Hydromagnetic Equations Research Articles

Hydromagnetic Waves in Cold Nuclear Matter

I consider a proton–neutron fluid mixture placed in an ultra-strong external static magnetic field and derive the spin-independent, small-amplitude disturbances in infinitely extended systems. As a theoretical framework I adopt a hydrodynamical model for the proton and neutron fluids moving in a Skyrme mean-field derived from the time-dependent Hartree Fock formulation of the many-body nuclear problem. From the mass, momentum balance, and Maxwell equations, I set up a system of equations governing the electromagnetic field and the continuum-mechanical fields of the mixture. Next, the hydromagnetic equations are linearized, and the occurrence of small-amplitude distortions of the velocity field is analyzed for various orientations of the constant external magnetic induction with respect to the wave propagation vector. The derivation of the above equations is carried out for the inviscid case.

The influence of dissipative effects on dynamic processes in a rotating electrically conductive liquid medium

The present study is concerned with dynamical processes in a rotating layer of electrically conducting incompressible liquid located in a magnetic field which is parallel to the normal vector to the boundary surfaces. We take into account not only the convective terms, but also the diffusion terms in the magnetic field induction equations. This problem, as well as the geophysical hydrodynamics problem, calls for the construction of approximate variants of the principal hydromagnetic equations and rigorous mathematical analysis of these approximate equations. Moreover, advancements in the above problems depend on the approximations to be introduced. To this end, by introducing characteristic scales of variation of the variables in the original equations and estimating the magnitude orders of the terms involved in the equations, one can single out the principal and secondary terms, simplify the equations, and build a model of the process under consideration. By introduction of auxiliary functions, the system of partial differential equations is reduced to one scalar equation. This suggests the conclusion about the analytical structure of magnetohydrodynamic characteristics. From the results obtained it follows that the magnetic field generation in an electrically conducting liquid stems from the instability characterized by the corresponding relations between the gravitation force, the Coriolis force, the magnetic force, and the peculiarities of the relief topography.

Unsteady hydromagnetic flow and heat transfer of a viscous fluid near a suddenly accelerated flat surface

The unsteady boundary layer flow and heat transfer near a suddenly accelerated flat surface in an unbounded mass of hydromagnetic viscous fluid with the combined influence of the magnetic field, viscous dissipation, internal heat generation/absorption and thermal radiation have been investigated. A new similarity transformation is recommended, which transforms the hydromagnetic boundary layer equations into a set of non-linear ordinary differential equations. These equations are then solved numerically using the finite-difference method for some values of the governing parameters that involve unsteadiness parameter \(\beta \), heat source/sink parameter \(\lambda \), Eckert number E, magnetic interaction parameter M, radiation parameter N and Prandtl number Pr. The influence of these parameters on the velocity as well as the temperature field is investigated in detail. In particular, the impact of viscous dissipation (measured through E), which is a strong function of constant reference temperature \(T_{ref}\), on the temperature field has been investigated in different situations. This entails considering the value of \(T_{ref}\) as positive or negative depending on whether the surface temperature is higher or lower than the fluid temperature. The analysis reveals that there exists a critical value of E depending upon the values of the other physical parameters for which the surface heat flux vanishes. Below this critical value, heat flows either from the surface to the fluid or from the fluid to the surface depending on whether \(T_{ref}>\) or \(< 0\).

Quaternionic Approach to Dual Magnetohydrodynamics of Dyonic Cold Plasma

The dual magnetohydrodynamics of dyonic plasma describes the study of electrodynamics equations along with the transport equations in the presence of electrons and magnetic monopoles. In this paper, we formulate the quaternionic dual fields equations, namely, the hydroelectric and hydromagnetic fields equations which are an analogous to the generalized Lamb vector field and vorticity field equations of dyonic cold plasma fluid. Further, we derive the quaternionic Dirac-Maxwell equations for dual magnetohydrodynamics of dyonic cold plasma. We also obtain the quaternionic dual continuity equations that describe the transport of dyonic fluid. Finally, we establish an analogy of Alfven wave equation which may generate from the flow of magnetic monopoles in the dyonic field of cold plasma. The present quaternionic formulation for dyonic cold plasma is well invariant under the duality, Lorentz, and CPT transformations.

Double-diffusive convection of compressible rotating Walters’ (B′) fluid with Hall currents saturating a porous medium

Keeping in view the conflicting tendencies of rotation and Hall currents (magnetic field) while acting together; combined effects of Hall currents and rotation are considered on the hydromagnetic stability of a compressible Walters’ (Model B′) elastico-viscous fluid heated and soluted from below saturating a porous medium. Boussinesq approximation is used to simplify the complex hydromagnetic equations and the perturbation equations are analyzed in terms of normal modes. A dispersion relation governing the effects of visco-elasticity, salinity gradient, rotation, Hall currents and medium permeability is derived. It has been found that for stationary convection, Walters’ (Model B′) fluid behaves like an ordinary Newtonian fluid due to the vanishing of the visco-elastic parameter. Compressibility, solute gradient, rotation and magnetic field postpone the onset of instability as such their effect is to stabilize the system. Hall currents and medium permeability are found to hasten the onset of instability for permissible range of values of various parameters. The dispersion relation is analyzed numerically and the effects of various parameters for permissible range of values are depicted graphically. The visco-elasticity, solute gradient and Hall currents (hence magnetic field) introduce oscillatory modes in the system which were non-existent in their absence. Also the case of overstability is discussed and sufficient conditions for the non-existence of overstability are derived.Keywords: Walters’ (Model B′) fluid, rotation, Hall currents, thermosolutal instability, compressibility, porous medium.

Spin paramagnetic deformation of a neutron star

Quantum mechanical corrections to the hydromagnetic force balance equation, derived from the microscopic Schr\"{o}dinger-Pauli theory of quantum plasmas, modify the equilibrium structure and hence the mass quadrupole moment of a neutron star. It is shown here that the dominant effect --- spin paramagnetism --- is most significant in a magnetar, where one typically has $\mu_{B}|\boldsymbol{B}|\gtrsim k_B T_e$, where $\mu_{B}$ is the Bohr magneton, $\boldsymbol{B}$ is the magnetic field, and $T_e$ is the electron temperature. The spin paramagnetic deformation of a nonbarotropic magnetar with a linked poloidal-toroidal magnetic field is calculated to be up to ${{\sim 10}}$ times greater than the deformation caused solely by the Lorentz force. It depends on the degree of Pauli blocking by conduction electrons and the propensity to form magnetic domains, processes which are incompletely modelled at magnetar field strengths. The star becomes more oblate, as the toroidal field component strengthens. The result implies that existing classical predictions underestimate the maximum strength of the gravitational wave signal from rapidly spinning magnetars at birth. Turning the argument around, future gravitational-wave upper limits of increasing sensitivity will place ever-stricter constraints on the physics of Pauli blocking and magnetic domain formation under magnetar conditions.

The perturbation of gravitational waves in plasma in the FRW space-time

In this paper we study the perturbation of gravitational waves in plasma,using the relativistic hydro-magnetic equation in the so-called Cowling approximation considering a Friedman-Robertson-Walker (FRW) cosmological model. It has been assumed the gravitational field and the weak magnetic field do not break the homogeneity and isotropy of the considered FRW space time. Applying the formalism proposed for Zel’dovich and Novikov (The structure and evolution of the universe, Volume II, 1983), Brandenburg et al. (Phys. Rev. D 54:1291, 1996) and Weinberg (Gravitation and Cosmology, 1972). We verify that density fluctuation may be obtained.

Magnetically controlled stellar differential rotation near the transition from solar to anti-solar profiles

Late-type stars rotate differentially owing to anisotropic turbulence in their outer convection zones. The rotation is called solar-like (SL) when the equator rotates fastest and anti-solar (AS) otherwise. Hydrodynamic simulations show a transition from SL to AS rotation as the influence of rotation on convection is reduced, but the opposite transition occurs at a different point in the parameter space. The system is bistable, i.e., SL and AS rotation profiles can both be stable. We study the effect of a dynamo-generated magnetic field on the large-scale flows, particularly on the possibility of bistable behavior of differential rotation. We solve the hydromagnetic equations numerically in a rotating spherical shell for a set of different radiative conductivities controlling the relative importance of convection. In agreement with earlier findings, our models display SL rotation profiles when the rotational influence on convection is strong and a transition to AS when the rotational influence decreases. We find that dynamo-generated magnetic fields help to produce SL differential rotation compared to the hydrodynamic simulations. We do not observe any bistable states of differential rotation. In the AS cases we get coherent single-cell meridional circulation, whereas in SL cases we get multi-cellular patterns. In both cases, we obtain poleward circulation near the surface with a magnitude close to that observed in the Sun. Moreover, both differential rotation and meridional circulation have significant magnetic cycle-related variations that are similar in strength to those of the Sun. Purely hydrodynamic simulations of differential rotation and meridional circulation are shown to be of limited relevance as magnetic fields, self-consistently generated by dynamo action, significantly affect the flows.

Competition of rotation and stratification in flux concentrations

In a strongly stratified turbulent layer, a uniform horizontal magnetic field can become unstable to spontaneously form local flux concentrations due to a negative contribution of turbulence to the large-scale (mean-field) magnetic pressure. This mechanism, called the negative effective magnetic pressure instability (NEMPI), is of interest in connection with dynamo scenarios where most of the magnetic field resides in the bulk of the convection zone, and not at the bottom. Recent work using the mean-field hydromagnetic equations has shown that NEMPI becomes suppressed at rather low rotation rates with Coriolis numbers as low as 0.1.}{Here we extend these earlier investigations by studying the effects of rotation both on the development of NEMPI and on the effective magnetic pressure. We also quantify the kinetic helicity from direct numerical simulations (DNS) and compare with earlier work.}{To calculate the rotational effect on the effective magnetic pressure we consider both DNS and analytical studies using the $\tau$ approach. To study the effects of rotation on the development of NEMPI we use both DNS and mean-field calculations of the 3D hydromagnetic equations in a Cartesian domain.}{We find that the growth rates of NEMPI from earlier mean-field calculations are well reproduced with DNS, provided the Coriolis number is below about 0.06. In that case, kinetic and magnetic helicities are found to be weak. For faster rotation, dynamo action becomes possible. However, there is an intermediate range of rotation rates where dynamo action on its own is not yet possible, but the rotational suppression of NEMPI is being alleviated.}{Production of magnetic flux concentrations through the suppression of turbulent pressure appears to be possible only in the upper-most layers of the Sun, where the convective turnover time is less than 2 hours.}

Derivation of two-dimensional couple-stress hydromagnetic squeeze film Reynolds equation and application to wide parallel rectangular plates

On the basis of Stokes couple-stress fluid model together with the hydromagnetic flow equations, a two-dimensional curved squeeze-film Reynolds equation has been derived. This equation can be applied to the study of squeeze film characteristics including the non-Newtonian hydromagnetic effects, in which the general film shape is h=h(x,z,t). To guide the application of the equation, an example of one-dimensional parallel rectangular plates lubricated with a non-Newtonian couple-stress, electrically conducting lubricant is illustrated. According to the results, the effects of couple stresses and external magnetic fields provide an increase in the load capacity and the response time as compared to the classical Newtonian hydrodynamic rectangular squeeze-film plates.

Toroidal hydromagnetic waves in an axi‐symmetric magnetic field

Hydromagnetic wave equations are derived for toroidal Alfvén waves in a background, axi‐symmetric magnetic field. In the case, where the spatial variations in the wave equations are along the background magnetic field, the equations can be re‐cast in Klein‐Gordon equation forms, with cut‐off frequency ωc. For harmonic, constant frequency solutions, the Klein Gordon equation reduces to a Sturm‐Liouville equation. We compute the eigenvalues and eigenfunctions appropriate for the Earth's dipole magnetic field, which are relevant for geomagnetic pulsations. It is suggested that the breather type Alfvén eigenmodes, obtained from the analysis may describe ultra low frequency Alfvénic pulsations (e.g. Pc and Pi pulsations) observed in space and on the ground with frequencies ranging from a few milli‐Herz (mHz) to a few Hz). The wave equations for the velocity and magnetic field perturbations can be re‐written in terms of Elsässer variables. The resultant wave equations are a special case of the wave mixing equations for Alfvénic fluctuations used to describe locally incompressible turbulence in the solar wind. The canonical wave energy equation for the backward and forward waves, and wave reflection and transmission coefficients are briefly discussed. The equations can also be cast in the form of a Dirac equation, in the Weyl spinor form, in which the mass in the Dirac equation is identified with the wave mixing coefficient for the backward and forward waves.

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed. The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations. Making use of this solution, analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. As a result, exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.

Stochastic flux freezing and magnetic dynamo

Magnetic flux conservation in turbulent plasmas at high magnetic Reynolds numbers is argued neither to hold in the conventional sense nor to be entirely broken, but instead to be valid in a statistical sense associated to the "spontaneous stochasticity" of Lagrangian particle trajectories. The latter phenomenon is due to the explosive separation of particles undergoing turbulent Richardson diffusion, which leads to a breakdown of Laplacian determinism for classical dynamics. Empirical evidence is presented for spontaneous stochasticity, including numerical results. A Lagrangian path-integral approach is then exploited to establish stochastic flux freezing for resistive hydromagnetic equations and to argue, based on the properties of Richardson diffusion, that flux conservation must remain stochastic at infinite magnetic Reynolds number. An important application of these results is the kinematic, fluctuation dynamo in nonhelical, incompressible turbulence at magnetic Prandtl number (Pr(m)) equal to unity. Numerical results on the Lagrangian dynamo mechanisms by a stochastic particle method demonstrate a strong similarity between the Pr(m)=1 and 0 dynamos. Stochasticity of field-line motion is an essential ingredient of both. Finally, some consequences for nonlinear magnetohydrodynamic turbulence, dynamo, and reconnection are briefly considered.

Klein-Gordon equations for toroidal hydromagnetic waves in an axi-symmetric field

Abstract. In this paper we develop the hydromagnetic wave equations for toroidal Alfvén waves in a background axi-symmetric magnetic field. In the case where spatial variations are directed along the ambient magnetic field direction, the equations can be cast in a Klein-Gordon form in which the adiabatic-geometric amplitude factor of the perturbations varies as √ρL5sin5θ along a magnetic field line (where θ is colatitude and L the L-shell number) and the cut-off frequency, associated with the Klein-Gordon form, displays an astonishing variation with distance along a field line (see Eqs. 35 and 37 of the text), in the case of a dipole magnetic field. We compute the eigenvalues and eigenfunctions for the Earth's dipole field which are relevant to geomagnetic pulsations.

Transient evolution of C-type shocks in dusty regions of varying density

Outflows of young stars drive shocks into dusty, molecular regions. Most models of such shocks assume that they are steady and propagating perpendicular to the magnetic field. Real shocks often violate both of these assumptions and the media through which they propagate are inhomogeneous. We use the code employed previously to produce the first time-dependent simulations of fast-mode, oblique C-type shocks interacting with density perturbations. We include a self-consistent calculation of the thermal and ionisation balances and a fluid treatment of grains. We identify features that develop when a multifluid shock encounters a density inhomogeneity to investigate whether any part of the precursor region ever behaves in a quasi-steady fashion. If it does the shock may be modelled approximately without solving the time-dependent hydromagnetic equations. Simulations were made for initially steady oblique C-type shocks encountering density inhomogeneities. For a semi-finite inhomogeneity with a density larger than the surrounding medium, a transmitted shock evolves from being J-type to a steady C-type shock on a timescale comparable to the ion-flow time through it. A sufficiently upstream part of the precursor of an evolving J-type shock is quasi-steady. The ion-flow timescale is also relevant for the evolution of a shock moving into a region of decreasing density. The models for shocks propagating into regions in which the density increases and then decreases to its initial value cannot be entirely described in terms of the results obtained for monotonically increasing and decreasing densities. For the latter model, the long-term evolution to a C-type shock cannot be approximated by quasi-steady models.

Thermal instability of compressible Walters’ (Model B′) fluid in the presence of Hall currents and suspended particles

Original scientific paper UDC: 532.582:536.22:66.011 DOI: 10.2298/TSCI1102487G Effect of Hall currents and suspended particles is considered on the hydromagnetic stability of a compressible, electrically conducting Walters’ (Model B ) elastico-viscous fluid. After linearizing the relevant hydromagnetic equations, the perturbation equations are analyzed in terms of normal modes. A dispersion relation governing the effects of visco-elasticity, magnetic field, Hall currents and suspended particles is derived. It has been found that for stationary convection, the Walters’ (Model B ) fluid behaves like an ordinary Newtonian fluid due to the vanishing of the visco-elastic parameter. The compressibility and magnetic field have a stabilizing effect on the system, as such their effect is to postpone the onset of thermal instability whereas Hall currents and suspended particles are found to hasten the onset of thermal instability for permissible range of values of various parameters. Also, the dispersion relation is analyzed numerically and the results shown graphically. The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection are obtained and the behavior of various parameters on critical thermal Rayleigh numbers has been depicted graphically. The visco-elasticity, suspended particles and Hall currents (hence magnetic fieldintroduce oscillatory modes in the system which were non-existent in their absence.

THE TOPOLOGICAL CHANGES OF SOLAR CORONAL MAGNETIC FIELDS. I. SPONTANEOUS CURRENT SHEETS IN THREE-DIMENSIONAL FIELDS

We present mathematical models to demonstrate the inevitability of current-sheet formation in a cylindrical magnetic field governed by the ideal hydromagnetic induction equation, as described by the Parker theory. This process, central to the heating of the solar corona, is radically different in fully three-dimensional fields as compared with two-dimensional fields. Magnetic neutral points or separatrix flux surfaces are necessary for sheet formation in two-dimensional fields. In fully three-dimensional fields, current sheets form readily even in the complete absence of neutral points and separatrix surfaces, and these sheets can form densely throughout the field in response to changes in the magnetic volume. This general result is established for cylindrical fields that are topologically untwisted, including the first direct demonstration of sheet formation in the absence of any magnetic neutral point. We discuss simple implications of this basic result for solar flares and coronal heating. In subsequent papers, we apply our result to coronal mass ejections and solar flares.

Implementation of the CIP algorithm to magnetohydrodynamic simulations

An implementation of the Constrained Interpolation Profile (CIP) algorithm to magnetohydrodynamic (MHD) simulations is presented. First we transform the original momentum and magnetic induction equations to unfamiliar forms by introducing Elsässer variables [W.M. Elsässer, The hydromagnetic equations, Phys. Rev. (1950)]. In this formulation, while the compressional and pressure gradient terms remain as non-advective terms, the advective and magnetic stress terms are expressed in the form of an advection equation, which enables us to use the CIP algorithm. We have examined some 1D test problems using the code based on this formula. Linear Alfvén wave propagation tests reveal that the developed code is capable of solving any Alfvén wave propagation with only small numerical diffusion and phase errors up to k ▵ h = 2.5 (where ▵ h is the grid spacing). A shock tube test shows good agreement with a previous result with less numerical oscillation at the shock front and the contact discontinuity which are captured within a few grid points. Extension of the 1D code to the multi-dimensional case is straightforward. We have calculated the 3D nonlinear evolution of the Kelvin–Helmholtz instability (KHI) and compared the result with our previous study. We find that our new MHD code is capable of following the 3D turbulence excited by the KHI while retaining the solenoidal property of the magnetic field.

Role of on-board discharge in shock wave drag reduction and plasma cloaking

In the present paper, a physical model is proposed for reducing the problem of the drag reduction of an attached bow shock around the nose of a high-speed vehicle with on-board discharge, to the problem of a balance between the magnetic pressure and gas pressure of plane shock of a partially ionized gas consisting of the environmental gas around the nose of the vehicle and the on-board discharge-produced plasma. The relation between the shock strength and the discharge-induced magnetic pressure is studied by means of a set of one-fluid, hydromagnetic equations reformed for the present purpose, where the discharge-induced magnetic field consists of the electron current (produced by the discharge)-induced magnetic field and the partially ionized gas flow-induced one. A formula for the relation between the above parameters is derived. It shows that the discharge-induced magnetic pressure can minimize the shock strength, successfully explaining the two recent experimental observations on attached bow shock mitigation and elimination in a supersonic flow during on-board discharge [Phys. Plasmas 9 (2002) 721 and Phys. Plasmas 7 (2000) 1345]. In addition, the formula implies that the shock elimination leaves room for a layer of higher-density plasma rampart moving around the nose of the vehicle, being favourable to the plasma radar cloaking of the vehicle. The reason for it is expounded.

Magnetorotational Instability around a Rotating Black Hole

The magnetorotational instability (MRI) in the Kerr spacetime is studied on a 3 + 1 viewpoint. Maxwell's equations are expressed in a circularly orbiting observer's frame that corotates with matter in Keplerian orbits. The hydromagnetic equations are represented in a locally nonrotating frame (LNRF). There exist large proper growth rates in the MRI around a rapidly rotating black hole. The large centrifugal force and the rapid variations of magnetic fields are caused by the rotation of the spacetime geometry. As a result, in the extreme Kerr case the maximum proper growth rate at r = rms becomes about 12 times as large as that in Schwarzschild case, where rms is the radius of a marginally stable orbit. The unstable range of the wavenumber expands according to the rotational speed of the spacetime geometry. The overstable mode of the instability becomes remarkable when the circular motion of the disk is quasi-relativistic and the strength of the magnetic field is so large that the Alfven velocity is vA ≥ 0.1c. When the waves of perturbations propagate in the radial direction, these waves oscillate, and their amplitudes grow exponentially. This instability is caused by the differential rotation of the circularly orbiting frame. The universality of the local Oort A-value of the disk is discussed in curved spacetime. In the extreme Kerr geometry, the amplitude of the maximum growth rate in the dynamical shear instability (DSI) reaches to infinity at r = rms. The behaviors of maximum growth rates in MRI and DSI are remarkably different from each other.