Effects of ion to electron temperatures on electrostatic solitons with applications to space plasma environments

Space environments span a wide range of plasma parameters, such as the composition, plasma $β$⁠, number density, and temperatures. Due to the lack of sufficient collisions, different species may possess distinct temperatures and even the same species may have different origins with different temperatures. A variety of nonlinear plasma phenomena may be generated under various plasma environments. The electrostatic (ES) solitons and solitary waves have been widely observed in various space plasma environments, including the magnetosheath, magnetotail, and aurora zones.^1–7 Theoretical studies of the electrostatic solitons and solitary waves are usually based on two approaches, steady mathematical models and full particle-in-cell simulations. The former ones include the Korteweg–De Vries (KdV) methods for weakly nonlinear waves and the Sagdeev potential formulation for fully nonlinear structures.^8–10 These approaches have been widely adopted to develop the acoustic solitons in electron–ion plasmas and their extensive models such as the dust acoustic solitons in electron–ion-dust plasmas.^8–15 In these models, the light and hot species, such as electrons, are treated by the kinetic theory while the heavy and cold species, such as protons, are described by the fluid models. The relevance of these models to the observed ES waves and solitons, however, is not clear.⁷ On the other hand, the particle simulations have been adopted to study the formations of ES solitary waves via the electron streaming instabilities. In these studies, the ions are usually regarded as an immobile background and the electrons play the major role in the formation of solitary waves.^16–22 The simulation results with electron holes in phase space show some features similar to the observations.^16–24 Since the steady mathematical models do not incorporate the physics of electron holes, two types of ES structures based on two different approaches seem incompatible. In particular, the steady models are developed mostly for the ion acoustic solitons while the kinetic simulations describe the electron solitary waves. The issue remains of which types of solitons are relevant to the observed ES structures.

Recently, the observational analyses based on the Magnetospheric Multiscale (MMS) Mission spacecraft seem to show no evidence of electron holes in the ES solitons observed in a magnetotail reconnection jet.²⁵ It is important to develop the theoretical models for ion acoustic solitons with the parameter values suitable for the observational analyses. In particular, the cold ion assumption is a major constraint for the applications of space plasma environments for which the ion to electron temperature $Ti/Te$ may vary in a wide range of values with $Ti/Te$ being 0.5–3 and 3–10 in the ionosphere and magnetosphere, respectively.^26–28 The purpose of this paper is to develop the theoretical models for the ion acoustic solitons with finite ion temperature. In particular, the ratio of ion to electron temperature is a free parameter adjustable for different space plasma environments. The hot components may possess two types of nonthermal velocity distributions including the non-monotonic Cairns et al.-like distribution and the monotonic Kappa distribution function with the Maxwellian case being a special limit. The Cairns et al.-like distributions have been adopted to explain the anomalous features of acoustic solitons observed in space environments,²⁹ while the Kappa distributions are frequently adopted to fit the nonthermal profiles of the observed particle distributions in the solar wind and magnetosphere.^30–32 Both nonthermal velocity distributions and their modified versions, such as the regularized Kappa functions, have widely been adopted in the study of ion and dusty acoustic solitons, which, however, are mostly limited to cold fluids.^{11–13,32,33} The derived model is the generalization of the earlier works by Chuang and Hau¹¹ and Wang and Hau¹³ by including the effects of ion temperature in the fluid equations. In particular, the dispersion relations are first analyzed for the characteristic speed of acoustic waves modified by the finite ion temperature, which is essential for the occurrence conditions of acoustic solitons. The weakly nonlinear KdV solitons are then developed, and the fully nonlinear acoustic solitons are solved numerically based on the new methods of analyses. Note that there exist a few studies on the ion or dusty acoustic solitons in warm plasmas, which, however, do not include the linear and KdV analyses and may be solved based on the standard Sagdeev potential method due to the assumption of one degree of freedom for the charged particles.^34–38 The present theoretical models may describe various thermodynamic conditions with the existing works as the special limits. In addition, the charges of each species are regarded as free parameters such that the model equations are applicable for the ion, dusty, and electron acoustic waves, solitons, etc.

The generalized model equations for fully nonlinear solitons become highly nonlinear to be solved by the standard Sagdeev potential method widely adopted in the study of acoustic solitons.^11–13 A new method of analyses similar to those applied by Hau and Sonnerup³⁹ and Hau⁴⁰ for the study of nonlinear hydromagnetic shocks, and solitons is adopted for solving the fully nonlinear solutions. It is shown that the effect of $Ti/Te$ may lead to larger phase speed of acoustic waves and weaker potential drops for the same propagation speed. The family of solutions characterized by the Mach number may exhibit self-similar profiles. For sufficiently large $Ti/Te$⁠, the ions may be treated by the kinetic theory and the electric potential become opposite to the standard models for ion acoustic solitons, which shall be referred to as the electron acoustic solitons. The ES solitons with negative electric potentials have also been found in many space environments and attributed to the features of ion hole structures generated by the ion streaming instability.^2,3 Our models may provide an alternative theory for the observed ES structures and can suitably be applied to the space environments by using the observed parameter values as to be illustrated by an MMS event reported by Fu et al.²⁵ 

In this paper, we consider a plasma system consisting of one relatively cold fluid component (with the subscript $s=c$⁠) with finite temperature and two hot kinetic components (with the subscripts $s=h1, h2$⁠). The relatively cold component is described by the fluid equations such as the continuity, momentum equations and the adiabatic energy law with the following forms. For the ion acoustic waves and solitons, the cold and hot components are referred to the ion fluids and kinetic electrons, respectively, though the developed formulations are not restricted to the ion acoustic waves,

where $mc$⁠, $qc$⁠, $nc$⁠, $pc$⁠, $u⇀c$⁠, and $γc$ are the mass, charge, number density, thermal pressure, flow velocity, and ratios of specific heats of the cold fluid component, respectively. In Eq. (2), the relation $E⇀=−∇Φx$ for electrostatic cases has been used. Using the ideal gas law $pc=nckBTc$⁠, the term $−∇pc$ in Eq. (2) can be expressed as $−kBTc∇nc$ for the isothermal condition. Poisson's equation is

As for the hot components, we adopt the Kappa distribution and the Cairns et al.-like distribution for components h1 and h2 with the following forms:¹³ 

for which the constants $Nh1$ and $Nh2$ are

The quantity $n0,s$ (⁠$s=h1, h2$⁠) denotes the unperturbed density. In Eqs. (5) and (7), $κh1$ is the Kappa parameter of the Kappa function, which may recover the Maxwellian distribution for $κh1→∞$⁠. The Cairns et al.-like nonthermal distributions shown in Eq. (6) are the modified Maxwellian distributions and may recover the distribution of Cairns et al. for $a1,h2=0$ and $a2,h2=a$ and the Maxwellian distribution for $a1,h2=a2,h2=0$⁠.¹³ The Cairns et al.-like distributions may incorporate the highly nonthermal cases with non-monotonic profiles, which cannot be achieved by the monotonic Kappa velocity distribution.^11,12

Based on Eqs. (5) and (6), the number density can be derived from $ns=∭fsv,xd3v$ as

Here, $A1,h2=a1,h2/1+(3a1,h2/2)+(15a2,h2/4)$ and $A2,h2=a2,h2/1+(3a1,h2/2)+(15a2,h2/4)$⁠. The corresponding temperatures for hot components may be obtained by the relations of $p⃡=m∭v⇀v⇀fv,xd3v$ and $p=nkBT$⁠,

The plasma model under study comprises Eqs. (1)–(4), (9), and (10). By writing the charge as $qs=Zse$ (⁠$s=c,h1,h2$⁠), the charge-neutrality condition is imposed by the relation of $Zcn0,c+Zh1n0,h1+Zh2n0,h2=0$⁠.

Before solving the nonlinear model equations for the electrostatic solitons, it is essential to calculate the characteristic speed of the plasma system. Assuming the uniform background with the perturbed quantities to have the form of $expikx−ωt$⁠, linearization of the model equations (1)–(4), (9), and (10) yields

The dispersion relation can then be obtained as follows:

where $T0,c$ is the temperature of the background cold component and the background temperature $T0,h$ for hot components is defined as¹³ 

Here, $λD,h2=ε0kBT0,h/n0,cZc2e2$ is the Debye length of the hot components, and $CS,c2=γckBT0,c/mc$⁠, $CS,h2=kBT0,h/mc$ are the sound speeds of the cold and hot components, respectively. The cases with Maxwellian distributions for hot components correspond to $κh1→∞$⁠, $n0,h2=0,$ and $T0,h=T0,h1$ (or $a1,h2=a2,h2=0$⁠, $n0,h1=0,$ and $T0,h=T0,h2$⁠). For $T0,c=0$⁠, Eq. (19) recovers the standard cases with cold fluids.¹³ For the long wavelength limit, the phase speed of the acoustic waves becomes the effective sound speed defined as $CS,eff2=CS,c2+CS,h2$⁠. As indicated, the effects of finite temperature $T0,c$ may lead to larger phase speeds of the acoustic waves, which will be further verified by the nonlinear calculations. The acoustic speed corrected by the warm fluid components is indispensable for the definition of the Mach number and the subsequent nonlinear analyses.

We first study the weakly nonlinear solitons by adopting the standard reductive perturbation technique^13,41 to derive the corresponding KdV equations. The dimensionless variables include $x′≡x/λD,h$⁠, $t′≡tωp,c$⁠, $Φ′≡Φ/kBT0,h/e$⁠, $uc′≡uc/CS,h$⁠, $pc′≡pc/n0,ckBT0,h$⁠, $ns′≡ns/n0,c$⁠, and $σs≡T0,s/T0,h$ (⁠$s=c,h1,h2$⁠). The two stretched coordinates are $ξ=ϵ1/2x′−u0t′$ and $τ=ϵ3/2t′$⁠, where $ϵ$ is a smallness parameter and $u0$ normalized by $CS,h$ is the speed to be determined in the following. The plasma frequency of the cold fluid component is defined as $ωp,c2=n0,cZc2e2/mcε0$⁠. The dimensionless variables $nc′$⁠, $uc′$⁠, $pc′$⁠, and $Φ′$ can be expanded in terms of the smallness $ϵ$ as $nc′=1+ϵnc′1+ϵ2nc′2+⋯$⁠, $uc′=ϵuc′1+ϵ2uc′2+⋯$⁠, $pc′=σc+ϵpc′1+ϵ2pc′2+⋯$⁠, and $Φ′=ϵΦ′(1)+ϵ2Φ′(2)+⋯$⁠, respectively. By substituting Eqs. (9) and (10) into (4), we obtain the following equations for the first and second order variables:

where the parameters $b1=−1$ and $b2$ is

Note that $b2=0.5$ for the cases with Maxwellian distributions.

From the first-order equations (21)–(24), we obtain the normalized phase speed of acoustic waves $u0=1+γcσc1/2$⁠, which is consistent with the linear wave analyses shown in Sec. III and may increase with increasing temperature $Tc$ of the cold fluid component. In particular, by writing $γcσc=CS,c2/CS,h2$⁠, it is easy to see that $u0$ is also the normalized propagation speed of acoustic waves in the linear theory [see Eq. (19)]. For the boundary conditions of $nc′=1$⁠, $uc′=Φ′=0$⁠, and $pc′=σc$ at $ξ→∞$⁠, the relations of $nc′1=ZcΦ′(1)$⁠, $uc′1=u0ZcΦ′(1)$⁠, $pc′1=γcσcZcΦ′(1)$ may be derived. By taking the above first-order identities into the second-order equations (25)–(28), the KdV equation can be obtained as

for which $c1=Zcγc2σc+γcσc+3+2b221+γcσc12,$ and $c2=1/21+γcσc1/2$⁠.

Equation (30) can be further transformed by the independent variable $ζ=ξ−v0t′$ with $v0$ being a constant normalized by $CS,h$ along with the boundary conditions of $Φ′(1)=dΦ′(1)/dζ=∂2Φ′(1)/dζ2=0$⁠, the result being

Equation (31) can be solved for

where $Δ=4c2/v01/2$ and $Φm=3v0/c1$⁠. The electric field $E′(1)=−dΦ′(1)/dζ$ can be inferred as

The above relation implies that the amplitude of the bipolar electric field is $Em=43Φm/9Δ$ at $ζm=Δarsech(2/3)$⁠. Note that the normalized propagation speed of the electrostatic soliton is $U0′=U0/CS,h=u0+v0$⁠. It is clear that the electric potential $Φm$ and the width $Δ$ increases and decreases with increasing $v0$⁠, respectively, and $Em∝v03/2$⁠. The analyses of $c1$ and $c2$ show that for the Maxwellian distribution and for the same $v0$⁠, the electric potential $Φm$ and the width $Δ$ increases and decreases with increasing $σc$ values, respectively. Figures 1(a) and 1(b) show examples of the spatial profiles of the KdV ion acoustic solitons for $U0′=1.2$ and $MS$ = 1.1, respectively, with various values of $σc$ (⁠$T0,i/T0,e$⁠), where $MS=U0/CS,eff$⁠. As indicated, for the same $U0′$⁠, the electric potential $Φm$ and the width $Δ$ decrease and increase with increasing $T0,i/T0,e$ values, respectively. Indeed, for the same $U0′$⁠, the solitons may nearly vanish for $T0,i/T0,e$ exceeding certain values. Figure 1(b) shows the similar spatial profiles by fixing the upstream Mach number $MS$ with various $T0,i/T0,e$ values, indicating that the soliton solutions may well be characterized by the dimensionless Mach number based on the generalized phase speed of acoustic waves, which will further be confirmed by the fully nonlinear calculations presented in Secs. V and VI.

In this section, we discuss the exact solutions of the acoustic solitons by solving Eqs. (1)–(4), referred to as the fully nonlinear equations, in contrast to the weakly nonlinear KdV model. By using the coordinate transformation $ζ=x−Uot$⁠, we may write Eqs. (1)–(4) in the steady state form as

With the boundary conditions of $nc=n0,c$ and $uc=0$ at $ζ→∞$⁠, Eq. (34) gives rise to

For cold temperature cases (⁠$Tc=0$⁠) with the boundary conditions of $Φ=0$ at $ζ→∞$⁠, Eq. (35) yields

Substituting Eqs. (9), (10), and (39) into (37) and performing the integration then yields

where $VΦ=VcΦ+Vh1Φ+Vh2Φ$ is the Sagdeev potential,¹³ and

As shown by Mendoza-Briceño et al.,³⁸ for $γc=3$ corresponding to one degree of freedom for charged particles, Eqs. (34)–(36) may be combined and integrated to obtain the following explicit form for $nc=ncΦ$⁠:

where $qcφ=qcΦ−γckBT0,c/γc−1−mcU02/2$⁠. The corresponding Sagdeev potential can then be obtained for the special case of $γc=3$ for the fluid components; the forms of $VcΦ$ are referred to the papers by Mendoza-Briceño et al.³⁸ and Singh et al.,³⁷ where the distributions of Caines et al. and Kappa distributions are adopted for the hot components, respectively, for the dusty and ion acoustic solitons.

For general cases with finite temperature and arbitrary ratios of specific heats for the fluid component, there exists no explicit form for $nc=ncΦ$ such that the fully nonlinear equations shown in Eqs. (34)–(37) cannot be solved based on the Sagdeev potential method. By using the ideal gas law of $p=nkBT$ and defining $y=dΦ/dζ$⁠, Eqs. (34)–(37) can be rewritten as the following sets of four first-order differential equations, which need be solved numerically:

Note that for the isothermal case, $γc=1$ and $dTc/dζ=0$⁠, such that Eq. (44) is not needed in the model calculations.

Equations (43)–(46) with arbitrary ratios of specific heats cannot be solved without performing the fixed point analysis in a manner similar to the hydromagnetic shock and soliton problems analyzed by Hau and Sonnerup³⁹ and Hau.⁴⁰ Linearizing Eqs. (43)–(46) around the upstream or downstream states of the nonlinear structures and assuming the perturbations to have the form of $eλζ$⁠, the eigenfunctions and the eigenvalues can be solved as

As indicated, the eigenvalues are real only for

which implies that the effects of finite temperatures for the fluid component yield larger critical velocity $Uc=CS,eff$ for the existence of steady solitons. For $U0<Uc$⁠, the eigenvalues become imaginary implying no steady solutions.^39,40 By writing $MS2=U02/CS,eff2$⁠, Eq. (50) becomes

which is the necessary condition for the existence of steady solitons. Since $CS,eff2$ increases with increasing sound speeds of the cold and hot components, the propagation speeds of the acoustic solitons increase with increasing $T0,c$ and $T0,h$⁠. The fully nonlinear equations shown in Eqs. (43)–(46) can be solved numerically by using the linear eigenfunctions as the initial perturbations,^39,40 and the calculations are shown in Sec. VI.

To study the effects of finite temperatures for ion fluids, we first discuss the cases with $T0,i<T0,e$ such that the ions and electrons are treated as cold fluid and hot kinetic components (subscripts $c=i$ and $h=e$⁠), respectively. For the hot electrons, the Maxwellian distributions are considered such that the electron temperatures are constant. Figure 2 shows the spatial profiles of electric potential, electric field, electron density, and ion density from the nonlinear calculations assuming isothermal ions (⁠$γi=1$⁠). Note that in the calculations, the electric potential, electric field, density, temperature, and spatial length are normalized by $kBT0,h/e$⁠, $kBT0,h/eλD,h$⁠, $n0,c$⁠, $T0,h$⁠, and $λD,h$⁠, respectively. In all cases, with different $T0,i/T0,e$ values shown in panel (a) of Fig. 2, the propagation speed normalized by the electron sound speed $U0′=U0/CS,e$ is fixed. The dotted curve for the case of $T0,i/T0,e=0$ is the analytical solution from solving Eqs. (41) and (42), indicating the consistent results between nonlinear calculations and analytical solutions. It also shows that the steady solutions exist only for $U0′>Uc′=Uc/CS,e=1+γiT0,i/T0,e1/2$ in accordance with Eq. (50). In particular, the amplitude of the ion acoustic solitons decreases with increasing $T0,i/T0,e$ and becomes nearly diminished for $γiT0,i/T0,e∼U0′2−1$⁠. Figure 2(b) shows that for the same $T0,i/T0,e$ the amplitude of the ion acoustic solitons increase with increasing $U0′$⁠. Comparisons between Figs. 1(a) and 2 for the same $U0′$ show consistent results between the KdV and fully nonlinear calculations except that the peak electric potential are about $40%$ smaller than the corresponding fully nonlinear solutions.