Some exact solutions of the ideal MHD equations through symmetry reduction method

Picard, Philippe

doi:10.1016/j.jmaa.2007.03.100

Cited by 23 publications

(18 citation statements)

References 25 publications

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“…A similar reduction is proved in [28] for the equations of gas dynamics in the absence of a magnetic field. Since the set of invariant solutions of the equations of magnetohydrodynamics (1.1) has generally been studied [18,29], this solution is not investigated further. 3.3.…”

Section: Finally From (320) We Obtainmentioning

confidence: 99%

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

Головин

2009

J Appl Mech Tech Phy

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All irreducible regular partially invariant submodels with one noninvariant function for the equations of ideal magnetohydrodynamics are constructed. The submodels are completed to involution, and partially integrated. The submodels specify Ovsyannikov vortex type motion or motion with homogeneous deformation in some spatial directions.Key words: ideal magnetohydrodynamics, partially invariant solutions, overdetermined systems of differential equations.Introduction. The notion of a partially invariant solution as a natural generalization of invariant solutions of differential equations was first proposed by Ovsyannikov [1,2]. The usefulness of this generalization is indicated by numerous examples of partially invariant solutions constructed for the equations of gas dynamics (see [3][4][5][6] and [7] and the references therein), hydrodynamics [8][9][10], the dynamics of a viscous heat-conducting gas [11], magnetohydrodynamics [12,13], plasticity equations [15,16], and the equations of other models of mechanics and physics [17][18][19]. In contrast to invariant submodels, partially invariant submodels are given by overdetermined systems of equations, which complicates their analysis but allows one to obtain classes of solutions with greater arbitrariness compared to invariant solutions.The general theory of partially invariant solutions of differential equations is set forth in [20]. The notion of the regularity of a partially invariant solution used in practice is introduced in [21]. An important property of partially invariant solutions is reducibility: in some cases, partially invariant solutions coincide with invariant solutions of the same rank. Finding the reduction of a solution is important since this eliminates need to perform the extra work of completing the equations of the submodel to involution. The known sufficient tests of reduction of some special classes of partially invariant solutions are given in [20,22].The notion of the hierarchy of partially invariant solutions was introduced in [23] to simplify and systematize the study of the set of partially invariant submodels of a given system of differential equations. The existence of the hierarchical structure reduces the construction of the set of all partially invariant submodels to the analysis of only irreducible submodels, from which all the remaining submodels are obtained by invariant reduction, which considerably simplifies the calculations.The present paper analyses the regular partially invariant solutions of the equations of ideal magnetohydrodynamics. The analysis is performed only for nonbarochronic submodels in which the pressure is a function of spatial coordinates. The set of barochronic submodels is supposed to be studied separately by analogy with the barochronic submodels in gas dynamics [24]. Eight types of irreducible submodels are identified. The equations of all submodels with one noninvariant function are completed to involution; the obtained equations of the submodel are simpler than the equations of the initial model. The fir...

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Section: Finally From (320) We Obtainmentioning

confidence: 99%

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

Головин

2009

J Appl Mech Tech Phy

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“…Finding three-dimensional solutions with no spatial symmetry to the MHD equation is a daunting task. Only a few analytical solutions are known (Titov, Tassi & Hornig 2004;Hayat 2006;Hoernel 2008;Picard 2008;Al-Salti, Neukirch & Ryan 2010;Rajotia & Jat 2014;Shehzad, Hayat & Alsaedi 2016), and even numerical methods are often not straightforward.…”

Section: Introductionmentioning

confidence: 99%

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

Moawad

2020

J. Plasma Phys.

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This paper investigates the motion of three-dimensional ideal magnetohydrodynamics with incompressible flows. The governing equation is performed at steady state, with the magnetic field parallel to the plasma flow. The equations of stationary equilibrium are derived and described mathematically in Cartesian space. Two approaches for derivation of general three-dimensional solutions for Alfvénic and non-Alfvénic flows at constant and variable fluid densities are constructed. The general vector and scalar potentials of the velocity field are used to derive general formulas of general three-dimensional solutions for Alfvénic and non-Alfvénic flows. To verify the general results we have obtained, some examples are presented. An application that may be of interest for coronal loops and solar prominences is presented.

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“…MHD equilibria of ideal plasmas with incompressible flows and translational as well as axial symmetry were investigated by many authors. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] The fundamental concept behind MHD equilibria with flow is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. The set of equations which describe the MHD equilibria with flow are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.…”

Section: Introductionmentioning

confidence: 99%

Trigonometric and hyperbolic functions method for constructing analytic solutions to nonlinear plane magnetohydrodynamics equilibrium equations

Moawad

2015

Physics of Plasmas

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In this paper, we present a solution method for constructing exact analytic solutions to magnetohydrodynamics (MHD) equations. The method is constructed via all the trigonometric and hyperbolic functions. The method is applied to MHD equilibria with mass flow. Applications to a solar system concerned with the properties of coronal mass ejections that affect the heliosphere are presented. Some examples of the constructed solutions which describe magnetic structures of solar eruptions are investigated. Moreover, the constructed method can be applied to a variety classes of elliptic partial differential equations which arise in plasma physics. V C 2015 AIP Publishing LLC.

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Some exact solutions of the ideal MHD equations through symmetry reduction method

Cited by 23 publications

References 25 publications

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

Trigonometric and hyperbolic functions method for constructing analytic solutions to nonlinear plane magnetohydrodynamics equilibrium equations

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