Particle Orbits in a Force-Balanced, Wave-Driven, Rotating Torus

TL;DR

This paper explores the physics of particle orbits in a novel wave-driven rotating torus fusion device, emphasizing force balance, magnetic and electric field effects, and potential impacts on plasma transport.

Contribution

It introduces a detailed analysis of particle orbits and force balance in the wave-driven rotating torus, highlighting the role of plasma current and magnetic fields for stability.

Findings

Force balance requires a small plasma current and vertical magnetic field.

Particle orbits show velocity-space anisotropies due to electric and magnetic field interactions.

Potential neoclassical effects on conductivity and transport are identified.

Abstract

The wave-driven rotating torus (WDRT) is a recently proposed fusion concept where the rotational transform is provided by the E x B drift resulting from a minor radial electric field. This field can be produced, for instance, by the RF-wave-mediated extraction of fusion-born alpha particles. In this paper, we discuss how macroscopic force balance, i.e. balance of the thermal hoop force, can be achieved in such a device. We show that this requires the inclusion of a small plasma current and vertical magnetic field, and identify the desirable reactor regime through free energy considerations. We then analyze particle orbits in this desirable regime, identifying velocity-space anisotropies in trapped (banana) orbits, resulting from the cancellation of rotational transforms due to the radial electric and poloidal magnetic fields. The potential neoclassical effects of these orbits on the perpendicular conductivity, current drive, and transport are discussed.

Tables

Table 1. Table 1: Sample reactor parameters for a force-balancing WDRT scenario.

Parameter	Value
$a$	0.3 m
$R_0$	10 m
$\epsilon$	0.03
$E_r$	$3\times {10}^6$ V/m
$B_{\varphi }$	5.0 T
$B_{\theta }$	0.1 T
$B_z$	0.02 T
$v_{E\times B}$	$6\times {10}^5$ m/s
$v_{thi}$	$1\times {10}^6$ m/s
$v_{the}$	$4\times {10}^7$ m/s
${\beta }_p$	10
${\beta }_E$	10

Equations

$${ϵ}_{\alpha }\approx 2e{E}_{r}a,$$

$${\beta }_p=\frac{\int Pda}{{\mu }_0{I}^2/8\pi }$$

$${\beta }_E\equiv \frac{⟨P⟩}{{ϵ}_0⟨{ϵ}_{rr}{E}_r^2⟩/2}.$$

$${ϵ}_{rr}=S=1+{\left(\frac{c}{v_A}\right)}^2,$$

$${\beta }_E$$

$$=\frac{{\sum }_s{n}_s{m}_s⟨v_{ths}^2⟩}{n_i{m}_i⟨v_{E\times B}^2⟩/2}$$

$$=(Z_i+1)\frac{2⟨n_{i}T⟩}{m_i⟨n_i{v}_{E\times B}^2⟩}$$

$${\beta }_E\sim {\left(\frac{2v_{thi}}{v_{E\times B}}\right)}^2.$$

$$B_{\theta a}>B_z.$$

$$F_h=4\pi \int Pda.$$

$$B_z=-\frac{2\int Pda}{R_0{I}_{\varphi }}.$$

$$B_z=-\frac{{\mu }_{0}I}{4\pi R_0}\left(\log \left(\frac{8R_0}{a}\right)+\Lambda -\frac{1}{2}\right),$$

$$\Lambda$$

$$v_{thi}$$

$$\gg \frac{1}{2}\frac{E_r}{B_{\varphi }}$$

$$\gg \frac{1}{2}\frac{{ϵ}_{\alpha }/2ae}{B_{\varphi }}.$$

$$a\gg a_0=\frac{1}{4v_{thi}}\frac{V_{\alpha }}{B_{\varphi }},$$

$$I_0\equiv \sqrt{\frac{\int Pda}{{\mu }_0/8\pi }}=\sqrt{\frac{8{\pi }^2⟨P⟩a^2}{{\mu }_0}},$$

$$B_{\theta a}=\frac{{\mu }_{0}I}{2\pi a}=\frac{{\mu }_0{I}_0}{2\pi a}{\beta }_p^{-1/2}=\sqrt{\frac{2{\mu }_0⟨P⟩}{{\beta }_p}}.$$

$$B_z\approx \frac{2\int Pda}{R_{0}I}=\frac{2\pi a^2⟨P⟩}{R_0{I}_0{\beta }_p^{-1/2}}=\frac{a}{R_0}\sqrt{\frac{{\mu }_0⟨P⟩{\beta }_p}{2}}$$

$${\beta }_p<2\left(\frac{R}{a}\right).$$

$$P=nT=1.6\times 10^4{n}_{20}{T}_{\text{keV}},$$

$$⟨P⟩=\frac{{\int }_0^a{P}_{\max }\left(1-{\left(\frac{r}{a}\right)}^2\right)2\pi rdr}{\pi a^2}=\frac{1}{3}{P}_{\max }.$$

$$⟨P⟩=5.3\times 10^3{n}_{20}{T}_{keV}.$$

$$B_{\theta a}$$

$$B_z$$

$$ϵ$$

$$\mu$$

$$p_{\varphi }$$

$$0$$

$$+\frac{1}{2}\left[{\left(\frac{R_i}{R}{v}_{\varphi i}^2-\frac{q}{mR}\Delta \Phi \right)}^2\frac{|B{|}^2}{B_{\varphi }^2}-v_{\varphi }^2\frac{|B_i{|}^2}{B_{\varphi i}^2}\right]$$

$$+q\left(V(\Phi )-V({\Phi }_i)\right),$$

$$V(\Phi )-V({\Phi }_i)\approx \frac{\partial V}{\partial \Phi }\Delta \Phi .$$

$${\hat{R}}_i$$

$${\chi }_M$$

Full text

††preprint: AIP/123-QED

I Introduction

In a tokamak or stellarator, the rotational transform necessary for toroidal confinement is produced by the twist in the field lines introduced by the poloidal magnetic field. However, motion along twisted field lines is not the only way to mitigate the vertical drifts which result in a toroidal magnetic field. For instance, in low-temperature, single-species plasmas often of interest in particle physicsJanes et al. (1966), the minor-radial electric field due to space-charge results in a poloidal $E\times B$ drift which offsets the vertical driftAvinash (1991). This magnetoelectric confinement has been demonstrated to confine a cold electron plasma for thousands of poloidal rotation periodsZaveri et al. (1992).

The potential for magneto-electric confinement in fusion plasmas has been less thoroughly studied. In the early days of the fusion program, T. H. Stix pointed out that fast ion losses near the edge of the plasma could produce a narrow, $E\times B$-rotating region, resulting in confined, D-shaped orbits when combined with the vertical drifts(Stix, 1970, 1971a, 1971b). The experiments that seek to manipulate the electric fields in tokamaks via electrodes, such as the electric tokamak at UCLATaylor et al. (2002) and the TCABR tokamak Nascimento et al. (2005) at the University of Sao Paulo have similarly focused on a thin edge region, where they have often had success in controlling the poloidal flows Taylor et al. (2005).

Recently, it has been proposed to replace the poloidal magnetic field entirely with a minor radial electric fieldRax, Gueroult, and Fisch (2017), with the requisite volumetric space charge produced by the RF wave-driven extraction of positively-charged fusion productsFisch and Rax (1992); Fetterman and Fisch (2008). This wave-driven extraction is predicted to be more efficient than classical RF-driven current drive in tokamaksFisch (1978, 1987). In addition, such a confinement system could produce the same rotational transform with less available free energy, suggesting that turbulent transport and instabilities could be reduced compared to the tokamak case.

There are many research questions to be addressed in considering this new confinement scheme, as it is adapted for a multi-species plasma. These include, but are not limited to, analyses of MHD and kinetic stability, demonstration that the perpendicular conductivity will allow such a large radial field to be sustained, estimates of the viscous power dissipation due to the poloidal flow, and elucidation of the mechanisms for $\alpha$ particle extraction. In this paper, we leave aside most of these questions for now, only aiming to demonstrate that single-particle confinement and macroscopic force balance (i.e. balance of the hoop force) can be simultaneously achieved for a hot fusion plasma.

In the case of cold, single-species, non-neutral plasmasJanes et al. (1966); Avinash (1991); Zaveri et al. (1992), the hoop force was balanced by the electric field from image charges in the conducting wall of the chamber. However, we find (Appendix A) that force balance is likely far more easily achieved in a fusion plasma with a small plasma current and vertical field than with an electric field.

We begin in Section II by establishing constraints on reactor design, allowing us to identify desirable, macroscopically-force-balancing field configurations based on typical fusion plasma pressures. Then in Section III, we analyze particle orbits using constants of motion, showing that we recover known tokamak results in the limit of no electric field, in particular the trapped, banana-shaped orbits of the neoclassical regime. In Section IV, we analyze these orbits in detail for the WDRT with purely toroidal flux surfaces, demonstrating how similarly trapped orbits can result from the cancellation of $E\times B$ motion with thermal motion along the small poloidal component of the magnetic field. Interestingly, we find that only electrons are likely to experience these trapped orbits, due to anisotropy of the trapping region in velocity space. In Section V, we confirm our analytical results with single-particle simulations, and examine the impact of including relativistic effects and vertical fields consistent with force balance, which are minor. Finally, in Section VI, we examine the consequences of non-equipotential flux surfaces in a simple field configuration, showing how too large an electric field parallel to the magnetic field can lead to trapped orbits for the entire electron population. The consequences of our results for fusion reactor design are discussed in Section VII.

II Defining the Reactor Regime

II.1 Constraints on Reactor Design

A wave-driven rotating torus has fundamentally different drives and sources of free energy than a tokamak, giving rise to different constraints on the system. In this section, we outline these constraints.

The first constraint arises from the fact that an optimal WDRT should be able to almost directly convert $\alpha$ particle birth energy into radial potential energy. Although it is in theory possible to confine the plasma via an electric field pointing in either direction, an inward-pointing field is far more attractive, since then fast ions will be cooled as they leave the device, converting their kinetic energy into $E\times B$ rotation. Thus $\alpha$ particles should lose most of their $\epsilon_{\alpha}\approx 3.5$ MeV birth energy via direct conversion as they leave the plasma, i.e.:

[TABLE]

where $E_{r}$ is the (minor) radial electric field, $e$ is the elementary charge, and $a$ is the minor radius. The coordinate systems used throughout the paper are described in Figure 1.

It is also desirable to have sources of free energy in the plasma-sustained fields to be small compared to a conventional tokamak; this is one of the main advantages of the WDRT. For the magnetic field, this is accomplished via the condition $\beta_{p}\gg 1$, where

[TABLE]

is the ratio between the thermal energy and the pressure in the poloidal magnetic field.

In addition to the poloidal magnetic field, there is also a large radial electric field, which gives rise to two additional sources of free energy, from the electric field itself and from the plasma rotation. Both of these are accounted for in the dielectric tensor, so that the energy, so the “electric $\beta$” is given by

[TABLE]

If $\beta_{E}\gg 1$, we will also ensure that the hoop force from the electric field pressure is negligible compared to the thermal plasma pressure.

Now because $\hat{r}\perp\hat{b}$, the dielectric tensor component $\epsilon_{rr}$ is given by the low-frequency limit of the Stix $S$ term:

[TABLE]

where $c/v_{A}$ is the ratio of the speed of light to the Alven ($E\times B$) speed. Since we expect the rotation velocity to be substantially less than the speed of light, most of the energy is stored in the kinetic energy of the $E\times B$ rotation. In this limit, we can express $\beta_{E}$ straightforwardly in terms of the $E\times B$ velocity $v_{E\times B}$:

[TABLE]

where we have used $n_{e}=Zn_{i}$ and assumed $T_{i}=T_{e}$. For hydrogen isotopes,

[TABLE]

Finally, to have minimal shifting of the magnetic axis, so that our flux surfaces approximately enclose our assumed current profile, we will want

[TABLE]

Our goal will be will be to combine these constraints with the requirement of macroscopic force balance.

II.2 Macroscopic Force Balance and Reactor Regime

Macroscopic force balance in a toroidal system requires that we balance the hoop force that causes the plasma to expand, given by:

[TABLE]

If a total current $I_{\phi}$ runs toroidally through the plasma, then this hoop force can be balanced by the addition of a vertical magnetic field,

[TABLE]

This corresponds to the $\beta_{p}\rightarrow\infty$ limit of the tokamak vertical field in large aspect ratioWesson and Campbell (2011):

[TABLE]

where

[TABLE]

II.3 Reactor Regime

Now that we have related the vertical field to the plasma current, we are almost in a position to define the reactor regime. Note that all terms in the estimates which follow are taken to be positive, so that it is only the magnitude of the terms which matters.

We will start by combining the conditions on rotational free energy ($\beta_{E}\gg 1$) and marginal $\alpha$ confinement (Eq. 1).

[TABLE]

This can be rearranged to give an expression for the minor radius:

[TABLE]

where $V_{\alpha}$ is the voltage corresponding to the easily extractable $\alpha$ particle birth energy—say 2 MV. Thus we see that there is a fundamentally minimum minor radius if we want to convert most of the $\alpha$ particle birth energy into electric potential, without having more rotational than thermal energy. Taking $T_{i}=10$ keV, we have $v_{thi}\approx 10^{6}$ m/s, so if we take a typical tokamak field of $\sim 5$ T, we find $a_{0}\approx 10$ cm. Noting that $\beta_{E}\sim(a/a_{0})^{2}$, this means that at a minor radius of about 30 cm, we would have $\beta_{E}\approx 10$. Thus quite small sources of free energy seem to be consistent with reasonable device dimensions.

Now we must impose our force balance condition. For simplicity, we will assume a constant current profile. We start by expressing the magnetic fields in terms of $\beta_{p}$, subject to force balance. For the poloidal magnetic field, this is straightforward. First we define

[TABLE]

so that when $I=I_{0}$, $\beta_{p}=1$, and in general $\beta_{p}=\left(\frac{I_{0}}{I}\right)^{2}$. Then we have, assuming a constant current profile:

[TABLE]

The vertical field is then obtained from the force balance condition (Eq. 11):

[TABLE]

Finally, we include our last constraint, that $B_{\theta a}>B_{z}$, which gives:

[TABLE]

Since we want large $\beta_{p}$, this indicates that the favorable reactor regime lies at large aspect ratio. However, since we wish to minimize viscous damping of the poloidal rotation, large aspect ratio was the regime of interest anyway.

It is instructive to consider the requisite fields for typical fusion temperatures and densities. First, note that the pressure is given by

[TABLE]

where $n_{20}$ is the density normalized to $10^{20}$ m*-3*, and $T_{\text{keV}}$ is the temperature in keV. If we additionally assume parabolic profiles $n,T\sim 1-(r/a)^{2}$, then

[TABLE]

Thus

[TABLE]

With these assumptions, we can write the fields in SI as:

[TABLE]

For a reactor with a major radius of 10 m, a minor radius of 30 cm, confining a 10 keV, $10^{20}$ m*-3* plasma at a $\beta_{p}$ of 10, we thus have $B_{\theta a}=0.12$ T, $B_{z}=0.019$ T.

III Particle Orbits

Now that we have a rough picture of how to introduce macroscopic force balance in a WDRT, we will re-examine particle trajectories with the consistent poloidal and vertical fields for force balance. We will assume that there is infinite conductivity parallel to the magnetic field, so that flux surfaces are equipotentials.

Our constants of motion are then

[TABLE]

Eliminating the final velocities using the last two equations, and taking $v_{\parallel}=|B|/B_{\phi}v_{\phi}$, we thus have the orbit constraint equation

[TABLE]

where we have defined $\Delta\Phi=\Phi-\Phi_{i}$.

Now we make a couple approximations, based around the assumption that the particle will not deviate far from the flux surface. The first consequence of this is that the relative strength of the poloidal and toroidal fields should not change dramatically; thus, we take $\frac{|B_{i}|^{2}}{B_{\phi i}^{2}}\approx\frac{|B|^{2}}{B_{\phi}^{2}}$.

The second consequence is that we can Taylor expand the potential around the flux surface, i.e.

[TABLE]

Here, we assume that $\Delta\Phi$ is not large enough that $\frac{\partial V}{\partial\Phi}$ changes significantly.

Now we will define a few ratios:

[TABLE]

The first of these terms simply relates the current and initial major radii; the second involves the mirror force, the third is a centrifugal term, and the fourth involves the pitch angle of the magnetic field. It should be noted that

[TABLE]

The orbit constraint equation can then be written in terms of these variables:

[TABLE]

where

[TABLE]

The solution is, of course,

[TABLE]

Now, much of the useful physics is contained in

[TABLE]

Typically, this will be small, since $\chi_{M},\chi_{C}\ll 1$. In this case, we can Taylor expand to find $\Delta\Phi$. The exception will be when the denominator goes to zero with a finite numerator, in which case we can expect trapped (banana) orbits.

III.1 Verification of Tokamak trapping

To show that the limit $D>1$ corresponds to tokamak trapping, we will first consider $\frac{\partial V}{\partial\Phi}=0$, which describes a large-aspect-ratio tokamak, and show that we recover the banana orbit condition. We will then extend these results to large $E_{r}$ in the subsequent sections.

If we let $\frac{\partial V}{\partial\Phi}\rightarrow 0$, then we simply have a (non-inductively-driven) tokamak. Then our condition for particle trapping becomes

[TABLE]

i.e.

[TABLE]

We can recognize this as the mirror trapping conditionWesson and Campbell (2011); Rax (2011).

IV Trapped Orbits in WDRT

Once the electric field is added, the denominator of $D$ contains two terms, and the trapping condition becomes (in part) a resonance condition. The center and width of this resonance will determine how much of the particle population lives on trapped orbits. However, first we must have an explicit form for $\frac{\partial V}{\partial\Phi}$.

IV.1 Field setup

For simplicity, we take $B_{z}$ constant, and

[TABLE]

where $R$ is the major radial coordinate, $R_{0}$ the center of the magnetic axis, $r$ is the minor radial coordinate, and $a$ is the plasma minor radius at $\theta=\pi/2$. Then our flux function $\Phi=RA_{\phi}$ is given by

[TABLE]

As we show in Appendix B, this choice of $B_{\theta}$ gives us concentric flux surfaces centered around

[TABLE]

By taking the electric field to be constant at $R=R_{v}$, it is also shown that

[TABLE]

where $z_{v}(\Phi)$ is the height of the flux surface $\Phi$ at $R=R_{v}$ (Figure 2).

IV.2 Resonance center for trapped orbits

To better understand the resonance condition, take $B_{z}=0$, so that

[TABLE]

Then $z_{v}=r$, and Eq. (48) becomes

[TABLE]

Now, the particle will start moving across many flux surfaces when the denominator of $D$ is 0. Setting the denominator of Eq. (42) to 0, we have

[TABLE]

where in the last line we made use of the definition of $B_{\theta}$.

Now, we multiply the last line by $\frac{B_{\phi}B_{\theta}}{|B|^{2}}$, and also recall the definition of $\chi_{\alpha}\equiv\frac{|B_{i}|^{2}}{B_{\phi i}^{2}}$. Then we have

[TABLE]

where

[TABLE]

is the parallel velocity consistent with momentum conservation if the particle remains on its initial flux surface, and

[TABLE]

are operators which project the parallel and $E\times B$ velocities respectively onto $\hat{\theta}$.

Thus we can see that the particle moves across many flux surfaces when the rotational transforms due to velocity along the field line and $E\times B$ drift across the field line cancel. We can summarize this condition as $v_{\theta,RT}=0$, where

[TABLE]

It is worth noting that in the case where $E_{r}$ points inward, which is the case of interest for the WDRT, electrons which support the plasma current are more likely to be trapped; thus there should be an enhanced resistivity even beyond the normal neoclassical term.

It is worth noting that trapped partical effects are not necessary deleterious. For instance, absorption of Alfven waves by trapped particles has been shown to produce sheared poloidal rotationTsypin et al. (2002), which is important in the formation of transport barriers for accessing H-mode.

IV.3 Resonance Width

The boundaries of the resonance are given by $D=1$. The following analysis will become simpler with the following normalized definitions:

[TABLE]

Note that $v_{E\times B}>0$ if $E_{r}>0$, and $v_{E\times B}<0$ if $E_{r}<0$. In general we will also have $\hat{\Pi}\gg 1$.

In these new variables, our resonance boundaries are given by:

[TABLE]

When we solve this for $\hat{v}_{\parallel i}$, we find

[TABLE]

Thus the trapping region is defined by a hyperbola in velocity space:

[TABLE]

Interestingly, a similarly shaped trapping region was calculated for impurities in a toroidally-rotating plasma, where it resulted from the inclusion of the Coriolis forceWong and Cheng (1989).

Now our relatively small poloidal field ensures $\hat{\Pi}\gg 1$, while $v_{E\times B}$ is some large fraction of the ion thermal velocity. Our resonant trapped velocity will thus be much larger than the ion thermal velocity, making ion trapping rare. Thus we only expect a significant number of trapped orbits for electrons in the WDRT, as well as potentially for fusion-born $\alpha$ particles.

IV.4 Banana width

To find the banana width, let the discriminant in Eq. (41) be 0. Then,

[TABLE]

Now, we make use of the approximation

[TABLE]

which gives

[TABLE]

where $\Omega_{p}=qB_{\theta}/m$.

The “fattest banana” is generally the most marginally-trapped orbit, given from inequality (64) by:

[TABLE]

So, plugging this in above,

[TABLE]

The first part, in parentheses, is the conventional banana width. In general, for electrons, we will have $v_{the}\gg v_{E\times B}$, so the denominator of the second term in the square root should be large–on the order of an electron-ion mass ratio. However, the numerator is determined by $\hat{\Pi}=B_{\phi}/B_{\theta}\gg 1$, so it is easy to envision this being the dominant term. In that case, the banana scaling will be given by (assuming $\Pi_{\hat{r}\times\hat{B},\hat{\theta}}\approx 1$):

[TABLE]

Assuming $\chi_{C}\approx\chi_{M}$, this means the main difference is that we replace the thermal velocity with

[TABLE]

when calculating the banana widths.

IV.5 Large aspect ratio

For the purpose of clarity and easy comparison to the transport literature, it is instructive to take the large-aspect ratio limit of our results. In a large-aspect-ratio WDRT, we will have

[TABLE]

Here we have adopted the convention from neoclassical transport literature, denoting the inverse aspect ratio $\varepsilon\equiv a/R$.

Our trapping region is given at large aspect ratio from Eq. (64) by:

[TABLE]

Or, more dimensionally,

[TABLE]

where

[TABLE]

is the critical parameter which determines the degree of deviation from tokamak-type trapping. In a WDRT, $\xi\gg 1$. Thus, compared to the tokamak case, the trapping region both shifts and expands, with the degree of each determined by both the ratio of $v_{E\times B}$ to $v_{the}$, and $B_{\theta}$ to $B_{\phi}$. Unlike in tokamak banana transport, the toroidal field plays a role in determining the shape of the trapping region.

In its two limits:

[TABLE]

Once the trapping region is known, the fattest banana width is given by

[TABLE]

Or, more dimensionally,

[TABLE]

In its two limits,

[TABLE]

In contrast to the shape of the trapping region (Eq. 76), the banana width is always independent of the toroidal field.

For the values in Table 1, the electron fattest banana width is on the order of 1 mm. Thus, assuming banana diffusion, trapping should not lead to any sudden loss of confinement due to instantaneously lost orbits, though it could lead to enhanced conductivity perpendicular to the magnetic field.

V Simulations and Finite $B_{z}$

To test our analytical predictions, we performed single-particle full-orbit simulations for the device parameters in Table 1 using either a non-relativistic Boris Boris (1970) or relativistic Vay Vay (2008) particle pusher and the field configuration described in Appendix B. Particles were initialized on the low-field side of the outer flux surface at $z=0$, as described in Appendix B.1. To map the trapped region in phase space, the initial velocity was swept across a range of initial values $v_{\perp i}$ and $v_{\parallel i}$ in the approximate $E\times B$-drifting rest frame; i.e. the particle with $(v_{\perp i}=0,v_{\parallel i}=0)$ was initialized at $\mathbf{v}_{i}=\frac{\mathbf{E}\times\mathbf{B}}{B^{2}}$.

Simulation results for the trapping region for $B_{z}=0$ are shown in Figure 3 for both particle pushers. The trapping region is well described by the analysis, and is largely unaffected when relativistic effects are included.

When we add a finite $B_{z}$ that ensures macroscopic force balance (0.021 T for the parameters in Table 1), the trapping region is largely unaffected, but the banana orbits get slightly wider, especially at low $v_{\perp i}$ (Figure 5). Thus macroscopic force balance in a WDRT with equipotential flux surfaces seems fairly achievable from a single-particle perspective.

Finally, an example of a simulated trapped electron orbit is shown in Fig. 6. Apart from the finite banana width at negligible initial $vpi$, the orbits are clearly similar to the banana-trapped orbits in a tokamak, of which they are a generalization.

VI Non-equipotential flux surfaces

The extremely high $q$ (low $B_{\theta}$) anticipated for the WDRT means that an electron must traverse a large distance along a field line in order to traverse a short distance poloidally. Thus it is not trivially guaranteed that the flux surfaces will be equipotential, and it is worth considering the consequences of non-equipotential flux surfaces.

As a simple model for this situation, consider that we have the same flux function $\Phi$ as in Eq. (46), but now have an electric field $\mathbf{E}=E_{r}\hat{r}$ that points purely radially. Taylor expanding Equation (47), we see that this will offset flux surface center from the electric center by a distance

[TABLE]

If we carry out the same constants of motion analysis for this situation, then we find that wide, trapped orbits will result even for cold particles unless (in the limit $B_{\theta a}\gg B_{z}$):

[TABLE]

In contrast to the equipotential flux surface case, the resulting trapped orbits can occur on either the high- or low-field side of the device. For a derivation of this result from the constants of motion, see Appendix C.

The trapping condition can be understood heuristically as follows. For cold electrons, the rotation transform is provided entirely by the $E\times B$ rotation, so an electron at radius $a$ traverses from the high-field to low-field side of the device on a timescale

[TABLE]

As the electron traverses this orbit, there is a component of $E_{r}$ parallel to the magnetic field due to $B_{z}$. The force along the field line due to the radial $E$ field when $B_{z},B_{\theta a}\ll B_{\phi}$ is thus

[TABLE]

Over the half orbit, the electron thus gains a velocity parallel to the magnetic field

[TABLE]

If the projection of this velocity onto $\hat{\theta}$, i.e. $v_{\theta}=v_{\parallel}\frac{B_{\theta}}{B_{\phi}}$, is opposite in sign and larger in magnitude than the $E\times B$ velocity, then the electron will reverse direction, becoming trapped. The rotational transforms will oppose, for instance, for electrons when $E_{r}<0$ and $B_{z},B_{\theta a}>0$. Then the condition for trapping is

[TABLE]

Multiplying through by $B_{\phi}$ and taking $\langle\sin\theta\rangle=\frac{2}{\pi}$, we have

[TABLE]

which is within a factor of two of our exact condition from COM.

We can rewrite Eq. (83) in terms of the flux surface shift:

[TABLE]

Thus we see that the condition for passing cold-electron orbits introduces a maximum allowable deviation from equipotential flux surfaces.

VII Conclusion

We demonstrated via a constants-of-motion approach that fusion-pressure electrons and ions can be well confined in a macroscopically force-balancing wave-driven rotating torus (WDRT) configuration. In so doing, we identified several possible sources of difficulty for the WDRT, which require further research.

Most of these potential problems arise from the fact that the banana-trapping region is not symmetric in phase space along $v_{\parallel}$. The first consequence of this asymmetry is the trapping only of electrons, but not of ions. This differential confinement could have large effects on the perpendicular conductivity, which must be low to sustain the MV-scale potentials needed for efficient direct conversion of $\alpha$-particle energy.

The second consequence of the asymmetry is the preferential trapping of electrons which support the plasma current, thus potentially increasing the resistivity far in excess of the standard neoclassical increase due to symmetric electron trapping. This effect could dramatically lower the current drive efficiency.

Finally, we showed that deviation of flux surfaces from equipotential surfaces could lead to large-scale trapping of the electron population, emphasizing the importance of having large parallel conductivity despite the low perpendicular conductivity.

By confirming that a small toroidal current can lead to a force-balanced plasma, without destroying the WDRT confinement, our results demonstrate the potential promise of the WDRT as a fusion concept with far lower free energy than a tokamak. However, the interesting trapped orbit effects uncovered as a result also point to the important research avenues ahead in evaluating its feasibility.

Acknowledgements.

The authors would like to thank Elijah Kolmes, Renaud Gueroult, and Jean-Marcel Rax for helpful discussions. This work was performed under U.S. DOE contract DE-SC0016072. One author (IEO) also acknowledges the support of the DOE Computational Science Graduate Fellowship (DOE grant number DE-FG02-97ER25308).

Appendix A Force Balance with an Electric Field

It could also be possible to balance the hoop force with an electric field, given the plasma space charge necessary to produce the radial electric field. The size of this space charge can be approximated from Gauss’ law, by assuming a cylindrical plasma:

[TABLE]

Solving for a linear charge density $\lambda$, this becomes

[TABLE]

We now need to produce a force per unit length equal to the force provided by the vertical field above. Thus

[TABLE]

which, plugging in $\lambda$, gives

[TABLE]

Now, plugging in $\langle P\rangle=5\times 10^{3}n_{20}T_{\text{keV}}$, then

[TABLE]

Combining this with our constraint on marginal alpha confinement (Eq. 1),

[TABLE]

where all lengths are measured in meters. Now, in order for $E_{R}\ll E_{r}$ (a condition for closed poloidal orbits), we can see that we will have to push to very large aspect ratio and small $a$; i.e. for $a=0.3$, we would need $R_{0}\gg 100$ m.

This requirement for extremely large aspect ratio becomes clear when we note that that Eq. (93) can be written

[TABLE]

where $\beta_{E}$ is defined in Eq. (3) as the ratio of the thermal to electric field energy. Thus have the rotational energy small compared to the thermal energy (one of the weaker of our free energy constraints), we must go to extremely large aspect ratio. This difficulty makes electrostatic force balance an unattractive option.

Appendix B Electric field for equipotential flux surfaces

For $B_{\theta}=B_{\theta a}\left(\frac{r}{a}\right)^{p}\frac{R_{0}}{R}$, our vector potential is given by

[TABLE]

where $r=\sqrt{(R-R_{0})^{2}+z^{2}}$. Thus our flux $\Phi=RA_{\phi}$ is given by

[TABLE]

The gradient of this, along which the poloidal projection of the electric field will point, is given by

[TABLE]

For $p=1$, corresponding to constant current density and linearly increasing $B_{\theta}$,

[TABLE]

Note that as $B_{z}\rightarrow 0$, the gradient points purely along $\hat{r}=\cos\theta\hat{R}+\sin\theta\hat{z}$.

Now, in general, the constant-flux surfaces will be elongated along $\hat{z}$ and compressed along $\hat{R}$. They will also no longer be centered around $R=R_{0}$. Instead, we can easily identify the point at which $\nabla\Phi$ points purely vertically from Eq. (101). Setting the $\hat{R}$ term to 0, we find that this occurs at $R=R_{v}$, where

[TABLE]

Now, we choose to define our $E$ field profile along this line where $\nabla\Phi$ is vertical. We will call the field along this line $E_{r}(z_{v}(\Phi))$, where $z_{v}(\Phi)$ is the unique positive value of $z$ where a given flux surface $\Phi$ intersects this line. We can find $z_{v}(\Phi)$ simply by plugging $R=R_{v}$ into Eq. (100), and solving for $z_{v}$. This gives

[TABLE]

Now

[TABLE]

At $R=R_{v}$, $\nabla\Phi\parallel\hat{z}$, and this becomes

[TABLE]

Inverting to find $\frac{\partial V}{\partial\Phi}$:

[TABLE]

Plugging this back into Eq. (104), and also plugging in our definition of $\nabla\Phi$ from Eq. (101), we find

[TABLE]

B.1 Outer flux surface

We define our outer flux surface by $z_{v}=a$. However, when calculating banana widths, we generally start by considering a particle at $z=0$. For small $\frac{a}{R_{0}}\frac{B_{z}}{B_{\theta a}}$, the minor radius as measured from the flux axis $r_{F}=R-R_{v}$ at $z=0$ is

[TABLE]

Initializing particles at this radius allows us to compare results between different $B_{z}$ values.

Appendix C Cold particle trapping for non-equipotential flux surfaces

The conserved quantities along the particle orbit are:

[TABLE]

For the following discussion, we will assume $E_{r}$ is constant and $B_{\theta}=B_{\theta a}\left(\frac{r}{a}\right)$ at $R=R_{0}$.

We first note that

[TABLE]

and of course

[TABLE]

where

[TABLE]

Now our potential is given by

[TABLE]

And our vector potential is given by

[TABLE]

Our orbit constraint equation is then given by

[TABLE]

where the COM’s are calculated from the initial conditions.

With our COM’s in hand, we now turn to the question of electron trapping. Consider Eq. (120) with $v_{\perp i},v_{\parallel i}=0$. Also take $|B|\approx B_{\phi}$, $R_{0}/R\approx 1$. Then, plugging Eqs. (118-119), we find

[TABLE]

Now, take

[TABLE]

Then, by noting $R_{i}^{2}-R^{2}=(R_{i}+R)(R_{i}-R)=-2\bar{R}\Delta$ we can rewrite the above as

[TABLE]

Now we will consider the case where $z=0$, so that $R=R_{0}\pm r$. For an untrapped orbit which starts at $R_{i}<R_{0}$, we must have $R>R_{0}$ at the next midplane crossing, so

[TABLE]

Plugging this into Eq. (127) and expanding the square, we find

[TABLE]

For the moment, assume $\delta\ll r_{i}$, so that $\bar{r}\approx r_{i}$. Then we have a simple quadratic equation, the solution to which is

[TABLE]

where

[TABLE]

Because $4AC/B^{2}>0$ always, the magnitude of $\delta$ will go as $C/B\leq aB_{z}/B_{\theta a}$. So as long as $B_{z}/B_{\theta a}\ll r_{i}/a$, then $\delta\ll r_{i}$, and everything is consistent.

Because $A$ and $C$ are strictly positive, this equation will only have a real solution when $4AC/B^{2}<1$, i.e. when

[TABLE]

Now force balance requires $B_{z}B_{\theta a}>0$. Thus, we can see that when $qE_{r}>0$, as is the case for electrons in a negatively-biased WDRT, the LHS is strictly smaller until the second term is twice as large as the first term. The minimum electric field to satisfy this condition is

[TABLE]

This is our first passing orbit condition.

To find the second condition, consider the limit $B_{\theta a}\rightarrow 0$. Now we can no longer take $\delta\ll r_{i}$; instead, $\bar{r}=\delta/2+r_{i}$. Thus

[TABLE]

So

[TABLE]

Similarly to before, the answer is simply quadratic, given by

[TABLE]

where now

[TABLE]

So for a real solution to exist, we must have $4AC/B^{2}<0$, i.e. (for $qE_{r}>0$),

[TABLE]

Thus, our two conditions on $E_{r}$ are

[TABLE]

To test these predictions, we ran single-particle simulations (Boris scheme) for fixed values of all parameters besides $E_{r}$; if the orbit was trapped, $|E_{r}|$ was increased, while if it was untrapped, it was decreased. Once both a trapped and untrapped orbit were identified, this process became a binary search for the minimum $|E_{r}|$ required for orbit untrapping.

The comparison of these simulations with Eq. (144) is shown in Figure 7. Each theoretical prediction agrees well with the simulation in its regime of applicabity.