors: Spiegel, E. A.
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.
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.
1960IAUS...12..319S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: The Onset of Thermal Convection in a Radiating Atmosphere.
Authors: Spiegel, Edward Alexander
1959PhDT.........2S Altcode:
No abstract at ADS

---------------------------------------------------------
Title: The Smoothing of Temperature Fluctuations by Radiative
Transfer .
Authors: Spiegel, Edward A.
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.
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.
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.