The eikonal (WKB / geometrical-optics) method for multi-component waves in inhomogeneous anisotropic media was formulated for magnetohydrodynamics (MHD) by Steven Weinberg in 1962, who derived ray equations, zeroth-order amplitude transport, and formal second-order expressions for diffraction corrections, and applied the machinery to hydromagnetic wave propagation in the Earth's ionosphere. The paper explicitly leaves the second-order program unexecuted: the diffraction-loss terms given by the right-hand sides of his exact identities (63) and (94), the general diffraction term in his second-order amplitude equation (145), the explicit integration of the pure-MHD fast-mode amplitude (58) and of the coupled-mode amplitudes (105)-(106), the quantitative solution of the latitude-dependent three-dimensional ionosphere ray problem (34)-(40), and dissipative energy calculations below 200 km. This problem asks for the completion of that program for explicitly specified backgrounds. The zeroth-order amplitude theory has since been generalized (covariant eikonal formulations; action-conservation reformulations), the ionospheric physical questions are now answered by full-wave numerics, and the physics of diffraction for MHD waves has been approached by other routes (energy-balance reformulations, complex-eikonal Gaussian-beam ray tracing for general anisotropic media, wave-packet mixing analyses). The second-order diffraction corrections to MHD eikonal amplitudes do not appear to have been evaluated anywhere in the subsequent literature; the precise surviving open core is the evaluation of these second-order terms and the associated unexecuted amplitude integrations for a concrete system such as the ionosphere model.

Background

The eikonal method — known in wave optics as geometrical optics and in quantum mechanics as the WKB (Wentzel–Kramers–Brillouin) approximation — describes waves in media whose properties vary slowly on the scale of a wavelength. Each field is written as a slowly varying amplitude multiplying a rapidly oscillating phase factor, with a small parameter ε\varepsilon given by the ratio of wavelength to inhomogeneity scale. Leading order in ε\varepsilon yields a dispersion relation (the eikonal equation) for the phase and ray (characteristic) equations for the path and wave vector of the wave energy; the next order yields transport equations for the amplitudes. Magnetohydrodynamics (MHD) is the fluid description of a conducting plasma coupled to magnetic fields; in ideal (perfectly conducting) MHD, linear waves about a uniform background split into three modes: the transverse Alfvén mode, propagating along the magnetic field at the Alfvén velocity vA=B0/4πρ0v_A = B_0/\sqrt{4\pi\rho_0} (in cgs units, with B0B_0 the background field and ρ0\rho_0 the mass density), and the fast and slow magnetosonic modes, compressive hybrids of sound and Alfvén propagation. In an inhomogeneous medium these modes refract, and where the eikonal ordering breaks down — at turning points, where a ray reverses direction, and at caustics, where neighboring rays cross and the leading-order amplitude formally diverges — they can couple or reflect.

Weinberg (1962) formulated the eikonal method for general multi-component wave systems in inhomogeneous anisotropic media and applied it to MHD. For each mode he derived ray equations for the wave vector and zeroth-order transport equations for the leading amplitudes (f0f_0 for the Alfvén mode, g0g_0 for the fast mode in his notation). For the pure Alfvén mode the transport integrates to a known conservation law (f0ρ1/2f_0\rho^{1/2} constant along a ray); for the fast mode, Eq. (58), d(ln⁡g0)/dt=−(V/2k)∇⋅kd(\ln g_0)/dt = -(V/2k)\nabla\cdot\mathbf{k}, he notes that "there is no trick available" and the integral must actually be done. Beyond leading order he derived exact identities — Eq. (63) in the pure-MHD section, and Eq. (94) in the coupled-mode (magneto-acoustic-electro-hydrodynamic) section, ∇⋅(E0×B0)=E0⋅∇×B1−B0⋅∇×E1\nabla\cdot(\mathbf{E}_0\times\mathbf{B}_0) = \mathbf{E}_0\cdot\nabla\times\mathbf{B}_1 - \mathbf{B}_0\cdot\nabla\times\mathbf{E}_1 — whose right-hand sides, when the first-order field corrections E1,B1\mathbf{E}_1, \mathbf{B}_1 are inserted, give the diffraction losses: the leakage of wave energy across ray tubes caused by transverse spreading of the disturbance, the direct analogue of optical diffraction. He also gave the general second-order amplitude equation (145), whose extra term "represents the change in f0f_0 due to diffraction of the disturbance from one ray path into another." He then applied the leading-order machinery to the propagation of hydromagnetic waves — the standard ionospheric-physics term for MHD waves — in the Earth's ionosphere, computing energy fractions versus altitude in a model where the Alfvén velocity depends only on altitude (finding, e.g., trapped oscillation near 450 km altitude), setting up but not solving the full latitude-dependent three-dimensional ray problem, and stopping the analysis at 200 km altitude because finite conductivity (ohmic dissipation) becomes important below.

At each of these points the paper explicitly declines the computation: "We will not actually do any of this here, though in principle it is straightforward" (on evaluating the right side of Eq. (94)); "We will not venture on an adequate treatment of diffraction in this paper" (on Eq. (63)). Full-wave numerical treatments of ionospheric hydromagnetic waves already existed (Francis and Karplus 1960, cited by Weinberg), but the asymptotic, diffraction-corrected amplitude theory was left as a formal skeleton. The unresolved target is the execution of that skeleton.

Problem Statement

Complete the second-order eikonal program for magnetohydrodynamic waves that Weinberg (1962) set up but explicitly left unexecuted. The single scientific objective is a second-order-accurate eikonal account of MHD wave amplitudes — including diffraction — for an explicitly specified background medium, with quantitative predictions. The requested results, which jointly support that objective, are:

  1. Evaluate the right-hand sides of the exact identities (63) (pure MHD) and (94) (coupled magneto-acoustic-electro-hydrodynamic system) by inserting the first-order field corrections, obtaining the explicit diffraction losses to the zeroth-order amplitudes along ray paths — the computation the paper declines with "we will not actually do any of this here, though in principle it is straightforward".
  2. Evaluate the general second-order diffraction term of amplitude equation (145) for at least one specific case.
  3. Actually integrate the fast-mode amplitude equation (58), d(ln⁡g0)/dt=−(V/2k)∇⋅kd(\ln g_0)/dt = -(V/2k)\nabla\cdot\mathbf{k}, for a specified pure-MHD background, and the coupled-mode amplitude equations (105)–(106) for a specified system, going beyond the formal integral expressions given in the paper; the cited covariant-eikonal and action-conservation reformulations state the transport law in general form but do not carry out this evaluation for a specified background.
  4. Quantitatively solve the three-dimensional, geomagnetic-latitude-dependent ionosphere ray problem (equations (34)–(40)) for the energy fractions, extending Weinberg's altitude-only model, and extend the energy calculations to include dissipative (finite-conductivity) effects below 200 km altitude.

The program is to be carried out for an explicitly specified background medium (all profiles and parameters, e.g., an Alfvén velocity vA(z,λ)v_A(z,\lambda) with altitude and latitude dependence); the listed quantities (result groups 1–4 above) are derived or computed from the stated equations; and the domain of validity of the eikonal ordering is characterized, including behavior near turning points, caustics, and the trapping region near 450 km altitude.

References

  1. Steven Weinberg, "Eikonal Method in Magnetohydrodynamics", Physical Review 126, 1899–1909 (1962), DOI: 10.1103/PhysRev.126.1899, https://doi.org/10.1103/PhysRev.126.1899
  2. H. Heintzmann and E. Schrüfer, "Lorentz covariant eikonal method in magnetohydrodynamics I. The dispersion relation", Physics Letters A 60, 79 (1977), DOI: 10.1016/0375-9601(77)90328-0, https://doi.org/10.1016/0375-9601(77)90328-0
  3. E. Schrüfer and H. Heintzmann, "Lorentz-covariant Eikonal method in magnetohydrodynamics", Physics Letters A 81, 501 (1981), DOI: 10.1016/0375-9601(81)90451-5, https://doi.org/10.1016/0375-9601(81)90451-5
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  5. Aaron Barnes, "Theory of magnetohydrodynamic waves: The WKB Approximation revisited", Journal of Geophysical Research 97, 12105 (1992), DOI: 10.1029/92ja00996, https://doi.org/10.1029/92ja00996
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  10. Z. E. Musielak, J. M. Fontenla, and R. L. Moore, "Klein–Gordon equation and reflection of Alfvén waves in nonuniform media", Physics of Fluids B 4, 13 (1992), DOI: 10.1063/1.860452, https://doi.org/10.1063/1.860452
  11. W. E. Francis and Robert Karplus, "Hydromagnetic waves in the ionosphere", Journal of Geophysical Research 65, 3593 (1960), DOI: 10.1029/jz065i011p03593, https://doi.org/10.1029/jz065i011p03593
  12. Lee Tepley and R. K. Landshoff, "Waveguide theory for ionospheric propagation of hydromagnetic emissions", Journal of Geophysical Research 71, 1499 (1966), DOI: 10.1029/jz071i005p01499, https://doi.org/10.1029/jz071i005p01499
  13. W. J. Hughes and D. J. Southwood, "The screening of micropulsation signals by the atmosphere and ionosphere", Journal of Geophysical Research 81, 3234 (1976), DOI: 10.1029/ja081i019p03234, https://doi.org/10.1029/ja081i019p03234
  14. Anders Västberg and Bengt Lundborg, "Signal intensity in the geometrical optics approximation for the magnetized ionosphere", Radio Science 31, 1579 (1996), DOI: 10.1029/96rs02630, https://doi.org/10.1029/96rs02630
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  16. Yu. A. Kravtsov and Yu. Ya. Yashin, "Complex geometric optics of inhomogeneous anisotropic media", Radiophysics and Quantum Electronics 12, 537 (1969), DOI: 10.1007/bf01033133, https://doi.org/10.1007/bf01033133
  17. Yoshiaki Sofue, "MHD lensing in inhomogeneous ISM for qualitative understanding of the morphology of supernova remnants", arXiv:2408.00260 (2024), DOI: 10.48550/arXiv.2408.00260, https://arxiv.org/abs/2408.00260