Characteristics of plasma Beltrami states

Beltrami states in several models of plasma dynamics – incompressible magnetohydrodynamic (MHD) model, barotropic compressible MHD model, incompressible Hall MHD model, barotropic compressible Hall MHD model, electron MHD model, barotropic compressible Hall MHD with electron inertia model, are considered. Notwithstanding the diversity of the physics underlying the various models, the Beltrami states are shown to exhibit some common features like – certain robustness with respect to the plasma compressibility effects (albeit in the barotropy assumption), the Bernoulli condition. The Beltrami states for these models are deduced by minimizing the appropriate total energy while keeping the appropriate total helicity constant.     

Blowup Criterion for Viscous Baratropic Flows with Vacuum States

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.     

On some properties of multidimensional hyperbolic quasi-gasdynamic systems of equations

We study a multidimensional hyperbolic quasi-gasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ²). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ²) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it.     

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.     

Viscous Multipolar Fluids-Physical Background and Mathematical Theory-

We introduce the theory of multipolar fluids in which constitutive laws depend linearly not only on the first spatial gradients of velocity as in classical Navier-Stokes theory of newtonian fluids but also on its higher order spatial gradients up to the order 2k − 1, k = 2, 3,… Such fluids are called k-polar fluids. A thermodynamic theory of the constitutive equations satisfying the second law of thermodynamics and the principle of material frame indifference is developed. Special thermodynamic processes as isothermal, barotropic, adiabatic and general heat-conductive motion for compressible multipolar fluids are studied. It is well known that there does not exist adequate existence theory for compressible newtonian fluids. We given a consistent theory for compressible multipolar fluids in two or three dimensions, i. e. we prove the global in time existence of weak solutions for the initial boundary value problems in bounded domains for the systems of partial differential equations describing isothermal, barotropic, adiabatic and general compressible motion. Under some assumptions on the regularity of the initial data and external forces, we prove existence of strong solutions, uniqueness and regularity. Some other properties as e. g. cavitation of density are discussed. We put stress on the lowest possible polarity of the fluid. In the isothermal case we consider the polarity k ≧ 2 and in barotropic and heat-conductive gas the polarity k ≧ 3.     

Global Weak Solutions to One-Dimensional Non-Conservative Viscous Compressible Two-Phase System

In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 < γ_± < 6 assuming non-zero surface tension. In our study, we strongly improve the results by taking advantage of the one space dimension. More precisely, we obtain global existence of weak solutions without using capillarity terms and for pressure laws with the same range of coefficients as the degenerate barotropic mono-fluid system, namely γ_± > 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit.     

Spin degree of freedom in the nu=1 bilayer electron system investigated by nuclear spin relaxation

The nuclear-spin-relaxation rate 1/T(1) has been measured in a bilayer electron system at and around total Landau level filling factor nu=1. The measured 1/T(1), which probes electron spin fluctuations, is found to increase gradually from the quantum Hall (QH) state at low fields through a phase transition to the compressible state at high fields. Furthermore, 1/T(1) in the QH state shows a noticeable increase away from nu=1. These results demonstrate that, as opposed to common assumption, the electron spin degree of freedom is not completely frozen either in the QH or the compressible states.     

A fluid-mixture type algorithm for barotropic two-fluid flow problems

Our goal is to present a simple interface-capturing approach for barotropic two-fluid flow problems in more than one space dimension. We use the compressible Euler equations in isentropic form as a model system with the thermodynamic property of each fluid component characterized by the Tait equation of state. The algorithm uses a non-isentropic form of the Tait equation of state as a basis to the modeling of the numerically induced mixing between two different barotropic fluid components within a grid cell. Similar to our previous work for multicomponent problems, see [J. Comput. Phys. 171 (2001) 678] and references cited therein, we introduce a mixture type of the model system that consists of the full Euler equations for the basic conserved variables and an additional set of evolution equations for the problem-dependent material quantities and also the approximate location of the interfaces. A standard high-resolution method based on a wave-propagation formulation is employed to solve the proposed model system with the dimensional-splitting technique incorporated in the method for multidimensional problems. Several numerical results are presented in one, two, and three space dimensions that show the feasibility of the method as applied to a reasonable class of practical problems without introducing any spurious oscillations in the pressure near the smeared material interfaces.     

Formation of large‐scale structures in four‐component dusty plasmas

A possibility of self‐organization of magnetized four‐component dusty plasmas to double Beltrami (DB) state is explored. It is found that for a specific set of Beltrami parameters, the four‐component dusty plasma self‐organizes to DB state. The DB state characterized by two scale parameters may represent a paramagnetic or diamagnetic field structure. The impact of Beltrami parameters, charge and densities of dust grains on formation of self‐organized structures has also been investigated. This study has potential relevance to the formation of large‐scale structures in astrophysical plasmas.     

Non-hydrodynamic modes and general equations of state in lattice Boltzmann equations

The lattice Boltzmann equation is commonly used to simulate fluids with isothermal equations of state in a weakly compressible limit, and intended to approximate solutions of the incompressible Navier–Stokes equations. Due to symmetry requirements there are usually more degrees of freedom in the equilibrium distributions than there are constraints imposed by the need to recover the Navier–Stokes equations in a slowly varying limit. We construct equilibria for general barotropic fluids, where pressure depends only upon density, using the two-dimensional, nine velocity (D2Q9) and one-dimensional, five velocity (D1Q5) lattices, showing that one otherwise arbitrary function in the equilibria must be chosen to suppress instability.     

Vortex lattices in rotating atomic Bose gases with non-local interactions

We study the groundstates of rotating atomic Bose gases with non-local interactions. We focus on the weak-interaction limit of a model involving s- and d-wave interactions. With increasing d-wave interaction, the mean-field groundstate undergoes a series of transitions between vortex lattices of different symmetries (triangular, square, “stripe” and “bubble” crystal phases). We discuss the stability of these phases to quantum fluctuations. Using exact diagonalization studies, we show that with increasing d-wave interaction, the incompressible Laughlin state at filling factor ν=1/2 is replaced by compressible stripe and bubble states.     

Observation of soft magnetorotons in bilayer quantum Hall ferromagnets

Inelastic light-scattering measurements of low-lying collective excitations of electron double layers in the quantum Hall state at total filling nu(T)=1 reveal a deep magnetoroton in the dispersion of charge-density excitations across the tunneling gap. The roton softens and sharpens markedly when the phase boundary for transitions to highly correlated compressible states is approached. The findings are interpreted with Hartree-Fock evaluations that link soft magnetorotons to enhanced excitonic Coulomb interactions and to quantum phase transitions in the ferromagnetic bilayers.     

Nuclear-spin-lattice relaxation in a bilayer quantum Hall system

We examined the electron spin degree of freedom around the total Landau-level filling factor ν=1 in a bilayer system via nuclear spins. In a balanced bilayer system, nuclear-spin-lattice relaxation rate 1/T₁, which probes low-energy electron spin fluctuations, increases gradually as the system is driven from the quantum Hall (QH) state through a phase transition to the compressible state. This result demonstrates that the electron spin degree of freedom is not frozen either in the QH or compressible states. Furthermore, as the density difference between the two layers is increased from balanced bilayer to monolayer configurations, 1/T₁ around ν=1 shows a rapid yet smooth increase. This suggests that pseudospin textures around the bilayer ν=1 system evolves continuously into the spin texture for the monolayer system.     

Large-Time Behaviors of Solutions to an Inflow Problem in the Half Space for a One-Dimensional System¶of Compressible Viscous Gas

The “inflow problem” for a one-dimensional compressible barotropic flow on the half-line R ₊= (0,+∞) is investigated. Not only classical waves but also the new wave, which is called the “boundary layer solution”, arise. Large time behaviors of the solutions to be expected have been classified in terms of the boundary values by [A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, to appear in Proceedings of IMS Conference on Differential Equations from Mechanics, Hong Kong, 1999]. In this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave. Received: 25 April 2000 / Accepted: 20 April 2001     

Bilayer coherent and quantum Hall phases: duality and quantum disorder

We consider a fully spin-polarized quantum Hall system with no interlayer tunneling at total filling factor nu = 1/k (where k is an odd integer) using the Chern-Simons-Ginzburg-Landau theory. Exploiting particle-vortex duality and the concept of quantum disordering, we find a large number of possible compressible and incompressible ground states, some of which may have relevance to recent experiments of Spielman et al. [Phys. Rev. Lett. 84, 5808 (2000)]. We find interlayer coherent compressible states without Hall quantization and interlayer incoherent incompressible states with Hall quantization in addition to the usual (k,k,k) Halperin states, which are both interlayer coherent and incompressible.     

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.     

Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit–explicit Runge–Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier–Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge–Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge–Kutta methods.     

Properties of warm dense matter at low entropies

We describe the qualitative properties of warm dense matter on adiabats, paying particular attention to their behavior in the vicinity of phase transitions. The equation of state of matter on the adiabat corresponding to an entropy of 1 k_B per nucleon is calculated within the compressible liquid drop model for nuclei.     

Distinctive features of random local motions in critical and supercritical fluids

Equations for large-scale local fluctuations in fluids, from an ideal gas to an incompressible fluid, including the critical and supercritical state are derived for the first time based on the first principles. The modern phenomenological representation of the critical state of fluids is confirmed and essentially refined; in particular, it is demonstrated that that local density fluctuations in a compressible fluid are accompanied by nonthermodynamic fluctuations in the collective velocity and temperature of the fluid. Distinctive features of the development of these fluctuations near the critical point determine the specific behavior of fluids in the critical and supercritical states.