The feedback-diffraction theory of ionospheric heating

Jourmdof

Atmospheric

and TerrestrialPhysics,

Printed inGreatBritain.

Vol.44,No. 12,pp.106-1074,

OoZl-9169/82/121061-14$03.00/O 8 1982 Perzqmon Press Ltd.

1982.

The feedback-diffraction theory of ionospheric heating P. A. BERNHARDT and L. M. DUNCAN Atmospheric Sciences Group, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. Abstract-Plasma fluid equations describing concentration, velocity and temperature are coupled to the nonlinear wave equation in a numerical model of F-region heating by high power radio waves. Self-focusing of radio waves is calculated for plane waves and beams propagating in an underdense ionosphere. Ponderomotive forces are found to be negligible in comparison with thermal forces for self-focusing in the ionosphere. The structure of plasma irregularities near focal points is determined by diffraction of the radio waves. 1.INTRODUffION

When high power electromagnetic waves propagate through the ionosphere, a wide variety of nonlinear wave-plasma interactions may occur. Examples of

these interactions are (1) self-focusing, (2) parametric wave-interaction and (3) neutral breakdown. This paper describes self-focusing due to thermal and ponderomotive forces. Analytic, linearized treatments of the thermal selffocusing instability have been studied by PERKINS and VALEO (1974). CRAGIN and FEJER (1974), GUREVICH (1978) and PERKINSand GOLDMAN (1981). These works provided estimates of instability threshold and growth but do not examine the complete, nonlinear evolution of the instability. A numerical model of resistive heating without selffocusing has been described by PERKINS and ROBLE (1978). Their treatment predicted changes in plasma concentrations due to chemistry and transport and in electron and ion temperatures due to conduction, compression, convection, collisional cooling and ohmic heating. The focusing of the incident ratio beam by the plasma gradients was not included. MELTZ et al. (1974) compute the heating of the ionospheric plasma by a perturbation approach which couples equations of plasma heating with equations describing wave propagation using geometric optics. The effects of diffraction were not included. In this work, we have attempted to construct a model of ionospheric heating which involves a minimum of approximations and assumptions. The equations of continuity, motion and energy for the ionosphere are solved numerically in conjunction with the nonlinear wave equations. Linearization or perturbation techniques are not employed. Consequently, complete nonlinear growth of plasma irregularities to saturation can be determined. The ionospheric heating model is used for simulation

experiments. The frequency, power level and spatial structure of the radio waves are varied and the ionospheric response is computed. Specific parameters of interest include (1) irregularity scale size, (2) nonlinear instability threshold and (3) time delay for plasma response to high power radio waves. The remainder of this paper is outlined as follows : Section 2 contains the theory for the feedback model describing the formation and evolution of the heated ionosphere. Techniques for the numerical calculations are described in Section 3. In Section 4 we present results for underdense heating by a uniform plane wave. In Section 5 we present the effects of underdense heating by a radio wave beam, emphasizing the diffraction of the beam. Conclusions are presented in Section 6.

2. THEORY

A self-consistent model of radio wave heating of the ionosphere requires theoretical treatments of (1) electromagnetic wave propagation, (2) ion and electron temperatures and (3) plasma concentration. The mathematical representation of these processes is as follows. The derivation of the wave propagation equations is an extension of the one given by YEH and LIU (1972). For a single-frequency time variation da’, Maxwell’s equations are written as : V x E = -jopOH,

(1)

VxH =jWsE,

(2)

V *(E’E)= 0,

(3)

V-H=O,

(4)

where E and H are the electric and magnetic field vectors, p0 is the permeability of free space and Eis the dielectric constant. In an isotropic lossy medium, E is

1061

1062

P. A. BERNHARDT and L. M. DUNCAN

given by E

=

E’ _ j&" =

E.

(I-$-)-ja,(%),

(5)

where E,,is the dielectric constant of free space, w is the wave frequency (rad s- ‘), o, = n,e’/m,&,, is the plasma frequency (rad s- ‘), n, is electron concentration (mm3), e is the electron charge, m, is the electron mass, and v, is the electron collision frequency (s - ‘). This expression is derived assuming that w >>v, and that the wave frequency is large compared to the electron gyrofrequency. The loss of power in the radio wave by absorption is represented by E”,a source of heat for the plasma. Equations (1) and (2) combine to yield the wave equation V’E - V(V - E) + ~‘p(,eE = 0.

(7)

The rapid, spatial variations in the complex vector E are replaced by slowly varying functions E = a exp (-jk,Y),

=

n*+ 2, 0

VY *Va = -f(V*Y + k,m*)a,

+[~+(~~+$+(~~]/k&

ay ay

ay aA ay aA

ax%+--=

-k

(lla)

89

g+E+k,m*(x,y)

1 ,

(lib) where A = In a. The power density in the wave is related to the amplitude of the electric field by

p=!!C 2

(12)

?’

where P is power density in W m-*, a is in V m- ’ and r] = ,/(P~/E’) is the impedance in ohms. The radio wave directly heats the electron gas. The electron temperature is described by the heat equation

(8)

where a is an amplitude vector, Y is the phase path and k, = w/c. Both a and Y are real quantities. Substitution of equation (8) into equation (7) and separation of real and imaginary parts yields two coupled equations (my

(gc (g =n*(x,Y)

(6)

Equation (3) is expanded and substituted into equation (6), yielding V’E + V(E - V In E’)+ w*~,+E = 0.

vertical axis, respectively. The wave parameters are assumed to be constant in the third dimension. In rectangular, two-dimensional coordinates, equations (9) and (10) become

(10)

where n = &c(E’~~)] is the refractive index and M = ,/[c(~“p~)] is the absorptive index. To derive equations (9) and (lo), wave polarization is neglected (see GUREVICH,1978) and the amplitude is taken to be a scalar rather than a vector quantity. The two terms on the right-hand side of equation (9) represent changes in the phase front due to refraction (because of variations in n) and due to diffraction (for example, at focal points or at caustics). If the second term on the right-hand side of equation (9) is neglected, the resulting equation describes ray optics. Equation (10)describes the amplitude fluctuations of the propagation wave. The first term on the right-hand side of equation (10) accounts for the change in wave intensity arising from a non-planar phase front. The second term gives the reduction in wave amplitude due to absorption. In this paper, only a two-dimensional geometry is considered. The x and y axis are the horizontal and

+n.kT.(2)

- $-e(z)]

= Q,--L

(13)

where T, is electron temperature, n, is electron concentration, k is Boltzmann’s constant, w, is the electron velocity, K, is the electron thermal conductivity, Qe is the heat input to the electrons, L, is the heat loss, and t is time. The coordinate q is along the magnetic field. Assuming straight field lines in the x-y plane, the magnetic coordinate is given by q = -x

cos I+y sin I,

(14)

where I is the magnetic dip angle. Equation (13) is a time-dependent expression for the electron temperature including the effects of convection, compression, conduction, heat production and loss. The electron cooling occurs via elastic collisions with neutrals, via rotational and vibrational excitation of molecular species, via fine structure transitions in atomic oxygen and via collisions with ions. Detailed expressions for the loss (L,) term can be found in the text by BANKSand KOCKARTS(1973), Vol. B. During daytime, photoelectrons are produced by solar ionization of the neutral atmosphere. These photoelectrons provide a heat source for the electron gas. This heat input is contained in the term Q, in equation (13). The numerical expressions for the

1063

The feedback-diffraction theory of ionospheric heating photoelectron production and heating are described by ANTONIADIS (1976). Another contribution to the electron heating comes from the radio wave (IS) where the variables have been previously defined in equations (5) and (8). The electron collision frequency is the sum of the electron-ion and the electron-neutral collision frequencies as given by BANKSand KOCKARTS (1973), Vol A. The ions are, in general, heated by collisions with electrons and cooled by collisions with neutrals. The ion heat equation is

;nik 2 =Q,-_L,,

(>

(16)

where ni is the ion concentration, T is the ion temperature, Qi is the heat input to the ions and Li is the ion heat loss. The expressions for Qi and Li are found in BANKSand KOCKART~(1973), Vol. B. Indirect heating of the ions occurs via direct radio wave heating of the electrons which subsequently transfers thermal energy to the ions. The radiation pressure of the radio wave acting on the plasma is called the ponderomotive force. In regions where the wave frequency is much greater than the electron gyrofrequency, the ponderomotive acceleration is given by Pi = &

Va2,

@,k7J a4

+

1 a(n,kT,) -~ nemm a4

+ (g - p,,) sin Z+ pxm cos Z

= - v,,(w, - IJ,) - v,o(w,-

1 a(n,kl--;) + 1 @r&T,) -7 ncmo7 nOrnO + (g - pro) sin Z+ pXocos I (18)

wo), (19)

where m, and m, are the ion masses, g is the acceleration of gravity (positive downward), pxo, pyo, pxm and pym are components of the ponderomotive acceleration, van and v,, are the ion-neutral collision frequencies, vom and v,,,~ are the ion-ion collision frequencies, U, is the component of the neutral wind along the magnetic field and w. and w, are the ion drift velocities parallel to the magnetic field. We found it necessary to include both monoatomic and diatomic ions to give reasonable electron concentrations at low altitudes. Equations (18) and (19) can be solved simultaneously to yield the ion velocities in terms of gradients of plasma temperatures and concentrations. Note that the time-dependent, inertial terms in the momentum equations have been dropped. Also we have not considered the effects of external electric fields for transport of plasma across magnetic field lines. By employing fluid equations, parametric decay instabilities (FEJER, 1979) and soliton formation (WEATHERALL et al., 1982) are not included. The plasma concentration is found from the continuity equations

2+-=abow,) a4

(17)

where pi is the acceleration of ion species Y’and mi is the ion mass (CHEN, 1974). The ponderomotive force acts on the electrons which carry the ions along via a charge separation electric field. The effect of the force is to move plasma out of regions where large radio waveamplitude gradients are present. The electron and ion temperatures and the ponderomotive force influence the plasma concentration through the momentum and continuity equations. Here, the plasma is assumed to be composed of two species, one monoatomic and one diatomic. The concentration of the monoatomic species (i.e. O+) is represented by n,. The concentration of the diatomic (or molecular) ions (representing NO+, O:, Nl, etc.) is denoted n,. The steady-state equations for the two ion velocities are (BANKSand KOCKARTS,1973, Vol. B) :

= - vo,(wo - ZJ”)- vonl(w0- wnl),

1

-~ n,m,

p

_L

0

09

ahd4 =Pm-L,. 2+ ~ a4 The ion velocities and concentrations those of the electrons by n, = n,+n,, w, =

won0

w,n, n,

+

are related to

(22)

(23)

The production and loss terms (PO, P,, Lo and L,) represent photochemical processes in the upper atmosphere. Specifically, O+ is produced by photoionization of atomic oxygen, and the molecular ions are formed by ion-molecule reactions between O+ and N, or 0,. The molecular ions are lost via recombination with electrons. The temperature dependent rates for the ion chemistry are found in BANKS and K~CKART~ (1973), Vol. A. The electron recombination rate for the molecular ions is taken to be the representative value of 2 x lo-’ crnw3 s-l. Equations (9)-(23) are a complete set for describing radio wave heating of the ionosphere. The coupling

P. A. BERNHARD T and L. M. DUNCAN

1064

Radio Wave Amplitude

Electron Heating

Ponderomotiva

Electron Temperature

Ion Velocity

Fig. 1. Feedback

processesproducing self-focusing.

between the variables of this system is illustrated in Fig. 1. The refractive index of the medium modifies the phase and amplitude of the propagating wave. The wave amplitude provides a heat input into the plasma. The temperature changes in the heated plasma cause a redistribution of the plasma affecting the refractive index. The wave also acts directly on the plasma via ponderomotive force. Other effects(not includedin Fig. 1 but included in the mathematical description) are the reduction in the amplitude by absorption, and the changes in collision frequencies and in chemical reaction rates with temperature. There are some limitations to the theory. The radio wave propagation is considered to be isotropic. The anistropy due to the magnetic field is not considered. Consequently, the radio wave splitting into ordinary and extraordinary modes is neglected. Also equations (9) and (10) were derived neglecting the vector-like nature of the propagating fields. The plasma equations do not contain any description of plasma motion across the magnetic field lines (see BERNHARDTet al., 1982). Finally, this paper only considers a two-dimensional description of ionospheric, radio wave heating. 3. NUMERICALCOMPUTATIONS The solution of the equations in the previous section is accomplished numerically. Second-order, finitedifference approximations to derivatives yield nonlinear coupled equations for the unknown variables located on a grid in space and time. The neutral atmosphere and neutral temperature

profiles are analytical models taken from WALKER (1965). Figure 2 illustrates the neutral parameters used in the computations. The neutral wind, U,, is set to zero. The plasma equations for concentration, velocity and temperature are one-dimensional, the dimension being the magnetic field coordinate. These equations are nonlinear, parabolic partial differential equations. They require boundary conditions at the top and the bottom of the field-line coordinate space. At the bottom (100 km altitude), we assume chemical equilibrium for the plasma concentration and thermal equilibrium (i.e. T, = ‘&= 7’,‘,) for the electron and ion temperatures. At the top (600 km), we initially specify the O+ concentration and electron temperature. The concentration at the upper boundary is allowed to vary as plasma flows from the protonospheric reservoir above 600 km altitude (see PARK and BANKS,1974,1975). The plasma model used in this paper is simplified, yet a modernized version of the one used by BERNHARDTand PARK (1977). The equations are solved using implicit, iterative techniques similar to those described by HASIINGS and ROBLE (1977). The initial plasma Of and molecular ion (Mf) concentrations and the ion and electron temperatures are computed for daytime assuming steady-state conditions. Their profiles are illustrated in Fig. 2. These steady-state quantities are used as initial conditions for the time varying computations of plasma in the 20& 300 km region. The equations for radio wave phase and amplitude [equations 11(a) and 1l(b)] are solved on a two-

1065

The feedback-diffraction theory of ionospheric heating (4

NEUTRRLS

PLASMA

(b)

600 550 So0 450 g 400 g 350 !z I- 300 d 250 200 150 100 w. ma lg r-ooooool(Dm PD m P = 8 _____I___ CBNCENTRRTIBN INUHBEWCHw3)

m.

8

g

ijjg 5 H $, $j 101 102 103 104 IOS 106 lo7 N CENCENTRRTIBN ~NUHBER/CHw31 ___I_ TEMPERATURE (K)

Fig. 2. Unperturbed profiles of (a) neutral concentrations, (b) temperatures, and (c) plasma concentrations. Molecular ions are labeled M+.

dimensional rectangular grid. Figure 3 illustrates the region in which wave propagation takes place in relation to the tilted magnetic field lines. The initial wave phase and amplitude are specified at the lower boundary. The problem is solved numerically using the explicit, ‘leapfrog’ scheme to march forward in the y-direction. Since the coupled equations are nonlinear, iterative techniques are required for convergence to the desired solution. The computation space used for the problems

described here is illustrated in Fig. 3. The magnetic field lines have a dip angle (I) of 50”. The RF heating is calculated between 200 and 300 km altitude. The dashed lines in Fig. 3 indicate the side boundaries for the radio wave propagation. For computations using plane waves (Section 4), the plasma is periodic at these dashed lines. For heating with a Gaussian beam (Section 5),the plasmaextends along the field lines, past the dashed lines as shown. The grid spacing (dx and dy) is approximately 0.5 km for all calculations.

TOP BOUNDARY (300 km)

BOTTOM BOUNOARY (200 km)

Fig. 3. Area of computation for ionospheric heating by high power radio waves.

P. A. BFRNHARDT and L. M. DUNCAN

1066

LOG (4 300

IO (POWER

TIME = 12.000 h

DENSITY

(W/m*)) fc)

(b)

’ ‘TIME=l2.069

TIME= 12.026 h

S-N

DISTANCE

h

(km)

Fig. 4. Power density contours for a 13 MHz plane wave propagating through a sinusoidal, plasma irregularity centered at 230 km altitude. The initial power density is low5 W m-‘. Local maxima are labeled ‘II’. 4. UNDERDENSE HEATING WITH A UNIFORM PLANE WAVE In this section we examine the heating ofthe F-region with a uniform plane wave propagating vertically. We use the ionospheric profile illustrated in Fig. 2 which has a critical frequency, foF2 = 12.18 MHz. On this ionosphere, we impose a 1% sinusoidal perturbation (period = 21 km) centered at 230 km altitude. This perturbation is field aligned, dropping off as exp [ -(y-230)2/2a2] where c is 10 km. The electron and ion temperatures are initially unperturbed. A plane wave with a frequency of 13 MHz is initiated at 200 km altitude with a power density of 10 PW mm2 and a phase(Y) of 200 km. The initial propagation is in the + y (upward) direction. Radio wave heating starts at 1200 LT in the model ionosphere. The evolution of the self-focused radio wave power is shown in Fig. 4. Initially, the ionospheric irregularities at 230 km focus the wave at 270 km altitude. The maximum power density is 14.7 PW rnm2. The power density is not large enough to sustain this level of focusing. The initial irregularity begins to diffuse away and consequently the radio wave power is reduced. After 4 min, the maximum focused power density has been reduced to 14.0 PW mm2 [Fig. 4(c)]. At this time, the system is in equilibrium and the focused region supports itself. Horizontal slices of power density at 300 km altitude are illustrated in Fig. 5. The power variation remains

nearly sinusoidal at all times. The phase shifts as the plasma is restructured by the heating wave. At equilibrium, the peak-peak power fluctuation is only 0.3 dB. The corresponding temperature variation at 300 km altitude is illustrated in Fig. 6. The initially uniform temperature distribution at 1030 K is heated to 1065 K with a 3.3 K peak-peak sinusoidal variation 1.7 min after the RF heating has started [Fig. 6(b)]. The effects

-25

-20

-15

-10

S-N

-5

0

DISTANCE

5

IO

I5

20

25

(km)

Fig. 5. Horizontal variation in power density at 300 km altitude. The initial power density at 200 km altitude is lo-’ W m-‘.

1067

The feedback-diffraction theory of ionospheric heating (a)

(b)

TIME=l2.0OOh

TIME=12.028

(c)

h

TIME=l2069h

P ~l400IY

-20

-10

0

IO

-20

-10

S-N D&NE

;Okrd

-20

20

0

-10

IO

Fig. 6. Horizontal variation in power density at 300km altitude due to the ohmic heating from the radio power density shown in Figs 4 and 5. of power density reduction and of thermal conduction cause this amplitude to reduce to 1.3 K. The difference in phase between the radio wave power [Fig. S(c)] and the electron temperature [Fig. 6(c)] at 300 km altitude is due to heat conduction from below. The example of heating with the 10 PW mm2 plane wave illustrated with Figs 4-6 produces perturbations of such small magnitude that they remain sinusoidal. The system never becomes nonlinear. The maximum perturbation in electron concentration is 0.17%. The next example uses a power density of 100 PW m-‘. The other initial parameters remain unchanged.

The evolution of the power density is illustrated in Fig. 7. The initial contours of logarithmic power [Fig. 7(a)] are identical to the ones in Fig. 4(a) except for the addition of unity. The higher power radio waves are strongly focused above 280 km altitude. As in the previous example, the initial sinusoidal irregularity at 230 km altitude gradually dissipates. The higher power radio signal is self-focused to a sinusoidal amplitude structure at 270 km and to a strongly diffracted pattern at 300 km altitude. The amplitude structure in a horizontal line at 300 km is shown in Fig. 8. Two cycles of the structure are

LOG IO (POWER DENSITY (a) TIME = 12.000 h

2

270: 3

,,,

260

(b) TIME = 12.006 h

(cl TIME = 12.028 h

(W/m2)) (e)

(d) TIME=l2.056

h

TIME= 12.083 h

c3 3 I-

250

t=

240 230

Fig. 7. Power density for a 13 MHz plane wave propagating through a sinusoidal, plasma perturbation. initial power density is 10T4 W m-‘. Local maxima are labeled 2-I’.

The

1068

P. A. BERNIUR~Tand L. M. DUNCAN

// TIME=12.056h 1

S-N DISTANCE (km1

Fig. 8. Self-focused structure in power density at 300 km altitude.

illustrated. The initial sinusoidal pattern with a wavelength of 21 km is distorted into a pattern with smaller scale sizes. Five minutes after the start of the simulation, the structure size approaches the mesh size of the computation [Fig. 8(f)]. The peak-peak

amplitude variations range from 0.8 dB initially to 33 dB after 5 min. The focusing at 300 km altitude is due to lens structures being formed at 270 km by the radio wave heating. Figure 9 illustrates contours of electron

ELECTRON TEMPERATURE (K) TIME = 12.000 h

TIME=l2.006

h

TIME=l2.028

300

h

TIME = 12.056 h

TIME =12.083

I 290 280 270 / 260

+

240

F -I

230

a

Fig. 9. Electron temperature enhancements due to radio wave heating by the power density illustrated and 8. Local maxima

and minima

are labeled ‘H’ and ‘L’, respectively.

in Figs 7

h

The f~back~~ract~on

theory of ionospheric heating

The temperature structures tend to be field aligned. Conductive and convective heat transport reduces the maximum temperature at 270 km altitude. The temporal variations in a horizontal slice at 300 km altitude are illustrated in Fig. 10. As with radio wave amplitude, the initially sinusoidal variation evolves to one with a smaller scale structure. A comparison of Figs 8 and 10 shows that there is approximately a 1.5 min time lag between radio amplitude and temperature effects. The modification of the electron concentration is shown in Fig. 11. At the F-region peak (270 km), the heating produces a localized reduction in electron concentration. This reduction is sinusoidal, following the pattern of the initial irregularity. The lens action at 270 km leads to irregular structure being formed at 300 km altitude. This structure is field aligned. The changes in electron con~ntration are delayed responses to the electron temperature variations. The calculations described in this section have been made with and without the inclusion of ponderomotive forces. Even though strong gradients of electric field are produced [see, for example Fig. 8(f)], the ponderomotive force had negligible influence on the plasma in comparison to the thermal effects.

5. UNDERDENSE

temperature.

WITH A RADIO BEAM

We again consider low (10 @W m-‘) and high (100 PW m- ‘) heating ofthe F-region plasma. Rather than a plane wave, a Gaussian-shaped beam with a cr, of 10 km is used. The daytime ionosphere is horizont~ly stratified ; no initial structure is introduced into the ionosphere. Otherwise the conditions are identical to the previous section : the ionospheric critical frequency is 12.18 MHz. The radio frequency is 13 MHz. The Gaussian-shaped beam with a maximum initial power of 10.~W rnw2 is self-focused by theionosp~eric plasma. The logarithmic power distribution at 300 km altitude is illustrated in Fig. 12. The power density at the center increases by 8 dB due to the self-focusing [Fig. 12(c)]. The focusing process reaches equilibrium 6 min after the power is turned on. The phase fronts for the radio beam are illustrated in Fig. 13. The phase path is retarded and curved by the modified refractive index in the ionosphere. The electron temperature enhancements are beam aligned below 290 km altitude. Above 290 km altitude and to the right of the beam the temperature tends to be field aligned. Electron heat conduction controls the formation of field-aligned temperature enhancements

TIME * 12.006 A

HEATING

1069

h A

Fig. 10. Electron temperature structure at 300 km altitude.

P. A. BERNHARDTand L. M. DUNCAN

1070

ELECTRON CONCENTRATION (10’0m-31 TIhlE=l2.000

TIME= 12.083 h

TIME= 12.056 h

TIME= 12.028 h

TIME = 12.006 h

h

300

1

-160

160 t-

‘.

-120

120

-80

1

60 I

i

Fig. I 1.Modification of electron concentration by a 10e4 W m-‘, 13 MHz plane wave. Local maxima are labeled ‘IT.

extending from the point of power density focus (Fig. 14). PERKINSand ROBLE(1978) also show that enhanced

heating is beam aligned below and at the F-layer peak and field aligned above the peak. The combination of both beam-aligned and field-aligned perturbations also occurs for the electron concentration (Fig. 15). For a radio wave power of 10 PW m- ‘* the ~rturbations remain unimodal. Increasing the power to 100 PW mm2 causes the Gaussian beam to filament near the focus. The evolution of the Gaussian beam during the 30 s after turn on is illustrated in Fig. 16. The beam rapidly self-

(a)

TIME= 12 006

h

(b)

focuses producing an irregular structure at the focal region. The distribution of power at 300 km altitude is illustrated in Fig. 17. The initial cross section of the beam is shown in Fig. 17(a). Twenty seconds after turn on, energy in a 25 km radius about the beam has been focused near the beamcenter [Fig. 17(b)]. After 40s the beam filaments to a scale size approaching the spatial resolution of the computations [Fig. 17(c)]. The 60 dB fluctuations in power may be unrealistic because of the limited mesh size. The electron temperature contours are illustrated in Fig. 18. Initially the thermal layer is horizontally

(c)

TIME= 12.053h

S-N

DISTANCE

TIME:

(km)

Fig. 12. Self-focusing of a 13 MHz Gaussian beam at 300 km altitude.

12.106 h

1071

The feedback-diffraction theory of ionospheric heating TIHE = 12.006 neuIS PHASE PATH (KM,

TIME = 12.053 N0LRS PHRSE PATH IKMl

TINE : 12.106 N6l.m PHASE PRTH IKKI

270

E 260 g 250 t; 2 240 230

200

-40

-30

-20

-10 0 S-N OISTRNCE

IO ,KN,

20

30

40

-40

-30

-20

-10 0 S-N OISTRPTE

IO 20 IKNI

30

40 -40

-30

-20

-IO 0 S-N OISTANCE

10 20 IKHI

30

40

Fig. 13. Phase front for the self-focused,13 MHz beam shown in Fig. 12.

stratified [Fig. 18(a)]. After theradio beam is turned on, the temperature between 230 and 290 km altitude becomes beam aligned. The region below 230 km and above 290 km is field aligned due to the influence of thermal conductivity. The evolution of the electron population is shown in Fig. 19. After 40 s of radio wave heating, the plasma has not had enough time to reach an equilibrium distribution [Fig. 19(c)]. At this time, both beamaligned and field-aligned structures are forming. If the computation could be carried out for longer times, the plasma structure would eventually mimic the irregular structure of the radio wave heating beam.

TINE = 12.006 nems ELFCTRBN TEMPERRTURE tKl

6. DISCUSSION

In both plane wave and beam simulations, the focused power pattern becomes irregularly structured for 100 PW m-’ power levels. This structure is due to diffraction of the radio beam near the focus. Initially, the radio wave energy is deposited near the peak of the F-layer forming a reduction in plasma concentration which acts like a convergent lens. The size and focal length of the lens are a function of the plasma concentration, the radio frequency, the power density and the dimension of the radio wave beam. The pattern near the focus of a lens may be estimated

TIHE = 12.106 WJRS ELECTRBN TEMPERRTURE ,Kl

TIK = 12.053 mJRs ELECTRBN TENPERRTURE [Kl

270 z 260 9 250 t 2 240 230

-40

-30

-20

-10 0 S-N OISTRNCE

\VS.W tKNl

30

40

L -40

,

,

-30

-20

V4P. ,

. * (L I . 144F.l

-10 0 ‘W4yo S-N OISTANCE IK”,

3J

40

t -90

, -30

, -20

v40. I 0 -10 S-N DISTRNCE

*Y I w.4Rl IKNl

Fig. 14.Electron temperature enhancements produced by the 10e5 W m- 2beam illustrated in Figs 12 and 13.

I4SO.J M 40

P. A. BEXNHARDT and L. M. DUNCAN

1072

300

II* ELECTRiW

= 12.m IemS C9NCENlRRTI9N (IO’O,~~

11% = 12.053 li9lBS ELECTRBN CENCENTRRTl9N OO’Om-3)

TIBIE = 12.lrs Iam ELECTRBN CBCENTRRTI~

(tdom-3)

290

10 IIW

S-N OISTRNCE

2O

m

40

-40

-m

-20

-10 0 S-N (IISTIWZE

10 IKHI

20

SO

40

-40

-30

-20

-10 0 S-N OISTfWtE

IO ItWl

2O

30

10

Fig. 1.5.Modi~~ation of electron concentration by lo-’ W m-’ beam. Local maxima are labeled ‘H’.

from classical diffraction by a slit and collimating lens. The intensity at the focus of a lens is proportional to [sin (x)/xl2 (Fig. 20). The location of the first zero in the pattern is A-o=---,

%fo D

(24)

where 1 is radio wavelength,f, is the focal length of the lens, D is the diameter of the lens, and x0 is the position of the zero at the focal plane. For the example the focused beam at 100 PW m-‘, the diffraction parameters are I = 23 m,

f.= 50 km,

and D = lOkm,

yielding a diffraction scale size x0 = 230 m. Radio wave heating of the ionosphere produces an imperfect lens. The sidelobes of the diffraction pattern for an imperfect lens are more intense than the lobes of the pattern for a perfect lens (see BERNHARDT and ROSA, 1977). Equation (24) provides only an approximate estimate of the spacing between the lobes in a diffraction pattern. It does, however, provide a DA

means of estimating small-scaie plasma structure produced by focused diffraction during underdense heating. The major uncertainty with using equation (24) is the focal length (fo).The focal length depends on

TIUE : 12.006 HEURS LEGIOfPBHER JENSITI tY/H..2))

S-N

DISTRNCE

iKN!

5-N

OISTRNCE

TIME : 12.011 LEGlOLPBUER DENSilr

HEURS CU,tt~.Zl,

W‘t

Fig. 16.Self-focusing and diffraction of a 10V4W m- * Gaussian beam. Radio maxima are labeled ‘I-I’.

wave

frequency is 3 MHz. Local

The feedbackdiffraction TlME*l2.000

h

theory of ionospheric heating

TIME=

12.006

h

-i

1073

(cl

TIME=IZ.OII

h

-z -! -4 -5 -6 -7 -8 -9 -10 -li

Fig. 17. Power density at 300 km altitude. TINE = 12.006 HWRS ELECTRBN 1E~ERRT~ tK1

(b)

-40

-30

-20

-10 0 S-N OISTRNCE

10 IKIV

M

TIME = 12.011 HRWS ELECTRBN I~~Eff~lURE fK1

(c)

30

40 S-N OISTANCE

,KHl

Fig. 18. Electron temperature enhancements produced by the low4 W rnwz beam. (aI JaO

TIME = 12.000H&IRS

ELECTRON WNCENTRRT iBN GdO m-31

(bf

1IW = 12.006 H0ttRS ELECTRBN CENCENTRRTIBN iiO’%~-~i

fc)

TltlE = 12.011 H0uRs ELECTRBN CBNCENTRRTIBN (lOlO,-3)

290

zoo

-40

-30

-20

-10

0

S-N OISTANCE

10 IKMI

20

SO

40 -40

-30

-20

-10 0 S-N OlSTRNCE

IO 20 IKHI

30

40 -40

-30

-20

-10 0 S-N DISTANCE

IO 20 iKFtk

Fig. 19. Electron concentration changes resulting from heating with a 10v4 W m-’ beam. Local maxima are labeled ‘H’.

30

40

P. A. BERNHARDT and L. M. DUNCAN

1074

(sin+

Fig. 20. Diffraction pattern at the focus of a perfect lens.

In all of our computer simulations, the effects of ponderomotive forces were completely negligible in comparison with thermal forces. This is consistent with the results of GUREVICH (1978). Gurevich states that striction (i.e. ponderomotive) forces are about a factor of lo4 weaker than thermal forces. The computer simulations showed that, for the two specific problems in a fixed ionosphere, a threshold existsforpowerlevels between lOand lOOpWm_*.The simulations at 10 ptw m-’ were carried out to the point of equilibrium. The high power (100 pW m-‘) computations had to be halted prior to the attainment of equilibrium because the scale size of the diffraction pattern was becoming comparable to the computation mesh size. Future use of the computer model will concern overdense radio wave heating. Heating with a beam frequency below the critical frequency should produce a concave reflector in the bottomside ionosphere. The reflected beam may focus producing a diffraction pattern similar to the one produced by underdense heating. Consequently, diffraction may play a determining role for the generations of plasma irregularities during overdense radio wave heating. These irregularities may be aligned with the filamented, reflected radio beam.

the relative values of wave and plasma frequency which governs (1) the energy deposition

(2) the refraction

in the ionosphere

and

of the wave.

Acknowledgement-This Department of Energy.

work was supported by the U.S.

REFERENCES BANKSP. M. and K~CKARTSG. BERNHARDT P. A. and DA ROSA A. V. BERNHARDT P. A. and PARK C. G. BERNHARDT P. A., PONGRATZM. B., GARY S. P. and THOMSEN M. F. CHEN F.F.

1973 1977 1977 1982

Aeronomy, Vol. A and B. Academic Press, New York. Radio Sci. 12, 327. J. geophys. Res. 82, 5222. J. geophys. Res. 81,2356.

1974

CRAGIN B. L. and FWER J. A. FEJER J. A. GUREVICHA. V.

1974 1979 1978

HASTINGS J. T. and ROBLER. G. MELTZ G., HOLWAYL. H., JR. and TOMUANOVICH N. M. PARK C. G. and BANKSP. M. PARK C. G. and BANKSP. M. PERKINSF. W. and GOLDMAN M. V. PERKINSF. W. and ROBLER. G. PERKINSF. W. and VALEOE. J. WALKERJ. C. G. WEATHERALL J. C., SHEFXINJ. P., NICHOLSON D. R., PAYNE G. C., GOLDMANM. V. and HANSENP. J. YEH L. C. and LIU C. H.

1977 1974

Introduction to Plasma Physics, pp. 256259. Plenum Press, New York. Radio Sci. 9, 1071. Rev. Geophys. Space Phys. 17, 135. Nonlinear Phenomena in the Ionosphere. Springer, New York. Planet. Space Sci. 25, 209. Radio Sci. 9. 1049.

1974 1975 1981 1978 1974 1965 1982

J. geophys. Res. 79,466 1. J. geophys. Res. SO, 2819. J. geophys. Res. 86,600. J. geophys. Res. 83,161l. Phys. Rev. Left. 32, 1234. J. atmos. Sci. 22,462. J. geophys. Res. 87, 823.

1972

Theory of Ionospheric Waves, pp. 228-231. Press, New York.

1976

Tech. Rep. No. 18, SU-SEL-76-013

Reference is also made to the following unpublished material: ANT~NIADIS D. A.

Academic