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Wave induced supersonic rotation in mirrors Abraham Fetterman and Nathaniel Fisch Princeton University

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

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Rotation is produced by crossed electric and magnetic fields W W Centrifugally confined Mirror confined Loss cone FcFc E B The confinement condition is then: Electrodes Mirror coils Magnetic field lines F c|| 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). Lehnert B. Nucl. Fus. 11 485 (1971)

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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

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Supersonic rotation provides many benefits to mirror traps One finds by energy conservation, assuming Thus the parallel energy confinement time, Bekhtenev AA, et al, Nucl. Fus. 20 579 (1980), Pastukhov VP, Rev Plasma Physics 13 203 (1987),, The centrifugal force is stabilizing to interchange modes for Sheared rotation and parallel conductivity can also stabilize MHD modes. Because the confining potential is much larger than the ion thermal energy, the ion population is Maxwellian, and not subject to loss cone instabilities. Ionized neutrals enter the plasma with the rotation energy, providing a natural heating mechanism. Lehnert B, Phys Scripta 13 317 (1976) Volosov, V. Plasma Physics Rep 35 719 (2009).

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Using endplates to drive rotation introduces technical difficulties 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 Electrodes Mirror coils Magnetic field lines Externally driven current Dissipated current Power Source

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

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Alpha channeling can be useful for rotating plasmas No waves Alphas ElectronsFuel ions Regular alpha channeling Wave energy Tail ions Rotation energy 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. Fisch NJ Phys Rev Lett 97 225001 (2006) Zhmoginov AI and Fisch NJ Phys Plasmas 15 042506 (2008)

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Alpha channeling in rotating mirrors introduces the radial potential In the frame rotating with the ions, the RF wave appears Doppler shifted, B r 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 101 205003 (2008)

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Three options are apparent for energy transfer (a) Kinetic and potential energy are reduced; the wave is amplified; the particle exits through the loss cone. (c) Kinetic and potential energy are increased; the wave is damped; the particle is removed at the outer wall. (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. Path (b) allows us to convert alpha particle energy to potential energy. Paths (a) and (c) exist for a wave with positive phase velocity in the rotating frame. Path (b) requires a negative phase velocity. W0W0 W ||0 r 0 =0r 0 =r w ~ (a) (b) (c) High potential (+) Low potential (-) W llres ~ Fetterman AJ and Fisch NJ. Phys Rev Lett 101 205003 (2008)

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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,so that In order for the particle to satisfy the resonance condition we must then use a wave with azimuthal mode number, 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 101 205003 (2008) Fetterman AJ and Fisch NJ. Phys Plasmas 17 042112 (2010)

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Multiple wave regions are used to cover phase space Phase space: Real space: Wave properties Device properties Loss cone Alpha particle birth energy Magnetic field lines Fetterman AJ and Fisch NJ. Phys Plasmas 17 042112 (2010)

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The perturbation appears as a fast Alfvén wave in rotating coordinates 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. In rotating coordinates, the stationary ripple appears with a wave with,

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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.

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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

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Rotation may also be produced by waves in plasma centrifuges Injection cellEnriching sectionStripping section Heads (Product) Tails (Waste) Rotation Drag Drift In the rotating frame: 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 18 045003 (2009)

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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.

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End

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Power balance is achieved without end electrodes Power recovered Alpha particles lost: radially (19%): 1.4 MeV, axially (49%): 640 keV channeled (32%): 2.5 MeV Axial fuel loss:Deuterium: 320 keV Tritium: 480 keV Power consumed Fuel used for fusion Fuel lost axially 550 kW 440 kW -400 kW -550 kW D: 400 keV, T: 600 keV Net power to potential40 kW Total fusion power: 9 MW Consider an example reactor with the simulation parameters and L=40 m (not optimized for power production) Particles that leave axially must overcome the centrifugal potential and leave with less rotation energy

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Aspect ratio and centrifugal beta The expression for beta is modified to include the centrifugal pressure, Bekhtenev AA, et al, Nucl. Fus. 20 579 (1980). Lehnert B. Physica Scripta 9 229 (1974) 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. r1r1 r0r0 a Plasma cross section at midplane: Fusion power production is maximum for fixed beta if,

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In rotating mirrors, the midplane resonance depends on radius The resonant energy including centrifugal effects is: For perpendicular diffusion, the particle stays in resonance as it moves radially, but diffusion paths are not parallel to resonance lines at fixed radius Lines of resonance Diffusion Path W0W0 W ||0 r 0 =0r 0 =r w ~ W llres ~ The radius changes with perpendicular energy as:

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