JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. A8, PAGES 12,073-12,091, AUGUST 1, 1992

Numerical Test of the Weak Turbulence Approximation to Ionospheric Langmuir Turbulence

ALFRED HANSSEN AND E. MJOLHUS

Institute of Mathematical and Physical Sciences, University of Troms½, Troms½, Norway

D. F. DuBoIS AND H. A. ROSE

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico

The standard weak Langmuir turbulence approach to explain the artificial plasma line in ionospheric radio modification experiments is examined. We compare solutions of a weak turbulence approxima- tion (WTA) derived from a version of the one-dimensional driven and damped Zakharov system of equations (ZSE) with solutions to the same full ZSE. The electromagnetic pump field is modeled as a long-wavelength parametric driving term. We found that from a certain distance below the O mode reflection level the wave number saturation spectra computed from the WTA agree qualitatively with those from the ZSE for weak driving strengths, in the sense that the number of cascade lines increases with increasing pump strength. However, in general, the number of cascades apparent in the WTA solutions is larger than that predicted from the full ZSE. At higher intensities of the driver the saturation spectra from the ZSE differ from the WTA cascade spectra, in that a truncation of the cascade sets in, with a subsequent filling in of the bands between the cascades. This truncation takes place far before the ZSE cascade spectra reach the so-called "Langmuir condensate," contrary to earlier conjectures based mainly on dimensional analysis arguments. In the reflection region a qualitatively different process takes place: temporal cycles of large ensembles of localized events; nucleation of cavitons, collapse, and burnout constitute the basic elements of the turbulence in this region of space. No WTA exists for this region. Our findings are discussed with respect to the experiments performed at Arecibo and Troms½, the conclusion being that the ZSE yields results closer to observations than does the WTA, in all regions of space.

1. INTRODUCTION

The conventionally accepted theory for the plasma pro- cess responsible for the artificial plasma line backscatter in ionospheric radio modification experiments [e.g., Carlson et al., 1972; Hagfors et al., 1983] has been the weak Langmuir turbulence approximation, or cascade theory, as developed by, for example, Kruer and Valeo [1973], Fejer and Kuo [1973], and Perkins et al. [1974]. This interpretation has recently been debated. First, certain features are observed in the experiments that are not consistent with this kind of theory, in particular, the observed height in the ionosphere from which the plasma line comes [Muldrew and Showen, 1977; Isham et al., 1987], the broad spectral background in which the discrete lines are embedded, and the "free-mode" spectral features observed at early times of cold start exper- iments at Arecibo [Djuth et al., 1986; Cheung et al., 1989, 1991]. Second, the above mentioned picture has recently been challenged theoretically, by Doolen et al. [1985], Russell et al. [1988], and DuBois et al. [1988, 1990, 1991], who claim that the weak Langmuir turbulence theory has a very limited validity range close to the reflection level and must be replaced by a theory of strong Langmuir turbulence in this region. According to the latter theory, the turbulence in the reflection region consists of a large number of local- ized cycles of nucleation of cavitons, collapse of Langmuir wave packets, and burnout.

The objective of this paper is to report a study of the

Copyright 1992 by the American Geophysical Union.

Paper number 92JA00874. 0148-0227/92/92JA-00874505.00

question of internal consistency mentioned above. The set of equations known as the Zakharov model [Zakharov, 1972] is a well-studied mathematical model that contains the basic elements which are of importance in Langmuir turbulence driven by an electromagnetic field oscillating near the plasma frequency. With respect to the present application of iono- spheric radio modification, this model was first invoked by Weatherall et al. [1982], and the same model was also used by the Los Alamos group [Doolen et al., 1985; Russell et al., 1988; DuBois et al., 1988, 1990, 1991]. This model certainly contains the relevant initial parametric instabilities [Nichol- son and Goldman, 1978; Nicholson, 1983], and furthermore it allows the construction of a weak turbulence approxima- tion (WTA) in a way completely similar to that developed by Kruer and Valeo [1973], Fejer and Kuo [1973], and Perkins et al. [1974], as will be demonstrated below.

DuBois et al. [1990] showed that strong caviton-caviton correlations in the overdense region (above the reflection height) could produce power spectra having a line structure similar to that of the observed radar spectra. This model was based on the then accepted experimental results, indicating that the RF enhanced radar backscatter came from the reflection height [e.g., Muldrew and Showen, 1977; Isham et al., 1987]. Recent high-resolution experiments [Djuth et al., 1990; Fejer et al., 1991] have shown that the observed echoes in many cases start at the reflection height and then spread downward. These findings, together with relaxation measurements of decaying turbulence, have ruled out the caviton-lattice model of DuBois et al. [1990] as a plausible explanation of the line structure in the observed ionospheric spectra. These discrepancies were pointed out by the same authors [DuBois et al., 1991]. Thus in the present paper, as

12,073

12,074 HANSSEN ET AL..' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

in the work by Hanssen and MjOlhus [1990] and DuBois et al. [1991], we examine from a theoretical point of view the distribution of RF-generated turbulence extending a few kilometers down from the O mode reflection height.

Our specific aim will be to numerically solve the Zakharov [1972] system of equations (ZSE) and the corresponding WTA derived from the ZSE, in order to compare the wave number spectra in the saturated state for the two models. A study of this kind was recently reported by Payne et al. [1989]. However, these authors considered the initial value problem for a conservative version of the Zakharov model. The present work differs by considering a version of the Zakharov model which contains damping terms and which includes a parametric driving oscillating electric field.

The emphasis will be on the effect of the parameter

an() = a0- where &0 is the frequency of the driving field and &p (z-) is the local plasma frequency at height • in the ionospheric plasma. Since &p depends on height, All is a measure of the height relative to the reflection level (&0 = &•) of the ordinary polarized driver, being positive (negative) below (above) the reflection level. (Note that DuBois et al. [1990, 1991] re- ferred to Af/ as •o 0 = o) m -- O)p, where o) m denoted the heater frequency.)

For a positive value of all, corresponding to a height below the reflection level, a decay instability will start up around a certain wave number k l (all) selected uniquely by the resonance condition and the dispersion relation. A natural expectation is that when all is large enough, this instability will saturate by cascading into successively smaller wave numbers of opposite signs -k'• < •l, k-• < - if2, where I•j+l I < I,1, each satisfying the resonance condition for parametric decay from the former mode, and the dispersion relation. This picture was indeed qualitatively verified by our solutions to the full ZSE, although the number of cascades turned out to be consistently smaller than that predicted by the corresponding WTA.

On the other hand, in the reflection region all = 0 (see section 4 for a precise definition of the reflection region) the expectation is that the caviton nucleation process described by Doolen et al. [ 1985] will dominate. Our numerical results confirm these earlier findings and also show that the wave number spectrum produced in the all = 0 region is broad enough to contain the wavelengths detected by the European Incoherent Scatter (EISCAT) VHF (224 MHz) radar, in northern Norway (k•vH F = 9.4 m-l), and the Arecibo (433 MHz) radar, in Puerto Rico (k•Arecib o = 18 m-l). Our simulations indicate that the wavelength detectable with the EISCAT UHF (933 MHz) radar (k--uHF = 39.1 m -1) is probably not excited in this region for most naturally occur- ring geophysical conditions. However, for large background plasma densities the Debye length is lowered, possibly permitting the excitation of Langmuir turbulence at UHF wave numbers, in which this process could possibly be detected by this radar. For all radars, also nonresonant heights will scatter at the radar wave numbers and add signals to the total height-integrated spectrum. These non- resonant signals are, however, orders of magnitude weaker than those from resonant heights, but they carry a large amount of the total power contained in the plasma lines (especially for the smaller values of kAD), as the broad background in which the discrete lines are embedded. This

broad background is not accounted for in weak turbulence approximations.

A preliminary version of this investigation was presented by Hanssen and Mj½lhus [1990]. However, all the numerical data have been recomputed in the present paper, using a larger number of Fourier modes, i.e., with much better spectral and spatial resolution. Also, the temporal resolution is improved in this paper, for those simulations where this proved to be necessary. We have also expanded the paper by adding many new diagnostics and considered a large number of new cases, compared to what appeared in the work by Hanssen and Mj½lhus [1990]. Unfortunately, there was an error in the WTA wave kinetic model in that paper, such that all appearances of/2 i should read 2 vi. The correct version is given by equations (8)-(10) in this paper. The missing factor of 2 led to a larger number of cascades from the weak turbulence theory than should have been the case.

2. ZAKHAROV'S MODEL AND ITS CASCADE THEORY

The version of the one-dimensional Zakharov system of equations that are appropriate for the present work reads [e.g., Payne et al., 1984; Hanssen and MjOlhus, 1990]

•-• + 12 e * + (all -- n) + 0--• E = nE o - {nE) (2)

+ 2vi * ' IE + Eo 2 Ot •x D n = • (3) Ox 2

where E(x, t) is the complex envelope of the potential electric field associated with the Langmuir waves, E0 is the infinite wavelength driving electric field, and n(x, t) is the ion density perturbation, or the quasi-neutral slow time scale electron density perturbation. The total electric field is represented as «(E + E 0) exp (-io•ot) + c.c. (Note that some authors choose to envelope around %, instead of •o0, which causes the all to disappear from the left-hand side of (2) and instead to reappear in the driver as E0 exp (-iallt), as the work by Payne et al. [1984].)

In (2) and (3), dimensionless variables are used, and they are related to their dimensional counterparts (denoted by a tilde) by [Nicholson, 1983] t = /7T, x = Z/X, E = t•/O, and n = t•/N, where

T= (3/2•q)(M/m)(1/&•,) X = (3/2)(M/m•q)1/2• D

0 = rl (m/M) 1/2(16n0K Te/3 e o) N = 4rlmno/3M

in SI (MKS) units. Here r/ = (TeTe + TiTi)/Te, where T e and T i are the electron and ion temperatures, respectively, 'Ye and Yi are the ratios of specific heats (we assume isothermal electrons, 'Ye = 1, and adiabatically responding ions, Yi = 3), m and M are the electron and ion masses, respectively, •0 = 8.854 x 10 -12 F/m is the dielectric permittivity of free space, K = 1.38 x 10 -23 J/K is Boltz- mann's constant, no is the background plasma density, and AD is the Debye length.

In this paper we have chosen the model ionosphere (F region) to consist of 60% O + ions, 30% O•- ions, and 10% NO + ions, leading to an effective mass ratio of M/m = 4 x 10 4 (effective ion mass number 22).

In Table 1 we list the scaling values T, X, and O, together

HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,075

TABLE 1. Scaling Values for Time (T), Space (X), and Electric Field (O), for the Set of Plasma Parameters Applied in This Paper

Set rte, m -3 Te/T i f•, MHz •D, mm T, ms X, m O, V/m I 2 x l0 II 2 4.01 6.90 0.95 1.31 0.72

II 8 x l0 TM 1 8.02 2.44 0.30 0.37 1.63

We applied T i = 1000 K in both sets I and II.

with the plasma frequency and the Debye length, for the plasma parameters applied in this paper, and with M/m = 4 x 10 4. Henceforth, we will refer to a plasma with parameters pertaining to the first or second row in Table 1 as I or II, respectively.

The dimensionless electron and ion damping operators re(X) *, vi(x) *, are nonlocal (convolutions) in real space but local (multiplications) in wave number space. There is no need to specify these damping operators in x space, because we solve the linear parts of these equations in Fourier transform space. Hence we need only know the functional form of these operators in wave number space. The param- eter All is the before mentioned frequency mismatch (1), measured in units of T.

The angle brackets in the last term of (2) mean spatial mean value. We have chosen (n) = (E) = 0 by definition, which means that the term (nE) must be included for consistency. This formulation ensures that the driving field E0 is externally controlled. If E were allowed to have a nonvanishing mean value, the driving field would be E0 + (E), i.e., it would be self-modifying, which would not be appropriate for the process that we wish to investigate.

As shown by Zakharov [1972], the dimensionality of the problem plays a crucial role in the analysis of collapse, because of the way the ponderomotive force scales with the volume of cavities. For the undriven case an inertial collapse occurs only in two and three dimensions. In two dimensions the initial energy must exceed some threshold value to initiate the collapse, whereas in three dimensions, no such threshold exists. In the forced case, however, such as the one studied here, a driven collapse can occur also in one dimension. The important mechanism in this context, which is also present for the one-dimensional dynamics, is the nucleation of cavitons [Doolen et al., 1985]. Once the nucleation process is initiated, the full nucleation-collapse- burnout cycle can take place. Comparison between spectra from one- and two-dimensional solutions to the driven Zakharov system shows a remarkably good agreement, at least on a qualitative level. A major advantage of computing one-dimensional rather than two-dimensional solutions is that the one-dimensional case permits use of the full physical mass ratio with excellent spectral, spatial, and temporal resolution. It has so far proven prohibitively expensive and time-consuming to obtain the same resolution in two dimen- sions for the ionospheric mass ratio.

A recent work by Mounaix et al. [ 1991] shows analytically that the Zakharov system has a wider range of applicability, •/noKTe < (•D)-2, than what is usually stated, l•/noK T e < 1, for T e >> T i. Here l•/nogT e is the ratio of Langmuir wave energy to thermal energy, and k is the characteristic wave number of low-frequency perturbations. This criterion was also discussed by Newman et al. [1990] and verified numerically in the same paper. These authors showed nu-

merically that the extended validity criterion holds also when T e --• Ti, the case being studied in this paper. Another important validity criterion for the Zakharov system, worth mentioning, is I&•/ti0 << 1, where /tr• is the ion density fluctuation and r• 0 is the background plasma density. This criterion might be violated during short time intervals, when strong collapse takes place. When applying the Zakharov model (equations (2) and (3)) including a frequency mis- match, l($0 - $•,)/•,1 << 1 must also be added to the validity criteria.

A dispersion relation which determines the growth rate of the parametric decay instability [DuBois and Goldman, 1965] can easily be deduced from the Zakharov model (equations (2) and (3)): Assuming the existence of a pump wave (subscript 0), high-frequency daughter waves (sub- script 1), and low-frequency daughter waves (subscript 2), we can decompose E and n according to

E = E © exp (iOo) + E (1) exp (iOl) + E (-•) exp (-iOn)

n = n2 exp (i02) + n• exp (-iOn) (4)

where the phase Oj = kjx - cojt (j = 0, 1, 2) satisfies the usual resonance condition for frequencies and wave num- bers,

= 00- (5)

In (4), coj are assumed complex, while kj are real; 0y denotes complex conjugate of Oj. By substituting E and n from (4) into (2) and (3) and eliminating the low-frequency terms, we derive the following dispersion relation for the Langmuir waves:

CO 1 + iv e(kl) + All- k• 2 = (ko- k•)2l E(ø) 2[(COo- CO•)2

- 2i(CO0- COl)vi(ko- k•) - (ko- kl)2] -1 (6) When we put E © = E0, k0 - 0, and COo = 0, (6) becomes the dispersion relation for the parametric decay of the electromagnetic pump wave. However, the formulation (4) also allows a Langmuir wave of amplitude E © , frequency shift COo, and wave number k0 to be the mother wave. One could also use COo % 0 in the pump, but as easily seen from (2) and (3), this frequency can be included in All, so one can with no loss of generality put COo - 0 in the external pump.

Neglecting the nonlinear coupling and damping terms and invoking the resonance condition (5) in the dispersion rela- tion (6), one arrives at the expression

kl(Af/) = _+«[(4Af/+ 1) •/2- 1] (7)

for the wave number of the primary decay. Following Perkins et al. [1974], we solve for the imaginary

part of the Langmuir wave frequency COx from the dispersion

12,076 HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

relation (6), in order to obtain an expression for the growth rate of Langmuir waves due to the parametric decay insta- bility. Following Kruer and Valeo [1973] and Perkins et al. [1974], the additional approximation of neglecting the imag- inary part of w• when it occurs in the denominator of (6) is applied to obtain a closed form for the growth rates. On the basis of IM(w•), and assuming that the wave phases are random and statistically independent, we can formulate the following statistical one-dimensional weak turbulence ap- proximation for the Langmuir waves, in a way parallel to that of Perkins e! al. [1974] (see also Kadomtsev [1965] and Zakharov [1989]):

OI(k, t) •=2

Ot

where

To(k)- re(k)+ Z M(k, k')I(k', t)]I(k, t) k'

(8)

2k2(A• _ k2)lEo 2 To(k)--12i(k) [(Al•_k2)2_k212+4(Al•_k2)2v•(k ) (9) M(k, k') = 2vi(k- k')(k '2- k2){(k- k')2[(k + k') 2- 1] 2

+ 4(k- k')2v/2(k- k')} (10)

where I(k, t) = (IE(k, t)l 2) is the ensemble-averaged spectral intensity at time t.

We obtain To(k) by assigning k0 = 0, E © --> E0, and k• -• k, while 7(k, k') = M(k, k')I(k', t) is obtained by assigning E © -• E(k'), k o -• k', and k• -• k. We identify To(k) as the growth rate of the primary decay mode given by (7), due to the electromagnetic pump wave, and M(k, k') describes the coupling between a Langmuir wave at wave number k and another Langmuir wave at k'. Note that in (6), % and vi are considered k dependent, permitting us to include kinetically determined Landau damping rates in the model. By inspection we see that the coupling elements M and 70 have the correct symmetry properties [Kruer and Valeo, 1973], M(k, k') = -M(k', k) and 70(k) = 70(-k), ff we assume vi(k ) = •i(Ikl).

In deriving the present WTA (equations (8)-(10)) from the ZSE (equations (2) and (3)) we make a transition from a dynamical to a statistical description of the excited turbu- lence [e.g., Kadomtsev, 1965; Nicholson, 1983; Zakharov, 1989]. In the statistical description (WTA) we have no information about the Langmuir wave phases, since the WTA is given in terms of the wave intensities alone. A crucial difference between the full ZSE model and its corre- sponding reduced WTA description is that in the latter we have to assume that the linear dispersion relations for Langmuir waves and ion acoustic waves always apply, whereas the ZSE model is not limited by such a serious constraint. The turbulence described by a WTA therefore consists of a superposition of plane monochromatic waves with different wave numbers. It should be obvious that this representation can never describe any localization phenom- ena. To derive our WTA, we also invoked the assumption, commonly used in ionospheric applications, that the damp- ing of ion acoustic waves is so large that no significant enhancement of ion fluctuations occurs [e.g., Perkins et al., 1974; Zakharov, 1989]. Mathematically, this is expressed as

max {[7o(k)- v•(k)], f M(k, k')I(k') dk'}<<vi(k) This condition is not necessarily met at moderate to high pump levels. Violation of the above condition indicates that it is necessary to include a wave kinetic equation also for the ion acoustic waves.

We also remind the reader that the threshold electric field for onset of the parametric decay instability is readily obtained from (8) by neglecting the nonlinear term and assuming OI(k)/Ot = 0 [DuBois and Goldman, 1965]. The result is

Et2h = 2ViOVe(kl) (11) where the ion acoustic damping decrement is assumed to be vi(k) = •i01kl, where vi0 is a constant coefficient depending on the plasma parameters and k•, the wave number of the primary cascade mode, is given by (7). For later reference we define the ratio P between the pump power and the threshold power for onset of the parametric decay instability as

P = Eo2/Et2h (12)

3. NUMERICAL RESULTS

The set of coupled nonlinear partial differential equations (2) and (3) were solved numerically by a second-order explicit pseudospectral method [Payne et al., 1983; Canuto et al., 1988]. The linear parts were solved in Fourier trans- form space, whereas the nonlinear parts were solved in real space, before transforming back to k space. Thus the solu- tions are the time dependent Fourier coefficients i•(km, tn), h(km, tn), k m -- mAk (m = -N/2, ..., N/2 - 1), and t n -- nat (n = 0, 1, 2, ...), for fixed k resolution (Ak) and time step (At). N is the number of discrete equidistant grid points, where the grid spacing is Ax = L/N and Ak = 2,r/L, in real and transform space, respectively, and L is the length of the (periodic) turbulence cell.

The corresponding wave kinetic equation, equations (8)- (10), was solved by means of a standard vectorial explicit Runge-Kutta method [e.g., Canuto et al., 1988], also of second order. In order to obtain the saturation spectra from the weak turbulence approximation, we invoked the averag- ing procedure introduced by Fejer and Kuo [1973]. It con- sists of restarting the integration procedure several times, after time averaging the results over a subinterval of the total integration time, and with the last obtained time average as the new initial value. In this way, a very fast convergence to the stationary turbulent state is achieved.

From a computational point of view the discretized Za- kharov model is faster to solve than its discretized WTA, for the same number of modes. The Zakharov model require --•N log2 N multiplications per time step, while the WTA require --•N 2 multiplications, where N is the number of modes. One can reduce the number of operations needed in solving the WTA, by utilizing the fact that no wave number upshift mechanism or spectral broadening effects exist in this approximation. This permits us to truncate Fourier space at kma x = k• + 8k, where 8k -> 8k•, where 8k• is the half width of the primary cascade spectral line.

The solutions presented are well resolved; we applied 2048 and 4096 modes for the pseudospectral method, using the

HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,077

larger number of modes when the wave number spectra were expected to be very broad. The large number of modes ensured that we could resolve all important spectral features and at the same time probe down to small spatial scales. Aliasing errors were first eliminated in the standard way by zero padding of 1/3 of the Fourier modes [e.g., Canuto et al., 1988]. Later tests have shown that the aliasing errors are negligible if only some of the modes that extend far into the Landau damping regime are included.

We applied initial conditions which correspond to a cold start, and hence at time t = 0 we start from a low background noise level in the Langmuir electric field Fourier components, E(k, t = O) = A exp (2•ri0k), where A << E 0 (typically, A --- 10-4-10 -2) and 0k is a random variable uniformly distributed on [0, 1]. The spectra we find in the saturated state are not affected by the particular value A we assign to the initial fluctuations. The ion density fluctuations at t = 0 are assumed to vanish, n(k, t = 0) = 0, and the same assumption is made for their time derivative, 0 trt(k, t = 0) = 0. Tests with nonzero initial fluctuations in one or both of the ion density variables show no differences in the saturated state of the turbulence. Our finding is that the saturated turbulent state does not have any memory of its initial state or the linear growth phase. The particular choice of initial values affects only the transient development of the turbulence, on a time scale of the order of 1 Zakharov time unit.

We would like to remind the reader of the role played by the initial conditions in the Zakharov model' The initial plasma state does not need to contain preexisting cavitons in order to produce a caviton collapse-dominated turbulence, and the initial state does not need to contain ion acoustic waves in order to produce propagating ion acoustic waves at later times. In the underdense region AI/>_ 0, the saturated solutions to the ZSE are not sensitive to the particular choice of initial conditions if the oscillating two-stream instability and/or the parametric decay instability are allowed to de- velop. If a nucleation-collapse-burnout state is reached (by any scenario), the dynamics contained in the Zakharov model creates the right number of nucleation centers per unit length, and they behave in a self-organized fashion. If the initial condition consists of preexisting cavitons, the solu- tions to the ZSE exhibit a transient phase with coalescence, decay (splitting), or creation of cavitons, in order to produce the required number of cavitons for the applied driving field. By rewriting the ZSE as a set of coupled nonlinear integral equations [e.g., Payne et al., 1983], the initial conditions appear explicitly, and it is easy to see that nonzero initial conditions in at least one of the quantities l•(km, t = 0), h(k m, t = 0), or O th(km, t = 0) are required, in order to generate oscillations in both E and n at later times. The numerical results of this paper act as good examples of both scenarios, and section 3 of DuBois et al. [1990] contains a lengthy discussion of the initial development of Langmuir turbulence. The Los Alamos group [Russell et al., 1988; DuBois et al., 1988, 1990, 1991] sometimes applied initial conditions containing cavitons in their two-dimensional so- lutions to the ZSE, only for numerical convenience. As argued, the results they obtained are not affected in any way by this particular choice of initial state, as long as all results are produced after the transient phase is over.

All averaged spectra presented in this paper are taken after the linear growth phase is over and hence when the

nonlinear effects are dominant. Besides that, in the standard weak turbulence theory that we aim to test, the ion acoustic waves do act only as effective scatterers of Langmuir waves, so there is no equation describing the ion dynamics in that particular model. It is, however, possible to include ion acoustic waves in weak turbulence theory (see, for example, Kadorntsev [1965] or Payne et al. [1989]).

In the Runge-Kutta code for (8) we solved the same number of coupled equations as there were nonzero-padded modes in the pseudospectral code solving (2) and (3). The initial condition was taken as I(k, t = 0) = A 2, for the same choice of A as used in the ZSE initial condition.

Our solutions are long-time simulations of the dynamics. A typical run represents between 20 and 50 Zakharov time units in the fully developed turbulent state, or about 20-50 ms for parameter set I of Table 1 or 6-15 ms for parameter set II, in physical units. The larger of these time intervals is rather long compared to some of the experiments being performed. We have applied long integration times in our simulations in order to obtain good statistics, i.e., to get a better representation of the ensemble averages. This is necessary, since our numerical simulation cell is much less than any experimental radar volume.

As phenomenological models of the damping operators in Fourier space, we use for the ions the kinetically determined linear Landau damping of ion acoustic waves (in dimension- less units),

vi(k ) = l•io k (13)

where the coefficient rio is given by

(•) 1/2[(•//)1/2 (r.•te.) 3/2 (r e •) rio = + exp 2Ti For the electrons we apply the kinetically determined linear Landau damping of Langmuir waves, and we also include collisional damping [Ichirnaru, 1973],

b' cø11 (•) 1/2 (•) 4 (•mm) 5/2 1 Pe(k) = +

ß exp 8 •/m k 2 (14)

where we throughout this paper use Vcoll = 1.4 for the collisional damping. Comparison between solutions to a Zakharov model and a hybrid particle simulation [Clark et al., 1990] indicates that the ion kinetics is adequately mod- eled by ion acoustic damping operators of the form applied in this paper, for the parameters of interest here. Note that formula (13) for the linear ion damping decrement is derived under the assumption Te/T i >> 1, whereas we apply it for T e --• T i. It does, however, serve its purpose in this paper, where the comparison of two models is the main purpose. For the few cases where very strong collapse occurred (implying broad k spectra), we arbitrarily modeled the high-k (kA D > 0.5) part of the Langmuir wave Landau damping as --•k 2. This high-k extension to the Landau damping ensured that all Langmuir wave energy was dissipated at small scales [Zakharov and Shut, 1981; Russell et al., 1986], thus avoid- ing an unphysical explosive buildup of energy in the turbu- lence cell. The choice k s, for some exponent a, is suggested by analytic analysis of collapse [Zakharov and Shut, 1981],

12,078 HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

4OOO

3O00

2000

1 ooo

o o

LINEAR LANDAU DAMPING ......... i ......... i ......... I .........

•(iv) / • (i) M/m=1836 / • (ii) M/m= 10000 / • (iii) M/m=20000

100 200 300 400 k

Fig. 1. Linear Landau damping rate (in Zakharov frequency units) versus wave number (in Zakharov units), for four different mass ratios: (curve i) M/m = 1836, (curve ii) M/m = 10,000, (curve iii) M/m = 20,000, and (curve iv) M/m = 40,000. Case iv corresponds to the cases considered in this paper, for r• -- 1 + 3Ti/T e = 2.5 (parameter set I of Table 1).

numerous particle-in-cell (PIC) simulations [Zakharov et al., 1988, 1989; D'yachenko et al., 1988; Newman et al., 1990], theoretical work on nonlinear transit time damping [Robin- son, 1991], and direct observations of collapse in laboratory experiments [Wong et al., 1977]. The self-consistent treat- ment of nonlinear kinetic effects is, however, an important and fundamental problem; in particular, it could explain the observed enhanced airglow effects [Bernhardt et al., 1991, and references therein] during RF heating of the ionosphere. The fast electrons responsible for the observed airglow are presumably produced by collapsing cavitons, parametric decay instabilities, and multiple acceleration of electrons passing near or through collapsing cavitons [Gurevich et al., 1985]. The self-consistent analysis of nonlinear kinetic ef- fects is, however, beyond the scope of the present paper. PIC simulations show that the turbulent processes examined in this paper are generic and that the full kinetic description might modify the details but not the gross properties of the excited turbulence. Therefore at the present level of analysis it is fully justified to model the dissipation operators as described above. If no turbulent wave numbers are excited beyond the maximum of equation (14), it is sufficient to use the linear Landau damping formula as it stands. This is indeed the case for most of the results presented in this paper.

It is important to note that in the dimensionless Zakharov model (equations (2) and (3)) the mass ratio enters only through the damping operators. In Figure 1 we show how the linear Landau damping of Langmuir waves (without the high-k extension) behaves for four different mass ratios. The case considered in this paper, M/m = 4 x 10 4 , corresponds to curve iv in Figure 1. Note the shift toward larger wave numbers as we increase the mass ratio. It is because of this shift toward larger k values with increasing mass ratio that two-dimensional well-resolved solutions to the Zakharov system are difficult to obtain for the physical mass ratio.

The amount of Langmuir energy dissipated per time unit is given by [Doolen et al., 1985]

OL = 2 • ve(k)([E(k)l 2) (15) k

A convenient measure •of the amount of .: Langmuir wave energy dissipated by collisions is then given by the ratio

R = 2Vec • <IE(k)I2>/DL (16) k i

We routinely calculate this ratio from our ZSE solutions, and the numbers obtained are listed in the figure captions.

Solutions to a linear multidimensional magnetized wave propagation model [Leyser and Thidd,. 1988] show that the pump electric field E0 attains a maximum value of about 1-2.5 V/m in the F region above Troms0, including swelling, ionospheric D region absorption of 10 dB, and collisional damping. For Arecibo the typical maximum field strength is about 1 V/m [Fejer, 1979].

We will compare solutions to the Zakharov model and its weak turbulence approximation for four different choices of the frequency mismatch. Set I of plasma parameters from Table 1 (corresponding to a heater frequency of about 4 MHz) were applied to the mismatch values All = 25, All = 163.59, and All = 570.96. For this set of parameters, All = 25 represents a nonmatching height close to the reflection height, with the primary decay line at k• = 4.5 and with k•Ao = 0.02. Our second case, All = 163.59, corresponds to the case where the primary decay mode, kl = 12.3, is equal to the EISCAT VHF radar wave number and where ffl •o -• 0.06. This defines the "matching height" for the radar [Fejer et al., 1991]. The third choice of frequency mismatch for this plasma state, All = 570.96, corresponds to the matching height for the Arecibo radar; i.e., the primary decay mode k l = 23.4 occurs at the wave number detect- able with this radar, and • •o = 0.12. Set II of parameters from Table 1 (corresponding to a heater frequency of about 8 MHz) were applied to the case All - 219.1. In this case, All - 219.1 corresponds to the matching height for the EISCAT UHF radar, with k• - 14.3 and •o = 0.1. Note that we have chosen rather untypical F region parameters for the EISCAT UHF case, in order to test the limit of small Landau damping for this radar and to save computational time.

We also present solutions to the Zakharov model for All = 0, using both sets of plasma parameters, but in this case, of course, there is no WTA equivalent to our equations (8)- (10).

At this point, one should be aware that in the radar experiments, one actually observes height-integrated fre- quency spectra of time-varying fluctuations at the radar backscatter wave number, k•, rather than the wave number spectra presented in Figures 2-18. In weak turbulence theory, one does, however, use the linear dispersion relation for Langmuir waves to map spectra from wave number space to frequency space, since for weak turbulence, I(k, •o) - I(k)8[•o - •o(k)], where w(k) is the dispersion relation for the wave mode under consideration and I is the wave intensity [e.g., Kadomtsev, 1965]. As mentioned in section 2, the ZSE is not limited to linear dispersion relations, so we also present examples of frequency spectra ka2( E(k R, f; AII)I 2) and (In(k•, f; All)[2), for All = 25 and for All = 163.59 at the EISCAT VHF backscatter wave number kR = kvH F. These spectra thus represent quantities proportional to the radar-measured plasma lines and ion lines, at one single height, without any spatial integration. In contrast to the WTA, we do in this case in general excite a large number of frequency components, or even a continuum of frequencies, at each wave

HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,079

0.25

0.00 0.25

0.00 0.25

,h,

WTA

AFt - 25

,h

ZAK

E 0 -- 0.5

ZAK

P- 1.1

0 5 10 k

Fig. 2. Saturation wave number spectra computed from the

WTA and the ZSE, for All = 25, 1•0_ = 0.5 (P = 1.1), N = 2048, Ak = (1011/2 _ 1)/92, and dt = 3, for plasma parameters I of Table 1, corresponding to a heater frequency of about f0• -• 4 MHz: (top) I(k) = (IE(k)l 2) from the WTA, (middle) ( E(k)I •) from the ZSE, and (bottom) (In(k)l 2) from the ZSE. This case corresponds to a nonmatching altitude just below the O mode heater wave reflection height. The ratio of collisional dissipation to the total dissipation is R= 1.

number. This constitutes a major difference between predic- tions from the full dynamical ZSE description and any WTA prediction. As discussed in the next section, the number of visible cascade lines in the height-integrated plasma line fre- quency spectrum will be roughly the same as the number of

0.50

0.25-

0.00 0.50

0.25

WTA AFt - 25

ZAK

E o -- 0.7

0.00 1.2

I [' ZAK I • 0.6

0.0 0 5 10

k

Fig. 4. Same as in Figure 2, but E 0 = 0.7 (P = 2.2). R = 0.9999.

cascades in the k spectra at matching height. A detailed examination of such power spectra for the available radar facilities will be dealt with in a later publication.

Figures 2-17 all depict wave number spectra computed from the applied ZSE and its WTA. In all figures the top panel shows the saturated ensemble-averaged wave number spectrum I(k) = ( E(k)[ 2) from the WTA, the middle panel shows the corresponding (IE(k)l 2) spectrum from the ZSE (these are the two plots that are to be compared), and the bottom panel shows the saturated ion acoustic spectrum (In(k) 2) from the ZSE. Figures 2-5 show spectra for the

0.4

WTA AFt - 25

ZAK

'-• 0.2- 0=0'6

0.0 1.8 '

•. • ZAK • 0.9

0.0 0 5 l0

k

Fig. 3. Same as in Figure 2, but E o = 0.6 (P = 1.6). R = 0.9999.

0.8

0.0 0.8

WTA A• - 25

ZAK

P-4.5

0 15 30 k

Fig. 5. Same as in Figure 2, butE0 = 1 (P = 4.5). R = 0.83.

12,080 HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

0.50

0.25

0.00 0.50

0.25

0.00 4

WTA A•- 163.59

ZAK

0.7

ZAK P- 2.2

_

0 10 20 k

30

Fig. 6. Saturation wave number spectra computed from the WTA and the ZSE, for AI• = 163.59, E0 = 0.7 (P = 2.2), N = 2048, Ak = [(4AI• + 1) 1/2 - 1]/250, and dt = 5 x 10 -4, for plasma parameters I of Table 1, corresponding to a heater frequency of about f0 = 4 MHz: (top)l(k) = (IE(k)l 2) from the WTA, (middle) <lE(k)lZ> from the ZSE, and (bottom) <ln(k)lZ> from the ZSE. This case corresponds to the matching height for the EISCAT VHF radar for this set of plasma parameters. R = 0.9999.

case Af/= 25' Eo = 0.5 (P = 1.1) in Figure 2, E o = 0.6 (P = 1.6) in Figure 3, Eo = 0.7 (P = 2.2) in Figure 4, and E o = 1 (P = 4.5) in Figure 5. In Figures 6-9 we present spectra at the EISCAT VHF radar matching height Af/= 163.59, for a heater frequency of 4 MHz. We applied Eo = 0.7 (P = 2.2) in Figure 6, E o = 1 (P = 4.5) in Figure 7, E o = 1.25

1.0

0.0 1.0

0.0 15

Fig. 7.

lO

5

WTA A•- 163.59

ZAK

. E0--1

ZAK P-4.5

0 10 20 30 k

Same as in Figure 6, but Eo = 1 (P = 4.5). R - 0.9999.

1.5

o.5

0.0 1.5

•.o

0.5

WTA - A•- 163.59 -

-- _

ZAK

- E o-- 1.25 -

0.0 25.0

12.5-

0 10

ZAK P--7

30

Fig. 8. Same as in Figure 6, but Eo = 1.25 (P = 7). R = 0.9999.

(P = 7) in Figure 8, and Eo = 1.75 (P = 13.7) in Figure 9. Figures 10-13 are the results for the Arecibo matching height, Af/= 570.96, with a heater frequency of 4 MHz. We applied E o = 0.7 (P = 2.2) in Figure 10, E o = 1 (P = 4.5) in Figure 11, Eo = 1.5 (P = 10) in Figure 12, and E o = 2 (P = 17.9) in Figure 13. Finally, Figures 14-17 are our results for the EISCAT UHF matching height, Af/= 219.1, for a heater frequency of about 8 MHz. We applied Eo = 0.3 (P = 2.4)in Figure 14, Eo = 0.44 (P = 5.2)in Figure 15,

80

0 0.75

WTA A•- 163.59

,J,J.J.J.J,J ZAK

E 0 -- 1.75 •-- 0.50

• 0.25

0.00

120 • 80

o o 20 40 60

Fig. 9. Same as in Figure 6, butEo = 1.75 (P = 13.7). R = 0.64. Note that in this case the depicted WTA spectrum does not represent a true ensemble-averaged saturation spectrum, since the k = 0 feature increases with time.

HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,081

0.6

r• 0.2 0.0 0.6

0.0 6

WTA 57O.96

ZAK

- E0=0. 7 -

ZAK p- 2.2 -

20 40 60 k

Fig. 10. Saturation wave n9mber spectra computed from the WTA and the ZSE, for Aft = 570.96, E0 = 0.7 (P = 2.2), N = 2048, Ak = [(4Aft + 1) 1/2 - 1]/574, and dt = 5 x 10 -4, for plasma parameters I of Table 1, corresponding to a heater frequency of about f0 -• 4 MHz: (top)l(k) = (IE(k)l z) from the WTA, (middle)

2 2 (IE(k)l) from the ZSE, and (bottom) (In(k)l) from the ZSE. This case corresponds to the matching height for the Arecibo radar for this set of plasma parameters. R = 0.9994.

Eo = 0.55 (P = 8.2) in Figure 16, and Eo = 0.65 (P = 11.4) in Figure 17.

From these spectra, note a few characteristic features: The WTA only produce cascade spectra, but saturation becomes impossible for the WTA spectra when very long

1.0

0.0 1.0

0.0 16.0

10.7-

5.3-

0.0 0

Fig. 11.

WTA 570,96

ZAK E0=I

ZAK P-- 4.5

2O 40 60 k

Same as in Figure 10, but Eo = I (P = 4.5). R = 0.9972.

Fig. 12.

50

25

WTA Aft - 570.96

ZAK

E0 = 1.5 --

,

ZAK - P-10

0 20 40 60 k

Same as in Figure 10, but Eo = 1.5 (P = 10). R = 0.9779.

wavelengths ([k[ < 1/2) are excited. This shows up as nondecreasing cascade lines in the spectra and a very strong excitation around k = 0. When the k = 0 feature is excited, the WTA fails completely, and we see a strong buildup of energy in the condensate as time increases. Therefore the time-averaged WTA spectra are not representative of the ensemble-averaged spectra in this case. The ZSE Langmuir cascades look very similar to the WTA cascades for low

18

WTA • All = 570.96 "• 9- -

0 ll. illlllllli_11

•-- o., •o=2

• o.• j• .... 0.0 ...... ......... , ......... _ . _

--= 20

. . - 0 20 •

Fig. 13. Same as in Figure 10, butEo = 2 (P = 17.9). R = 0.9300. Note that in this case the depicted •A spectrum d•s not represent a true ensemble-averaged saturation spectrum, since the k = 0 feature increases with time.

12,082 HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

0.12

•-- 0.08

• 0.04

0.00 0.12

0.08-

0.o4-

0.00 0.8

0.4-

0.0 0

WTA Af•- 219.1

ZAK

E0-- 0.3

ZAK P-- 2.4

15 30 k

Fig. 14. Saturation wave number spectra computed from the WTA and the ZSE for AI• = 219.1, E0 = •0.3 (P = 2.4), N = 2048, Ak = [(4AI• + 1) 1/2'- 1]/416, and dt = 10 -•, for plasma parameters II of Table 1, corresponding to a heater frequency of about f0 = 8 MHz: (top) l(k) = <lE(k)l 2) from the WTA, (middle) <lE(k)l 2) from the ZSE, and (bottom) (In(k)l 2) from the ZSE. This case corre- sponds to the matching height for the EISCAT UHF radar for this set of plasma parameters. R = 0.9998.

pump strengths. When increasing the value of Eo, the cascade from the ZSE is truncated after just a few cascade steps. Increasing Eo even more results in a rapid broadening of the spectra, with a gradual disappearance of the cascade lines. For a fixed set of plasma parameters, the larger the

0.4

0.2

0.0 0.4

0.2

0.0 5.0

2.5-

0.0_ 0 15

k

WTA Af]- 219.1

ZAK

-

ß

ZAK P-- 8.2

30

Fig. 16. Same as in Figure 14, butE0 = 0.55 (P = 8.2). R = 0.9986.

value of All we apply, the larger the value of E0 is needed before the truncation and broadening sets in. Note also that the ZSE cascades look "cleaner" for the larger values of All.

In the cascade-dominated case we always see a weak oscillating two-stream instability (OTSI) spectral feature at the wave number koTsi (for Eo2/All << 1),

0.2

0.1

0.0 0.2

0.1

0.0 2

1 -

15 k

WTA Af]- 219.1

ZAK E o -- 0.44

ZAK P- 5.2

Fig. 15. Same as in Figure 14, butE0 = 0.44 (P = 5.2). R = 0.9995.

A

5.0

2.5

0.0 0.4

0.2

0.0 10

_

WTA Af•- 219.1

ZAK

- o.s

ZAK P- 11.4

15 30 k

Fig. 17. Same as in Figure 14, but E0 = 0.65 (P = 11.4). R = 0.9972. Note that in this case the depicted WTA spectrum does not represent a true ensemble-averaged saturation spectrum, since the k = 0 features increases with time.

HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,083

0.0025 0.0020 0.0015 o.oolo[

o

t=3

20

Aft= 25

0.050

0.040

0.030

0.020

0.010

0.000 0

t=5

10 20

1.0

0.8

0.6

0.4

0.2

0.0 0

t=7

20

t=8

0 • o 1'o 20

1.0

0.8

0.6

0.4

0.2

0.0 0 lO

k

t=9

20

0.8

0.6

0.4

0.2

0.0 0

t=l 1

lO 20 k

Fig. 18. Time evolution of the wave number spectrum E(k, t)l 2 of Langmuir waves, from the ZSE, for ACt = 25, Eo = 1 (the case considered in Figure 5), at t = 3, 5, 7, 8, 9, and 12.

kOTSi = k 1 + • 1 + 2(All)1/2' (17) but only during the linear growth phase, and it never shows up in the saturated state.

In Figure 18 we show the time evolution of the instanta- neous Langmuir wave number spectrum IE(k, t)l 2 for the case All = 25, E0 = 1 (corresponding to the average spectra of Figure 5). Note that the spectrum starts with a decay instability that excites the primary cascade but that the spectrum rapidly broadens and takes on the character of a collapse type broad spectrum.

In Figure 19 we display typical time histories of the total electrostatic Langmuir and ion acoustic energies, wL(t) = Y. elE(k, t)l 2 and wS(t) = Y. eln(k, t)l 2, for All = 163.59 and E0 = 0.7 and E0 = 1.75. The first case (corresponding to Figure 6) is in the cascade regime, whereas the second case (corresponding to Figure 9) is in the collapse regime. As a general remark, notice that we get a pronounced overshoot in the total electrostatic energies only in the cascade regime and not when the system is driven into the collapse regime.

In Figure 20 we present wave number spectra of Langmuir and ion acoustic waves at exact reflection, All = 0, com- puted from the Zakharov model. In Figure 20a we present the "typical" F region case, parameter set I of Table 1, and pump fields of E0 = 1 (E0 = 0.7 V/m) and E0 = 2.3 (E0 = 1.65 V/m). Figure 20b is included to illustrate the limiting case of the smallest obtainable F region Landau damping (parameter set II of Table 1) and strong drive, E0 = 2 (E0 = 3.2 V/m) and E0 = 3 (E0 = 4.8 V/m). It is important to

notice how strongly the width of these collapse type spectra depends upon the pump strength.

Figure 21 shows the fields E(x, t) and n(x, t) in configu- ration space; Figure 2 l a is at the matching height for the EISCAT VHF radar, with parameters All = 163.59, E0 = 0.7, as in Figure 6, and Figure 21 b is at the reflection height, with parameters All = 0, E0 = 1, as in Figure 20a, curve i. By examining a series of snapshots (animation) like those in Figure 21 b, we note that the saturated turbulent state in the reflection region consists of nucleation-collapse-burnout cy- cles in a "sea" of propagating ion sound pulses and Lang- muir waves. From Figure 21 b we note one of the character- istics of this process: Most of the burnt-out cavitons bifurcate into leftgoing and rightgoing ion sound pulses, some of which again recombine after propagating a certain distance. Some of these will later act as nucleation centers for new cycles of the process.

Figures 22-26 all present ensemble-averaged frequency spectra calculated from the time series E(k, t) and n(k, t), at a given wave number k and for a given frequency mismatch All. In all five figures the top panel displays k2(IE(k, f; A)l 2) ("plasma line"), and the bottom panel shows (In(k, f; A)l 2) ("ion line"), at a wave number k that is detectable by one of the available radar facilities. Here f = w/2 •r is the frequency shift measured in terms of dimensionless Za- kharov time units, T. Figures 22 and 23 are spectra from the matching height for the EISCAT VHF radar, Af• = 163.59, for E0 = 0.7 (Figure 22) and E0 = 1 (Figure 23). Note that both spectra consist of narrow intense spikes, downshifted by the ion acoustic frequency shift f = --kvHF/2•r = --2, resembling the monochromatic frequency spectra assumed in a WTA. For the stronger driving case, we also see a weak non-WTA anti-Stokes excitation at f = 2. The frequency axis in this case can be read in units of kilohertz, according to set I of Table 1. We observe that the width of the ion acoustic spectra is about 1 kHz in this case. In Figures 24 and 25 we show power spectra of fluctuations at the same

lOOO.OOO

,•, lO.OOO

• 0.100

o.ool 0 20 40

..... ;--, ....... • 10 -2 I•'•,• '•';;'"'- tO lO -6

0 20 40 t

Fig. 19. Total electrostatic Langmuir and ion acoustic energies from the ZSE, for ACt = 163.59, the EISCAT VHF matching height:

2 S (top) wL(t) = •'-k IE(k, t)[ and (bottom) W (t) = Y-re In( k, t)l 2, for (curve i) E0 = 0.7 (corresponding to Figure 6) and (curve ii) E0 = 1.75 (corresponding to Figure 9).

!2,084 HANSSEN ET AL..' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

0.8:•• Aft=0 .

.•. 0.6

•' o.4 •%,,.•, • (ll) v 0.2 :- (i)',,,,,.,.__•.• o.o

o 5o

O) Co=

(ii) Eo = 2.3

100

15

0 -10 -5 0 5 10

x

800

..

50 100 k

Fig. 20a. Saturation wave number spectra (IE(k)l 2) and (In(k)l 2) from the ZSE at the reflection height, All = 0, for plasma parameters I of Table 1, (curve i) E0 = 1 and (curve ii) E0 = 2.3, and N = 2048, Ak = 0.0984, and dt = 5 x 10 -4.

EISCAT VHF wave number but for a case much closer to the reflection height, All = 25. Figure 24 displays the case E0 = 0.7, and in Figure 25 we study E0 = 1. In this case the width of the ion acoustic spectra is increased to about 10 kHz. By comparing the signal strengths in Figures 24 and 25 with those of Figures 22 and 23, we note that the observable fluctuations at the radar backscatter wave number are orders of magnitude less close to reflection than at the exact matching height. Figure 26 shows power spectra at the wave number k = 19, close to the Arecibo backscatter wave number, at a height just below reflection, All = 15, for a driving field of E0 = 1. Note the distinct "free-mode" feature in the plasma line, upshifted approximately by the

0 50 100

1500

• 500

0 0 50 100

k

Fig. 20b. Saturation wave number <lE<k>l> and <ln<k)l•> from the ZSE at the reflection height, All = 0, for plasma parameters II of Table 1, (curve i) Eo = 2 and (curve ii) Eo = 3, and N = 4096, Ak = 0.15, anddt = 10-4.

20

-10 -5 0 5 10 x

Fig. 21a. Fields E(x, t)2 and n(x, t) in real space at the matching height for ElSCAT VHF, All = 163.59, E0 = 0.7, corresponding to the spectrum of Figure 6.

amount Af = 192/2•r = 57.5, as predicted by the theory. It is also important to note that close to reflection, the power spectra are broad, showing that the linear dispersion rela- tions for the modes are no longer valid. Observe also the nonvanishing excitations at zero-frequency shift in the ion line, similar to effects often observed in radar measurements during ionospheric heating. Similar zero-frequency features were also reported from two-dimensional simulations by DuBois et al. [1991], and they identified this feature with ion acoustic waves generated by collapsing cavitons. Both of these features (broadening of resonances and zero-frequency excitations) are not described within the standard WTA.

4. DIscussioN

Our numerical study clearly demonstrates two different mechanisms by which a Langmuir wave spectrum can be produced by a long-wavelength electromagnetic pump.

The first process, which might be termed cavitation (nu- cleation of cavitons), takes place under the condition

E0=l Aft=0

i IE(z,e)l e

2000 •! [[ • .,

-2ooo -4000 .......

-5 0 5 x

Fig. 2lb. Fields IE(x, t)l 2 and n(x, t) in real space at the reflection height, All = 0, E0 = 1, corresponding to curve i of Figure 20a. Both snapshots are from the saturated regime.

HANSSEN ET AL..' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,085

8.108 6.108 4.108 2.108

0

Af• = 163.59

Eo = 0.7 k= 12.3

-20 0 20

1200

• lOOO <l 800 42, 600

, , , -

I Aft = 25 Ill o=O.7

---- ,

-20 0 20

g--, 2.5.107 •' 2.0.107

1.5.107 ...

• 1.0.107 e 06 • 5.0.1

-20 0 20 FREQUENCY SHIFT

Fig. 22. Ensemble-averaged power spectra (top) k2(lE(ok • f; AfDI2) and (bottom) (n(k, f; AfD[2), for Af/= 163.59, E 0 = . at the EISCAT VHF backscatter wave number, kl = 12.3 (the primary cascade of Figure 6). The ensemble consists of 14 spectra, each having Ns = 4096 samples, and the sampling interval is At = 5 x 10-4, giving a frequency resolution of about Af = 0.5, or about 0.5 kHz in real units.

An < max {3/4, 2Eg(2Eg + 1)} (18)

that is, near the region of reflection (40 = 4p) of an ordinary polarized incident pump. The first criterion in (18) defines the region for which parametric decay is impossible, the so-called "Langmuir condensate." It is determined by the smallest possible real wave number solution of the three- wave resonance conditions, 40(k'•) = 4• (•) + 42(k'•), k• = • + •'2, where 4j(•.)- 4p(1 + 3•.2Xo2/2), forj = 0, 1, and 42(k-2) = eslœ21, where the ion sound speed is defined by 5s - (•o•:Te/M) •/2. The second criterion in (18) is derived by determining the region for which the growth rate of the

2000

1500 1000

500 0

ß , , ,

-20 0 20 FREQUENCY SHIFT

Fig. 24. Same as Figure 22, but for A• = 25, E0 = 0.7. These spectra represent fluctuations at the EISCAT VHF backscatter wave number, but at a nonresonant height close to the O mode reflection height. The ensemble consists of 14 spectra, each having Ns = 2048 samples, with a sampling interval of At = 10 -3 .

oscillating two-stream instability (OTSI) is larger than that for the parametric decay instability (PDI), when neglecting dissipation (see, for example, Nicholson [1983], chapter 7.17, for the appropriate dispersion relations). Equation (18) is supported by a large number of numerical simulations. The cavitation process can start from an OTSI (also called modulational instability, MI), but in the fully developed stage it consists of many localized cycles of nucleation of cavitons, (one-dimensionally driven) collapse, and burnout, similar to what was reported by Doolen et al. [1985]. In physical units the condition (18) reads

4 o-wp<max •-•4p, m'r/

•-- 2.0.109 •' 1.5.109 ,;2, 1.0.109 • 5.0.108

0

All = 163.59

Eo-1 k = 12.3

-20 0 20

• 2.0.106 •' 06 All = 25 <1 1.5.1 Eo=l •,• 1.0.106 k = 12.3 •_• 5.0.105

-200 - 100 0 100 200

1.2'108 - 1.O, lO 8 8.0.107 6.0.107 4.0.107 2.0.107

0 0

FREQUENCY SHIFT

•-- 5.0.107 •' 4.0.107 .•, 3.0.107 • 2.0.107 • 1.0.107

, , , .

-20 0 20 FREQUENCY SHIF•

Fig. 23. Same as in Figure 22, but E 0 = 1. Fig. 25. Same as in Figure 24, but E0 = 1.

12,086 HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

8.105 6.105 4.105 2.105

-200 - 100 0

AQ= 15 E0=i

k=19

100 200

•'- 2.5.107

• 2'0'107 • 1.5.107 •,• 1.0.107 • 5.0.106

0 -20

ß .

0 FREQUENCY SHIFr

Fig. 26. Ensemble-averaged power spectra (top) k2( E(k, f; All) 2) and (bottom) (In(k, f; All) 2), for All = 15, E0 = 1, close to the Arecibo radar backscatter wave number, k = 19. The ensemble consists of 14 spectra, each having Ns = 2048 samples, with a sampling interval of At = 10 -3 . Note the pronounced "free-mode" feature upshifted by the amount of Af = 19 2/2rr --• 57.5.

where we have defined

0œ0: W 0 -=

4norl t( Te

The saturated turbulence dynamics in this range is qualita- tively the same as that at the reflection level.

With the assumption of a linear density profile, av2(D = ao2[ 1 + (•- 17)/17], where/_7 is the plasma scale length and I(g- E)/EI << 1, equation (19) gives the typical width of the collapse-dominated region (in physical units) as

-- + œ (20) rn

when the second term in (19) is the larger. For the typical range of experimental parameters this shows that the thick- ness of the caviton collapse layer(s) ranges from tens of meters to the full widths of the first few Airy maxima, depending strongly on the pump strength and the plasma scale length. For conditions where the pump strength is so weak that the first term in (19) is the larger, the thickness of the caviton collapse layer becomes much less,

A•--• L (21) 3M

or only a few meters thick. For most experimental condi- tions, however, equation (20) is the applicable estimate.

For practical purposes, this shows that we can apply fairly large frequency detunings and still obtain cavitation as the elementary dynamical process. In comparing the linear estimate (18) with our full nonlinear numerical solutions, it appears that we may apply even larger detunings than this expression suggests and still obtain nucleation of cavitons.

Hence the actual extent of the collapse-dominated region is even larger than suggested by (18). This is as expected, since we have neglected the dissipation in the derivation of (18), since several approximations lie behind the expressions for the growth rates leading to (18), and since (18) is based on linear instability arguments. The ensemble-averaged k spec- trum of Langmuir waves in the saturated state in this case is broad, starting around k -• 0 and expanding toward higher k, where Landau damping is important [see also Doolen et al., 19851.

The second mechanism takes place under the condition All >> 1 or, more precisely,

I• rn (• 3// I•02 ) a0- av>> max • at,, + I•0 a t, (22) in dimensional quantities, which is satisfied some distance below the reflection level. It has the characteristics of a

cascade process: the wave number spectrum has a line structure, the highest wave number is the one that satisfies the selection rule for the decay instability of the pump, and the lower lines satisfy the resonance conditions for the further decays of parametrically driven Langmuir waves. The total number of cascades increases with pump intensity up to a point where the cascade is truncated and a broaden- ing of the spectrum occurs. In comparing the results of the cascade spectrum from the Zakharov model with those of the WTA (equations (8)-(10)), it is in general found that a smaller number of cascades is produced in the ZSE model than that predicted by the WTA.

The present work, together with Hanssen and Mjg)lhus [1990] and DuBois et al. [1991], demonstrates that the two processes under discussion are not "conflicting theories" nor "competing processes." In their pure form, the two processes exist only in separate height regions. In addition, they can coexist in the transition region between the "col- lapse" layer and the "cascade" layer. In this respect, the height relative to the reflection level, All, and the strength of the driver, E0, are the control parameters that determine the nature of the excited turbulence. Earlier works solved driven

and damped ZSE models, including a frequency mismatch: for example, Payne et al. [1984], Zakharov [1989], and DuBois et al. [1990]. Payne et al. [1984] applied All - 18 and E0 = 1.6, which, according to our criterion (18), must eventually saturate at a collapse type turbulence. Note that the time evolution of their spectra starts with a limited cascade but that it ends up with a broad collapse type spectrum, similar to the scenario seen in our Figure 18. Zakharov [1989] applied All - 50 and E0 - 3, which again, according to (18), is far into the collapse regime. This is indeed confirmed by their numerical results. DuBois et al. [1990] report on a computation with All = 40 and E0 = 1.2, which is in the transition region where coexisting collapse and cascading turbulence is expected. They did not show spectra for this set of parameters.

We consistently find that the cascade spectra from the Zakharov equations are truncated after a few steps, long before the so-called Langmuir condensate (Ikl < 1/2, or Ik• < (•m/M)1/2•1/3) is excited. Furthermore, after the truncation sets in, the bands between the cascade lines start to fill in. Increasing the pump strength even more results in a gradual transition to a broadened spectrum, indicating that we enter a regime where collapse of Langmuir wave packets

HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,087

takes place. In configuration space the cascade regime consists of large-amplitude propagating Langmuir and ion acoustic waves, whereas the collapse regime consists of localized collapsing wave packets in addition to propagating waves.

Similar truncation effects were also reported from two- dimensional numerical calculations in the work by DuBois et al. [ 1991], where the following description was proposed: the long-wavelength Langmuir waves act as pump waves for cavitation in already generated large-amplitude density fluc- tuations. This intermediate process, with coexisting cas- cades and cavitation, is, of course, most easily generated at low values of All. In particular, Figure 18, for All = 25 and E0 - 1, shows that the process starts with a limited cascade, while it ends with the filling in and expansion of the spec- trum.

Possible truncation mechanisms include (1) enhanced ef- fective damping of Langmuir waves by scattering on en- hanced short-wavelength ion fluctuations, (2) modulational instability of the cascade spectrum resulting in localized caviton excitations, and (3) direct resonant nucleation of cavitons in density cavities by long-wavelength Langmuir waves [DuBois et al., 1991]. The details of the truncation mechanism are, however, at present not known.

The observed truncation of the cascade train from the

Zakharov model is encouraging, if we compare with exper- imental data. The Troms0 radar measurements seldom show more than three spectral lines displaced by 12325 ion sound frequency shifts [Stubbe et al., 1992]. A recent theoretical examination of weak turbulence theory by Hanssen [1991] shows that the inclusion of resonance broadening effects will reduce the number of cascades detectable by the EISCAT UHF radar, but not for the VHF radar. This weak turbu- lence theory did not consider the possible enhancement of ion fluctuations. In this respect, it is promising to see that the one-dimensional Zakharov model, at the matching heights for the radars under discussion, truncates the cascade after a few steps and that this effect occurs as a natural conse- quence of the nonlinear coupling of Langmuir waves and ion acoustic waves. This truncation mechanism will work more

effectively in higher dimensions, since also an angular broad- ening takes place in two and three dimensions. In two- dimensional solutions to the ZSE we have never seen more

than four distinct spectral lines in the wave number spectra. Hence this might be the simplest possible explanation of the fact that the Troms0 experiments never show more than three cascade lines. Another plausible explanation could be in terms of a strongly nonlinear D region absorption, causing only a fraction of the transmitted heater power to reach the F region.

It is important to note that for a fixed pump strength E0 this model gives a continuous distribution of the turbulence, from cascade type spectra some distance below the reflec- tion level (All >> 1) to a gradually broader spectrum as we approach the reflection height from below (All --) 0 +). Also, for fixed All, we find a continuous transition from weak turbulence type cascade spectra to strong turbulence type broad spectra, as we increase the pump strength E0. The weak turbulence and strong turbulence are thus continu- ously interconnected and are both included in the same nonlinear mathematical model, the version of Zakharov's equations applied in this paper.

It is interesting to note the persistence of the primary

cascade mode k• in the ZSE solutions, even for strong driving at All >> 1. This is typified by Figure 9, for a driving field of E0 = 1.75 at All = 163.59, representing the matching height for EISCAT VHF. We see a sharp knee in the (E(k)l 2) spectrum around k•, in an otherwise continuous caviton collapse type spectrum. Another good example is provided by Figure 13, for a driving field of E 0 = 2 at All = 570.96, the matching height for the Arecibo radar. In this case the <lE(k)12> and ( n(k)12> spectra are broad continua but with sharp well-defined peaks at k•. One daughter cascade is barely visible in the Langmuir wave spectrum but clearly distinguishable in the ion acoustic spectrum.

As seen from our numerical results, we always excite finite amplitude ion acoustic waves. This was also seen in two-dimensional simulations reported by DuBois et al [1991], and they applied a stronger damping on ion acoustic waves than the one in this paper, and in the work by Clark et al. [1990], where a still stronger ion damping was applied. This is contrary to one of the key assumptions in the conventionally accepted WTA to ionospheric heating, where the ion waves are assumed to be so heavily damped that they do not take part in the dynamics but only act as passive scatterers of Langmuir waves. We are thus I•d to the conclusion that this particular assumption o f strongly damped noninteracting ion acoustic waves is not applicable to RF-generated turbulence in the ionosphere. Experiments have always indicated this', where a strongly enhanced ion line is seen during heating, and often a strong zeroifrequency shift feature appears in th e ion line spectrum [e.g., Hagfors and Zamlutti, 1973; Frey, 1986; Djuth et al., 1987; Stubbe et al., 1985, 1992]. On the basis of these experiments, it is therefore obvious that a better weak turbulence model (in the region of space and for the range of pump strengths for which it is justified to apply the WTA) would be one coupling the ion dynamics to the dynamics of the Langmuir waves. Conservative weak turbulence models coupling ion acoustic waves and Langmuir waves already exist in the literature [e.g., Kadomtsev, 1965; Payne et al., 1989], and numerical solutions were provided by Payne et al. [1989].

Payne et al. [1989] derived the following validity criteria for their conservative WTA, in our notation (see also Za- kharov [ 1989]):

(Wn •) << (Akn•) 2 (23)

rc2(WS)ak s << 1 (24)

Superscripts L and S refer to Langmuir and ion acoustic waves, respectively, and the subscript n refer to the nth cascade line. Here (Wn •) = (•-•k•kn IE(k)l 2) is the average total electrostatic Langmuir energy in the modes constituting cascade line number n, and (W s) = (Y•k In(k)l 2) is the average total electrostatic ion acoustic energy in the turbu- lence cell. The Akn • is the width of cascade line number n, and Ak s is the total width of the ion acoustic wave number spectrum. The parameter r c is the correlation time for ion acoustic fluctuations. We can estimate r c by rc -• 1/Bs, where Bs is the width of the frequency spectrum of ion acoustic fluctuations (In(k, f; AI)12), e.g., from Figures 22-25. The rest of the parameters entering (23) and (24) are readily obtainable from our numerical solutions. Physically, equation (23) is the criterion preventing the wave packet forming cascade number n from becoming modulationally

12,088 HANSSEN ET AL.' NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

unstable. Equation (24) states that no significant amount of Langmuir waves must be trapped in density perturbations, or that "the Langmuir wave trapping time must be larger than the correlation time for ion acoustic fluctuations" [Payne et al., 1989, p. 1799]. Note that it is only (23) that applies directly to our WTA, since we do not include a wave kinetic equation for the ion acoustic waves. We can, how- ever, apply (24) to the ion acoustic spectra calculated from our ZSE, to see whether they satisfy the WTA validity criteria of Payne et al. [ 1989]. A further approximation of the validity criterion (23) can be made by noting that the cascade lines are approximately triangular in shape, permitting us to formulate (23) as

I(kn) <<Akn L (25) where I(kn) is the peak value of cascade number n, com- puted from (8). Our finding is that the validity criteria are only marginally satisfied for very weak driving when the wave intensities I(kn) and (In(kn)l 2) are low. At stronger driving, the ZSE primary cascade line yields I(k•) •- Ak•, while the corresponding WTA cascade has I(k•) > Ak•. This difference occurs mainly because the ZSE spectra broaden as we increase the driver, while no broadening mechanism exists in the WTA. We also notice that the

daughter cascades from the ZSE are such that Akn•+• > Akn • , i.e., the lines get successively broader. We do not offer any explanation for this successive broadening effect, at the present time.

Looking at the above stated validity criteria, we see that the standard WTA suffers from an inherent contradiction:

The validity criteria of the theory require broad k spectra, while the model itself describes a narrowly peaked k spec- trum. Furthermore, the construction of a WTA requires the use of the random phase approximation (RPA), while the inclusion of a coherent pump introduces a certain definite phase correlation between the cascade modes. Bezzerides and DuBois [ 1976] and DuBois and Bezzerides [ 1977] studied the effect of the correlation (induced by a coherent pump) between the daughter decay waves, in the wave kinetic equations for the turbulence. They concluded that when eot•o•/no•Te << 12Vio(ff•o) :, which is typically the case for ionospheric heating conditions, there are no important modifications of the results of kinetic equations such as (8)-(10). In this regime their analysis also showed that because of the discrete nature of the cascade spectrum, the RPA did not play an essential role in the derivation of the weak turbulence equations. (In their works also, the effect of enhanced ion acoustic fluctuations was ignored.)

In the experiments that have been performed in Arecibo, Puerto Rico, and at Ramfjordmoen, near TromsO, Norway, one wave vector component selected by the Bragg condition for the probing radar is observed. According to the theory presented above, the conventional interpretation in the form of a cascade process requires that the echo comes from the matching height. This is in conflict with earlier observations at Arecibo, where the conclusion had been drawn that the echoes came from the reflection region [Muldrew and Sho- wen, 1977; Isharn et al., 1987]. Recent high-resolution mea- surements by Djuth et al. [1990] and Fejer et al. [1991] reveal another picture: the turbulence at the wavelength observable by the radar is seen to start at the reflection level and then spread to lower altitudes. The radar observes enhanced turbulent fluctuations, where the backscattered echoes come

from a layer about 1-2 km thick. Djuth et al. and Fejer et al. also report structure in the turbulent layer, corresponding more or less to the standing heater wave, that is, to the Airy pattern. The interpretation of these experiments is, how- ever, still a problem: first, they see a broad continuous power spectrum at the reflection level. After some time they also see line structure in the enhanced plasma line spectrum at the matching height. At later times the line structure appears to come from a number of heights, including heights close to the reflection level. This indicates that the matching condition is satisfied at a number of points in the ionosphere, at later times.

Therefore the complete description of the turbulence should contain the detailed structure of the pump electric field. A self-consistent solution to the Zakharov system including a nonlinear propagation model for the electromag- netic field is in preparation by the present authors. The nonlinearity in the above mentioned theory appears as a consequence of the adiabatic density change, due to the ponderomotive force of the excited electrostatic Langmuir waves. It can be shown that this ponderomotive term depends upon the strength of the incident heater wave.

A crucial point in the preceding discussion of these results has been that the polarization of the O-polarized pump wave under Arecibo conditions (magnetic field forms --•40 ø to the vertical) is along the magnetic field. This is exact at the reflection level and a good approximation for the whole range of altitudes of interest here. In addition, multidimen- sional solutions to the WTA [Chen and Fejer, 1975; Perkins et al., 1974; Das et al., 1985] show that the angular spread of the cascade from this model is not large enough to be observed with the Arecibo radar. Instead, the old interpre- tation has been in terms of an enhanced thermal fluctuation spectrum [Perkins et al., 1974], or refraction in density ducts [Muldrew, 1978; Fejer, 1979; Rypdal and Cragin, 1979].

The new interpretation seeks to explain the observations by multidimensional versions of the All -• 0 process [DuBois et al., 1990] and the All >> 1 process; in particular, the matching height spectra for Arecibo are examined by DuBois et al. [1991] for a mass ratio of M/rn •- 4000. The numerical calculations of Russell et al. [ 1988] and DuBois et al. [ 1988, 1990] show that spectral features observable (in the reflec- tion region) by the Arecibo radar can be produced by this process, because the multidimensional collapse stage leads to a broadening of the Fourier spectrum in both magnitude and angular distribution. The two-dimensional calculations ofDuBois et al. [1991] indicate that also at the Arecibo radar matching height the cascade spectrum from the two- dimensional ZSE has a larger angular width than the corre- sponding width computed from multidimensional WTAs [Perkins et al., 1974; Chen and Fejer, 1975; Das et al., 1985]. In addition to giving an explanation of the observed height of the plasma line, this model explains other observations [Cheung et al., 1989, 1991], in particular, the so-called "free-mode" signature of free Langmuir waves produced by wave collapse.

Turning to the Troms0 observations, we see that the magnetic field geometry is such that both "types" of turbu- lence should be observable with the EISCAT VHF radar. It seems most reasonable that it is observations of coexisting weak and strong turbulence that have been reported so far or that the scattering volume for the VHF radar contains both the matching height cascade spectra and regions where the

HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION 12,089

collapse dynamics dominates. The pronounced line structure of the VHF spectra [Stubbe et al., 1992] suggests weak turbulence, whereas the high-level continuous background on which the cascades ride, in addition to the observation that only the broad spectral background increases with increasing pump strength, suggests that collapse phenomena are also present. For EISCAT VHF the height separation between the reflection level and the matching height is only a few hundred meters, so the VHF spectra presented by Stubbe et al. [ 1992] are most likely integrated over the whole turbulent region, covering both the caviton collapse and the cascade turbulence parts. The Troms0 experiments have unfortunately not so far verified the matching height unam- biguously, but there are simultaneous UHF and VHF exper- iments [Stubbe et al., 1992] that show a height separation between the plasma line echoes measured by the two radars, consistent with the matching height concept of this paper. Also for the Arecibo case, it appears from recent high- resolution experiments [Djuth et al., 1990; Fejer et al., 1991] that one can observe a combination of both types of turbu- lence, from pure cascades via coexistence spectra to pure broad collapse type spectra.

For the EISCAT VHF and the Arecibo radars our numer- ical work has indeed verified that the cavitation spectrum extends out to the wave numbers detectable. These two radars should therefore be able to "see" signatures of both the strong and the weak turbulence simultaneously for poor spatial resolution, or in separated height bins for good spatial resolution.

Our calculations have shown that the spectrum of the cavitation process at All = 0 is probably not broad enough to extend out to the wave numbers detectable with the EISCAT UHF radar, under usual quiet ionospheric conditions. The width of the spectru m is, however, very dependent upon the strength of the applied electric field. We find that a pump electric field of at least a few volts per meter is needed in order to produce a Spectrum broad enough to be detected by this radar. D region absorption will thus have a strong effect on the possible detectability of the cavitation process by the UHF radar. Signals from this region will unfortunately be much weaker than those from the matching height, as seen by comparing Figures 22 and 23 with Figures 24 and 25. Our prediction is thus that the EISCAT UHF radar under usual circumstances (nonpreconditioned ionosphere, cold start, and quiet geophysical conditions) is able to observe only a cascade type turbulence at All(kuH F) = kUHF(kuH F + 1) according to (7), where kuH F is the wave number of waves detectable by the UHF radar (k-uH F -- 39.1 m-•.) As demonstrated, one can expect to detect backscattered sig- nals from Langmuir turbulence excited in the reflection region also with the UHF radar when the plasma density is large and the electron temperature is low.

Finally, by studying frequency spectra from solutions to the Zakharov model applied in this paper, we have con- structed plasma lines and ion lines in the manner used in radar experiments. In particular, our computations have led us to the following Prediction for the "free-mode" [Djuth et al., 1986; Cheung et al., 1989, 1991; DuBois et al., 1990] feature' Under normal conditions, only the Arecibo radar should be able to detect and resolve this spectral feature. EISCAT VHF will not be able to see this spectral feature because kR AD is too small, implying that the frequency shift (Af= 3(k-• A-o) 2fp/2) for this line is very small. The effect is

that the weak signature of the free Langmuir waves gener- ated by collapsing cavitons becomes indistinguishable from the upshifted caviton continuum part of the spectrum. EIS- CAT UHF will not be able to see it, because kRAo is too large. As shown, this has the effect that we normally would not be able to detect the caviton collapse type turbulence at the large wave number observable with this radar. In order to test our prediction, one should measure plasma line spectra with a radar facility with kRA o = 0.1 (the optimal choice for detecting the free mode) or having a frequency of about f'n = 600 MHz, possibly along the magnetic field line in the auroral region. A preliminary study of height- integrated frequency spectra at radar wave numbers shows that the number of cascade lines in the integrated plasma line is roughly the same as the number of cascades in the wave number spectrum at the matching height. Further details on our frequency spectrum predictions will appear in a later publication.

5. SUMMARY AND CONCLUSIONS

In this paper we have examined the standard weak turbu- lence approximation to ionospheric Langmuir turbulence. We compared solutions to a version of the one-dimensional Zakharov system of equations (ZSE) with a weak turbulence approximation (WTA) obtained from the very same Za- kharov model. The parameters determining whether the generated turbulence will be collapse-dominated or cascade- dominated are the height relative to the reflection height for the O mode driver (All) and the strength of the driver (E0), in addition to the damping parameters.

We found that for very weak driving and far below the reflection height (large All), there is a qualitative agreement between Langmuir wave number spectra from the ZSE and the WTA. Increasing E 0 or moving closer to the reflection height (lowering All) causes predictions from the full ZSE and its WTA to differ strongly: The ZSE cascade is truncated and broadened, while the WTA continues to produce a pure cascade continuing straight into the so-called "Langmuir condensate." This truncation of the ZSE cascade is ob- served to occur far before its wave number cascade enters the Langmuir condensate, contrary to what has earlier been assumed. We do not understand the details of the truncation mechanism at this time.

The findings of this paper are best summarized by Figure 27. This figure shows the locations of collapse-dominated turbulence (where TPDI < TOTSI, marked "Collapse") and cascade-dominated turbulence (where TPDI > 'YOTSI, marked "Casc.") in (E0, All) space. With reference to the iono- sphere, All = 0 is the reflection point, and increasing All represents moving closer to the transmitter. The solid curve separating the two regions is based on our equation (18), i.e., the location of the curve ¾PDI -- ¾OTSI, derived from a linear analysis. The sudden jump at E0 _ •/2 marks the OTSI --' threshold. In comparing this curve with our numerical full wave solutions from the Zakharov model, it is evident that the true curve based on nonlinear analysis lies above the curve drawn and hence the collapse region is actually larger than suggested by Figure 27. We have marked a portion of (E0, All) space with "Coex," meaning that in this region we observe coexisting cascades and collapse in our numerical solutions to the ZSE. The dashed curve marking the "bor- der" of this region is based on solutions from the Zakharov

12,090 HANSSEN ET AL.: NUMERICAL TEST OF WEAK TURBULENCE APPROXIMATION

1000.0

100.0

• 10.0

1.0

10 -2

10 -3

10 -4

0.1 .... • .................. I .......... 10 '5 0 1 2 3 4

Fig. 27. Demarcation of the locations of collapse-dominated turbulence (marked "collapse"), cascade-dominated turbulence (marked "casc."), and coexisting collapse and cascade turbulence (marked "coex."), in the two-dimensional parameter space (Aft, E0). The suggested demarcation curve is based on our equation (18), the criterion 3tPDI = TOTSI derived from collisionless, linear insta- bility analysis. The "Langmuir condensate" is marked "cond.," and the region where the E0 is below the PDI threshold is marked "no inst." The dashed curve suggesting the "boundary" of the coexistence layer represents the parameters for which 50% of the Langmuir energy is dissipated by noncollisional damping. The dotted line is a typical linear Airy pattern, and the right-hand vertical axis shows the distance below reflection in terms of the plasma scale length.

model, and it represents the location of parameters (E 0 , AI•) for which 50% (arbitrary choice) of the Langmuir energy is dissipated by noncollisional damping. The so-called Lang- muir condensate is marked by "Cond." and is more pre- cisely the region where the kinematic matching conditions for parametric decay cannot be satisfied. The region marked "No inst." is the region for which the pump electric field does not exceed the PDI threshold and hence no instability is possible there. One should interpret the curves drawn as being "fuzzy," or having a finite width. Not that higher- dimensional models would most certainly predict a larger coexistence layer due to the inertial collapse, meaning that collapse phenomena would take place far into the cascade regime.

To discuss the ionospheric heating experiments, we have included a typical linear Airy heater profile (the dotted curve). On the right-hand vertical axis we show the distance below the reflection point, in terms of the plasma scale length /7,, obtained from the relation AZ//7, = (4•q/3)(m/ M)AI•. (Note that this estimate is based on the assumption of a linear density profile.) From this diagram we see that the first few Airy maxima may be strong enough to excite a collapse-dominated or coexistence type turbulence but that the number of collapse layers is a very sensitive function of the strength of the driver. It is important to notice that the first Airy maximum extends far into the collapse regime for most experimental parameters, so applying a WTA descrip- tion for the Langmuir waves would be totally meaningless in this region of parameter space. We also observe that the

matching heights for the radars are in the cascade-dominated part of parameter space for a large range of experimental parameters but with the possibility of being driven into the collapse regime for stronger driving.

The cascade physics dominates in the upper left-hand sector of the region marked "Casc." in this diagram, and our finding is that even there, the differences between solutions to the standard WTA and the full ZSE can be quite large. In comparing numerical solutions from our version of the ZSE and the standard WTA with experimental radar spectra, it is obvious that the extended Zakharov model yields results resembling a larger number of key features of the spectra (truncated cascades, a broad spectral background, a zero- frequency shift feature in the ion spectrum, free modes for Arecibo wave numbers, and a height distribution of the turbulence) than does the WTA (only cascade spectra). This is as expected, if we examine Figure 27, since the WTA is a highly specialized approximation to the full theory, valid only in a small region of parameter space.

Note added in proof. This scenario has been reported in many experiments performed under night or evening condi- tions [Djuth et al., 1990; Fejer et al., 1991]. However, from other experiments using the chirp technique [Isham et al., 1987] and performed during daytime, it is reported that the echoes stay within a narrow height interval near the critical height throughout the heating period (T. Hagfors, private communication, 1992).

Acknowledgments. We want to thank W. Tom Armstrong, Noralv Bj0rnfi, Frank T. Djuth, Kristian B. Dysthe, Jules A. Fejer, Alex V. Gurevich, Tor Hagfors, Cesar LaHoz, Dwight Nicholson, David Russell, Kristoffer Rypdal, and Peter Stubbe for helpful discussions and criticism during the process of this work. The numerical calculations were performed on the Norwegian CRAY X-MP in Trondheim, and on various CRAY X-MP's and Y-MP's at Los Alamos National Laboratory, and we made use of scientific visualization and data processing software at the Advanced Com- puting Laboratory (ACL) at Los Alamos. One of the authors (A.H.) wants to thank Don DuBois for providing partial economical sup- port, and for the kind hospitality met during my year-long stay at the Los Alamos National Laboratory, where this work was completed. This work is supported by the Norwegian Research Council for Science and the Humanities (NAVF/RNF), under project 426.90/ 002, by the U.S. Department of Energy, and by National Science Foundation-Air Force Geophysical Laboratory Joint Services con- tract ATM-9020063.

The Editor thanks G. L. Payne for his assistance in evaluating this paper.

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(Received November 4, 1991; accepted March 11, 1992.)