Radio Science Bulletin 325 - June 2008 - URSI

Ray Tracing ofMagnetohydrodynamic Wavesin GeospaceA.D.M. WalkerAbstractThe method of ray tracing is reviewed for MHDwaves in stationary media and for media in the steady state.The method is generalized to allow for media that changearbitrarily but slowly in space and time. This requires anadditional equation representing the change in frequencyalong the ray as a result of Doppler shifts. Explicit expressionsfor ray tracing of transverse Alfvén waves and magnetosonicwaves are presented. Some illustrative examples arepresented.1. IntroductionFor more than fifty years, since its introduction byHaselgrove [1], ray tracing has been a powerful tool forstudying radio propagation in the magnetosphere [2]. Thishas not been the case for lower-frequencymagnetohydrodynamic waves, the chief reason being thatoften the wavelength is comparable to the size of themagnetosphere, and in no sense can the medium be regardedas slowly varying – a sine qua non for the validity of the raytracingapproximation. For shorter-period waves in the Pc3range (periods between 10 and 45 seconds), ray tracing maybe valid in the magnetosheath and solar wind, while in thesolar wind it may be valid for much longer periods. Anotherdifficulty arises in these regions: the medium is moving.Ray tracing in a moving medium has been discussed forsound waves by Lighthill [3], for example, and his discussionwas generalized for MHD waves by Walker [4, pp. 429-430.]. The treatment in this paper follows these treatmentsfor steady-state motions, but goes beyond them in allowingfor non-steady flow and the resulting Doppler shifts. We donot apply the methods to specific problems of ULF wavesin geospace, which will be the subject of a later paper, butinstead provide some illustrative numerical examples.As an example of the issues that are likely to arise insuch geospace applications, consider a transverse Alfvénwave. It is well known that the group velocity of such awave in a stationary medium is directed precisely along themagnetic field, and, since the medium is nondispersive, thegroup velocity is equal to the Alfvén velocity. The problemof ray tracing in such a case is trivial: the ray paths coincidewith the magnetic field lines. However, in a moving mediuma wave packet travels with a velocity that is the resultant ofthe Alfvén velocity and the velocity of the medium. Thesolar-wind velocity is generally much larger than the Alfvénvelocity, so that the ray paths are more nearly radial thanalong the magnetic-field direction. The case of themagnetosonic waves is even more complicated.In this paper, we first discuss the geometrical optics ofMHD waves in uniform media. We then review thederivation of the ray-tracing equations in moving media inthe steady state. Finally, we show how these equations canbe generalized to include arbitrary flows, provided changesin the background flow with time are slow compared to theperiod of the wave. We present explicit equations to allowthe numerical computations of MHD rays, and performsome computations to illustrate the techniques.2. Geometrical Optics of MHDWaves in Uniform Media2.1 Stationary MediaIn a uniform stationary compressible plasma, atfrequencies much less than the lowest ion gyrofrequency,there are three characteristic magnetohydrodynamic wavesthat can be propagated. These can be identified as thetransverse Alfvén wave, with the dispersion relationA. D. M. Walker is with the School of Physics, UniversityA. D. M. Walker is with the School of Physics, Universityof KwaZulu-Natal, Durban 4000, South Africa;e-mail: walker@ukzn.ac.za.24TheRadio Science Bulletin No 325 (June 2008)