Caviton dynamics in strong Langmuir turbulence

Recent studies based on long time computer simulations of Langmuir turbulence as described by Zakharov's model will be reviewed. These show that for strong to moderate ion sound damping the turbulent energy is dominantly in non-linear "caviton" excitations which are localized in space and time. A local caviton model will be presented which accounts for the nucleation-collapse-burnout cycles of individual cavitons as well as their space-time correlations. This model is in detailed agreement with many features of the electron density fluctuation spectra in the ionosphere modified by powerful HF waves as measured by incoherent scatter radar. Recently such observations have verified a prediction of the theory that "free" Langmuir waves are emitted in the caviton collapse process. These observations and theoretical considerations also strongly imply that cavitons in the heated ionosphere, under certain conditions, evolve to states in which they are ordered in space and time. The sensitivity of the high frequency Langmuir field dynamics to the low frequency ion density fluctuations and the related caviton nucleation process will be discussed.


Introduction
In this lecture we want to summarize the results of a research program which has focussed on the "long time" development of strong Langmuir turbulence (SLT) which is driven by intense coherent radiation in long scale length plasmas near critical density. This problem was chosen for its theoretical simplicity and because of its relevance to ionospheric modification experiments using powerful H F radiation. These experiments probably provide the best physical realization of the theoretical ideal, long scale length, quasi stationary plasma.
The strong Langmuir turbulent system which we will try to describe is a fascinating system in which locally coherent, nonlinear "wave" structures or excitations undergo chaotic or turbulent motion. These elementary excitations which we call "cavitons" consist of high frequency Langmuir waves trapped in self-consistently evolving density cavities. The low frequency density response which describes the cavities evolves under the influence of the ponderomotive force of the localized Langmuir waves. These excitations have a finite lifetime; they are born by gaining energy from the external fields in a process of "nucleation"; they "collapse" to small dimensions where they die or "burnout" by giving up their electrostatic energy to accelerated electrons. These excitations also interact by the radiation of ion acoustic waves and "free" Langmuir waves. The study of this interaction is in its infancy but we have examples in which the caviton gas apparently undergoes a phase transition to states which are highly ordered in space and time. These phenomena are a significant departure from our concepts of linear wave excitations and weak turbulence or renormalized turbulence theories * Research supported by USDOE with their often uncontrolled assumptions such as random phase approximations.
Data from the ionospheric modification experiments has accumulated for about 15 years. In the early days these data seemed consistent with the conventional theory involving linear parametric instabilities and weak turbulence cascades. However, with the passage of time it was realized that these data were manifestly inconsistent with the conventional theory. A more detailed discussion of this is given in Ref. [l] from which much of the material here is derived.
During the same time period new theoretical insights into SLT were developed centering around the seminal work of V. E. Zakharov [2]. He developed a very useful unified description of SLT and concluded that collapsing localized Langmuir states would play a central role in the turbulent state. In recent years it has been possible to carry out long time computer simulations of Zakharov's model equations in two [3, 41 and three [5] spatial dimensions. We believe our work has led to a new set of paradigms for the understanding of SLT and its application to the ionospheric modification experiments [ 11.
The results of this research can be summarized as follows: (1) States of SLT can be excited for heater (pump) intensities only marginally above the threshold for parametric instabilities. Thus we expect the ionospheric heating experiments, which are estimated to be well above the threshold for these parametric instabilities will be in the SLT regime.
(2) In these states of SLT a major part of the power in high frequency density fluctuations is contained in localized states in the case of strong ion sound wave damping which is appropriate to the ionosphere [1, 3, 41. These localized states, which we will call cavitons, consist of a high frequency Langmuir field trapped in a self-consistent density cavity (i.e., density depletion). The dynamics of these cavitons will be a major concern of this paper. It is important to emphasize that these localized states are not wavepackets of plane linear Langmuir waves, but new nonlinear Langmuir states and consequently cannot be described by perturbation arguments such as weak turbulence theory based plane wave states.
(3) This state of SLT is sustained by a local nucleation process [see (6) below], and not by linear parameteric instabilities [3, 6, 71. The developed turbulent state is stable to the excitation of these global parametric processes because of the level and localized nature of the turbulent fluctuations. Parametric instabilities may play a role in the transient excitation of the SLT state from quiescent initial conditions; for ionospheric parameters this would be the first ms following the turn-on of the heater.
(4) The localized states are trapped in self-consistently evolving density wells which collapse to small dimensions because of the dominance of the nonlinear ponderomotive force over the linear pressure force. The evolution from nucleation to collapse is discussed by Rose and Weinstein [SI and the collapse process follows the self-similar scaling discussed by several Soviet authors [2, 91.
(5) As the caviton's spatial dimension decreases to the order of 5-10 electron Debye lengths, %De, the electrostatic energy trapped in the caviton is rapidly given up in the acceleration of electrons resulting in the sudden dissipation or "burnout" of electrostatic energy.
(6) The electrostatic burnout process leaves an empty density cavity, no longer supported by a ponderomotive force, which then evolves as a free ion sound pulse. These residual ion density wells provide nucleation centers for the excitation of new collapsing cavitons [3, 6, 71. For strong ion sound damping, appropriate to the ionosphere, the burntout density wells relax in place.
One purpose of this lecture is to review several years of research on strong Langmuir turbulence. Highlights of this research have been reported in several short articles [3, 4, 6, 71. Limiting the scope of this discussion to ionospheric heating greatly reduces the "volume" of the potentially very large parameter space of SLT which is considered, although what remains is still very rich in phenomena. Other applications such as laser-plasma interactions (e.g. see Rose, DuBois and Bezzerides [lo, 111) involve many of the same or related phenomena. The same conditions of excitation near the critical density in weak density gradients might be approximated in long scale length laser produced plasmas with weak collisionality or in laboratory microwave-plasma experiments with sufficiently long-lived plasmas uneffected by boundaries. We will not treat such applications in detail here.
A second purpose of this paper is to sketch out the application of this strong turbulence scenario, which we believe represents the most credible description of Langmuir turbulence, to the experimental facts of ionospheric heating. This SLT approach represents a significant departure from the accepted or conventional ideas associated with parametric instabilities and weak turbulence cascades.
Our most complete current understanding of SLT is based on simulations of a homogeneous, isothermal model described by Zakharov's equations [2]. This situation is best realized in ionospheric modification for early times (several ms) after heater turn on before large scale (several m) density and temperature fluctuations have had time to develop. The most relevant data for this situation seems to be that obtained by Wong et al. [12], Djuth et al. [13] and recently by Cheung et al. [14] in low duty cycle pulsed heating of the ionosphere.
A more detailed comparison of the SLT theory to ionospheric modification experiments is given in Ref. [l].

Zakharov's model of nonlinear Langmuir wave-ion sound
The calculations to be reported here will be based on solutions of Zakharov where w i = 47c2n0/me where no is the mean plasma electron Phj rica Scrtpta T30 density. It is assumed that

I3,Ei 4 lw,El.
The total ion density is written as where A is the fluctuation about the mean density; the spatial average of A is then zero. The equations of Zakharov's model are where V x E = 0. Here iD is the electron Debye length and cs = (qTe/mi)"2 is the ion acoustic speed which is often expressed in terms of specific heat parameters. Eo is the possibly time-dependent pump which is assumed to be spatially uniform. This is the "heater" field in ionospheric modification experiments. Tildes are used to denote conventional dimensional quantities to distinguish them, where necessary, from dimensionless quantities introduced below.
The damping operators 3 , . and 3 , . which are nonlocal in coordinate space are local in Fourier space. In Fourier space it is also simple to include a weak background geomagnetic field Bo based on the modified Bohm-Gross dispersion relation for Langmuir waves [l, 151.
where v,' = Te/me, w, = eBo/mc and 8 is the angle between Bo and k.
In this paper we adopt the convention for spatial Fourier transforms: where L is the linear dimension of the system and D is the dimensionality of space.
The ionospheric heater or "pump" field Eo(!) is included, ignoring pump depletion, by assuming that the spatially uniform, k = 0, Fourier component is a given function. We will generally take E0(t) = Eo exp [-i&,t] where h0 = wHcop is the difference between the heater frequency and the average plasma frequency. The Langmuir wave dumping term is taken to be collisional damping plus Landau damping.
for k" < 0.3kD; this function is continued smoothly to increase as k"z for large L. The latter step is necessary in order to arrest collapse at small scales as discussed by Zakharov and Shur [16] and Russell er al. [3, 71, it is essential for numerical resolution. This damping is an ad hoc addition to the model which is justified by comparing with particle in cell simulations [17][18][19][20] which show nearly complete dissipation of the trapped electrostatic field at the burnout stage of collapse. For the work reported here, where we treat heater intensities well above the collisional thresholds for parametric instabilities, we will take Cc = 0. This is valid provided all physically important rates are much larger than Cc [l].

Caviton Dynamics in Strong Langmuir Turbulence 139
For ionospheric conditions we expect the ratio of electron to ion temperatures, T,/Tl, to be of order unity for early times after the onset of heating. Fluid description of the ion density response is then expected to be quantitatively inaccurate because of the important role of Landau damping on ions [21]. Since kinetic simulations of the ion response are prohibitively expensive for the problems we treat here we have adopted the following strategy: We use the fluid description of eq. (2.4b) but the sound velocity cs and the ion Landau damping used in this equation are chosen to coincide with the least damped poles of the linear kinetic response. Using this procedure we find for 6 < kDe that ?,(6)/&,(@ = v, where GI(k) = &qT,/ml)"2. The values of v, and q are found from the least damped roots of the full kinetic dispersion relation We have found the qualitative features of the nucleation process to be unaffected by the values of v, in the regime 0.9 > v, > 0.4 for systems driven well above the nucleation threshold discussed in Section 3.
It is well-known [2], that the linearized form of these equations contains the parametric decay instability (PDI) [22] and modulational instability [23,241 (MI or OTSI) of the pump wave. Furthermore, when weak turbulence analysis [25, 261 is applied to these equations it yields the usual wave kinetic type of equations which lead to the weak turbulence cascade. However, the validity conditions for the weak turbulence approximations are very limiting [26].
We have studied examples of the solution of these equations for parameters relevant to ionosphere heating in which the system is initially excited by a linear parametric instability and evolves to a state of SLT [l]. In this paper, however, we will consider only the developed turbulent state.
In carrying out numerical solutions of these equations it is convenient to use dimensionless untilded quantities which are related to dimensional tilded quantities in the following way [25]: [ll.
significant for k" > 0.2kD(kD = 3,;'). Thus in dimensionless units this dissipation becomes significant for k greater than the dissipation scale kd : Since the dynamics of the decay instability involves 6s on the scale of 6* = (2/3) (qm,/m,)''' k , we need Fourier components at least as small as this, if the parametric processes are important, and this sets the linear dimension of the simulation cell to be L, = LL > 2n/k,. In dimensionless units k , = 1 and L, = LL. > 2n. The number of Fourier modes must be sufficient to probe deep within the dissipation range of k,,, > kd in order to resolve collapse. This sets a limit on the value of M which can be accommodated in a reasonably sized simulation of say 128 x 128 Fourier modes in two dimensions. In view of these limitations we have chosen M = 1836 for our simulations. At various points in the text to follow, especially in Section 3, we will discuss the scaling of physical quantities with the mass ratio. This scaling will allow us, at least roughly, to translate the simulation results to the larger mass ratios.
The validity conditions for Zakharov's model have been discussed elsewhere [2, 251 and include the condition.
(2.1 la) no A discussion of the degree to which these conditions are satisfied in our numerical simulations is given in Ref. [l].
Our simulations are carried out on a 128 x 128 square grid of sides L , = L , = 2n, with periodic boundary conditions in x and y. In physical units this implies LA = L, = 4O4iDe and a grid-point spacing Ax = Ay = 3.15LDe. The Debye wavenumber in these units is 64.3 and the maximum wavenumber is 91. Spot checks with a dealised code with a nominal 256 x 256 grid were used to confirm the validity of our simulations. Typically the spectrum ( 1 n(k) 1 ' ) decreases by 4 orders of magnitude between the peak of the spectrum and the largest k values. The test of temporal spatial resolution is energy conservation as expressed by the balance between the average dissipation and injection rates [ 11.

The local caviton model
The scaled equations then have the familiar simple form The accumulated evidence from many computer simulations Here a caviton event i is localized at the space time point x,, t, . The single event function 2,(x, t ) has its maximum at x = 0, t = 0 with a spatial width 6, ( t ) and a temporal width parametric instability space and time scales which increase or lifetime zi; from the simulations we find this lifetime to be of the order of 5 to lops for ionospheric parameters. At a given time t, the number of events N ( t ) which contribute to the sum in eq. (3.1) are those for which 0 < I rtil < zi which is clearly proportional to the volume of the system if the cavitons are roughly uniformly distributed. For example, if the portion of the heated volume observed by the radar is ionospheric heating experiments is (200 m) [3], the mean caviton spacing is 0.25 m which is about 501. , , as observed in our simulations, and the mean time between events is roughly t, N 10 t i as observed in our simulations, then we find N(r) -IO' which is crude but representative. In eq. (3.1) the term Enonlocal(x, t ) represents the nonlocalized part of the envelope field which we will assume is relatively negligible. This local caviton model can be put into a more formal setting by introducing the instantaneous vector eigenfunctions e,(x, t ) of the operator on the left-hand side of eq. (2.9a). These satisfy (for B, = 0) The complete description of these states for an arbitrary n(x, t), especially for D 2 2 is beyond our capability. In D = 1 it is relatively easy to compute these states from an arbitrary realization of n(x, t ) obtained from the complete numerical simulation of eqs. (2.9a, b) [6]. We note that a subset { i } of the states { v } are localized at x = xi in density depressions of n(x, t ) and some of these, which have the proper symmetry to couple to the pump E,, evolve to collapse. This subset of states can be viewed as local ground states of the "potential" n(x, t ) .
The electric field envelope can be resolved in this complete set of states E ( x , t ) = C h,(t)e,(x, t ) exp (-io,t). (3.7)

F'
Here we have taken the e, to be a complete orthonormal set with  To understand the various terms in eq. (3.7) first consider the case where n ( x ) is independent of time and therefore (d/dt)e,(x) = 0. Then the amplitude h, is driven directly by the source term E, (e, 1 n ) . If e, where a plane wave state proportional to exp ik x; then this source term is proportional to E,k n ( k ) the so called direct conversion source term [l, 121. However, the important states are the localized states. The coefficient (ev + veev) couples states because of the nonlocal nature of the Landau damping. This term becomes important in the time dependent case only in the burnout phase. Note that by introducing the spatial Fourier transform of the eigenstates e,(k, t ) we can write (e, + veev > = 1 e,*(k, t> * ev,(k, t)Ve(k). (3.10) For v = v' this certainly becomes important in the burnout phase. For v # v' this is less important if one of the states is not localizede.g., noncollapsingor is localized at a different space-time point.
In the time-dependent case of interest the coefficient (e, + 12, ) can provide a coupling between rapidly collapsing states, say v' and a nonlocalized state v. This is one of the mechanisms responsible for the excitation of the "free mode" states observed in the spectra. We will return to this in more detail below.
We can now make a tentative connection between the localized event functions 8,(xx,, tt , ) of eq. (3.1) and the subset {i} of localized eigenstates. It is reasonable to identify where E , ( X -x I , tt , ) = h,(t)e,(x, t). (3.11b) The contributions from the remaining nonlocalized states in the set {v} make up the term Enonloca,(x, t ) in eq. (3.1).
The eigenstates e,(x, t ) are in a sense the natural basis or coordinates for describing the turbulent system. Unfortunately, they can only be obtained by first solving eq. (2.9) for n(x, t). In spite of this they are conceptually useful and some observed properties of the turbulence can be related to general properties of these states. In effect the use of the states e,(x, t ) represents a huge reduction in the effective dimensionality of the problem. While we use (128)2 Fourier modes for the simulation there may be of the order of 10 collapse sites in the cell and therefore roughly 10 localized states.
We have gained useful insight into the nature of these eigenstates and their connection to the observed turbulence by considering the scalar Zakharov model. In this model E(r, t) and Eo are scalar fields are in place of eq. (2.9) we have In this model only spherically symmetric collapsing cavitons are allowed and the three dimensional problem for an isolated collapse reduces to one in which E and n depend only on the radial coordinate r. This scalar model has several properties in common with the physical three-dimensional vector model (2.9): threshold and maximum growth rate for the modulational instability, collapse scaling exponents which are discussed below, no threshold energy for collapse and the possible failure of a density well to support a localized eigenstate.
Spherical symmetry is imposed by representing all fields in terms of the Fourier modes sin (k,r), k , = d / r o , 1 = I , 2, . . . , with r, chosen large compared to a typical caviton size. In these scalar studies we have observed for v,(k)/k = 0.9 that at the nucleation site, E(r, t ) is dominated by its projection, h,(t), on the localized ground state eo(r, t). In nucleation e,(r, t) remains localized; at every time step eo(r, t) can be computed from n(r, t). Here we will adopt a simplified model in which h,(t) is evolved neglecting the excited state contributions v' # 0 in eq. (3.7). The density evolves according to eq. (3.12b) with the ponderomotive force replaced by V 2 I E, + h,(t)e,(r, t)I2. The solution to this model is insensitive to boundary conditions (i.e., the choice of r,).
Let us restrict our attention to the case where vi is large enough so that after burnout, the relaxing density fluctuation is essentially nonpropagating. Immediately after burnout, energy absorption is minimal because the eigenvalue is large and negative, implying a far from resonant coupling to So [see eq. (3.9c)l. The ponderomotive force is negligible, and the density fluctuation evolves according to the acoustic Green's function. A simple model for this phase of the dynamics is obtained by replacing the r.h.s. of eq. (3.12b) by where the "impulse" Z E j dt J dx I E ( X , t) 1 ' = j I h,(t) l2 dt z (I ho 12)5,. In three dimensions, the response of n is where G(e) = --e 1. (3.13b) Even though n is evolving self similarly, e, is not. The figure of merit, ,U, for the ground state is simply expressed in terms of the width w, w ( t )t, of n, and its depth, d, d -Zit4, as ,U N dw2 -Z/t2. If p is too small, there is no localized state. Since 3D solitons are unstable, as t increases, either enough energy will be accumulated so that another collapse follows, or the bound state will be lost. In the latter case, the bound state will be localized in the immediate vicinity of the expanding density fluctuation, until just before the bound state is lost. So that during the time when energy is being injected, one may be able to ignore the coupling between states localized at different collapse sites. At a particular collapse site there is a lowest lying localized state which has a nonzero coupling to the pump. Excited states at the same site typically have a smaller source term because they are oscillatory while E,n(x, t) is essentially uniform in direction. Also at a particular site there may only be a small number of localized states. This motivates the study of a model for the evolution of a caviton in a previously existing density fluctuationa process we call caviton nucleationin which only one localized state is present. For a given ion fluctuation the lowest lying state, e,, with nonvanishing source, So, (we shall call it the ground state) is calculated from eq. (3.9c), ho is evolved according to eq. (3.7) without the coupling to other amplitudes.
In Fig. 2 we show some typical results from the scalar model driven by a spatially uniform field E, at the plasma frequency (coo = 0). For a range of E,, a stable nucleation cycle is observed, with a complete cycle over the interval 0 < t < t,. We expect that in a turbulent environment of other such nucleation sites, the strict periodicity of this cycle may be lost, but at a given site there may be strong correlations over a few cycle times for strong ion acoustic damping.
In Figure 3 we show snapshots of the lowest two eigenstates e,(r, t ) and e, (r, t ) and the density n(r, t ) as they evolve during one of these cycles for a case where coo < 0. In Fig. 2(a) we show the time evolution of 1E(r = 0, t) 12 and n(r = 0, t), in Fig. 2 collapse. This illustrates that we believe to be the typical behavior: As the relaxing density well becomes shallower and broader its eigenvalue approaches resonance with the pump causing a rapidly increasing PMF which initiates the next collapse. It is, of course, important that the state remain localized so that it maintains a significant PMF. Under some conditions for D 2 2 a localized bound state can be lost, i.e., A, crosses zero before sufficient P M F is built up to initiate collapse. For the D = 3 scalar model discussed here we find a finite nucleation threshold (Rose et al. [l 11). For E, below this value the cycle cannot be maintained. There are certain qualitative features of the scalar model which enable one to make scaling predictions. This has been discussed by Rose et al. [l 11 Consider a regime in which E, is time independent, and dissipative effects are significant during a time very short compared to z, the caviton cycle time, with the cycle beginning at t = 0, and ending at t = T . We shall call this ideal nucleation. One dimensionless parameter of nucleation is the degree of localization, p, at the turnaround time, t , , when 6 begins to shrink. This can also be expressed in terms of the impulse I , eq. (3,13a), which drives the density well: p = Zit' where t is the time elapsing since the previous collapse. For 0 < t < t , , the ponderomotive force is negligible, and one obtains the estimates d ( t = t , ) _ _ At , and p -I / ( . Near turn-around, 1 A, 1 attains its smallest value, A (for an isolated nucleation site, localization requires %, < 0), and I h, 1 ' attains its maximum value, N,,, .
If one makes the hypothesis, justified by the results of  The sonic regime is Obtained for EO ET where ET is the threshold value of EO, above which stable caviton cycles are eo@, t ) and e , (r, t ) vs r at the same times. The qualitative behavior is the same for the case coo = 0.

Physica Scripta T30
obtained. At turn-around, the cavity is closest to resonance, and the oscillations in I h, /* are potentially large. However, there is not sufficient time between turn-around and burnout for these large oscillations to take place, so that N,,, 2 N ( t ) , the amount of energy taken into burnout, which is also the amount of energy dissipated, 9. In this regime, one finds that (T -t , ) / t 6 1, and therefore as E0A3", because the oscillation amplitude of a nonresonantly driven oscillator -(source/frequency mismatch). Therefore N,,, -EiA3. (3.16) Combining eqs. (3.14) and (3.16) one obtains (3.14) The evolution of the caviton parameters for E,, = 3.40, and no damping until the very end of the cycle, are shown in Fig. 4. Note that the major variation in time of I h,, l2 is on the time same T , which may be reinterpreted as the sonic transit time over A, and that is why we call this sonic nucleation. Relations (3.14) do not depend explicitly on E,, and so we shall call them internal scaling relations. If one assumes that collapse sites are typically situated a distance A from each other, then the mean energy density (I E l 2 ) W -N,,,/ A3 -r l/A2, and the mean dissipation rate (per unit time per unit volume) R -W / tl/A3. Therefore R -r W3I2. (3.15) Another set of scaling results is obtained by relating the above quantities, i.e. N,,, and A, to E,,. From dimensional analysis the source term in eq. ( 3 . 9~) can be estimated as E03.,,83'2. The maximum amplitude 1 h,, 1 is then simply estimated (3. 17) In words, the stronger the drive, the weaker, smaller, and briefer are individual collapses, in a way such that the overall dissipation rate is greater. As far as we can tell, the results in eq. (3.17) are quite different from those predicted from a modulational instability analysis.
As E,, increases, (t -t , ) A becomes large and we encounter the supersonic nucleation regime, there are large oscillations in N for t , < t < r , and 9 decreases faster than N,,,. The evolution of N for E,, = 8.0 and 32.0 is shown in Fig. 4. In In order to physically realize ideal nucleation, dissipative effects must be negligible when energy is being absorbed. Peak absorption rate occurs roughly at the same time that the caviton is fattest with width A, In dimensionless units the Debye wavenumber -[ M mi/~m,]Ii2. Therefore, if AM"' -M"'/EJi6 % 1, nucleation will be essentially nondissipative, Ignoring the difference between 7/6 and 1, and recalling the units of electric field strength, this means the energy density of the external field must be small compared to the particle kinetic energy density.
For example, if Eo = 8.0, numerical studies indicate that M > 40000, i.e. field energylkinetic energy < 0.003, is required in order that 9 be within 20% of the ideal value.
This estimate of a few tenths of one percent does not depend sensitively on M .
has an interesting consequence if it is assumed that the scale of the caviton G(diss) at the end of collapse, i.e. at the beginning of dissipation is independent of Eo (say 5ADe in physical units). Then since 9 is the energy dissipated at the end of collapse we have 9 N j El2 peak d3(diss)const. as the electron-ion collision rate, or frequency mismatch between the external electric field and the average plasma frequency, becomes more important as M increases. This isolated collapse model is oversimplified in several ways, one is its neglect of the turbulent environment of the collapse site. The density fluctuation n(r, t ) was constrained so that there was locally no net change in particle numberi.e., 1 dr r'H(r, t) = 0i.e., the local averaged plasma frequency is the same as the global average plasma frequency which is the zero of frequency in our envelope approximation. This constrains all bound or localized states to have A,,(t) < 0. However, locally on a scale larger than a single caviton but macroscopically small there can be fluctuations in the background plasma frequency away from the global average. If the local plasma frequency is different from zero this is equivalent to replacing n(r, t ) in eq. (5.12a) by dn, + n(r, t ) and bound states can occur if AV < dn,. Since there are local domains or "patches" of positive and negative dn, we conclude that in a large multicaviton system localized states can occur for A,(t) < (6n0)max where (dnO)max > 0 and depends on the parameters E,, wo, etc. which determine the turbulent state. Simulations with wo > 0 are consistent with this picture. The "localized" states for 6no > 0 are not strictly localized from the mathematical point of view; their eigenfunctions may have extended tails which are exponentially small but do not decay at large distances. Such states are better described as resonance states as discussed at the end of this section.
The self similar scaling of the parameters of the eigenstates during the collapse phase are well-known (e.g., Galeev et al. [9]). A self-similar ansatz for e,(x, t ) can be written (3.18b) The collapses observed in our D = 2 simulations are not cylindrically symmetric as can be seen in Figure 4.2, but have a pancake shape with the narrow direction mainly aligned along the pump polarization (the x direction in Fig. 1). The aspect ratio of the y dimension to the x dimensions appears to be in the range of 2 to 3. The ansatz of eq. (3.18) implies that although the aspect ratio is not necessarily unity all dimensions scale with G(t). Simulations of collapse in D = 3 for isolated cavitons [27] and for multicaviton states [5] display these pancake cavitons whose orientation arises either as a result of initial conditions or by coupling to a drive source such as we have used. The scaling of the parameters of the self-similar, supersonic, collapse depend on the spatial dimension D as follows: where t, is the time of collapse. The self-consistent density behaves as  (t)]. The scalar model collapse behavior is consistent with these scalings for D = 3. We will use the self-similar formula (3.18) for e,(x, t ) in other regimese.g. nucleation but where d ( t ) does not satisfy the scaling of eq. (3. 19). Examples of the evolution of h(t) in the scalar model are shown in Fig. 4. In the scalar model results, presented above, the contributions from nonlocalized states or from localized excited states are neglected. Such localized excited eigenfunctions have one or more nodes in the region of the confining density well and would be expected to couple less efficiently to the pump in the overlap integral of eq. (5.9(c)). Localized excited states which evolve to collapse are not observed in the simulations.
For an isolated density well the localized state eigenfrequencies lie below those of the nonlocal (or continuum) states. In the "patchy" model with fluctuating domains of differing mean plasma frequency, it appears that the eigenvalue ranges of localized and nonlocalized states may overlap.
The definition of a localized state as one of the localized eigenfunctions e,(x, t ) is actually too restrictive. A wavepacket of nonlocalized states whose I,, lie just above the localization limit can be a resonance state analogous to those known in quantum mechanical scattering theory if it is a superposition of states with a sharp peak in the density of states. Such a resonance state will appear spatially coherent and localized for a time At -(AI")-' where AA is the frequency width of the resonance. The resonance state will then have a ponderomotive force over a time which may be sufficiently long to depress the density so that a strictly localized eigenstate can again appear. The effective source terms E,, * (e, n) are nearly the same for all the states comprising the resonance. A narrow resonance state therefore cannot be distinguished from a strictly localized eigenstate and so the definition of the states e,(x, t) which are identified in eq. (3.11) should be extended to include such narrow resonance states. It can be shown that for a sufficiently narrow resonance the equation of motion for its amplitude h,(t) is indistinguishable from the equation of motion discussed above for a localized state. The existence of such resonances is another reason why localized states appear to exist for A, > 0. As discussed above, the random density environment of a caviton can also raise the eigenvalue limit for localization to positive values. Such localized states are also best described as resonance states.

Energy spectra and power spectra in the local caviton
It is well-known that the power spectrum of electron density fluctuation, A,(x, t), contains information concerning the elementary excitations of a plasma and can be measured by incoherent Thomson scatter techniques. For frequencies w near the electron plasma frequency wp (or its negative) the ions cannot respond significantly and so one can relate the electron density fluctuation directly to the total electric field 47ieii,(~, t) = v * ETOT(z, t) = +V E(z, t")e-'"p' + C.C.
(4.1) From this it is easy to see that the power spectrum of ne is directly related to that of the envelope field. For (3 N wp we ( 4 4 To obtain the low-frequency spectrum associated with the ion line we note that for ci, 6 wp that quasineutrality is obtained so ne(l, 63) N n,(E, 63) and the low frequency electron density spectrum is obtained directly from the density fluctuation which appears in the Zakharov model: I &(E, 63 12 = I fi(E, 63) 12. (4. 3) The power spectra can be found by taking the temporal Fourier transform of eq. (3.1).
is the single event Fourier coefficient. If we make the assumption, which is probably invalid, that all events are uncorrelated we obtain the power spectrum as a sum over single event spectra:

(4.5)
The important effect of correlations will be discussed below.
Simulation parameters can be chosen so that the collapse events are so well separated in time that we were able to compute the single event spectra 1 ~( k , w) 1 ' for this case.
These spectra, shown in Fig. 6, have a surprisingly rich structure including the following features: (1) Essentially all the spectra energy occurs for w < w,; in this case wo = 0; where -U,,, N k, i.e. wp -G , , , -I&~C,; (5) There is a weak 'free mode" peak at w -k2, i.e., roughly at the Bohm-Gross frequency. These single event spectral properties are similar to those shown in Fig. 8 for the power spectra from multicaviton states in a magnetic field [l]. Caviton-caviton correlations also can have a strong effect on the spectral shape and will be discussed below.
We have obtained some insight into the sources of this structure from the scalar model discussed in Section 3. A realization of the single event spectrum I ~( k , 0) 1 ' for the scalar model is constructed by taking the temporal transform of the function f ( t ) = h,(t)eo(k, t) for 0 < t < T , and f ( t ) = 0 for z, < t < Twhere Tis chosen to give the desired frequency resolution and e,(k, t ) is the spatial Fourier transform of the numerically obtained eo(r, t). The results are shown in Fig. 7. These model spectra contain the features listed above for the D = 2 Zakharov model (Fig. 6) except for (4) and (5). The predominance of negative frequencies arises because the phase velocity 6 is in Fig. 2  The peaks arise from a modulation of the spectrum with an angular frequency Aw = 2 4 2 , where z, is the caviton lifetime as measured by the width in time of the total electrostatic energy pulse, I h,(t) 12, shown in Fig. 4. This is the same kind of modulation that arises in the spectrum of a single square wave pulse. Let us assume that the early-time spectrum from low-duty-cycle experiments can be identified with the incoherent average (1 ~( k , w ) I*) of single-event spectra.
In this averaged spectrum the individual spectral peaks may be smeared out but it is reasonable to assume that the halfpower frequency width is approximately that of the first and strongest maximum of the single-event spectra. Application Physica Scripta T30 of this argument to the data of Djuth, Gonzales, and Ierkic [13] in this regimee.g. their The free-mode peak observed in Fig. 6 is, of course, not seen in this simplified scalar model calculation since it neglects all of the excited states. The free mode excitation in the scalar model is treated below. The behavior near w = 0, including the shift of the maximum peak with k, is also different in the scalar model than in the D = 2 Zakharov simulation of Fig. 6. The inclusion of excited states in the scalar model brings the results into closer qualitative agreement. In the case of overdense drive wo < 0, this single state model agrees well with the complete Zakhavov simulations such as those shown in Fig. 13.
The laboratory experiments such as those of Cheung and Wong [27] and in local rocket-borne diagnostics of the heated ionosphere.
In the ISR experiments, however, the spectrum is the result of about 10' events and the question of caviton-caviton The second or "coherent" term in this equation has N 2 potential contributions and so could have a potent effect on the spectrum if events are correlated. We have a limited theoretical understanding of correlations at this point of our research, but recent experimental evidence from Cheung et al. [14] supports the importance of correlations. (4.9) Equation (4.8a) shows that in this approximation the single event spectrum 1 S(k, w ) 1 ' is modulated by the correlation or structure factor: (1 ~( k , 0) 1 ' ) .
We will argue in more detail in Section 6 that the increasing sharpness of the ISR spectrum observed by Djuth et al. [13] which develops a few ms after heater turn on or for low duty cycles is a result of the development of caviton correlations. The wave length of 35 cm probed by the ISR radar is about 702, and is comparable to our estimates of intercaviton spacing. Thus the spatial structure factor 1 exp ik * (x, -x,) might be probed near a maximum if the cavitons have some quasi-regular arrangement. If the events are also correlated in time, the effect of correlations are more potent, as discussed in Section 6. I]

Sensitive dependence on oHup: overdense drive and
As discussed previously the principal effects of a slowly varying density profile can be modeled by letting the heater amplitude E, be a function of space, reflecting the Airy pattern of the heater, and by letting bo = wHwp be a function of space to account for the mismatch between the heater frequency and the local plasma frequency. In this section we will examine the dependence of the turbulence properties on 63,.
If we assume that the electron density profile is smooth and monotonic we can relate changes in bo to changes in altitude. In Fig. 9 we plot the total time-averaged Langmuir energy per unit volume (area) ( W ) (4.7), and the effective dissipation rate R, = vefl Eo 1 ' density versus wo for Eo = 1.2, w c / o p = 0.1, v, = 0.9. We see a dramatic increase in Rd as w,, becomes negative. Negative values of wo are only of interest in the evanescent region of the pump wave, z > z,, or if there are domains of over-dense plasma whose spatial dimensions are smaller than the skin depth of the heater wave, 1c/ wp(l -~; / 6 3 ' ) -' '~ so that the heater can penetrate.
This increase of R, is a sign of greatly increased caviton activity. This is easily understood from the nucleation picture: For coo < 0 the relaxing density well remaining from a previous burnout comes earlier into resonance with the pump and therefore at a relatively deeper depletion compared to the wo = 0 case. Thus at the time of closest resonance i.,wo the eigenfunction e,(x, t ) is more confined. The caviton cycle presumably will be more stable and less effected by neighboring cavitons in this more confined caviton cycle. Thus we expect more rapid caviton cycles, i.e. smaller t c r with less energy carried into collapse. The fact that ( W ) increases by 50% and R, increases by about 300% as wo goes from + 5 to -5 is consistent with this picture. The overdense drive Go < 0 is much more efficient in the nucleation of cavitons.
An important observation of the scalar, local caviton model discussed in Section 3, is that the single event functions El (xx, . tt , ) are phase locked to the pump. Thus it was more convenient to replace these functions in eq. (3.1) by exp (-iiw,t)~,(.~ -x,, tt , ) ; that is to explicitly separate out the pump phase. [See also eq. (3.11).] Another way to look at the problem is to rewrite eq. (2.9) in terms of g(x, t ) = exp (iwot)E(x, t). i.e.. to envelope around the pump fre- where the PMF reverses the relaxation of the density well. This picture is confirmed in detail in the isolated scalar model calculations. In Fig. 10 we show the behavior of the dissipation rate Rd = veffI E, 1 ' and the total Langmuir energylunit volume ( W ) versus wo in this model. Again we see a dramatic increase in caviton activity as measured by Rd as wo becomes increasingly negative. Since this model contains only the ground state it becomes unreliable for wo > 0. In these calculations we also find that the energy per caviton taken into collapse, N,,, , decreases as wo decreases; however, the density of cavitons eh P 3 increases in such a way that ( W ) -6-3Nmax increases as wo decreases.
These observation have led us to examine more closely the

Physica Scripta T30
properties of the turbulence for wo < 0. A very important new property emerges which is not evident in Fig. 9. In the regime wo < 0 there are significant hysteresis effects; this means that the turbulent state depends very much on initial conditions or at least that the memory of initial conditions decays in time much more slowly than in cases where wo < 0.
For example, ifwo > I Eo 1 ' the system is modulationally stable and cannot be excited to a nonlinear or turbulent state from quiet (e.g. thermal) initial conditions. On the other hand, the turbulence is sustained by nucleation from turbulent initial conditions for the data points at wo = -5 in Fig. 9 where E, = 1.2. This behavior is not unexpected in light of the arguments made above for caviton cycles which are more stable and less effected by neighboring cavitons when w,, < 0. To obtain the data points in Fig. 9 we chose initial conditions in which the initial caviton locations and cycle phases did not have any particular order. (In fact, the initial state was the final state of another simulation with wo = 0 which also had disordered turbulent initial conditions.) This hysteresis greatly increases the parameter space which needs to be studied since now we need to specify the detailed initial conditions. We are far from a complete understanding of the basins of attraction of various initial conditions. However, we wish to present here one case which has interesting properties and has led us to consider the perfectly correlated, alternating lattice models postulated in the following section. This example is one of a class of simulations in which cavitons initially arranged in a regular array of sites persist at these sites, their cycles become regular and become phase locked to one another in various temporal patterns.
In Fig. 1 1 we show the initial locations of two density cavities which resulted from previous collapses and in which the initial electric field fluctuation is set to zero. This is the initial state for a simulation with Eo = 1.2 and w, = -25 in scaled units. This value of wo corresponds to a domain in physical units which is about 1% overdense. In Fig. 12 we show the time series of the maximum value of lE(x, t)I2 versus time as the system evolves into a periodic pattern. As time increases each evolves into a strict limit cycle with period 5 = 0.59 with the cycles at the two sites becoming 5 / 2 out of phase with one another. We will generalize this behavior in the alternating lattice models postulated in the next section.
In Fig. 11 we also show the extended spatial periodicity implied by our periodic boundary conditions. This shows that the simulation is equivalent to two interpenetrating square lattices in which the 0 positions are all in phase but   Fig. 13 the single event spectra are also shown for the same values of (k, e). These are easy to isolate from a single cycle at a given site. According to eq. (4.8a) the single event spectrum modulates the spectrum of the structure factor. Comparison of the complete spectra and the single caviton spectra in Fig. 13 verifies this. In Fig. 14 we show the ion line spectra obtained from the same simulations. Note, in the cases in which the plasma line spectra in Fig. 9 has peaks at odd multiples of 2n/t (i.e. cases a and d) structure, the ion line consists of a symmetrical peak around w = 0 (corresponding to C I = coH) and two displaced peaks at co = k 2 4 2 . These two displayed peaks are shifted by exactly the same frequency as the "decay line" peak in the plasma line spectra in Fig. 13(a) and (d). This correlation of frequency shifts in the plasma line and ion line spectra has often been offered as evidence of the parametric decay instability; here we see that the same spectral correlation can arise from completely different physics!
The very sharp spectra in these examples arise from nearly perfect correlations which persist over many caviton cyclesabout 50 cycles in the simulations. Note in Figs. 9.5(a) and (d), which have prominent peaks in the 1 : 3 : 5 pattern, that there are also weak peaks at w = 0 and w = m(2n/5) where YM is odd. These arise from slight deviations from the perfectly correlated model. The weak peak at w = 0 would contribute to the "OTSI" line. We will argue in the next section that these spectral components could also arise from higher order (e.g., I v I = 2 Bragg resonances from domains with larger lattice spacings, a, corresponding to regions with weaker E,, which produce weaker plasma line signals). We also argue  1 n(k, o) 1' for the coherent alternating lattice simulation. Both the total spectrum and the single caviton spectrum are again shown.
there that the "OTSI line" would also have important contributions from domains which are temporally correlated but not spatially resonant. These models again suggest how caviton correlations might produce spectra which mimic the "decay-line'' (and its anti-Stokes partner), the cascade and the 'ZOTSI line".
These preliminary results encourage us to believe that the SLT model with correlations might explain all the observed features of the ISR spectra. We believe there is overwhelming theoretical and experimental evidence that the conventional parametric instability theory cannot do the same.
Much remains to be done to understand theoretically how (or even if) such correlations develop from physically realistic initial conditions in SLT. Until this is done or until experiments directly measure such caviton correlations these correlation models must be regarded as postulates which appear to be consistent with ISR observations and with our limited theoretical understanding. The theretical understanding, based mainly on simulation results for the case of overdense driving, can be summarized as follows: (i) Temporal behavior at a given site develops, on the time of a cycle period, into stable limit cycles.
(ii) Temporal correlations between sites evolve more quickly than spatial correlations between sites.
(iii) Some spatial patterns appear to be stable equilibrium arrangements with definite temporal correlations between sites.
(iv) Those patterns which are not stable equilibia evolve to stable patterns on an experimentally relevant time scalesay tens of ms.

Physica Scripla T30
We have discussed (i) at some length above. An example of (ii) is the simulation discussed in relation to Fig. 11 in which cavitons at neighboring sites became anti-correlated in time, Points (iii) and (iv) (as well as other examples of (i) and (ii) are based on simulations which we will not present here. This work is part of our continuing research.
Other questions remaining to be addressed concern how the overdense regions are formed and how the turbulence is excited in these regions. We note that the existence of a modulational instability (or PDI) is inconsistent with overdense drive ifcoo > 1 E, 1 ' . Therefore, in such a preexisting overdense region the only way the heater can excite the turbulence is through direct nucleation in preexisting density cavities as discussed in Section 3. [The direct nucleation by the pump might be enhanced by free modes driven up by direct conversion of the heater on preexisting density fluctuations. These free modes can also provide a nucleation source for localized states as discussed in Section 7.1 Another scenario is that the turbulence is initially excited in underdense regions which subsequently become overdense in intermediate scale (10-100 m) density modulations resulting perhaps from thermal instabilities that develop on an intermediate time scale (say tens of ms). The adiabatic increase of the density to an overdense condition would then "freeze in" local caviton cycles which might then evolve into a spatially correlated state. Density increases of 1% or less can lead to overdense regions in parameter ranges comparable to the simulations discussed above.
The space and time scales (lO-lOOm, tens of ms) of the density modulations or irregularities invoked above are Caviton Dynamics in Strong Langmuir Turbulence 15 1 smaller than usually calculated for thermal self-focussing or related instabilities which usually consider ordinary Ohmic heating [28-301. However, in the turbulent region the effective dissipation rate (into hot electrons), v ,~, is much larger than the ordinary collision frequency. The space and time scales of the thermal instabilities depend on collision frequency as v, I , (e.g., see Rubenchik and Turitsyn [31]) which would imply that they scale roughly as v, ' in the case of turbulent dissipation. Thus the space-time scales of the thermal instabilities in the regime of turbulent dissipation could be one or two orders of magnitude smaller than in the case of Ohmic dissipation. A more detailed analysis of these effects will be given elsewhere.
Our current research is focussed on gaining a better perspective on the parameter regimes of E,, wo and v, for which correlations persist. We have verified, for example that the behavior seen in Figs

Caviton correlations
Evidence consistent with the premise that cavitons are strongly correlated with one another is contained in the data obtained by Cheung et al. [14] in low duty cycle experiments, described above, at Arecibo. Plasma line data was obtained from single radar pulses of about 1 ms duration. Figure 15 shows the typical time evolution of the total plasma line power over a radar pulse duration with a time resolution of about 1 p s . It shows a strongly modulated signal with peaks about 0.025 ms wide separated by about 0.05 ms. The widths of these peaks are comparable to that of the peaks in total electrostatic energy, 1 h,(t) I *, versus time in Fig. 2 This signal is not consistent with a system of uncorrelated cavitons. Suppose the caviton "firing" times t, [see eq. (3.1)] are uniformly distributed over the radar pulse interval T and that the mean temporal spacing between events, TjN, is much less than the caviton period z,. Then at each instant of time the signal can be approximately represented as E ( k t ) z 2, 1 a,(t) 1 exp i[4[(t)] where the sum is over the many overlapping events at time t which have slowly varying amplitudes 1 a,(t) I = e,@, t -I,) I h,(tt,> I and rapidly varying t l ) . The phase is composed of the spatial part Y, = kx, which for this argument we take to be uniformly distributed in [0, 2711 and the phase of the caviton @(tt , ) which we studied in Section 3. Considering the rapid variation of O with time found in these studies it is reasonable to consider that the total phase dl is also uni-  tt,), which are independent of the spatial phase @[ and are therefore taken to be uniformly distributed independently from the $,'s, we can compute the following averages: This signal is consistent with the local caviton model where there are significant correlations between events. If out of a total of N there are N , correlated events the ISR signal from the correlated events is proportional to N,' while that from the uncorrelated events is proportional to N -N,. If the ratio of the modulation peaks in the power signal in Fig. 15 to the background are estimated to be 100 to 1 then we have roughly   N,'/(N -N , ) -lo2 so with N -10' we find N, -10'.
Thus if only 0.1% of the total events are correlated a dramatic effect on the spectrum can be expected! This argument seems quite general and independent of the details of the correlations. The observed real-time backscattered power signal in a single radar pulse, from the perspective of the local caviton model, leads then to the conclusion that many Physrra Scripta T30 collapse events are coincident in time. This conclusion rests on our estimates for N and for the caviton lifetime t,. From this point of view each peak in the time series is a resultant macrofluctuation of many coincident collapse events.
Further evidence of correlations is obtained in these experiments from the power spectra of single radar pulses. The single pulse observed spectra show a much finer frequency structure. We can associate this fine "line" structure with a discrete set of temporal events; each "event" now includes the correlated collapse of many cavitons with an average event spacing in time of about 0.04-0.05ms. Our simple estimate above implies that each event involves the order of 10JNcorrelated cavitons. A detailed analysis of the single radar pulse spectrum is given in Ref. [l].
We can summarize the comparison between the local caviton model, the D = 2 simulations and the short time scale ISR experiments as follows [I]: (1) The many pulse averaged observed spectra agree in detail with the smoothed simulation spectra. The main energy containing portion of these spectra occur for o < 0 and there is a free mode peak for w > 0. Examples of experimental spectra from Cheung et al. [14] are shown in Fig. 16. ( 2 ) The local caviton model accounts for the w < 0 spectral features as arising from the nucleation-collapse-burnout caviton cycle as discussed in Section 4.
(3) Associated with each caviton cycle a nearly free Langmuir wave packet is radiated away from each caviton site. This is discussed in detail in Section 7. The free mode peak occurs at a frequency Gf = op[l + (3/2)(k1.,)2 + (1/2) (G,/w,)~ sin2 191 associated with a free Langmuir wave for which Uif > ojH yet is a distinct signature of the collapse process.
In several sets of observations [12][13][14]311 sharp spectral peaks are observed to develop as the time delay of the radar pulse is increased following the onset of the heater pulse. In Fig. 17, a 50-pulse average spectrum is shown in which the radar pulses occur 29ms following the onset of a 30ms heating pulse with a 150ms IPP. This spectrum shows features observed in many previous long-time experiments [32] consisting of a main "decay line" peak lying about 3.0 k 0.5kHz below the heater frequency and two "cascade" peaks lying further below the heater frequency by 10.0 f 0.5kHz and 16.0 k 0.5kHz, respectively. This approximate "1 : 3 : 5" pattern of frequency displacements is sometimes associated with a weak turbulence cascade [32].
The question to concern us next is whether such a spectral pattern can be explained in terms of caviton correlations? We will first postulate a class of strongly-correlated models which are motivated by the simulation results discussed in the previous section for the case of overdense drive. We found that for overdense driving where Ui, > wp very stable cycles can be found with cavitons in ordered spatial arrays. We also will assume for reasons discussed in Section 3 that the spacing between cavitons decreases like E;' where E, is the local strength of the heater electric field.
We will assume that the cavitons tend to order themselves in a regular three-dimensional lattice. In the ionosphere application, because the heater field varies within the radar observed region, due to the antenna pattern and the altitude dependent Airy pattern, the spacing, a, of cavitons in the lattice varies on a scale large compared to a. In regions of most intense E,, a will be smallest. The orientation of the lattice may also vary within the observed region. Our scalar model simulations show that for E, N 0.5 V/m the maximum isolated caviton size is about 40iD so we might expect an intercaviton spacing a > 80-100&, or about 40-50cm. At Arecibo the radar wavenumber is about 27r (35 cm)-' which means that low order Bragg scattering from such a lattice is possible with smaller adjustments of the lattice spacing which could arise through variations in E,,. We imagine that the observed region of the heated ionosphere contains domains of ordered cavitons whose lattice spacing and orientation varies from domain to domain. These spatially and temporally ordered domains take some time to organize themselves after the onset of heating. We have some preliminary simulation evidence for this which we presented in Section 5.
In each domain the cavitons are located at sites in a spatially periodic lattice. Each caviton undergoes a periodic cycle of nucleation-collapse-burnout, etc. with period 2 but nearest neighbor cavitons have their cycles displaced by 2 / 2 from each other. (We have presented evidence that in certain regimes nearest neighbors tend to be out of phase in Section 5. ) We can consider a domain to consist of two interpenetrating synchronous lattices whose cycle time differs by 2/2. Thus we can write where n l , n,, n3 are integers. For simplicity we chose the displacement of the second lattice to have the symmetric form: (6.5) We assume the two lattices are the same except for the displacement d which must have a special relation to the symmetry of these lattices. It is easy to carry of the sum over m and to find (6.6) w = 2nm (m = 0 or any integer) (6.7) and spatial (Bragg) resonances at a set of wave vectors k = K, (6.8) This structure factor has temporal resonances at for which exp iK,, * x, = 1 for all x, in the lattice. These resonant wave vectors K,, are known as "reciprocal lattices" vectors in solid state physics. (All of the lattice concepts which we use here are treated in texts on elementary solid state physics, e.g., Kittel [33]. When both of these resonance conditions are satisfied it is easy to see that I Q(k, U ) 1 ' reaches it maximum correlated value proportional to where MN, = N, is the total number of events in the observed space-time volume. In the specific models we will discuss below we find where v is an integer. Therefore when both the spatial and temporal resonance conditions are satisfied the modulating factor cos' 1/4(wz -2kd ) has its maxima at (6.1 1) Thus we have two cases: (1) v = odd integer where I e(k, w ) 1 ' has peaks at w = (2x/t)m where m is any odd integer (2) v = even integer where I Q @ , w)12 has peaks at w = (2n/t)m where m is any even integer or zero.
In case (1) i~( k , w I 2 has spectral peaks at w equals (. . . -5, -3, -1, 1, 3 . . .) x (247). When multiplied by the single event spectrum, (I E(k, w ) I*, which has most of its energy for w < 0 the total spectrum will consist of a major line at w = -27c/z and succeedingly weaker lines at w = -3(2n/t), -5(2n/t), etc., which is similar to the 1 : 3 : 5 structure in the observed spectra mentioned above. In addition, Physica Scripta T30 depending on the strength of the single event spectrum for w > 0, there may be a weak line at w = 2n/t which would correspond to the often observed "anti-Stokes" line [32].
Of course, when v is zero or an even integer the spectral pattern is different with peaks at w equal to (-6, -4, -2, 0, 2, 4, 6 . . .)2n/t. The two classes of spectra are illustrated in Fig. 18.
For the spatial Bragg resonance the radar k must equal one of the reciprocal lattice vectors. The length of k = x/%radar is fixed and we can always write the length of a reciprocal lattice vector in the form where a is the smallest lattice dimension. The Bragg resonance condition then provides a condition on a: a = -

r d d d i A I '
(6.13) The direction of K, which must also be in the direction of the radar k . We will allow the orientation of the lattice to vary somewhat in our further discussion. The smallest lattice spacing, which corresponds to the smallest value of A , consistent with the angular constraints, should correspond to the domains with the largest E,. It is shown in Ref.
[l] that the averaged plasma line power, ( P ( k ) ) = (I E(k) I*), increases rapidly with Eo in parameter regimes of interest. Therefore, we expect the strongest correlated signal to come from domains with the smallest resonant values of a.
A reciprocal lattice vector (RLV) can also be represented in terms of three elementary lattice vectors IC,, IC*, and I C~. From these results it is easy to show IC, -U, = 2n6, from which it follows using eq. (6.10) that (6.14) where v = vi + v* + v 3 . correspond to the values of the triad (vl, v2, v2) of ( I , 0, 0), (0, 1, 0) and (0, 0, l), respectively. Thus for the three shortest RLV's v = 1 and so f o r k = IC,, I C~ or I C~ a 1 : 3 : 5 type of spectrum is found. The domains whose lattice spacing is small enough to achieve the strongest resonance must also have their "crystal" orientations such that the direction of k measured by the radar is nearly along the direction of one of the basis vectors I C~, I C~ or I C~. There are at least two ways to achieve this: The crystal orientations as well as the spacing may vary throughout the heated region allowing the resonance conditions to be satisfied in some local region. Another possibility is that the orientation of the caviton lattice does not vary much and the particular directions, 8 -45" for the Arecibo radar and 6 -0" for the Tromss radar, both correspond to directions of reciprocal lattice basis vectors. A more complete discussion is given in Ref. [l].
Domains of coherent cavitons whose lattice spacing is not

Radiation of "free" Langmuir waves by collapsing
Our studies have shown that the collapse process invariably excites "free" Langmuir waves. These manifest themselves in the "free mode" peak which occurs in all the power spectra, I E(k, w ) 1 2 , which we have computed. As discussed in Section 6 there is strong evidence that the free mode peak has been observed in the short time scale experiments of Cheung et al. [14].
These free mode states are extended states whose energy is not localized at a particular point in space. A single collapsing caviton will radiate a wave packet of free modes which spread out and whose amplitudes decay geometrically (as r -I ) away from the excitation center of the caviton. [we use the term extended in a physical sense; to be mathematically precise boundary conditions at "infinity" must be specified. Because we are studying transient dissipative phenomena, these Langmuir fluctuations cannot explore "infinity."] Outside of the spatial region of an isolated collapsing caviton these radiated Langmuir waves are asymptotically free Langmuir waves obeying the dispersion relation of eq. (2.5). In a many-caviton experiment, the large density fluctuations generated by collapse distort their propagation. The free mode frequencies appear to approach the dispersion relation (2.5) as k increases. For lower k values the frequencies are shifted to somewhat higher values due to the perturbation of the density fluctuations.

cavitons
The generation of free modes can be understood in terms of the coupled mode amplitude equations (4.7). For simplicity we ignore the dissipative coupling (4.9b) which is not important for the k values measured by most radars. The coupling between states is given in eq. (4.7) involving the matrix (4.9a) Mvv, = i(e, + e,.). it is easy to see that the scattering terms do not change the total free mode energy while the terms on the right-hand side of eq. (7.4) do.
We have explored the details of the free mode radiation in the spherical scalar model discussed in Section 4. The numerical solution of eq. (7.4) can be broken into two parts. First, for given % > 0, calculate the associated eigenfunction and matrix elements M; second, choose an appropriate discretization of the integral over 1. In three dimensions it is convenient to work with Y = re,, which satisfies (for the (7.6) There are two linearly independent solutions which may be taken as having either unit value or slope at r = 0. In order that e, have finite I Ve, 12, the solution for Y with unit value is inadmissable. Except for overall normalization, for given n(r, t ) eq. (7.6) is integrated out from r = 0 with Y(0) = 0, dYldr(0) = 1. Note that for kr large enough such that n(r) 4 i, Y ( r )sin (kr + 6) where k 2 = 1 and 6 is the so called phase shift which depends on n. Therefore e,sin (kr + 6)jr for r large. This is the form expected from scattering theory where 6 determines the " S matrix" and everything that you know about scattering. For our purposes, however, the detailed variation of e,.(r) for r small enough such that n matters is required to calculate M .
To determine the choice of discrete R it is useful to think of the caviton sitting in the center of a very large sphere of radius R, with the boundary condition e , ( r = R) = 0. It can be shown that the density of states at eigenvalue 1, e(L), For large enough R the density of states is indistinguishable from the case of eq. (7.6) with n = 0. That is, the discrete set ofpositive eigenvalues is lo, 410, 9A0, . . . where A, = (n/R)*.
With this choice of states, a consistent normalization is 4n loR r2el dr = 1. (7.8) For large R, this integral is dominated by regions of space where n is ignorable, and then (7.9) It can be shown that the discretization of eq. (7.4), under the transformation A. --f A0/4 (equivalently, R + 2R), transforms in such a way that the energy density of the free modes (the amount of energy in the free modes per unit 3.) is invariant in the limit where R is large, so that this choice of discretization should converge to the exact solution of eq. (7.4) in the limit of Lo -+ 0.
r : Ji On physical grounds, the between a very energetic free mode and the collapse process should be weak. Numerical convergence is indicated when, for fixed do, increasing Amax has only a small effect on the total evolution, and for fixed E, , , , , a decrease in E. , has little effect. This is the case for the  Fig. 19.
We turn now to some results from these numerical studies. During the first part of nucleation, there is relatively little energy, I ho 12, in the local ground state, e.g. see Fig. 19(c) and the coupling to the heater, (the last term on the right-hand side of eq. (7.4)), is the dominant source of free mode energy. This coupling is a dynamic generalization of direct conversion, and so we call it dynamic conversion. Deep into collapse, coupling to the ground state is again unimportant because the coupling coefficient is going to zero and the coupling is more and more nonresonant. At intermediate times during collapse the ground state coupling is dominant.
To determine which source term is overall most important, the scalar local nucleation model was used to calculate all the terms in eq. (7.4) for the case of an overdense plasma with parameters M,/M, = 20000, E, = 5.0, v, = 0.8, wo = -40.0. In Fig. 19(a) the evolution of the total energy in the free modes, dA I hi, 12, is shown, while in Figs. 19(b) and (c) the evolution of the ground state parameters is shown. Since the energy in the free modes is a small fraction of the collapsed energy, the coupling of the free modes in the equation of motion for h, was ignored. In Fig. 19(d) for the same collapse as in Figs. 19(a-c), the evolution of the free mode energy is shown with the ground state coupling term removed from eq. Physica Scriptu T30 (7.4). We see that the coupling to the time dependent collapsing Langmuir ground state mode is the dominant source of Langmuir wave radiation, at least in the case of an overdense plasma.
In Ref.
[l] we have also shown that the hot electrons emitted in the burnout phase of collapse do not produce a significant rate of free Langmuir wave radiation by Cerenkov radiation. Thus it is the wave-wave coupling of the collapsing field to the nearly free Langmuir waves which is the dominant source of free mode emission.
In the experiments of Cheung et al. (1989), the ratio of the strength of the "collapse continuum" portion of the spectrum for w < wo to the strength of the free mode line increases as the time delay following the onset of the heating increases. It is observed that the strength of the free mode line does not change by as much as an order of magnitude while the strength of the collapse continuum increases by several orders of magnitude. This is consistent with the increase of the w < wo spectrum due to the onset of correlations, i.e. a signal proportional to N,' rather than N,. The free mode line is not strengthened by correlations except in the unlikely case that the free mode peak at w = wf in the single caviton spectrum coincides with one of the correlation peaks at w = 2nm/z,, Caviton Dynamics in Strong Langmuir Turbulence 151 which were discussed in Section 8. In addition, we know that concluded [ 11 that the strongest plasma line signal should free mode emission is weaker from overdense regions where occur near the altitude of strongest E,, which is the first Airy coo < 0 which are likely to be correlated. Note, for example, maximum in an undisturbed profile. that the free mode peaks in the single caviton spectra from the (2) The shapes of the SLT spectra are much less sensitive correlated simulations of Fig. 14 were coo = -25 are not to the angle, 8, between the wave vector, k , observed by the very strong and do not produce prominent peaks in the radar and the geomagnetic field. Therefore, there is no procorrelated spectra. blem in understanding why 1 : 3 : 5-structured spectra are often observed both at Arecibo (8 -45") and at T r o m s~ (8 -0). WTT, on the other hand predicts the turbulent
One goal of this research has been to explore in detail the (3) The short time scale spectra observed by Wong et al. implications of strong Langmuir turbulence theory for iono- [12], Djuth et al. [13] and Cheung et al. [14] can be explained spheric heating experiments. The short time scale data from in detail by SLT. This includes a new prediction of SLT, the these experiments provide the best test of the theory available free mode peak, which results from the radiation of Langmuir today. A major conclusion of our work is that weak turbu-waves from collapsing cavitons (Cheung et al. [14]). The lence theory (WTT) cannot be valid for the conditions of broad featureless spectrum for 6 < GH is explained by the ionospheric heating. Our conclusion is based, first of all, on dynamics of the local caviton cycle. None of these features is extensive numerical solutions of Zakharov's model encom-explained by WTT. passing many generations of collapsing cavitons. The fact The success of SLT in explaining these short time scale that the numerical solutions are dominated by coherent, results leads to the conclusion that if the turbulence is concollapsing cavitons proves that the nonlinear state is far from fined to an altitude layer near the heater critical density and the regime of WTT. Recently Payne, Nicholson and Shen [26] , if SLT is to explain the sharp spectral features observed on a have explored in detail the limit of WTT in numerical solulong time scale then cavitons must evolve to a state with tions of Zakharov's equations in one dimension and have strong spatio-temporal correlations. Single radar pulse data established rough criteria for the validity of WTT. These obtained by Cheung et al. [14] are consistent with strong stringent criteria are not satisfied for the conditions of ionotemporal correlations even at relatively early times after the spheric heating. onset of heating. We have postulated correlated caviton The strong Langmuir turbulence theory has developed on models which predict that when spatial Bragg resonance two levels. The first level is based on solutions of Zakharov's conditions which depend on the radar k , are satisfied that model equations. From the properties of these solutions we sharp spectral features including the "decay line" the have proposed the local caviton model which is a more 3 : 5 : 7 : . . . "cascade lines", the "anti-Stokes" line and the ''phenomenological'' level. The local caviton model is built "OTSI line" are present in the spectrum. The possibility of on single caviton properties. Cavitons go through cycles of very narrow lines [34, 351 such as the "OTSI line" emerges as nucleation, collapse and burnout. Associated single caviton a result of very long lived caviton cycles.
properties include their lifetimes (or cycle times) z,, the single At this stage we can only make preliminary guesses as to caviton field fluctuation E ( X , t ) and its power spectrum how such correlated caviton states could develop. We believe I &(k, co) 1 ' . These single caviton properties are not necessarily that overdense regions of relatively high heater strength those of isolated cavitons, although the isolated caviton are necessary to obtain strong temporal correlations. If the approximation is at least qualitatively useful in many cases. heater is intense enough to excite modulational instability in As the driving becomes increasingly overdense (coo < 0) we such regions, then some spatial order might be imposed by have evidence that caviton interactions decrease but the the periodicity of the instability. We do know that initially residual interactions can lead to coherent caviton states. It is correlated states in such regions can be very stable and have a great challenge to understand the mechanism(s) for selfbeen able to set up such states which lead to spectra very organization of this weakly interacting caviton gas. For these similar to the postulated strongly correlated models. In such highly correlated states, the name turbulence hardly seems regions the spacing of an ordered array of cavitons is expected appropriate. to decrease as € ; ' which led us to conclude that the lowest We believe that the qualitative properties of the local order Bragg resonance condition can only be satisfied for caviton model will be those deduced from the Zakharov relatively strong E, and that this might explain why the sharp model. More complete and accurate descriptions of single featured spectra are the strongest spectra. caviton properties are needed to treat the end stages of In situ rocket or satellite diagnostics in the heated ionocollapse and the burnout processes, whereas the nucleation sphere appear to be the most direct way to observe the and early collapse stages should be accurately described by presence of Langmuir energy in the form of localized the model. We anticipate that these improvements will make cavitons. Perhaps even correlation properties could be quantitative but not qualitative changes in the picture measured. developed in this paper.
The problem of fast (or hot) electron generation in We believe that the SLT model has at least three apparent collapse and the airglow excited by these electrons in a successes in explaining the ionospheric heating data [ 11: problem at the frontier of this subject. The recent airglow (1) The altitude dependence of the plasma line signal experiments of Bernardt, Duncan and Tepley [40] give [36,37] is easily explained because the localized caviton states increased incentive for a more complete understanding of the are not tied to the linear dispersion relation, (2.5). Based on flux energy distribution of hot electrons. As discussed in the sensitive dependence of the turbulence level on Eo we Ref.
[I], in regard to Cerenkov emission from hot electrons, we believe that the model discussed in this paper can give 11. Rose, H. A., DuBois, D. F., Russell, D. and Bezzerides, B., Physica D fairly accurate values for the hot electron flux but allows no estimate of the energy distributions. Kinetic (PIC) simulations and transit time damping calculations can provide information here. This is a topic for future study.
As mentioned in Section 5, density compression regions associated with irregularities excited by thermal instabilities might be the source of the postulated overdense domains in which cavitons are correlated. The coupling of the small scale Langmuir turbulence to the larger scale thermal fluctuations is another important area for new research.
These questions are a challenge for future work. We believe that the new strong Langmuir turbulence model presented here is considerably more successful in describing the early-time behavior of the heated ionosphere than the conventional theory. We hope this paper will stimulate new experimental and theoretical tests of these ideas.