Hydromagnetic Equations Research Articles

Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

AbstractThe rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.

Hydromagnetic acceleration in relativistic outflows

We demonstrate that hydromagnetic acceleration can be the driving mechanism of relativistic outflows in AGNs as well as in Gamma-ray burst sources and Crab-like pulsars. Using semianalytical solutions of the full set of the steady, axisymmetric, ideal hydromagnetic equations in flat spacetime—i.e., solving the momentum equation along the flow as well as in the transfield direction—we model the acceleration of relativistic outflows. We find that a Poynting-flux dominated flow near the source reaches equipartition between matter and magnetic energy-fluxes, or even becomes completely matter-dominated (as in the case of Crab-like pulsar winds).

Plasma waves driven by gravitational waves in an expanding universe

In a Friedmann–Robertson–Walker (FRW) cosmological model with zero spatial curvature, we consider the interaction of the gravitational waves with the plasma in the presence of a weak magnetic field. Using the relativistic hydromagnetic equations, it is verified that large-amplitude magnetosonic waves are excited, assuming that both the gravitational field and the weak magnetic field do not break the homogeneity and isotropy of the considered FRW spacetime.

Comptonization and synchrotron emission in 2D accretion flows

A numerical approach is presented for solving the hydro-magnetic radiative transfer equation in 4D, in which the action of the Kompaneets operator and synchrotron cooling are taken into account. The equation is solved using the pre-conditioned Black-White-Brown line Gauss-Seidel semi-iterative method. This alternating approach enables the use of vector and parallel machines without loss of efficiency. The method can be used to constrain and search for density-, temperature- and magnetic-field distributions with or without outflows. This is appropriate for displaying spectra of 2D accretion flows that are consistent with observations of low-mass X-ray binaries.

Fast magnetosonic waves driven by gravitational waves

The propagation of a gravitational wave (GW) through a magnetized plasma is considered. In particular, we study the excitation of fast magnetosonic waves (MSW) by a gravitational wave, using the linearized general-relativistic hydromagnetic equations. We derive the dispersion relation for the plasma, treating the gravitational wave as a perturbation in a Minkowski background space-time. We show that the presence of gravitational waves will drive magnetosonic waves in the plasma and discuss the potential astrophysical implications.

Neutrino conversions in solar random magnetic fields

We consider the effect of a random magnetic field in the convective zone of the Sun super-imposed to a regular magnetic field on resonant neutrino spin-flavor oscillations. We argue for the existence of a field of strongly chaotic nature at the bottom of the convective zone. In contrast to previous attempts we employ a model motivated regular magnetic field profile: it is a static field solution to the solar equilibrium hydro-magnetic equations. These solutions have been known for a long time in the literature. We show for the first time that in addition they are twisting solutions. In this scenario electron antineutrinos are produced through cascades like ν eL → ν μL → V ̃ eR , The detection of V ̃ eR at Earth would be a long-awaited signature of the Majorana nature of the neutrino. The expected signals in the different experiments (SK, GALLEX-SAGE, Homestake) are obtained as a function of the level of noise, regular magnetic field and neutrino mixing parameters. Previous results obtained for small mixing and ad-hoc regular magnetic profiles are reobtained. We confirm the strong suppression for a large part of the parameter space of the V ̃ eR- flux for high energy boron neutrinos in agreement with present data of the SK experiment. We find that MSW (Mikheyev-Smirnov-Wolfenstein) regions ( Δm 2 ≅ ]0 −5 eV 2, both small and large mixing solutions) are stable up to very large levels of noise ( P = 0.7−0.8) but they are acceptable from the point of view of antineutrino production only for moderate levels of noise ( P ≅ 0.95). For strong noise and a reasonable regular magnetic field, any parameter region ( Δm 2, sin 2 2 θ) is excluded. As a consequence, we are allowed to reverse the problem and to put limits on the r.m.s. field strength and transition magnetic moments by demanding a particle physics solution to the SNP in this scenario.

Solutions of the hydromagnetic equation with magnetic null points: uniform-resistivity case

A general class of solutions is found for the resistive hydromagnetic equation. The results are applied to the case of a prescribed stagnation point flow. It is shown that general solutions with magnetic null points exist. Scaling laws for the length and the width of the current sheet of the solutions are given for the general case. It is shown that the length of the current sheet increases with magnetic Reynolds number unless the outflow boundary conditions prevent the sheet length from growing. The latter would result in the appearance of boundary layers in the outflow region.

Stability analysis of the ideal m=n=1 kink mode in toroidal geometry by direct expansion of the hydromagnetic equations

The stability condition and growth rate of the ideal, internal m=n=1 kink mode in a toroidal plasma are calculated by means of a direct inverse-aspect-ratio expansion of the magnetohydrodynamic (MHD) equations, rather than by using the energy principle. The analysis is based on the use of computer-aided algebra. The general equation for incompressible MHD oscillations in a low-β circular tokamak is derived to lowest order in the inverse aspect ratio. This derivation generalizes the Pfirsch-Schlüter factor for the kinetic energy in toroidal geometry.

Solutions of the hydromagnetic equation with magnetic null points

Numerical simulations of magnetic reconnection with uniform resistivity show that the length of the current sheet increases for increasing magnetic Reynolds number. In order to prevent the current sheet from growing in general the resistivity is assumed to be localized. For uniform resistivity the reconnection proceeds much slower than for localized resistivity. In this paper analytical solutions of the hydromagnetic equation are presented for localized and uniform resistivity. It is shown that there exists an essential singularity in the behaviour of a solution of magnetic reconnection in the limit of large magnetic Reynolds number: For prescribed boundary conditions the solution for localized resistivity does not approach the solution for uniform resistivity in the limit of large magnetic Reynolds number.

Surface wave excitation of local field line oscillations

The well‐behaved solution of the hydromagnetic wave equation for a general class of initial conditions is used to study the ringing, phase‐mixing, transient amplification of field line resonances. An expression for the phase‐mixing time in terms of the amplitude of initial impulse at the magnetopause and the gradient of Alfven speed in the magnetosphere is derived. The features of free oscillation of field lines studied here compare well with the previous numerical studies for particular types of impulse excitations of the magnetosphere.

Stability of contrasting solutions of nonlinear hydromagnetic dynamo equations and magnetic fields reversals in galaxies

Abstract It is known that steady solutions of the hydromagnetic dynamo problem describe sufficiently well the Global Magnetic Patterns (GMP) in spiral galaxies. This paper is devoted to nonlinear quasi-steady state GMP in a thin disc and the stability of these patterns. It is shown that for a majority of spiral galaxies (with a flat rotation curve and with the disc thickness being a linear function of the radius) any stable or long-living axisymmetric GMP with reversals of magnetic field do not exist. For Milky Way it is shown that the analytical results are in a qualitative agreement with the observational information revealing several stable or quasi-steady reversals. The main mathematical instrument is the method of boundary functions of the theory of singular perturbations. The ratio of thickness of the galactic disc to its radius is used as an asymptotic (small) parameter.

A closed set of conservation laws and the evolution of the electron magnetic moment in the collisionless solar wind

A hydromagnetic equation system for the interplanetary collisionless solar wind is used to derive a set of conservation laws for that medium. It is found that every equation of the original system, including the closure relation, is related to one conservation law. The set that has been derived does not only include the traditional laws, but also a new one for the magnetic moment of the electrons. The conservation set is then used to obtain the space constants for the solar coronal expansion. The new law yields a constant that has not been predicted by other models.

Numerical analysis of coupled hydromagnetic wave equations with a finite difference scheme

A finite difference scheme offering second-order accuracy is introduced to solve numerically a system of two mixed-type coupled partial differential equations with variable coefficients. The stability conditions of the scheme have been examined by both the Fourier method and the matrix method. The Fourier method via the local transform is first used to investigate parametrically the stability conditions of the proposed scheme. The stability conditions are checked point by point for the entire domain of interest without involving the convolution of the Fourier transform. These conditions are further verified by the matrix method. Since two different methods are employed, one can ensure that the stability conditions are achieved consistently. Moreover, the optimum parameters increasing the accuracy of the numerical solutions can be determined during the stability analysis. The proposed numerical algorithm has been demonstrated by a boundary value problem which considers the coupling and propagation of hydromagnetic waves in the magnetosphere.

Reconstruction of global micropulsations in the magnetosphere

The coupling of hydromagnetic Alfvén waves is studied numerically in a dipolc-field model of the magnetosphere. The two coupled hydromagnetic equations derived by Radoski are solved as an implicit boundary-value problem, namely the boundary conditions at the magnetopause are determined self-consistently. Thus the calculated wave-field distribution inside the magnetosphere can match all known linear characteristic features of the stormtime Pc5 waves observed on 14/15 November 1979 from satellites. A set of proper boundary conditions is found, excellent agreement between the numerical results and observations is demonstrated. Based on the very limited spatial coverage (L ≈ 6·6 and within a latitudinal region ( −10°, 10°)), of the data provided by the satellites, the theoretical model can successfully reconstruct the global micropulsations in the magnetosphere and identify the source regions of hydromagnetic waves.

Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics

A prescription is given for conservatively integrating generalized hydromagnetic equations using flux-corrected transport (FCT) techniques. By placing the magnetic-field components at the interface locations of the finite-difference grid, the field is kept divergence-free to within machine roundoff error. The use of FCT techniques allows an integration scheme of high accuracy to be employed, while the numerical ripples associated with large dispersion errors are avoided. The method is particularly well suited for problems involving magnetohydrodynamic shocks and other discontinuities.

Exact nonlinear helically symmetric solutions of hydromagnetic equations

Exact solutions of nonlinear helically symmetric hydromagnetic equations for a highly conducting fluid in a cylindrical geometry are obtained. These solutions may be useful because they can be expressed in terms of arbitrary scalar functions, and are applicable to astrophysics and laboratory plasmas.

Nonlinear MHD Models of Astrophysical Winds

The solutions obtained by Tsinganos (1982), Hu and Low (1989), and Tsinganos and Trussoni (1991) for the nonlinear coupled partial hydromagnetic equations are examined in a brief analytical review. The derivation for collimated hydromagnetic outflows is outlined, and the topologies of the solutions are presented in graphs. It is suggested that these solutions can be used in the testing and physical interpretation of numerical simulations of solar and stellar winds and astrophysical outflows and jets (based on models which account for nonspherically symmetric, MHD, and nonpolytropic phenomena). 13 refs.

Symmetry breaking bifurcations of a current sheet

Using a time evolution code with periodic boundary conditions, the viscoresistive hydromagnetic equations describing an initially static, planar current sheet with large Lundquist number have been evolved for times long enough to reach a steady state. A cosh2 x resistivity model was used. For long periodicity lengths Lp, the resistivity gradient drives flows that cause forced reconnection at X point current sheets. Using Lp as a bifurcation parameter, two new symmetry breaking bifurcations were found: a transition to an asymmetric island chain with nonzero, positive, or negative phase velocity, and a transition to a static state with alternating large and small islands. These states are reached after a complex transient behavior, which involves a competition between secondary current sheet instability and coalescence.

Numerical analysis of the coupled hydromagnetic wave equations in the magnetosphere

Inside the magnetosphere two basic hydromagnetic modes, the poloidal and toroidal, are always coupled. The problem of coupled waves is of fundamental importance for nontrivial geometries and general field configurations. As a first step the box model is investigated here, using the ideal MHD equations for the cold plasma. The spectrum and eigenfunctions are calculated for several cases by applying the finite element method. Both the improper continuum eigenmodes as well as the discrete solutions are computed. Existing analytical results are confirmed, and it is found that the discrete modes situated inside the Alfvén continuum are the best candidates for explaining observed hydromagnetic resonances. Other properties of these global modes are derived and compared with experimental data, and satisfactory agreement is obtained despite the simple geometric model employed. Generally, damping of the discrete modes is found to be considerably lower than that of the continuum. The results indicate that the weakly damped global modes lose their energy directly through ionospheric Joule dissipation, and not via mode conversion to field line resonances.

Macroscopic Stability of Rotating and Propagating LIB

The linear stabilities of a rotating and propagating light ion beam in a low-density plasma are studied numerically on the basis of the hydromagnetic equations. The centrifugal force acts on the beam and the self-magnetic fields are induced by the rotating and propagating charged particles. To clarify the physical mechanism, a simple model characterizing a rotating and propagating beam is utilized, and it is found that the centrifugal force and the magnetic field in the azimuthal direction play a role of stabilization. Under such beam parameters which are practically required for the energy driver for inertial confinement fusion, the numerical results indicate that the beam in the equilibrium state is stable if the ratio of the beam-rotating velocity to the propagating one is more than the value of 0.8.