Boundary Magnetic Field Research Articles

Control Problems for the MHD Equations under Inhomogeneous Mixed Boundary Conditions

The new control problems for the stationary magnetohydrodynamics equations under inhomogeneous boundary conditions for the magnetic field are considered. In these problems we use velocity and magnetic field boundary controls to minimize functionals depended on velocity and pressure. We study uniqueness and stability of solutions to these control problems and discuss some computational results.

Multiscale convection in a geodynamo simulation with uniform heat flux along the outer boundary

AbstractIt is generally expected that Earth's magnetic field, which is generated by convecting liquid metal within its core, will substantially alter that convection through the action of Lorentz forces. In most dynamo simulations, however, Lorentz forces do very little to change convective flow, which is predominantly fine‐scaled. An important exception to this observation is in dynamo models that employ uniform heat flux boundary conditions, rather than the usual uniform temperature conditions, in which multiscale convection is observed. We investigate the combined influence of thermal boundary conditions and magnetic fields using four simulations: two dynamos and two nonmagnetic models, with either uniform temperature or heat flux fixed at the outer boundary. Of the four, only the fixed‐heat‐flux dynamo simulation produces multiscale convective flow patterns. Comparison between the models suggests that the fixed‐flux dynamo generates large patches of strong azimuthal magnetic field that suppress small‐scale convective motions. By allowing temperature to vary along the outer boundary, the fixed‐flux dynamo generates stronger azimuthal flow and, in turn, stronger magnetic field, and the resulting Lorentz forces alter the nature of convective flow. Extrapolation of the analyses presented here suggests that magnetic fields may also suppress small‐scale convection in the Earth's core.

A Plane Magnetothermoelastic Waves Reflection and Refraction Between Two Solid Media with External Heat Sources and Initial Stress

In this article, an estimation to illustrate the reflection and transmission of thermoelastic waves at the interface of two different thermoelastic solid half-spaces under initial stress, magnetic field, and external heat sources is made. The governing equations are considered for an isotropic and homogeneous generalized thermoelastic model under initial stress, heat sources, magnetic field, and perfect boundary condition. The reflected and transmitted waves amplitudes are obtained for the incidence of SV-waves. Approximate expressions are obtained as a special case from the present problem. The results obtained are computed numerically for the amplitude ratios under initial stresses and are displayed graphically. It appears that the effects of initial stress, magnetic field, and external heat sources are very pronounced on the propagation phenomena. If the magnetic field and heat sources are neglected, the results obtained then lead to the same results obtained by relevant authors.

Analysis of Combined Convection in an Open Cavity under Constant Heat Flux Boundary Conditions and Magnetic Field Using Finite Element Method

This study investigated the effects of the aspect ratio of the cavity for average fluid temperature at exit port, average Nusselt number, maximum temperature of the fluid in the domain, drag coefficient, isotherms and streamlines on behalf of different Hartmann numbers and Rayleigh numbers. Solution of governing equations of momentum and energy has been made by finite element technique. Above mentioned parameters such as an aspect ratio which is cavity height to cavity length change from Ar = 0.5 to 2 for different Rayleigh numbers and Hartmann numbers which change from Ra = 103 to 105 and Ha = 0 to 50 respectively. Prandtl number Pr = 7 and Reynolds number Re = 100 is fixed in this simulation. It is found that variation of the aspect ratio makes an important effect for higher values of Rayleigh numbers. Heat transfer enhances with increasing of aspect ratio. Increasing of Hartmann number decreases the heat transfer inside the cavity. Keywords: Temperature boundary conditions; Open cavity; Aspect ratio; Finite element methods. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.14505 J. Sci. Res. 6 (2), 243-256 (2014)

Influence of Magnetohydrodynamic and Thermal Radiation Boundary Layer Flow of a Nanofluid Past a Stretching Sheet

The problem of laminar fluid flow which results from a permeable stretching of a flat surface in a nanofluid with the effects of heat radiation, magnetic field, velocity slip and convective boundary conditions have been investigated. The transport equations used in the analysis took into account the effect of Brownian motion and thermophoresis parameters. The solution for the velocity, temperature and nanoparticle concentration depends on parameters viz. thermal radiation parameter R, magnetic parameter M, Prandtl number Pr, Lewis number Le, Brownian motion parameter Nb, thermophoresis parameter Nt, velocity slip parameter A convection and Biot numbe Bi. Similarity transformation is used to convert the governing non-linear boundary-layer equations into coupled higher order non-linear ordinary differential equations. These equations are numerically solved using fourth order Runge-Kutta method along with shooting technique. An analysis has been carried out to elucidate the effects of governing parameters corresponding to various physical conditions. Numerical results are obtained for distributions of velocity, temperature and concentration, as well as, for the skin friction, local Nusselt number and local Sherwood number for several values of governing parameters. Keywords: Nanofluid, Boundary layer flow; Stretching sheet; Thermal radiation; MHD; Velocity slip; Convective boundary. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.17233 J. Sci. Res. 6 (2), 257-272 (2014)

Thermodynamic limit and surface energy of the XXZ spin chain with arbitrary boundary fields

In two previous papers [26,27], the exact solutions of the spin-12 chains with arbitrary boundary fields were constructed via the off-diagonal Bethe ansatz (ODBA). Here we introduce a method to approach the thermodynamic limit of those models. The key point is that at a sequence of degenerate points of the crossing parameter η=ηm, the off-diagonal Bethe ansatz equations (BAEs) can be reduced to the conventional ones. This allows us to extrapolate the formulae derived from the reduced BAEs to arbitrary η case with O(N−2) corrections in the thermodynamic limit N→∞. As an example, the surface energy of the XXZ spin chain model with arbitrary boundary magnetic fields is derived exactly. This approach can be generalized to all the ODBA solvable models.

Nonlinear magnetic field diffusion in a substance metallized by shock compression

The problem of nonlinear magnetic field diffusion in a substance metallized by shock compression is considered. The problem is solved numerically for various time dependences of the current through the conducting region (direct current, linearly increasing current, and the current increasing as the square root of the time). Nonlinear diffusion leads to qualitative changes in the structure of the current in the substance being metallized. In this case, the maximal current density is shifted from the conducting boundary (at which it is located in the case of linear diffusion) to the bulk of the conducting material. For strong nonlinear diffusion and an increasing boundary magnetic field, the current density peak may approach the shock front. The numerical solution obtained here is compared with the analytic solution obtained earlier for linear diffusion.

Ion beam driven resonant ion-cyclotron instability in a magnetized dusty plasma

Electrostatic ion cyclotron waves are excited by axial ion beam in a dusty plasma via Cerenkov and slow cyclotron interaction. The dispersion relation of the instability is derived in the presence of positively/negatively charged dust grains. The minimum beam velocity needed for the excitation is estimated for different values of relative density of negatively charged dust grains. It is shown that the minimum beam velocity needed for excitation increases as the charge density carried by dust increases. Temperature of electrons and ions, charge and mass of dust grains, external static magnetic field and finite boundary of dusty plasma significantly modify the dispersion properties of these waves and play a crucial role in the growth of resonant ion cyclotron instability. The ion cyclotron modes with phase velocity comparable to the beam velocity possess a large growth rate. The maximum value of growth rate increases with the beam density and scales as the one-third power of the beam density in Cerenkov interaction and is proportional to the square root of beam density in slow cyclotron interaction.

ALFVÉN WAVE SOLAR MODEL (AWSoM): CORONAL HEATING

We present a new version of the Alfven Wave Solar Model (AWSoM), a global model from the upper chromosphere to the corona and the heliosphere. The coronal heating and solar wind acceleration are addressed with low-frequency Alfven wave turbulence. The injection of Alfven wave energy at the inner boundary is such that the Poynting flux is proportional to the magnetic field strength. The three-dimensional magnetic field topology is simulated using data from photospheric magnetic field measurements. This model does not impose open-closed magnetic field boundaries; those develop self-consistently. The physics includes: (1) The model employs three different temperatures, namely the isotropic electron temperature and the parallel and perpendicular ion temperatures. The firehose, mirror, and ion-cyclotron instabilities due to the developing ion temperature anisotropy are accounted for. (2) The Alfven waves are partially reflected by the Alfven speed gradient and the vorticity along the field lines. The resulting counter-propagating waves are responsible for the nonlinear turbulent cascade. The balanced turbulence due to uncorrelated waves near the apex of the closed field lines and the resulting elevated temperatures are addressed. (3) To apportion the wave dissipation to the three temperatures, we employ the results of the theories of linear wave damping and nonlinear stochastic heating. (4) We have incorporated the collisional and collisionless electron heat conduction. We compare the simulated multi-wavelength EUV images of CR2107 with the observations from STEREO/EUVI and SDO/AIA instruments. We demonstrate that the reflection due to strong magnetic fields in proximity of active regions intensifies the dissipation and observable emission sufficiently.

Peristaltic Flow with Inclined Magnetic Field and Convective Boundary Conditions

Peristaltic flow of viscous fluid in an asymmetric inclined channel with heat transfer and inclined magnetic field is examined. The convective boundary conditions have been handled. Complexity of emerging equations is simplified by utilizing long wavelength and low Reynolds number approximation. Variation of emerging parameters embedded in flow system are discussed. It is observed that an increase in Brikman number increases the temperature profile. Further, it is seen that temperature distribution is an increasing function of Biot number at lower wall.

Peristaltic Flow with Inclined Magnetic Field and Convective Boundary Conditions

Peristaltic flow of viscous fluid in an asymmetric inclined channel with heat transfer and inclined magnetic field is examined. The convective boundary conditions have been handled. Complexity of emerging equations is simplified by utilizing long wavelength and low Reynolds number approximation. Variation of emerging parameters embedded in flow system are discussed. It is observed that an increase in Brikman number increases the temperature profile. Further, it is seen that temperature distribution is an increasing function of Biot number at lower wall.

Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields

The one-dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly via the Bethe ansatz methods. With the coordinate Bethe ansatz in the charge sector, the second eigenvalue problem associated with the spin sector is constructed. It is shown that the second eigenvalue problem can be transformed into that of the inhomogeneous XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.

An MHD simulation model of time‐dependent global solar corona with temporally varying solar‐surface magnetic field maps

We present a model of a time‐dependent three‐dimensional magnetohydrodynamics simulation of the sub‐Alfvenic solar corona and super‐Alfvenic solar wind with temporally varying solar‐surface boundary magnetic field data. To (i) accommodate observational data with a somewhat arbitrarily evolving solar photospheric magnetic field as the boundary value and (ii) keep the divergence‐free condition, we developed a boundary model, here named Confined Differential Potential Field model, that calculates the horizontal components of the magnetic field, from changes in the vertical component, as a potential field confined in a thin shell. The projected normal characteristic method robustly simulates the solar corona and solar wind, in response to the temporal variation of the boundary Br. We conduct test MHD simulations for two periods, from Carrington Rotation number 2009 to 2010 and from Carrington Rotation 2074 to 2075 at solar maximum and minimum of Cycle 23, respectively. We obtained several coronal features that a fixed boundary condition cannot yield, such as twisted magnetic field lines at the lower corona and the transition from an open‐field coronal hole to a closed‐field streamer. We also obtained slight improvements of the interplanetary magnetic field, including the latitudinal component, at Earth.

Drift of domain walls in a harmonic magnetic field

It is shown that a two-step form of the dynamic magnetization curve (and the hysteresis loop) established for a multiaxial ferrite-garnet wafer with a low quality factor (Q < 1) and considerable anisotropy in the plane (K p /K u = 14) in the frequency range of 25–1000 Hz is explained by the reconstruction of the dynamic domain structure, particularly by the established features of the drift of domain boundaries in the harmonic magnetic field.

Generalized Strategic Dual Image Method for Open Boundary Axisymmetrical Magnetic Field Problems

During the early 1990s, the strategic dual image (SDI) method for axisymmetric open boundary magnetic field problems was proposed. Although the method is powerful, the specific axis ratio of the ellipsoidal boundary has not been clarified, and therefore no further research has been done so far. Our aims are to clarify the relationship between the specific axis ratio of the ellipsoidal boundary and the harmonic solutions of Laplace equation, and to extend the strategic dual image method in general form. We have investigated the harmonic solutions of the Laplace equation in the oblate spheroidal coordinate to derive the explicit formula regarding the axis ratio and the averaging factor of Dirichlet and Neumann boundary value problems. Numerical analyses have also been carried out to verify the formula. We have determined the explicit formula regarding the axis ratio and averaging factor of Dirichlet and Neumann solutions, thus extending the SDI method in general form. Utilizing the derived formula, one can obtain the open boundary solutions of axisymmetrical magnetic field problems without alternating the existing software which are widely distributed through the Internet.

Characteristic of Gravity and Magnetic Anomalies in the Daba Shan and the Sichuan basin, China: Implication for Architecture of the Daba Shan

Abstract:The Dabashan nappe structural belt links the Hannan block to the west with the Huangling block to the east between Yangxian and Xiangfan. The Dabashan arc‐shaped fold belt formed during late Jurassic and was superposed on earlier Triassic folds. To achieve an improved understanding of the deep tectonics of the Dabashan nappe structural belt, we processed and interpreted the gravity and magnetic data for this area using new deep reflection seismic and other geophysical data as constraints. The results show that the Sichuan basin and Daba Mountains lie between the Longmenshan and Wulingshan gravity gradient belts. The positive magnetic anomalies around Nanchong‐Tongjiang‐Wanyuan‐Langao and around Shizhu result from the crystalline basement. Modeling of the gravity and magnetic anomalies in the Daba Mountains and the Sichuan basin shows that the crystalline basement around Nanchong‐Tongjiang‐Wanyuan‐Langao extends to the northeast underneath the Wafangdian fault near Ziyang. The magnetic field boundary in the Zhenba‐Wanyuan‐Chengkou‐Zhenping area is the major boundary of the Dabashan nappe thrusting above the Sichuan Basin. This boundary might be the demarcation between the south Dabashan and the north Dabashan structural elements. The low gravity anomaly between Tongjiang and Chengkou might be partly caused by thickened lower crust The local low gravity anomaly to the south of Chengkou‐Wanyuan might result from Mesozoic strata of low density in the Dabashan foreland depression area.

Validation of single-fluid and two-fluid magnetohydrodynamic models of the helicity injected torus spheromak experiment with the NIMROD code

We present a comparison study of 3-D pressureless resistive MHD (rMHD) and 3-D presureless two-fluid MHD models of the Helicity Injected Torus with Steady Inductive helicity injection (HIT-SI). HIT-SI is a current drive experiment that uses two geometrically asymmetric helicity injectors to generate and sustain toroidal plasmas. The comparable size of the collisionless ion skin depth di to the resistive skin depth predicates the importance of the Hall term for HIT-SI. The simulations are run with NIMROD, an initial-value, 3-D extended MHD code. The modeled plasma density and temperature are assumed uniform and constant. The helicity injectors are modeled as oscillating normal magnetic and parallel electric field boundary conditions. The simulations use parameters that closely match those of the experiment. The simulation output is compared to the formation time, plasma current, and internal and surface magnetic fields. Results of the study indicate 2fl-MHD shows quantitative agreement with the experiment while rMHD only captures the qualitative features. The validity of each model is assessed based on how accurately it reproduces the global quantities as well as the temporal and spatial dependence of the measured magnetic fields. 2fl-MHD produces the current amplification ItorIinj and formation time τf demonstrated by HIT-SI with similar internal magnetic fields. rMHD underestimates ItorIinj and exhibits much a longer τf. Biorthogonal decomposition (BD), a powerful mathematical tool for reducing large data sets, is employed to quantify how well the simulations reproduce the measured surface magnetic fields without resorting to a probe-by-probe comparison. BD shows that 2fl-MHD captures the dominant surface magnetic structures and the temporal behavior of these features better than rMHD.

FZZT brane relations in the presence of boundary magnetic fields

We show how a boundary state different from the (1,1) Cardy state may be realised in the (m,m+1) minimal string by the introduction of an auxiliary matrix into the standard two hermitian matrix model. This boundary is a natural generalisation of the free spin boundary state in the Ising model. The resolvent for the auxiliary matrix is computed using an extension of the saddle-point method of Zinn-Justin to the case of non-identical potentials. The structure of the saddle-point equations result in a Seiberg-Shih like relation between the boundary states which is valid away from the continuum limit, in addition to an expression for the spectral curve of the free spin boundary state. We then show how the technique may be used to analyse boundary states corresponding to a boundary magnetic field, thereby allowing us to generalise the work of Carroll et al. on the boundary renormalisation flow of the Ising model, to any (m,m+1) model.

OHMIC DISSIPATION IN THE INTERIORS OF HOT JUPITERS

We present models of ohmic heating in the interiors of hot jupiters in which we decouple the interior and the wind zone by replacing the wind zone with a boundary temperature Tiso and magnetic field Bphi0. Ohmic heating influences the contraction of gas giants in two ways: by direct heating within the convection zone, and by heating outside the convection zone which increases the effective insulation of the interior. We calculate these effects, and show that internal ohmic heating is only able to slow the contraction rate of a cooling gas giant once the planet reaches a critical value of internal entropy. We determine the age of the gas giant when ohmic heating becomes important as a function of mass, Tiso and induced Bphi0. With this survey of parameter space complete, we then adopt the wind zone scalings of Menou (2012) and calculate the expected evolution of gas giants with different levels of irradiation. We find that,with this prescription of magnetic drag, it is difficult to inflate massive planets or those with strong irradiation using ohmic heating, meaning that we are unable to account for many of the observed hot jupiter radii. This is in contrast to previous evolutionary models that assumed that a constant fraction of the irradiation is transformed into ohmic power.

Comparison of self‐consistent simulations with observed magnetic field and ion plasma parameters in the ring current during the 10 August 2000 magnetic storm

We assess whether magnetically and electrically self‐consistent ring current simulations can account simultaneously for in situ magnetic field and ion flux measurements in the inner magnetosphere during the large 10 August 2000 storm (min Dst = −107 nT). We use the Rice Convection Model–Equilibrium (RCM‐E) and drive it with time‐dependent magnetic field, electric field, and plasma boundary conditions that are guided by empirical and assimilative models. Comparisons of the simulated and observed magnetic field from Geostationary Operational Environmental Satellites (GOES) and observed proton differential flux spectra from Los Alamos National Laboratory (LANL) satellites are made at geosynchronous orbit (GEO). Similarly, simulated and observed magnetic field and proton density and temperature are compared along the orbit of Polar (r ∼ 1.8–9 RE) for the event. The simulated and observed magnetic field components agree reasonably well at GEO and along the orbit of Polar. However, since the effects of substorm dipolarizations are not explicitly modeled, the simulation fails to reproduce observed sawtooth fluctuations in the magnetic field. Over energies from 1 to 150 keV, the RCM‐E reproduced well the ion dispersion features in the LANL 1994‐084 ion differential flux spectra over energies at GEO and proton densities and temperatures calculated from Polar proton flux measurements. Thus, the RCM‐E simulations can account simultaneously for in situ magnetic field and ion flux measurements for the 10 August 2000 storm. This demonstrates that a self‐consistent model can produce realistic features of the storm time inner magnetosphere.