90 research outputs found
The role of zero-clusters in exchange-driven growth with and without input
The exchange-driven growth model describes the mean field kinetics of a
population of composite particles (clusters) subject to pairwise exchange
interactions. Exchange in this context means that upon interaction of two
clusters, one loses a constituent unit (monomer) and the other gains this unit.
Two variants of the exchange-driven growth model appear in applications. They
differ in whether clusters of zero size are considered active or passive. In
the active case, clusters of size zero can acquire a monomer from clusters of
positive size. In the passive case they cannot, meaning that clusters reaching
size zero are effectively removed from the system. The large time behaviour is
very different for the two variants of the model. We first consider an isolated
system. In the passive case, the cluster size distribution tends towards a
self-similar evolution and the typical cluster size grows as a power of time.
In the active case, we identify a broad class of kernels for which the the
cluster size distribution tends to a non-trivial time-independent equilibrium
in which the typical cluster size is finite. We next consider a non-isolated
system in which monomers are input at a constant rate. In the passive case, the
cluster size distribution again attains a self-similar profile in which the
typical cluster size grows as a power of time. In the active case, a surprising
new behavior is found: the cluster size distribution asymptotes to the same
equilibrium profile found in the isolated case but with an amplitude that grows
linearly in time
A model differential equation for turbulence
A phenomenological turbulence model in which the energy spectrum obeys a
nonlinear diffusion equation is presented. This equation respects the scaling
properties of the original Navier-Stokes equations and it has the Kolmogorov
-5/3 cascade and the thermodynamic equilibrium spectra as exact steady state
solutions. The general steady state in this model contains a nonlinear mixture
of the constant-flux and thermodynamic components. Such "warm cascade"
solutions describe the bottleneck phenomenon of spectrum stagnation near the
dissipative scale. Self-similar solutions describing a finite-time formation of
steady cascades are analysed and found to exhibit nontrivial scaling behaviour.Comment: April 10 2003 Updated April 22 2003, 9 pages revtex4, 9 figures Added
some figures, additional references and corrected typo
Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence
We examine the probability distribution function (PDF) of the energy injection rate (power) in numerical simulations of stationary two-dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating 2D turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured PDFs with the theoretical calculations and briefly discuss how the power PDF might change with other forcing mechanisms
The life-cycle of drift-wave turbulence driven by small scale instability
We demonstrate theoretically and numerically the zonal-flow/drift-wave
feedback mechanism for the LH transition in an idealised model of plasma
turbulence driven by a small scale instability. Zonal flows are generated by a
secondary modulational instability of the modes which are directly driven by
the primary instability. The zonal flows then suppress the small scales thereby
arresting the energy injection into the system, a process which can be
described using nonlocal wave turbulence theory. Finally, the arrest of the
energy input results in saturation of the zonal flows at a level which can be
estimated from the theory and the system reaches stationarity without damping
of the large scales.Comment: 4 pages, 2 figure
Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method
Models and methods that are able to accurately and efficiently predict the
flows of low-speed rarefied gases are in high demand, due to the increasing
ability to manufacture devices at micro and nano scales. One such model and
method is a Fokker-Planck approximation to the Boltzmann equation, which can be
solved numerically by a stochastic particle method. The stochastic nature of
this method leads to noisy estimates of the thermodynamic quantities one wishes
to sample when the signal is small in comparison to the thermal velocity of the
gas. Recently, Gorji et al have proposed a method which is able to greatly
reduce the variance of the estimators, by creating a correlated stochastic
process which acts as a control variate for the noisy estimates. However, there
are potential difficulties involved when the geometry of the problem is
complex, as the method requires the density to be solved for independently.
Importance sampling is a variance reduction technique that has already been
shown to successfully reduce the noise in direct simulation Monte Carlo
calculations. In this paper we propose an importance sampling method for the
Fokker-Planck stochastic particle scheme. The method requires minimal change to
the original algorithm, and dramatically reduces the variance of the estimates.
We test the importance sampling scheme on a homogeneous relaxation, planar
Couette flow and a lid-driven-cavity flow, and find that our method is able to
greatly reduce the noise of estimated quantities. Significantly, we find that
as the characteristic speed of the flow decreases, the variance of the noisy
estimators becomes independent of the characteristic speed
Rossby and Drift Wave Turbulence and Zonal Flows: the Charney-Hasegawa-Mima model and its extensions
A detailed study of the Charney-Hasegawa-Mima model and its extensions is
presented. These simple nonlinear partial differential equations suggested for
both Rossby waves in the atmosphere and also drift waves in a
magnetically-confined plasma exhibit some remarkable and nontrivial properties,
which in their qualitative form survive in more realistic and complicated
models, and as such form a conceptual basis for understanding the turbulence
and zonal flow dynamics in real plasma and geophysical systems. Two idealised
scenarios of generation of zonal flows by small-scale turbulence are explored:
a modulational instability and turbulent cascades.
A detailed study of the generation of zonal flows by the modulational
instability reveals that the dynamics of this zonal flow generation mechanism
differ widely depending on the initial degree of nonlinearity. A numerical
proof is provided for the extra invariant in Rossby and drift wave turbulence
-zonostrophy and the invariant cascades are shown to be characterised by the
zonostrophy pushing the energy to the zonal scales.
A small scale instability forcing applied to the model demonstrates the
well-known drift wave - zonal flow feedback loop in which the turbulence which
initially leads to the zonal flow creation, is completely suppressed and the
zonal flows saturate. The turbulence spectrum is shown to diffuse in a manner
which has been mathematically predicted.
The insights gained from this simple model could provide a basis for
equivalent studies in more sophisticated plasma and geophysical fluid dynamics
models in an effort to fully understand the zonal flow generation, the
turbulent transport suppression and the zonal flow saturation processes in both
the plasma and geophysical contexts as well as other wave and turbulence
systems where order evolves from chaos.Comment: 64 pages, 33 figure
Constant flux relation for diffusion-limited cluster-cluster aggregation
In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d(c)=2, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes
Non-stationary Spectra of Local Wave Turbulence
The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is
studied in the limit of strongly local interactions where the usual kinetic
equation, describing the time evolution of the spectral wave-action density,
can be approximated by a PDE. If the wave action is initially compactly
supported in frequency space, it is then redistributed by resonant interactions
producing the usual direct and inverse cascades, leading to the formation of
the K-Z spectra. The emphasis here is on the direct cascade. The evolution
proceeds by the formation of a self-similar front which propagates to the right
leaving a quasi-stationary state in its wake. This front is sharp in the sense
that the solution remains compactly supported until it reaches infinity. If the
energy spectrum has infinite capacity, the front takes infinite time to reach
infinite frequency and leaves the K-Z spectrum in its wake. On the other hand,
if the energy spectrum has finite capacity, the front reaches infinity within a
finite time, t*, and the wake is steeper than the K-Z spectrum. For this case,
the K-Z spectrum is set up from the right after the front reaches infinity. The
slope of the solution in the wake can be related to the speed of propagation of
the front. It is shown that the anomalous slope in the finite capacity case
corresponds to the unique front speed which ensures that the front tip contains
a finite amount of energy as the connection to infinity is made. We also
introduce, for the first time, the notion of entropy production in wave
turbulence and show how it evolves as the system approaches the stationary K-Z
spectrum.Comment: revtex4, 19 pages, 10 figure
Percolation transition in the kinematics of nonlinear resonance broadening in Charney–Hasegawa–Mima model of Rossby wave turbulence
We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney-Hasegawa-Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterize how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a non-trivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system
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