Electrostatic weak turbulence theory for plasmas immersed in an ambient magnetic field is developed by employing a hybrid two-fluid and kinetic theories. The nonlinear susceptibility response function is calculated with the use of warm two-fluid equations. The linear dispersion relations for longitudinal electrostatic waves in magnetized plasmas are also obtained within the warm two-fluid theoretical scheme. However, dissipations that arise from linear and nonlinear wave–particle interactions cannot be discussed with the macroscopic two-fluid theory. To compute such collisionless dissipation effects, linearized kinetic theory is utilized. Moreover, a particle kinetic equation, which is necessary for a self-consistent description of the problem, is derived from the quasilinear kinetic theory. The final set of equations directly generalizes the electrostatic weak turbulence theory in unmagnetized plasmas, which could be applied for a variety of problems including the electron beam–plasma interactions in magnetized plasma environments.

Among the methods in nonlinear plasma theory, the weak turbulence theory occupies a special place. It was developed by early pioneers of modern plasma physics—see, e.g., Refs. 1–11. The recent monograph by one of the present authors (P.H.Y.) expounds on such a theory from a modern perspective.12 Its usefulness has been demonstrated by numerous examples including the non-thermal electrons measured in the solar wind13–17 and the solar type II and type III radio bursts.18–21 The kappa distribution22 was introduced in order to empirically fit the observed non-thermal solar wind electron distribution, but the weak turbulence theory provided the first-principle based explanation. Specifically, Refs. 12, 23, and 24 demonstrated that the formation of electron kappa distribution is intimately related to the long-time evolution of Langmuir turbulence. The weak turbulence theory is also successfully employed to explain the solar radio bursts.25–33 The validity of weak turbulence theory was recently confirmed against the particle-in-cell (PIC) simulation.34 Several PIC simulations of electron beam-generated Langmuir turbulence and the ensuing electromagnetic (EM) radiation emission have been carried out in the literature.35–48 However, Ref. 34 stands out in that the PIC simulation and weak turbulence theory was compared quantitatively.

Despite its successes, the standard weak turbulence theory found in the literature is mostly limited to unmagnetized plasmas. A fully general version of such a theory for magnetized plasmas does not yet exist. Some early efforts49–51 attempted to formulate such a theory from a fully general kinetic plasma theory, but the usefulness of such efforts is obscured by the inherent complexity. The more concrete weak turbulence theories for magnetized plasmas that readily lend themselves to theoretical and/or numerical analyses, instead, have been developed by making certain simplifying assumptions at the outset. For instance, by assuming a low-frequency and long-wavelength regime, the magnetohydrodynamic weak turbulence theory was formulated and solved.52–60 The mode-coupling process among electrostatic cyclotron-harmonic waves was discussed within the framework of weak turbulence ordering.61–64 Recently, a weak turbulence theory that involves the whistler-mode and lower-hybrid waves was formulated and applied to a number of space plasma situations.65–68 The weak turbulence theory was extended to interpret the polarization of solar coronal type III radio bursts.51,69–75 A weak turbulence theory for general magnetized plasmas was formulated and solved but under the strict assumption of either parallel or perpendicular propagation.76–80 

A major obstacle for extending the weak turbulence theory to fully magnetized plasmas is the computation of nonlinear susceptibility, as evidenced by the above-referenced early attempts.49–51 A method was recently proposed to overcome such a difficulty. In a recent work,81 one of the present authors (P.H.Y.) noted that one can partially reformulate the kinetic weak turbulence theory by resorting to the warm two-fluid theory. Specifically, it was noted that the nonlinear decay interactions among Langmuir and ion-sound waves could be fully discussed by resorting to the warm two-fluid approach. The cold two-fluid theory was pointed out as being inadequate since certain decay interaction coefficients are inversely proportional to the electron temperature. Obviously, the cold plasma theory is inapplicable for such processes. While Ref. 81 pointed out the usefulness of the warm two-fluid approach under a general situation, including the magnetized plasmas, for actual demonstration, Ref. 81 only considered the unmagnetized plasma problem as an example that proves the basic concept.

The weakly turbulent processes in unmagnetized plasmas that involve the interaction of Langmuir wave, ion-acoustic waves, transverse radiation, and the particles form the basic building blocks for the so-called plasma emission, which is the fundamental radiation emission mechanism responsible for the solar coronal and interplanetary type II and type III radio bursts. However, for type III radio bursts close to the solar active regions, the effects of finite background magnetic field can be an important factor in the interpretation of data. A recent particle-in-cell simulation of the plasma emission process in magnetized plasmas shows that the assumption of unmagnetized plasmas may be valid under certain conditions, particularly, when the medium is characterized by a high ratio of electron plasma frequency to electron–cyclotron frequency; but as the same ratio is reduced, say to order ten or less, the underlying wave–particle interaction as well as the mode-coupling (that is, the weakly turbulent plasma) processes undergo some dramatic shifts in their characteristics, which call for further theoretical development that reflects plasma magnetization.82 

The purpose of this paper is to consider the first example of utilizing the warm two-fluid formalism to derive the basic equations of weak turbulence theory in magnetized plasmas, with a long-term focus of extending the existing unmagnetized plasma theory of plasma emission to that of magnetized plasma theory of plasma emission. To simplify the analysis, however, we first consider the electrostatic problem. Obviously, a fully electromagnetic version should follow, but it is a subject of future tasks. Without the electromagnetic effects, the radiation emission cannot be discussed, but the wave–particle interaction between the type III-emitting electron beams and electrostatic turbulence can be discussed with the electrostatic weak turbulence theory. For an unmagnetized plasma, the type III electron beams interact with the Langmuir waves, which undergo nonlinear interaction with the ion-sound waves and the background protons. The magnetized plasma analog of such a process will involve the electron beam interacting with the upper-hybrid waves, which undergo nonlinear interaction with the low-frequency sonic type of modes as well as the protons. The electrostatic weak turbulence theory in warm magnetized plasmas to be discussed herein is meant to provide a quantitative description of these processes.

As the remainder of this paper will illustrate, we begin the discourse based on the warm two-fluid equations (Sec. II). Section II is subdivided into subsections that deal with the first- and second-order iterative solutions, which are then combined into a nonlinear wave equation. Section III discusses the generic form of an electrostatic wave kinetic equation under weak turbulence ordering. A detailed derivation of the equations for electrostatic weak turbulence theory in magnetized plasmas is given in Sec. IV. This section is also subdivided into subsections, where each subsection deals with various subtopics, which includes adding kinetic effects for the complete descriptions of wave–particle interaction. Finally, the findings of the present paper are summarized in Sec. V, and some discussions related to future directions of the research are presented therein.

In the present analysis, na denotes the fluid density for species a; va denotes the fluid velocity; ma, ea, and c denote the mass, unit electric charge, and speed of light in vacuum, respectively; E is the electrostatic field vector; and B0 is the ambient magnetic field. We start from the electrostatic two-fluid equation in magnetized plasmas as follows:

nat+·(nava)=0,manadvadt+Paeana(E+1cva×B0)=0,·E=a=e,i4πeana,
(1)

where d/dt=/+va·, and a=e,i denotes electrons and ions, respectively. Assuming that the pressure is given by the product of density and temperature, Pa=naTa, and separating density into an average and fluctuation term, na=n0+δna, while also denoting the velocity and electric field with δ preceding them to indicate that they are fluctuating quantities, we have

δnat+n0·δva+·(δnaδva)=0,δvatΩaδva×beamaδE+δnan0vTa2+δva·δvaδnan0δnan0vTa2=0,·δE=a4πeaδna,
(2)

where vTa2=Ta/ma represents the square of fluid thermal speed and Ωa=eaB0/mac is the cyclotron frequency for species a. Here, we assume that the temperature is defined in units of energy; hence, the Boltzmann constant is set equal to unity, kB = 1. Note that vTa2 differs from the kinetic counterpart where it is defined by vTa2=2Ta/ma. We employ an iterative method to obtain the solution, δna=na(1)+na(2)+ and δva=va(1)+va(2)+, where na(1) and va(1) are proportional to O(δE),na(2) and va(2) are proportional to O(δE2), etc. That is, we follow the standard weak turbulence ordering where particle quantities are expanded in power series with each term proportional to the power of the field intensity. We then organize the resulting equations for each order. We write down the result in spectral form where the spectral transformation is defined by fk,ω=(2π)4drdtf(r,t)eiωtik·r, together with the inverse transformation, f(r,t)=dkdωfk,ωeiωt+ik·r=k,ωfk,ωeiωt+ik·r. Let us adopt the following shorthand notations:

q=(k,ω),rq=nk,ωan0,uq=vk,ωa,η=Ωaω.
(3)

We then have the following for each order:

rq(1)=k·uq(1)ω,uq(1)iηuq(1)×bkvTa2ωrq(1)ieamaωEq=0,rq(2)=k·uq(2)ω+kω·qrq(1)uqq(1),uq(2)iηuq(2)×bkvTa2ωrq(2)12ωq[(kk)·uq(1)uqq(1)+k·uqq(1)uq(1)]+kvTa22ωqrq(1)rqq(1)=0,k·Eq=ia4πean0(rq(1)+rq(2)).
(4)

Let us pay attention to the momentum equations. One may rearrange these equations to obtain

(1η2)uq(1)=vTa2ω(kiηb×kη2bk·b)rq(1)+ieamaω(Eqiηb×Eqη2bb·Eq),(1η2)uq(2)=vTa2ω(kiηb×kη2bk·b)rq(2)+12ωq(kk)·uq(1)×(uqq(1)iηb×uqq(1)η2bb·uqq(1))+12ωqk·uqq(1)×(uq(1)iηb×uq(1)η2bb·uq(1))vTa22ωq(kiηb×kη2bk·b)rq(1)rqq(1).
(5)

Let us define

qωij=δij+iΩaωεijkbkΩa2ω2bibj.
(6)

Then, we obtain a compact notation

uq(1)i=qωijω(1η2)(kjvTa2rq(1)+ieamaEqj),uq(2)i=1ω(1η2)(qωijkjvTa2rq(2)+12q{[qωijkk+qωik(kk)j]uq(1)juqq(1)kqωijkjvTa2rq(1)rqq(1)}).
(7)

Making use of the velocity fluctuations given by Eq. (7), we may construct the density fluctuations as follows:

rq(1)=kiqωijω2(1η2)(kjvTa2rq(1)+ieamaEqj),rq(2)=1ω2(1η2)(qωijkikjvTa2rq(2)+12q{ki[qωijkk+qωik(kk)j]uq(1)juqq(1)kqωijkikjvTa2rq(1)rqq(1)})+1ωqrq(1)k·uqq(1).
(8)

The first-order solution can be obtained from coupled equations given by Eqs. (7) and (8) by ignoring the nonlinear terms and second-order terms. This results in the following:

rq(1)=ieamakiqωijEqjω2(1η2)[k2η2(k·b)2]vTa2,uq(1)i=ieamaω(1η2)([ω2(1η2)[k2η2(k·b)2]vTa2]qωijEqjω2(1η2)[k2η2(k·b)2]vTa2+qωijkjkkqωklvTa2Eqkω2(1η2)[k2η2(k·b)2]vTa2).
(9)

We make note of the fact that the fluid correction becomes important near resonances. Thus, the various terms associated with vTa2 affect the denominators but can be ignored in the numerator. Adopting such an approximation scheme, we have the first-order solution as follows:

uq(1)i=ieama|k|ωqωijkjEqω2(1η2)[k2η2(k·b)2]vTa2,rq(1)=ieama|k|[k2η2(k·b)2]Eqω2(1η2)[k2η2(k·b)2]vTa2,qωij=δij+iηεijkbkη2bibj,
(10)

where we have expressed the electric field vector as

Eqj=kj|k|Eq
(11)

since we are dealing with an electrostatic problem.

The second-order solution can be discussed with the density rq(2) only since the Poisson equation involves only rq(1) and rq(2). Upon making use of the first-order solution given by Eq. (10), it is possible to show, after some tedious but otherwise straightforward algebraic manipulations, that rq(2) is given by

rq(2)=q12ea2ma2ω(ωω)k|kk|RqRqRqq×[kiqωilkmkjqωlj(kk)kqωωmk+kiqωimkjqωlj(kk)l(kk)kqωωmk]EqEqqq12ea2ma2ω2Ωa2ωk|kk|RqRqRqq×[(ωω)[k2η2(k·b)2]kl(kk)kqωωlk+ω{(kk)2η2[(kk)·b]2}klkjqωlj]EqEqq,
(12)

where

Rq=ω2Ωa2[k2η2(k·b)2]vTa2,Rq=ω2Ωa2[k2η2(k·b)2]vTa2,Rqq=(ωω)2Ωa2{(kk)2η2[(kk)·b]2}vTa2,η=Ωaω,η=Ωaω,η=Ωaωω.
(13)

It is instructive to carry out the vector multiplications associated with the following quantities: kiqωilkmkjqωlj(kk)kqωωmk,kiqωimkjqωlj(kk)l(kk)kqωωmk,kl(kk)kqωωlk,klkjqωlj. After carrying out explicit manipulations of these quantities, we have

rq(2)=q12ea2ma2ω(ωω)k|kk|RqRqRqq×({(1+ηη)(k·k)(η2+η2+ηηη2η2)(k·b)(k·b)+i(η+η)(k×k)·b}×{k·(kk)iη(k×k)·bη2(k·b)[(kk)·b]}+{(1+ηη)[k·(kk)](η2+η2+ηηη2η2)(k·b)[(kk)·b]i(η+η)(k×k)·b}{k·(kk)+iη(k×k)·bη2(k·b)[(kk)·b]})EqEqqq12ea2ma2ω2Ωa2ωk|kk|RqRqRqq×((ωω)[k2η2(k·b)2]{k·(kk)iη(k×k)·bη2(k·b)[(kk)·b]}+ω{(kk)2η2[(kk)·b]2}{k·k+iη(k×k)·bη2(k·b)(k·b)})EqEqq.
(14)

In what follows, we restrict k and k to lie in the xz plane, k=x̂k+ẑk and k=x̂k+ẑk, while assuming b=ẑ. Then (k×k)·b=0. This is a reasonable assumption since the nonlinear term in the wave equation is associated with the integral q=dk, which includes the average over azimuthal angle associated with the k vector. Therefore, physical quantities orthogonal to both k and k will be averaged over the azimuthal angle. The terms associated with k×k are such quantities. As such, it is reasonable to assume that both k and k lie in the xz plane and the resultant nonlinear terms can be averaged over the azimuthal angle in the end. This simplifies the expression as follows:

rq(2)=q12ea2ma2ω(ωω)k|kk|RqRqRqq×([(1+ηη)(k·k)(η2+η2+ηηη2η2)kk]×[k·(kk)η2k(kk)]+{(1+ηη)[k·(kk)](η2+η2+ηηη2η2)k(kk)}×[k·(kk)η2k(kk)])EqEqqq12ea2ma2ω2Ωa2ωk|kk|RqRqRqq((ωω)(k2η2k2)×[k·(kk)η2k(kk)]+ω[(kk)2η2(kk)2]×(k·kη2kk))EqEqq.
(15)

We substitute the densities, rq(1) and rq(2) to Poisson equation in Eq. (4). The result can be written in long-hand notation as

ε(k,ω)Ek,ω=k,ωχ(k,ω|kk,ωω)Ek,ωEkk,ωω,
(16)

where ε(k,ω) is the fluid version of the linear dielectric constant and χ(k,ω|kk,ωω) is the (second-order) nonlinear susceptibility, which are defined, respectively, by

ε(k,ω)=1aωpa2k2k2η2k2Rk,ω,
(17)
χ(k,ω|kk,ωω)=ai2eamaωpa2kk|kk|Rk,ωRk,ωRkk,ωω×[ω(ωω)([(1+ηη)(k·k)(η2+η2+ηηη2η2)kk]×[k·(kk)η2k(kk)]+{(1+ηη)[k·(kk)](η2+η2+ηηη2η2)k(kk)}×[k·(kk)η2k(kk)])+ω2Ωa2ω((ωω)(k2η2k2)[k·(kk)η2k(kk)]+ω[(kk)2η2(kk)2](k·kη2kk))].
(18)

In the above, the resonance denominators are now expressed in long-hand notation as Rk,ω=ω2Ωa2(k2η2k2)vTa2,Rk,ω=ω2Ωa2(k2η2k2)vTa2,Rkk,ωω=(ωω)2Ωa2[(kk)2η2(kk)2]vTa2.

Note that Eq. (16) is given by the standard form of electrostatic weak turbulence theory. We start from the general form of nonlinear wave equation given by Eq. (16), regardless of the specific form of ε(k,ω) or χ(k,ω|kk,ωω), which is compactly rewritten as

0=ε(K)Ej(K)+1+2=Kχ(1|2)Ej(1)Ek(2),
(19)

where K=(k,ω),1=K1=(k1,ω1), and 2=K2=(k2,ω2). We multiply this equation with Ei(K) and take the ensemble average

0=ε(K)E2Kδ(K+K)+1χ(1|K1)E(K)E(1)E(K1),
(20)

where we have made use of the property of homogeneous and stationary turbulence, E(K)E(K)=E2Kδ(K+K). The third-body correlation E(K)E(1)E(K1) can be obtained in the customary way. That is, we iteratively solve the wave equation by writing E(K)=E(0)(K)+E(1)(K), where E(0)(K) satisfies the linear dispersion relation, ε(K)E(0)(K)=0. Then, the next-order correction is obtained via the nonlinear term. Specifically, we have E(1)(K)=ε1(K)2χ(2|K2)E(0)(2)E(0)(K2),E(1)(1)=ε1(1)2χ(2|12)E(0)(2)E(0)(12), and E(1)(K1)=ε1(K1)2χ(2|K12)E(0)(2)E(0)(K12). Since odd cumulants of E(0) vanish, the desired third-body cumulant E(K)E(1)E(K1) can be obtained by adding contributions from E(1)(K)E(0)(1)E(0)(K1),E(0)(K)E(1)(1)E(0)(K1), and E(0)(K)E(0)(1)E(1)(K1). This process leads to the four-body cumulants, but we close this hierarchy by writing the four-body cumulants as products of two-body cumulants, while ignoring the irreducible four-body correlation function. In manipulating the products of two-body cumulants, we ignore the correlations when the argument becomes zero, since such quantities represent spatial correlations separated by an infinite distance, and temporal correlations separated by an infinitely long interval. Such terms are clearly unphysical. These procedures are well-described by the recent monograph,12 but also discussed in standard literature, including Ref. 83. This method of closure is known as the quasi-normal closure in the literature. The result is the following:

E(K)E(1)E(K1)=2δ(K+K)(χ(1|K+1)E21E2K1ε(K)+χ(K|1K)E2KE2K1ε(1)+χ(K|1)E2KE21ε(K1)).
(21)

From this, we obtain the formal nonlinear spectral balance equation, but in doing so, we make note of the symmetry property associated with the linear and nonlinear response functions, ε(K)=ε*(K),χ(1|2)=χ(2|1), and χ(1|2)=χ(1+2|2). We also take the customary approach of replacing the leading linear dielectric constant by a term that retains the slow time derivative, ε(k,ω)E2k,ωε(k,ω)E2k,ω+(i/2)[ε(k,ω)/ω](E2k,ω/t). The result is the following:

0=i2ε(k,ω)ωE2k,ωt+ε(k,ω)E2k,ω+2dkdω{χ(k,ω|kk,ωω)×χ(k,ω|kk,ωω)×(E2kk,ωωε(k,ω)+E2k,ωε(kk,ωω))E2k,ωχ(k,ω|kk,ωω)χ*(k,ω|kk,ωω)×E2k,ωE2kk,ωωε*(k,ω)}.
(22)

By taking the real part of Eq. (22) while ignoring the nonlinear part, we have Reε(k,ω)=0, from which we obtain the wave dispersion relation, ω=ωkα, where α denotes the possibility of multiple roots. This also leads to E2k,ω=σ=±1αIkσαδ(ωσωkα). Upon substituting this back to Eq. (22), we obtain the final form of weak turbulence wave kinetic equation under electrostatic approximation as follows:

Ikσαt=2Imε(k,σωkα)ε(k,σωkα)Ikσα4πε(k,σωkα)β,γσ,σ=±1×dk|χ(k,σωkβ|kk,σωkkγ)|2×( IkkσγIkσαε(k,σωkβ)+IkσβIkσαε(kk,σωkkγ)IkσβIkkσγε(k,σωkα) )δ(σωkασωkβσωkkγ)4ε(k,σωkα)Imβσ=±1dk×P2{χ(k,σωkβ|kk,σωkασωkβ)}2ε(kk,σωkασωkβ)IkσβIkσα,
(23)

where P denotes the principal value and

ε(k,σωkα)=Reε(k,σωkα)(σωkα).
(24)

We remind the readers that Eq. (23) is a generic wave equation for electrostatic weak turbulence. In the context of the present warm two-fluid theoretical result, Eqs. (16)–(18), however, the imaginary part of ε(k,ω) is absent. However, as we will discuss later, we complement the formulation of electrostatic weak turbulence theory for magnetized plasmas by computing the imaginary part of ε(k,ω) from the kinetic theoretical calculation. For this reason, we leave Imε(k,ω) intact in the subsequent formulation.

We are now ready for a concrete formulation of electrostatic weak turbulence theory in magnetized plasmas. The first step in the discussion is the dispersion relation. We then make use of the linear dispersive properties of the normal mode to simplify the nonlinear susceptibility, which leads to the desired wave kinetic equation. The warm two-fluid theoretical approach is thus adequate for formulating the nonlinear wave kinetic equation in magnetized plasmas. However, to complete the analysis, one must supplement the formalism by computing the imaginary part of the dielectric function as well as to provide a self-consistent description of the dynamical evolution of the particle distribution function. The kinetic equation for the particles will thus be provided by invoking the quasilinear kinetic theory.

We start from a discussion of the wave dispersion relation and the related properties of the normal modes. If we ignore nonlinear terms in Eq. (16), the dispersion relation is given by

0=ε(k,ω)=1ωpe2ω2Ωe2k2vTe2+(Ωe2/ω2)k2vTe2(1Ωe2ω2k2k2)ωpi2ω2Ωi2k2vTi2+(Ωi2/ω2)k2vTi2(1Ωi2ω2k2k2).
(25)

Figure 1 plots the numerical solution of the dispersion relation given by Eq. (25) for ωpe/|Ωe|=5 and Ti/Te=0.1. The numerical solution was obtained for low values of Ti/Te since ion-sound waves damp for high Ti. Although the fluid dispersion relation given by Eq. (25) does not have an imaginary part, the ion-sound waves damp for high values of Ti when compared with the electron temperature when we include the collisional damping effects by adding the kinetic effects. Figure 1 (top-left) corresponds to the Langmuir mode solution for quasi-parallel angle of propagation, which gradually turns into the upper-hybrid mode for quasi-perpendicular angle of propagation. In Fig. 1 (top-left), the cases of θ=5° and θ=80° correspond to quasi-parallel and quasi-perpendicular angles of propagation, with the intermediate value θ=45° also plotted. Figure 1 (top-right) shows the electron–cyclotron mode, whose frequency is close to the electron–cyclotron frequency for quasi-parallel propagation, which gradually decreases as θ increases. The bottom-left panel of Fig. 1 plots the ion-acoustic branch of the solution, which shows that the curves for all three angles of propagation almost overlap. Finally, Fig. 1 (bottom-right) plots the ion–cyclotron mode. As with the electron–cyclotron mode, the ion–cyclotron mode frequency ω is close to Ωi when θ is low, but as θ increases, the frequency decreases. For θ=90°, both cyclotron modes reduce to zero frequency. The numerical solution shows that the Langmuir/upper-hybrid mode and the ion-acoustic mode are the propagating thermal modes that directly generalize the unmagnetized plasma modes. The cyclotron modes are not modified by thermal effects. We are interested in the extension of the unmagnetized plasma turbulence—where the Langmuir and ion-sound waves participate in the wave–particle and wave–wave interactions—to the magnetized plasma turbulence. For this purpose, we focus on the Langmuir/upper-hybrid mode and the ion-acoustic mode participating in the wave–wave and wave–particle resonances. As such, we will consider these two thermal modes as the fundamental normal modes of the magnetized plasma. For Langmuir/upper-hybrid and ion-acoustic modes, we superpose the analytic solutions of these modes (to be discussed later) on top of numerical solutions. The analytic Langmuir/upper-hybrid mode is so close to the numerical solution that the curves almost completely overlap. For the ion-acoustic mode, the analytical solution is shown with a dotted line, which overlaps almost perfectly for most ranges of kvTe/|Ωe| until kvTe/|Ωe| becomes quite high. For high kvTe/|Ωe|, the ion-acoustic mode will be heavily damped.

FIG. 1.

Numerical solution, ω/|Ωe| vs kvTe/|Ωe|, to the dispersion relation given by Eq. (25) for ωpe/|Ωe|=5 and Ti/Te=0.1. Top-left panel corresponds to the Langmuir/upper-hybrid mode, top-right plots the electron–cyclotron mode, bottom-left shows the ion-acoustic mode, and bottom-right displays the ion–cyclotron mode. For the Langmuir/upper-hybrid and ion-acoustic modes, we superpose the analytical dispersion curves. For the Langmuir/upper-hybrid mode, the analytic solution is so close to the numerical solution that the curves almost completely overlap. For the ion-acoustic mode, the analytical solution is shown with a dotted line, which overlaps almost perfectly for most ranges of kvTe/|Ωe| until kvTe/|Ωe| becomes quite high.

FIG. 1.

Numerical solution, ω/|Ωe| vs kvTe/|Ωe|, to the dispersion relation given by Eq. (25) for ωpe/|Ωe|=5 and Ti/Te=0.1. Top-left panel corresponds to the Langmuir/upper-hybrid mode, top-right plots the electron–cyclotron mode, bottom-left shows the ion-acoustic mode, and bottom-right displays the ion–cyclotron mode. For the Langmuir/upper-hybrid and ion-acoustic modes, we superpose the analytical dispersion curves. For the Langmuir/upper-hybrid mode, the analytic solution is so close to the numerical solution that the curves almost completely overlap. For the ion-acoustic mode, the analytical solution is shown with a dotted line, which overlaps almost perfectly for most ranges of kvTe/|Ωe| until kvTe/|Ωe| becomes quite high.

Close modal

In the present analysis, we generally consider ωpe, which is sufficiently higher than |Ωe|. In the example shown in Fig. 1, we chose ωpe/|Ωe|=5. Generally, we assume ωpe2/Ωe21. This means that ωpe/|Ωe| can be as low as 2 or 3, but generally values higher than these are to be considered. For ωpe/|Ωe|=2, the square of the frequency ratio is ωpe2/Ωe2=4, which can be marginally satisfying the requirement ωpe2/Ωe21, but for ωpe/|Ωe|=3, the square of the frequency is ωpe2/Ωe2=9, which is certainly significantly higher than unity. Approximate, analytical solution to dispersion relation given by Eq. (25) is of relevance. For the high-frequency, Langmuir/upper-hybrid mode, we assume

ω2Ωe2,ω2k2vTe2.
(26)

We also ignore the ion response. If we ignore thermal effects altogether, then we have

0=ω2(ω2ωpe2Ωe2)+ωpe2Ωe2ωpe2Ωe2k2k2
(27)

whose solution is

ω2=12[ωuh2+((ωpe2Ωe2)2+4ωpe2Ωe2k2k2)1/2],
(28)

where ωuh2=ωpe2+Ωe2 is the square of the upper-hybrid frequency. To simplify further, we replace the above by an approximate form

ω=(ωpe2+Ωe2k2k2)1/2=(ωpe2+Ωe2sin2θ)1/2.
(29)

Figure 2 plots both Eqs. (28) and (29) vs θ, for ωpe/|Ωe|=5 and 2. As Fig. 2 shows, the agreement is excellent.

FIG. 2.

Comparison between the exact cold-plasma solution given by Eq. (28) vs approximate solution given by Eq. (29) for ωpe/|Ωe|=5 and 2.

FIG. 2.

Comparison between the exact cold-plasma solution given by Eq. (28) vs approximate solution given by Eq. (29) for ωpe/|Ωe|=5 and 2.

Close modal

Making use of the solution given by Eq. (29) as the basis, we add the thermal correction

0=ω2Ωe2ωpe2(1Ωe2ω2k2k2)k2vTe2(1Ωe2ω2k2k2).
(30)

We write down the approximate solution by

ω=ωkU=(ωpe2+Ωe2sin2θ+k2vTe2)1/2.
(31)

We have superposed this solution to the numerical solution in the top-left panel of Fig. 1, and as already noted, the two overlap almost completely. In short, the dispersion relation for the Langmuir/upper-hybrid wave, which we simply call “upper-hybrid” or U mode, is given by Eq. (31). The approximate solution given by Eq. (31) amounts to replacing the dielectric constant for the U mode as follows:

Reε(k,ω)=1ωpe2ω2Ωe2sin2θk2vTe2,
(32)

where ω=σωkU. This also means that the derivative is given by

ε(k,σωkU)=Reε(k,σωkU)(σωkU)=2σωkUωpe2.
(33)

Next, we consider the low-frequency ion-acoustic mode. For this mode, we ignore the thermal and magnetic effects. We also assume

k2vTe2ω2
(34)

for the electrons. This leads to the approximate dispersion relation

Reε(k,ω)ωpe2k2vTe2ωpi2ω2=0.
(35)

From this, we obtain

ω=ωkS=kcS,cS=Temi,Reε(k,ω)=ωpe2k2vTe2ωpi2ω2,ω=σωkS,ε(k,σωkS)=2σωkSωpe2k2vTe2.
(36)

We have also superposed this solution to the bottom-left panel of Fig. 1 and found that the comparison with the numerical solution was excellent.

We now write down the wave kinetic equation for α=U and α=S, respectively. In doing so, we note that only the three-wave interaction of the type U+SU is allowed from the viewpoint of wave energetics and that we only retain the induced scattering of the type U+iU in the wave kinetic equation for the U mode, while we ignore the induced scattering for the S mode. Such considerations are a direct analogy with the case of unmagnetized Langmuir/ion-acoustic turbulence situation. We thus write down the specific wave kinetic equations for the U and S modes as follows:

IkσUt=σωkUμkUImε(k,σωkU)IkσU2πσωkUμkU×σ,σ=±1dk|χ(k,σωkU|kk,σωkkS)|2×( σωkUμkUIkkσSIkσU+σωkkSμkkSIkσUIkσUσωkUμkUIkσUIkkσS )δ(σωkUσωkUσωkkS)4σωkUμkUImσ=±1dkIkσUIkσU×p{χ(k,σωkU|kk,σωkUσωkU)}2ε(kk,σωkUσωkU),
(37)
IkσSt=σωkSμkSImε(k,σωkS)IkσSπσωkSμkSσ,σ=±1dk|χ(k,σωkU|kk,σωkkU)|2×(σωkUμkUIkkσUIkσS+σωkkUμkkUIkσUIkσSσωkSμkSIkσUIkkσU)δ(σωkSσωkUσωkkU),
(38)

where we have defined

μkU=ωpe2(ωkU)2,μkS=k2vTe2ωpe2.
(39)

A quantity of relevance is the nonlinear susceptibility, which is determined entirely by the electron response. This is because the lighter and more mobile electrons readily respond to the perturbation, while the heavier ions remain much less mobile. We make note of the fact that we are generally concerned with the weakly magnetized situations exemplified by the condition ωpe2Ωe2. We also note that the U mode is a fast mode, while the S mode is a slow mode in the following sense:

ωkUkvTe,|Ωe|ωkU1,ωkSkvTe,|Ωe|ωkS1.
(40)

Because of this, let us approximate the susceptibility by first making use of the relative magnitudes of η, η, and η; but in doing so, we retain terms that survive in the unmagnetized limit. Thus, we approximate the following depending on various limits:

χe(k,ω|kk,ωω)=i2emeωpe2kk|kk|Rk,ωRk,ωRkk,ωω×(ω(ωω){(k·k)[k·(kk)η2k(kk)]+[k·(kk)η2k(kk)][k·(kk)]}+ω(ωω)k2[k·(kk)η2k(kk)]+ωω[(kk)2η2(kk)2](k·k))
(41)

for η1,η1,η1,

χe(k,ω|kk,ωω)=i2emeωpe2kk|kk|Rk,ωRk,ωRkk,ωω×(ω(ωω){(k·kη2kk)[k·(kk)]+[k·(kk)][k·(kk)η2k(kk)]}+ω(ωω)(k2η2k2)[k·(kk)]+ωω(kk)2(k·kη2kk))
(42)

for η1,η1,η1, and

χe(k,ω|kk,ωω)=i2emeωpe2kk|kk|Rk,ωRk,ωRkk,ωω×(ω(ωω){(k·kη2kk)[k·(kk)]+[k·(kk)η2k(kk)][k·(kk)]}ωη2{(ωω)k2[k·(kk)]+ω(kk)2(k·k)})
(43)

for η1,η1,η1.

Next, we retain the dominant terms in relations to the magnitudes of ω, ω, and ωω, i.e.,

χe(k,ω|kk,ωω)=i2emeωpe2(k·k)ωω[(kk)2η2(kk)2]kk|kk|Rk,ωRk,ωRkk,ωω
(44)

for ωkvTe,ωkvTe,ωω|kk|vTe

χe(k,ω|kk,ωω)=i2emeωpe2[k·(kk)]ω(ωω)(k2η2k2)kk|kk|Rk,ωRk,ωRkk,ωω
(45)

for ωkvTe,ωkvTe,ωω|kk|vTe, and

χe(k,ω|kk,ωω)=i2emeωpe2[k·(kk)]ω(ωω)(k2η2k2)kk|kk|Rk,ωRk,ωRkk,ωω
(46)

for ωkvTe,ωkvTe,ωω|kk|vTe.

Next, we approximate the resonant denominators accordingly as follows:

χe(k,ω|kk,ωω)=i2emeωpe2ωωk·kkk|kk|×(kk)2η2(kk)2Ωe2+[(kk)2η2(kk)2]vTe2
(47)

for ωkvTe,η1,ωkvTe,η1,ωω|kk|vTe,η1

χe(k,ω|kk,ωω)=i2emeωpe2ω(ωω)k·(kk)kk|kk|×(k2η2k2)Ωe2+(k2η2k2)vTe2
(48)

for ωkvTe,η1,ωkvTe,η1,ωω|kk|vTe,η1, and

χe(k,ω|kk,ωω)=i2emeωpe2ω(ωω)k·(kk)kk|kk|×k2η2k2Ωe2+(k2η2k2)vTe2
(49)

for ωkvTe,η1,ωkvTe,η1,ωω|kk|vTe,η1.

Making use of all this, we now approximate the following nonlinear susceptibilities of interest:

|χ(k,σωkU|kk,σωkkS)|2=14e2me2μkUμkU(k·k)2k2k2|kk|2×|(kk)2(ηkkS)2(kk)2Ωe2+[(kk)2(ηkkS)2(kk)2]vTe2|2,ηkkS=|Ωe|σωkkS,
(50)
{χ(k,σωkU|kk,σωkUσωkU)}2=14e2me2μkUμkU(k·k)2k2k2|kk|2×|(kk)2(ηk,kU)2(kk)2Ωe2+[(kk)2(ηk,kU)2(kk)2]vTe2|2,ηk,kU=|Ωe|σωkUσωkU,
(51)
|χ(k,σωkU|kk,σωkkU)|2=14e2me2μkUμkkU[k·(kk)]2k2k2|kk|2×|k2(ηkS)2k2Ωe2+[k2(ηkS)2k2]vTe2|2,ηkS=|Ωe|σωkS.
(52)

This leads to the following provisional weak turbulence wave kinetic equations, where the imaginary parts related to the linear dielectric constant are yet to be determined:

tIkσUμkU=ωpe2σωkUImε(k,σωkU)IkσUμkU+2σωkUσ,σ=±1dkVk,kU[σωkUIkσUμkUIkkσSμkkS(σωkUIkkσSμkkS+σωkkSIkσUμkU)IkσUμkU]×δ(σωkUσωkUσωkkS)+σωkUImσ=±1dkIkσUμkUIkσUμkU×PMk,kε(kk,σωkUσωkU),
(53)
tIkσSμkS=σωkSμkSImε(k,σωkS)IkσSμkS+σωkSσ,σ=±1dkVk,kS[σωkSIkσUμkUIkkσUμkkU(σωkUIkkσUμkkU+σωkkUIkσUμkU)IkσSμkS]×δ(σωkSσωkUσωkkU),
(54)

where

Vk,kU=π4e2me2(μkU)2(μkU)2μkkS(k·k)2k2k2|kk|2×| (kk)2ηkkS(kk)2Ωe2+[(kk)2ηkkS(kk)2]vTe2 |2,Vk,kS=π4e2me2(μkU)2(μkkU)2μkS[k·(kk)]2k2k2|kk|2×| k2ηkSk2Ωe2+[k2ηkSk2]vTe2 |2,Mk,k=e2me2(μkU)2(μkU)2(k·k)2k2k2|kk|2×| (kk)2ηk,kU(kk)2Ωe2+{(kk)2ηk,kU(kk)2}vTe2 |2.
(55)

Note that the nonlinear terms in Eqs. (53) and (54) associated with the three-wave resonance delta function condition represent the decay interactions. In the U mode wave kinetic equation, the last term on the right-hand side of Eq. (53) denotes the induced scattering terms. The linear terms associated with the imaginary parts of the dielectric constant in Eqs. (53) and (54) correspond to the quasilinear growth/damping (or induced emission) terms.

Let us consider the inverse dielectric constant with shifted argument

1ε(kk,σωkUσωkU).
(56)

Since the shifted frequency σωkUσωkU is small, the low-frequency expression of the real part is applicable, i.e.,

ε(kk,σωkUσωkU)ωpe2|kk|2vTe2+iImε(kk,σωkUσωkU).
(57)

Thus, we have

ImP1ε(kk,σωkUσωkU)=(μkkS)2Imε(kk,σωkUσωkU).
(58)

Consequently, the induced scattering term in the U mode wave equation can be alternatively written as

t|ind.scatt.IkσUμkU=σωkUσ=±1dkMk,k×Imε(kk,σωkUσωkU)IkσUμkUIkσUμkU,Mk,k=e2me2|kk|2ωpe4(μkU)2(μkU)2(k·k)2k2k2×|(kk)2vTe2ηk,kU(kk)2vTe2Ωe2+{(kk)2ηk,kU(kk)2}vTe2|2.
(59)

The provisional wave kinetic equation given by Eqs. (53) and (54) with the modified induced scattering term given by Eq. (59) are almost complete, except that the imaginary part of the dielectric constant is undetermined. Under the strict warm two-fluid formalism, the dielectric constant is purely real. However, the present warm two-fluid formalism does not provide a complete description of the electrostatic weak turbulence in magnetized plasmas. We thus turn to kinetic theory in order to supplement the missing information as far as the warm two-fluid approach goes. Also, the wave kinetic equation must be solved in conjunction with the particle kinetic equation. For that, we resort to the quasilinear theory.

We start from the Vlasov–Poisson system of equations given by

[t+v·+eama(E(r,t)+vc×B0)·v]fa(r,v,t)=0,·E(r,t)=4πaeanadvfa(r,v,t).
(60)

Separating the physical quantities into average and fluctuating parts, fa(r,v,t)=Fa(v,t)+δfa(r,v,t) and E(r,t)=δE(r,t), and considering only linear equation for the perturbation, we obtain

Fat=eamavidkdωkikEk,ωfkωa,Ekω=4πikaean0dvfkωa,fkωaφ+i(ωk·v)Ωafkωa=eamaΩaEkωkikFavi,
(61)

where we have assumed the gyrotropy for Fa, and we have expressed the results in spectral representation.

Solving for the perturbed distribution function—third equation in Eq. (61)—following the standard textbook method, we have

fk,ωa=ieamakn=Jn(b)eibsinφinφωkvnΩa+i0×(nΩavv+kv)FaEk,ω.
(62)

Substituting Eq. (62) to the Poisson equation—second equation in Eq. (61), we have the kinetic version of the wave dispersion relation, together with the kinetic definition for the dielectric response function

0=ε(k,ω)Ek,ω,ε(k,ω)=1+aωpa2k2dvn=Jn2(b)ωkvnΩa+i0×(nΩavv+kv)Fa.
(63)

From this, we obtain the desired expression for the imaginary part of dielectric constant given by

Imε(k,ω)=aπωpa2k2dvn=Jn2(kvΩa)×δ(ωkvnΩa)×(nΩavv+kv)Fa.
(64)

Inserting Eq. (62) to the particle kinetic equation in Eq. (61), we also obtain the desired quasilinear velocity–space diffusion equation as follows:

Fat=πea2ma2dkdωn(nΩavv+kv)×Jn2(kvΩa)δ(ωkvnΩa)×δE2k,ωk2(nΩavv+kv)Fa.
(65)

We are now in a situation to write down the complete set of equations that can describe electrostatic turbulence in magnetized plasmas. The onset of turbulence may be initiated by some free energy source associated with the particles. For instance, an electron beam traveling along the ambient magnetic field may excite primarily Langmuir instability, but during the course of nonlinear mode coupling, the beam-generated Langmuir waves may undergo backscattering and decay that involves upper-hybrid waves and low-frequency ion-sound waves. The set of equations to be summarized here can be solved either by analytical means or by fully numerical means to describe such processes. To present the final result, we take the expression for the imaginary part of dielectric constant computed from kinetic theory, namely, Eq. (64). We also make use of the particle kinetic equation (65) to provide a self-consistent dynamical description of the particle distribution function. We next discuss the incorporation of these kinetic effects.

First, we note that the upper-hybrid mode is a high-frequency mode. As such, we may ignore ions in the linear growth/damping (or induced emission) term given by

γkσU=σωkUμkUπωpe22k2dvn=Jn2(kv|Ωe|)×δ(σωkUkvn|Ωe|)×(n|Ωe|vv+kv)Fe.
(66)

The Langmuir/upper-hybrid mode is characterized by ωkUωpe, ωuh. Consequently, the resonance condition leads to v(σωkUn|Ωe|)/k. If the instability is driven by the electron beam, then it is seen that the harmonic mode number corresponding to

nσωkUkvb|Ωe|[σ(ωpe2Ωe2+k2k2+k2vTe2Ωe2)1/2kvbΩe2],
(67)

where vb is the average electron beam speed, is expected to make the most important contribution. If ωpe/|Ωe| is, say, 3 or so, then n could also be close to 3. If ωpe/|Ωe|5, then n5, etc. On the other hand, the Bessel function factor Jn2(kv/|Ωe|) decreases for increasing n. This means that there is a trade-off between the resonance condition and the Bessel function multiplicative factor. In general, many harmonic terms need to be included in the summation.

For the S mode, on the other hand, the frequency is low so that one may keep only the lowest harmonic term in the electron Bessel function series. In fact, only the n =0 term (the Landau resonance) will be sufficient. The S mode, however, is affected by the ions as well, but since many higher-harmonics of ion terms need to be kept, we approximate the problem by treating the ions as unmagnetized. Thus, the S mode damping rate can be approximated by

γkσS=σωkSμkSπωpe22k2dvJ02(kv|Ωe|)×δ(σωkSkv)kFev+σωkSμkSπωpe22k2dvδ(σωkSk·v)k·Fiv.
(68)

For the induced scattering (nonlinear Landau damping) term in the U mode wave equation, we retain only the ion (proton) contribution. As such, we also replace the expression by its unmagnetized counterpart given by

t|ind.scatt.IkσUμkUσωkUσ=±1dkUk,kdvIkσUμkUIkσUμkU×δ[σωkUσωkU(kk)·v](kk)·Fiv,
(69)

where

Uk,k=mameπωpe2e2me2(μkU)2(μkU)2(k·k)2k2k2×|(kk)2vTe2ηk,kU(kk)2vTe2Ωe2+{(kk)2ηk,kU(kk)2}vTe2|2.
(70)

Finally, we treat ions as a stationary background so that we only solve for the electron velocity–space diffusion equation. For the electrons, only the U mode waves contribute to the velocity–space diffusion, since the low-frequency S mode is generally unimportant for the electrons especially in the range of electron beam.

The final set of equations are thus summarized as follows: The wave kinetic equation for the U mode is given by

tIkσUμkU=2γkσUIkσUμkU+2σωkUσ,σ=±1dkVk,kU[σωkUIkσUμkUIkkσSμkkS(σωkUIkkσSμkkS+σωkkSIkσUμkU)IkσUμkU]×δ(σωkUσωkUσωkkS)+σωkUσ=±1dkUk,kdvIkσUμkUIkσUμkU×δ[σωkUσωkU(kk)·v](kk)·Fiv,
(71)

where γkσU is given by Eq. (66); Vk,kU is defined in Eq. (55); the dispersion relations ωkU and ωkS are defined in Eqs. (31) and (36), respectively; the quantities μkU and μkS are given in Eq. (39); and the coefficient Uk,k is defined by Eq. (70).

The S mode wave kinetic equation is a slight modification of Eq. (54) where

tIkσSμkS=2γkσSIkσSμkS+σωkSσ,σ=±1dkVk,kS[σωkSIkσUμkUIkkσUμkkU(σωkUIkkσUμkkU+σωkkUIkσUμkU)IkσSμkS]×δ(σωkSσωkUσωkkU),
(72)

where γkσS is defined in Eq. (68), and Vk,kS is defined in the same manner as in Eq. (55). The ions are treated as quasistationary, but the electron distribution Fe evolves according to the dictates of the quasilinear velocity–space diffusion equation given by

Fet=πe2me2σ=±1dkk2n(n|Ωe|vv+kv)×Jn2(kv|Ωe|)δ(σωkUkvn|Ωe|)IkσU×(n|Ωe|vv+kv)Fe.
(73)

This completes the formulation of weak turbulence in magnetized plasmas under the assumption of electrostatic interaction. If we eliminate the correction that arises from the presence of ambient magnetic field, then the set of equations that we have derived thus far reduce to that of unmagnetized plasmas, which had been derived with kinetic theory and solved for one- and two-dimensional (or three-dimensions with azimuthal symmetry) situations.33,84–95

To summarize the present paper, we have made use of a hybrid technique that involves a warm two-fluid theory to compute the linear dispersion relation, nonlinear susceptibility, and the basic form of nonlinear wave equation under the weak turbulence ordering. We have then formulated the general weak turbulence analysis to derive the wave kinetic equation that describes linear wave–particle interaction, or induced emission, nonlinear wave–wave interaction, or decay/coalescence, and nonlinear wave–particle interaction, or induced scattering process. Among these, the decay term is adequately described by the warm two-fluid approach, but the processes that involve particles cannot be discussed with macroscopic theory. We have thus employed the linear and quasilinear kinetic theory to provide the mathematical expressions for induced emission and induced scattering terms. We have also derived the quasilinear diffusion equation for the particles, thereby completing the formalism.

As noted in Sec. I, the present state of matter regarding the weak turbulence theory in magnetized plasmas is not at a mature state. Instead, weak turbulence theory for magnetized plasmas is discussed under various simplifying assumptions. Despite some early efforts,49–51 completely general kinetic theory of weak turbulence in magnetized plasmas is not practical. The purpose of the present paper has been to derive a reduced theory of weak turbulence in relatively weakly magnetized plasmas under the assumption of electrostatic interaction. Unlike the early works,49–51 the present paper has taken a more pragmatic approach in that, we started from the warm two-fluid theory. Recently, one of us (P.H.Y.) demonstrated that the warm two-fluid theory is capable of partially reproducing the weak turbulence wave equation for unmagnetized plasmas, which is normally derived from full kinetic theory.81 The present paper adopted such an approach, which is combined with quasilinear kinetic theory, and succeeded in formulating the weak turbulence theory for magnetized plasmas under the assumption of electrostatic interaction.

In the future, the set of equations derived in this paper will be analyzed/solved for a practical problem. Further, the present formalism will be extended to a fully electromagnetic formalism. Such tasks are beyond the scope of the present work, however.

P.H.Y. acknowledges NASA Grant No. NNH18ZDA001N-HSR and NSF Grant No. 1842643 to the University of Maryland. L.F.Z. acknowledges partial support by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001, and support from CNPq (Brazil), Grant No. 302708/2018–9.

The authors have no conflicts to disclose.

The theoretical plots are in normalized units, and the equations are clearly explained in the text. Therefore, no actual data are generated by the theory.

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