Use of Boltzmann distributions

The physical assumption underlying the use of the Boltzmann description for a given species is that its thermal speed is very much greater than the wave speed (the ratio approaching infinity). As a result, during the wave motion, heat flow between the particles locally involved in the wave and those that are a long way away, in the undisturbed region of the plasma, can take place ‘instantaneously’. That requirement is reflected in the basic isothermal assumption underpinning the Boltzmann distribution, and aspects of the physics involved are discussed in more detail in our recent paper (Hellberg, Verheest & Mace Reference Hellberg, Verheest and Mace2018).

A Boltzmann species is in effect assumed to have an infinite thermal speed, that is, the particles are assumed to be effectively inertialess. It is clear that this is a mathematical statement, because, for given energy and mass the thermal velocity is, of course, finite. Evidently, in the presence of other plasma species, described by a fluid description, the thermal velocities of the Boltzmann species are the highest velocities in the system, much higher than phase or acoustic wave speeds, and much higher than the other species’ thermal velocities. In principle, when the Boltzmann assumption is not self-evident, this needs to be checked before the basic equations are suitably normalized and a full-fledged analysis is embarked upon.

Let us start from the density of a Boltzmann (ion) species in a normalized form as

(1) $$\begin{eqnarray}n_{i}(x,t)=\exp [-\unicode[STIX]{x1D711}(x,t)]\quad \Rightarrow \quad \frac{1}{n_{i}(x,t)}\frac{\unicode[STIX]{x2202}n_{i}(x,t)}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}(x,t)}{\unicode[STIX]{x2202}x}=0,\end{eqnarray}$$

where in the second equation the logarithmic derivative has been taken with respect to  $x$ . Comparing this with normal fluid momentum equations, one recognizes the pressure term for an isothermal species, and the electric field. What is automatically absent is the mass $\times$ acceleration term (i.e. ion inertia has been neglected), and, for the case of a magnetized plasma, the magnetic part of the Lorentz force. The pressure is assumed to be isotropic with a scalar temperature, hence, whether the plasma is magnetized or not does not play a role at this level.

Moreover, acoustic modes have phase speeds or solitary wave velocities that lie between the thermal speeds of the species (inter alia, to avoid Landau damping). In fact, a basic requirement is that there needs to be at least one subsonic and at least one supersonic species (Verheest et al. Reference Verheest, Cattaert, Lakhina and Singh2004). Thus, the phase speed should exceed the thermal speeds of those species with a low thermal speed (in the model of Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018), the electrons). The Boltzmann assumption for the ions also implies that the two ‘fast’ species (high thermal speed, ions) should have thermal speeds that are many orders of magnitude greater than those of the two slow species (electrons).

A very important aspect to bear in mind is that, for wave studies, one is interested in the thermal speeds of the different species, rather than the temperatures (whether measured in Kelvin or eV), as it is the relative thermal speeds that play an important physical role. Thus, the statement on page 4 of the paper (that $T_{i}>T_{e}$ ) does not provide any physical guidance, as the two species have very different masses. The physical parameters of importance are the ratios $V_{\text{tij}}/V_{\text{phase}}$ , $V_{\text{tej}}/V_{\text{phase}}$ and $V_{\text{tij}}/V_{\text{tej}}$ .

Numerical ‘justification’ of the model?

Let us begin by determining the thermal velocities of the different species, before normalization. To obtain correct estimates for the ion thermal velocities, we use the authors’ definition on page 7 of $V_{\text{tij}}=\sqrt{T_{\text{ij}}/m_{\text{ij}}}$ . In the original version of the paper, Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018) indicated that the protons were the hotter species at 6000 eV and the singly ionized oxygen ions the cooler species at 150 eV. In that case one obtains $V_{\text{tic}}=3\times 10^{4}~\text{m}~\text{s}^{-1}$ and $V_{\text{tih}}=7.6\times 10^{5}~\text{m}~\text{s}^{-1}$ . In the corrigendum (Kamalam, Steffy & Ghosh Reference Kamalam, Steffy and Ghosh2019) it is the oxygen ions which are the hotter species, and the protons the cooler ones. That modifies the ion thermal velocities to $V_{\text{tic}}=1.2\times 10^{5}~\text{m}~\text{s}^{-1}$ and $V_{\text{tih}}=1.9\times 10^{5}~\text{m}~\text{s}^{-1}$ , as given in the corrigendum. Remarkably, these values correspond to the square roots of the clearly unphysical ion thermal velocities in table 1 of the published paper (Kamalam et al. Reference Kamalam, Steffy and Ghosh2018). In what follows, we will base our estimates on the ion velocities as given in the corrigendum.

From the definition on page 5 of an effective ion temperature one finds $T_{\text{iff}}=360$  eV, in both versions, because this temperature depends on the relative densities but is independent of the masses. Hence, the typical electron-acoustic wave speed, used to normalize the velocities, is $C_{\text{iff}}=\sqrt{T_{\text{iff}}/m_{e}}=8\times 10^{6}~\text{m}~\text{s}^{-1}$ . These values agree with the numbers given in the paper, after appropriate rounding. We note immediately that for both versions the Boltzmann ion thermal velocities are an order or more smaller than $C_{\text{iff}}$ , but we will return to that point later.

The electron thermal velocities are not given in the paper, but are obviously important, as mentioned above. They may easily be computed from $C_{\text{iff}}$ using the given values of $\unicode[STIX]{x1D70E}_{\text{eb}}=T_{\text{eb}}/T_{\text{iff}}=1/40$ and $\unicode[STIX]{x1D70E}_{\text{es}}=T_{\text{es}}/T_{\text{iff}}=1/30$ . This yields the electron thermal speeds $V_{\text{teb}}=\sqrt{T_{\text{eb}}/m_{e}}=C_{\text{iff}}\sqrt{\unicode[STIX]{x1D70E}_{\text{eb}}}=1.26\times 10^{6}~\text{m}~\text{s}^{-1}$ and $V_{\text{tes}}=\sqrt{T_{\text{es}}/m_{e}}=1.46\times 10^{6}~\text{m}~\text{s}^{-1}$ . These are smaller than the measure for the acoustic speed $C_{\text{iff}}$ , thus, the electron species are indeed fluid. The authors use adiabatic pressures for the electrons, so that the definition of $T_{\text{eb}}$ and $T_{\text{es}}$ is not immediate, but might possibly include a factor of  $\sqrt{3}$ . However, this will not greatly affect the estimates.

Taking all this together, the thermal and acoustic velocities are thus ordered as

(2) $$\begin{eqnarray}V_{\text{tic}}<V_{\text{tih}}<V_{\text{teb}}<V_{\text{tes}}<C_{\text{iff}}.\end{eqnarray}$$

Not only is the normalizing speed larger than all four thermal velocities, but the two ion thermal speeds are, in fact, both smaller than the two electron thermal speeds.

The inescapable conclusion is that the ions cannot possibly be described as Boltzmann, and the whole model is incorrect. Unfortunately, that qualification completely undermines the validity of the subsequent analysis and conclusions, in particular, the claims about electron-acoustic supersolitons in magnetized plasmas.

Lest one thinks that this unpleasant conclusion might be avoided by a ‘better’ choice of numerical values, we have to remember that an average temperature like $T_{\text{iff}}$ is constructed in such a way that $T_{\text{ic}}<T_{\text{iff}}<T_{\text{ih}}$ , assuming merely that $T_{\text{ic}}<T_{\text{ih}}$ and that the two density fractions satisfy $\unicode[STIX]{x1D707}_{i}+\unicode[STIX]{x1D708}_{i}=1$ , regardless of the specific parameter values considered. Consequently, it is easy to prove that

(3) $$\begin{eqnarray}\frac{T_{\text{ic}}}{m_{\text{ic}}}<\frac{T_{\text{iff}}}{m_{\text{ic}}}\ll \frac{T_{\text{iff}}}{m_{e}}\quad \Rightarrow \quad V_{\text{tic}}\ll C_{\text{iff}},\end{eqnarray}$$

so that the cooler ion thermal velocity can never exceed the assumed measure for the acoustic velocity. Hence, at least for the cooler ions, no matter their temperature, a Boltzmann description can never hold for what are thought of as electron-acoustic structures with speeds of the order of $\sqrt{T_{\text{iff}}/m_{e}}$ .

For the hot ions, there is no such analytical proof, but one would need to push $T_{\text{ih}}/T_{\text{ic}}$ above 73 000, say, before the hot ion thermal velocity $V_{\text{ih}}$ would just exceed the acoustic speed $C_{\text{iff}}$ for a plasma with $\unicode[STIX]{x1D707}_{i}=0.4$ . For the Boltzmann assumption to hold for the hot species, one would need $V_{\text{tih}}\gg C_{\text{iff}}$ , translating into unphysically large ion temperature ratios.

Another way to see this goes as follows. Since we are interested in nonlinear solitary waves, we suppose that in the original units we have a soliton at speed $V_{\text{sol}}$ . How that might be determined is difficult to ascertain: we cannot trust the theoretical analysis in the paper, given the flaws in the model. Nevertheless, the assumption that the ions are Boltzmann, means that the thermal velocity of the cooler ions (and a fortiori of the hotter ions) needs to exceed the soliton velocity by far, thus yielding

(4) $$\begin{eqnarray}V_{\text{sol}}\ll V_{\text{tic}}=\sqrt{\frac{T_{\text{ic}}}{m_{\text{ic}}}}<\sqrt{\frac{T_{\text{iff}}}{m_{\text{ic}}}}.\end{eqnarray}$$

Normalizing all velocities with respect to $C_{\text{iff}}$ , as done in the paper, leads to an effective Mach number

(5) $$\begin{eqnarray}M_{\text{eff}}=\frac{V_{\text{sol}}}{C_{\text{iff}}}\ll \left.\sqrt{\frac{T_{\text{iff}}}{m_{\text{ic}}}}\right/\sqrt{\frac{T_{\text{iff}}}{m_{e}}}=\sqrt{\frac{m_{e}}{m_{\text{ic}}}}.\end{eqnarray}$$

That is, use of the Boltzmann assumption for the cooler ions implies that $M_{\text{eff}}$ should actually be $\ll 0.023$ . From such extremely small Mach numbers, far from what is used in the paper, we can only conclude that $C_{\text{iff}}$ is a very poor measure for the acoustic speeds studied here and hence does not describe the physics of the wave. In other words, this is yet another hint that the model is not correct.

We recognize that the indicator, $C_{\text{iff}}$ , is not necessarily an accurate value of the actual phase or solitary wave speed. An improved value might be chosen by using the minimum value for solitary waves on page 9, thus concentrating on the nonlinear and essential part of the paper. That value, 0.7 in normalized form, leads to a value of $5.6\times 10^{6}~\text{m}~\text{s}^{-1}$ . Unfortunately, it is derived from the unphysical model underpinning the calculation and is therefore not reliable. However, one notes that this value, too, exceeds all the species thermal speeds, and in particular the ion thermal speeds. Even when $\cos \unicode[STIX]{x1D703}=0.5$ is factored in, for $\unicode[STIX]{x1D703}=60^{\circ }$ , this still leaves us with a value of $2.8\times 10^{6}~\text{m}~\text{s}^{-1}$ , that is still the highest speed of all.

Electron-acoustic (solitary) waves

There are two different definitions of what are called ‘electron-acoustic’ modes in the literature, based on very different physics and hence with different phase speeds. The more common form occurs in a plasma with two-temperature electrons and effectively immobile cold ions, the hot electron pressure providing the driving force. The other form is driven by hot ion pressure and occurs in a magnetized plasma, at almost perpendicular propagation. Amongst the latter studies, two papers having Boltzmann ions are cited in the paper, those of Buti (Reference Buti1980) and of Devanandhan et al. (Reference Devanandhan, Singh, Lakhina and Bharuthram2012), with models closest to the one used by Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018). Buti (Reference Buti1980), using a fluid approach, first discussed a nonlinear electron-acoustic mode in a plasma composed of one background electron and two ion components. The assumption of Boltzmann-distributed ions and quasineutrality leads to an inequality that is found to be easily satisfied if $0<\cos \unicode[STIX]{x1D703}<(m_{e}/m_{i})^{1/2}$ , i.e. for wave propagation almost across the magnetic field.

We note that in the more recent work of Devanandhan et al. (Reference Devanandhan, Singh, Lakhina and Bharuthram2012), too, oblique propagation in a plasma of electrons and a hot $\unicode[STIX]{x1D705}$ -distributed ion component is similarly restricted to a very narrow angle from perpendicular, related analogously to the mass ratio, but reduced slightly by a $\unicode[STIX]{x1D705}$ -based term. The condition on the obliquity arises from the typical acoustic wave requirement that $V_{\text{ti}}>V_{\text{phase}}>V_{\text{te}}$ . Noting that  $V_{\text{phase}}$ includes a factor $\cos \unicode[STIX]{x1D703}$ , it is precisely the very smallness at near perpendicular propagation of $\cos \unicode[STIX]{x1D703}$ which renders the ordering possible. In contrast, the model of Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018) considers $\unicode[STIX]{x1D703}=60^{\circ }$ , translating to $\cos \unicode[STIX]{x1D703}=0.5$ . Hence, it does not relate to this form of electron-acoustic wave in a magnetized plasma.

Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018) have adopted the model discussed by Ghosh et al. (Reference Ghosh, Pickett, Lakhina, Winningham, Lavraud and Décráu2008) and have also based their parameter values on the data from that source, including the angle of propagation. That model was introduced to explain some magnetospheric observations. Unfortunately, however, the paper did not provide a quantitative check on the validity of the Boltzmann assumptions for the hot ions. The ion energies appear only on pages 12–14 of that paper, long after the analytical treatment has been developed. Regrettably, there is no evidence that they considered the actual values of the ion thermal velocities in relation to the model, and there is no a posteriori check on the validity of the assumptions. Hence, it appears that in that paper, as in Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018), the assumption of Boltzmann ions is invalid.

Lessons from acoustic speeds in unmagnetized hydrogen plasmas

An illustration of the discrepancies found in the paper under criticism can be seen when examining a simpler model, for which the algebra is straightforward and transparent. As discussed below, this gives a clear indication that first assuming the ions to be Boltzmann and on the basis of that computing the linear phase velocities, as done on page 6 of Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018), is bound to give completely unreliable results, unless a careful post hoc calculation is carried out to confirm the validity of the assumption.

Let us examine the linear dispersion relation for acoustic-type modes in an unmagnetized hydrogen plasma, obtained by assuming that the phase velocity $\unicode[STIX]{x1D714}/k$ remains finite in the limit $k\rightarrow 0$ ,

(6) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}_{\text{pe}}^{2}}{\left({\displaystyle \frac{\unicode[STIX]{x1D714}}{k}}\right)^{2}-V_{\text{te}}^{2}}+\frac{\unicode[STIX]{x1D714}_{\text{pi}}^{2}}{\left({\displaystyle \frac{\unicode[STIX]{x1D714}}{k}}\right)^{2}-V_{\text{ti}}^{2}}=0.\end{eqnarray}$$

This contains inertial and thermal effects for both the electrons and the ions, without any preferred ordering at this stage (Verheest, Hellberg & Lakhina Reference Verheest, Hellberg and Lakhina2007). Solving this equation for the phase speed yields

(7) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}=\frac{T_{e}+T_{i}}{m_{e}+m_{i}},\end{eqnarray}$$

and one can show that this acoustic phase velocity always lies between the smaller and the larger of the two thermal speeds. As indicated earlier, this is of fundamental importance.

Take first the case of ordinary ion-acoustic modes, where the electrons are much hotter than the ions, $T_{i}\ll T_{e}$ , with of course the usual mass relation $m_{e}\ll m_{i}$ , so that (7) becomes

(8) $$\begin{eqnarray}\frac{T_{i}}{m_{i}}\ll \frac{T_{e}}{m_{i}}\simeq \left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}\ll \frac{T_{e}}{m_{e}}.\end{eqnarray}$$

This amounts to treating the ions as almost cold, and the electrons as an inertialess Boltzmann species, a well-studied model from the literature.

For the reverse case, a so-called electron-acoustic mode from the paper, the electrons are much cooler than the ions, $T_{e}\ll T_{i}$ , but the usual mass relation $m_{e}\ll m_{i}$ remains, of course, so that some care is needed to simplify (7) to

(9) $$\begin{eqnarray}\frac{T_{e}}{m_{e}}<\left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}\simeq \frac{T_{i}}{m_{e}+m_{i}}\lesssim \frac{T_{i}}{m_{i}}.\end{eqnarray}$$

The left inequality might work, provided $T_{e}/T_{i}<m_{e}/m_{i}$ , a very small and maybe unrealistic temperature ratio. In contrast, the right inequality clearly indicates that the acoustic phase velocity lies very close to the ion thermal velocity. In fact, it is so close that the mode will be Landau damped (but this requires kinetic theory and cannot be dealt with in a fluid description) and, moreover, that ion inertia cannot possibly be omitted, i.e. the ions cannot be treated as Boltzmann. A full fluid description of the ions is needed.

Summing this up, crudely omitting $V_{\text{ti}}$ from the ion term and $(\unicode[STIX]{x1D714}/k)$ in the electron term in (6) gives for ion-acoustic modes the phase velocity squared as $(T_{e}/m_{i})$ , as expected. Doing the same for electron-acoustic modes, literally a ‘translation’ exercise, i.e. omitting $V_{\text{te}}$ from the electron term (‘ice-cold’ electrons) and $(\unicode[STIX]{x1D714}/k)$ in the ion term (as required for Boltzmann ions) gives the phase velocity squared as $(T_{i}/m_{e})$ . This, however, is much larger than the Boltzmann ion thermal velocity squared $(T_{i}/m_{i})$ , that is, there is a contradiction in the model assumptions.

We realize, of course, that Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018) investigate acoustic modes in magnetized plasmas, but extending ideas garnered from acoustic modes in an unmagnetized hydrogen plasmas to magnetized plasmas and oblique wave propagation does not help in making the Boltzmann assumption for the ions more palatable, even though that assumption might render the ions effectively unmagnetized, but their inertia cannot possibly be ignored.

Keeping ion inertial and isothermal terms on board would mean that in the model of Kamalam et al. (Reference Kamalam, Steffy and Ghosh2018) we now have three species with such effects, including possible magnetic forces, which completely blocks the analytical treatment used for oblique propagation. For that, one may have at most one species for which the magnetic part of the Lorentz force in the momentum equation is retained, and, moreover, overall charge neutrality is required, not only in equilibrium but also in the wave (Verheest Reference Verheest2009).

Conclusions

The net result is that table 1 (even modified by the corrigendum) does not provide any justification for the use of the model of Boltzmann ions and ‘sluggish’ electrons. In fact, as we have seen, for the listed plasma parameter values the electrons have higher thermal speeds than the ions. Also, the typical wave speed exceeds the ion thermal speeds, too, thus negating the use of Boltzmann ions. And unfortunately, we believe that it is not possible to find a plausible scenario in which this model does apply. Importantly, it is clear from the above arguments that the claim by the authors that they have found electron-acoustic supersolitons propagating obliquely in a magnetized plasma is invalid.