The influence of coriolis force on MHD waves

The influence of coriolis force on the propagation of MHD waves in compressible medium is investigated by using linear stability analysis. In a combined effect we find that, while compressibility gives rise to an acoustic wave, coriolis force results in a modified Alfvén wave.     

Approximate Deconvolution Models for Magnetohydrodynamics

We consider the family of approximate deconvolution models (ADM) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove the existence and uniqueness of solutions to the ADM-MHD equations, their weak converge to the solution of the MHD equations as the averaging radii tend to zero, and derive a bound on the modeling error. We demonstrate that the energy and helicity of the models are conserved, and the models preserve the Alfvén waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models.     

Irregular modulation of non-linear Alfvén/Beltrami wave coupled with an ion-sound wave

A singularity of a system of differential equations may produce “intrinsic” solutions that are independent of initial or boundary conditions—such solutions represent “irregular behavior” uncontrolled by external conditions. In the recently formulated non-linear model of Alfvén/Beltrami waves [Commum Nonlinear Sci Numer Simulat 17 (2012) 2223], we find a singularity occurring at the resonance of the Alfvén velocity and sound velocity, from which pulses bifurcate irregularly. By assuming a stationary waveform, we obtain a sufficient number of constants of motion to reduce the system of coupled ordinary differential equations (ODEs) into a single separable ODE that is readily integrated. However, there is a singularity in the separable equation that breaks the Lipschitz continuity, allowing irregular solutions to bifurcate. Apart from the singularity, we obtain solitary wave solutions and oscillatory solutions depending on control parameters (constants of motion).     

THE GLOBAL L~2 STABILITY OF SOLUTIONS TO THREE DIMENSIONAL MHD EQUATIONS

In this paper,we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains.Under suitable conditions of the large solutions,it is shown that the large solutions are stable.And we obtain the equivalent condition of this stability condition.Moreover,the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.     

Hydromagnetic instability of two coaxial streaming cylinders with double perturbed interfaces

The magnetohydrodynamic (MHD) stability of a double interface perturbed streaming liquid cylinder coaxial with a streaming fluid mantle acting upon capillary, inertial, pressure gradient and electromagnetic forces has been developed. The problem is formulated, solved and the stability criterion of the model is estabilished. The latter is discussed analytically and the results are confirmed numerically and interpreted physically. Some reported works are recovered as limiting cases. The capillary force is stabilizing or not according to restrictions. The magnetic field has a strong stabilizing influence. The radii (liquid–fluid) ratio plays an important role in increasing the MHD stabilizing domains. The density of liquid–fluid ratio has a little stabilizing effect. The streaming has a destabilizing influence for all kinds of (non-) axisymmetric perturbation modes. However, if the magnetic field strength is so strong such that the Alfvén wave velocity is greater than the streaming velocity, then the destabilizing character due to capillary force or/and streaming is completely suppressed and stability sets in. In the absence of the magnetic field and we neglect the fluid inertial force, the present results are in good agreement with the experimental results of (Kendall J.M. Phys Fluids 1986;29:2086).     

Small Alfvén number limit for incompressible magneto-hydrodynamics in a domain with boundaries

For any fixed Alfvén number, the local well-posedness is proved for the equations of threedimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfvén number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfvén number goes to zero.     

微分方程右函数间断及其处理方法

§1.引言 现代电气、化工、生物工程和航天飞行器、航空飞行器、核反应工程等都是复杂的系统工程.在这类工程的科学研究和工程设计中,经常需要用常微分方程组的初值问题来建立系统运动行为的数学模型.为了获得研究和设计的数据,需要进行科学工程计算和动     

MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory

We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl-type equations that are derived from the incompressible MHD system with non-slip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-i-time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. © 2018 Wiley Periodicals, Inc.     

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfvén and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

Stability of contact discontinuity for the Boltzmann equation

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and Navier-Stokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation corresponding to the contact discontinuities for the Euler equations, we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate. In our analysis, an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates.     

Hyperbolic boundary value problems for symmetric systems with variable multiplicities

We extend the Kreiss-Majda theory of stability of hyperbolic initial-boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a “totally nonglancing” or a “nonglancing and linearly splitting” condition. At the same time, we give a simple characterization of the block structure condition as “geometric regularity” of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner-Kruskal and Blokhin-Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin-Trakhinin by direct “dissipative integral” methods, of stability in the zero-magnetic field limit. We also discuss extensions to the viscous case.     

Relationships between discontinuities of derivatives on characteristics and trajectories

We investigate the discontinuities of normal derivatives on characteristics and on trajectories, which arise in nonisentropic gas flow computations. Such computations require knowledge of the relationships between the derivative discontinuities. A specific feature of the numerical solution of the Euler equations is noticed while investigating the effects associated with the appearance of vorticity. In real cases (unsteady flow, flow past a body, nonhomogeneous medium) the derivative of entropy has a discontinuity along the normal to the trajectory, and this effect should be taken into consideration in numerical work. The discontinuity of the entropy derivative may be obtained from relationships linking the discontinuities of the derivatives of fluid-dynamic functions. These relationships are derived from the dynamic consistency conditions of the Euler equations. In this article we derive relationships linking the discontinuities of the normal derivatives on characteristics and trajectories for functions describing particle velocities, the velocity of sound, and entropy.     

Stability of discontinuity structures described by a generalized KdV–Burgers equation

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form φ(u) = u⁴–u² is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.     

The method of potential functions in problems of the theory of elasticity for bodies with a defect

The method of potential functions using a Fourier transformation in the class of slowly increasing distributions, corresponding to the classical method of complex potentials, is proposed for solving well-known problems of the theory of elasticity for bodies with a defect. It is shown that when a Fourier transformation with respect to all the spatial variables is used, the solution of the dynamic problem of the theory of elasticity can also be represented in terms of a jump in the stresses and displacements at the defect. The correctness of the transformed problem is considered (in terms of an analogue of the Lopatinskii condition). The solution of the system of Helmholtz equations, to which the system of Lamé equations is reduced in the case of the two-dimensional dynamic problem, is expressed in terms of the jump in the stresses and displacements at the defect as a result of solving the corresponding singular integral equations.     

A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics(MHD)boundary layer system,it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by G′erard-Varet and Dormy(2010).This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.     

Large eddy simulation for turbulent magnetohydrodynamic flows

We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ₁,δ₂ tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD.     

Long time behavior of stochastic {MHD} equations perturbed by multiplicative noises

In this paper, 2-dimensional (2D) magnetohydrodynamics (MHD) equations perturbed by multiplicative noises in both the velocity and the magnetic field is studied. We first considered the stability, or the upper semi-continuity, for equivalent random dynamical systems (RDS), and then applying the abstract result we established the existence and the upper semi-continuity of tempered random attractors for the stochastic MHD equations. This result shows that the asymptotic behavior of MHD equations is stable under stochastic perturbations.     

Viscous quantum hydrodynamics and parameter‐elliptic systems

The viscous quantum hydrodynamic model derived for semiconductor simulation is studied in this paper. The principal part of the vQHD system constitutes a parameter‐elliptic operator provided that boundary conditions satisfying the Shapiro–Lopatinskii criterion are specified. We classify admissible boundary conditions and show that this principal part generates an analytic semigroup, from which we then obtain the local in time well‐posedness. Furthermore, the exponential stability of zero current and large current steady states is proved, without any kind of subsonic condition. The decay rate is given explicitly. Copyright © 2010 John Wiley & Sons, Ltd.