Radio Science Bulletin 325 - June 2008 - URSI

decreasing to half their value in a time such that T = 1000 s.This was a much larger change than would be typical of thereal magnetotail, but illustrated the physics of the problem.1The initial frequency used in the computation was 0.5 s − .However, since all frequencies have the same group andphase velocity, the ray paths calculated were independentof ω , and the change in frequency was proportional to theinitial frequency. Thus, the results applied to any frequency,provided that it was not so low that the slowly-varyingcriterion was violated.The results are shown in Figure 5 for a wave packetthat started at 22.5R E from the current sheet. Figure 5ashows the ray path, and Figure 5b shows the change in thefrequency ω . Ticks are shown along the ray pathrepresenting equal intervals of time. As the medium wascompressed, the magnitudes of the magnetic field and thedensity were doubled, and the Alfvén speed increased by afactor of 2. Since the wavelength was unchanged, thefrequency f = V λ was increased by the same factor.AIn interpreting this diagram, it is important tounderstand that there are significant differences from thebehavior shown in a stationary or a steady-state medium. Ifthere is no dependence on t, successive parts of a wave of agiven frequency follow the same path. The ray path isindependent of time. When the medium varies in time, partsof the wave passing through a given point follow differentpaths as the medium changes. To clarify this, considerFigure 6. Imagine that at time t = 0 , just as the compressionis becoming appreciable, we have a continuous transverseAlfvén wave limited in space, forming a thin pencil alignedwith the magnetic field and represented by the line ABCD.Assume also that the wave normal is in the x direction. Thepoints A, B, C, and D are equally spaced and separated byan integral number of wavelengths. We consider the locationof the signal at a succession of times: t 1 , t 2 , etc., separatedby equal intervals. Each portion of the pencil moves alonga ray path like that of Figure 5a. Wave packets that initiallyfollowed each other along a single ray path parallel to thefield now follow different paths as the medium changes intime. At consecutive equal time intervals, the signal occupiesa pencil defined by the change in position of the wavepackets at A, B, C, and D. The horizontal distance betweenthese points remains constant, and thus the wavelengthcannot change. However, the velocity is increasing, as canbe seen from the fact that the horizontal distance betweensuccessive positions of each point increases. Thus, thefrequency must also increase. The net effect of thecompression is that the wave, initially confined to a fieldline and propagated along it, continues to be confined to thesame frozen-in field line as it is carried inwards by thecompression. As this happens, the velocity and frequencyincrease.This is an illuminating example that keeps clear thedistinction between temporal and spatial effects. In morecomplicated cases, spatial and temporal effects areinextricably interconnected.5. Discussion and ConclusionsRay tracing has not often been used as a technique foranalyzing the behavior of MHD waves in geospace.Frequently, the wavelengths are so large that the mediumcannot be regarded as slowly varying, so that full-wavetreatments are necessary. In addition, characteristic waveFigure 5. Ray tracing in amodel in which the magneticfield is increasing with time:−1V A ,0 = 0.01 REs,−1ω 0 = 0.5 s , T = 1000 s ,f = 1.0 . (a) The ray pathof a wave packet. (b) Thechange in frequency withmagnetic field increase as afunction of time.34TheRadio Science Bulletin No 325 (June 2008)