3,420 research outputs found
Surface Quasigeostrophic Turbulence : The Study of an Active Scalar
We study the statistical and geometrical properties of the potential
temperature (PT) field in the Surface Quasigeostrophic (SQG) system of
equations. In addition to extracting information in a global sense via tools
such as the power spectrum, the g-beta spectrum and the structure functions we
explore the local nature of the PT field by means of the wavelet transform
method. The primary indication is that an initially smooth PT field becomes
rough (within specified scales), though in a qualitatively sparse fashion.
Similarly, initially 1D iso-PT contours (i.e., PT level sets) are seen to
acquire a fractal nature. Moreover, the dimensions of the iso-PT contours
satisfy existing analytical bounds. The expectation that the roughness will
manifest itself in the singular nature of the gradient fields is confirmed via
the multifractal nature of the dissipation field. Following earlier work on the
subject, the singular and oscillatory nature of the gradient field is
investigated by examining the scaling of a probability measure and a sign
singular measure respectively. A physically motivated derivation of the
relations between the variety of scaling exponents is presented, the aim being
to bring out some of the underlying assumptions which seem to have gone
unnoticed in previous presentations. Apart from concentrating on specific
properties of the SQG system, a broader theme of the paper is a comparison of
the diagnostic inertial range properties of the SQG system with both the 2D and
3D Euler equations.Comment: 26 pages, 11 figures. To appear in Chao
Shocks and Universal Statistics in (1+1)-Dimensional Relativistic Turbulence
We propose that statistical averages in relativistic turbulence exhibit
universal properties. We consider analytically the velocity and temperature
differences structure functions in the (1+1)-dimensional relativistic
turbulence in which shock waves provide the main contribution to the structure
functions in the inertial range. We study shock scattering, demonstrate the
stability of the shock waves, and calculate the anomalous exponents. We comment
on the possibility of finite time blowup singularities.Comment: 37 pages, 7 figure
Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation
In this first paper we present a Lagrangian framework for the description of
structure formation in general relativity, restricting attention to
irrotational dust matter. As an application we present a self-contained
derivation of a general-relativistic analogue of Zel'dovich's approximation for
the description of structure formation in cosmology, and compare it with
previous suggestions in the literature. This approximation is then
investigated: paraphrasing the derivation in the Newtonian framework we provide
general-relativistic analogues of the basic system of equations for a single
dynamical field variable and recall the first-order perturbation solution of
these equations. We then define a general-relativistic analogue of Zel'dovich's
approximation and investigate its implications by functionally evaluating
relevant variables, and we address the singularity problem. We so obtain a
possibly powerful model that, although constructed through extrapolation of a
perturbative solution, can be used to put into practice nonperturbatively, e.g.
problems of structure formation, backreaction problems, nonlinear properties of
gravitational radiation, and light-propagation in realistic inhomogeneous
universe models. With this model we also provide the key-building blocks for
initializing a fully relativistic numerical simulation.Comment: 21 pages, content matches published version in PRD, discussion on
singularities added, some formulas added, some rewritten and some correcte
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
Element sets for high-order Poincar\'e mapping of perturbed Keplerian motion
The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key
topic in celestial mechanics and astrodynamics, e.g. to study the stability of
orbits or design bounded relative trajectories. The high-order transfer map
(HOTM) method enables efficient mapping of perturbed Keplerian orbits over many
revolutions. For this, the method uses the high-order Taylor expansion of a
Poincar\'e or stroboscopic map, which is accurate close to the expansion point.
In this paper, we investigate the performance of the HOTM method using
different element sets for building the high-order map. The element sets
investigated are the classical orbital elements, modified equinoctial elements,
Hill variables, cylindrical coordinates and Deprit's ideal elements. The
performances of the different coordinate sets are tested by comparing the
accuracy and efficiency of mapping low-Earth and highly-elliptical orbits
perturbed by with numerical propagation. The accuracy of HOTM depends
strongly on the choice of elements and type of orbit. A new set of elements is
introduced that enables extremely accurate mapping of the state, even for high
eccentricities and higher-order zonal perturbations. Finally, the high-order
map is shown to be very useful for the determination and study of fixed points
and centre manifolds of Poincar\'e maps.Comment: Pre-print of journal articl
Gravito-inertial waves in a differentially rotating spherical shell
The gravito-inertial waves propagating over a shellular baroclinic flow
inside a rotating spherical shell are analysed using the Boussinesq
approximation. The wave properties are examined by computing paths of
characteristics in the non-dissipative limit, and by solving the full
dissipative eigenvalue problem using a high-resolution spectral method.
Gravito-inertial waves are found to obey a mixed-type second-order operator and
to be often focused around short-period attractors of characteristics or
trapped in a wedge formed by turning surfaces and boundaries. We also find
eigenmodes that show a weak dependence with respect to viscosity and heat
diffusion just like truly regular modes. Some axisymmetric modes are found
unstable and likely destabilized by baroclinic instabilities. Similarly, some
non-axisymmetric modes that meet a critical layer (or corotation resonance) can
turn unstable at sufficiently low diffusivities. In all cases, the instability
is driven by the differential rotation. For many modes of the spectrum, neat
power laws are found for the dependence of the damping rates with diffusion
coefficients, but the theoretical explanation for the exponent values remains
elusive in general. The eigenvalue spectrum turns out to be very rich and
complex, which lets us suppose an even richer and more complex spectrum for
rotating stars or planets that own a differential rotation driven by
baroclinicity.Comment: 33 pages, 14 figures, accepted for publication in Journal of Fluid
Mechanic
Regularization of the triple collision in the collinear three body problem
This is a work about the collision of 3 celestial bodies which are aligned in a straight line, by following Newton's Gravitational Law. There is an introduction which shows the most important items of the general N-Body Problem in order to introduce the main concepts to work on with the main problem, which is the collinear 3-Body Problem. After this, at the conclusions we will see how the dynamics work for more than 3 bodies
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