Wave induced supersonic rotation in mirrors

Abraham Fetterman and Nathaniel Fisch Princeton University

Outline

• Introduction to centrifugal confinement • Advantages to driving rotation with waves • Computational results

Rotation is produced by crossed electric and magnetic fields Mirror coils F c F c|| E  Electrodes B Magnetic field lines If we assume the potential is constant along a field line, we find the rotation frequency is also constant, by conservation of flux (particles are isorotating).

W  The confinement condition is then: W  Lehnert B. Nucl. Fus.

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485 (1971)

Conclusions

• Alpha particle energy can be used to maintain the radial electric field by driving a current volumetrically, eliminating the need for electrodes.

• A group of waves was found and simulations show 2.5 MeV may be recovered for each resonant alpha particle.

• Using these waves, mirror devices with supersonic rotation can be self-sustaining.

• Rotation in plasma centrifuges may also be driven by waves

Supersonic rotation provides many benefits to mirror traps One finds by energy conservation, assuming , Thus the parallel energy confinement time, Because the confining potential is much larger than the ion thermal energy, the ion population is Maxwellian, and not subject to loss cone instabilities.

The centrifugal force is stabilizing to interchange modes for Sheared rotation and parallel conductivity can also stabilize MHD modes.

Ionized neutrals enter the plasma with the rotation energy, providing a natural heating mechanism. Bekhtenev AA, et al, Nucl. Fus.

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579 (1980), Pastukhov VP, Rev Plasma Physics 13 203 (1987), Lehnert B, Phys Scripta 13 317 (1976) Volosov, V. Plasma Physics Rep 35 719 (2009).

Power Source Electrodes Using endplates to drive rotation introduces technical difficulties Mirror coils Dissipated current Externally driven current Magnetic field lines • • Electrodes must support extremely high voltage drops (several million volts overall) Plasma density at the mirror throat must be sufficient to provide conductivity between bulk plasma and electrodes

Lines intersect an insulator Driving the current volumetrically eliminates some of these problems Mirror coils RF driven current Dissipated current Magnetic field lines • • • Field lines intersect an insulating surface This surface can be designed to support high voltages and avoid limitation by the Alfv én critical ionization velocity There is no condition on conductivity between the bulk plasma and mirror throat

Alpha channeling can be useful for rotating plasmas • The radial electric field provides an extra energy source/sink. We want to convert alpha particle energy directly into electric potential energy.

• Prompt loss of alpha particles is a benefit as in other fusion devices: alpha particles take up plasma pressure that could be used to confine fuel.

• To keep a low ambipolar potential, the electron temperature must be kept low; alpha particles are an important electron heat source.

Regular alpha channeling Alphas Wave energy Tail ions No waves Rotation energy Electrons Fisch NJ Phys Rev Lett

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225001 (2006) Zhmoginov AI and Fisch NJ Phys Plasmas

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042506 (2008) Fuel ions

Alpha channeling in rotating mirrors introduces the radial potential In the frame rotating with the ions, the RF wave appears Doppler shifted, The change in the particle’s midplane coordinates are then, including the centrifugal potential, The change in radius implies to a change in lab-frame potential energy, Fetterman AJ and Fisch NJ. Phys Rev Lett

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205003 (2008)

B

r

~ W  0 Three options are apparent for energy transfer r 0 =0 r 0 =r w W llres (a) (b) (c) ~ W ||0 (a) Kinetic and potential energy are reduced; the wave is amplified; the particle exits through the loss cone.

(b) Kinetic energy reduced and potential energy increased; the wave may be amplified or damped; the particle may leave through loss cone or at the outer wall.

(c) Kinetic and potential energy are increased; the wave is damped; the particle is removed at the outer wall.

Paths (a) and (c) exist for a wave with positive phase velocity in the rotating frame . Path (b) requires a negative phase velocity.

Path (b) allows us to convert alpha particle energy to potential energy. Fetterman AJ and Fisch NJ. Phys Rev Lett

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205003 (2008)

Stationary perpendicular waves have a branching ratio of 1 We define the

branching ratio

to be the ratio of potential energy gained to kinetic energy lost. Assuming Note that for waves that are stationary in the lab frame, In order for the particle to satisfy the resonance condition we must then use a wave with azimuthal mode number, so that For practical values of  and  c , this implies a mode number of 20 or higher. The difficulty of launching these modes is mitigated by the fact that the plasma is localized near the cylinder boundary.

Fetterman AJ and Fisch NJ. Phys Rev Lett

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Fetterman AJ and Fisch NJ. Phys Plasmas

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205003 (2008) 042112 (2010)

Multiple wave regions are used to cover phase space Phase space: Real space: Magnetic field lines Alpha particle birth energy Loss cone Wave properties Device properties Fetterman AJ and Fisch NJ. Phys Plasmas

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042112 (2010)

The perturbation appears as a fast Alfvén wave in rotating coordinates In rotating coordinates, the stationary ripple appears with a wave with, The electromagnetic fields inside the plasma are related to B 1z , which is, where k r is the solution to the two fluid cold plasma dispersion relation. This solution is matched to vacuum solutions and a current layer at the wall.

Alpha particles are removed quickly compared to the slowing down time 26% of particles exit promptly, 32% are removed by alpha channeling in 1 s. The slowing down time is 4 s.

Alpha particles exit at low energy after alpha channeling Particles that undergo alpha channeling return 2.5 MeV to the potential, or 64% of their total energy

Rotation may also be produced by waves in plasma centrifuges Stripping section Injection cell Enriching section Tails (Waste) Heads (Product) •The separative power is proportional to (rotation speed)^4 •Electrodes can react strongly with separation products--removing them simplifies design •Power consumption is comparable to gas centrifuges Fetterman AJ and Fisch NJ. Plasma Sources Sci Tech

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045003 (2009) In the rotating frame: Drift Drag Rotation

Conclusions

• The concept of a branching ratio was developed to describe the unique interaction of particles with the wave and potential energy.

• Alpha particle energy can be used to drive a radial current in centrifugal mirror machines, so that end electrodes are not necessary for rotation.

• Simulation of a group of stationary waves shows that an average of 2.5 MeV may be recovered per resonant alpha particle, allowing a reactor to be self-sustaining.

• The branching ratio and alpha channeling concepts were also applied to plasma centrifuges.

End

Power balance is achieved without end electrodes Consider an example reactor with the simulation parameters and L=40 m (not optimized for power production) Power recovered Alpha particles lost: radially (19%): 1.4 MeV, axially (49%): 640 keV channeled (32%): 2.5 MeV 550 kW Particles that leave axially must overcome the centrifugal potential and leave with less rotation energy Axial fuel loss: Deuterium: 320 keV Tritium: 480 keV 440 kW Power consumed Fuel used for fusion Fuel lost axially D: 400 keV, T: 600 keV -400 kW -550 kW Net power to potential 40 kW Total fusion power: 9 MW

Aspect ratio and centrifugal beta The expression for beta is modified to include the centrifugal pressure, The second term in this expression is proportional to the plasma size r 0 -r 1 , unlike the first term. Thus, for fixed beta there is an optimum plasma layer thickness a.

Plasma cross section at midplane: a Fusion power production is maximum for fixed beta if, r 1 r 0 Bekhtenev AA, et al, Nucl. Fus.

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579 (1980).

Lehnert B. Physica Scripta

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229 (1974)

In rotating mirrors, the midplane resonance depends on radius The radius changes with perpendicular energy as: The resonant energy including centrifugal effects is: r 0 =0 r 0 =r w ~ W  0 Lines of resonance Diffusion Path W llres ~ W ||0 For perpendicular diffusion, the particle stays in resonance as it moves radially, but diffusion paths are not parallel to resonance lines at fixed radius