2. Derivation of the counter-differential rotation equilibria

Figure 1 depicts a solution of the counter-differential rotation equilibria. An infinitely long lithium-ion (Li$^+$) plasma column contains an infinitely long $e^-$ plasma confined radially through $\boldsymbol {B}_z=B_0\hat {z}$, where $\hat {z}$ is the unit vector. The origin of the cylindrical coordinate system ($r$, $\theta$, $z$) is located at the midplane of the coaxial plasmas, and the $z$-axis is selected to be parallel to $\boldsymbol {B}_z$. Both Li$^+$ and $e^-$ plasmas have corresponding thermal equilibria and finite pressure $p_\sigma (r)$. The fluid velocity $\boldsymbol {v}_\sigma$ is assumed to be non-relativistic. Moreover, the plasma current $e(n_i\boldsymbol {v}_i-n_e\boldsymbol {v}_e)$ is insufficient to change $B_z$ because of the low $n_i$ and $n_e$. One of the possible states likely to exist is the rigid-rotation equilibrium of the two-fluid plasma in which both pure Li$^+$ and $e^-$ plasmas can be independently relaxed into their corresponding thermal equilibria. Thus, the $\omega _{r\sigma }$ values are constant. In this case, the counter differential rotation equilibrium can be derived as follows.

Figure 1. Illustration of the differential rigid-rotation equilibrium of a two-component (two-fluid) plasma model with finite $T_\sigma$. To find solutions in the realistic case of laboratory plasmas, we refer to the beam experiment upgrade (BX-U) linear trap experiment (Himura Reference Himura2016; Akaike & Himura Reference Akaike and Himura2018, Reference Akaike and Himura2019; Yamada et al. Reference Yamada, Himura, Kato, Okada, Sanpei and Masamune2018; Kato et al. Reference Kato, Himura, Sowa and Sanpei2019), where Li$^+$ and $e^-$ plasmas constitute the two-component plasma.

Because each plasma rotates as a rigid body around the $z$-axis, the $\theta$ component of $\boldsymbol {v}_\sigma$ ($v_{\theta \sigma }$) is proportional to $r$ and therefore, $v_{\sigma \theta }=\omega _{r\sigma } r$. The term $\boldsymbol {\nabla } p_\sigma$ is equivalent to $k_BT_\sigma \boldsymbol {\nabla } n_\sigma$ because $T_\sigma$ is spatially uniform at thermal equilibrium. Hence, the equation of steady-state motion for both plasmas can be expressed as $m_\sigma n_\sigma (\boldsymbol {v}_\sigma \boldsymbol {\cdot } \boldsymbol {\nabla })\boldsymbol {v}_\sigma = n_\sigma q_\sigma (\boldsymbol {v}_\sigma \times \boldsymbol {B}_z-\boldsymbol {\nabla } \phi _s)- k_B T_\sigma \boldsymbol {\nabla } n_\sigma$, where $m_\sigma$ and $q_\sigma$ represent the mass number and elementary charge of each species, respectively. Solving this equation for $n_\sigma (r)$,

(2.1)\begin{equation} n_\sigma(r)=n_{\sigma0}\exp\left(-\frac{\psi_{\sigma}}{k_BT_\sigma}\right), \quad (\sigma=i,e),\end{equation}

where

(2.2)\begin{equation} \psi_\sigma(r)\equiv q_\sigma\phi_s(r)-\tfrac{1}{2}m_\sigma r^2(\textrm{sgn}(q_\sigma)\omega_{c \sigma}\omega_{r\sigma}+\omega_{r\sigma}^2).\end{equation}

The coefficient $n_{\sigma 0}$ on the right-hand side of (2.1) represents the value of $n_\sigma$ on the $z$-axis, where $r = 0$. In addition, $\psi _\sigma$ are the corresponding effective potential energies (Davidson Reference Davidson2001) of the singly ionized ions and $e^-$ plasmas. Substituting them in Poisson's equation, the rotation equilibrium equation with finite $T_\sigma$ can be expressed as

(2.3)\begin{equation} \frac{1}{r}\frac{\textrm{d}}{\textrm{d} r}\left(r\frac{\textrm{d}}{\textrm{d} r}\phi_s\right)= \sum_{\sigma=i,e}-\frac{q_\sigma }{\epsilon_0}n_{\sigma0}\exp{\left(-\frac{\psi_\sigma}{k_BT_\sigma}\right)}.\end{equation}

To numerically determine the solutions of (2.3), we apply the measured values in the beam experiment upgrade (BX-U) linear trap experiments (Himura Reference Himura2016; Akaike & Himura Reference Akaike and Himura2018, Reference Akaike and Himura2019; Yamada et al. Reference Yamada, Himura, Kato, Okada, Sanpei and Masamune2018; Kato et al. Reference Kato, Himura, Sowa and Sanpei2019), as examples for the calculation. The boundary condition of $\phi _s$ is the same as that of the BX-U as well, as listed in table 1. Although the value of $B_0$ is variable, it is fixed to 0.13 T in the presented calculation. Lithium (Li$^+$) is employed as the singly ionized ion. The value of $n_{i0}$ can be varied in the $10^{11}\text{--}10^{12}$ m$^{-3}$ range, whereas $n_{e0}$ is in the $10^{12}\text{--}10^{13}$ m$^{-3}$ range. For $T_\sigma$, we assume $T_i = T_e = 2$ eV because the confinement time is considerably greater than the binary collision time. This observation implies two-fluid rotational equilibrium. To determine solutions within the $n_i$ and $n_e$ ranges in table 1, the coefficients of $n_{i0}$ and $n_{e0}$ are set to $1\times 10^{11}$ and $5\times 10^{12}$ m$^{-3}$, respectively. Thus, $n_{i0}/n_{e0} = 0.02$. Under these conditions, the Gauss–Seidel method was employed to solve (2.3). Values of $\omega _{ri}$ and $\omega _{re}$ are also computational parameters. First, we obtain $\phi _s (r)$ from (2.2) and (2.3) by substituting independent values into $\omega _{ri}$ and $\omega _{re}$ one by one. Then, the obtained $\phi _s (r)$ is utilized to calculate the corresponding $n_\sigma (r)$ from (2.1). Using these numerical schemes, we systematically find self-consistent sets of solutions of $\psi _s$, $n_i$ and $n_e$ that satisfy (2.1)–(2.3) simultaneously even with finite $T_i$ and $T_e$, as shown below.

Table 1. Nominal parameters of the BX-U machine and assumed boundary condition for $\phi _s$ in this calculation.

$^{\textrm {a}}$All the collision times are calculated using the values of $n_i$ and $n_e$ listed above.

3. Possible $\omega _{re}$ and $\omega _{ri}$ with which counter-differential rigid-rotation equilibria exist

Figure 2 shows the dependency of $\omega _{r\sigma }$ on $n_{i0}/n_{e0}$, where $\omega _{r\sigma }$ is normalized by the cyclotron frequency $\omega _{c\sigma }$. For the three cases where $n_{i0}/n_{e0} = 0.02$, 0.5 and 0.9, the possible ranges of $\omega _{re}$ and $\omega _{ri}$ in which rigid-rotation equilibria of the two-fluid plasma exist are denoted by the six solid-line sections, where the red and blue colours represent $\omega _{re}$ and $\omega _{ri}$, respectively.Footnote ¹ For the reader's understanding, it should be noted that the value of $\omega _{r\sigma }$ of a single-component plasma such as pure $e^-$ plasma must be either $\omega _{r\sigma }^+$ (fast mode) or $\omega _{r\sigma }^-$ (slow mode) if $T_\sigma$ is zero. For $T_\sigma \neq 0$, $\omega _{r\sigma }$ of a single-component plasma can take any value between $\omega _{r\sigma }^+$ and $\omega _{r\sigma }^-$. However, for two-fluid plasmas with finite $T_\sigma$, the possible ranges of both $\omega _{re}$ and $\omega _{ri}$ are limited. This is noticeable for $\omega _{ri}$, as depicted in figure 2. The sign of $\omega _{re}$ is always positive. On the other hand, the sign of $\omega _{ri}$ is always negative, contrary to the case of one-component pure ion plasmas. The different signs of $\omega _{re}$ and $\omega _{ri}$ physically imply that the Li$^+$ and $e^-$ plasmas rigid-rotate in opposite directions. As previously mentioned, $B_z$ is along the positive direction of the $z$-axis, whereas $E_r (=-\nabla _r \phi _s)$ is from the plasma edge toward the plasma axis, inward. This can be deduced from the fact that $n_{e0}>n_{i0}$. Overall, it is recognized that the $e^-$ plasma rotates in the direction of $E_r \times B_z$, whereas the Li$^+$ plasma counter-rotates in the opposite direction of $E_r \times B_z$. Because $\omega _{ri} \neq \omega _{re}$, this can be considered as the counter-differential rotation equilibrium of two-fluid plasmas. As example solutions, we present extraordinary cases. When $\omega _{re}/\omega _{ce}$ takes a minimum value of $1.6\times 10^{-4}$, $\omega _{ri}/\omega _{ci}$ can take any value in the $-0.97<\omega _{ri}/\omega _{ci}<-0.05$ range. Such arbitrariness is provided by the fact that changes in the profiles of $n_\sigma (r)$ and $\phi _s (r)$ occur self-consistently to satisfy (2.1)–(2.3).

Figure 2. Dependency of $\omega _{r\sigma }$ on $n_{i0}/n_{e0}$. Here $\omega _{r\sigma }$ is normalized by the corresponding cyclotron frequency $\omega _{c\sigma }$. The dashed (red) curve shows the possible solutions ($\omega _{ri}^+$ and $\omega _{ri}^-$) for a two-component plasma with $T_i = T_e = 0$ eV, whereas the dashed (blue) lines indicate the possible solutions ($\omega _{re}^+$ and $\omega _{re}^-$) for a two-component plasma with $T_i = T_e = 0$. $T_e = 0$ eV. The values of $n_{e0}$ are set to $5\times 10^{12}$ m$^{-3}$. As can be observed, for a two-component plasma with finite $T_\sigma$, the possible ranges of $\omega _\sigma$ are limited. These are denoted by the corresponding solid-line sections, where the blue colour represents $e^-$ plasma and the red represents Li$^+$ plasma.

4. The finite temperature effect

The counter-rotation of Li$^+$ plasma at rigid-rotor equilibrium is attributed to the finite $p_\sigma$. Figure 3 shows the radial profiles of the azimuthal components of $\boldsymbol {E\times B} \, (\equiv v_\phi = (1/B_0)\, \textrm {d}\phi _s/ \textrm {d} r)$ and the diamagnetic ($\equiv v_{d\sigma } = -(k_B T_\sigma /n_\sigma q_\sigma B_0)\, \textrm {d} n_\sigma / \textrm {d} r$) drift terms along with $v_{\sigma }$. These are calculated from a typical set of equilibrium solutions of $\phi _s(r)$ and $n_\sigma (r)$, as depicted in figure 4. Figure 3 shows that the sign of $v_\phi$ is positive along the entire $r$-axis. However, $|v_\phi |$ is one order of magnitude smaller than the absolute value of $v_{di}$, which is negative in the entire plasma, causing counter-rotation. Here, we note that $v_\phi$ and $v_{di}$ change nonlinearly along the $r$-axis, which can be clearly recognized in the inset of figure 3. However, $v_{i}$, composed of $v_\phi$ and $v_{di}$, increases linearly along the $r$-axis, resulting in rigid-body rotation.

Figure 3. Radial profiles of the azimuthal components of $v_\phi$ (black dashed curves), $v_{d\sigma }$ (red dotted curves for Li$^+$ plasma and blue for $e^-$ plasma) and $v_\sigma$ (two solid red and blue lines) for a typical set of equilibrium solutions obtained for the case where $\omega _{ri} = -3.3\times 10^{5}$ and $\omega _{re} = 3.6\times 10^{6}$ rad s$^{-1}$. Both Li$^+$ and $e^-$ exhibit counter-differential rigid-rotation equilibrium.

Figure 4. Radial profiles of $\phi _s(r)$ and $n_\sigma (r)$ for the set of equilibrium solutions shown in figure 3. Here $r_i$ and $r_e$ are never equal but are different when the two-component (two-fluid) plasma is in counter-differential rigid-rotation equilibrium. In addition, the lengths of $r_i$ and $r_e$ in the two-fluid equilibrium are smaller than those calculated for the pure Li$^+$ and $e^-$ plasmas.

The same linearization occurs for $v_{e}$ as well, as depicted in figure 3. Counter-differential rigid-rotation equilibrium is caused by the balance between $p_\sigma$ and $\phi _s$ perpendicular to $B_z$, which is qualitatively similar to the study of non-uniform $p_\sigma$ and $\phi _s$ on toroidal magnetic surfaces (Pedersen & Boozer Reference Pedersen and Boozer2002; Himura et al. Reference Himura, Wakabayashi, Yamamoto, Isobe, Okamura, Matsuoka, Sanpei and Masamune2007).

At counter-differential rotation equilibrium, $n_\sigma (r)$ assumes the corresponding bell-shaped profile, which is qualitatively the same as that in the cold plasma case. However, because of finite $T_\sigma$, the pressure-gradient terms ($k_BT_\sigma \boldsymbol {\nabla } n_\sigma$) play dominant roles in maintaining the corresponding rotational equilibria, as mentioned above. In figure 4, the value of $n_{i}(0)$ is approximately $1.4\times 10^{11}$ m$^{-3}$, which is greater than $n_{i0}$, whereas $n_{e}(0) \approx 3.5\times 10^{12}$ m$^{-3}$ is smaller than $n_{e0}$. The difference between $n_i$ and $n_e$ indicates that the two-fluid plasma is electrically non-neutral. Because $n_{i}(0) < n_{e}(0)$, $\phi _s$ becomes negative at $r =0$. However, in addition to the plasma axis, the negative $\phi _s$ extends over the entire plasma, regardless of $n_i$. The minimum value of $\phi _s$ is at $r = 0$, which is approximately $-0.7$ V in this case. The curvature of $\phi _s(r)$ becomes convex toward the top, as observable in figure 4.

For $n_\sigma (r)$, their maxima appear at the plasma centre ($r=0$) and decrease monotonically, consistent with $\phi _s(r)$. However, the remarkable result inferred from the profiles of $n_i(r)$ and $n_e(r)$ is that $r_i$ and $r_e$ never become equal and always remain different. Defining $r_\sigma$ as the distance between the plasma centre and the coordinate, where $n_\sigma$ decreases to 1/10 of $n_\sigma (0)$ (i.e. $n_\sigma (r_\sigma ) / n_\sigma (0) = 1/10$), $r_i$ and $r_e$ are approximately 2.5 and 0.5 cm, respectively, in the presented case.

The obtained bell-shaped profiles shown in figure 4 may be due to the finite-temperature effect (Davidson & Krall Reference Davidson and Krall1969) to some extent. However, the past study assumed that the Debye length $\lambda$ was sufficiently short compared with the plasma radius $r_p$. This assumption implied that either $n_e$ was relatively high or $r_p$ was relatively long. Contrary to these, the present result is obtained from a different parameter regime in which $n_\sigma$ is relatively lower and $T_\sigma$ is finite. As a result, $\lambda$ has the same order as that of $r_p$.

5. On the radii of single-component and two-fluid plasmas

The lengths of $r_\sigma$ reduce when the pure ion as well as $e^-$ plasmas with finite $T_\sigma$ are in counter-differential rotation equilibrium together. Substituting the values of $n_{\sigma }(0)$ in figure 4 in Davidson's formulaFootnote ² derived for a single-component plasma, $r_i$ and $r_e$ are expected to be approximately 6 and 1 cm, respectively. Here, $r_\sigma \approx -\{{\sqrt {k_BT_\sigma /m_\sigma }}/{\omega _{p\sigma }}\} \ln {[({{2(\omega _{r\sigma }\omega _{c\sigma }-\omega _{r\sigma }^2)}/{\omega _{p\sigma }^2}})-1]}$. This discrepancy is caused by the increase in $\psi _i$ of the two-fluid plasma. When a single-component ion plasma is in rotational equilibrium, $\phi _s$ is estimated to be of the order of $er_i^2n_{i0}/\epsilon _0$. In addition, $n_{i0}$ must always be smaller than the Brillouin density (Davidson Reference Davidson2001) such that $\omega _{ri}^-\approx -\omega _{pi}^2/2\omega _{ci}$, in the case where $\omega _{ri}^-\ll \omega _{ci}$. Substituting these in (2.2), we estimate $\psi _i^0$ of the single-component ion plasma as

(5.1)\begin{equation} \psi_i^0 = e\phi_s - \frac{1}{2}m_ir^2\{\omega_{ci}\omega_{ri}^-{-}(\omega_{ri}^{-})^2\}\approx \frac{5}{4}\frac{e^2n_{i0}}{\epsilon_0}r^2_i. \end{equation}

Here, we used the following relationship: $m_ir_i^2\omega _{pi}^2=e^2r^2_in_{i0}/\epsilon _0$. However, in the case of a two-component plasma with $0< n_{i0}< n_{e0}$, $\phi _s\approx er_i^2n_e(0)/\epsilon _0$ and $\omega _{ri}^-\approx (n_{e0}/n_{i0})\omega _{pi}^2/2\omega _{ci}$. Thus, for the counter-differential rotation equilibrium example shown in figures 3 and 4, $\omega _{ri} \approx -(n_{e0}/n_{i0})\omega _{pi}^2/2\omega _{ci}$ because $\omega _{ri}=-0.19\times \omega _{ci}=-1.1\times \omega _{ri}^-$. Therefore, $\psi _i$ of the two-component plasma is derived as

(5.2)\begin{equation} \psi_i \approx \frac{5}{4}\frac{e^2n_{e0}}{\epsilon_0}r^2_i=\frac{n_{e0}}{n_{i0}}\psi_i^0 \quad (>\psi_i^0). \end{equation}

According to (2.1), an increase in $\psi _i$ causes a rapid decrease in $n_i$ as $|r|$ increases, resulting in a narrower $n_i(r)$ as seen in figure 4. In general, for $0< n_{i0}< n_{e0}$, $r_i$ of a two-component plasma becomes approximately $\sqrt {n_{i0}/n_{e0}}$ times smaller than that of a single-component ion plasma.

The shorter $r_e$ is also explained by the increase in $\psi _e$. In the equilibrium depicted in figures 3 and 4, $\omega _{re}/\omega _{ce}$ is $\sim 10^{-4}$, which is an order of magnitude greater than the slow mode: $\omega _{re}^-/\omega _{ce}\sim 10^{-5}$. In addition, $(1-n_{i0}/n_{e0})\approx 1$ in this case. Thus, $\phi _s$ is almost the same for the pure $e^-$ as well as two-component plasma. We compare the two effective potentials of the two cases. The effective potential of the pure $e^-$ plasma is $\psi _e^0$. Substituting these in (2.1) and (2.2), we estimate $\psi _e$ and $\psi _e^0$ as

(5.3)\begin{equation} \psi_e \approx \frac{\omega_{re}}{\omega_{re}^-}\psi_e^0>\psi_e^0. \end{equation}

Based on these considerations, it is concluded that the two-component plasma becomes narrower overall.

6. Summary

In summary, the 2-D rigid-rotation equilibria of electrically non-neutral two-component (two-fluid) plasma with finite $T_\sigma$ were presented in this study, for the first time. Furthermore, self-consistent solutions of the differential rigid-rotation equilibria were determined. However, the possible range of $\omega _\sigma$ becomes narrower than that of the two-component plasma with $T_\sigma = 0$. Remarkably, in contrast to the cold plasma case, the ion plasma is only permitted to counter-rotate because of its diamagnetic drift. In the future, we intend to investigate the following. In this study, three cases of $e^-$ rich plasmas ($n_{i0}/n_{e0}= 0.02$, 0.5, and 0.9) were presented to straightaway show the existence of counter differential rigid-rotation equilibria. A complete set of possible ranges of $\omega _\sigma$ for different values of $n_{i0}/n_{e0}$ will be considered. Cases with $T_i\neq T_e$ will be investigated as well. Moreover, in the BX-U experiment, there is no constraint that the two-fluid plasma must rotate rigidly. A more general solution would be to use $\omega _i (r)$ and $\omega _e (r)$. In fact, the axial length of actual plasmas is finite so that three-dimensional computations are suitable for comparison between experiments and simulations.

Finally, in the case of a small fraction of positive ions in an otherwise pure electron plasma, the ion resonance instability has been observed to emerge not only theoretically (Levy, Daugherty & Buneman Reference Levy, Daugherty and Buneman1969) but also experimentally (Marksteiner et al. Reference Marksteiner, Pedersen, Berkery, Hahn, Mendez, de Gevigney and Himura2008). Therefore, a stability analysis would be required for the counter differential rigid-rotation equilibrium with a minimal value of $f$. Since $\lambda \approx r_\sigma$ in the presented parameter regime, collective plasma effects are not expected to be significant. Perhaps, such an instability might not grow as much.