• •q RUN-TIt-76-297

S) Final Technical ReportOctober 1976

DEVELA)PM4T OF A COMPUTER MODEL FOR SCATTERINGOF EEICTROMAGNETrIC WAVES BY A TURBULIT .WA•X

General Electric CompanyIs-ontry & Eiviromsntal Syetema Division

Approved for public release; distribution unliuited.

JROME AIR DEVELOPMENT CENTER D D CAIR FOR.CE SYSTE1MS COMKAZND Yl 34 011GRIPFISS AIR FORCE BASE, NEW YORK 13441

HOY 8 1976 ,

% :.I COPY AVAILA']RV lj£gIE ~lF PEN"IT [ULU( L.."'ii' + A

A

Paul E. 3imbing is the Principal Investigatotr for Lda~i 4aontract.Ronald L. Fantt! is the RADC Project Engineer (Cu•rract Monitor).

This rep irt has been reviewed by the RADC Information Office (01)and is releasable to the National Technical Information Service (NTIS).At NTIS it will be releasable to the eneral public, including foreignnations.

This technical report has been reviewed and approved for publication,

APPROVED: pRONALD L. FANTEProject Engineer

APPROVED: ~(K-~ '~LDv~oALLAN C. SCHE:LLActing ChiefElectromagn~etic Sciuncts Division •.-

FOR THE COMMANDER;

Plans Office.

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20. (CONTINUED)

THIS THEORETICAL MODEL ASSUMES THAT THE ELECTRICAL PROPERTIES OF THE WAKEPLASMA ARE KNOWN IN TERMS OF CERTAIN STATISTICAL AND SPATIAL DISTRIBUTIONS.THE TECHNIQUE USED BY THE MODEL IS A DISTORTED WAVE BORN APPROXIMATION IN WHICHTHE PROPAGATION IN THE PACKSROUND MEDIUM IS CALCULATED RIGOROUSLY FOR A LAYEREDCYLINDER OF MEAN WAKE PACKGROUN MODEL PROVIDES PREDICTIONS OF THE BACKSCAT-TERING AMPLITUDE AND THE FIRST AND SECOND MOMENTS OF THE DOPPLER SPECTRUM, BOTHAT EACH INPUT WAKE AXIAL STATION AND AS A FUNCTION OF APPARENT AXIAL STATIONIN THE BACKSCATTERED PULSE. DIRECT AND CROSS POLARIZED SCATTERING ARE INCLUDEDFOR BOTH LINkAR AND CIRCULAR POLARIZATION RADARS.

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UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE(Wlton Data Atntaesd)

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EVALUATION

1. This report is tha Final Report on the contract. It coversresearch done on the radar cross section of turbulent wakes duringthe twelve month period 1 Jul 75 to 30 Jun 76. The objective of thiswork is the generation of a computer code which can accuratelypredict the radar cross section of turbulent reentry wakes of arbitrarydiameter, electron density and strength of turbulence.

2. The above work is of value since it provides a method forcomputing the radar cross section of ballistic missiles and decoys,including the doppler spectrum of the return. It will be used by theUSAF to evaluate the possibility of discriminating between the reentrywakes of large and small vehicles.

!

RONALD L. FANTEProject En'gineerMicrowave Detection Techniques Br.Electromagnetic Sciences Division

I.

I'

i.i

Itit.

P RE A 4C E

The research reported on in this document warn

conducted by Paul E, Bisbing. It is related to

work done under subcontract No. A00621-DT-A-E

I to TRW Systems, TRW Inc. for prime contract No.

DAHC 60-72-C-0096, sponsored by the U. S. Army

Ballistic Missile Defense Advanced Technology

Center.

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at II

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4,, cI?

.. .. . ...-.. .. ....

CONTENTS

Page

I. INTRODUCTION 1

II, GENERAL DESCRIPTION OF THE MODEL 2

A. Background 2

B. A Green's Theorem 2

C. Representation of the Background Medium 4

D, Representation of Turbulence Effects 7

E. Representation of Axial Variations 10

III, DETAILED INFORMATION FOR THE USER 11

A. General 11

B. Description of the Program 15

1. a15n Program

2. Subroutines 18

a) FIT 18

b) PROPS 21

c) WFP 24

d) HANK 28

e) BIN 28

f) PROP 29

)BC0 31

h) STORF 34

i) PHINT 35

, )FFSUM 35

k) AVGS 37

1) RINT 38

m) ZINT 39

vii

Ati

CONTENTS (Continued)

PazeC. Input and Output

40D1 Summary of Checkout Procedures Used 42IV. SAMPLE RIrEULTS 45

A. Comparisons with Experimental Data 45B. Some Parametric Results 50V. CONCLUSION 558::

Appendix

A. FORTRAN LISTINGS 59

B. EXAMPLE OF OUTPUT 89II

vii

MAi

. . . . . .- , . *

I. INTRODUCTION

The subject of this report is a computer model for the radar backscatter-ing from a re-entry induced ionized wake. The report gives both the theoreti-cal derivations and the FORTRAN coding for the model, and thus it serves asboth a final report and a user's manual.

The model is based on a previously existing model', which applied thesame basic technique but in a simpler way. Improvements in both theory andmethod of application have been made during this study.

The basic technique of this model is to use a distorted wave Born approx-imation in which the nonrandom background medium is represented as a radiallynonuniform cylindrical plasma. The propagation of electromagnetic (em) wa'esin this background medium is formulated rigorously and solved numerically to ahigh degree of accuracy. The results of these calculations are used to "dis-tort", or weight, the Born approximation for the scattering by the turbulentfluctuations. Section II give, a more detailed general description of themodel and Section III gives the full details in a manner which relates direct-ly to the computer code. Section IV discusses the results of some calcula-tions intended to demonstrate the model's agreement with published data andto show the significance of certain phenomena included in the model. Section V,which concludes the report, gives recommendations for future development.

1. P.E. Bisbing, "Fina-lReport'. The EM Scattering Model for Re-entry Wakes.Volume 1. The Wake Field Penetration Model of Radar Scattering, "General

Electric Company, RESD, Report No. 75SDR2386-I, December 1975.

I

11. GENERAL DESCRIPTION OF THE MODEL

A. Background

This section is intended to give the theoretical basis for this model.This theory fits within the general topic of the interaction of waves withrandom media. In particular it relates to am wave scattering by turbulentplasma. Inasmuuh as this report is not intended as a survey, we refer thereader to the review of this topic by Granatstein and Feinstein2 . Also, t.heclosely related topic of em wave propagation through random media has been •jsurveyed recently by Fante3 . Thus we will refer only to works which haveestablished those points which are pertinent to the approach used herein.

The most basic part of the model is the Born approximation, which ap-pears to have been originally applied by Pekeris" to acoustic scattering inturbulent media. Its first application to radio waves was by Booker andGordon5 and to ionized media, by Booker 6 . Tatarski 7 gave it a generalizedform in terms of the turbulence spectrum function. The name "Born approxima-tion" is borrowed from quantum theo~y to denote single scattering; i.e., itis assumed that waves propagate undistorted in the medium. Improvements tothis approximation have accounted for various aspects of the medium whichtend to distort the wave before (and after) it reaches the scatterer inquestiun, Thus the name "distorted wave Born approximation".

Refraction in the mean wake plasma is a very significant source of dis-tortion for em waves in ionized media because it tends to bend the path ofpropagation away from the interior of the medium. In fact when the line ofincidence makes a low angle to the wake axis this effect is important atelectron densities which are sufficiently low that the undistorted Born approx-imation would apply at normal incidence. Thus propagation in the backgroundmedium is the main source of distortion used in this model. Other distortingeffects are included in ways which are similar to previous theories.

•. A Green's Theorem

In mathematicul terms the Born approximation is the first term in a aer-turbation-iteration series called the Neumann series. A survey by Keller gives

2. V.L. Grenatatein and D.L. Feinstein, "Multiple Scattering and Transport ofMicrowaves in Turbulent Plasmas", Advances in Electronics and ElectronPhysics, Vol. 32, pp. 311-381, Academic Press, N. Y., 1973.

3. R.L. Pants, "Electromagnetic Beam Propagati.,n in Turbulent Media", Proc.IEEE, Vol. 63, No. 12, pp 1669-1692, Dec. 1975.

4. C.L. Pekeris, "Note on the Scattering of Radiation in an InhomogeneousMedium", Physical Review, Vol. 71, No. 4, pp. 268-269, 15 February 1947.

5. H.G. Booker and W.E. Gordon, "A Theory of Radio Scattering in theTroposphere", Proc. IRE, Vol. 38, pp 401-412, April 1950.

6. H.0. Booker, "Radio Scattering in the Lower Ionosphere", J. GeophysicalResearch, Vol. 64, No. 12, 2164-2177, December 1959.

7. V.I. Tatarski, Wave Propagation in a Turbulent Medium, tvanslated byR.A. Silverman, McGraw-Hill, N.Y., 1961 (especially Chapter 4).

8. J.B. Keller, "A Survey of che Theory of Wave Propagation in ContinuousRandom Media", Proceedings of the Symposium on Turbulence of Fluids andPlasmas, New York-1968, Polytechnic Press, Brooklyn, 1969 (pp. 131-142).

2

the derivation of this series in very simple terms. Letting M and V representthe nonrandom and purely random differential operators, respectively, p the"unknown wave function and g the known source function, the equntion to besolved is

(M + V)P g . (1)

Operate on both sides by the inverse of M.

(1 + M-1V) M- g (2)

This equation can be written as follows:

u - -M-'Vjj + M-'g (3)

Returning to (2), multiply both sides by the inverse of the binomial, using thebinomial theorem on this inverse, giving

= - V)n- I (4)nuo

which is the Neumann series. This result can be written more explicitly asfollows:

E p- n (5)n-O

SM'zg (6)

-n M-VIn-l n>o (7)

where ý'n represents the nth order contribution to W.

Now let us specify the above system of equations for em waves. Assumeall quantities have the time dependence, exp(-iwt), where i is the imaginarynumber, w is the radian frequency and t is the time. Then the vector electricfield E obeys the differential equation

VXVXE - k2 (K+K')E + ikJ (8)

where k is the free-space wave number (2n divided by the wavelength), K and K'respectively are the mean and random components of the complex dielectricconstant of the plasma and J is the source (transmitter) current density dis-tribution. This equation is the analog of (1), where p is E, 14 is k2 K-VXVX,V is k2 K' and g is -ikJ. The inverse of M is customarily expressed in termsof a dyadic Green's function G, which satisfies Maxwell's equations for theelectric field of a unit point vector source in the mean medium only.

VxVxG - k2 KG + 4nI6(r-r') (9)

In this equation I is the identity dyadic, 6 is the Dirac delta function andr and r' respectively are the radius vectors of the field and source points.Apply the divergence theorem of Gauss to the combination [GX(VxE)-Ex( VXG)].Use of certain vector identities gives

3

JiG.(VXVXE) - E.(VxVxG)]dV I f(VxE).(oxdS)

-f(VxG).(ExdS) (10)

where dV is the element of volume enclosed within the surface having outwardnormal element dS. Let this surface be an infinitely large sphere containingboth the source and the scatterer relatively near its origin. Then both Eand G are outgoing radiation on the surface and the curl is proportioned tothe vector product with the normal, so the right hand side of (10) vanishes.Substitution of (8) and (9) in (10) gives

O4E - k 2fK'E-GdV + ikfJ.GdV (11)

which is the analog of (3), so M9- is an integral over a scalar product with G.The analog of the Neumann series is the following:

E = E En (12)n~.o

4rEo w ikfJ*GdV (13)

4••tEn k 2 fK'FR._'GdV, n>o (14)

In summary, (11) is a Green's theorem, in which G represents propagationin the background medium (the mean or nonrandom component of the total medium),it leads to the perturbation - iteration bolution represented by (12) through(14), provided this series converges.

C. Representation of the Background Medium

Almost all previous theorieo ignore the background mediuml i.e., K isassumed to be unity everywhere. Then G is the free-space Green's function andthe usual Born series comes about. Accounting for the background mediumobviously is not easy, but in the present problem we can at least represent itsuch that G can be calculated. Thus let th.. mean wake be an infinitely lonscylinder of plasma in which the electron density and collision frequency dependonly on the radial distance from the axis. (The effect of axial gradientswill be evaluated later.)

Consider (13), the equation for the zero-order approximation. If thebackground medium were free space, EO) would be the field radiated by the trans-mitter because G is the free-space Green's function. The effect on G of addingthe inhomogeneous background medium is to add a scattered wave, which alsocouples to J in the integral. But if the transmitter is in the far field, thecoupling of the scattered part of G to J can.be neglected. Thus the signifi-cance of the right hand side of (13) is that Eo is the vector field induced inthe background medium by the transmitter. Note, however, that it Is the totalinduced field, incident plus scattered, even though the latter component isassumed not to couple to the transmitter.

The dyadic naLure of G is more essential in (14), since, when the pertur-bation-iteration scheme is carried out, En must be evaluated locally, at allpoints in the medium. The dyadic qualities of G can be obviated, however, whenthe nth order far SCAttered field is evaluated. Simply take the dot product of

4

P . .. _ -• " •••• : ,*:• , •::,•,...i: : •: ''" ::•. i •'• , i': :•• ",:i •

...................................................i , I .

'I

both sides of (14) with the unit vector representing the receiver polarization.This means that En on the left represents this approximation to the field whichcouples to the receiver and that G in the integral is the total (incident plusscattered) vector electric field induced in the background medium by the re-ceiver if it were operated as a transmitter. This last result comes about be-cause of reciprocity; i.e., the coupling between the receiver and a small testantenna in tho background medium is independent of which antenna is used totransmit. Therefore the effect of the background medium is quite simple in thefirst-order approximation of the far field since one needs only to calculatethe vector fields induced by incident plane waves.

In order to lead up to the radar cross section calculation let us put (14)in the form of the first-order approximation to the fields scattered by anincident, uni.t amplitude, plane wave. Then E. in the integr• is the totalvector elect ic field induced in the mean medium by this in( -nt plane w.ave.The fact that G is the fields induced by radiation from the L .eeiver impliesthat

4rrE = k2eikr fK'E 0 dGodV (15)

In this equation r is the distance from the receiver to the originof thescatterer and Go is the total electric field induced in the mean medium by aunit amplitude plane wave propagating away from the receiver and having itsphase reference at the origin of the scatterer. The incident waves foL E0 andGo have the polarization of the transmitter and the receiver, respectively.

We now give the problem of calculating EO and Go precise definition. Theincident wave Ei propagates at an angle a to the wake axis, as shown in Figure1. The axis is also the z-axis of a right handed cylindrical coordinate system(p, $, z), where the direction 0 - o lies in the plane of incidence and pointsAn the forward direction. The complex amplitude of the incident wave has thefollowing form: (The time variation will be suppressed in all quantities.)

I, = oxp [ik(p sin a cos ( + z cos (x) (16)

Arbitrary polarization of tho VOCC'lver or transmi.t Ir i,; represontiLd by super-position of two principal plane polarizations. The eluctric field of theincident wave for perpendicular (often called transverse below) polarizationis taken, by arbitrary choice, to point into the plane of Figure I when thewave is at the origin. Thus

hita EI(r') sin ( + $ cos () (17)

where E1 denotes the vector incident field for transverse polarization anddenotes a unit vector. Let the direction of parallel polarization be such thatat normal incidence the electric vector at the origin would be aligned with thepositive Z-axis.

Eip Eilz sin a - (0 cos (F- $ sin )cos a] (18)

With the assumption that the electron density and collision frequency dependonly on p, the problem of the background medium is completely specified. Itssolution is worked out in detail below.

5

S.. .. .: . .. .. .. .. ,.. . , .. . .... . .. L . .

~ 1800 0-

Direction of incident A0

electric vector atorigin:

ParalltelPolarization I4

TransversePolarization

FIGURE 1. PROBEM~ DEFINITION

6

D. Representati1on O1f TurbulencLe Ufecti-

O ce' we o , '%.l 0 U1 O tlt d tI h LociL I It x vlp' I\ rok r i, and Gy 0L . 111 ju I•lt; I II H 11 ,wake, thim t irst-ordor appiOxi•ilmti L .on LO IIt, c Lllrh ii ence sen Iter L' g Is h ',; II by(15). HoWeVer, a number of dllfi'culties arc encountered at this point

Perhaps the greatest difficulty with (15) is the statistics since K' is arandom variable. The most direct approach, if the statistical distribution ofK' were known, would be a Monte Carlo analysis in Which the fixed distribution

* of Eo.GO would be multiplied by each realization of K'. Perhaps such anapproach would be feasible, but it has not been attempted in this study. In-stead we have relied on the correspondence between (15) and the Born approxima-tion, which occurs when the background medium vanishes. Thus we assume thatthe statistics can be handled in a manner similar to the Born approximation.Except for a spreading owing to the finite volume, the Born solution is propor-tional to the three-dimensional Fourier transform of the turbulence autocorrela-tion function at a wavenumber which belongs to the vector resultant of theproduct Eo.Go, each member of which is a unit amplitude plane wave. This wave-number is in fact equal to the gradient of the argument* of the dot product ofthe two plane waves. Therefore we approximate the statistical effect of turbu-lence by taking the spectrum function at the wavenumber which is equal to thegradient of the argument of Eo.Go.

The Born result is proportional also to the volume integral of the meansquare fluctuation of the dielectric constant. It is obvious, however, that ifeither the incident wave or the scattered wave were attenuated in the medium,this volume integral would have to be appropriately weighted. Suppose, for amoment, that EO and Go have the same phase as the incident wave but differentamplitudes at various points in the medium. Then the expectation of the radarcross section, which is proportional to the mean square of (15), would beequal to the Born result with the volume integral weighted by 1Eo'Gol 2 . There-fore we include this weighting in our model. Of course the phase of EoGovaries throughout the medium, so the spectrum function at the appropriate wave-number weights the volume integral also, rather than factoring out as in theBorn approximation for homogeneous turbulence. The result to this point maybe written in terms of radar cross section a as follows:

j!N'jFGo'%l S(ke)dV

o a 4 7 r (19)1j + (••2

S (k) fD(r)eik@rdV (20)

where re is the classical electron radius (2.81777 X 10 1 5 m), N' is the electrondensity fluctuation (overbar denotes mean), and v is the electron collison fre-quency. In the definition (20) for the spectrum function S(k), k is a vectorwavenumber, r is the radius vector and D(r) is the autocorrelation function ofthe electron density fluctuations. The effective vector wavenumber ke is thegradient of the argument (phase) of Eo'GO,

Application of (19) is not done without difficulty in certain instances

The term "Wrgument" denotes the mathematical term for a component of acomplex variable; i.e., it is the angle in radians between the realaxis and the complex vector.

7Ki

\* j I

*.,-*. . ... 1

owing to the plasma resonance effect, in which the electric field tends toinfinity at the surface where K tends to zero. Thus the term 1E.GC 12 causesthe integral to diverge when the mean wake iii overdense (electron densitygreater than r/(A:'r,,), where A Its Lh frroc-slMaC wovOlength) and suflficlentlyco.1.ll Ision less. In the pri.or+ study (see Rel". 1) we did not aecotlnt l'or kL,usilng instead the argumeiiL that at 1oW aspect angle t00 wIvLIUlmlber was nutgreatly perturbed from that for free space so the Born spectrum functionfactors out. Unfortunately many of the numerical results were several ordersof magnitude greater than the Born approximation. This was especially true incases for which the diffraction effect allows the electric field to penetrate

* where refraction would otherwise stop it. One avenue of investigation wasfollowed in that study in great detail. This was based on the fact that themagnetic field and the product of the dielectric constant and the electric fieldare not resonant even when the electric field is. Going back to the derivationof a Green's theorem (see Section IIA), the analogs of (8) and (9) can be writ-ten for magnetic fields and (10) can be written to include electric and/ormagnetic fields, Also, (10) can include K explicitly by applying the divergencetheorem to Kn times the type of combination originally used above to derive(10), where n is an arbitrary numerical power to which K is raised. One canderive an unlimited number of Green's theorems which have various combinationsof K and either electric or magnetic field quantities in place of E and C in(11). Although all but the purely electric field theorem (11) have additionalterms involving K and sometimes VK', the term in K' is always similar to thefirst term on the right hand side of (11). In addition to the result of (11)we studied the results of theorems using HoF, KEF, and KE.KG, where H and Fare the magnetic field and its Green's function, respectively. The above-men-tioned additional terms were neglected and radar cross section was calculatedparametrically for each of the four theorems. The conclusion was reached thatnegligible benefit could be obtained from these alternatives. In all caseswhere the resonance effect was absent for any reason, all four theorems gaveessentially the same results.* In other cases even most of the nonresonanttheorems tended to diverge because of other effects, and none of the theoremsagreed on the answer.

The resonance effect has been used by Ruquist 9 to derive a surface scatter-ing effect. But we find a few difficulties in that approach also. In the dif-fraction regime the resonance is diffuse in that the electric field is high atnot only the critical surface but also over a large fraction of the volume. Inthe geometric optics regime the critical surface is inside the surface whererefraction causes ray turning, so one would not normally associate surfacescattering with the critical surface except at normal incidence.

The approach which we have adopted to avoid the resonance effect is toneglect the contribution of the critical surface to the integral in (19) atall points where lEo.G 0 12 is greater than unity on that surface, The ration-ale for this is based on the fact that the electric field has a phase reversalat the resonant point; i.e., the phase varies infinitely rapidly through the

* The exception to thin conclusion is in the cross polarized backscattering,which was generally considerably smaller when the magnetic field was used.

9. R.). Ruquist, "Resonant Scattering from Weak Fluctuations in PlaneStratified Dielectric Media", Proc. IEEE, Vol. 63, No. 1, pp. 205-206,Jan. 1975.

' '/ " , , ,. , ... • ' ., . , . ' ,. ,,,,. . " , ' " ,, " , ,'"""8

;ii

critical surface. As a result the wavenumber goes to infinity, where thespectrum function vanishes exponentially. This argument is admittedly 11 bitweak since the phase of E0*Go does not have a discontinuity at the criticalsurface, because the product of two functions which chrnge sign at a point doesnot itself change sign. On the other hand it is found in numerical studies thatthe effective wavenumber, defined as the gradient of the phase of Eo'GO, doesincrease rapidly in the neighborhood of the critical surface. An example ofthis result is given below.* It is interesting to note that in some theoreticalstudies appearing in the litarature the effective wavenumber is taken as thefree-space wavenumber times the square root of the dielectric constant of thebackground medium. In that approximation one would have zero wavenumber at thecritical surface, which would not help cancel the resonance effect. Of coursethe literature which uses that approximation also uses geometric optics, whichis bothered not by resonance but by caustics.

It goes without saying that the second and higher order approximations forthe turbulence scattering are very complicated, Thus at this time we have notfr developed the technique for rigorously applying the model to the multiplescattering problem. However, in order to include the main effects of higherorder terms, we have opted for an approximate representation within the frame-work of the first-order theory. Two phenomena, attenuation owing to randomscattering and depolarization from multiple random scattering, are included inthis representation. The equations for both of these effects are taken directlyfrom the literature, especially in the case of depolarization. A small excep-tion is the fact that scattering attenuation is not integrated along ray paths,since rays are not used in this model. The local scattering attenuation is usedinstead to define a plasma having equivalent attenuation owing to absorption.This plasma is then linearly added to the original mean plasma distribution inorder to simulate the effect of scattering attenuation on the coherent wave.

The doppler frequency spectrum of the scattering is implicitly representedby (15) when the theories o; Lanelo and Schapkeril are invoked. These authorsshowed that the spectrum of the electron velocity fluctuations in the wake isGaussian, which implies that the first and second moments are sufficient to des-cribe it. Also in the Born approximation the doppler spectrum is a linear com-bination of the electron velocity spectra in the wake, so each of the dopplermoments is the same linear combination of the corresponding moments of thevelocity distribution. Thus the generalized form of (19) is

Sn- 4rf2 n (21)

Swhere Sn a the nth moment of the doppler spectrum and un is the nth moment of ithe electron velocity spectrum at a given point in the wake. Of course the radar

cross section o, as given by (19), is the zeroth moment (uo0El).

*See Section III C, top of page 42.

10. F. Lane, "Frequency Effects in the Radar Return from Turbulent, WeaklyIonized Missile Wakes", AIAA J., Vol. 5, No. 12, pp 2193-2200, December1967.

11. R.L. Schapker, "Frequency Effects of Electromagnetic Scattering fromUnderdense Turbulent Plasmas", AIAA J., Vol. 8, No. 9, pp. 1582-1590,September 1970. 9

9i.

E. Representation of Axial Variations

Up to this point the wake has been treated as an infinite cylinder, soprovision must be made for axial variations in its properties. This is donesimply by breaking the wake up into a series of axial stations, each of whichis assumed to fit the unifQour c~finder representation. When the scatteringfrom any given station is calculated, we carry out the volume integration in(21) over only the radial and angular coordinates. The result is the quantitydSn/dz, which represents, for example if n - o, the radar cross section perunit length of wake.

An integration must be performed over the axial coordinate z to get thetotal scattering, In doing this we assume that the radar signal is pulsed andthat the entire wake lies within the main beam of the radar antenna. Then theacattering is a function of the apparent wake station (time delay), which isgiven by a convolution integral. Thus

Sn(z) • n )P(z - z')dz' (22)

where Sn(z) is the nth doppler moment of the scattered pulse shape functionversus apparent wake station z and P(z) is the effective transmitted radar pulseshape. For backucattering P(z) is equal to the relative transmitted pulse powerat a time (2z/c) cos a during the pulse, normalized such that the peak value isunity. In this model, the principal lobe of a cosine squared function is usedto represent P(z). Its width is defined in terms of the parameter range resolu-tion, which is the distance between half-power points when a - o.

.1

.,. .K'. . , ,

III. DETAILED INFORMATION FOR THE USER

A. General

This computer program was developed and checked out using the Honeywell6060 at the General Electric Company, Space Systems Division Computer Center,Valley Forge, Pa. A complete deck of punched cards (in 26-punch and fullyinterpreted in column alignment) has been supplied to the contract monitor

*, (RADC/ETEP). A corresponding listing is included herewith in Appendix A. Thislisting contains the same statements as the punched cards. The deck and thelisting are both in the form developed for the Honeywell 6060, however nocontrol cards are included. This subsection gives certain general informationabout the program, including how to revise its central memory requirements todo the problem at hand economically and how to make it run on the CDC computers.

The language is FORTRAN-Y and complex, single precision arithmetic is used,The only kind of type statement used is COMPLEX; all integer variables are typedby the initial letter, from I through N, of their names. No mixed mode arith-metic expressions involving complex variables are used, despite the fact thatthe Honeywell 6060 FORTRAN-Y compiler permits them, in order to avoid any ambi-guity for the user and to avoid difficulties with other compilers. All state-ment numbers increase as one reads down in each routine, and none is super-fluous. Format statements are all numbered 1000 or greater. Output formats'contain at least one space or a new page at the beginning of each line, to leaveroom for carriage control, even though the Honeywell 6060 does not need it.With certain exceptions noted below the program calls only the built-in andlibrary functions listed in Table 1.

All common blocks are segregated according to variable type and labeledwith a name which indicates the type as well as the routines containing thecommon. The first letter of the label indicates the variable type (C for com-plex, I for integer, and R for real) with one exception, ABCKLW, which wouldbe too long if the R were prefixed. The other letters in the label denotethe routines in which the common appears. The code letter of each routine ap-pears in column 8 of its initial comment card. The 13 subroutines are denoteda through,m, and the main program is denoted w for "wake". Thus, for example,COMMON/RBCW/ consists of real variables and appears in subroutines PROFS andWFP as well as in MAIN.

Most of the dimensions for array variables, and hence to a certain extentthe central memory requirements, are dictated by the choices of certain more or

i;, less arbitrary limits on the problem description. The coding is designed tomake it very easy to change these limits. The limiting factors are the numberof layers, the number of angular modes, the number of axial stations, and thedetail with which radial profiles can be input. The radial distribution of wakeproperties is currently represented by a series of 50 concentric cylindricallayers. The variable LAYERS which appears in MAIN, FIT, PROFS, RINT, WFP, BCO,STORF, PHINT, FFSUM, and AVGS has this value. The dimension of any subscript in.the entire program which now has the value 50 also is determined by this quan-tity. The data statements for the arrays ROR and YIP must of course conform tothis number; but ree the detailed description below (subsections IIB2a) andIIB2c)). Note that YIP is dimensioned one less than this number.

t 11

41

Table 1. Mathematical Functions Called By The Program

NUMBER TYPE OFOF

NAME DEFINITION ARGUMENTS ARGUMENT FUNCTION

AEB(A) JAI 1 real real

AIMAG(C) Imaginary part of C 1 complex real

ALOG (A) Natural log of A 1. real real

ALOGlO (A) Coimmon Log of A 1 real real

CABS(C) ICI 1 complex real

C)PLX(AOB) A+iB 2 real complex

COS(A) coo(A), for A in radian. I real real

CSQRT (C) V. - 1 complex complex

EXP(A) 6A 1 real real

FLQAT(I) Convert integer to real 1 inLeger real

IABS(1) I 1integer integer

IFIX (A) Truncate real to 1 real integerinteger

MXO (IJ11.,I Maximum vleof al>1 integer integer

L410D(I,J) I minus the product of 2 integer integerJ with the truncatedvalue of (I/J)

REAL(C) Real part of C 1 complex real

FSIN(A) sin(A), for A in radians 1 real real

SQRT (A) 1 real real

.12

The snfields In the sean wake are-currently reprasented by up-to 64.angular modes. Thus the variable HOW)1 has this value, which aunt be an ints-

pal ~erof two,0 in subroutines. VIP and HANN. All variables which aremow d Imeniond 64 correspond to tRhis requirement., The variable CN(65) in

subroutine HANK asut be dimensioned at least'one greater than KODIS * The dimen-sio'n 256 for the variable. 0, R and 1W in 'subroutines W3'SIJ and AMO must Le.tout tioo~e au o 0S In AVGS the va I~wlii I08, U and V iie dimen-sioned one gwesior than twice M1ODBS.

Currently one can input..up to 20 axial stat ions, which..relates to allquantities now dimensioned 20. Note also that the data statement for MUTZ inMWI Wast.conform to the maximum number of axial stations,, which is denoted,

S1TAT. in subrout.itio SINT. Also.,' in ZINTO the variable P muut be dimensionedat least- it or One loess -than ISTAT, * hichever is greater.' The quantitiesdim~mstsd.n4240 relate, not only~ to the maximum number 'of axial stations but alsoto the fact that, currentlyý',a given radiaV-prof ii. function can be input in-tonms of- up to slaese nusibrers Thus the dimensions of all quantities now di-Umesioned 240 must .be the maximum number of axial stations times one more thanthe number of q14ntities@leich may be aned to specify an input radial. profile.

it t ' osdsivrsd to input -thi iii p~of iles, tit tereofmstn elevenmustsi '06o *6hang t8e 4deson Of, and the data'ststesnt frthe

Var~abl a it, suroutino VIr (see s~usectilon 11320P).

'This piolp should require only slight modification to make it run onmachineqi other than the Honeywell 6060.* A preliminary version has been convert-ad end run successfully on the CDO6600 at AWOL# Hanscom APB, MA. The forervumnnerof-this program was run extensively on the CDC7600 at the Army WE1ATu-ARE InYkintsvi.lle, AL. The CDC '(FORTRAN extended) compiler is a bit more particularthgn the Honeywell (IFORTRAN-4)o in that 'certain changes had to be made to avoidwarnings or fatal error* whore nonedadcdpreviously occurred. All such changesOre Compatible with I aud they have alrnady been incorporated into the'O~de as a general pola~y.

teCertain statements (in NAZINCO MpSTOI, and MTUN) must be omitted to saketeprogram run'on the CDC machines. All, such statements are identified in

columns 73 to 80 by a comeent, which denotes the reason for their presence in%he Honeyw'ell version2. This ioomputer uses a word of only 36 bits limited tomagnitudes between approximately 10"'~. Thus exponent underf low occurs octa-.aionally, although it has a negligible effect .on the accuracy of Lthe Output..21% library subprogriam YXOPT(1.7 ,ll .0) .U~pprM135G8 the' printing Of this error

~** message and proceed's as usual with execution,* A second peculiarity in theHoneywell version is the use of scratch random disc for storage of certainquantties. Among the reasons for this are the facts that the number of lot&-Stone required depends on the input and that the Honeywell execution cost isproportional to both central meory and times while the latter is hardly affect-ed by disc access. The library subroutine RANSZZ(Ifn,n) defines the local fileIfn as a random disc having a voids per records The read and write functionsare IUD(lfun')list and WRIT3(ltu'n)liat,, where u is the location of the record.

W A caveat Is in order regarding any changes to FIT which involve either the*'numiber of 'inputablia profile points or the number LAYUS., If any of the

possible input points is exactly the same aso one of the values ofth inde-pendant variable for either mesh point or layer profiles, special programingtechniques sust be used to avoid an indefinite condition.

G." J.13

The third and final peculiarity is the use of overlays to minimize core storagerequirements*. The library routine LLINK(Iname') loads all routines denoted bycertain control cards as belonging to "name" into an area where previous callsmay have 'loaded other routines. (The program has been run with LINKA consist-ing of subroutines WFP, HANK, BIN, PROP, BCO, and STORF and LINKB containingPHINT, FFSUM, and AVGS.) In addition there is one labeled common block whichis common only to the two overlaid areas, so it is included in the main ptogramto avoid its loss during overlaying. This card may be removed from MAIN 'ifoverlays are not used. In general only those cards having an identifier incolumns 73 to 80 need to be removed for the CDC computers. No other' cards haveanything in these columns.

Certain cards must be added to make the program run on the CDC machines.The Honeywell computer does nop take a PROGRAM card. Also, a common block' mustbe defined to replace the variable stored on disc. The following is a suggestedprogram:

PROGRAM WAKE (INPUT,OUTPUT, TAPE5 INPUT, TAPE6=OUTPUT)

COMMON/RGHJN/EE (x)

CALL MAIN

STOP

END

The quantity x must be specified according to the inputs for the case being run,so this approach permits one to control the amount of central memory requiredwithoutc recompiling the entire program. The value of x must be at least12*N,(LAYERS+I), where N is defined by the fourth comment card of MAIN. Thepresent main program should be headed by the card SUBROUTINE MAIN; and the cardCOMMON/RGHJN/EE(l) should be placed in subroutines BCO, STORF, and FFSUM.Also, the appropriate do-loop, transferring the array E into the array BE (orvice versa) replaces the statements writing (or reading) E on disc. For example,

L - 12*NFILE - 11

DO 300 1-1,12

EE(L) - E(I)

300 L - L + I

could be used in both BCO and STORF in place of the card identified by "DMSC",in columns 73-76 although the index I need run only to 8 in BCO. If the middlestatement is reversed, the same loop could be used in FFSUM. It is estimatedthat approximately 100000 octal central memory locations will be needed for thisprogram in its present form on the CDC machines, not counting the space neededfor the EE array. The latter could be anywhere from 3454 to 114400 octal as afunction of the inputs, provided that 50 layers and up to 64 modes are used.

* The program requires 30K memory in the Honeywell 6060 when disc and overlays

are used, where one K is defined as 1024 decimal locations.

14

Although it is probably not necessary to do so, one may want to change thevalues of certain threshold quantities which are ured to guard against errorslike exponent overflow or exponentials with outsiDA argument, in view of thefact that 1010 is roughly the limit for the Honeywell 6060. Such quantities areTHRESH in MAIN and subroutiros HANK, AVGS, FFSUM and ZINT; EPSO, STP and TEST insubroutine WFP; DBT in'subroutines FMSUM and ZINT; and EXA in subroutine AVGS.

B. Description of the Program

In this subsection we give detailed accounts of the mathematical and logi-cal techniques used. The listings in Appendix A may be followed alone withthese accounts.

1. Main Program

The principal function of this routine is to direct the calculations bycalling the major subroutines, WFP, which calculates the em fields in the mean

* wake, PHINT and RINT, which perform the integrations over p and 0 in (21), andZINT, which does the convolution (22). Another function, which occupies a largeportion of the executable statements, is to generate the radial profiles ofcollision frequency, electron density, and velocity on the standard grid oflayers and boundaries between layers. The logic is very simple. There are twoprincipal nested loops. The outer one is an implicit loop over input cases,between statement number l and the statement immediately before the STOP state-ment. The inner loop iterates over all axial stations. Let us now go throughthe listing explaining each part in order.

After reading and writing the title and input data, certain constant quan-tities are calculated. CAY represents k, the free-space wavenumber (per meter).

WO and VO rapresent the absolute values of the sine and cosine, respectively, ofthe aspect angle a. CS, SB, SC, and SMC represent certain combinations of thesine and cosine of the polarization angle, which are needed in subroutine AVGS.RMC is the critical electron density Nc (par cubic centimeter).

In the loop over the axial stations, EMNCO is the axial mean electron den-sity divided by critical, Ne/Nc, RW is the radLuc of the wake r , CAYR is krw,and both VCH and VCOM are the axial collision frequency ratio v.

Shkarofsky 1 2 has defined a generalized spectrum function for isotropic tur-bulence which, when defined by (20), has the following equations:

() (21r ''r(ri/ro)'K , (U) (23)s(k) - (23) .UýK (ri/ro)

U - (ri/ro)/ I + (kro) 2' (24)

In these equations ri and r are the inner and outer scales of. turbulence, re-spectively, KL(x) is the mo ified Bessel function of the second kind of order y,and the parameters y and * are interrelated with the parameter w, which definesthe shape of the spectrum function.

12. I.P. Shkarofsky, "Generalized Turbulence Space - Correlation and Wave-Number Spectrum - Function Pairs", Can. J. Phys., Vol. 46, No. 19, pp.2133-2153, 1 October 1968.

15

*' -.. ,. ,*.I,

Y - u-1)/2 (25)

-+ 2)/i (26)Now in order to obviate both the need for calculating modified Bessel functions

and the evaluation of an indeterminate form when the inner scale vanishes, wehave used the following equation in place of (23):

sti r I rr Y•:•,S (k) -]e [ (27)s E[1 + (kro)2J expi(kri/•)']

"A comparison betwoen (23) and (27) is shown in Figure 2.

The program has the spectrum function (27) built in for six values of p.The input quantity MUTL and its representative MUT in the loop over axialstations are defined as 3(p - 1). Thus one can specify P from 4/3 to 3 insteps of 1/3. The Kolmogorov Ppectrum function, which is obtained by default,is represented by p - 5/3. The variable GAMR and its representative GAR are thequantity q(p)/r(y). The meanings of the other FORTRAN names for the spectrumfunction parameters are obvious.

One can sea from the listing that the five sets of profile functions areeach generated in the same manner. An integer, for example LC for collisionfrequency, keeps count of the location in the input array where the data for theprofile begin, If the initial data point is zero the process is skipped, leav-ing the profiles as specified at the previous axial station. Most of the pro-files are generated both for mesh points (IND>O) and for layers (IND<O), theneed for which will be apparent below.

At statement number 6 the variables WS and EMNC represent the values ofmin2 a and Ne/Nc to be used by subroutine WFP. The reason for these new variablesis that subroutine PROFS changes their values and also the value of VS, whichis cos 2a. TZCS is used by subroutine AVGS.

Statement numbero 8 through 70 represent output of the parameters for thecurrent axial station including all profile functions. After the wake fieldpenetration calculation, MINL (and also L) is the number of the innermost layerreached, all em fields in layers within it being set equal to zero, AN is(Ne/(Ncsin2a)]', and BN is the denominator of (27) for S(2k), where k is the free-space wavenumber. Thus BN represents the reciprocal of the non-constant partsof the backscattering spectrum function in the Born approximation.

After integration over ý and p (the comment, "integrate over 2RDR", refersto the latter integral), the quantities, SO, S1, S12, and S2, represent thenormalized moments of the scattering per unit length for each of five polariza-tion combinations. There are two kinds of normalization, which are removed bythe "do 80". A represents the Born approximation for a collisionless uniformwake of radius r in which the electron density fluctuation equals the mean elec-tron density. AZso, each normalized spectrum moment, when calculated by (21),

uses the appropriate electron velocity profile, all of which are normalized bythe axial wake velocity/body velocity, VVBZ. The absolute first moment, as seenby the radar, requires multiplication by the cosine of the aspect angle as wellas by this normalizing velocIty. The second moment comes from cos 2 a times the"first squared" moment plus the true second moment of the fluctuations. The

16

iI

0i

Shkarof sky

A Equation (27)

I I"

a I

-1 0 1 2 3

Log 1 0 (kro)

FIGURE 2. COMPARISON OF SHKAROFSKY'S SPECTRUM FUNCTION WITHEQUATION (27). THE SPECTRUM FUNCTION PARAMETERSARE AS FOLLOWS: • - 5/2, ri " 0ro. !,

17

-__ __ __ __ ___ __ . . . . . .. ~~*,A~,A... ... .I .. -.. . , , ~~ .6c". *Al,.., . .....

quantity S12, representing the "'irf't squared" moment, is the result of inte-grating over the square of the mean velocity profile. In the "do 90" loop, ,;hemoments are convarted into velocity mean and spread, where the square of thelatter is defined as the difference between the mean square velocity (secondmoment divided oy the zeroth moment) and the square of the mean.

2. Subroutines

In this subsection we follow an outline in which the nomenclature is exact-ly the same as used in the program listing; i.e., the code letter for each sub-routine denotes the paragraph heading for its description here.

Sa) Subroutine FIT(N,M,INDF,Y) has the purpose of fitting the inputdata to a profile on a grid of either layers or layer boundaries.N is the number of points input, M is the location in the inputarray where the last of these N points is located (M - N + 1 isthe location of the first data point), IND is a control variable,F is the array of input data, and Y is the array of output data.The array F must be dimensioned in the calling program at leastas large as the value of M. IND controls two aspects of theroutine. If IND>0 the profile is calculated for points locatedin the layers; otherwise the points are at mesh points (boundar-ies between layers). IfIINDI<2 it is assumed that the inputdata are the values of a smooth function at N equally spacedpoints starting at the axis and ending at the outer edge of thewake. Otherwise the data represent a step function defined bythe height of the first step (at the axis), the relative radiusof the first step, the height of the next step, the relativeradius of the next step, etc. Except, of course, for Y, thevalues of all variables in the calling sequence are returnedunaltered,

A second function of this routine is to define the array ROR,which represents the relative radius, p/rw, of each layer bound-ary. Note that the layers are not uniform in thickness. Infact they are defined so as to place a definite ceiling qn therelative thickness of any given layer. Thus, except of coursefor the axial layer, the relative thickness, (pi+l-Pi)/Pt isnot greater than 0.28519903. If the value of LAYERS werechanged, the data for ROR would have to be changed. Let crepresent the limit on the relative thickness. (See the die-cussion of subroutine PROP, where it is shown that c should notexceed 0.3.) The value of c is minimized if there is an, in-teger j for which

(c + l)J - (c + 1)/c (28)

Then

(c + l)i-1b, O<i<j + 2

Pi " i1_ + b , J<i<n (29)

rw ,i-ii

18

' '.. . . .E. . .. ., . ,,..' u,. .. . ,> >, ,

where

b rwI(n - J + 1/c) (30)

and where n is the total number of layers. Of course j is thenumber of layers having uniform relative thickness and the (J+l)th isthe first layer to have the uniform absolute thickness, b, possessedby the remaining layers. In the present program, j - 6 and b -0.021049829rw. Thus relatively few layers have nonuniform thicknessand of the fifty layers none has a thickness greater than about 2.1%of the wake radius.

Figure 3 gives a block diagram of this routine. As indicated by thecomuent card, the location of the independent variable for profilesbelonging to points in the layers is defined to be at the midpoint ofp2 for each of the layer boundaries (rms value of the radius of thelayer). This convention more or less guarantees conservation of linedensity in the layered cylinder approximation of a smooth profile.Note also that, for profiles on mesh points, the last data point isautomatically the value of the last profile point.

As the comment card indicates, N-point Lagrange interpolation isused to give the output values of Y when the input represents asmooth function of the independent variable X. This approach wasadopted only after abanioning one which consisted of a least squares"fit to a polynomial oe delree (N - 1). The least squares approachencountered numerical di. iculties, resulting in rapid divergence ofthe result, when more than seven points were used to define the in-put function. This difficulty occurred even when the input functionwas actually an Nth degree polynomial. The Lagrange interpolationtechniqua, on the other hand, is numerically quite stable. Forexample when the input represents a typical Gaussian at elevenpoints, the output is no more than one unit off in the third decimalplace at any of the values of the array ROR.

The theory of the Lagrange interpolation for uniformly spaced datais right out of Abramowitz and Stegunla. A bit of manipulation re-duces those formulas to the following:

NZY- £ FjP(X)/[GjAj(X)] (31)J- 1

Aj(X) -- (-)JEj(X) (32)

Ej(X) - J - 1 - (N- l)X (33)

NP(X) - E K(X) (34)

K-

13. M. Abramowitz and L. A. Stegun, editors, "Handbook of MathematicalFunctions", U.S. National Bureau of Staudards, Washington, D.C.,

P. .1964 (pags 878).

19

"," i!ýf 4v61JWl

ENTER

NOYE

INDEPENDENTINENDT

FIGUR 3SILYIFLOWDAGRANG FO URUINE F>T.

.................................

t 4

The quantity G3 is a symmetric function of J about the midpoint ofSl,... , where

Gj - (J - 1)1(N - J)l , J < 1 + N/2. (35)

Except for P(X) the FORTRAN names follow the above notation. Thearray G takes advantage of the symmetry of Gi and stores the datafrom (35) in triangular form. The numbers 1,2,2,3,J,4,4,5,5,6 count,uf the locations of Gj for N - 2,...,li, respectively. The vari-able L is the location immediately prior to the first one neededin a given calculation. The "do 10" loop covers all values of theindependent variable X defined above. The symmetry of G is utilizedin breaking up the sum into the sequential "do 7" and "do 9" loops.The variable B is used to accumulate the dependent variable Ypwhich is not permitted to be negative since it represents a profilefunction for a non-negative physical quantity. The routine couldin principle give division by zero at the two locations where D isset equal to F divided by the quantity G times A. This will happenif any of the values of X is precisely equal to the value corre-sponding to one of the uniformly spaced input data. This cannothappen in the present program because the data for ROR and the rmsvalues of adjacent pairs are not rational fractions for numbers upto ten. However, it is not likely to happen even if LAYERS werechanged as long as equations (28) to (30) are used to define thevalues for ROR and c is not allowed to exceed 0.3. Obviously, ofcourse, &ven if it happens, one would simply code the routine totake the input data value (Y a F) at such singular points.

The step function routine simply searches for X values which fiton a given step (the if statement in the do-loop), setting Yequal to the height of the step for all those that do. The stepscan go up or down or both.

b) Subroutine PROFS(IREKRRT,MUT,PSI) has the primary purpose ofcomputing the effects of scattering by turbulence on the propa-gation of the coherent wave in the mean wake. Other functions ofthis routine, whtch are included because of their logical rela-tion to either this function or the point in MAIN where PROFS iscalled, are the coefficient of depolarization by second scattering,the adjustment of the aspect angle and electron density to makethe critical points lie at layer boundaries, and the profile ofthe reciprocal of the dielectric constant at layer boundaries.

The variables IRE, MUT and PSI, in the calling sequence, areinputs, which are unaltered by the routine. IRE - I implies thatthe profile of mean electron density is a smooth function cudthat readjustment of the axial electron density and the aspectangle is necessary for reasons given below. MUT and PSI pertainto the turbulence spectrum function in precisely the same wayas in MAIN. KRIT is returned as zero if IRE 0 1 or if there isno point at which the real part of the adjisted complex dielectricconstant vaninhes. Otherwise it is the number of the layerboundary at which this critical point occurs, The logic of thisroutine is very simple. It starts with a do loop over all layers,in which the scattering attenuation and depolarization are

21

I I I I 'Ill. .l l l l HI , i -l 'l I

calculated and the effect of scattering attenuation on the meanplasma is accounted for. Then the adjustment of aspect angleand electron density are made, if required, in two do loops overthe layers, Finally there is a do loop generating the profileof the reciprocal of the dielectric constant at layer boundariesand placing the adjusted values of plasma properties in theappropriate arrays for use by subroutine WFP. The five taskscovered by the last three sentences are each denoted by coimnentcards.

The coefficients of attenuation by scattering and of depolari-sation by second Born are calculated from, for example, thetheory given by Shkarofsky 14 . Thus

re.Ne, 2 cos~ma s (1 + x')S(keV*2(l-x))dx (36)

4[1 + (v/w)0] -1

L .xI (lx 2 ) S(2kex)dx (37)1 + (V/W) 2 0

where a. is the scattering attenuation and B is the depolarizationrate.* In the argument of the spectrum function S, the symbol ke

represents the effective wavenumber in the mean medium, which could beevaluated as half the gradient of the argument of Eo°Go. But thiswould lead to an iterated problem, so we let ke equal the free-spacewavenumber k times the real part of the square root of the complexdielectric constant. In (36) the angle Om represents a cutoff anglefor omitting the effect of forward scattering from the extinctioncoefficient 01.. The idea for, and the form of, $m are taken fromduthart and Graf 15 . Thus we use the following equation:

cos8 - 1 - [I + (k + ke)(rw - p)/(27rina)]" (38)

where ke is k times the real part of the square root of the complexdielectric constant at the local value of the radius p in the mean wake.

Of course the spectrum function (27) is used in the above integrals,except that the exponential cutoff involving the inner scale ri isreplaced by a sharp cutoff. The integrals reduce to the following:

XUs/k w D/ (167r)f (l - 2x + 2x2) (1 + ax) dx (39)

B= kD��x 2 (l-x)2 (I + ax)-*dx (40)

0

14. I.P. Shkarofsky, "Modified Born Backscattering from Turbulent Plasmas.Attenuation Leading to Saturation and Cross-Polarization", Radio Science,Vol. 6, ips. 8, 9, pp. 819-831, August-September 1971.

15. H. Guthart and K. Graf, "Attenuation Due to Scatter in Random Media",Physics of Fluids, Vol. 15, No.7, pp. 1292-1299, July 1972.

"* Vote that B is the two way depolarization; i.e., it is twice Shkarofsky's B.

22

. . ............. .............. .... . ..... ., ....

Fr (i, 21ro) 3Ne 2

D -(41)r(y)NC[1 + (\l)/W]2Y - (1 - cOS~m)/2 a - (2kero)2 (42)

The upper limit X is equal to either unity or the quantity, [v/(2keri)]2whichever is smaller, the latter quantity representing the cutoff atthe inner scale. The integrals in (39) and (40) are evaluated, if the

quantity a is not too small, by repeated integration by parts to thepoint where the polynomial in x is reduced to a constant by the re-peated differentiations. These formulas are represented in theFORTRAN statements prior to statement number 2, and they differ whenSis integral (MUT - 3), as at statement number 1. These equationslead to indefinite conditions when a is small, so in that case weintegrate by replacing the binomial in ax by (1 - aix), which is donebetween statements 2 and 3. In all statements FJ denotes the integralin (39) and GJ denotes that in (40). The quantity D at statement

number 3 is D in the above formulas. AK represents as/k and 3OK,8/(kainc),

The attenuation owing to scattering, a., is a concept from geometric

optics theories. As such it is not easy to fit to the present theory,which is a full wave theory. In optical theories one integrates a.along ray paths to get the total scattering attenuation to a givenpoint, which is in close analogy with the phenomenology of collisionalattenuation. However in this theory we get the local field at a givenpoint by rigorous expansions of solutions of Maxwell's equations. Inorder to use a representation of scattering attenuation which is con- 1sistent with this theory, we adopt the approach of adding to the mean d

plasma properties in such a way thaz scattering attenuation is auto- '

matically accounted for. Ideally it would be preferable simply toadd a quantity to the collision frequency since collisions causeattenuation, but it qan be shown that this is not possible to do ina consistent way. Therefore we add quantities to both the electrondensity and collision frequency. These quantities must give anattenuation coefficient which is identical with os and they must van-ish when as vanishes, There are many different sets of equationswhich have these properties, but we use the following:

G- l/(l + Va5/) (43)

D - 1+(c/k)2 - (44)

v/w - 2G0t/(kD) (45)

Ne/Nc - [1 + (v/w)2]D (46)

These equations are used, down to statement number 6, to give EMS andCPS, which represent normalized profile function arrays for theseadded electrons and collisions. The arrays EM and CFL are temporarilyused for the total profile functions at mesh points (layer boundaries).

VIn previous studies it was found that wave propagation in a layered

23

plasma can often be seriously affected by the method in which thelayers are chosen to represent a given smooth distribution. Thiseffect is related to the fact that the smooth distribution hasnritical density at only an isolated point, while the layered approx-Imation, if selected at random, might have critical density in anentire layer. It was found that all spurious effects of layeringcould be avoided if the two critical density points are made to lieat layer boundaries. These two critical densities are defined asthat for which the real part of the dielectric constant K vanishes(intrinsically critical density) and that for which the real partof K is equal to cos 2o (critical density for refraction). The "do

10" loop adjusts DfNC to meet the first criterion and the "do 15"loop adjusts WS (defined as sinZa) to satisfy the second. Addition-al conditions satisfied by these loops are that the electron densitybe a decreasing function of radius p at each critical point and thatthe outer edge be underdense.

It will be seen below that the radial component of the electric fieldis related to the tangential magnetic field components and to thereciprocal of the complex dielectric constant. The arrays ROK andAOK are the real and imaginary parts of 1/K at mesh points.

At the and of the routine the arrays EK and CFL are given theirpermanent meaning, for use in subroutine WFP, as the profile func-tions for electron density and collision frequency in the layers.24O and CFO are the corresponding quantities without enhancementowing to scattering attenuation.

a) Subroutine WFP(JMIN,NHP) calculates the em fields in the mean wakeplasma*. In the calling sequence, JMIN is the number of the inner-most layer boundary reached, all fields within this point being assumedto vanish. NHP is the total number of angular modes used to repre-sent the fields. Both variables are, of course, outputs of theroutine.

The array YIP is defined as the thickness of a given layer dividedby its inner radius, starting, of course, at the second layer (i.e.,excluding the axial layer). Its values depend on the values of RORin subroutine FIT. The first executable statements define the arrayRK as kp for the outer boundaries of all layers and set up a fewconstants for later use.

The problem of em wave propagation in the cylindrical mean wake plasmais solved by separation of variables and superposition. Thus the vec- ptor electric field E at an arbitrary point (p,o,z) is written as follows:

E - exp(ikscosa) I e(n,p)einO (47)

"The quantity e(n,p) is the partial electric field vector for the nthorder mode and is a spatial function of only thn radial coordinate."The magnetic field H has the same representation except that h replaces

* The name of this routine derives from the name Wake Field Penetration.

24

e. In free space the functions e and h involve Bessel functions ofargument kpsina and order n. The incident wave includes only theordinary Bessel functions and the scattered wave has the Bessel func-tion of the second kind as well The asymptotic behavior of thesefunctions at large orders is such that the function of the second kindincreases whilk the ordinary Bessel function decreases. If the modalseries (47) is to converge at all,its convergence must depend on theratio of these two functions. Therefore, in consonance with the36-bit computer word,(approximately eight place accuracy) we truncatethe series when the ratio of the two functions of argument kr sinais greater that 109. In the section, "determine the number oT modesto use", A represents this argument and NHP is the number of modeswhich satisfies this criterion; The equation for NHP is a good fitto the above criterion for values of the argument between about 0.5and 35, and it is conservative at all other values. (Note that nega-tive orders are %tot distinguished from positive ones in this dis-cussion because of the synmmetry properties which obtain. Thus NHP isone greater than the largest value of n used in (47).)

The complex dielectric constant K and the propagation constant ydetermine the solution for em wave propagation. Their forms, whichare used in the "do 100" loop, are as follows:

K a 1 - Ne/[N (l + iv/w) (48)

y - kA - cosv (49)

In these equations N and v are the electron density and collisionfrequency in a given layer. The array DIEL represents K and the arrayGAM, y/k.

The logic of this subroutine is very simple. It consists of twonested loops over the angular modes and the cylindrical layers. Theseparation of variables (47) allows the solutions • and h to be ob-tained as functions of p for each n independently, so we iterate overmodes (NP - n + 1) in the outer loop ("do 600"). In this loopEYEN~in, SGN=(-)n, and ENEn.

The solutions for e and h for a given value of n come from substitut-ing both (47) and its magnetic analog into Maxwell's equations, whichfor exp(-iwt) time dependence are

WVE ikH ; VxH - ikKE (50)

where we have assumed a system of units in which the characteristicimpedance of free space is unity, for convenience*. These equationsgive six linear first order ordinary differential equations in eachof which are three of the six cylindrical coordinates components ofe and h. These six equations reduce to four by elimination of theradial components, but each of these four equations contains in gene-ral three of the four tangential components. The easiest way to

•The system of units is immaterial since all results will eventually be

expressible in nondimensional terms.

25

solve these is to suppose that all fields are the sums of two kinds,TE and TM, where T denotes transverse to the axis and E and H referto electric and magnetic. Thus, for TE, when ez - 0 is substitutedone gets, by elimination of the ' - components, a second order differ-ential equation in hz. A similar equation for ez results fromassuming hz = 0. These two equations differ only in terms of theeffect of gradients of K, so in our layered approximation they areidentically the same equation, which is Bessel's equation for ordern and argument •, where

EMyp (1

The four simultaneous differential equations can also be solved forthe *- components as functions of the z - components. The resultsare expressible in terms of the following variables:

p-:ik/y (52)

q-nkcosc%/ (yf,) (53)

Thus

s. - - qe3 - phi' (54)

ho - - qhz + Kpe 2 ' (55)

where prime denotes differentiation with respect to •.

Each tangential field component must be continuous across boundariesbetween layers. A matrix representation is a convenient way to keepsatisfying this condition from layer to layer. Thus we arbitrarily

adopt the following tangential fields vector F:

(56),hz

Now, for the time being, we restrict the discussion to the behaviorof F within any given layer. Since the z - components are eachindependent linear combinations of two independent solutions ofBessel's equation, let F w UA, where A is a column vector of undeter-mined coefficients and the square matrix U comes from the equationsfor each component of F. Let

u V 0 0

Kpu' Kpv' -qu -qv (57)

0 O U V A

qu qv pu' pv' :

26

where u and v are the tw6 solutions of Bessel'. equation. Let sub-script 0 denote the value of a-quantity at the inner boundary of agiren layer, unuubestripted variables denoting either constants orthe values of the .rrbles at the outer edge of the layer. There;ant exist a square watrix P such that F a PF 0 , which implies thatUK - PUDA since A is csntstant throughout the layer. Thus U w PU0 or

M'.- Now if we require of the two solutions that v0 U- .0,then U0 has only two nonuser off-diagonal elements and is easy toinvert. The result isI 0

u/uo v/ (Kpvo' ) qoV/(KpVo' ) 0,:.

.[pu' /uo+qqov/(pvo' ) v'/vo' (qov,/vo'-qu/uo) -qv/(pvo' 5)P0 (58)-qov/(pvo') 0 u/u 0 v/(Pvo')

* "(qu/uo-qov'/vo') qv/(Kpvo 1 ) .qqov/(Kpvo')+pul/uO] V'/vO''

Three of the fourteen nonzero elements of this matrix arirepeated,so the arrays PP, PQ, .. ,p PZ represent the eleven unique elements intheir order of appearance reading across rows. The variables P, Q, R9,and .0, which are calculated in subroutine PROP, represent u/uOv/vol, uO/u, and v'/vo*, respectively.

Now the boundary condition between layers ip automatically satisfiedif 7 is matched, Therefore 7 at the outer edge of a given layer isidentical with F at the inner edge of the next layer. Thus, since Prepresents the propagation of V through a layer, cumulative matrixmultiplication of the P matrices for each layer gives the propagationof V through all layers. This cumulative product has the TORTRAM nameqq(ij) for the element of the ith row and jth column. Initiallyit is the identity matrix (the "do 200" loop) and it is updated in '.he"do 300" loop. One will note that QQ is replaced by QQ times P as thelayers are iterated from the outside in. Thus, since P representspropagation of F from the inside to the outside of the layer, QQ isthe propagation of F from the currently reached inner boundary to theoutermost boundary.

The overall propagation matrix (let us call it Q instead of qq inthis discussion) tends to diverge in cases in which there is stronscutoff of the em fields in the central regions of the layered cylinder.Owing to the nature of the Wronekian, the inverse representation Q-'1which gives propagation inward from the outside edge, also diverges.Thus, no matter how one calculates propagation, there can be a pointat which exponential overflow will occur, especially in case of alimited computer word. In the computer model we insert a perfectlyconducting boundary just before that point is reached. Three criteriaare used. The first is the if statement near the beginning of the"do 500" loop, which guards against the fact that the layer propaga-tion matrix P tends to increase exponentially with the argument •.The second is the two if statements in the "do 400" loop, which guardalainst overflow of JQ12. The third criterion, which is exercisedjust after the latter loop, is based on the supposition that t (calledEPSO) represents the smallest field intensity of any practical

,27

vignificance. T.cn the greatest value of Q of any interest is suchthat IQiac bas about the magnitude of the field intensity at theouter odge, which is approximately proportional to the square ofthe ordinary Bassel function of argument kr sina. The Bessel func-tion is in the neighborhood of unity for small orders and at largeorders it drops off according to well known asymptotic forms.Combining these limits into a single expression gives the criterion

IQI•2 (l + VF[2n/(ekrwsina)ln.Cl (59)

where e is the base of natural logarithms. The magnitude squared ofQ is estimated from its average value for all sixteen elements.

The section of this routine which involves boundary tonditions isclosely related to subroutines BIN and BCO. The theory is detailedunder those routines rather than here because this routine merelycalculates certain parameters for, or from, those routines.

d) Subrountine HANK(NP,X,HNK,HNKP) calculates the Hankel functions ofthe first kind and their derivatives for use in the outer boundarycondition, NP is the number of orders to be calculated (n a o toNP - 1) and X is the argument, a real variable. The complex arraysHNK and HNKP contain the output Hankel functions and their deriva-tives and must be dimensioned at least NP by the calling program.The method used is downward recurrence on the Bessel function,normalization based on the value unity for the generating function,the Neumann expansion of Y0 (x) in terms of Ju(x) for all even valuesof n, the use of the Wronskian to get Y1 (x), and finally upward re-currence on the function of the second kind, Yn(x), Refer tochapter 9 of Abramowitz and Stegun (op. cit., rif. 13) for theequations, which are readily located in the listing with the aid ofthe comnent cards.

The form of the convergence criterion, which identifies the order atwhich to begin the downward recurrence, is based on the behavior ofBessal functions, for which it can be shown from tabulated valuesand asymptotic forms that J (x)<1O-Y if n is approximately equal toor greater than [y - 4.5 + 31x1(IxI + 8)]. The value 12 fp- y wasderived from experimental (checkout) runs using the 36-bit computerword.

e) Subroutine BIN(X,NP,Bl) calculates Jn'(X)/Jn(X) for complex X and fororders n - 0 to NP - 1. The results are placed in the complex arrayBl, which must be dimensioned at least NP in the calling routine.The method used is downward recurrence; i.e.,

Jn'(X)/Jn(X) - n/X - an+l(X)/Cn(X) (60)

Cn(X) - 2(n + l)Cn+l(X)/X - Cn+2 (X) (61)

where the Cn 's are started with arbitrary values at the order givenby the convergence criterion, which was derived as in oubroutineHANK. Of course normalizatiou using a generating function is notrequired. Also, there is no danger of division by zero as long asthe imaginary part of X does not vanish (collision frequency not

28

identically zero in the axial layer), because all zeros of Jn(X)are' real.

The raison d'etre of this routine is the inner boundary condition.In case, in subroutine WFP, the axial layer is reached in thepropagation calculation, the boundary condition at its surface mustaccount for the presence of only the Bessel function of the firstkind. We state this boundary condition in the following form interms of the two unknown constants A1 and A2 :

| - q AaPi NO (62-)

LA2312' + qA1E11 j

B'1 " ikB1 /Y (63)

'11 - 1/(BIIK) (64)

i1 in, - 0in (65)

where the dielectric constant K and the quantities y, • and q, whichare defined by (49), (51) and (53), are for the axial layer at itsouter edge. Of course Pi represents the tangential fields vector Iat this boundary. This normalization is'uged, rather than one inwhich Jn and Jn' appear separately in all terms of (62), because B,is easier to calculate and because this form is compatible with theperfectly conducting boundary condition, in which B1, and B,, arezero to account for the vanishing of the tangential electric field.,

f) Subroutine PROP(ENXZR,XZI,Y,C,DCP,DP) calculates the four basicquantities which comprise the no'.6tro elements of the propagationmatrix P for a given layer. Tne first four variables in the callingsequence represent input5 and the last four, outputs. EN is thefloating point value oi the mode order n. XZR and XZI are the realand imaginary parts of the quantity go, which represents yp at theinner boundary of the layer. Y is the relative layer thickness. forthe purposes of the derivations, let C, D, 010. and DI representC, D, CP, and DP, which are called P, Q, R, and S in subroutine WFP0

In the limit of zero layer thickness the propagation matrix mustapproach the identity matrix. In this limit the quantities C and D',which are on the diagonal, must So to unity and the quantities D andC' must vanish. Supposing that this limit is approached continuously,we use series expansions in the layer thickness to calculate thesequantities. The limiting forms serve to determine the forms ofthese series up to the second order. Thus let

C- I+ ýOYIE, aj(6

29

00

D - oy[1 + E. dj] (67)

ooo

C'-•E (i + 1)c1 (68)

D' -1 + (j + 1)dj (69)

In these equations c and d are each equal to a complex number, whichdepends on J, times The j thpower of the relative layer thickness Y.The recurrence relation for these terms is derived by substitution ofeither the C or the D series in Bessel's equation, for which the inde-pendent variable is • - o(i + Y), and requiring that the result applyindependently for each power of Y. Thus

1(j + l)aj + (2j - l)jYaj. 1 + [(j - 1)' - n 2 ]yaj_2 (70)

+ EY2 (aj. 2 + 2Yaj_ 3 + Y 2 a•Ja 4 ) - 0

where aj represents either cj or d1. This recurrence relation appliesliberally as long as the following initial conditions are used:

a -0 , J< - 1 (71)

co d. 1 - 0 (72)

C.-1 1/(0Y) (73)

do -1 (74)

In the program the approximate convergence criterion represents a fitto the minimum number of terms needed in the infinite series toguarantee convergence in all of more than 500 runs of this routinewhich were made during checkout. These sums also have an internalconvergence criterion following the "do 8" loop. The array AJ(I,J,K)represents cj for I a 1 and dj for I - 2. The real and imaginaryparts are for J u 1 or 2, respectively. The five locations in thethird subscript represent the values needed for recurrence. The valuesof these locations are denoted L, LA, LB, LC, and LD and are updatedin each iteration, The first three terms of (70) have real coeffi..cients and these are represented temporarily in terms of the array P.The coefficient of the complex variable • is temporarily placed inthe array Q. After updating the a1 the sums used in C and D areaccumulated in U and those used in C' and D', in W, but not before Pand Q are temporarily used for the old values of U and W to facili-tate convergence testing. At the end the outputs are calculated asin (66) to (69),

P The present form of this routine is believed to be the optimum for amulti-layered medium. These four matrix elements are, of course,identical with the cross products of the first and second kinds ofBessel functions for the complex arguments for either boundary of thelayer. But the method of calculation based on this fact would re-quire much more memory and would have great inefficiency in thin

30

Allv--~- - - - -- -

* !layers. The recurrence relations for these cross products were tried,but no way was found to eliminate the fatal round off error which wasaccumulated. A detailed study of series expansions whose leadingterms are designed to fit the various asymptotic forms of Bessel func-tions was done also, however all of these failed in some regime wherethe above scheme worked. With the constraint that Y be not greaterthan 0.3 this algorithm works in the Honeywell 6060 for values of thegouter-edge argument bea(s + Y)j as large as 88. This is exceptionallygood performance because the result tends to be exponential in this

* •argument and exp(88) is at the limit of ýhe computer word, Notethat the implicit series in Y, in which YJ is part of c and dj, islargely responsible for this good performance. -When ex licit powerseries were used we could not get past about 50, instead of 88, with-

* •out exponent overflow.

g) Subroutine BCO(M) applies the boundary conditions co get the localfields at all mesh points from layer M out to the outer edge.

The outer boundary condition is the sum of the tangential fields forthe incident and scattered waves. Standard Fourier-Bessel expansionsof the vector incident fields given by (16), (17) and (18) produceexpressions for the tangential components of e in the incident wave,The incident waves are so defined that the magnetic field for trans-verse.polarization is identical with the electric field for parallelwhile the magnetic field for parallel is the negative of the electricfield for transverse. The result is

'F1 0

F2 -b FPi" or (75)

0 F j

parallel transverse

F1 = in Jn(krw sin a) sin a (76)

F 2 - in + I Jn'(krw %in a,) (77)

where Fi denotes the contribution of the incident wave to the tangen-tial fields vector, defined by (56), at the outer edge and b is q,defined by (53), evaluated in free space at the outer edge.

b - n cos at(krw sin2a) (78)

The axial components of the scattered waves must be proportional tothe Hankel function of the first kind, so we let the scattered tangen-tial fields vector FG have the following form:

31 31

-ggi- g -w g.

As

Fs " (79)As2

A.82 B2 + b As,

B2 ij H(krw sin a)/[ Hn(krw sin a) sin a] (80)

where A' and AS are unknown scattering coefficients.

The boundary conditions are expressible as the following matrix equa-tion:

Fi + F - Q P, (81)

where one will recall that Fi is the inner boundary condition, givenby (62), and that Q represencs propagation of P from the inner boundarypoint to the outer edge. Because it involves the four unknown co-efficients in Fi and FP, this equation is equivalent to another matrixequation as follows:

1 8 -1 0 Al

S21 S22 b A2 i (82)

S 3 S 0 -1 AG-

[S, S4 2 -b -B2 A' ,

whare

sl- B11 (QjI + q Qj4) + Qj 2 (83) ,

SJ2 QJ3 + B1 2 QJ4 - l QJ2 (84)

It is not necessary to solve this equation for AS and As because A1 andA determine the fields at all points via the propagation matrix P foreach layer, Equation (82) represents the equation SA - Fi, which istransformed by the matrix "

T.[B2 -l -b 01ol5

[b 0 B2 -lJ(5

into the following equation of rank two:

32

Dr l .:

S1)

TS " or (86)SA2 0 FS •

parallel transverse

The quantity F3 is (B2F1 -_ F2), but the Wronskian gives

F3 - - 2in/[r krw Hn(krw sin cx) sin a] (87)

The two systems of equations represented by (86) are readily solvedfor the two alternative sets of values of A, and A2. Note that allretults are proportional to the quantity Fs. The complex exponentialeinf is used in the modal expansion (47) because it represents allfield components simultaneously. However each cylindrical coordinate

* component of each (electric and magnetic) field for each incidentV wave polarization has either even or odd symmetry in n and *. Thusj (47) is actually a summation over positive n involving either 2 cos(n4)

or 2i sin(n$), as well as the term for n - 0. Therefore we takecare of the factor 2 for n>0 by multiplying Fs by two in subroutine

1 'WFP, where this quantity is calculated. (The quantity F1 , which isneeded to give the coherent scattering coefficients, is likewise doubledat the same point.)

Now, to interpret the FORTRANý the variable C7 represents b and Brepresents q. C5 and C6 are temporarily used for S,, and S,, and C1and C2 are the elements in the first column of TS. Then C5 and C6 areS and S and C3 and C4 are the second column of TS. The two setsok simultane..ius equations (86) are then solved and, after the "co 10"loop the arrays FJA(K), FJB(K), FJC(K), and FJD(K) are the four ele-ments, in order, of the tangential fields vector F. The firet set(K - 1) is for a parallel polarization incident wave and the aecond(K - 2) is for transverse. After this the "do 200" loop generates thetangential fields at each layer boundary by iterated matrix multipli-cation by the stored propagation matrix P for each layer.

The final function of this routine is to generate, output, and store(on dibc) the coherent scattering coefficients, starting at statementnumber 210. Note from the definition (56) of F that according to (79)and (75) the scattering coefficients are fully charactexized by the

* z-components of the fields at the outer boundary after F, has beensubtracted from ez for parallel polarization and hz for transverse.

N Now in free space the four z-components each propagate as Hn(kp sin a),so their values at any point may be gotten by multiplying the ratioof this quanticy to its value at the wake edge by their values at the

4i. wake edge. In the far field this ratio has the asymptotic. form

2 exp[i(-r/4 t kp qtin a)]

/T2uikp: sifaf in Hn(krw sin a)

Now see subroutine WFP, where Z is 0.5 1n aina Hn(krw sin a). Alesonoting that the glant range r from the origin of thW wake is equal top/sin a, this implies, for example, that in the far scattered field for

33

...

parallel polarization

sin a exp[i(kr - iT/4)]Z E C3cos n (88)

S2ir ~r_ n-o o

where Q3 is the FORTRAN variable which is output and stored (on disc)at this point in the program. Now, seeing that the scattered wavepropagates in the direction making angle a to the z-axis,

Escattered parallel _e ,•kr - / C o (89)

.'27rkr w

This is a compunent of the coherent far electric field scattered by awave incident in the plane of the axis (as defined above). This com-ponent also lies in the plane of the axis, where the positive directionis as defined for the incident wave. The same component of the scat-tered magnetic field for transverse incident wave polarization hasidentically the same equation with C6 replacing C3. These two fieldcomponents have even symmetry. The other two, having odd symmetry,use i sin no in place of coo no. The axial plane magnetic field forparallel incident wave polarization has C4 in the summation and theaxial plane electric field for transverse polarization has C5. Thefact that half of these results are magnetic fields presents no prob-lemm conceptually since in the far field the electric vector is equalto the croes product Hxf, where f is a unit vector in the direction of"scattering, Thus the transverse (4 - component) electric fieldscattered by a parallel incident field is the sum in C4 and the trans-verse electric field for transverse incident field uses C6. The writestatement, then, gives the coefficients for parallel and transversein order, first for parallel incident field and then for transverse.For reasons of convenience below, the real and imaginary parts of thefield components are stored in odd, even order, where the odd compon-ents have the coefficient multiplied by i so that the Fourier sums areover either sin no or cos no. This order is Etp Eft, Ept, E wherethe first cubscript is the incident wave polarization and the-second isthe scattered. The storage location implies that these are scatteredfields in that one extra "layer" is saved for this (LAYERS + 1).

h) Subroutine STORF(NP,J,ENKR,V) stores in odd, even order the localelectric fields components. All elements of the calling sequence are

- inputs, which are unaltered by the routine. NP is (n + 1), J is thenumber of the layer boundary, ENI(R is n/(kp) for this boundary, and Vis cos a. All components having odd symmetry are multiplied by i inorder that all total fields be straight Fourier sums. The ovder ofstorage is real, imaginary; p, 4, z - component; transverse, parallelincident wave; where the first set of things varies the most rapidlyand the last varies the least rapidly.

The radial electric fields are given by the following equation, which-is one of the six resulting :From substitution. of the modal expansions(47) in Maxwell's equations:

34-" I'

•;..,.

,, 1 W - ....

ep- { hcp cos ct - [n/(kp)] hzI/K (90)

The general storage location denotes both the layer boundary and themode order,

i) Subroutine PHINT(LSMM) directs the analysis ot the 4 - dependence of-- all field quantities for the mean wake. One of the most important

aspects of these calculations is the integration over 4 for equation(21), whence the name PHINT. In the calling sequence LS is the number

S ,! .of the innermost layer boundary reached in the field penetration cal-culation. MM Is the total number of modes used in the expansions ofthe fields,

The first part of this routine sets up the grid of angles to use forrepresenting the variation of all fields with 0. The criterion isthat the number of uniformly spaced angles be a power of two which isat least four times the number of modes used in the field expansions.This criterion derives from the fact that the integral over all anglesSof any mth nrder Fourier sum is precisely represented by the trape-zoidal rule for evenly apaced values of 0 as long as the number ofangles is greater than or equal to m. The product IEo'CG012 representsa Fourier sum of order 4m if E0 and G, are each of order m. In theroutine, IM is the number of angles and M is the power of 2 to whichIt is equal.

The routine calls for the summation of the coherent scattered fieldsbefore going to the loop over layer boundaries, at each of which the$ - dependence is analyzed,

J) Subroutine FFSUM(LYR,MM,N,M,LTH) uses the FFT algorithm to evaluate thevarious Fourier sums which represent the components of the electricfield in the mean wake. LYR is the number of the layer boundary or,if greater than LAYERS, an indication that the coherent scatteredfields are to be evaluated. HM is the number of modes for which theFourier coefficients of the fields have been calculated. N is thenumber of angles, which must be the Mth power of 2, If LTH - 0 theroutine assumes that sin $ and cos 4 for all 0 on the grid of anglesihave not been calculated before, These quantities are placed in thearray WW during execution and LTH - 1 is returned. If LTH - 1 it isassumed that these values already are stored in WW. (These assumptionsare valid as long as the calling routine (PHINT) does not change thevalues of N and M without resetting LTH - 0 and as long as the arrayWW is not changed by the routine (AVOS) with which it has this variablein common.)

This routine is based in greater part on the subroutine FFCPLX byMartin!' In the present notation, the arrays G and H represent both

* input and output. Thus the FFT, between "*bit inversion" and statementnumber 40, represents the following:

16, M.A. Ma-rtin, _"Frequen~cy Analysis of Real Sampled Data by the Fast Fourier, Transform Technique", General Electric Co., Technical Information Series,

Report No. 69SD267, May 1,969.

-. .35

N '$G(IK) + iH(I,K) -Z EG(I,J) + iH(IJ)]ei(J (91)Jul

where * - 2T(K 1)/N and K also runs from 1 through N. The bitinversion part of tho routine, in which the locations of all variablesare changed to the location whose binary digits are the reverse, per-mits the output to occupy the same places as the input. The econo-mized Fourier summation is essentially the Cooley and Tukey17 method.

Note that the imaginary part of the right hand side of (91) is acosine series if the G's are all zero or a sine series if the H's arezero. Thus we apply the fast Fourier sum by inputting all even func-tions to the H array and all odd ones to G; the output is then in Hin all cases. All members of both arrays are initially set to zero

and then the data for the modes calculated by WFP are read in. Thedata are missing, of courue, if the innermost layer boundary reachedby WFP, which is denoted INLAY, is outside of the point currentlybeing considered. The arrays G and H are doubly subscripted to permitthe fast Fourier sums to be evaluated simultaneously for all 8 or 12field components which were stored by BCO or STORF. Alternatingpairs of these comporents are odd and even, so alternating pairs areinput to G and H in the "do 2" loops.

If the point in question is within the wake boundaries, the routineproceeds to the evaluation of the integral over 0 involving 1E0 'G0 I2

Otherwise it outputs the coherent scattering versus 0, which goes from0 to 1800, the other half of the space being symmetrical with thishalf. Referring back to g) Subroutine BCO, the complex parallelpolarization scattering ia in the arrays H(7, J) and H(8, J). Like-wise elements 3 and 4 are for transverse. Linear orthogonal, whichis X(3), is taken as the rms value of Et and Etp. Circular polariza-tion is straightforward also since the transmitted field is composedof one linear polarization plus i multiplied by the other. We thendefine principal circular polarization scattering as the conjugatecotibination; i.e.,

E (Ett + i Ept - i (Etp + i Epp)] (92)X, ~ ~ principal.yp

and orthogonal is therefore

Eorthogonal [Ett + i Ept + i (Etp + i Ep)] (93)

These equations are used to give X(4) and X(5) in the program. Ofsome interest in calculating far field diffraction into the shadow ofthe wake is the coherent forward scattering. The complex coefficientsof E p and Ett for --0 are output, after accounting for e-i¶1/4 in (89),in t~e variables A, B, C, and D.

17. J.W. Cooley and J.W. Tukey, "An Algorithm for the Machine Calculationof Complex Fourier Series", Mathematica of Computation, Vol. 19, pp 297-301, 1965.

36

_____

k) Subroutine AVGS(H,IK,N,LAYR) his the main purpose of performing the

integral in (21) over all angles *. The variable R in the callingsequence represents the IK componenti of the electric field at Nequally spaced values of 0 from 0 to 3600, IK must be equal to 12but N can be as large as 256 in the current version of the program,LAYR is the number of the layer boundary where the field points arelocated, and it is the only one of the variables which may be alteredon return to the calling program. This is the only subroutine inthis program which relies on memory for quantities calculated during

' previous calls of it.

The logic of this routine is not complicated when it is broken downj" •iinto the parts headed by coment cards. The first part evaluates

I. the line integral of 0 given by (40). This line integral is takenalong a straight line parallel to the direction of incidence to thepoint (p, ý) in question. A given path of integration intercepts agiven radius p at two values of 0 which are symetric about 900, sothe range of * covered by "do 6" is 900, The line integrals at agiven value of ý are c,.-•ulated in implicit loops over the radialgrid points. Between aiLatements 2 and 3 the line integral for * < 900

' I. is accumulated in the variable A as p goes through all radial gridpoints from rw down to the one in question, The variable B accumu-lates the line'integral from this point down to but not including thepoint of closest approach, which is represented by the variableaRS.(One will recognize that the equations for A and B have the form ofthe line integral in polar coordinates when the radial coordinate isthe variable of integration.) The contribution up to the point ofclosest approach is added after statement number 5. Statement number6 takes account of the fact that the integral for 0> 900 is that for0< 900 plus twice the integral from that point to the point of closestapproach. Multiplication by CAYR (krw) removes the normalization byrw, and the fact that BOK is 0/(k sin a) implies that the array BDSis IB do for the grid of angles from 0 to 1800,

In the volume integral in (21) we need the effective wavenumber ka,which is the gradient of the phase angle of E0 ,G . This derivative -r

is obtained numerically in terms of the values ol this scalar productat five points in the radial and angular grids. Thus the point inquestion in the integration for (21) must always be retarded by one inboth directions as we iterate p and *, or else we would not have therequired five points about the point in question, Thus LM is thelayer boundary in question and if it is not inside of MlIN, the inner-most point at which the fields were calculated, NH is set equal toone greater than the number of angles needed to cover the region 0 to1800 in the average over angles, In the "do 30" loop over theseangles we usually (unless it is logically *uperfluous) start byevaluating the scalar products E0,#0 for five polarization combinations.The arrays U and V are the real and imaginary parts of these productsand their middle subscript denotes the polarization in the ordervertical, horizontal, linear orthogonal, circular, and circular ortho-gonal. The last subscript pertains to the angle grid and the firstin designed to save the value for the current radial gkid point, theprevious one and the one before that in locations L, LA, and LB,reoslectively. To interpret further, the ten values of 8, after comple-tion of the "do 9" loop, are the ten possible dot products between

37

the real and imaginary parts of the fields induced by each of the twoprincipal plane incident wave polarizations. Refer to the definitionof the 12 components of E give. in h) '%broutine STORF, which appliesalso to the components of the h array in this routine, to decode theseequations. In the "do 25" loop UA and VA are the real and imaginaryparts of E *G at the point in question for evaluation of the integral.over 0. TRe variable T is the square of the magnitude of this quantity.

The effective wavenumber ke is the gradient of the argument of

E0 OG0 (U + i V) e21kz con a (94)

where U and V have the meaning of the corresponding FORTRAN variables.Elementary analysis gives

ke (2k cos a)2 + [(U -V ) 2) + (U i

V v ._52/p2lJ/EO.GoI' (95)

DUR and DVR represent rw times the partial derivatives with respect top; DUP and DVP represent rw/p times the partial derivatives with re-spect to 0. Note that the radial grid is nonuniform, so generallythree-point numerical differentiation, except when only two points arehad at the edges, is used for the radial variation. The angular gridbeing uniform, the two points surrounding the point in question areused and A4 * 4-/N. As we indicated above, the contribution of thecritical point (LM - KRIT) is ignored if iE0*G0 j is possibly resonant(T>l).

When "accumulate averages" is reached, A represents 1E0"G012 times thenonconstant part of S(ke). The variable FGS accumulates that quantity'sintegral over 4 by the trapezoidal rule and the variable DEP accumulatesthe same quantity weighted by the depolarization by second Born. Multi-plication by V" at completion of the integrals normalizes them by S(2k),the free-space spectrum function. Both are printed out layer by layer,so one can see the relative magnitude of the second Born depolarization.Then DEP is added to FGS using the logic that the depolarization in theorthogonally polarized wave scattering adds to the principally polarizedwave scattering and vice versa.

Finally, if the layer boundary called is the outermost (LAYR - LAYERS),we add one more layer and run through almost the entire routine againbecause of the lag in the layer boundary of interest.

1) Subroutine RINT(JM) is designed to calculate the normalized momentsof the scattering per unit length of wake, by performing the integra-tion in (21) over the radial coordinate. The argument JM is thenumber of the innermost layer boundary at which the integral overhas been completed by subroutine PHINT.

"The output of this routine is a set of 20 integrals, defined asfollows:

38

1 ll/

Su% aw" • d(p/rw) (96• ,, ,I + ( U m 2 )

where

', { ) - l / [E0G0ll S(ke) (1 + d) dý (97):i • 2TS (2k)

N is the axial value of the mean electron density and, Lu the defini-t1o of {(}, d is symbolic of the depolarization by second Born. Theprecise definition of (97) is such that IE .G012 multiplying d is for

-i the opposite polarization to that multiplying unity (See paragraph k)Subroutine AVGS). The quantity {4}, which is the input, is repre-sented by the array FGS. The output SO takes un w 1 in (96), S1 takesv/v8, S2 takes v'1/vog, and B12 takes (v/v ) 2, The quantity v is themean velocity profile, v' is the velocity Iluctuation, and v 0 is theaxis velocity. Each output has five values, one for each of the fivepolarization combinations carried through from AVGS. Before returningto the calling routine we print out these 20 integrals for each oftwo normalizations. The first represents normalization by the firstBorn approximation for principal polarization, in terms of the quanti-ties BN, CN, DN and EN; i.e., these normalizing factors are the resultsof (96) if { 1} .1. Since it is interesting to examine refractioneffects, the 20 integrals are also printed out in terms of their nor-malization by the values of the first Born at critical density for re-frarcion, (Recall that in MAIN, AN is [Ne/(Nc sin 2*)

m) Subroutine ZINT(NZ,V) performs the convolutions defined by (22), whereNZ is the number of axial stations at which input values of dSn/dzexist and V is cos ot.

The variables DSDZ, DTDZq and DVDZ are the input values of dSn/dz forn = 0, 1, and 2, respectively. Referring back to MAIN,whance thesequantities come, one. can see that essentially all normalizations havebeen removed, DSDZ is radar cross section per unit length of wake inunits of meters. The other two quantities, when divided by D9DZ, arethe mean and mean square doppler velocity at the axial station inquestion, in units of the body velocity and the square of the bodyvelocity.

The logic of this routlne is based on the various situations whichcan arise. Thus there is a special form of result for normal inci-dence (I cos al<10-), as follows:

Sn(O) d9

where only one output axial station is required since the backscatteredpulse shape is identical with the transmitted pulse. If only oneaxial station is input, including the case of normal incidence, thenthe output is simply set equal to the input since the convolution ismeaningless. In all, other situations the convolution is representedas follows:

'. .39

S -(z) (zS- Z ) co°2( ) dz (99)

where a is the range resolution (distance between half-power pointsfor the pulse backscattered from a point target) divided by cos x.Note that the pulse is assumed to have the shape of the cosinesquared function. This integral is evaluated at a set of uniformlyspaced values of z regardless of whether or not the input is on auniform grid. The number of points used to calculate it by thetrapezoidal rule depends on the relative widths of the scatteringfunction (the distance between the first ana last input stations)and the pulse (between zero points). If thu pulse width dominateswe use the larger of 20 or the maximum allowable number of axialstations. If the scattering function dominates we use an odd numberwhich approaches unity as the relative width of the pulse approacheszero and which is idential with the number used in the other, owsewhen the pulse width appruoches the width of the scattering function.The limits of the range oi z for the output generally cover the pointswhere the result goes to zero, where it is assumed that the inputscattering function drops to zero at either end of its range ofstations.

Interpolation of the scattering function dSn/dz is required in generalbecause the grid of input will not necessarily match the integrationgrid. The form of the interpolation has been designed to try toaccount for situations where the function varies either linearly orexponentially. If either interpolated value is zero or if they arewithin an order of magnitude of each other, linear interpolation isused. Otherwise the interpolation is exponential; i.e., it is linearwith respect to the logarithm.

C. Input and Output

The listing for MAIN gives a complete definition of the input requirements,so very little need be added. A title card, which can contain any symbols onewishes in all columns, must precede each set of input data. One will noticethat certain conventions are used in naming the input variables. All dimensionalquantities contain a reminder of their units at the end of their name. With theexception of FMHZ, all names which contain Z refer to axially varying quanti-.ties and all but NZ are arrays. MUTZ is defined as (6* - 9) in the spectrumfunction; thus * can take any of the values 5/3, 11/6, 2, 13/6, 7/3, and 5/2.The program will retnrn to the beginning of MAIN and look for new data if, aftercompletion of all calculations for a given case, the number of cases so fardone in this run is less than whatever value KASES has been given. In theHoneywell 6060 one need not repeat namelist inputs which are no different fromwhat is currently in memory, so KASES needs to be input only with the first setof data. In the definition of POLAD, "plane of incidence" is the plane contain-ing both the wake axis and the radar line of sight. All profile functions,which are denoted by PRO at the end of the name, are defined as the given quan-tity at some radius divided by a normalizing quantity. For both EMPRO and EPPROthe normalizing quantity is the axial value of the mean electron density. Thecollision frequency and velocity profiles are normalized by the axial value ofcollision frequency and velocity, respectively. in general, however, it doesao harm to use inconsistent normalizations; e.g., one could input a set of

40

• ,. • , ' ' . ' ., ' ... • ',." , " : • . ... ., ' ;,. ,,, .... •; ,• ,. L • ,•_ •1•

values of E4PRO which represents a profile which is different from unity atthe axis with no adverse effect on program operation. The EPPRO often willdiffer from unity at the axis if N1 0 Ne. The methods of inputting varioustypes of profile functions are detailed on the listing, Note that when asmooth profile function is to be 1tput, the data correspond to evenly spacedradial points starting with the axis and ending with the outer edge. Whenusing this option it is best to go to the limit and input eleven points inthe profile, Obviously the meaning of the variable VV5Z is arbitrary and someother velocity than, that of the re-entry body could be used as a normalizingfactor.

Other than those already mentioned, there are relatively few restrictionson the values of the input quantities. ASPECD must not be zero (or such thatits sine is zero). CZPSEC, EMZPCC, FMHZ, RESM, and RZM must be greater thanmero. The values of EPPRO may not be such that the profile of electron densityfluctuation is identically zero, Finally, the array ZM must have unequal, in-creasing values.

Appendix B is a copy of the complete output for a sample case, which canserve as a user's test case. The output starts on a new page. The top linegives the case number in this run and the title card characters. The formatfor this line is given in statement number 5000 of MAIN. We have used a titlecard spanning all 80 columns, to illustrate where they appear on this line. Alldata for the input variables are then written in NAMELIST format,

The next new page gives the data used for the first axial wake station.The meanings of most of the items should be obvious. Each "changed to" entryrefers to the item to its left, which has been adjusted by PROFS to avoid spur-ious effects of nearly critical density in a layer (format, including columnheadings: 1000, MAIN), In the profile functions the relative radius refers tothe layer boundaries, which are called "mesh pt," In the listing, The nextthree columns fall under the heading of electron density. All other columnsare self-explanatory (format for each row of data: 2000, MAIN). Note thatvarious possible kinds of input profiles are illustrated by the inputs used forthis case. The mean electron density is approximately equal to a Gaussian andthe rms fluctuation is a step function to half the wake radius. The collisionfrequency is uniform and the two velocity profiles linearly decrease by afactor of ten,

The next page of output gives the summary of the results of the calcula-tion of em fields in the mean wake (format for heading: 3000, MAIN). The firstcolumn shows the mode orders used. The next wavet columns give each of the twoscattering coefficients for each incident wave polarization. In each pair ofcolumns the first is the real part and the second, the imaginary, These scat-tering coefficients are defined in the discussion on subroutine BCO concerningequation (89). The last column shows that the wave propagation was not cutoff in this case (format for each row of data: 1000, BCO).

The Lnext page gives the far field coherent scattering pattern, where"azimuthal angle" refers to c and the first row of the table is for the forwardscattering (format for headingsi 2000, PHINT; format for rows of numbers$ 1000,FFSUM). The complex forward scattered fields, with the phase relative to theincident wave, are written at the bottom of the table (format: 2000, FFSUM).

41

~ !

The next page is the output of subroutine AVGS. It shows the averageover 0 of 1E #G; 2 S(ke)/S(2k) for each polarization combination. The fivecolumns of depotarization factors include the second Born in this average aswell (format for headings: 4000, PHINT; format for data: 1000, AVGS). Thedata for this sample case show a very interesting phenomenon on this page.The critical point in the mean electron density profile is at point #33,where AVGS has set all results equal to zero to avoid the resonance effect.However this wake is in the diffraction regime, which permits wave propaga-tion without cutoff all the way to the axis, Diffraction also spreads theresonance effect over a band of radii about critical. But, as can be seonfrom the data either side of critical, the phase varies so rapidly in thisregion that the spectrum function drops off as te 'critical point is approached.

The next page gives the lowest three moments of the scattering spectrum(format for page headings: 1000, RINT). In the two normalizations by the Bornapproximation, even the cross polarized restlts are divided by the first Bornapproximation for direct polarization. Also, "first squared" refers to thecontribution to the second moment of the profile of the square of the meanvelocity (formats: 2000 and 3000, RINT). The heading "absolute" means, forthe meroth moment, radar cross section per unit length of wake in meters. Forthe first or second moment the units are meters times the body velocity or itssquare, respectively (format: 6000, MAIN). In "dopplar/body velocity", theterm "mean" is self explanatory. "Spread" is defined as the standard devia-tion of the doppler spectrum in units of the body velocity; i.e.,

"spread" _ /Vbody (100)

The above five pages of output are repeated for each axial station whichwas input. Note that in the sample run we have illustrated a three-step func-tion radial profile in the second axial station. Also, note that in the secondand third stations we have used electron densities which decrease by fourorders of magnitude. The results show that, indeed, the Born approximation isapproached in this limit,

The final pages of output give the backscattered pulse shape functionsfor each of the five polarizations (format for headings: 1000, ZINT). Thepulse shapes are expressed in terms of radar cross section (dBom) and dopplermean and spread (as defined above) versus apparent axial station (format 2000,ZINT). A page is turned after the first 13 stations and then after each 14more (format 3000, ZINT).

D. Summary of Checkout Procedures Used

The reliability of this computer code for doing the calculations forwhich it was designed has been proven by a series of checkout runs during itsdevelopment, The checkout was done at various levels in the program, startingwith subroutines which do not cuall other subroutineo and working up to theentire progr-am,

The Lagrange interpolation in subroutine FIT was checked by exercising iton two profilw functions, a Gaussiau and a Gaussian multiplied by a quadratic.The input data were taehn from the appropriate points or the true underlyingfunction. The number of points used was varied from on,? throuugh eleven, andthe accuracy and reasonablAness of the outpuL were verified by Calcomp plots.as wall as numerical princout. For each of these ftunctions, one can not

42

visually detect any difference from the true curve, on a graph scale of about13 em height, as long a ,eight oT more points are input. The sample outputdiscussed above verifies that the step-function type of output is correct also.

Subroutine PROPS was checked out by hand calculation of its outputs fora few selected cases.

"Subroutine HPANK was checked out by comparing its output with tabulatedvalues for a number of argmnents ranging f om 0.2 to 50. The results wereaccurate to within one part in at least 109 or within an absolute value of 10-6or less if the result is much less than 10-6. In the process of these checkoutruns we developed the convergence criterion which is built into this routine.These same statements apply to subroutine BIN as well, except that imaginaryand complex arguments also were used.

Subroutine PROP was checked out by comparing its results with the directcalculation of the cross-products of Bessel functions, Pn, qn, rn, and an in9,1,32 on page 361 of Abramowitz and Stegun (op. cit.), (C - - V&0rn/2,D 0pn/2, C' u - rTosn/2, D' - ir'0qn/2, a * 0, b - t) Subroutine HANKwas the standard of comparison for real argumnents; library routines for modi-fied Bessel functions were the standard for imaginary arguments; and selectedtabulated values were the standard for certain real, imaginary, and complex(using Kelvin functions) arguments. Initially the Serierm expansions of thematrix elements in PROP were formulated in terms of power series in Y with thecoefficient of YJ calculated by recurrence relations, This approach began tolose accuracy when 1ý01 exceeded 10 or 20, and it encountered exponent overflowsin the Honeywell 6060 for arguments greater than 50, The formulation describedabove, in which the power of Y in included in the coefficient which is calcu-lated by recursion, significantly improved the performance. Now, as statedabove, we can go to values of [II0(l + Y)] up to 88, which is the ultimatesingle precision limit of the Honewell 6060, with accuracy generally of 10"6or better. Values of Y used in these runs wore 0.02, 0.05, 0.1, 0.2, and 0,3,An earlier series of experimental runs using a more primiti.ve formulation hadshown that the series would not always convorge if Y was greater than 0.3, how-ever this conclusion may now be conservative.

Subrout-Ine WFP and the routines it calls (HANK, BIN, PROP, BCO, and STORF) werechecked out by doing a s9t of calculations for a uniform cylinder. 'Me stan-dard of comparison was a specially written routine using Bessel functions in-side the cylinder, It was found that, even for zero collision frequency, theaccuracy is consistent with the computer word size.

Sitbroutine FFSUM was checked out by putting in the Fourier expansion co-efficients for the incident plane wave and noting that its output faithfullyreproduced the plane wave.

Subroutine PHINT, which calls FFS•M and AVGS, was checked by doing the caseof zero electron density in subroutine WFP and calling PHINT. This, of course,ppToduces the incident wave as the output of FFSUM. AVGS should produce resultswhich are unity for the three principal polarizations and zero for 'he ortho-genal. This result was obtained except that zero is about 10-15 for linearorthogonal and 10-8 for circular orthogonal.

Subroutine RINT ;;.a checked out by input of analytical functions for whichthe integrals are known in closed form.

... . . . . . .. .43

Subroutine ZINT was checked out by putting in various functions of z whichcAn be convolved with the pulse shape function in closed form. These runs in-cluded cases in which the scattering function ranged from much shorter to muchlonger than the radar pulse. They included cases in which the scattering func-tien was specified in terms of as few as one point to as many as 20. Caseswere done also to verily the program's logic for normal incidence.

The sample run discussed above serves as a checkout for the entire program.At the third axial station the electron density is less than l107 of criticaldensity and the principal polarization results are 1.0005 times the Born approxi-mation while those for linear and circular orthogonal are down by at least 1016and 1013, respectively. The factor 1.0005 comes from the numerical differentia-tion to obtain the effective wavenumber, which therefore doss not exactly equal 2k.

A\

,Ip~ 44

, I. I III

... . . ...... . .....

IV. SAMPLE RESULTS

A. Comparisons with Experimental Data

Shkarofsky at al. 16 have reported experimental data on microwave scatteringfrom a turbulent arc jet. Their paper is an excellent subject for analysis byour theory since it gives all the necessary information on the plasma. Radial.and axial variations of mean and rms electron den..,ity are given in terms of graphsof the profiles at nine axial stations, and the turbulence spectrum function alsois given. Figure 4 shows the comparison of this theory with Shkarofsky's data.

* Note that they reported the absolute backscatter power while our model givesradar cross section per unit length, so we have displaced their data to force itto agree with the Born approximation at low electron density. The polarizationis the one parameter not explicitly stated in ref. 18; but even in the worstcase that the polarization was parallel to the plane of incidence, the agreementbetween this theory and the data is excellent, (Note that in these calculationsfifty angular modes were used on a grid of 256 angles.)

Cross-polarised backscatter results are shown in Figure 5 in terms of theratio to the principal polarization. The agreement with the data is still good,although only within a factor of three. The most satisfying aspect of both setsof results is the very good agreement in the shapes of the curves all the waythrough critical density. Note that the background medium contributes to depolar-ization also, a fact which has been noted by Halseth and Sivaprasad. 1 8 The dashedlines in Figure 5 show that this contribution is quite significant in this case,being about equal to that from the second Born approximation when the plasma isoverdonse. (Special changes were made in the program to output these results.)

Graf at al. 2 0 have studied the effects of incidence angle on refraction ina turbulent laboratory plasma. Their data for backscattering at 400 aspect angle(between the line of incidence and the plasma axis) follow the first Born squarelaw dependence on the rms electron density. These data points extend up to anrma electron density of 0.7 times critical density. They do not show backacatterdata for 200 aspect angle although they state, 'At an aspect angle of 200, thebackscatter cross sections deviated ... from the slope-of-two curve ... about3db when n' was 0.15 nc." Regrettably, they do not report much detail about theconditions of their experimrnt, especially regarding the background (average)electron density distribution, which is so important in this theory. We havemade calculations using radially Gaussian electron density distributions, wherethe standard deviation of the Gaussian is 2.1 cm for the average electron den-sity and 2.5 cm for the fluctuations. The results of calculations using thistheory are shown in Table 2.

18. I.P. Shkarofsky, A.K. Ghosh, and E.N. Almey, "Direct and Crosu-PolarizpdBackecatter Power from a Turbulent Plasma", Plasma Physics, Vol. 14, pp,935-950, October 1972.

19. M.W. Halseth and K. Sivaprasad, "Depolarization of Singly Scattered* Radiation in a Turbulent Plasma", Phys. of Fluids, Vol. 15, No. 6, pp.

1164-4166, Juno 1972.20. K.A. Graf, H. Guthart, and D.C. Douglas, "Refraction Effects in Turbulent

Media", Radio Science, Vol, 9, Nos. 8, 9, pp. 777-787, August-September1974.

.45

,•I,

'II

,,-1 1 1' 11,111 i 111 -1 :

First Born

/ I

Transvorse polarization / 00

0Parall0

Shkarofsky's data Parallel

--

1 _- Theory

i.I

m io

1011 1012 1013

rme electron density fluctuation, cm- 3

FIGURE 4. COMPARISON OF THIS THEORY WITH SHKAROFSKY'SDATA FOR BACKSCATTERING AT 16GHz FROM ATURBULENT ARC JET.

46

.........

T -TT

1i0-

! 1 Shkarofskyls data 0 0 0 0 -

-0o

U0

.00

4, 0- .= her4.4

0e o

S- / -4ckground -medium ..

10- m i li I ,I,!L"LLU

101 101 101

)". i. rms electron density fluctuation, cm-3:

DATA FOR THE RELATIVE DEPOLARNZATION OF THE,BACKSCATTERING.

4. .

A&C ,IM AMPL I:M1jDELBORPi(d.B)ASPECT PARALLEL IUhASVERSEANGLE N' /N ~ N'Iff POLARIZATION POLARIZATION

40o 0.7 0.5 -8.4 -5,3

400 0.7 1 .4,7 -2,4

200 0.15 0.5 -7.0 -6.0

200 0.15 1 -3.5 -2.8

The parameter N1/9 represents the axial value of the ratio of the rms electrondensity fluctuation to the mean electrun density, and it can be seen that thisis an important parameter in the theory. Although the nominal value is givenas 0.5, the theory agrees with the experiment better for a ýarger value.

Graf et al. 20 have given reqults of coherent field measurements also.Figure 6 reproducen their resu. for meeaur~d fie3d intensity at a distance of156 cm' from the transmitter when the plasma axis was a perpendicular bisectorof the line of eight. The theory gives the coherent forward scattered fieldas a byproduct, which falls off as the invier.' square root of the far-fielddistance because .f tho assumption of cylindrical syn~vetry. The points plottedon the graph show thu resultsof the theory for the total field intensity,aswuming that the incident field fells off as I/r. The square symbol denotesparallel polarization and the circle$ transverse. The open symbols show theresults of accounting tor only the mean electron density, neglecting the effectof scattering attenuatioa. The filled symbols show the results when the meanplasma properties are etshanced to acwount for scattering attenuation, assumingthat N'/N - 0.5. The theory gives more diffraction than the data show, althoughthe trend with electron density is similar. It would appear to be anamalousthat the transmitted field is iacreased when scattering attenuation is accountedfor; but this result is caused by the manner in which the calculation is done,in that quantities are added to the mean electron donsity and collision fre-quency. Apparently, more diffraction than attenuation is added.

AttwoodRI,2 2 has reported the resultc of backscattering experiments for awide range of aspect angles and electron densities either side of critical.But this theory comes nowhere near explaining the extreme degree of dependenceof his data on these parameters, especially at and above critical density. Thisis probably caused by the fact that his pl&ama does not fit the descriptionassumed herein, of a random medium embedded in a steady, cylindrical backgroundmedium.

21.D. Xttwood, "Microwave Scattering from Underdense and Overdense TurbulentPlasmas", Physics of F-uids, Vol, 15, No. 5, pp. 942-944, May 1972.

22. D. Attwood, "Microwave Scattering from an Overdense Turbulent Plasma",Physics of Fluids, Vol. 17, No. 6, pp. 1224-1231, June 1974.

48

S... + + .,+ • •,.. ,.,YO:+ •

*i 0'° I1IIJ I I, 1"', '11'i, I' I l

I: ITheory

10

-201 iiil I I i ii0.1 1

Axial mean electron density/critical, N*/Na

FIGURE 6. COMPARISqON OF THIS THEORY WITH ORAl' S DATA ON THECOHERENT FIELD INTENSITY TRANSMITTED THROUGH A

* TURBULENT FLAME AT NORMAL INCIDENCE.

49

'p 49

,. 1, • . . h' . . .

Y

B. Some Parametric Results

In the previous study (see Ref. 1) the importance of finite wavelengtheffects was evaluated in some detail. These effects refer to many of theconsiderable differences which are found when the results of rigorous em wavepropagation in a cylindrical plasma are compared with the results of raytracing. Thus that study identified the diffraction effect, which was sonamed because it represents the departure from geometrical optics laws as thewavelength is increased. Furthermore, in that study, it was shown that thediffraction regime occurs when the parameter kOe sin a is small, where Ce isthe width (standard deviation) of the radial profile of the mean electron densitydistribution. In other words the standard of comparison for the wavelength isthe width of the electron density distribution projected onto the line of in-cidence. An important corollary of this result is the fact that low radaraspect angle can be a sufficient condition for diffraction, Our objective inthis subsection is to determine how significantly the improvements embodiedin the present model affect the results both in the diffraction regime and in

the transition between diffraction and geometrical optics.STo permit direct comparisons we have used the same radial profile functionsas for most of the prior parametric studies. Thus the mean electron density

is a Gaussian with ae a O.3 rw; i.e., exp [- p2 /(0.18rw2 )]. The rms electrondensity fluctuation is equal to the mean multiplied by (1 + p2 /(0.18rw2 )]; i.e.,

¶ it has an incipient off-axis peak., The mean velocity profile is Gaussian withav - 0.9408 ate. The rms velocity fluctuation has an incipient off-axis peakand the standard deviation of its Gaussian part a 1.0787 ae. We use amoderately collisionless condition (V/w - 0.1) an• a low aspect angle (sin a -

1/18). The scales of turbulence are kri - 0.021 and kro - 0.21.

Figures 7 through 10 show the RCS (radar cross section), the ratio oforthogonal/principal RCS, the mean doppler and the contribution of velocityfluctuations to tVe doppler second moment as functions of axial electron densityfor circular polarization. Theae figures are all for kae sin a - 0.03, sothey represent the diffraction regime. The dashed line in each figure repre-sents the results from the previous study and we conclude that the improvementswhich the new model incorporates (effective wavenumber calculation, scatteringattenuation, and depolarization from turbulence) make little difference in thediffraction regime.

Figures 11 through 14 show the same four quantities as functions of kaesin a at a value of Ne/(Nc sin a) - 100. The unly significant difference be-tween the old and the new results is the polarization ratio for kae sin agreater than about 2. Thus it appears that in the diffraction regime thedepolarization by the meai. wake plasma dominates that by second Born scatteringfrom the turbulence.

A

50

"•,• , . ," ' u". .,

* - 40

30

20

10-

"-10

20 2 34

Normaliz~ad axi~al electron density

FIGURE 7. RCS VERtSUS ELECTRON DENSITY IN THE DIFFRACTION REGIME.THE NORMALIZATION IS WITH RESPECT TO THE REFRACTIONEFFECT; I.E., THE NORMALIZING ELECTRON DENSITY ISN sin a~ AND THE NORMALIZING RCS IS THE BOR~N APPROXIMA-TION AT THIS ELECTRON DENSITY. (FIGURES 7 THROUGH 14

* ARE FOR CIRCULAR POLARIZATION AND THE DASHED CURVE ISF... 1

I; • z

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-60

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-70 , , ,, I liii, , ,,I , i mil I II1

10"i 100 101 102 103 104

Normalized axial electron density,N,/ (NSin2cs)

FIGURE 8. ORTHOGONAL/PRINCIPAL POLARIZATION RCSIN THE DIFFRACTION REGIME.

52

J,, , .

Normalized mean 211,,,doppler velocity ... ,

10'1 I i° I 1i02 103 104 1 I

100 101 0

Normalized axial electron density,Ne/ (Ncein 2 a)

FIGURE 9. MEAN DOPPLER VELOCITY IN THE DIFFRACTION REGIME.THE NORMALIZING VELOCITY IS THE BORN APPROXL7•TION; I.E,, THE VELOCITY PROFILE WEIGHTED BY N'7.

2i• , 2 i vII '-, I I i" i""i I i "' I ' ' i

Normalized doppler/second moment from 1velocity fluctuations

.40 010"1 100 101 102 103 10;

Normalized axial electron density,NO/ (N~cfin 2Ct)

FIGURE 10. CONTRIBUTION OF VELOCITY FLUCTUATIONS TO THEDOPPLER SECOND MOMENT IN THE DIFFRACTIONREG IME.

V 5

!U.

50 ' 'w' t i i'7 'I i T

40

.30

20

0 L ..... i i i Ii t1 i I I I *,i1i1

0.01 0.1 1 10

kaesinc%

FIGURE 11. NORMALIZED RCS AS A FUNCTION OF THE ELECTRONICWAKE WIDTH PROJECTED ONTO THE LINE OF INCIDENCEAND COMPARED WITH THE WAVELENGTH. (IN FIGURES11 THROUGH 14 N /(Nce*n2a) 100.)

54...... .

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-30

-40 -

-50

I,,S-6o - N

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FIGURE 12. ORTHOGONAL/PRINCIPAL POLARIZATION RCS VERSUSNORMALIZED ELECTRONIC WAKE WIDTH.

55

S'' i '' ' ', I, , , ' ,., *. . . . ... . . ..

10

0.1

0,01

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FIGURE 13. NORMALIZED MEAN DOPPLER VELOCITY VERSUSNORMALIZED ELECTRONIC WAKE RADIUS.

56

kliA .- ' . .....I..... ... ~tifvA M

10.

0.

0.01

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V. CONCLUSION

A sophisticated computer model for the radar backscattering from turbu-lent re-entry-induced ionized wakes has been described in sufficient detail topermit the user to modify the program as desired, This model uses the basicconcept that propagation in the background, average electron density distribu-tion determines both the local fields and the Green's function fur scattering;and this effect is formulated in a rigorous and general way by assuming thatthis background is a cylindrical medium. At this stage of development, however,the application of this model to the turbulence scattering is zalatively crude,being of the type generilly used in so-called distorted wave Born theories.Thus much room for improvement exists,

Many improvements and modifications are possible without basic theorychanges. For example, one could generalize the-program for bistatic scattering.Thin would take major programming changes but it does not require any newtheory. Generalization for anrtotropic turbulence would be relatively straight-forward, and allowance for various types of radar pulses is almost trIvial,The same can be said for including the effects of antenna patterns; i.e., non-uniform illumination.

This model probably could also be applied to the probleta of radar scatter-ing from the turbulent plume of a rocket exhaust. The main constraint wouldbe that it be in a regime of flow that simulate& the turbulent wake flow. Athigh altitude, where the plume Is laminar and spheripal, the canonical problemto be solved would be scattering from spheroidal media. The preasucve of atrong,nonrandom irregularities in the plume, such as those created by shock diamonds,would require yet different techniques such as, perhaps, the geometric theoryof diffraction,

Various basic theory Improvement3 are possible in the problem of turbulent,cylindrical wake scattering. There is every reason to expect that, with theknown rigorous solution for the effect of the background, one could do thestatistics of the scattering from the random component of the medium in a rigor-ouu way, At least one could do a Monte Carlo calculation. Also, the quasi-optical approximations (and their accompanying ad hoc assumptions) borrowedfrom the distorted wave Born approximation could ba removed by carrying outhigher order iterations in the Neumann series. Of course the difficulties ofdoing this are not trivial because one would have to get near-field solutionsfor propagation in the background medium. Obviously this is easier to do inthe diffraction regime, where fewer angular modes are needed, but this is wherethe significance of this model is the greatest.

In conclusion, this model provides not only a valuable tool for applicationsbut also a framework for future research in uhe theory of scattering fromrandom media.

58I

".1I

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APPENDIX A

FORTRAN LISTINGS

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