Stationary MHD structures

Woodward, T. I.; McKenzie, J. F.

doi:10.1016/0032-0633(93)90061-6

Cited by 10 publications

(4 citation statements)

References 28 publications

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“…The dispersion equations can be regarded as first-order partial differential equations for the singular manifold function φ(x, t), with Cauchy characteristics that define the ray equations of MHD geometrical optics. For standing waves in a steady flow, the fixed frame frequency ω = −φ t = 0, and in this case the ray equations define the group velocity surface and Mach cone for the magnetoacoustic waves and Alfvén wings, described in earlier analyses of Kogan (1960), Sears (1960), Bazer and Hurley (1963), Crapper (1965) and Woodward and McKenzie (1993). However, these earlier analyses did not emphasize the connection between the theory of Cauchy characteristics of first-order partial differential equations and the ray equations used in the present analysis.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…Illustrative examples of the group velocity surface for standing magnetoacoustic waves in a background flow with velocity u are given for example by Sears (1960), and Woodward and McKenzie (1993). In general, there are four roots of the characteristic equation (4.52), but depending on the magnitude and direction of the background flow relative to the background magnetic field, not all roots of (4.52) will necessarily be real.…”

Section: The Mach Conementioning

confidence: 99%

“…The Alfvén waves also have a Mach cone for super-Alfvénic flow which are commonly referred to as Alfvén wings (e.g. Woodward and McKenzie (1993)). …”

Section: The Mach Conementioning

confidence: 99%

“…Thus, the characteristics for steady flows correspond to standing, MHD waves in a steady background flow (e.g. Crapper (1965);McKenzie (1991) and Woodward and McKenzie (1993)). Kogan (1960) and Cabannes (1970) consider, steady (ω = 0), linear MHD waves in two-dimensional flows, in Cartesian geometry, in the xy-plane, in which the z coordinate is ignorable, and the unperturbed background flow velocity u = (u x , 0, 0) T is directed along the x-axis.…”

Section: Steady Flow Characteristicsmentioning

confidence: 99%

See 3 more Smart Citations

Magnetohydrodynamic waves in non-uniform flows I: a variational approach

Webb¹,

Zank²,

Kaghashvili³

et al. 2005

J. Plasma Phys.

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Abstract. The interaction of magnetohydrodynamic (MHD) waves in a nonuniform, time-dependent background plasma flow is investigated using Lagrangian field theory methods. The analysis uses Lagrangian maps, in which the position of the fluid element x * is expressed as a vector sum of the position vector x of the background plasma fluid element plus a Lagrangian displacement ξ(x, t) due to the waves. Linear, non-Wentzel-Kramer-Brillouin (WKB) wave interaction equations are obtained by expansion of the Lagrangian out to second order in ξ and ∆S, where ∆S is the Lagrangian entropy perturbation. The characteristic manifolds of the waves are determined by consideration of the Cauchy problem for the wave interaction equations. The manifolds correspond to the usual MHD waves modes, namely the Alfvén waves, the fast and slow magnetoacoustic waves and the entropy wave. The relationships between the characteristic manifolds, and the ray equations of geometrical MHD optics are developed using the theory of Cauchy characteristics for first-order partial differential equations. The first-order differential equations describing the singular manifolds are the dispersion equations for the MHD eigenmodes, where the wave vector k = ∇φ and frequency ω = −φ t correspond to the characteristic manifolds φ(x, t) = constant. The form of the characteristic manifolds for both time-dependent and steady MHD flows are developed. The bicharacteristics for steady MHD waves in a steady background flow are related to the group velocity surface and Mach cone for the waves, and determine when the flow is elliptic, hyperbolic, or of mixed hyperbolic-elliptic type. The wave interaction equations are decomposed into coupled equations for the compressible and incompressible perturbations.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: The Mach Conementioning

confidence: 99%

“…The Alfvén waves also have a Mach cone for super-Alfvénic flow which are commonly referred to as Alfvén wings (e.g. Woodward and McKenzie (1993)). …”

Section: The Mach Conementioning

confidence: 99%