2. Theoretical consideration

The main difference between regular and moving multiple mirrors is non-zero axial velocity of the trapped particles. In the case of the helical corrugation this velocity is estimated as $v_z \approx \omega _{E\times B}h/2{\rm \pi}$, where $\omega _{E\times B}$ is the angular velocity and $h$ is the corrugation period. In the general case, this velocity can be chosen arbitrary. We assume later that the axial velocity of the trapped ions is of the order of the ion thermal velocity, $v_z \sim v_i$ (if $v_z\ll v_i$, the force pumping the plasma seems to be weak, in case $v_z\gg v_i$ it is too difficult to populate trapped ions). In our experiments axial velocity is about twice as high as the thermal velocity. Such non-equilibrium distribution function can be the source of free energy for instabilities.

The possibility of anomalous scattering depends on the energy transfer to the transiting ions. Building up anomalous scattering due to the intrinsic processes in the plasma requires the following energy transfer. External fields raise the kinetic energy of the trapped ions. They transfer the energy to the electrostatic or electromagnetic waves. The turbulent energy cascade forms. Chaotic electric fields scatter the transiting ions. Turning off the energy transfer from the trapped particles to the wave and from the wave to the transiting particles results in impossibility to build up anomalous scattering by the energy of the non-equilibrium distribution function. Here we focus mostly on the energy transfer from the trapped ions to the wave.

Let us consider now properties of the ion trajectories in the helical mirror and properties of the unstable wave. When ion motion is collisionless, the trajectories can be divided into trapped (axial velocity is close to the velocity of the helical corrugation in the rotating frame, the ion moves near the minimum of the magnetic field) and transiting. The guiding centre of the trapped ion moves along the banana-like trajectory on the plane $r \cos (\theta -2{\rm \pi} z/h)$, $r \sin (\theta -2{\rm \pi} z/h)$, where $r$, $\theta$ and $z$ are the cylindrical coordinates. Thus, the trajectory of the trapped ion coincides approximately with the helical line $r={\rm const.}$, $\theta =2{\rm \pi} z/h+{\rm const}$. This line lies on the magnetic surface and coincides with the line of the minimal magnetic field magnitude.

The unstable wave can interact effectively with the trapped ions if the phase of the wave is constant along the trajectory of the trapped ion. If the helical corrugation is weak and the plasma is approximately cylindrical, the electric field of the wave can be approximately written in the form

(2.1)\begin{equation} \boldsymbol{E}(r)=\boldsymbol{E}(r) \exp ({\rm i}k_{\|}z+{\rm i}m\theta-{\rm i}\omega t). \end{equation}

The wave phase along the trapped ion trajectory is approximately $(k_{\|}v_z+2{\rm \pi} mv_z/h-\omega )t$. The phase is constant when the wave frequency satisfies the condition

(2.2)\begin{equation} \omega \approx (k_{\|}+2{\rm \pi} m/h)v_z \approx (k_{\|}h/2{\rm \pi} +m)\omega_{E\times B}. \end{equation}

The wave frequency measured in the experiment is relatively low, $\omega \ll \omega _{E\times B}$. There are two possibilities. First, the wave can be axisymmetric, $m=0$, and the wavelength is large (of the order of length of helical mirror $L$)

(2.3)\begin{equation} k_{\|}v_z \approx (k_{\|} h/2{\rm \pi})\omega_{E\times B} \ll \omega_{E\times B}. \end{equation}

If the axial electric field component of the wave is not equal to zero and the wave period is of the order of the ion transit time $L/vt$, the electric field changes slowly on most of the ion trajectories. Then the wave can decelerate outflow ions and increase its own amplitude.

On the other hand, the condition $\omega \ll \omega _{E\times B}$ is possible if $m \neq 0$ and the wavelength is several times less than the helical pitch,

(2.4)\begin{equation} k_{\|} \approx-(2{\rm \pi}/h)m. \end{equation}

In this case, the wave frequency is approximately $\omega _{E\times B}$ and the axial phase velocity is $\omega _{E\times B}/k_{\|} \approx mv_z \sim mv_i$ in a frame of reference rotating with the frequency $\omega _{E\times B}$. In the case $m=1$ the phase velocity is of the order of the thermal velocity of ions. Thus, the wave electric field is approximately constant along the ion trajectories and such a wave also can decelerate ions.

In a chaotic electric field the transiting ion receives random additions to its transverse velocity

(2.5)\begin{equation} \delta v_{{\perp}} \sim\frac{q}{m_{\rm i}} \delta E \tau, \end{equation}

where $q$ and $m_{\rm i}$ are the ion charge and mass, $\delta E$ is the characteristic value of the electric field and $\tau \sim \delta /v_{\rm Ti}$ is the time of the flight through the region of the invariable field. The ion experiences $N \sim h / \delta$ small-angle scatterings during passing through the distance of a corrugation period. Then its pitch angle should be changed by $\Delta \theta \sim 1$:

(2.6)\begin{equation} \sqrt{\frac{h}{\delta}}\frac{\delta v_{{\perp}}}{v_{\rm Ti}} \sim \frac{q}{m_{\rm i}}\frac{\sqrt{\delta h}}{v_{\rm Ti}^2}\delta E \sim 1. \end{equation}

It can be satisfied if the mean electric field is

(2.7)\begin{equation} \delta E \sim \frac{m v_{\rm Ti}^2}{q} \frac{1}{\sqrt{\delta h}}. \end{equation}

Simple estimation of the length of the invariable field region can be based on the wavelength obtained in (2.4), $\delta \lesssim h/(2{\rm \pi} m)$. Helix pitch length equals $18\ {\rm cm}$ for the SMOLA device. Assuming $m = 1$, we obtain the characteristic electric field $\delta E \gtrsim 1\ {\rm V}\ {\rm cm}^{-1}$ in the SMOLA case.

3. Experimental set-up and parameters

The layout of the SMOLA helical mirror is presented in figure 1. The device was built for studies of a low-temperature hydrogen plasma flow through a 2.5-m-long transport section with the helically symmetric magnetic field. Plasma was generated in a source with a magnetically insulated heated LaB$_6$ cathode (Ivanov et al. Reference Ivanov, Ustyuzhanin, Sudnikov and Inzhevatkina2021). Then the plasma was injected into a compact mirror trap in the entrance tank. Hereinafter this is referred to as the confinement region. In the discussed experiments, these trap mirrors were asymmetric with mirror ratios $R \approx 8$ on the plasma source side and $R \approx 3$ on the helical mirror side. Plasma flows in the axial direction from the confinement region to the transport section. This section has two independent magnetic systems, a solenoid for a straight field and a bispiral helical winding that forms helical magnetic mirrors. The spiral has $N = 12$ corrugation periods. We denote the fieldline without the periodic variation of the magnetic field module as the magnetic axis of the transport section. This axis has a spiral shape, its radius depends on the helical to axial components of the magnetic field ratio. The last part of the device is the exit expander that contains an exit limiter and radially segmented plasma receiver endplates. The axially symmetric confinement region, the helically symmetric transport section and the axially symmetric exit expander are matched together by the special correction units. The centre of the cathode is projected by the magnetic fieldline to the magnetic axis of the transport section and to the centre of the endplates. A detailed description of the device can be found in Sudnikov et al. (Reference Sudnikov, Beklemishev, Postupaev, Burdakov, Ivanov, Vasilyeva, Kuklin and Sidorov2017).

Figure 1. Layout of the SMOLA helical mirror. Position of the main diagnostics is indicated. Probe: Set of electrostatic probes including two or more emissive probes and a double or Mach probe. D.Sp: Doppler spectroscopy. $\dot B$: 12-channel array of Mirnov coils. 38 GHz: microwave interferometer.

The distribution of the guiding magnetic field is shown in figure 2. Later in this paper, the magnetic configuration is referred to by the guiding magnetic field in the transport section; all other magnetic fields vary proportionally. Another parameter of the magnetic system is the mean corrugation ratio $R_{{\rm mean}}$, which is the ratio of the maximal and the minimal magnetic field along the fieldline within the transport section averaged over the plasma cross-section. This parameter defines the fraction of locally trapped particles in the plasma. In the discussed experiments, the magnetic field was in the range $B_z = 50\unicode{x2013}100\ \text {mT}$, and the mean corrugation ratio was $R_{{\rm mean}} = 1.52$.

Figure 2. Guiding magnetic field profiles for different radii.

The experiments described in this paper were focused on the oscillations of the electrostatic potential in the transport section with the helical magnetic field and their possibility of meeting the criteria of the anomalous scattering. The main diagnostics for these oscillations were radially movable sets of the electrostatic probes in the different positions along the transport section ($z = 2.04$ m, 2.4 m, 2.58 m, 2.94 m and 3.48 m). The sets had the following arrangement of the emissive probes. The probe sets at the coordinates 2.58 m, 2.94 m and 3.48 m consist of two radially shifted electrodes. The set at $z = 2.04$ m consist of two probe assemblies separated azimuthally by $180^{\circ }$ and located at same $r$ for discussed experiments. In both assemblies the electrodes are shifted radially. The set at $z = 2.4$ m consists of seven individual electrodes: five of them are distributed along the vertical chord and two are shifted radially and axially relative to the central electrode of the vertical chord. The distances between the neighbouring electrodes in all assemblies are 1 cm. The thoriated tungsten electrodes were heated by the plasma during the initial part of the discharge and then provided simultaneous measurements of plasma potential and radial electric field. Reaching the working temperature of the probe was verified by the thermal radiation of the wire.

Each of these sets included also a double or Mach probe. Double probes were used alternately in the ion saturation or in the $I$–$V$ characteristic regime. Plasma density in the confinement region was measured by the radially movable double probe at $z = 0.4$ m. Other parts of the diagnostic complex (imaging Doppler spectrometers, an array of Mirnov coils and a microwave interferometer) were also used to control the discharge parameters.

The plasma density between the mirrors in the confinement region was determined by the hydrogen flow rate from the gas feeding system into the plasma source. The density was in the range $n = (0.5\unicode{x2013}5.5)\times 10^{18}\ \text {m}^{-3}$ for the confinement region, the average ion temperature was $T_{\rm i} = 3.5\unicode{x2013}4.5\ \text {eV}$ and the electron temperature on the axis was $T_{\rm e} = 20\unicode{x2013}30\ \text {eV}$ (Ivanov et al. Reference Ivanov, Ustyuzhanin, Sudnikov and Inzhevatkina2021; Sudnikov et al. Reference Sudnikov, Ivanov, Inzhevatkina, Larichkin, Postupaev, Sklyarov, Tolkachev and Ustyuzhanin2022b). The density in the entrance region of the transport section was roughly half that in all regimes, giving $n = (0.2\unicode{x2013}2.5)\times 10^{18}\ \text {m}^{-3}$. The typical plasma density profiles for $B = 70\ \text {mT}$ at axial coordinates $z=0.4\ \text {m}$ and $z=2.04\ \text {m}$ are presented in figure 3(a,c). These values correspond to the mean free path of the ion in the helical field with respect to the binary collisions $\lambda = 0.3\unicode{x2013}2.5\ \text {m}$. Therefore, the ratio of the mean free path to the period of the helical corrugation was $\lambda /h \sim 1$ on the high-density bound (in most high-density cases $\lambda \approx 2h$) and $\lambda /h \sim N$ on the low-density bound. The first case meets the criterion of an efficient multiple-mirror confinement with binary collisions only, the second requires anomalous scattering for the flow suppression. The helical mirror concept requires fast rotation of plasma due to $E\times B$ drift. Plasma rotation velocity in the discussed experiments was $\omega = (1\unicode{x2013}1.1)\times 10^6\ \text {s}^{-1}$ in the entrance tank ($z = 1.15$ m). Radial electric field in the plasma (figure 3b,d) is consistent with the rotation velocity. Electric field increases roughly linearly in the inner part of the plasma column (inside the projection of the cathode of the plasma source) and decreases to zero on the plasma edge. According to the radial electric field measurements the plasma rotation velocity profiles in the transport section are identical within error bars.

Figure 3. Experimental profiles of (a) the plasma density $n$ in coordinate $z=2.04\ \text {m}$, (b) the electric field $E_r$ in coordinate $z=2.04\ \text {m}$, (c) the plasma density $n$ in coordinate $z=0.4\ \text {m}$ and (d) the electric field $E_r$ in coordinate $z=0.4\ \text {m}$.

4. Signal analysis procedure

The fluctuation properties were obtained by the Fourier-based cross-correlation function analysis. This method is widely used for the investigation of plasma fluctuation in various magnetic field geometries (Turner, Powers & Simonen Reference Turner, Powers and Simonen1977; Kim et al. Reference Kim, Edgell, Greene, Strait and Chance1999; Melnikov et al. Reference Melnikov, Eliseev, Jiménez-Gómez, Ascasibar, Hidalgo, Chmyga, Komarov, Kozachok, Krasilnikov and Khrebtov2010). It extracts the most significant spectrum components and its phase difference between spatially separated channels. According to the convolution theorem the cross-spectrum of two signals is equal to the pointwise product of the signals spectra:

(4.1)\begin{equation} \mathfrak{F}[\text{cov}(s_1,s_2)]=\mathfrak{F}[s_1]\times \mathfrak{F}[s_2]. \end{equation}

Here $\mathfrak {F}[.]$ denotes the Fourier transform, $s_i$ denotes processing signals and $\text {cov }(. , .)$ denotes the cross-covariation function. Thus, the frequencies with high amplitude in both spectra have a high one in cross-spectrum. The cross-coherence, $\gamma _{1,2}$, can be used to determine the significance of the component:

(4.2)\begin{equation} \gamma_{1,2}=\frac{|\langle\mathfrak{F}[\text{cov}(s_1,s_2)]\rangle|}{|\langle\mathfrak{F}[s_1]\rangle\|\langle\mathfrak{F}[s_2]\rangle|}. \end{equation}

Here angle brackets denote averaging over multiple signals’ samples. The argument of the product corresponds to the phase shift between two same-frequency Fourier components of the signals:

(4.3)\begin{equation} \Delta\phi_{1,2}=\text{Arg}(\mathfrak{F}[\text{cov}(s_1,s_2)]). \end{equation}

Knowing the phase shifts between two Fourier components and the distance between the diagnostics registered them one can explore spatial structure of these components. It is natural for cylindrically shaped plasma to treat fluctuations in terms of azimuthal ($k_\theta$), radial ($k_r$) and axial ($k_z$) wavenumbers. One has to use probes separated in the directions listed above to obtain these wavenumbers.

Plasma potential was measured by emissive probes. The probes formed the sets allowing $k_r$, $k_\theta$ and $k_z$ to be obtained. The arrangement of the electrodes is described in § 3. Since the probes arranged along the vertical chord are separated both by radial and azimuthal coordinates, the phase difference between any pair has both $k_r$ and $k_\theta$ contribution. Considering one of the probes as the point of reference we obtain a linear system of ordinary equations of the form $\Delta \phi _i = k_r\times r_i + m\times \Delta \theta _i$. Here $m$ is the mode number, $\Delta r_i$ and $\Delta \theta _i$ are the radial and azimuthal distances, respectively, between $i$th and base probes. This system is solvable by the least-squares method (LSM). This probe set was also rotated by $90^{\circ }$ for more precise measurement of $k_z$. The radial coordinate of each set was varied to obtain radial profiles.

5. Spatial and temporal spectra of the fluctuations

The typical experimental waveforms are shown in figure 4. Time $t=0$ corresponds to the discharge initiation. The stationary plasma discharge builds up during the first 40 ms. Stationary neutral gas distribution, which depends on the hydrogen flow rate in the plasma source, discharge current and configuration, is achieved in $\sim 80\ \text {ms}$. The emissive probe reaches working temperature in $t\leq 40\ \text {ms}$ at radial coordinates $r\leq 6\ \text {cm}$. At the outer region ($r=6\unicode{x2013}8\ \text {cm}$) the heating takes up to $t=90\ \text {ms}$. The temperature rise time is consistent with the estimations similar to (Hershkowitz et al. Reference Hershkowitz, Nelson, Pew and Gates1983). At $t=165\ \text {ms}$ the plasma source switches off to avoid damaging the probes. The parts of signals corresponding to the stationary stage of the discharge were used for analysis. The ADC sample rate (50 MS s$^{-1}$) allows us to treat the fluctuation with frequency up to 25 MHz. The ADC output of the full length signal is decimated in order to save storage space. The resulting sample rate allows us to analyse frequencies up to 390 kHz. In addition, the short samples of the steady phase signals ($t=131.1\unicode{x2013}136.3\ \text {ms}$) are recorded with original sample rate allowing us to investigate high-frequency fluctuations.

Figure 4. Typical waveforms of plasma parameters in discharges with $R_{{\rm mean}} = 1.52$, $B_z = 70\ \text {mT}$ and plasma density $n = 5.1\times 10^{18}\ \text {m}^{-3}$ at $z = 0.4\ \text {m}$ and $n = 2.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 2.04\ \text {m}$: (a) the discharge current; (b) the voltage between the anode and the cathode of the plasma source; (c) the radial electric field component at z = 2.04 m; (d) the potential of the emissive probes at z = 2.04 m and $r = 58$ and $48\ \text {mm}$. Solid lines on the panels (a–d) show moving average, shaded areas correspond to the amplitude of the observed oscillations. Panels (e–h) show waveforms of the oscillations in the short time fragment.

The oscillations of the potentials and the electric field are shown in figure 4(e–h). One can see the delay between the oscillations of the potential on the probes shifted radially. Their Fourier spectra possess peaks in the frequency range of tens of kilohertz. The high-frequency rest of the spectrum decreases as $f^{-5/3}$ (shown in figure 5).

Figure 5. Emission probe signal spectrum for discharge with $B=70\ \text {mT}$, $R_{{\rm mean}} = 1.52$ and density $n = 5.1 \times 10^{18}\ \text {m}^{-3}$ at $z=0.4\ \text {m}$ and $n = 2.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 2.04\ \text {m}$. The probe is located at $z = 2.04\ \text {m}$ and $r = 58\ \text {mm}$.

The outcome of the Fourier-based analysis is the dependencies of cross-coherence and cross-phase on time.

As an example, the cross-coherence and the cross-phase of the pair of two emission probes are presented in figure 6. The probes are located at $z = 2.04\ \text {m}$ and at $r = 28\ \text {mm}$ and azimuthaly separated by $180^{\circ }$. On the cross-coherence plot, two distinct high-coherent lines are pronouncedly visible. Cross-phase corresponding to these lines are constant during this part of discharge. The histograms are built by cross-phase sampling involving the entire investigated time period. They illustrate the statistical features of the discussed lines such as the mean value and width. Despite the fact that the spectra presented in figures 5 and 6 have two clearly visible peaks, the number of peaks can vary depending on experimental parameters. We discuss the fluctuation corresponding to the frequency about 30 kHz in figures 5 and 6 because this fluctuation was observed in wider range of experimental parameters.

Figure 6. (a) Coherence spectrogram, (b) cross-phase spectrogram and (c) cross-phase histogram of the emission probes separated by $180^{\circ }$. The plasma density $n = 5.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 0.4\ \text {m}$ and $n = 2.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 2.04\ \text {m}$, $B = 70\ \text {mT}$ and $R_{{\rm mean}} = 1.52$.

In the experiments discussed here, the frequency depends on the guiding magnetic field and the plasma density. To find out the form of this dependency, the fluctuation frequency was measured in the guiding magnetic field range $B_z=70\unicode{x2013}100\ \text {mT}$ at the plasma density $n = 2.17 \times 10^{17}\ \text {m}^{-3}-1.83 \times 10^{18}\ \text {m}^{-3}$ at coordinate $z = 2.04\ \text {m}$. The resulting dependency of the frequency on the Alfvén velocity is presented in figure 7. This dependency is approximated by a linear function with the slope $1.7 \times 10^{-4}\ \text {cm}^{-1}$ for $V_{\rm A} < 2.8 \times 10^6\ \text {m}\ \text {s}^{-1}$. The function fits experimental data points with $R^2 = 0.9$. The Alfvén waves in cold plasma possess the same dependency on $V_{\rm A}$ in the low-frequency limit. For $V_{\rm A} > 2.8 \times 10^6\ \text {m}\ \text {s}^{-1}$ the dependency is no longer considered as linear and goes to the plateau with frequency value 48 kHz.

Figure 7. Dependence of frequency on the Alfvén velocity. Different colours correspond to the different magnetic field values.

Calculating the cross-phase and varying the probe set's radial coordinate we have obtained the radial profile of the wavevector components (shown in figure 8). It is noteworthy that the plasma cross-section can be divided by radius into two zones. For $r$ values less than 40 mm, the radial and azimuthal wavenumbers undergo slight changes, remaining close to zero and between one and two, respectively. For larger radii, the wavenumbers’ absolute value increases but the signs are opposite. Meanwhile, the axial wavenumber has a global maximum at $r = 40$ mm. This radius is close to the position of the highest radial electric field. Inside this region, the electric field rises linearly. It corresponds to the rigid rotor rotation (angular velocity does not depend on the radius). The rotation of the plasma boundary has lower angular velocity.

Figure 8. (a) Radial component of the wavenumber, (b) mode number, (c) axial component of the wavenumber and (d) product of the wavevector axial component on the mode number of the wave for $B = 70\ \text {mT}, R_{{\rm mean}} = 1.52$ and $n = 5.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 0.4\ \text {m}$ and $n = 2.1 \times 10^{18}\ \text {m}^{-3}$ at $z = 2.04\ \text {m}$.

According to $k_r(r)$ (shown in figure 8), the fluctuation is in phase for $r < 45$ mm. The radial size of this area can vary depending on the plasma parameters. In the periphery, sufficient $k_r$ appears and the fluctuation phase velocity is directed towards the magnetic axis.

The azimuthal mode number lies between 1 and 2 for $r < 40$ mm. For larger radii, the mode number increases. Mode number is positive and then the fluctuation azimuthal phase velocity is aligned with the ion rotation. The axial wavenumber has a maximum at $r = 40$ mm.