2.1 The GDT device

The GDT device is an axisymmetric mirror machine with a high mirror ratio $R=35$ and a distance between the mirrors of 7 m (see figure 2).

Figure 2. The GDT device: 1 – magnetic system coils; 2 – mirror plugs; 3 – expansion tanks; 4 – initial plasma generator; 5 – neutral beam injectors; 6 – waveguides of the ECR heating system; 7 – plasma column.

The magnetic field in the central plane is 0.35 T. An arc-type plasma generator creates the initial plasma. The main means of heating is a neutral beam system, consisting of eight modules with a total power of up to 5 MW and an operation time of 5 ms. The energy of neutral deuterium atoms is 22–25 keV. Under conditions of heating by neutral beams, the plasma density lies in the range $(1\unicode{x2013}5)\cdot 10^{19}\ {\rm m}^{-3}$, the electron temperature reaches 250 eV. When auxiliary electron cyclotron resonance (ECR) heating is used, the electron temperature reaches $\sim 900$ eV (Bagryansky et al. Reference Bagryansky, Anikeev, Denisov, Gospodchikov, Ivanov, Lizunov, Kovalenko, Malygin, Maximov and Korobeinikova2015a,Reference Bagryansky, Shalashov, Gospodchikov, Lizunov, Maximov, Prikhodko, Soldatkina, Solomakhin and Yakovlevb). The ECR heating is carried out by two gyrotrons with a total power of $\sim 800$ kW, a frequency of 54.5 GHz and an operation time of up to 10 ms. The time sequence of operation of the main systems of the GDT device during the plasma discharge is organised as follows:

• • the magnetic system turns on and the magnetic field reaches the required values;

• • then the plasma generator is turned on and within 4 ms the trap is filled with warm target plasma;

• • 0.5 ms before the end of the plasma generator operation, the neutral beam system is turned on, and operates for 5 ms;

• • in the process of neutral beam injection, the balance of warm plasma particles is maintained by injection of hydrogen or deuterium gas into the peripheral region of the plasma column using two gas boxes located near magnetic plugs.

2.2 Key problem 1 – MHD stabilisation of plasma in steady state under conditions of high $\beta$ and significant limitation of axial losses

It is well known that the axisymmetric magnetic configuration of the mirror cell is not favourable for ensuring MHD stability of the plasma. To suppress transverse plasma losses associated with the development of MHD instabilities, we use the vortex confinement method (Beklemishev et al. Reference Beklemishev, Bagryansky, Chaschin and Soldatkina2010), which has proven itself well in experiments at the GDT facility. To implement vortex confinement, it is necessary to create a zone of differential plasma rotation in the radial peripheral zone of the plasma column. This is achieved by applying an electrical bias to the electrodes of the sectioned plasma absorber (see figure 3). In this case, a potential jump and differential rotation occur in the peripheral region of the plasma column due to $E\times B$ drift in crossed electric and magnetic fields. A necessary condition for the implementation of vortex confinement is also the effect of finite Larmor radius (FLR) effects (Rosenbluth, Krall & Rostoker Reference Rosenbluth, Krall and Rostoker1962). Differential rotation, together with the action of FLR effects, modifies the convective motion of plasma during the development of MHD modes, transferring MHD oscillations to a saturation mode with closed flow lines in the cross-section of the plasma and small oscillation amplitudes relative to its radius (Prikhodko et al. Reference Prikhodko, Bagryansky, Beklemishev, Kolesnikov, Kotelnikov, Maximov, Pushkareva, Soldatkina, Tsidulko and Zaytsev2011).

Figure 3. Method for implementing of vortex confinement.

The use of the vortex confinement method makes it possible to achieve relative pressure values up to $\beta =8{\rm \pi} \cdot P_{\perp }/B^2_v \approx 0.6$ (Simonen et al. Reference Simonen, Anikeev, Bagryansky, Beklemishev, Ivanov, Lizunov, Maximov, Prikhodko and Tsidulko2010), where $P_{\perp }$ is the transverse plasma pressure and $B_v$ is the vacuum magnetic field. An obvious disadvantage of vortex confinement is the need for direct electrical contact of the plasma with the absorber electrodes. If the plasma flow into the expander is severely suppressed, the electrical contact may be broken. Thus, the use of vortex confinement becomes problematic when implementing plans to limit axial losses in order to maximise the power gain $Q$ for potential fusion applications.

The solution to this problem is planned in two steps:

1. (i) Transition to high beta mode using vortex confinement.

2. (ii) Suppression of losses during the development of MHD instabilities due to the interaction of diamagnetic currents in the plasma with image currents in conducting structures surrounding the plasma.

The results of a recent analysis by Kotelnikov, Prikhodko & Yakovlev (Reference Kotelnikov, Prikhodko and Yakovlev2023) predict the effectiveness of this method of MHD stabilisation, starting from the $\beta$ values achievable using vortex confinement.

The first steps in this direction have already been taken:

• • The magnetic configuration was changed in such a way that the volume of the region occupied by hot ions was reduced by approximately half compared with the previously used basic magnetic configuration. In addition, the density of the warm target plasma was optimally increased to achieve the values of the captured power of atomic beams at which the plasma diamagnetism reached its maximum value. These measures made it possible to achieve an average beta value $\langle \beta \rangle = 20\,\%$.

• • Motional Stark Effect (MSE)-diagnostics (Lizunov et al. Reference Lizunov, Den Hartog, Donin, Ivanov and Prikhodko2011; Lizunov, Donin & Savkin Reference Lizunov, Donin and Savkin2013) were prepared to measure the radial profile of the magnetic field inside the plasma and to calculate local $\beta$ values.

• • A system of magnetic probes located outside the plasma column was also prepared for measuring the distribution of the magnetic field associated with diamagnetic currents inside the plasma.

Further steps in this direction are planned in the near future:

• • Operation time of neutral beam injection (NBI) will be increased from 5 to 10 ms in order to further increase the plasma diamagnetism.

• • A recently upgraded microwave heating system will be used to increase $\langle T_e \rangle$, the plasma diamagnetism and $\langle \beta \rangle$.

• • In modes with the maximum achieved value of diamagnetism, measurements will be made of the radial profiles of the magnetic field inside the plasma and the distribution of the magnetic field associated with diamagnetic currents near the plasma boundary.

• • Using the results of Kotelnikov et al. (Reference Kotelnikov, Prikhodko and Yakovlev2023) and the results of measurements, conductive structures surrounding the plasma with an optimal shape for MHD stabilisation will be designed.

• • Conductive structures will be fabricated and assembled at the GDT facility.

• • The efficiency of this method of MHD stabilisation will be studied in a real experiment.

2.3 Key problem 2 – stable confinement with respect to kinetic modes

As already noted in the introduction, since the times of the classics it has been known that Coulomb kinetics predicts the maximum value of power gain for a reactor based on a simple mirror cell at the level of $Q \approx 1$. For example, calculations presented in Yurov & Prikhodko (Reference Yurov and Prikhodko2016a) predict $Q = 1.35$ for the synthesis of D–T $\rightarrow ^4{\rm He}(3.5\ {\rm MeV}) + n(14.1\ {\rm MeV})$. With such a value of $Q$, the mirror cell could be efficient source of neutrons. However, the development of kinetic instabilities may prevent the achievement of such a value of $Q$.

The presence of a loss cone in phase space and associated anisotropy of the ion distribution function in the velocity space causes the generation of several types of electromagnetic waves in the plasma, which leads to an increase in the ion scattering rate into the loss cone and a radical decrease in the ion confinement time.

What we do know about kinetic instabilities from researches under the GDT program?

1. (i) Alfvén ion cyclotron instability is well studied and a recipe for how to avoid its influence is known – oblique neutral beam injection (Zaytsev et al. Reference Zaytsev, Anikeev, Bagryansky, Donin, Korobeinikova, Korzhavina, Kovalenko, Lizunov, Maximov and Pinzhenin2014).

2. (ii) The global acoustic mode does not lead to noticeable losses (Skovorodin, Zaytsev & Beklemishev Reference Skovorodin, Zaytsev and Beklemishev2013).

3. (iii) Drift cyclotron loss-cone instability (DCLC) is being actively investigated theoretically and experimentally.

According to the results of numerous studies in past years (Post Reference Post1987), DCLC poses the greatest danger to ion confinement in a mirror cell. In a number of works in past years (see, for example, Ioffe et al. Reference Ioffe, Kanaev, Pastukhov and Iushmanov1974; Kanaev & Yushmanov Reference Kanaev and Yushmanov1975; Kanaev Reference Kanaev1979), a method for DCLC stabilisation was proposed and justified. The stabilisation method consists in creating a population of ions with an isotropic distribution: $n_{isotr}/n\approx$ a few per cent is enough to suppress instability.

Using the results of numerical simulation (Yurov & Prikhodko Reference Yurov and Prikhodko2016b) for a neutron source (NS) based on a GDT (see figure 4), we will show how the use of this DCLC stabilisation method affects the power gain $Q$. In the NS model, intense heating of warm ions occurs due to collisions with hot ions. In this case, the frequency of Coulomb collisions of warm ions decreases and becomes insufficient to fill the loss cone, that is, their distribution function becomes anisotropic. To maintain an isotropic distribution, it is necessary to inject large flows of cold particles, which leads to an increase in longitudinal losses and a decrease in $Q$ to $\approx 4\,\%$ (Bagryansky et al. Reference Bagryansky, Ivanov, Kruglyakov, Kudryavtsev, Tsidulko, Andriyash, Lukin and Zouev2004). The search for the possibility of significantly increasing $Q$ in a NS based on GDT and, at the same time, avoiding the development of DCLC is currently the subject of our theoretical (Kotelnikov, Chernoshtanov & Prikhodko Reference Kotelnikov, Chernoshtanov and Prikhodko2017; Kotelnikov & Chernoshtanov Reference Kotelnikov and Chernoshtanov2018) and planned experimental studies.

Figure 4. Schematic view of a neutron source based on GDT: 1 – neutral beams (oblique injection); 2 – population of hot ions; 3 – warm ions with an isotropic distribution function; 4 – injected flow of cold particles.

2.4 Key problem 3 – axial energy confinement (axial thermal conductivity)

The problem of the longitudinal thermal conductivity of plasma in open traps has been studied theoretically in sufficient detail in a number of works in past years (see, for example Konkashbaev, Landman & Ulinich Reference Konkashbaev, Landman and Ulinich1978; Ryutov Reference Ryutov2004). Due to the special importance of this problem, we carried out a series of our own theoretical (Abramov et al. Reference Abramov, Gospodchikov, Shaposhnikov and Shalashov2019; Skovorodin Reference Skovorodin2019) and experimental (Soldatkina et al. Reference Soldatkina, Anikeev, Bagryansky, Korzhavina, Maximov, Savkin, Yakovlev, Yushmanov and Dunaevsky2017, Reference Soldatkina, Maximov, Prikhodko, Savkin, Skovorodin, Yakovlev and Bagryansky2020) studies, which demonstrated good agreement. Thus, the conclusion of previous works was confirmed that, with the correct design and operating mode, the expander is capable of limiting the longitudinal energy flow to a level corresponding to the energy $\varepsilon \approx 8kT_e$, which on average is transferred by one electron–ion pair leaving the trap. Here, $T_e$ is the electron temperature and $k$ is the Boltzmann constant. In this case, the influence on longitudinal losses of non-ambipolar effects associated with the electrical biasing on the plasma absorbers, which is used to organise vortex confinement, deserves special attention. Experiments in the near future will be devoted to studying this problem.

2.5 Key problem 4 – transverse transport

Although experiments at the GDT device currently demonstrate a lower level of transverse energy losses relative to axial ones, further research is necessary. This problem is closely related to the problem of maintaining the balance of warm ions as well as controlling the radial plasma density profile in order not only to minimise transverse losses, but also to achieve conditions under which the radial limiters are not subjected to excessive thermal loads. Let us recall that the balance of warm plasma particles in the GDT is maintained by gas injection into the peripheral region of the plasma column near magnetic mirrors and limiters (see figure 3). This leads to the fact that the plasma density near the limiters turns out to be comparable to the plasma density at the core of the column. To demonstrate the significance of this problem, let us estimate the energy flux density per limiter for a typical operating mode of a GDT device. According to measurement data, the plasma density near the limiter is $n \approx 10^{19}\ {\rm m}^{-3}$, and the electron temperature is $T_e \approx 50$ eV. Assuming that the temperatures of electrons and ions of warm plasma near the limiter are equal, the plasma is deuterium and that each electron–ion pair brings energy $\varepsilon \approx 8kT_e$ to the limiter, we calculate the energy flux density

(2.1)\begin{equation} q \approx nv_i\cdot8T_e \approx 45 \text{ MW m}^{{-}2}. \end{equation}

It can be seen that, even with such low values of $n$ and $T_e$, the energy flow to the limiter is unacceptably high.

A solution to the problem may be to change the cold particle injection method to maintain a balance of warm plasma particles. If we abandon injection into the peripheral region of the plasma column and implement injection directly into the internal region, then we can assume that, in this way, it will be possible to minimise the plasma density near the limiter and, accordingly, the thermal loads. We are currently considering two ways to implement such an injection:

1. (i) injection of neutral beams of low energy;

2. (ii) injection of high-energy plasma jets with a frequency of $\sim 1$ kHz.

The development of low-energy neutral beam injectors is included in the GDT research program. Beams with an energy of ${\approx }1$ keV and an equivalent current of ${\approx }100$ A are supposed to be formed according to the ‘acceleration–deceleration’ principle (Pincosy & Turner Reference Pincosy and Turner1987). Currently, the corresponding technology is being tested at a special stand.

To implement the second method of particle injection it is required to develop an injector with a gas efficiency of ${\sim }1$ and a pulse repetition rate of ${\approx }1$ kHz, which will ensure the flow of cold particles into the internal region of the plasma column at a level of several kA.

Figure 5 shows a possible implementation of this type of injection from the expander side. Currently, testing of the high-energy plasma jet injector on the stand has been completed and preparations are underway for its use at the GDT device. The injector is a coaxial system designed on the principle of a Marshall gun (Marshall Reference Marshall1960) and capable of generating a jet of hydrogen or deuterium plasma with a density of $10^{21}$–$10^{22}$ m$^{-3}$, a speed of $\approx 100\ {\rm km}\ {\rm s}^{-1}$ and a duration of 20 $\mathrm {\mu }$s.

Figure 5. Possible method of injection of a high-energy plasma jet from the expander side: 1 – central cell; 2 – central cell coils; 3 – mirror plug coils; 4 – plasma absorber; 5 – expander tank; 6 – plasma gun.

3.1 Multiple-mirror systems

3.1.1 The GOL-NB experiment

The idea of plasma confinement in multiple-mirror magnetic systems was proposed in the early seventies of the last century (Budker, Mirnov & Ryutov Reference Budker, Mirnov and Ryutov1971; Logan et al. Reference Logan, Lichtenberg, Lieberman and Makhijani1972). Figure 8 explains the principle of plasma confinement in such systems. If the ion mean free path $\lambda _i$ is less than the trap length $L$, but greater than the corrugation scale $l$, the plasma flow becomes diffusive. In the thermonuclear temperature range, paired Coulomb collisions provide the required dissipation rate at extremely high densities $\sim 10^{22}\unicode{x2013}10^{23}$ m$^{-3}$. Therefore, initially the concept of a multiple-mirror reactor assumed a pulsed mode of operation.

Figure 8. Schematic view of a multi-mirror magnetic plasma confinement system.

In experiments on the GOL-3 multiple-mirror trap (Astrelin, Burdakov & Postupaev Reference Astrelin, Burdakov and Postupaev1998), improved ion confinement was discovered at a moderate density of ${\sim 10^{20}\unicode{x2013}10^{21}}$ m$^{-3}$. Improved confinement at moderate plasma densities indicates anomalous ion scattering, which is interpreted as a result of interaction with collective plasma oscillations at the bounce frequency of the corrugation cells. This circumstance opens the way to the use of multiple-mirror sections in a stationary reactor to limit longitudinal losses of particles and energy.

The mission of the GOL-NB device (Postupaev et al. Reference Postupaev, Batkin, Burdakov, Burmasov, Ivanov, Kuklin, Lykova, Melnikov, Mekler and Nikishin2022) is direct demonstration of multiple-mirror confinement in a steady state mode under the condition when the ion path length relative to Coulomb scattering $\lambda _{ii\text { Coulomb}} \gg L$. The GOL-NB (see figure 9) is an axisymmetric mirror cell with additional magnetic sections connected to the ends. Additional sections are solenoids consisting of a set of individual coils that have independent power sources. By controlling the currents in the coils, it is possible to implement within the sections both a longitudinal field profile close to uniform and a multiple-mirror configuration (see figure 9b).

Figure 9. The GOL-NB installation. a – Top view of the device: 1 – central cell; 2 – additional sections; 3 – expander tank; 4 – neutral beam injectors; 5 – neutral beam dumps; 6 – vacuum pumps. b – Magnetic field configurations with solenoidal and multiple-mirror shape.

The status of the device at the time of writing this text is as follows:

• • target plasma with $n \approx 3\times 10^{19}$ m$^{-3}$ and $T = 5\unicode{x2013}10$ eV is obtained;

• • neutral beam injectors have been commissioned ($U = 25$ keV, $P_{\varSigma } \approx 1,1$ MW);

• • plasma stabilisation system (edge-biasing) has been worked out.

In the near future, work will be continued in the following directions:

• • study and optimisation of the physics of plasma heating by neutral beams;

• • study of fast ion population parameters;

• • implementation of a multiple-mirror configuration and studying the effectiveness of limiting longitudinal losses by multiple-mirror sections;

• • development of auxiliary heating systems (ICRH ion cyclotron resonance heating).

3.1.2 Helical mirror SMOLA

The idea of a multiple-mirror magnetic system with a helical configuration to limit longitudinal particle flows and plasma energy is based on plasma rotation in a helical magnetic field (Beklemishev Reference Beklemishev2016b). Figure 10 illustrates a possible design of a mirror cell with screw multi-mirror sections. Plasma rotation occurs due to $E\times B$ drift in crossed electric and magnetic fields. The radial electric field inside the plasma is created by an electrical bias on radially sectioned electrodes located in the area of the expanders. In a rotating reference frame, plasma is influenced by running mirror cells. Running mirror cells transfer momentum to the particles trapped between them. Friction between trapped and passing particles causes plasma to be ‘pumped’ along the system back into the trap. In this case, the movement of trapped particles is accompanied by radial drift. The advantage of such a helical system compared with a conventional multiple-mirror system is the more efficient confinement predicted by the theory and the ability to control radial drifts. In addition, the counter flow of trapped ions can stimulate anomalous scattering, which leads to the formation of a population of trapped ions even in conditions of insufficient Coulomb collision frequency.

Figure 10. Possible arrangement of a mirror trap with helical multiple-mirror sections.

To test experimentally the principle of helical confinement, a SMOLA device was developed (Sudnikov et al. Reference Sudnikov, Ivanov, Inzhevatkina, Larichkin, Lomov, Postupaev, Tolkachev and Ustyuzhanin2022), which is shown in figure 11. The components of the installation are: a plasma source, a confinement area (axisymmetric mirror cell), a transport section and an expander. The transport section can have the configuration of either a simple solenoid or a helical multiple-mirror system.

Figure 11. a – The SMOLA device; b – main experimental result: on the left – radial profiles of plasma density in regimes with straight and helical field in transport section; on the right – radial profiles of ion flux density in expander region in regimes with straight and helical field in transport section.

Figure 11 also shows the main experimental result obtained from the SMOLA set-up. It can be seen that, in the mode with a helical configuration of the transport section, the ion flow into the expander decreases by more than an order of magnitude compared with the solenoidal configuration. In this case, the plasma density in the confinement region increases significantly. Based on the results of experiments at the SMOLA installation, the following conclusions can be drawn:

• • Significant suppression of axial flux is achieved.

• • The effective mirror ratio exceeded $R = 10$.

• • A 1.6-fold increase in plasma density in confinement area was demonstrated.

• • The radius of the plasma jet decreases

• • A good agreement with the theory is shown.

Further research is aimed at studying the mechanisms of formation of a population of trapped ions under conditions of rare Coulomb collisions, as well as the development a next-generation installation, which will be a mirror cell with two screw sections at the ends.