Radio Science Bulletin 325 - June 2008 - URSI

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Figure 1a. The refractive-index surface formagnetohydrodynamic waves when the sound speedis less than the Alfvén speed.Figure 1b. The refractive-index surface formagnetohydrodynamic waves when the sound speedis greater than the Alfvén speed.points located a distance ± VAfrom the origin in thedirection of the magnetic field.Examples of ray surfaces for the magnetosonic wavesare shown in Figure 2. The surface for the fast wave in theupper panel is an ovoid, showing the magnitude of thegroup velocity (equivalent in this nondispersive case to theray velocity) as a function of direction. The surface for theslow wave is more complicated. It can be understood byconsidering Figure 1a, which shows the correspondingrefractive-index surface. For slow-wave propagation withthe wave normal parallel to the field in the positive direction,the ray direction normal to the surface is also parallel to thefield. As we increase the angle between the wave normaland the magnetic field anticlockwise, the angle between theray direction and magnetic field increases clockwise, untilit reaches a maximum value where there is a point ofinflection in the refractive-index surface. At this point,there is a cusp in the ray surface. As the wave-normal angleincreases further, the ray direction again decreases. Bothray- and refractive-index surfaces are surfaces of rotationabout the magnetic-field direction.the plasma rest frame be ω 0 . The frequency in the observer’sframe is then Doppler shifted so thatω0= ω − k • V . (10)The dispersion relations are then those for a stationarymedium, Equations (2) and (4), with ω replaced byω − k•VIn MHD propagation, a point source radiates waveshaving the shape of the ray surface. The ray surface alsorepresents the shape of a pulse spreading out from thesource, since the medium is nondispersive and wavefrontsand signal fronts have the same shape. The ray surface is thecorrect surface to use in Huygens’ construction.2.2. Moving MediaIn uniform moving media, the situation is complicated.Suppose the plasma moves in the reference frame of theobserver with velocity V. Let the frequency of the wave inFigure 2. Ray surfaces for the fast and slow waves.26TheRadio Science Bulletin No 325 (June 2008)
In the moving medium, there is no longer a single axisof rotational symmetry. There are two characteristic axesparallel to the magnetic field and the flow velocity. Therefractive-index surfaces are now reflection symmetricacross the plane defined by B and V.The ray is still normal to the refractive-index surface.The group velocity isV G =∇ k ω( )=∇ k ω 0 + k•V (11)=∇ k ω 0 +V= VG,0+ V ,giving the fairly obvious result that the group velocity in themoving medium is the resultant of the group velocity in therest frame and the velocity of the medium. In the particularcase of the transverse Alfvén wave,VG= VA+ V , (12)and the group velocity in the moving medium is not parallelto B. For example, in observations of the solar wind bysatellites at the solar libration point, such as WIND or ACE,since V >> V A , transverse Alfvén waves are propagatedalmost radially from the sun, rather than parallel to themagnetic field.3. Theory of Ray Tracing3.1 Stationary MediaRay tracing applies when a medium is slowly varying.By this it is meant that the medium does not changeappreciably within a wavelength. The ray-tracing equationsin a stationary medium – as shown, for example, by Walker[2] – are given byd r =∇kω, (13)dtdk =−∇ω ,dtor in Cartesian tensor notation,dxidtdkidt∂ω= , (14)∂ ki∂ω=− ,∂ xwhere the subscripts take the values 1, 2, 3, correspondingto the three Cartesian coordinate directions, and summationon the repeated subscript is understood. Precise evaluationsof the errors depend on the specific circumstances of theproblem. Examples of such error calculations were givenby Walker [4, Section 15.4.2] for a particular case, forexample. Such calculations show that even when the mediumchanges by as much as 10% per wavelength, the error inphase is less than one radian in 200π wavelengths. Theaccuracy of the ray tracing depends on the precision withwhich the phase is known, and in most ray-tracing situations,the medium varies far more slowly than this.Let us first consider the trivial case of transverseAlfvén waves, and show that the ray-tracing equations do,in fact, give rays following the field lines with the Alfvénspeed. From Equation (1), with the choice of the positivesign, we getidrk( A)Adt =∇ k • V = V , (15)dk( A )dt =−∇ k • V .As expected, the first of these shows that the wave packetmoves parallel to V A , and hence to B. (The choice of theother sign would have given a wave packet moving antiparallelto B). The second equation shows how thewavenumber, and hence the wavelength, change along theray but are independent of the first equation. Thus, ratherthan two simultaneous vector differential equations for theray path, we can evaluate the change of ray path and thechange of wavelength independently. For the magnetosonicwaves, we evaluate the expressions on the right-hand sideof Equation (15) explicitly using Equation (4). The result,in Cartesian tensor notation, is( + ) ω − ( , )2 2 2 2ω⎡2ω− ( A + S )⎤⎡k V V k k V Vdx ⎢i=⎣dt⎢k V V⎣⎥⎦2 2 2 2 2i A S i j A j S( kV )2 2j A, j kVSVA,i2 2 2 2− , (16)ω ⎡ 2ω− k ( VA+ VS) ⎤⎢⎣⎥⎦⎤⎥⎦TheRadio Science Bulletin No 325 (June 2008) 27
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