4,243 research outputs found
Mathematical models for erosion and the optimal transportation of sediment
We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem for the sediment flow, we show that this class of weak solutions implements the optimal transportation of the sediment
Numerical Tests of Fast Reconnection in Weakly Stochastic Magnetic Fields
We study the effects of turbulence on magnetic reconnection using 3D
numerical simulations. This is the first attempt to test a model of fast
magnetic reconnection in the presence of weak turbulence proposed by Lazarian &
Vishniac (1999). This model predicts that weak turbulence, generically present
in most of astrophysical systems, enhances the rate of reconnection by reducing
the transverse scale for reconnection events and by allowing many independent
flux reconnection events to occur simultaneously. As a result the reconnection
speed becomes independent of Ohmic resistivity and is determined by the
magnetic field wandering induced by turbulence. To quantify the reconnection
speed we use both an intuitive definition, i.e. the speed of the reconnected
flux inflow, as well as a more sophisticated definition based on a formally
derived analytical expression. Our results confirm the predictions of the
Lazarian & Vishniac model. In particular, we find that Vrec Pinj^(1/2), as
predicted by the model. The dependence on the injection scale for some of our
models is a bit weaker than expected, i.e. l^(3/4), compared to the predicted
linear dependence on the injection scale, which may require some refinement of
the model or may be due to the effects like finite size of the excitation
region. The reconnection speed was found to depend on the expected rate of
magnetic field wandering and not on the magnitude of the guide field. In our
models, we see no dependence on the guide field when its strength is comparable
to the reconnected component. More importantly, while in the absence of
turbulence we successfully reproduce the Sweet-Parker scaling of reconnection,
in the presence of turbulence we do not observe any dependence on Ohmic
resistivity, confirming that our reconnection is fast.Comment: 22 pages, 20 figure
Wavelet-based density estimation for noise reduction in plasma simulations using particles
For given computational resources, the accuracy of plasma simulations using
particles is mainly held back by the noise due to limited statistical sampling
in the reconstruction of the particle distribution function. A method based on
wavelet analysis is proposed and tested to reduce this noise. The method, known
as wavelet based density estimation (WBDE), was previously introduced in the
statistical literature to estimate probability densities given a finite number
of independent measurements. Its novel application to plasma simulations can be
viewed as a natural extension of the finite size particles (FSP) approach, with
the advantage of estimating more accurately distribution functions that have
localized sharp features. The proposed method preserves the moments of the
particle distribution function to a good level of accuracy, has no constraints
on the dimensionality of the system, does not require an a priori selection of
a global smoothing scale, and its able to adapt locally to the smoothness of
the density based on the given discrete particle data. Most importantly, the
computational cost of the denoising stage is of the same order as one time step
of a FSP simulation. The method is compared with a recently proposed proper
orthogonal decomposition based method, and it is tested with three particle
data sets that involve different levels of collisionality and interaction with
external and self-consistent fields
Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
We discuss the dynamics of zonal (or unidirectional) jets for barotropic
flows forced by Gaussian stochastic fields with white in time correlation
functions. This problem contains the stochastic dynamics of 2D Navier-Stokes
equation as a special case. We consider the limit of weak forces and
dissipation, when there is a time scale separation between the inertial time
scale (fast) and the spin-up or spin-down time (large) needed to reach an
average energy balance. In this limit, we show that an adiabatic reduction (or
stochastic averaging) of the dynamics can be performed. We then obtain a
kinetic equation that describes the slow evolution of zonal jets over a very
long time scale, where the effect of non-zonal turbulence has been integrated
out. The main theoretical difficulty, achieved in this work, is to analyze the
stationary distribution of a Lyapunov equation that describes quasi-Gaussian
fluctuations around each zonal jet, in the inertial limit. This is necessary to
prove that there is no ultraviolet divergence at leading order in such a way
that the asymptotic expansion is self-consistent. We obtain at leading order a
Fokker--Planck equation, associated to a stochastic kinetic equation, that
describes the slow jet dynamics. Its deterministic part is related to well
known phenomenological theories (for instance Stochastic Structural Stability
Theory) and to quasi-linear approximations, whereas the stochastic part allows
to go beyond the computation of the most probable zonal jet. We argue that the
effect of the stochastic part may be of huge importance when, as for instance
in the proximity of phase transitions, more than one attractor of the dynamics
is present
Particles and fields in fluid turbulence
The understanding of fluid turbulence has considerably progressed in recent
years. The application of the methods of statistical mechanics to the
description of the motion of fluid particles, i.e. to the Lagrangian dynamics,
has led to a new quantitative theory of intermittency in turbulent transport.
The first analytical description of anomalous scaling laws in turbulence has
been obtained. The underlying physical mechanism reveals the role of
statistical integrals of motion in non-equilibrium systems. For turbulent
transport, the statistical conservation laws are hidden in the evolution of
groups of fluid particles and arise from the competition between the expansion
of a group and the change of its geometry. By breaking the scale-invariance
symmetry, the statistically conserved quantities lead to the observed anomalous
scaling of transported fields. Lagrangian methods also shed new light on some
practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy
Correspondence between one- and two-equation models for solute\ud transport in two-region heterogeneous porous media
In this work, we study the transient behavior of upscaled models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the time scale of interest (t) is smaller than the characteristic time (T1) for the relaxation of the effective macroscale parameters (i.e., when t ≤ T1); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (T1) (i.e., when t » T1); and (3) a one-equation, time-asymptotic formulation (1eq∞). This model can be adopted when (t) is significantly larger than the time scale (T2) associated with exchange processes between the two regions (i.e., when t » T2). In order to obtain some physical insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in simple cases. The main result of this paper is to show that there is weak long-time convergence of the solution of (2eq) toward the solution of (1eq∞) in terms of standardized moments but, interestingly, not in terms of centered moments. Physically, our interpretation of this result is that the spreading of the solute is dominating higher order non-zero perturbations in the asymptotic regime
Order out of Randomness : Self-Organization Processes in Astrophysics
Self-organization is a property of dissipative nonlinear processes that are
governed by an internal driver and a positive feedback mechanism, which creates
regular geometric and/or temporal patterns and decreases the entropy, in
contrast to random processes. Here we investigate for the first time a
comprehensive number of 16 self-organization processes that operate in
planetary physics, solar physics, stellar physics, galactic physics, and
cosmology. Self-organizing systems create spontaneous {\sl order out of chaos},
during the evolution from an initially disordered system to an ordered
stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical
resonances, or gyromagnetic resonances. The internal driver can be gravity,
rotation, thermal pressure, or acceleration of nonthermal particles, while the
positive feedback mechanism is often an instability, such as the
magneto-rotational instability, the Rayleigh-B\'enard convection instability,
turbulence, vortex attraction, magnetic reconnection, plasma condensation, or
loss-cone instability. Physical models of astrophysical self-organization
processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra
equation systems.Comment: 61 pages, 38 Figure
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