Nonlinear development of electron-beam-driven weak turbulence in an inhomogeneous plasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a r Nonlinear development of electron beam driven weak turbulence in an inhomogeneousplasma
E.P. Kontar ∗ Institute of Theoretical Astrophysics, University of Oslo, Postbox 1029, Blindern, 0315 Oslo, Norway
H.L. P´ecseli † Plasma and Space Physics Section University of Oslo, Postbox 1048, Blindern 0316 Oslo, Norway (Dated: March 21, 2019)The self-consistent description of Langmuir wave and ion-sound wave turbulence in the presenceof an electron beam is presented for inhomogeneous non-isothermal plasmas. Full numerical solu-tions of the complete set of kinetic equations for electrons, Langmuir waves, and ion-sound wavesare obtained for a inhomogeneous unmagnetized plasma. The result show that the presence of inho-mogeneity significantly changes the overall evolution of the system. The inhomogeneity is effectivein shifting the wavenumbers of the Langmuir waves, and can thus switch between different processgoverning the weakly turbulent state. The results can be applied to a variety of plasma conditions,where we choose solar coronal parameters as an illustration, when performing the numerical analysis.
I. INTRODUCTION
Electrostatic electron plasma waves are easily excited in controlled laboratory experiments [1, 2], and in naturallyoccurring plasmas, by for instance low density electron beams [3–7]. The evolution of weakly nonlinear coherent wavephenomena has been extensively studied in simple geometries, and found to be well described by standard analyticalmodels [8, 9]. Cases where electron plasma waves, or Langmuir waves, are excited in plasmas made linearly unstableby electron beams [10] are particularly interesting as being one of the first kinetic instabilities where the nonlinearevolution described by quasi-linear theory [11, 12] was studied experimentally in a controlled laboratory experiment[13]. This beam-excitation of plasma waves has proved particularly interesting in space plasmas, where high energy,low density electron bursts or beams are abundant. Full-wave numerical solutions for such problems have been carriedout in unmagnetized as well as magnetized [14, 15] plasmas, based on a standard and widely accepted theoreticalmodel [16]. It is interesting that beam generated electron waves are observed by satellites in the interplanetaryplasma near 1 AU [4], and in the ionosphere at high as well as lower altitudes by sounding rockets as well as satellites[4–6, 17, 18]. In these and similar cases, the waves appear as irregular or bursty in nature. A description aimingat a general full wave analysis seems unrealistic in such cases, and a weak turbulence model has been suggested togive the evolution of the most important quantities, such as the averaged electron distribution function, and the wavespectrum [19–21]. The beam-plasma instability excites a spectrum of Langmuir waves, which in turn are damped orconverted via nonlinear plasma processes, such as scattering off ions, l + i → l ′ + i ′ (nonlinear Landau damping offions [19, 22]), and decay into a Langmuir and ion sound waves, l → l + s , discussed previously [19, 23]. Obviously,there are also other nonlinear processes that change the spectrum of Langmuir waves, but the processes mentionedare much more effective in affecting Langmuir waves in the applications of interest.In the present work, we consider the case where energy density of the electron beam is much smaller than thethermal energy density of background plasma. The plasma waves and electron beam are described self-consistentlyby weak turbulence theory [19]. Quasi-linear relaxation of an electron beam with the velocity v b >> v T e generate theprimary Langmuir waves with wavenumbers k ≈ ω pe v b /v b that causes electron beam electron distribution functionto relax toward a plateau in velocity space ranging from v ∼ v b down to v ∼ v T e . Nonlinear processes, l + i → l ′ + i ′ and l → l ′ + s , are effective in scattering of primary, beam generated waves with wavenumber k , into secondaryLangmuir waves with wavenumber k ′ ≈ − k . The decay of a Langmuir wave also leads generation of the ion-soundwaves with k s ≈ k . The process repeats and produce the next generation of Langmuir waves. Every elementarycascade decreases the absolute value of Langmuir wave number by a small value k ∗ d = 2 p m e /m i p T i /T e / (3 λ De )for decay and k ∗ s = 2 p m e /m i p T i /T e / (3 λ De ) for scattering. Therefore repeated scattering and decay processes leadto Langmuir waves being accumulated in the region of small values of k ≤ k ∗ , the so-called Langmuir wave condensate. ∗ [email protected] † [email protected] The decay processes are considered to be dominant process for non-isothermal plasma T e >> T i , whereas scatteringoff ions is more important for isothermal plasma where ion-sound waves are heavily damped.However, the presence of plasma density gradient can change the spectrum of Langmuir waves significantly. In thepresence of a plasma gradient, a given Langmuir wave with wavenumber k propagating in inhomogeneous plasmachanges its wave number to k ± ∆ k , where ∆ k is determined by a plasma gradient. This effect may significantlychange the Langmuir wave spectrum, and therefore to slow down the nonlinear processes. If the plasma gradientis opposite to the direction of beam propagation, and the drift rate due to inhomogeneity is comparable with rateof nonlinear processes, a steady state spectrum of Langmuir waves can appear. This case was considered in thecase of isothermal plasma when the scattering off ions is the only nonlinear process in [24]. It was shown that thecondensate of Langmuir waves appears, but in a shifted region of k and scattering off ions can be compensated bya plasma inhomogeneity. Moreover, a plasma inhomogeneity can change the quasi-linear relaxation rate by shiftingplasma waves from the unstable phase velocity region to the region where Langmuir waves are strongly dampedby the beam electrons [25]. In a number of studies [26] it was found that a plasma inhomogeneity may suppressquasi-linear relaxation, provided the beam density is sufficiently low. The positive plasma gradient can also leadto “self-acceleration” of beam electrons [26]. The importance of a plasma inhomogeneity demonstrated in previousstudies [27] and the presence of inhomogeneities in most plasmas found in nature as well as in laboratories stimulatesan investigation of the time evolution of beam-driven Langmuir turbulence in non-isothermal plasma. The principalaim of the present paper is to discuss the effect of density gradients on the wave dynamics and to demonstrate thateven weak gradients can be of profound importance.In the present study we perform a self-consistent investigation of the time evolution of weak turbulence in weaklyinhomogeneous plasma. It is shown that a plasma inhomogeneity significantly modifies the scenario of weak turbulence.The disposition of the paper is as follows. In Section II we present a formulation of the problem, and present thebasic equations. In Section III we present numerical results. In Section IV we summarize the main results of ourinvestigations, while Section V contains our conclusions. In the present paper we attempt to consider physicallyrealistic parameters, and have chosen those appropriate for electron beams and plasmas for the conditions in the solarcorona. The problem is extremely time consuming numerically, and for this reason our investigations are restrictedto one spatial dimension. II. FORMULATION OF THE PROBLEM
We consider a beam of fast electrons and plasma waves within the limits of weak turbulence theory, when the energydensity W of plasma waves with wave number k is much less then that of surrounding plasma WnκT e < ( kλ De ) (1)where n and T e , are the electron plasma density and temperature, respectively, λ De is the electron Debye length,while κ is Boltzmann’s constant. In weak turbulence theory the evolution of electrons and waves is described bykinetic equations for an electron distribution function and spectral energy densities of plasma waves. The equationsare essentially nonlinear, which significantly complicates the problem [19]. However, having in mind application tolow- β systems with relatively strong magnetic field when the energy density of magnetic field is much larger thanthe kinetic energy of fast electrons, but the magnetic field is not strong enough to magnetize the plasma waves, wecan treat the system in one spatial dimension. The electron beams in the solar corona plasma are typical examplesfor such systems [28]. In these and similar cases, the plasma inhomogeneity along the beam propagation can not beignored.We also assume that the variation d λ of the wave length λ of the Langmuir oscillations is a small, i.e. (cid:12)(cid:12)(cid:12)(cid:12) d λ d x (cid:12)(cid:12)(cid:12)(cid:12) ≪ v | L | ≪ ω pe ( x ) (cid:16) v T e v (cid:17) (3)where L ≡ ω pe ( x ) (cid:18) ∂ω pe ( x ) ∂x (cid:19) − (4)is the scale of the local inhomogeneity, ω pe is the local plasma frequency, and v T e , v = ω pe /k are the electron thermaland wave phase velocities, respectively.The time evolution of the average velocity distribution f ( v, t ) is described by quasi-linear theory, which basicallydescribes a diffusion process in velocity space, where the diffusion coefficient is self-consistently determined by thespectrum of the Langmuir waves [11, 12]. The velocity distribution is assumed to be the same all over the relevantpart of space, and in those cases where we deal with an inhomogeneous plasma, the density variation is contained ina coefficient. The growth and damping of plasma waves is accounted for by the standard Landau prescription andderived from the derivative of the velocity distribution at the phase velocity of the waves. The evolution of the wavespectra is described by the wave kinetic equation basically having the form ∂W k ∂t + v g ∂W k ∂x − ∂ω k ∂x ∂W k ∂k = St , (5)which for St = 0 is the Liouville equation. Equation (5) can be postulated from basic physical arguments [20, 21], or bederived [30] from more basic equations [16]. In the dynamic equation (5), we can heuristically interpret W k as a spacetime varying distribution of wavepackets, each with a carrier wavenumber k , propagating with a corresponding groupvelocity v g ≡ ∂ω k /∂k , and subject to an effective force − ∂ω k /∂x . In the present case we have ∂ω k /∂x ≈ ∂ω pe /∂x , andthis force is directly related to the gradient in plasma density through the density dependence of ω pe . The wavepacketsconstituting W k are accelerated in the direction towards smaller densities while the carrier wavenumber increases.The term St on the right hand side of (5) accounts for sources and sinks, together with effects that redistributeenergy within the spectrum. With the foregoing assumptions (2), we can assume ∂ω pe /∂x ≈ ω pe /L , where ω pe is nowa constant plasma frequency obtained at a reference position in the center of the system. Apart from a negligiblecorrection of the relative order kλ De , this effective force is independent of wavenumber, and it will not induce anyspatial variations in an initially uniform distribution W k . For that case we consequently have ∂W k /∂x = 0. Withthese assumptions we simplify (5) in the following. As far as St is concerned, we have the linear instability actingas a source, and Landau damping as a sink of wave energy. Decay and nonlinear Landau damping act as sinks ofwave energy, but equally important, these effects redistribute the energy within the spectrum W k . The mathematicalexpressions for these latter effects are standard [19, 23].The basic equations are treated as an initial value problem (just as in related recent studies [31–33]). This givesa simplified alternative to the full problem, retaining at the same time the physics being important here. Using theassumptions mentioned above we can write the system of kinetic equations of weak turbulence theory ∂f∂t = 4 π e m ∂∂v W k v ∂f∂v , (6) ∂W k ∂t − ω pe L ∂W k ∂k = πω pe nk W k ∂f∂v + St ion ( W k ) + St decay ( W k , W sk ) (7) ∂W sk ∂t = − γ sk W sk − αω sk Z (cid:18) W k ′ − k ω k ′ − k W sk ω sk − W k ′ ω k ′ (cid:18) W k ′ − k ω k ′ − k + W sk ω sk (cid:19)(cid:19) δ ( ω k ′ − ω k ′ − k − ω sk ) dk ′ (8) St decay ( W k , W sk ) = αω k Z ω sk ′ (cid:20)(cid:18) W k − k ′ ω k − k ′ W sk ′ ω sk ′ − W k ω k (cid:18) W k − k ′ ω k − k ′ + W sk ′ ω sk ′ (cid:19)(cid:19) δ ( ω k − ω k − k ′ − ω sk ′ ) − (cid:18) W k + k ′ ω k + k ′ W sk ′ ω sk ′ − W k ω k (cid:18) W k + k ′ ω k + k ′ − W sk ′ ω sk ′ (cid:19)(cid:19) δ ( ω k − ω k + k ′ + ω sk ′ ) (cid:21) dk ′ (9) St ion ( W k ) = βω k Z ( ω k ′ − ω k ) v T i | k − k ′ | W k ′ ω k ′ W k ω k exp (cid:20) − ( ω k ′ − ω k ) v T i | k − k ′ | (cid:21) dk ′ (10)where α = πω pe (1 + 3 T i /T e )4 nκT e , β = √ πω pe nκT i (1 + T e /T i ) , (11) γ sk = r π ω sk " v s v T e + (cid:18) ω sk kv T i (cid:19) exp " − (cid:18) ω sk kv T i (cid:19) (12)where γ sk is the damping rate of ion-sound waves, v s = p κT e (1 + 3 T i /T e ) /m i is sound speed, f ( v, t ) is the averagedelectron distribution function, W ( k, t ) and W s ( k, t ) are the spectral energy densities of Langmuir waves and ion-sound waves respectively. W ( k, t ) plays the same role for waves as the electron distribution function for particles.The system (6) and (7) describes the resonant interaction ω pe = kv of electrons and Langmuir waves. The last termon the left-hand side of (7) represent the shift in wave number induced by the plasma inhomogeneity.In the right-hand side of equations (6) and (7) we omitted terms accounting for spontaneous emissions, becausethey are small in comparison to the ones retained. It should be also noted that small collisional terms are not includedeither. The system reaches steady state before collisional effects become important. III. NUMERICAL RESULTS
We consider an initial value problem when all initial energy is accumulated in electron beam. The electron dis-tribution function of electrons at the initial time moment t = 0 is the combination of fast electrons and backgroundMaxwellian electrons f ( v, t = 0) = n b √ π ∆ v b exp (cid:18) − ( v − v b ) ∆ v b (cid:19) + n √ πv T e exp (cid:18) − mv κT e (cid:19) (13)where n b , ∆ v b are the beam density and the electron beam thermal velocity. The initial spectral energy density isthermal W ( k, t = 0) ≈ κT e π λ De , (14)where T e is the electron temperature of the surrounding plasma, and λ De is the electron Debye length. The system ofkinetic equations is reduced to dimensionless form and integrated using finite difference schemes. For the numericaltime integration we used a method similar to the one used in [34] and the quasi-linear terms are integrated usingmethods described in [35]. Previous results [34, 35] are found in full agreement with the corresponding limiting casesof our calculations.The electron beam and plasma parameters are taken typical for the conditions in the solar corona [36, 37]. We use v b = 12 . v T e = 5 × cm/s, ∆ v b = 0 . v b , T i /T e = 0 . T e = 10 K, n b = 50 cm − , and ω pe / π = 200 MHz. Theplasma inhomogeneity is given by a constant plasma gradient, which can have either sign. Six different cases havebeen considered: the strongest plasma inhomogeneity considered is L = ± × cm, a medium plasma inhomogeneity L = ± × cm, and a weak plasma inhomogeneity L = ± × cm. Such plasma inhomogeneity might exist inthe low corona [38], in ionosphere [39], and in the solar corona due to small scale inhomogeneity [40]. A. Homogeneous plasma
As a reference case we first considered the evolution of Langmuir turbulence in a homogeneous plasma, correspondingto L → ∞ . Results are shown in fig. 1, demonstrating that the fastest process in the system is quasi-linear relaxation,which drives energy out of a beam and into Langmuir waves. As a result of quasi-linear relaxation the electrondistribution function rapidly flattens, building a plateau from 15 v T e > v > v T e for t ≈ .
02 s, which is close to a quasi-linear time τ ≈ n/ω pe n b . The beam driven Langmuir turbulence has a bright maximum at kλ De = v T e /v b ≈ . W ( k ) and produce back-scattered Langmuirwaves with maximum at kλ De = − . t = 0 .
04 s. In their turnsecondary Langmuir waves produce a new generation of scattered Langmuir waves with maximum at kλ De ≈ . k decreases with each act of decay by k ∗ d λ De ≈ . k < k ∗ d /
2. However, since the spectrum of Langmuir waves generated by a beam is broad in wavenumbers, thedecay continues in wavenumber regions where the level of Langmuir waves is relatively low. The rate of three wavedecay is proportional to the intensity of Langmuir waves and therefore at a given moment we see a few generationsof Langmuir waves simultaneously (fig. 1). Thus, each generation of Langmuir waves appear as a parabolic structurein that part of fig. 1 which shows the wavenumber distribution as a function of time.The ion-sound turbulence has a low intensity due to the strong ion Landau damping in a plasma with T i /T e = 0 . kλ De ≈ .
15, which corresponds to k s = 2 k − k ∗ d . The maximum of ion sound waves due tothe decay of secondary waves is seen at k s λ De ≈ . E e ( t ) = ∞ Z v Te mv f ( v, t ) dv/ , (15)the total energy of waves E l,s ( t ) = ∞ Z −∞ W l,s ( k, t ) dk , (16)the energy of waves propagating along and against the beam E + , − l,s ( t ) = ± ±∞ Z W l,s ( k, t ) dk (17)The corresponding distributions are shown in fig. 1. The lower integration limit in (15) of course chosen somewhatarbitrarily, and the energy and density of a beam obtains a somewhat arbitrary value. At later times, E e maytherefore exceed its initial value, when some of the background electrons are accelerated to velocities above the value4 v T e chosen in (15). In the presentation here, the distribution function is truncated at a level of 3 in the normalizedunits, and therefore the background plasma appears as a black band. To interpret the grey-scale in, for instance, theLangmuir wavenumber distribution as a function of time, the energy distribution E l ( t ) /E can be used as guide.The energy distribution in the system follows the well-known scenario [23]. In the present case, one fifth of theinitial beam energy is transformed into wave energy, E + e , of Langmuir waves along the beam propagation during thequasi-linear relaxation. However, the decay redistributes the energy between the primary waves and scattered waves.As a result the typical oscillations of wave energy, E ± l , along and opposite to the beam direction are clearly seen infig. 1(b). The total energy of Langmuir waves is almost constant, displaying the conservation of energy in the three-wave decay. Due to the strong damping of ion-sound waves, the energy of density oscillations is a small fraction ofthe initial beam energy. Ion-sound waves generated in the decay are rapidly absorbed by linear ion Landau damping,which is significant for the temperature rations in the present problem. Therefore the amplitude of ion-sound waveoscillations is small, see fig. 1(f). In order to interpret the grey-scale intensity levels, the energy curves in figs. 1(b)and1(c) can be used for an estimate. Obviously, the system tends towards a steady state solution. The final energydistribution of Langmuir waves is not symmetric in k . The energy of waves propagating against beam direction, E − l ,is one half of that propagating along the beam, E + l . This is simply due to back absorption of Langmuir waves by thebeam.The scattering off ions seems to be negligible during the initial redistribution of Langmuir waves. This trivial resultproves the well-known fact that decay process is the fastest process changing the Langmuir spectrum in non-isothermalplasma. However, the scattering off ions plays a critical role at later stages, when the Langmuir waves are accumulatedin the region of | k | ≤ k ∗ d /
2. The reason for this is that decay effectively generates high level of Langmuir waves in theregion of small k where scattering off ions more effective. Moreover, since decay is impossible for k ∗ s / ≤ | k | ≤ k ∗ d / t > .
05 s.
B. Strong plasma inhomogeneity
Since overall evolution of the system strongly depends on the plasma gradient value we consider three differentlength scales for the inhomogeneity separately.The main results for strong plasma gradient are presented in figs. 3 and 4. In this case the shift of the spectrum,due to plasma inhomogeneity, is stronger than any nonlinear process in the system, and the plasma inhomogeneitysuppresses all nonlinear processes, but it is not strong enough to arrest the quasi-linear relaxation. Independent onthe sign of the plasma gradient, L , the evolution of the Langmuir turbulence is very limited in time, while the physicalprocesses are different for positive and negative gradients.In the case of a positive plasma density gradient, see fig. 4, all plasma waves generated during the relaxation areabsorbed back by the beam at time t = 0 .
02 s. We see that the drift of Langmuir waves in k space is so fast that FIG. 1: Homogeneous density plasma. The normalized energy of electrons E e ( t ) /E , is shown in (a), the normalized energyin the Langmuir waves E l ( t ) /E in (b), and correspondingly for the ion-sound waves E s ( t ) /E are shown in (c). The energyof waves along E + l,s /E and against E + l,s /E the beam propagation is given by dashed and dash-dotted lines, respectively. Thetime evolution of the spectral distribution of electrons f ( v, t ) v b √ π/n b is shown in (d), for Langmuir waves W ( k, t ) ω pe /mn b v b in (e), and ion-sound waves W ( k, t ) ω pe /mn b v b is shown in (f). The disposition of the following figures is the same as here. nonlinear processes are not observable. No signs of ion-sounds waves are observed neither. For the time comparablewith the quasi-linear time, Langmuir waves are shifted from the generation region to absorption region. As a result,accelerated electrons are seen in fig. 4. The plateau is now formed from 20 v T e > v > v T e . The amount of energyreleased from a beam is a small fraction of the beam energy. Therefore, this case is specially interesting in termsof stabilization of quasi-linear relaxation by a plasma inhomogeneity. This limiting case has been considered in theliterature [25, 26, 41] and a simplified solution can be found in this case. Thus, the strong positive gradient L = 1 × cm leads to self-acceleration of electrons in a beam, consistent with observations reported in [4], for instance.In the case of a negative plasma gradient (decreasing plasma density), Langmuir waves are also effectively absorbedback by electrons. Contrary to the case with L >
0, Langmuir waves are now absorbed by the background electronswith velocities close to thermal velocities (classical Landau damping). As a result of this Landau damping, we seethe appearance of accelerated electrons in the range 4 v T e < v < v T e . The electron distribution function of theseelectrons is a decreasing function of velocity. Thus, due to the inhomogeneity, plasma waves play a role of Dreicer fieldextracting electrons from the background Maxwellian distribution. Figure 4 demonstrates that the amount of energyof electrons with v > v T e is greater than it was before the quasi-linear relaxation. The spectral energy density ofLangmuir waves reaches its maximum value at kλ De ≈ .
15, which is quite different from the beam resonant regionof k , where generation of Langmuir waves takes place, see fig. 5. It is also worth noting that the amount of energypumped into Langmuir waves is much larger than in the case of a positive plasma gradient and comparable with thecase of a homogeneous plasma. The amount of energy obtained by Langmuir waves is about 13% of initial beamenergy. The maximum is reached at time t = 0 .
03 s. Similar to the case of a positive plasma gradient, there is nosign of either back-scattered Langmuir nor ion-sound waves. Thus, the plasma inhomogeneity suppresses any furthernonlinear development of weak turbulence.
FIG. 2: The same as fig. 1, but the scattering off ions is switched off.
C. Medium plasma inhomogeneity
Decreasing the value of the plasma gradient in our system we can stimulate a further development of weak tur-bulence. For | L | = 5 × cm a single decay is observable in figs. 5 and 6. The primary waves, as in the caseof homogeneous plasma, have their maximum close to kλ De = 0 . kλ De = − . k for L > k for L <
0. The sign of a plasma gradient determines the spectrum of Langmuirwaves and the operating physical processes. However, independently on the sign of the plasma gradient (figs. 5 and6) the system approaches a steady state faster than in case of a homogeneous plasma.We consider the case of decreasing plasma density. One of the conspicuous features of the turbulence evolutionin inhomogeneous plasma is the existence of quasi-steady states. The spectrum of Langmuir waves remains almostconstant at the time scale greater than the time required for a decay. To obtain steady state, one needs a constantsource of Langmuir waves [24]. In fig. 5 we see that during some time the values of E ± l do not follow the typicaloscillations associated with a decay, as seen for instance in fig. 1. In homogeneous plasma these values undertakeoscillations with a period defined by the intensity of waves. The formation of the quasi-steady state is explained bythe following observations: Due to three wave interaction, Langmuir waves spread over k space, with each scatteringresulting in lower intensities for a given k . The rate of decay is essentially determined by the intensity of the Langmuirwaves and generally decreases with each decay. At some instant, the rate of decay becomes comparable with the effectinduced by the plasma inhomogeneity. From this time, t ≈ .
05 s, the decay is suppressed by the shift due to theplasma density gradient. With time, this equilibrium is broken at t ≈ .
09 s, and the oscillations proceed. This quasi-steady state is also clearly seen in the spectrum of ion-sound waves. The ion-sound waves corresponding to the decay ofback-scattered Langmuir waves appear only after breaking of the equilibrium. It should be noted that the stabilizationof the decay instability by the plasma inhomogeneity has not been considered before. The stabilization of a decay is
FIG. 3: The same as fig. 1, but for decreasing density plasma along beam propagation. Strong plasma inhomogeneity L = − × cm. somewhat analogous to stabilization of scattering off ions, the only effective process for an isothermal plasma. Thestabilization of scattering off ions by a plasma inhomogeneity has been considered previously [24]. In a more generalcases, when both nonlinear processes are allowed, the compensation of a decay by a plasma inhomogeneity enablesscattering off ions to be seen at early stages of turbulence. The quasi steady state spectrum of Langmuir turbulenceslowly changes due to scattering off ions.The opposite case, with positive sign of plasma gradient, L >
0, leads to Landau damping of back-scattered waveson thermal electrons with negative velocity. Indeed, as a result of absorption, accelerated electrons are seen in therange − v T e < v < − v T e (fig. 6). Further development of wave turbulence is suppressed and a quasi-steady stateis not formed. Similar to the large plasma gradient case, with
L >
0, the significant part of primary Langmuir wavesare re-absorbed by the beam. The maximum velocity of the plateau, see fig. 6, is clearly larger than in the case of ahomogeneous plasma.
D. Weak plasma plasma inhomogeneity
When the plasma inhomogeneity is decreased, the overall pictures tends to be more and more similar to the caseof homogeneous plasma. However, there are some features, which makes the spectrum different from homogeneousplasma (figs. 7 and 8). The weak plasma inhomogeneity effectively governs the turbulence in the range of small k .The negative plasma gradient (fig. 7) again leads to some quasi-steady states, but for L = − × cm it requires afew acts of decay to reach compensation of three wave decay by plasma inhomogeneity. The noticeable part of primaryLangmuir wave energy goes directly to background plasma. The other waves experience decay and are then absorbedby the beam. We also see an appearance of accelerated electrons as in the case of increasing plasma density (fig. 7).However, the physics behind this process is different. Both primary Langmuir waves and back-scattered secondarywaves are now absorbed by the beam. Thus, a relatively weak plasma inhomogeneity, with L = 1 × , cm preventsfurther decay of Langmuir waves.The notable point is the influence of a plasma inhomogeneity in the region of small k , where decay is prohibited FIG. 4: The same as fig. 1, but for an increasing plasma density. Strong plasma inhomogeneity, with L = 1 × cm. but scattering off ions continues building of Langmuir wave condensate. This high level of plasma waves accumulatednear k ≈ k . Since the damping rate is smaller for lower k , the intensity of these waves can be quite high (figs. 7and 8).The plasma inhomogeneity also affects the spectral distribution of ion-sound waves. Comparing cases with homo-geneous and inhomogeneous plasmas, we conclude that a plasma inhomogeneity increases the efficiency of ion-soundwave generation. The other interesting feature is the appearance of small wavenumber ion-sound waves in case ofweak plasma gradients. The damping rate γ sk decreases with k , implying that long wavelengths are most likely to beobserved late in the evolution of the turbulence. The distribution of these waves is a result of both nonlinear processes(decay and scattering) acting differently at low k . IV. DISCUSSION AND MAIN RESULTS
As we see, due to a plasma gradient the Langmuir wave spectrum is shifted in k space with different processes beingactive. Thus, a plasma inhomogeneity is effective in switching between different processes in the system. Indeed,for sufficiently small L (strong gradient) the plasma inhomogeneity prohibits any nonlinear processes. Instead, theelectron-Langmuir wave interaction becomes important. For L >
L < | L | (reducing the gradient) we activated the next process, the three wave decay of a Langmuir waves. Depending onplasma gradient, the Langmuir turbulence makes a fixed number of oscillations. The plasma inhomogeneity controlsthe rate of scattering off ions and decay at small k , where decay and scattering becomes equally important.Our calculations demonstrate that the Langmuir wave spectrum in general reaches a quasi-stationary state morerapidly in an inhomogeneous plasma that in a homogeneous one.The positive plasma gradient case shows that the amount of energy that is released in form of Langmuir waves ismuch smaller than that in case of a homogeneous plasma. The positive plasma gradient also leads to appearance of0 FIG. 5: The same as fig. 1 but for a decreasing plasma density. Medium plasma inhomogeneity L = − × cm. accelerated electrons in the plasma [25].The ion-sound waves are heavily damped in plasmas with comparable ion and electron temperatures. First, thisleads to a low level of ion-sound waves in comparison to that of Langmuir waves. Indeed, the energy accumulated inion-sound waves is a small fraction of initial beam energy. Second, the ion-sound waves are seen as bursts with smalltime duration. Obviously, the plasma inhomogeneity can also influence the ion-sound turbulence. The interestingobservation is that the period of time the bursts of ion-sound waves exist is dependent on plasma gradient. Generally,the bursts of ion-sound waves are longer in time in inhomogeneous plasma than in homogeneous. Therefore Langmuirwave turbulence loses energy via ion-sound waves more intensively in comparison with the homogeneous plasma case.It may be that anomalous spectra of ion sound waves observed by incoherent radar scattering can be explained interms of decay from Langmuir waves [31], and we anticipate that naturally occurring plasma density gradients in theupper parts of the ionosphere can contribute also to these processes.In view of application to astrophysical plasma, the plasma turbulence is normally observed via non-thermal radioemission [37, 42, 43]. The efficient emission mechanism giving radio emission at double plasma frequency is thecoalescence of two Langmuir waves. In order to obtain high intensity of radio emission one has to supply high levelof waves propagating at an angle to primary Langmuir waves. In this view low wavenumber Langmuir waves are themost effective. As shown, a plasma inhomogeneity successfully governs the evolution of the small wavenumber plasmawaves, and provides us with a better understanding also of the Langmuir turbulence responsible for radio emission. V. CONCLUSIONS
We demonstrated that a plasma inhomogeneity can act as a control parameter and play a crucial role in thedevelopment of weak Langmuir turbulence, by triggering various processes that affects the turbulence. The plasmainhomogeneity generally leads to a decline of turbulence, therefore a steady state of weak turbulence is reached fasterthan in case of homogeneous plasma. Selective effects of Landau damping of Langmuir waves, or self acceleration ofelectrons in a beam can easily be activated by proper choice of a density gradient.The plasma inhomogeneity is the main process that is capable of switching between the decay and scattering off1
FIG. 6: The same as fig. 1 but for an increasing plasma density. Medium plasma inhomogeneity L = 5 × cm. ions at very low wavenumbers. This part of the Langmuir wave spectrum is important for radio emission, and aplasma inhomogeneity can therefore be significant in adjusting the flux of radio emission from the plasma. Generally,we expect that the radio emission should be stronger for inhomogeneous plasmas. It might be appropriate to mentionthat the relative importance of electron temperature gradients is negligible in comparison to density gradients withthe same length scale L . When considering the last term in (5) we have an effective force acting on a wave packetbeing ∂ω k /∂x ∼ ω pe /L for the density gradient, while it is ∂ω k /∂x ∼ ω pe ( kλ De ) /L for the homogeneous densitywith inhomogeneous background electron plasma temperature. For this term in (5), a temperature gradient givesa negligible contribution as long as ( kλ De ) ≪
1. The constraint implied in the WKB-approximation (2) must befulfilled also, but this is a rather weak requirement, considering the large values of L used in the present analysis. Wemust, however, keep in mind that both St decay and St ion are temperature dependent, and that temperature gradientscan affect these terms.The presence of inhomogeneity affects the decay process, and as a result ion-sound turbulence is generated moreeffectively. In particular, the plasma inhomogeneity can give rise to a quasi-steady state of decay interaction, whenthe shift of the Langmuir wave spectrum due to inhomogeneity suppresses further developments of decay. From anapplication point of view, it is interesting that in our case a weak plasma inhomogeneity stimulates generation of ionsound waves with very low wavenumbers. We took particular care to consider physically realistic parameter values,in the present case some relevant for the solar corona. We find it of particular interest in this context, that the timescale for the initial evolution of the waves and the electron beam which is driving the process is short, of the order of0.02 – 0.05 s, i.e. comparable to a typical electron–ion collision time, ν − ei ≥ /
50 s. A collisionless plasma model forthis type of processes is therefore justified.2
FIG. 7: The same as fig. 1 but for a decreasing plasma density. Weak plasma inhomogeneity L = − × cm. Acknowledgments
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