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

W

W

Centrifugally confined

Mirror confined

Loss cone

Fc

E

B

The confinement condition is then:

Electrodes

Mirror coils

Magnetic field lines

Fc||

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)

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,

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 forSheared 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).

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 currentPower Source

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 currentLines intersect an insulator

Alpha channeling can be useful for rotating plasmas

No waves

Alphas

Electrons Fuel ions

Regular alpha channeling

Waveenergy

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)

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)

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.

W0

W||0

r0=0 r0=rw~

(a)(b)

(c)

Hig

h po

tent

ial (

+)

Low

pot

entia

l (-)

Wllres

~

Fetterman AJ and Fisch NJ. Phys Rev Lett 101 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, 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)

Multiple wave regions are used to cover phase space

Phase space: Real space:

Wave properties

Device propertiesLoss cone

Alpha particle birth energy

Magnetic field lines

Fetterman AJ and Fisch NJ. Phys Plasmas 17 042112 (2010)

The perturbation appears as a fast Alfvén wave in rotating coordinates

The electromagnetic fields inside the plasma are related to B1z, which is,

where kr 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,

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

Injection cell Enriching sectionStripping section

Heads (Product)Tails (Waste)

RotationDrag

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)

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

Power recovered

Alpha particles lost: radially (19%): 1.4 MeV,

axially (49%): 640 keVchanneled (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 kWD: 400 keV, T: 600 keV

Net power to potential 40 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

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 r0-r1, unlike the first term. Thus, for fixed beta there is an optimum plasma layer thickness a.

r1

r0

a

Plasma cross section at midplane:

Fusion power production is maximum for fixed beta if,

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

W0

W||0

r0=0 r0=rw

~

Wllres~

The radius changes with perpendicular energy as: