513 research outputs found
The pulsations and the dynamical stability of gaseous masses in uniform rotation
A variational principle, applicable to axisymmetric oscillations of uniformly rotating axisymmetric configurations, is established On the assumption that the Lagrangian displacement (describing the oscillation) at any point is normal to the level surface (of constant total potential) through that point, it is shown how the variational expression, for the frequencies of oscillation, can be reduced to simple quadratures. The reduction is explicitly carried out for certain stratifications of special interest. Some new results on the oscillations of slowly rotating configurations are included; and a number of related observations on their stability are also made
On the ellipsoidal figures of equilibrium of homogeneous masses
This article does not have an abstract
Magnetoelliptic Instabilities
We consider the stability of a configuration consisting of a vertical
magnetic field in a planar flow on elliptical streamlines in ideal
hydromagnetics. In the absence of a magnetic field the elliptical flow is
universally unstable (the ``elliptical instability''). We find this universal
instability persists in the presence of magnetic fields of arbitrary strength,
although the growthrate decreases somewhat. We also find further instabilities
due to the presence of the magnetic field. One of these, a destabilization of
Alfven waves, requires the magnetic parameter to exceed a certain critical
value. A second, involving a mixing of hydrodynamic and magnetic modes, occurs
for all magnetic-field strengths. These instabilities may be important in
tidally distorted or otherwise elliptical disks. A disk of finite thickness is
stable if the magnetic fieldstrength exceeds a critical value, similar to the
fieldstrength which suppresses the magnetorotational instability.Comment: Accepted for publication in Astrophysical Journa
The equilibrium and the stability of the Jeans spheroids
The equilibrium and the stability of homogeneous masses distorted by the tidal effects of a secondary (of mass M' at a distance R) are re-examined on the basis of the second-order virial equations. In agreement with known results, it is shown that, under circumstances when the figure of equilibrium is a prolate spheroid, there is a maximum value of π( = GM'/R3) which is compatible with equilibrium. The problem of the small oscillations of these Jeans spheroids is next considered. The characteristic frequencies of oscillation belonging to the second harmonics are determined both in case the mass is considered incompressible and in case it is considered compressible and subject to the gas laws governing adiabatic changes. In the former case, instability sets in when μ attains its maximum value; and in the latter case it sets in before that happens
On the occurrence of multiple frequencies and beats in the β Canis Majoris stars
An explanation is suggested for the occurrence of two nearly equal frequencies and associated beats in the light- and in the velocity-variations of the β Canis Majoris stars. It is shown that if the ratio of the specific heats γ is 1.6 and the star is rotating, any disturbance will excite two normal modes with nearly equal frequencies
Shear-flow transition: the basin boundary
The structure of the basin of attraction of a stable equilibrium point is
investigated for a dynamical system (W97) often used to model transition to
turbulence in shear flows. The basin boundary contains not only an equilibrium
point Xlb but also a periodic orbit P, and it is the latter that mediates the
transition. Orbits starting near Xlb relaminarize. We offer evidence that this
is due to the extreme narrowness of the region complementary to basin of
attraction in that part of phase space near Xlb. This leads to a proposal for
interpreting the 'edge of chaos' in terms of more familiar invariant sets.Comment: 11 pages; submitted for publication in Nonlinearit
Classical Equations for Quantum Systems
The origin of the phenomenological deterministic laws that approximately
govern the quasiclassical domain of familiar experience is considered in the
context of the quantum mechanics of closed systems such as the universe as a
whole. We investigate the requirements for coarse grainings to yield decoherent
sets of histories that are quasiclassical, i.e. such that the individual
histories obey, with high probability, effective classical equations of motion
interrupted continually by small fluctuations and occasionally by large ones.
We discuss these requirements generally but study them specifically for coarse
grainings of the type that follows a distinguished subset of a complete set of
variables while ignoring the rest. More coarse graining is needed to achieve
decoherence than would be suggested by naive arguments based on the uncertainty
principle. Even coarser graining is required in the distinguished variables for
them to have the necessary inertia to approach classical predictability in the
presence of the noise consisting of the fluctuations that typical mechanisms of
decoherence produce. We describe the derivation of phenomenological equations
of motion explicitly for a particular class of models. Probabilities of the
correlations in time that define equations of motion are explicitly considered.
Fully non-linear cases are studied. Methods are exhibited for finding the form
of the phenomenological equations of motion even when these are only distantly
related to those of the fundamental action. The demonstration of the connection
between quantum-mechanical causality and causalty in classical phenomenological
equations of motion is generalized. The connections among decoherence, noise,
dissipation, and the amount of coarse graining necessary to achieve classical
predictability are investigated quantitatively.Comment: 100pages, 1 figur
Parabolic resonances and instabilities in near-integrable two degrees of freedom Hamiltonian flows
When an integrable two-degrees-of-freedom Hamiltonian system possessing a
circle of parabolic fixed points is perturbed, a parabolic resonance occurs. It
is proved that its occurrence is generic for one parameter families
(co-dimension one phenomenon) of near-integrable, t.d.o. systems. Numerical
experiments indicate that the motion near a parabolic resonance exhibits new
type of chaotic behavior which includes instabilities in some directions and
long trapping times in others. Moreover, in a degenerate case, near a {\it flat
parabolic resonance}, large scale instabilities appear. A model arising from an
atmospherical study is shown to exhibit flat parabolic resonance. This supplies
a simple mechanism for the transport of particles with {\it small} (i.e.
atmospherically relevant) initial velocities from the vicinity of the equator
to high latitudes. A modification of the model which allows the development of
atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities
are clearly observed
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