Radio Science Bulletin 325 - June 2008 - URSI

2 ⎡ ∂VAj, ∂VS⎤ω ⎢VAj, + VS⎥dki2∂xi∂xidt =−k⎣⎦2 2 2 2ω⎡2ω− k ( V A + V S )⎤⎢⎣⎥⎦∂VkkV V ( kV ) V−ω⎡2ω− k ( VA+ VS)⎤⎢⎣⎥⎦2 Ak ,2j k Aj , S + j Aj , S∂xi2 2 2 2∂V∂xThe model provides expressions for the variation of VAand V S with position. These six simultaneous first-orderequations can be integrated numerically by a suitable stepby-stepprocess, such as the Runge-Kutta method, to providethe path of the ray.3.2 Steady-State FlowIn steady-state flow, the velocity of the medium is afunction of position, but it is independent of time, so that atany point in space the velocity remains constant. In thiscase, we can use Equation (10) and write the ray-tracingequations in the formdr=∇ kω =∇ k( ω0 + k• V)=∇ kω0+ V , (17)dtor in subscript notation,dk=−∇ ω =−∇ ( ω0+ k•V ) ,dtdxi∂ω0= + V , (18)idt ∂kiSi.motion is steady state is ω and is constant. (This is justifiedin the more general treatment of the next section). Thefrequency in the local rest frame is Doppler shifted, andvaries from point to point. This has consequences for thewave energy: as the wave progresses, it exchanges energywith the kinetic energy of the background flow.3.3 The General CaseThe most general case that can be treated by raytracingmethods is one in which the properties of themedium are functions of both position and time, and noframe of reference can be found in which the medium hasa steady-state flow. The restriction applied is that thedefinition of a slowly varying medium must be extended.The medium is assumed to vary slowly in space such thatlength scales are long compared with the wavelength and,in addition, the time scale of variation is long comparedwith the wave’s period. The complication now introducedis that the frequency can no longer be taken as constant.There will now be Doppler shifts in the frequency as thewave progresses, arising from the change in the propertiesof the medium as a function of time.asThe dispersion relation can now be written formally( x , k , t)ω = ω , (19)where there is now an explicit dependence on the time, t, asa result of the dependence of the Alfvén velocity and soundspeed on time.We now repeat the derivation of the ray-tracingequations as given by Walker [2], but including thedependence of the dispersion relation on time.iThe waves are assumed to vary in space and time as∫i( )expiΦ≡exp i⎡kidxi − ω xi, ki,t dt⎤⎣⎦∫(20)dkVi ∂ω ∂=− −dt ∂x x0 jk ji ∂ i.⎡ dxi⎤= exp i ∫ ⎢ki −ω( xi, ki,t)dtdt⎥.⎣⎦This shows that – as for a uniform medium – the wavepacket moves with a group velocity that is the resultant ofthe group velocity in the local rest frame of the medium andthe velocity of the medium at that point. The ray-tracingequations are then given by Equation (16), with ω replacedby ω 0 and the addition of the terms V i and −kj ∂Vj ∂ xi,which can be evaluated from the model.This set of equations allows us to follow the path of awave packet. The frequency in the frame in which theThe variation of the dispersion relation with position andtime is assumed to be slow, so that∂Φ ≈ ω , (21)∂t∂Φ =− k .i∂xi28TheRadio Science Bulletin No 325 (June 2008)