Collisionless particle dynamic in an axi-symmetric diamagnetic trap

Newly proposed regime of diamagnetic confinement of plasma in a mirror machine [1] allows us to essentially reduce particles and energy losses from the trap and increase power density of thermonuclear reactions. The idea is to confine plasma with extremely high pressure equal to pressure of magnetic field of the trap. It leads to formation of central region with sharp boundary occupied by dense plasma with extruded magnetic field (so called diamagnetic “bubble”). The effective mirror ratio inside the “bubble” is extremely high, so in MHD approximation longitudinal losses of plasma from inner area of the diamagnetic trap are suppressed. In frame of the MHD approximation the plasma and energy losses are driven by diffusion of plasma through magnetic field on the border of the “bubble” [1, 2]. There losses grow linearly with increasing “bubble” length and radius and are reduced when plasma conductivity rises.

The structure of magnetic field in the diamagnetic trap is close to Field Reversal Configuration (FRC) [4] with zero field reversal. So particle dynamic in the diamagnetic trap has a lot in common with dynamic of fast ions in FRCs.

Investigation of single-particle dynamic in the diamagnetic trap is needed for development of kinetic models of diamagnetic confinement. Small magnitude of magnetic field results in non-conservation of magnetic moment $mv_{\perp}^{2}/(2B)$ and can lead to chaotic behavior and fast longitudinal losses of particles. From the other hand, particle energy and canonical angular momentum are integrals of motion due to stationarity and azimuthal symmetry of magnetic field. The aim of this work is investigation of regimes of particle confinement and lifetime of unconfined particles in the diamagnetic trap.

The magnetic field is assumed to be fully axisymmetric later. So influence of possible instabilities and non-accuracy of magnetic system of the trap are neglected. Non-symmetry of magnetic system is seems to result in slow Arnold diffusion of energy and angular moment. In presence of dense plasma the Coulomb collisions should to masque the slow diffusion. Mechanisms of anomalous losses due to plasma instabilities in diamagnetic trap are addressed for future investigations.

The article is organized as follow. Hamilton function of particle in the trap is discussed in the second section. Two mechanisms can provide particle confinement in unlimited time if collision scattering is absent, namely so-called absolute confinement [3] and conservation of radial adiabatic invariant. There mechanisms are discussed in the third section. The estimations for lifetime of unconfined particles and plasma confinement time in gas-dynamic regume are found in the fourth and fifth sections. Simple numerical example is presented in the sixth section. The results are discussed in the Conclusion.

We consider particle dynamic in the axisymmertic magnetic field without spirality. This field can be described by only one function, namely magnetic flux $\Psi(r,z)=rA_{\theta}(r,z)$, here $A_{\theta}$, $r$ and $z$ are azimuthal component of vector potential, radial and longitudinal coordinate. The Hamilton function can be written in the following form

here $\varphi(r,z)$ is electrostatic ambipolar potential. The particle energy and canonical angular momentum $p_{\theta}$ are integrals of motion due to stationarity and azimuthal symmetry. If structure of magnetic field is “bubble”-like that magnetic field (and magnetic flux) is small inside region in central part of the trap. Outside this region magnetic field is approximately vacuum (examples of structure of magnetic field are shown in [1, 2], see also figures 2 and 3). We assume that the ambipolar potential is approximately constant in region with very small magnetic field and that longitudinal gradient scale of the potential on “bubble” boundary coincides with longitudinal gradient scale of magnetic field.

It is convenient in numerical simulations to express all dimension quantities via particle mass, vacuum magnetic field in the center of the trap $B_{0}$ and cyclotron frequency calculated by vacuum magnetic field $\Omega\equiv eB_{0}/(mc)$. Only quantities with dimension of length remain after this expressing. So integrals of motion with dimension of length will be used later together with energy and angular momentum, namely Larmor radius calculated by full energy and vacuum magnetic field $\rho\equiv(2\varepsilon/(m\Omega^{2}))^{1/2}$ and minimal possible distance between particle and axis in case of zero magnetic field $r_{\min}=p_{\theta}/\sqrt{2m\varepsilon}$.

An example of trajectories in the transversal cross-section are shown in figure 1. Radial dependence of the magnetic field is chosen in the form $B(r)=B_{0}\{1+\tanh((r-a)/\Delta r)\}/2$, here $a/\Delta r=10$. The trajectories can be divided into three classes in dependence on sign of particle angular momentum. Orbits of the particles with $\Omega p_{\theta}\leq 0$ and $\Omega p_{\theta}>0$ correspond to betatron and drift orbits in FRC [6]. As in the FRC, confinement of particles essentially depends on sign and magnitude of angular momentum (see below).

Non-conservation of magnetic momentum due to smallness of magnitude of magnetic field inside the diamagnetic “bubble” results in changing of regimes of particle confinement and essentially modification of concept of loss cone. Two mechanisms can provide particle confinement in the diamagnetic trap for unlimited time (in absence of collision scattering and non-axisymmetry): well-known absolute confinement [3] and conservation of radial adiabatic invariant.

Let’s us now looks particles which move chaotically and are not confined absolutely. If particle moves regularly than particle arrival to mirror at the same longitudinal velocity after each excursion from mirror to mirror. If radial adiabatic invariant not conserves than longitudinal velocity changes chaotically with each approach to mirror so particle leave the diamagnetic trap after several excursions from mirror to mirror. To estimate lifetime of chaotically moving particles we consider population of particles with same energy $\varepsilon$ and angular momentum $p_{\theta}$ which move inside quasi-cylindrical diamagnetic “bubble” with radius $a$ and length $L$. Distribution function of this particles is $f(\mbox{\boldmath$v$},\mbox{\boldmath$r$})=\delta(mv^{2}/2+e\varphi-\varepsilon)\delta(m(xv_{y}-yv_{x})+e\Psi/c-p_{\theta})H(v_{z}^{2}-V_{z0}(\varepsilon,p_{\theta},z)^{2})$, here $V_{z0}$ is solution of equation $I_{r}(\varepsilon,p_{\theta},V_{z0},z)=I_{r}(\varepsilon,p_{\theta},v_{z0},z=0)$, $v_{z0}$ is value of longitudinal velocity corresponding margin of adiabaticity. Full number of particles in the trap $N$ and particle flow through mirrors $J$ are approximately

here $z_{m}$ is coordinate of the mirror. Particle lifetime is $\tau=N/J$.

In the simplest case when all particles with energy $\varepsilon$ and angular momentum $p_{\theta}$ move chaotically, $V_{z0}=0$, “bubble” radius is large, $a\gg\rho$, and electrostatic potential is neglectingly small, estimation of particle confinement time is:

here $\tau_{b}=L/(\Omega\rho)$ is of the order of particle transit time from mirror to mirror (period of bounce-oscillations).

Calculation of particle confinement time in regime of diamagnetic confinement requires sophisticated calculations including solving of kinetic equation together with equilibrium equation. This calculation can be essentially simplified if plasma is quite dense and angular scattering time is less than lifetime of unconfined particles. In this case particle distribution functions are locally Maxwellian and one can estimate particle confinement time by calculating full number of particles and their flow through mirrors.

Let’s assume that distribution function of particles of type $s$ approximately coincides with distribution of Maxwellian particles inside cylinder with radius $a$

This distribution allows particle density to be uniform inside the diamagnetic “bubble” with radius $a$, like in MHD models (see next section). One can estimate lifetime of particles with distribution $s$ as

here $v_{s}=(2T_{s}/m_{s})^{1/2}$ is thermal velocity of particles of type $s$, $\rho_{s}=v_{s}/\Omega_{s}$ is mean Larmor radius calculated by vacuum magnetic field. Estimation (19) gives same lifetime for ions and electrons with same temperatures so plasma outflow in such regime should not be accompanied by appearing essential ambipolar potential.

Combination $R_{v}L/v_{i}$ is of the order of time of gas-dynamic outflow from trap with vacuum magnetic field $\tau_{GDT}$. So transition to regime of diamagnetic confinement increases $a/\rho_{i}$ times particle confinement time in gas-dynamic regime. This estimation should be compared with estimation

of particle confinement time in MHD model [1, 2], here $\lambda$ is thickness of boundary layer in MHD approximation. This comparison demonstrates that kinetic effects are important when $\rho_{i}>\lambda$.

It should be noted that effects of adiabatic confinement of part of particles not taken into account in estimation (19). This effects can be important if plasma flow through mirror is collisionless (“short” mirrors). Effects of adiabatitity of motion are seems to decrease particle losses. In this sense the estimation (19) is most pessimistic estimation of particle confinement time in the gas-dynamic regime.

To illustrate influence of structure of magnetic system on collision-less particle dynamic some results of numerical simulation of ions movement in diamagnetic trap are presented in this section. Magnetic flux is calculated similarly article [6]. Namely, magnetic flux satisfies Amperes’s law $(\nabla\times\mbox{\boldmath$B$})_{\theta}=\partial_{r}r^{-1}\partial_{r}\Psi=4\pi j_{\theta}/c$ (here $j_{\theta}$ is azimuthal component of plasma current) which can be write in the form of integral equation

here $\Psi_{v}(r,z)$ is flux of vacuum magnetic field and

is Green function (magnetic flux of thin coil), $E(\xi)$ and $K(\xi)$ are complete elliptic integrals of first and second kinds, $\xi=4rr_{0}/((z-z_{0})^{2}+(r+r_{0})^{2})$.

The integral equation (20) can be solved numerically by iterations. To calculate plasma current $j_{\theta}$ we assume electrons to be cold and choose distribution function of ions (18). Dependence of density and azimuthal current of ions on radial coordinate and magnetic flux are given in Appendix B. Example of radial dependence of plasma density and longitudinal component of magnetic field on radius is shown on figure 2. Plasma density is constant inside the “bubble” like in the MHD model [1, 2].

Magnetic system of trap consists of two mirror coils and of set of equidistant coaxial coils which generate quasi-uniform magnetic field (see figure 3). In one case corrugation of vacuum magnetic field does not exceeds tenths of percent (smooth magnetic field). In second case distance between the coils is doubled and radius of the coils is reduced (currents in coils are changed correspondingly so that value of magnetic field in center is the same in both cases). It results in observable corrugation of the “bubble” boundary.

Numerical simulation allows us to found maximal value of longitudinal velocity at which ions confine in the trap. Ions move regularly in trap with smooth field and maximal critical velocity is restricted only by criterion of confinement (6). An example of dependence of critical velocity on angular momentum for ions moving in trap with corrugated field is shown on figure 4. Ions with small $|p_{\theta}|$ scatter due to corrugation so their critical velocity is relatively low. This velocity increases when $|p_{\theta}|$ rises (in according with criterion of adiabaticity (14)). Ions with negative and small $p_{\theta}$ are confined absolutely so their maximal velocity is restricted by full energy $(2\varepsilon/m_{i})^{1/2}$. Critical velocity of ions with large $p_{\theta}$ decreases because this ions move in region with finite magnetic field outside the “bubble”.

An example of number of unconfined ions in trap with corrugated field after $n$ bounce-oscillations is shown on figure 5. This number decreases approximately exponentially with time. Confinement time $18.3\tau_{b}$ is consistent with analytical estimation (17).

Smallness of magnetic field in central region of the diamagnetic trap results in non-conservation of magnetic moment so regimes of particle confinement are modified. Particle can confine inside the diamagnetic “bubble” either due to conservation of radial adiabatic invariant (this mechanism occurs if vacuum magnetic field is smooth) or in regime of adiabatic confinement (if particle rotates quickly around axis of trap in direction coinciding with direction of cyclotron rotation). Possibility of conservation of the adiabatic invariant depends strongly on geometry of magnetic field, especially on small-scale perturbation of vacuum magnetic field. Lifetime of unconfined particles increases with increasing the “bubble” radius and vacuum magnetic field in the mirrors of the trap and decreasing particle “Larmor radius” $\rho$. Even in the worst case when all particles move chaotically particle confinement time exceeds the gas-dynamic time in the vacuum field in ratio of the “bubble” radius to mean ion “Larmor radius”.

The author wish to thank all the members of the laboratories 9-0, 9-1 and 10 of BINP SB RAS who participated in discussion of the results of this work. Author especially grateful to the Dr. Alexei Beklemishev, Dr. Dmitriy Skovorodin and Mikhail Khristo for fruitful discussions.

Now we consider particle moving with velocity $v_{0}$ inside corrugated cylindrical surface and reflecting elastically from it. Let $z_{n}$ and $v_{\|n}$ to be longitudinal coordinate and longitudinal component of velocity of particle at time $t_{n}$. At this point of time radial velocity equal to zero and azimuthal component of velocity is $(v_{0}^{2}-v_{\|n}^{2})^{1/2}$. Radial coordinate of particle is $r_{n}=r_{\min}v_{0}/(v_{0}^{2}-v_{\|n}^{2})^{1/2}$, here $r_{\min}=|p_{\theta}|/(mv_{0})$. Particle will collide with surface at point of time $t_{w}$ which is solution of equation $(r_{n}^{2}+(v_{0}^{2}-v_{\|n}^{2})t_{w}^{2})^{1/2}=a+\delta a\cos(kz_{n}+kv_{\|n}t_{w})$. Longitudinal and radial coordinate of the particle at the moment of collision are $z_{w}=z_{n}+v_{\|n}t_{w}$ and $r_{w}=a+\delta a\cos(kz_{w})$. Radial, azimuthal and longitudinal components of velocity at the moment of collision are $v_{r}=(v_{0}^{2}-v_{0}^{2}r_{\min}^{2}/r_{w}^{2}-v_{\|n}^{2})^{1/2}$, $v_{\theta}=v_{0}r_{\min}/r_{w}$ $v_{\|}=v_{\|n}$. After collision radial and longitudinal components of velocity are $v_{\|n+1}=(2f^{\prime}v_{r}+(1-f^{\prime 2})v_{\|n})/(1+f^{\prime 2})$ $v_{r}^{\prime}=(2f^{\prime}v_{\|n}-(1-f^{\prime 2})v_{r})/(1+f^{\prime 2})$, here $f^{\prime}=-k\delta a\sin(kz_{w})$. Azimuthal component does not change. Radial coordinate of particle will reach minimal value $r_{n+1}=r_{\min}v_{0}/(v_{0}^{2}-v_{\|n+1}^{2})^{1/2}$ through time $\Delta t=(r_{w}^{2}-r_{\min}^{2}v_{0}^{2}/(v_{0}^{2}-v_{\|n+1}^{2}))^{1/2}/(v_{0}^{2}-v_{\|n+1}^{2})^{1/2}$ after collision. Longitudinal velocity is $z_{n+1}=z_{n}+v_{\|n}t_{w}+v_{\|n+1}\Delta t$ when radial coordinate is minimal.

If corrugation is weak $\delta a\ll a$ that time before collision is approximately $t_{w}\approx(\Delta r_{n}I_{n}/v_{\|n})\{1+(\delta aa/\Delta r_{n}^{2})\cos(kz_{n}+k\Delta r_{n}I_{n})\}$, here $\Delta r_{n}=(a^{2}-r_{\min}^{2}v_{0}^{2}/(v_{0}^{2}-v_{\|n}^{2}))^{1/2}$ and $I_{n}=v_{\|n}/(v_{0}^{2}-v_{\|n}^{2})^{1/2}$. When radial coordinate minimal longitudinal component of velocity and longitudinal coordinate of particle are described by expressions (10).

Density of ions with distribution function (18) is

here $\psi(r,z)=\Psi(r,z)/B_{0}$ is normalized magnetic flux, $\varphi$ is electrostatic potential, $w=(2T_{i}/m_{i})^{1/2}$ is thermal velocity and

Current of ions is