scholarly journals Stationary states of two-dimensional magnetohydrodynamic turbulence: non-dissipative limit

1979 ◽  
Vol 22 (3) ◽  
pp. 385-396 ◽  
Author(s):  
Ronald Calinon ◽  
Danilo Merlini
Keyword(s):  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Canonical Ensemble ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Stationary States ◽  
Two Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Magnetohydrodynamic Turbulence ◽  
Statistical Solutions

A class of exact stationary statistical states for the inviscid magnetohydrodynamic equations in two dimensions and in various geometries is found and the corresponding fluctuation spectra are calculated. Some solutions agree with previous computations in the canonical ensemble while other solutions are found. In particular, the Navier—Stokes limit is recovered and maximum cross helicity solutions exist in two dimensions. The difficulty of proving existence and uniqueness of statistical solutions for non-dissipative two-dimensional turbulence is quoted in terms of rugged constants and associated Gibbs measure.

Thermodynamic formalism for certain nonhyperbolic maps

1999 ◽  
Vol 19 (5) ◽  
pp. 1365-1378 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Number Theory ◽  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Periodic Points ◽  
Two Dimensional ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Conformal Measures ◽  
One Dimensional Maps

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

Turbulent resistivity in wavy two-dimensional magnetohydrodynamic turbulence

2008 ◽  
Vol 595 ◽  
pp. 173-202 ◽  
Author(s):  
SHANE R. KEATING ◽  
P. H. DIAMOND
Keyword(s):  
Large Scale ◽  
Mean Field ◽  
Magnetic Reynolds Number ◽  
Two Dimensions ◽  
Magnetic Potential ◽  
Turbulence Theory ◽  
Two Dimensional ◽  
Restoring Force ◽  
Constrained System ◽  
Magnetohydrodynamic Turbulence

The theory of turbulent resistivity in ‘wavy’ magnetohydrodynamic turbulence in two dimensions is presented. The goal is to explore the theory of quenching of turbulent resistivity in a regime for which the mean field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple two-dimensional problem to include body forces, such as buoyancy or the Coriolis force, which convert large-scale eddies into weakly interacting dispersive waves. The turbulence-driven spatial flux of magnetic potential is calculated to fourth order in wave slope – the same order to which one usually works in wave kinetics. However, spatial transport, rather than spectral transfer, is the object here. Remarkably, adding an additional restoring force to the already tightly constrained system of high Rm magnetohydrodynamic turbulence in two dimensions can actually increase the turbulent resistivity, by admitting a spatial flux of magnetic potential which is not quenched at large Rm, although it is restricted by the conditions of applicability of weak turbulence theory. The absence of Rm-dependent quenching in this wave-interaction-driven flux is a consequence of the presence of irreversibility due to resonant nonlinear three-wave interactions, which are independent of collisional resistivity. The broader implications of this result for the theory of mean field electrodynamics are discussed.

scholarly journals Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Hong Yin
Keyword(s):  
Incompressible Flow ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Finite Dimensional

The existence and uniqueness of adapted solutions to the backward stochastic Navier-Stokes equation with artificial compressibility in two-dimensional bounded domains are shown by Minty-Browder monotonicity argument, finite-dimensional projections, and truncations. Continuity of the solutions with respect to terminal conditions is given, and the convergence of the system to an incompressible flow is also established.

scholarly journals Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

2014 ◽  
Vol 750 ◽  
pp. 518-554 ◽  
Author(s):  
Dan Lucas ◽  
Rich Kerswell
Keyword(s):  
Stokes Equations ◽  
Type Equation ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Multiple Attractors ◽  
Navier Stokes Equations ◽  
Kolmogorov Flow ◽  
Long Wavelength ◽  
Collisional Dynamics

AbstractKolmogorov flow in two dimensions – the two-dimensional (2D) Navier–Stokes equations with a sinusoidal body force – is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn–Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions – a kink and antikink – which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

scholarly journals Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bo You
Keyword(s):  
Stokes System ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Time Behavior ◽  
Contact Lines ◽  
Trajectory Attractor ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
Long Time ◽  
Statistical Solutions

<p style='text-indent:20px;'>The objective of this paper is to consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. As we know, it is very difficult to obtain the uniqueness of an energy solution for this system even in two dimensions caused by the presence of the strong coupling at the boundary. Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [<xref ref-type="bibr" rid="b38">38</xref>], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.</p>

On the existence and uniqueness of solution to a stochastic 2D Allen–Cahn–Navier–Stokes model

2019 ◽  
Vol 19 (01) ◽  
pp. 1950007 ◽  
Author(s):  
Theodore Tachim Medjo
Keyword(s):  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Variational Solution ◽  
Uniqueness Of Solution ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Stochastic Version ◽  
Phase Parameter

We study, in this paper, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional (2D) bounded domain. The model consists of the Navier–Stokes equations (NSEs) for the velocity, coupled with a Allen–Cahn model for the order (phase) parameter. We prove the existence and the uniqueness of a variational solution.

Granular hydrodynamics and pattern formation in vertically oscillated granular disk layers

2008 ◽  
Vol 597 ◽  
pp. 119-144 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
THORSTEN PÖSCHEL ◽  
CLARA SALUEÑA
Keyword(s):  
Numerical Scheme ◽  
Granular Flows ◽  
Quantitative Agreement ◽  
Vertical Vibration ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Faraday Waves ◽  
Periodic Patterns ◽  
Hydrodynamic Description

The goal of this study is to demonstrate numerically that certain hydrodynamic systems, derived from inelastic kinetic theory, give fairly good descriptions of rapid granular flows even if they are way beyond their supposed validity limits. A numerical hydrodynamic solver is presented for a vibrated granular bed in two dimensions. It is based on a highly accurate shock capturing state-of-the-art numerical scheme applied to a compressible Navier–Stokes system for granular flow. The hydrodynamic simulation of granular flows is challenging, particularly in systems where dilute and dense regions occur at the same time and interact with each other. As a benchmark experiment, we investigate the formation of Faraday waves in a two-dimensional thin layer exposed to vertical vibration in the presence of gravity. The results of the hydrodynamic simulations are compared with those of event-driven molecular dynamics and the overall quantitative agreement is good at the level of the formation and structure of periodic patterns. The accurate numerical scheme for the hydrodynamic description improves the reproduction of the primary onset of patterns compared to previous literature. To our knowledge, these are the first hydrodynamic results for Faraday waves in two-dimensional granular beds that accurately predict the wavelengths of the two-dimensional standing waves as a function of the perturbation's amplitude. Movies are available with the online version of the paper.

Urbanicity, City Delegations, and Two-Dimensional Liberalism

2018 ◽  
Author(s):  
Thomas K. Ogorzalek
Keyword(s):  
National Level ◽  
Two Dimensions ◽  
Political Order ◽  
Two Dimensional ◽  
Local Institutions ◽  
Horizontal Integration ◽  
National Politics ◽  
United Front ◽  
Strategies For Success

This theoretical chapter develops the argument that the conditions of cities—large, densely populated, heterogeneous communities—generate distinctive governance demands supporting (1) market interventions and (2) group pluralism. Together, these positions constitute the two dimensions of progressive liberalism. Because of the nature of federalism, such policies are often best pursued at higher levels of government, which means that cities must present a united front in support of city-friendly politics. Such unity is far from assured on the national level, however, because of deep divisions between and within cities that undermine cohesive representation. Strategies for success are enhanced by local institutions of horizontal integration developed to address the governance demands of urbanicity, the effects of which are felt both locally and nationally in the development of cohesive city delegations and a unified urban political order capable of contending with other interests and geographical constituencies in national politics.

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