Radio Science Bulletin 325 - June 2008 - URSI

Haselgrove equations [2]. Furthermore, recastingEquation (16) in terms of q [2], we obtain⎡ 2( ) ( ) ( )2 2 2q 1− X − Y + q Y p 2 1 X Y⎤⎢• + − −⎥+ 1− X = 0⎣⎦(23)from which q can be derived (note that there are twopossible values of q, corresponding to two of the possiblesolutions to Equation (11)). The corresponding current,and hence the electric field through Equation (5), can beobtained from the homogeneous Equations (11) using thecalculated values of p on the ray. However, Equations (11)do not determine the magnitude of the field. This must becalculated by tracking the power out from the source. Thiscan be done by tracing out ray tubes, and noting that within20 2 0Ap E η , must remainconstant (A is the cross-sectional area of the tube). Themagnitude of the electric field can then be calculated fromthe power.these tubes, the power, ( )There remains the issue of solving the above raytracingequations. Starting with some initial value, w () 0 ,for w , the system of ordinary differential equations,dw dt = W( w), can be solved using Runge-Kuttatechniques. This was suggested in the Haselgrove andHaselgrove paper [2]. If we start a ray at the Earth’s surface,the initial values of p are given by the direction cosines atthis starting point. However, the extreme variations that canexist in the ionosphere often make it necessary to vary theincrement in t at each stage of the solution process, in orderto maintain accuracy. Haselgrove and Haselgrove [2]suggested the introduction of a scaling to produce thiseffect, but the Runge-Kutta-Fehlberg (RKF) method (see[8] for example) can also achieve this.Consider the solution at time t and prospectiveincrement ∆ t . Form the vectors k1, k2, k3, k4, k5, andk through the process6k1( )=∆ tW w , (24)( )k =∆ tW w+ k , (25)2 1 4(k =∆ tW w+ 439k 216 − 8k + 3680k5135 1 2 3(4)− 845k4104 , (28)k =∆tW w− 8k 27 + 2k −3544k25656 1 2 3+ 1859k4104 − 11k40 . (29)4 5The approximate solution, wt () +∆ w at t+∆ t , is obtainedusing∆ w = 25k 216 + 1408k25654 1 3+ 2197k4104 − k 5(29)4 5for the fourth-order Runge-Kutta method, and by∆ w = 16k 135 + 6656k128255 1 3+ 28561k 56430 − 9k 50 + 2k55 (30)4 5 6for the fifth-order method. The global truncation error will45be O( ∆ t ) for the fourth-order method and O( ∆ t ) forthe fifth-order method. The magnitude of the differencebetween the fourth- ( w 4 ) and fifth- ( w 5 ) order estimates,∆ w= w5 − w4, gives an estimate of the global truncationerror in the fourth-order method.The Runge-Kutta-Fehlberg method proceeds byadjusting the step at each stage so that this global errorremains close to a pre assigned value, ∆ E . That is, at eachstage we adjust ∆ t so that1⎛∆E⎞4∆ tnew=∆told⎜ ⎟⎝∆w⎠). (31)( 3 32 9 32)k =∆ tW w+ k + k , (26)3 1 2(k =∆ tW w+ 1932k 2197 −7200k21974 1 23)+ 7296k2197 , (27)This approach has been used in [4] to provide an efficientnumerical implementation of the Haselgrove equations.3. The Weak-Background-Field LimitFor the collision-free case ( U = 1), Equation (3) canbe inverted to yield20TheRadio Science Bulletin No 325 (June 2008)