2.1 The Hesse–Winske moment model

As anticipated above, the Hesse–Winske (HW) model involves a moment truncation of the electron kinetics. More specifically, the electron kinetic equation is truncated to the second-order moment thereby leading to the following equation for the electron pressure (e.g. see equation (2) in Kuznetsova et al. (Reference Kuznetsova, Hesse and Winske1998)):

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\widetilde{\mathbb{P}}_{e}}{\unicode[STIX]{x2202}t}+(\boldsymbol{V}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\widetilde{\mathbb{P}}_{e}+(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{V}_{e})\widetilde{\mathbb{P}}_{e}+\widetilde{\mathbb{P}}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e}+(\widetilde{\mathbb{P}}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e})^{\text{T}}=\frac{e}{m_{e}}(\boldsymbol{B}\times \widetilde{\mathbb{P}}_{e}-\widetilde{\mathbb{P}}_{e}\times \boldsymbol{B}).\end{eqnarray}$$

This equation neglects heat flux contributions and this approximation may or may not be physically consistent depending on the case under study. In a series of papers (Hesse & Winske Reference Hesse and Winske1993, Reference Hesse and Winske1994; Winske & Hesse Reference Winske and Hesse1994; Kuznetsova et al. Reference Kuznetsova, Hesse and Winske1998, Reference Kuznetsova, Hesse and Winske2000; Yin et al. Reference Yin, Winske, Gary and Birn2001; Yin & Winske Reference Yin and Winske2003; Hesse et al. Reference Hesse, Kuznetsova and Birn2004), the authors approximated heat flux contributions by an isotropization term involving ad hoc parameters. However, in this section we shall continue to discard the heat flux, whose corresponding effects will be completely included in our later discussion of fully kinetic models. We address the reader to Basu’s work (Basu Reference Basu2002) and the more recent results in Sarrat, Del Sarto & Ghizzo (Reference Sarrat, Del Sarto and Ghizzo2016), Ghizzo, Sarrat & Del Sarto (Reference Ghizzo, Sarrat and Del Sarto2017) for a complete description of the Weibel instability in terms of kinetic moments. In addition, we point out that the gyration terms on the right-hand side of (2.1) are discarded in Yin et al. (Reference Yin, Winske, Gary and Birn2001), Yin & Winske (Reference Yin and Winske2003) (strong electron magnetization assumption), while these terms are retained in the present treatment.

The electron pressure dynamics (2.1) is coupled in the HW model to Faraday’s law $\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\times \boldsymbol{E}$ and the ion kinetics

(2.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}f_{i}}{\unicode[STIX]{x2202}t}+\boldsymbol{v}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{i}}{\unicode[STIX]{x2202}\boldsymbol{x}}+\frac{Ze}{m_{i}}(\boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B})\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{i}}{\unicode[STIX]{x2202}\boldsymbol{v}}=0,\end{eqnarray}$$

where the electric field is given by Ohm’s law in the form (1.6). In addition, quasi-neutrality gives

(2.3a,b ) $$\begin{eqnarray}Zen_{i}-en_{e}=0,\quad Zen_{i}\boldsymbol{V}_{i}-en_{e}\boldsymbol{V}_{e}=\unicode[STIX]{x1D707}_{0}^{-1}\unicode[STIX]{x1D735}\times \boldsymbol{B}=\boldsymbol{J},\end{eqnarray}$$

so that $(n_{e},\boldsymbol{V}_{e})$ can be expressed in terms of the ion moments.

Since we are interested in the Weibel instability, we linearize the HW model around a static anisotropic equilibrium of the type

(2.4a-c ) $$\begin{eqnarray}\boldsymbol{E}_{0}=\boldsymbol{B}_{0}=\boldsymbol{V}_{e0}=\boldsymbol{V}_{i0}=0,\quad \mathbb{P}_{0}=p_{\bot }\mathbf{1}+(p_{\Vert }-p_{\bot })\boldsymbol{z}\boldsymbol{z},\quad f_{0}=f_{0}(v_{\bot }^{2},v_{z}^{2}),\end{eqnarray}$$

and we consider longitudinal propagation along the wavevector $\boldsymbol{k}=k\boldsymbol{z}$ (here, $\boldsymbol{z}$ denotes the unit vertical). Notice that we have dropped the species subscripts for convenience and we have retained both electron and ion anisotropies. The corresponding dispersion relation is found in § A.1 and it reads

(2.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k^{2}v_{e\Vert }^{2}}=1-\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}+k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}+Z\bar{\unicode[STIX]{x1D707}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right],\end{eqnarray}$$

where $\bar{\unicode[STIX]{x1D707}}=m_{e}/m_{i}$ .

In order to distinguish the various contributions from the ions and the electrons, it is useful to study the electron Weibel instability and the ion Weibel instability separately. In the first case, one can restrict to an isotropic ion equilibrium, so that $T_{\bot }^{(i)}=T_{\Vert }^{(i)}$ . In addition, upon adopting a cold-fluid closure for the ion dynamics one can write $W(\unicode[STIX]{x1D714}/kv_{i\Vert })\simeq 0$ to obtain

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k^{2}v_{\Vert }^{2}}=1-\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}+k^{2}\unicode[STIX]{x1D6FF}^{2}+Z\bar{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

A detailed discussion of the dispersion relation (2.6) is presented later in the paper. For the moment, we remark that $\unicode[STIX]{x1D714}$ is imaginary only in the range $k^{2}\unicode[STIX]{x1D6FF}^{2}<T_{\bot }^{(e)}/T_{\Vert }^{(e)}-Z\bar{\unicode[STIX]{x1D707}}-1$ , while purely oscillating modes emerge otherwise.

The ion Weibel instability can be studied in a similar way upon setting $T_{\bot }^{(e)}=T_{\Vert }^{(e)}$ in (2.5) so that, upon restoring the species index and by denoting by $v_{e}$ the electron thermal velocity, we have

(2.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k^{2}v_{e}^{2}}-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}=Z\bar{\unicode[STIX]{x1D707}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right].\end{eqnarray}$$

Again, this dispersion relation is discussed later in this paper.

2.2 The effect of Coriolis force terms

In Tronci (Reference Tronci2013), one of us showed how one can neglect the electron mean-flow inertia terms in (1.5) by using variational methods based on Hamilton’s principle. This approach has the advantage of preserving the total energy and momentum and in recent years there is an increasing amount of work in exploiting this approach for nonlinear plasma modelling (Cendra et al. Reference Cendra, Holm, Hoyle and Marsden1998; Morrison Reference Morrison1998; Brizard Reference Brizard2000; Holm & Tronci Reference Holm and Tronci2012; Tronci & Camporeale Reference Tronci and Camporeale2015). Essentially, in plasma physics this approach goes back to Low (Reference Low1958), Newcomb (Reference Newcomb1962) and it was later used in Littlejohn (Reference Littlejohn1983) in his theory of guiding-centre motion. When applied to the case under study, this method produces Coriolis forces in the electron kinetics that modify the electron pressure dynamics (2.1) in the HW model as follows:

(2.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\mathbb{P}}_{e}}{\unicode[STIX]{x2202}t}+(\boldsymbol{V}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\widetilde{\mathbb{P}}_{e}+(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{V}_{e})\widetilde{\mathbb{P}}_{e}+\widetilde{\mathbb{P}}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e}+(\widetilde{\mathbb{P}}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e})^{\text{T}}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad +\,\widetilde{\mathbb{P}}_{e}\times \left(\frac{e}{m_{e}}\boldsymbol{B}+\unicode[STIX]{x1D74E}_{e}\right)-\left(\frac{e}{m_{e}}\boldsymbol{B}+\unicode[STIX]{x1D74E}_{e}\right)\times \widetilde{\mathbb{P}}_{e}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D74E}_{e}=\unicode[STIX]{x1D735}\times \boldsymbol{V}_{e}$ denotes the electron hydrodynamic vorticity. As shown in Tronci (Reference Tronci2013), the vorticity terms arise by neglecting the electron mean-flow inertia after expressing the electron kinetics in the relative frame moving with the Eulerian velocity $\boldsymbol{V}_{e}$ ; this takes the dynamics in a non-inertial frame thereby producing Coriolis forces that shift the magnetic field by the electron vorticity. We remark that the terms involving the electron velocity (including the vorticity terms) combine into a fluid transport operator (Lie derivative) so that the electron pressure becomes frozen into the electron mean flow in the case of strong electron magnetization (so that $(\boldsymbol{e}/\boldsymbol{m}_{e})\widetilde{\mathbb{P}}_{e}\times \boldsymbol{B}-(\boldsymbol{e}/\boldsymbol{m}_{e})\boldsymbol{B}\times \widetilde{\mathbb{P}}_{e}\simeq 0$ ). At this point, the Coriolis forces in the electron pressure dynamics lead to a modified version of the HW model.

Upon linearizing the modified HW model around the equilibrium (2.4a-c ), one obtains the dispersion relation (see § A.1)

(2.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k^{2}v_{e\Vert }^{2}}=1-\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\left\{1-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-\frac{Zm_{e}}{m_{i}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right]\right\}.\end{eqnarray}$$

By proceeding analogously to the previous section, we consider the electron Weibel instability by setting $T_{\bot }^{(i)}=T_{\Vert }^{(i)}$ and $W(\unicode[STIX]{x1D714}/kv_{i\Vert })\simeq 0$ thereby obtaining

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}^{2}}{k^{2}v_{\Vert }^{2}}=1-\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\left(k^{2}\unicode[STIX]{x1D6FF}^{2}+Z\bar{\unicode[STIX]{x1D707}}\right).\end{eqnarray}$$

Again, we notice that $\unicode[STIX]{x1D714}$ is imaginary only in the range $k^{2}\unicode[STIX]{x1D6FF}^{2}<1-Z\bar{\unicode[STIX]{x1D707}}-T_{\Vert }^{(e)}/T_{\bot }^{(e)}$ , while purely oscillating modes emerge otherwise.

In the same way, we can specialize (2.9) to the case of the ion Weibel instability. In this case, Coriolis effects become irrelevant and one obtains again equation (2.7).

The next section presents a study of (2.5) and (2.9) in each considered case.

2.3 Discussion on moment models

Here and in the following discussions, we consider an electron–proton plasma, with typical solar wind parameters. For comparison, we report the following dispersion relation corresponding to full Vlasov–Maxwell dynamics (see e.g. Gary & Karimabadi (Reference Gary and Karimabadi2006)), as it is obtained by using the exact form of Ohm’s law (1.7):

(2.11) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=\frac{\unicode[STIX]{x1D714}^{2}}{\unicode[STIX]{x1D714}_{pe}^{2}}-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\left[1+W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right].\end{eqnarray}$$

Here, the electron Weibel instability is studied by adopting a cold-fluid closure for ion kinetics, so that $T_{\bot }^{(i)}=T_{\Vert }^{(i)}$ and $W(\unicode[STIX]{x1D714}/kv_{i\Vert })\simeq 0$ yield

(2.12) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=\frac{\unicode[STIX]{x1D714}^{2}}{\unicode[STIX]{x1D714}_{pe}^{2}}-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

The ratio between electron thermal velocity (in the cold direction) and speed of light $v_{e}/c=0.0318$ (this is also the ratio between Debye length and electron inertial length). The mass ratio is physical $m_{i}/m_{e}=1836$ . Figures 1 and 2 show the dispersion relations for electron and ion Weibel instabilities, respectively. The four panels are for values of temperature anisotropy equal to 2, 5, 10 and 100. The blue lines show the reference solutions derived from the Vlasov–Maxwell model (2.12) (involving a cold-fluid closure for ion kinetics), while red lines are for (2.6). In figure 1 one can notice how the HW model yields much larger growth rates than the correct values. The results for the modified HW model (2.10) are shown in yellow. They partially correct the discrepancies with the full Vlasov–Maxwell model, but they are still unsatisfactory, especially for wavevectors larger than the inverse electron inertial length.

Figure 1. Growth rate for the electron Weibel instability for the HW (2.6) and modified HW (2.10) models, for four values of temperature anisotropy $T_{\bot }^{(e)}/T_{\Vert }^{(e)}=2,5,10,100$ . The blue lines are the reference solutions derived from the Vlasov–Maxwell model, while red and yellow lines are for (2.6) and (2.10), respectively.

Figure 2. Growth rate for the ion Weibel instability for the HW (2.5) model, for four values of temperature anisotropy $T_{\bot }^{(i)}/T_{\Vert }^{(i)}=2,5,10,100$ . The blue lines are the reference solutions derived from the Vlasov–Maxwell model, while red lines are for (2.5). In this case the modified HW model (2.9) yields identical results.

As we mentioned, the Coriolis effects are irrelevant for the case of the ion Weibel instability. In this case, the reference Vlasov solution is obtained in figure 2 by solving the dispersion relation

(2.13) $$\begin{eqnarray}1+\frac{k^{2}v_{e\Vert }^{2}}{2\unicode[STIX]{x1D714}^{2}}=\frac{\unicode[STIX]{x1D714}^{2}}{\unicode[STIX]{x1D714}_{pe}^{2}}-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\left[1+W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right].\end{eqnarray}$$

This is derived upon adopting a warm-fluid closure for electron kinetics, that is by inserting $T_{\bot }^{(e)}=T_{\Vert }^{(e)}$ and $W(\unicode[STIX]{x1D714}/kv_{e\Vert })\simeq (1/2)k^{2}v_{e\Vert }^{2}/\unicode[STIX]{x1D714}^{2}$ in (2.11). Interestingly, for the ion Weibel instability the discrepancies between the HW and VM models are already significant for $k\unicode[STIX]{x1D6FF}_{e}<0.1$ .

3.1 Removing the electron inertia

A first approach to neglect electron inertia consists of simply removing the term $-\text{D}\boldsymbol{V}_{e}/\text{D}t$ in (3.3), thereby leading to the modified electron equation

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}t}+(\boldsymbol{c}+\boldsymbol{V}_{e})\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}\boldsymbol{x}}-\left\{\boldsymbol{c}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e}+\frac{e}{m_{e}}[\boldsymbol{E}+(\boldsymbol{c}+\boldsymbol{V}_{e})\times \boldsymbol{B}]\right\}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}\boldsymbol{c}}=0.\end{eqnarray}$$

Although this equation retains the acceleration term $-\boldsymbol{c}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e}$ , inertial forces are only partially considered since the term (3.4) has been entirely neglected. At this point, one can easily take the first moment of (3.5), so that using the constraint $\int \boldsymbol{c}\,\mathsf{f}_{e}(\boldsymbol{c})\,\text{d}\boldsymbol{c}=0$ leads to the reduced Ohm’s law (1.6) and a fully kinetic model is formulated by using ion kinetics (2.2), along with Ampère’s and Faraday’s laws.

The model obtained in this way provides the basis for the HW moment model in § 2.1, except that the HW model invokes the quasi-neutrality conditions (2.3). The idea of using quasi-neutrality in a fully kinetic model is not new. In later sections, we shall show how the quasi-neutrality assumption can be used successfully in fully kinetic theories, although it requires extra care. However, for the purpose of this section, we shall keep assuming quasi-neutrality in the present discussion. Thus, Ampère’s law in (2.3) can be used to eliminate entirely the variable $\boldsymbol{V}_{e}$ in favour of the ion velocity $\boldsymbol{V}_{i}$ , as it is computed from (2.2).

Combining (2.2), (3.5), (2.3) and Faraday’s law yields a fully kinetic model, whose moment truncation to second-order yields exactly the HW moment model from § 2.1. For later reference, we shall refer to this as the HW kinetic model. Then, one would hope that completing the HW moment model by retaining fully kinetic effects (while still neglecting electron mean-flow inertia) could capture more physics. As we shall see, this may not always be true and we explain this below by considering again the case of the Weibel instability.

Here, we linearize the HW kinetic model around the bi-Maxwellian equilibrium

(3.6) $$\begin{eqnarray}\boldsymbol{E}_{0}=\boldsymbol{B}_{0}=0,\quad f_{0}=f_{0}(v_{\bot }^{2},v_{z}^{2}),\quad \mathsf{f}_{0}=\mathsf{f}_{0}(c_{\bot }^{2},c_{z}^{2}),\end{eqnarray}$$

where $f_{0}$ and $\mathsf{f}_{0}$ denote the ionic and electronic equilibrium, respectively. As shown in § A.2, we obtain the dispersion relation

(3.7) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=\left\{k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}+Z\bar{\unicode[STIX]{x1D707}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right]\right\}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right).\end{eqnarray}$$

In order to study the electron Weibel instability, we follow the approach in § 2.1 and adopt a cold-fluid closure for the ions by setting $T_{\bot }^{(i)}=T_{\Vert }^{(i)}$ and $W(\unicode[STIX]{x1D714}/kv_{i\Vert })\simeq 0$ . This yields

(3.8) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=(k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}+Z\bar{\unicode[STIX]{x1D707}})\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right).\end{eqnarray}$$

On the other hand, the ion Weibel instability requires special care since Ohm’s law (1.6) requires pressure to balance the Lorentz force in electron dynamics. Indeed, as one can see especially in (A 14) in § A.1, adopting a cold-fluid closure for electron dynamics would lead to consistency issues. However, a warm-fluid closure can be performed by setting $T_{\bot }^{(e)}=T_{\Vert }^{(e)}$ and $W(\unicode[STIX]{x1D714}/kv_{e\Vert })\simeq (1/2)k^{2}v_{e\Vert }^{2}/\unicode[STIX]{x1D714}^{2}$ so that (3.7) becomes

(3.9) $$\begin{eqnarray}\frac{2\unicode[STIX]{x1D714}^{2}}{k^{2}v_{e\Vert }^{2}}-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}=Z\bar{\unicode[STIX]{x1D707}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right]-1.\end{eqnarray}$$

The contribution of the heat flux and higher moments can be understood by comparing the above equation to the corresponding equation (2.7) for the HW moment model.

While the discussion of the dispersion relations (3.8) and (3.9) is left for later discussion, the next section aims at extending the modified HW moment model from § 2.2 to a fully kinetic theory.

3.2 Coriolis force effects

A modified version of the HW kinetic model was presented in Tronci (Reference Tronci2013) (see equations (1)–(5) therein), by exploiting variational techniques based on Hamilton’s principle. As discussed in § 2.2, this approach produces the Coriolis force terms appearing in (2.8). In the fully kinetic treatment, the same approach leaves (2.2), (2.3) and Faraday’s law unchanged while (3.5) is modified as follows:

(3.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}t}+(\boldsymbol{c}+\boldsymbol{V}_{e})\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}\boldsymbol{x}}-\left\{\boldsymbol{c}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}_{e}+\boldsymbol{c}\times \unicode[STIX]{x1D735}\times \boldsymbol{V}_{e}+\frac{e}{m_{e}}[\boldsymbol{E}+(\boldsymbol{c}+\boldsymbol{V}_{e})\times \boldsymbol{B}]\right\}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\mathsf{f}_{e}}{\unicode[STIX]{x2202}\boldsymbol{c}}=0.\end{eqnarray}$$

Evidently, this differs from (3.5) by the Coriolis acceleration term $\boldsymbol{c}\times \unicode[STIX]{x1D735}\times \boldsymbol{V}_{e}$ . As discussed in Tronci (Reference Tronci2013), this term appears from the variational approach due to the fact that the change of frame performed to express the electron kinetics in the mean-flow frame affects the Lorentz force term, which now is written in terms of the effective magnetic field $\boldsymbol{B}+m_{e}\unicode[STIX]{x1D74E}_{e}/e$ . This is a typical feature of electrodynamics in non-inertial frames, as explained in Thyagaraja & McClements (Reference Thyagaraja and McClements2009). Notice that the Coriolis acceleration term is absent in (3.3), which also means that this term is produced to guarantee a consistent force balance after the mean-flow inertia term $\text{D}\boldsymbol{V}_{e}/\text{D}t$ is dropped in (3.3). At this point, the modified HW kinetic model is given by (3.10), (2.2), (2.3) and Faraday’s law.

For comparison with the HW kinetic model in the previous section, we study the effect of Coriolis forces by considering again the Weibel instability. Then, we linearize the modified HW kinetic system around the equilibrium (3.6) to obtain the dispersion relation (see § A.2)

(3.11) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\left\{k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}+Z\bar{\unicode[STIX]{x1D707}}\left[1+\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right]\right\}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right).\end{eqnarray}$$

By following the approach in the previous sections, we restrict to consider the electron Weibel instability by adopting a cold-fluid closure for the ions. This yields

(3.12) $$\begin{eqnarray}\frac{T_{\Vert }^{(e)}}{T_{\bot }^{(e)}}+W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=(k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}+Z\bar{\unicode[STIX]{x1D707}})\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right).\end{eqnarray}$$

On the other hand, upon assuming a warm-fluid closure for the electrons by replacing $T_{\bot }^{(e)}=T_{\Vert }^{(e)}$ and $W(\unicode[STIX]{x1D714}/kv_{e\Vert })\simeq (1/2)k^{2}v_{e\Vert }^{2}/\unicode[STIX]{x1D714}^{2}$ in (3.11), we obtain the same dispersion relation (3.9) for the ion Weibel instability.

3.3 Discussion on kinetic models with inertialess electrons

In first instance, this section compares the dispersion relations (3.8) and (3.11) for the electron Weibel instability with the corresponding result (2.12) for the case of the Vlasov–Maxwell system for cold-fluid ions. A typical limit that is often used to study the Weibel instability is

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D714}\ll kv_{e\Vert }\end{eqnarray}$$

so that $W(\unicode[STIX]{x1D714}/kv_{e\Vert })\simeq -1-i\sqrt{\unicode[STIX]{x03C0}}\,\unicode[STIX]{x1D714}/kv_{e\Vert }$ . In this limit, equations (3.8) and (3.11) become (upon dropping the species superscript for convenience)

(3.14) $$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}=\frac{kv_{\Vert }}{\sqrt{\unicode[STIX]{x03C0}}}\left(1+\frac{1}{T_{\bot }/T_{\Vert }-k^{2}\unicode[STIX]{x1D6FF}^{2}-Z\bar{\unicode[STIX]{x1D707}}}\right)\end{eqnarray}$$

and

(3.15) $$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}=\frac{kv_{\Vert }}{\sqrt{\unicode[STIX]{x03C0}}}\left(1-\frac{T_{\Vert }}{T_{\bot }}\frac{1}{1-k^{2}\unicode[STIX]{x1D6FF}^{2}-Z\bar{\unicode[STIX]{x1D707}}}\right),\end{eqnarray}$$

respectively. On the other hand, upon assuming $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D714}_{p}$ , equation (2.12) becomes

(3.16) $$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}=\frac{kv_{\Vert }}{\sqrt{\unicode[STIX]{x03C0}}}\left[1-\frac{T_{\Vert }}{T_{\bot }}(k^{2}\unicode[STIX]{x1D6FF}^{2}+Z\bar{\unicode[STIX]{x1D707}}+1)\right].\end{eqnarray}$$

Now, we observe that in the limit $k\unicode[STIX]{x1D6FF}\ll 1$ the results in (3.15) and (3.16) coincide thereby showing that the variational model from § 2.2 agrees well with Vlasov–Maxwell dynamics for length scales much bigger than the skin depth. In turn, in the same limit $k\unicode[STIX]{x1D6FF}\ll 1$ (3.14) disagrees with the Vlasov–Maxwell result (3.16) with growing anisotropies.

However, both results (3.14) and (3.15) suffer from the important drawback that a vertical asymptote emerges in the growth rate as length scales approach the skin depth. We remark that the assumption $\unicode[STIX]{x1D714}\ll kv_{e\Vert }$ is no longer valid near and after the asymptote and so the dispersion relation needs to be solved numerically, as presented below. After the asymptotes, for both kinetic models the least damped mode is not the Weibel mode, but one with non-zero real frequency, hence yielding a completely different result from the Vlasov–Maxwell theory, in which the Weibel (purely damping) mode is dominant. Figure 3 shows the dispersion relations for the electron instability, for four values of temperature anisotropy $T_{\bot }^{(e)}/T_{\Vert }^{(e)}=2,5,10,100$ . The blue lines are the reference solutions derived from the Vlasov–Maxwell model (2.12), while red and yellow lines are for the HW kinetic (3.8) and modified HW kinetic (3.12) models, respectively. The aforementioned asymptote for the reduced models is clearly visible, with the distinguishing features that while it always occurs at $k\unicode[STIX]{x1D6FF}=1$ for the modified HW model, it becomes a function of anisotropy for the HW model. Both models present large discrepancies with respect to the full Vlasov–Maxwell solution, with the wavevector approaching the inverse electron inertial length. Figure 4 shows the real frequency of the least damped mode for the modified HW kinetic model (solid lines) and for the HW kinetic model (dashed lines). Once again, in contrast with the correct VM solution, the mode’s real frequency becomes non-zero after the respective asymptote.

Figure 3. Growth rate for the electron Weibel instability for the HW kinetic (3.8) and modified HW kinetic (3.12) models, for four values of temperature anisotropy $T_{\bot }^{(e)}/T_{\Vert }^{(e)}=2,5,10,100$ . The blue lines are the reference solutions derived from the Vlasov–Maxwell model, while red and yellow lines are for (3.8) and (3.12), respectively.

Figure 4. Real part of the frequency for the electron Weibel instability for the modified HW kinetic model (3.12) (solid lines) and for the KW kinetic model (3.8) (dashed lines), for values of temperature anisotropy $T_{\bot }^{(e)}/T_{\Vert }^{(e)}=2,5$ .

The ion Weibel instability is, on the contrary, well captured by both models. This is shown in figure 5. Once again, in this case the Coriolis correction does not play any role and the two models become identical. One can notice that, for any value of temperature anisotropy, the solutions are indistinguishable from the correct Vlasov–Maxwell results, which are obtained from the dispersion relation (2.13) (adopting a warm-fluid closure for the electrons). In some sense, this is not surprising, since ion kinetics is not subject to any approximation in either the HW kinetic model and its modified variant.

Figure 5. Growth rate for the ion Weibel instability for the HW kinetic model (3.7), for four values of temperature anisotropy $T_{\bot }^{(i)}/T_{\Vert }^{(i)}=2,5,10,100$ . The blue circles denote the reference solutions derived from the Vlasov–Maxwell model, while red lines are for (3.7).

4.1 The Cheng–Johnson model

As anticipated above, Cheng & Johnson (Reference Cheng and Johnson1999) were the first to design an alternative fully kinetic model in the quasi-neutral limit. More specifically, they expanded Ohm’s law (1.1) by using Ampeère’s and Faraday’s laws to obtain (1.7). Then, after assuming quasi-neutrality to write $\boldsymbol{J}=\unicode[STIX]{x1D707}_{0}^{-1}\unicode[STIX]{x1D735}\times \boldsymbol{B}$ , they neglected all terms of the order of $m_{e}/m_{i}$ . This process leads to the reduced form of Ohm’s law in (1.8), which is then accompanied by the kinetic equations (2.2) and (3.1), the quasi-neutrality conditions (2.3) and Faraday’s law. We remark that originally the CJ model was called a kinetic-multifluid system because the kinetic equation for each species was written to accompany the equation for its first moment. However, this is totally equivalent to retaining only the kinetic equations.

In order to compare the CJ model to the systems formulated in the previous section, we studied the Weibel instability by linearizing again around the equilibrium (3.6). Upon linearizing the CJ model around the bi-Maxwellian equilibrium

(4.1) $$\begin{eqnarray}\boldsymbol{E}_{0}=\boldsymbol{B}_{0}=0\,,\qquad \qquad f_{s0}=f_{s0}(v_{\bot }^{2},v_{z}^{2})\end{eqnarray}$$

(where the subscript $s$ refers to the particle species), we obtain the dispersion relation (see § A.3)

(4.2) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right).\end{eqnarray}$$

The above dispersion relation can be compared directly with the relation (2.11) that is obtained for the full Vlasov–Maxwell dynamics. As expected, both electron and ion Weibel instabilities are reproduced by the CJ model in exceptional agreement with the full Vlasov–Maxwell theory.

A point about the CJ model that has been omitted so far (while it deserves some attention) is that the quasi-neutrality conditions (2.3) are not preserved in time exactly, as it can be verified by a direct calculation. More specifically, one can ask if the electron velocity as computed from the first moment of (3.1) is compatible with the corresponding expression arising from (2.3). In order to provide the answer to this question, we use (2.3) and the first moment equation of (2.2) to rewrite (1.8) as

(4.3) $$\begin{eqnarray}\boldsymbol{E}=-\boldsymbol{V}_{i}\times \boldsymbol{B}+\left(1+\frac{Z\bar{\unicode[STIX]{x1D707}}}{1-Z\bar{\unicode[STIX]{x1D707}}}\right)\left(\frac{1}{en_{e}}\,\boldsymbol{J}\times \boldsymbol{B}-\frac{1}{en_{e}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\mathbb{P}}_{e}-\frac{m_{e}}{e}\frac{\text{D}\boldsymbol{V}_{e}}{\text{D}t}\right).\end{eqnarray}$$

Then, we use (4.3) as the CJ closure equation for the electric field in (3.1). Taking the first moment (here denoted by $\boldsymbol{K}_{e}$ ) of the resulting electron kinetic equation yields

(4.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)+\boldsymbol{V}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\boldsymbol{V}_{e\,}\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\boldsymbol{V}_{e}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad =\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)\times \left(\unicode[STIX]{x1D74E}_{e}-\frac{e}{m_{e}}\boldsymbol{B}\right)+\frac{Z\bar{\unicode[STIX]{x1D707}}}{1-Z\bar{\unicode[STIX]{x1D707}}}\left(\frac{1}{m_{e}}\,\boldsymbol{J}\times \boldsymbol{B}-\frac{1}{m_{e}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\widetilde{\mathbb{P}}_{e}-n_{e}\frac{\text{D}\boldsymbol{V}_{e}}{\text{D}t}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where we recall that $\unicode[STIX]{x1D74E}_{e}=\unicode[STIX]{x1D735}\times \boldsymbol{V}_{e}$ is the electron vorticity and $(n_{e},\boldsymbol{V}_{e})$ are expressed by using (2.3). Then, we conclude that the consistency relation $\boldsymbol{K}_{e}=n_{e}\boldsymbol{V}_{e}$ fails to be preserved in time. This is a fundamental consistency issue that is intrinsic to the model and can lead to different drawbacks beyond the linear analysis of the Weibel instability. For example, charge conservation is dramatically affected, thereby preventing neutrality from being satisfied at all times. In the next section, we show how this issue is solved by simply retaining all terms in the exact Ohm’s law (1.7).

4.2 The neutral Vlasov model

Recently, upon retaining all terms in the exact Ohm’s law (1.7), we showed (Tronci & Camporeale Reference Tronci and Camporeale2015) how the quasi-neutral limit can be consistently implemented in the Vlasov–Maxwell system both directly (by formally letting $\unicode[STIX]{x1D716}_{0}\rightarrow 0$ ) and in Hamilton’s variational principle: a comparison with the full Vlasov–Maxwell system was presented and good agreement was found in the linear case for both Alfvèn and Whistler modes at different angles of propagation. Later, Burby (Reference Burby2015) showed how this model also possesses a Hamiltonian structure, while Degond, Deluzet & Doyen (Reference Degond, Deluzet and Doyen2017) provided an alternative mathematical footing by exploiting scaling and asymptotic techniques in the case of one particle species (by preventing ion motion). In the same work, the question of the numerical implementation was also discussed. As it was presented in Tronci & Camporeale (Reference Tronci and Camporeale2015), the quasi-neutral model consists of the kinetic equations (2.2) and (3.1), the quasi-neutrality conditions (2.3), Faraday’s law and Ohm’s law (1.1). Here, the electron velocity is expressed in terms of the ion velocity by using Ampère’s law.

A question that emerged at the end of § 4.1 concerned the possibility of consistency issues precisely with the second equation in (2.3). Again, one asks if the electron velocity as computed from the first moment of (3.1) is compatible with the corresponding expression arising from (2.3). A positive answer can be found by following a similar procedure as in the previous section. First, one replaces (1.5) in (3.1) and then one takes the first moment of the resulting kinetic equation. As a result, one obtains

(4.5) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)+\boldsymbol{V}_{e}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\boldsymbol{V}_{e\,}\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\boldsymbol{V}_{e}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad =\left(\boldsymbol{K}_{e}-n_{e}\boldsymbol{V}_{e}\right)\times \left(\unicode[STIX]{x1D74E}_{e}-\frac{e}{m_{e}}\boldsymbol{B}\right),\end{eqnarray}$$

where we recall that $\unicode[STIX]{x1D74E}_{e}=\unicode[STIX]{x1D735}\times \boldsymbol{V}_{e}$ is the electron vorticity and $(n_{e},\boldsymbol{V}_{e})$ are expressed by using (2.3). Then, we conclude that if $\boldsymbol{K}_{e}=n_{e}\boldsymbol{V}_{e}$ is verified initially, then it stays so at all times.

As a further remark on the neutral Vlasov model, we notice that Ohm’s law (1.1) is not suitable for the numerical implementation, since the electron mean-flow inertia produces an explicit time derivative in the expression of the electric field. In his thesis, Burby expanded (1.1) by using (2.3) to obtain (1.7) in the form

(4.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(1+\frac{Zm_{e}}{m_{i}}\right)\boldsymbol{E}+\frac{m_{e}}{\unicode[STIX]{x1D707}_{0}e^{2}n_{e}}\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D735}\times \boldsymbol{E}=-\left(1+\frac{Zm_{e}}{m_{i}}\right)\boldsymbol{V}_{i}\times \boldsymbol{B}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad +\,\frac{1}{en_{e}}\left[\boldsymbol{J}\times \boldsymbol{B}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\widetilde{\mathbb{P}}_{e}-\frac{Zm_{e}}{m_{i}}\widetilde{\mathbb{P}}_{i}\right)\right]+\frac{m_{e}}{e^{2}n_{e}}\left[\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\boldsymbol{V}_{i}\boldsymbol{J}+\boldsymbol{J}\boldsymbol{V}_{i}-\frac{\boldsymbol{J}\boldsymbol{J}}{en_{e}}\right)\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $\boldsymbol{J}=\unicode[STIX]{x1D707}_{0}^{-1}\unicode[STIX]{x1D735}\times \boldsymbol{B}$ . While on one hand this eliminates the time derivative in the closure for the electric field, one the other hand it involves inverting the operator $\unicode[STIX]{x1D713}+\operatorname{curl}^{2}$ for some function $\unicode[STIX]{x1D713}(\boldsymbol{x})$ . (Here we shall not dwell upon the question of the numerical costs involved in inverting this operator). We remark that here we did not set $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{E}$ to zero, as there is absolutely no reason for this to hold: indeed, quasi-neutrality is obtained by letting $\unicode[STIX]{x1D716}_{0}\rightarrow 0$ in Gauss’ law, while no hypothesis is made on $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{E}$ .

The linear stability of the neutral Vlasov model can be easily studied since, as already noticed in Tronci & Camporeale (Reference Tronci and Camporeale2015) it suffices to take the limit $\unicode[STIX]{x1D716}_{0}\rightarrow 0$ in the standard dispersion relation for the Vlasov–Maxwell system. Then, for example, the case of the Weibel instability can be studied by simply discarding the term $\unicode[STIX]{x1D714}^{2}/\unicode[STIX]{x1D714}_{pe}^{2}$ in (2.11) so that

(4.7) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}\left[1+W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)\right].\end{eqnarray}$$

In the next section we show the results obtained with the neutral Vlasov model for both the electron and the ion Weibel instabilities.

Figure 6. Growth rate for the ion Weibel instability for four values of temperature anisotropy $T_{\bot }/T_{||}=2,5,10,100$ . The blue circles are the reference solutions derived from the Vlasov–Maxwell model, the red lines are the solution of the CJ model, the blue lines are for the neutral Vlasov model.

4.3 Discussion on quasi-neutral kinetic models

This section compares the dispersion relations for the ion and electron Weibel instability derived from the CJ and the neutral Vlasov model. As it was done in previous sections, the case of the electron Weibel instability is studied by adopting a cold-fluid closure for ion kinetics. Thus, upon setting $T_{\bot }^{(i)}=T_{\Vert }^{(i)}$ and $W(\unicode[STIX]{x1D714}/kv_{i\Vert })\simeq 0$ , equations (4.2) and (4.7) become

(4.8) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}\end{eqnarray}$$

and

(4.9) $$\begin{eqnarray}1+\frac{T_{\bot }^{(e)}}{T_{\Vert }^{(e)}}\,W\left(\frac{\unicode[STIX]{x1D714}}{kv_{e\Vert }}\right)=-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}},\end{eqnarray}$$

respectively. We notice how the electron kinetics completely decouples from the ions in (4.8), while a minor coupling persists in (4.9). Similarly, the case of the ion Weibel instability can now be studied by adopting a cold-fluid closure for electron kinetics (not available in previous sections, which instead adopted an electron warm-fluid closure for the ion Weibel instability). In this case, setting $T_{\bot }^{(e)}=T_{\Vert }^{(e)}$ and $W(\unicode[STIX]{x1D714}/kv_{e\Vert })\simeq 0$ in (4.2) and (4.7) leads to

(4.10) $$\begin{eqnarray}Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)=-1-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}\end{eqnarray}$$

and

(4.11) $$\begin{eqnarray}Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}}W\left(\frac{\unicode[STIX]{x1D714}}{kv_{i\Vert }}\right)=-1-k^{2}\unicode[STIX]{x1D6FF}_{e}^{2}-Z\bar{\unicode[STIX]{x1D707}}\frac{T_{\bot }^{(i)}}{T_{\Vert }^{(i)}},\end{eqnarray}$$

respectively. We notice that certain differences between the two models may be appreciated in this case only for unusual anisotropy values of the order of $T_{\bot }^{(i)}/T_{\Vert }^{(i)}\simeq \bar{\unicode[STIX]{x1D707}}^{-1}$ .

The dispersion relations (4.10)–(4.11) and (4.8)–(4.9) for the ion and electron Weibel instability derived from the CJ and the neutral Vlasov model are presented in figures 6 and 7, respectively. The results for the electron instability are indistinguishable from the VM solutions (obtained upon suitably specializing (2.11) for the case of electron and ion Weibel instabilities). Neutral Vlasov solutions are not shown as they also overlap with the VM solutions. For what concerns the ion instability a very small discrepancy can be noticed between the VM and the CJ, with the latter typically overestimating the growth/damping rates. However, the values are very small.

Figure 7. Growth rate for the electron Weibel instability for four values of temperature anisotropy $T_{\bot }/T_{||}=2,5,10,100$ . The blue circles are the reference solutions derived from the Vlasov–Maxwell model, while the red lines are the solution of the CJ model. The neutral Vlasov solution coincides with both the VM and CJ solutions.