Isothermal Magnetostatic Atmospheres Research Articles

The G′/G-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres

The equations of magnetohydrostatic equilibria for plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magneto static atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. The distributed current in the model J is directed along the x-axis where x is the horizontal ignorable coordinate. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential u. This equation depends on an arbitrary function of u that must be specified with choices of different arbitrary functions, we obtain analytical nonlinear solutions of the elliptic equation using the G′G-expansion method. Finally, the hyperbolic versions of these equations will be solved by the travelling wave hypothesis method.

Exact Travelling Wave Solutions for Isothermal Magnetostatic Atmospheres by Fan Subequation Method

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential u. This equation depends on an arbitrary function of u that must be specified. With choices of the different arbitrary functions, we obtain analytical solutions of elliptic equation using the Fan subequation method.

The Role of Nonlinear Alfven Waves in Shear Formation during Solar Magnetic Flux Emergence

Observations show that magnetic flux is constantly emerging at the solar photosphere to expand into the corona. Magnetic buoyancy is essential in bringing the magnetic field from the solar interior to the surface and beyond. In the simulations of buoyancy instabilities reported here, it has been discovered that nonlinear Alfven waves may play an important and dramatic role in the rise of magnetic flux. For initial states, our two-and-a-half-dimensional (2.5D), time-dependent simulations capitalize on the availability of a family of two-dimensional (2D) analytical solutions of isothermal magnetostatic atmospheres threaded by a layer of sheared undulating magnetic field. The magnetic field supports the weight of the atmosphere in an unstable configuration that sets the stage for the demonstration of magnetic flux emergence in a stratified atmosphere. When the system is perturbed, magnetic loops buoyantly rise from the flux layer and shearing motions are found to naturally arise in conjunction with mixed-mode (interchange and undulating) instabilities. The shearing motions take the form of large-amplitude shear Alfven waves that are driven by a component of the magnetic tension force pointing in the invariant direction. The waves are significant in that they transport magnetic flux from the shear layer into the ascending magnetic loops, causing them to become greatly inflated. The presence of such shear Alfven waves in magnetic loops rising through the photosphere provides an explanation for the impulsive shearing motions observed in newly emerged bipolar active regions.

Chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation (NLPDE) for the magnetic flux u, known as the Grad-Shafranov equation. The Chebyshev spectral collocation methods (CSCMs) are described and applied to obtain numerical solutions for nonlinear boundary value problems modelling two classes of isothermal magnetostatic atmospheres, in which the current density J is proportional to the exponential of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height, for the first class and proportional to the sinh( u) for the second class. The accuracy and efficiency of this Chebyshev approach are compared favorably with those of the standard finite-difference methods.

Equilibrium and Stability of Magnetostatic Atmospheres. I. Dungey‐Type Isothermal States

Two families of analytical solutions are presented that describe isothermal magnetostatic atmospheres in uniform gravity and varying in two Cartesian dimensions. The solutions in each family share a common set of magnetic flux surfaces but have different profiles of field intensity, magnetic shear, and plasma distribution across these flux surfaces such that force balance is satisfied. A family of solutions with this mathematical degree of freedom had previously been found by Dungey. The construction and properties of the new solutions are described. The hydromagnetic stability of these solutions is discussed by using the sufficiency criteria of Hu to determine those profiles of magnetic field and plasma that assure stability for the whole system. Among the stable equilibria found are examples of sheared magnetic structures intruding into a uniformly magnetized, isothermal atmosphere. One of the two families of solutions is extended to equilibrium states that vary fully with all three Cartesian coordinates. These extended solutions allow the possibility of constructing complex structures by juxtapositioning discrete three-dimensional magnetic structures built separately. These two-dimensional and three-dimensional magnetostatic states may be useful as initial states for numerical simulation of time-dependent magnetohydrodynamic processes in the solar atmosphere.

Application of the sine-Poisson equation in solar magnetostatics

Solutions of the sine-Poisson equation are used to construct a class of isothermal magnetostatic atmospheres, with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry. The distributed current in the model j is directed along the x-axis, where x is the horizontal ignorable coordinate. The current j varies as the sine of the magnetostatic potential and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height. We investigate in detail solutions for the magnetostatic potential A corresponding to the one-soliton, two soliton, and breather solutions of the sine-Gordon equation. Depending on the values of the free parameters in the soliton solutions, horizontally, periodic magnetostatic structures are obtained possessing either (a) a single X-type neutral point, (b) multiple neutral X-points, or (c) solutions without X-points. The solution cases (b) and (c) contain two families of intersecting current sheets, in which the line of intersection forms flux concentration points (or singularities) for the magnetic field. The solutions illustrate the contribution of the anisotropic J × B force (B, magnetic field induction), the gravitational force, and the gas pressure gradient to the force balance.

On the stability of a class of isothermal, magnetostatic atmospheres

The stability of a class of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is explored. Special attention is given to the stability of nonpotential equilibria that have the same magnetic field geometry as the potential equilibrium of the solution family. It is found that equilibria in which the source currents for the potential field are buried at large distances below the photospheric base are stable. Also considered is the stability of configurations in which the source currents for the potential field are located at a finite depth below the base.

Isothermal magnetostatic atmospheres

An investigation of a family of isothermal magneto-static atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. The distributed current in the model J is directed along the x-axis where x is the horizontal ignorable coordinate. The current J is taken proportional to the square of the magneto-static potential A and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height. A range of solution examples are displayed depending on the four free parameters in the solutions. Three of these parameters determine the source currents for the potential field solution of the family, whereas the fourth parameter determines the magnitude of the distributed current. The solutions illustrate the contribution of the anisotropic J × B force (B, magnetic field induction), the gravitational force, and gas pressure gradient to the force balance.

Isothermal magnetostatic atmospheres. II - Similarity solutions with current proportional to the magnetic potential cubed

The paper presents a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry. The distributed current in the model J is directed along the x-axis, where x is the horizontal ignorable coordinate. The current J is taken to be proportional to the cube of the magnetostatic potential A and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height. A range of similarity solution examples are displayed depending on the values of the similarity parameters. Each similarity parameter corresponds to a symmetry of the underlying nonlinear elliptic equation for A. The similarity parameters also determine the source currents for the potential field solution of the family. The solutions show the interplay between the gravitational force, the J and B force (B, magnetic field induction) and the gas pressure gradient. 21 references.