# Advances towards pB11 Fusion with the Dense Plasma Focus

**ABSTRACT** The prospects for achieving net energy production with pB11 fuel have recently considerably brightened. Studies have shown that the multi-GG field potentially obtainable with modest dense plasma focus devices have the effect of reducing the flow of energy from the ions to the electrons and thus suppressing bremsstrahlung radiation that cools the plasma. We report here on new simulations that indicate that net energy production may be achievable in high-magnetic-field devices at peak currents as low as 2.3 MA. While these simulations only model the dense plasmoid formed in the focus, new simulation techniques can allow a full particle-in-cell simulation of DPF functioning over the wide range of time and space scales needed. Such simulations will be of great value in the next round of experiments that will use pB11 fuel.

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**ABSTRACT:**The high Q-value of some (p,α) fusion reactions is very important in the investigation that can lead to power production with controlled fusion using advanced fuels (hydrogen-lithium-7, hydrogen-boron-11). For this reason, it is crucial to know the rates of these fusion reactions. Unfortunately, in the fusion machines such as plasma focus device, the interaction energy is usually far below the Coulomb barrier. Because of that, direct measurements of the relevant reaction cross sections are practically impossible. A few different indirect approaches have been proposed. In this work the Trojan Horse Method (THM) will be described. On the basis of the results obtained from the THM method and data, which are well-known from our previous work (Banjanac et al. in Radiat Meas 40:483–485, 2005), the reaction rate for proton-induced reaction 7Li(p,α)α produced in the hydrogen plasma focus is calculated. This calculation will be compared with the measurements of α particles production rate using CR-39 detectors. KeywordsPlasma focus–Trojan horse methodJournal of Fusion Energy 01/2011; 30(6):487-489. · 1.00 Impact Factor

Page 1

ADVANCES TOWARDS PB11 FUSION

WITH THE DENSE PLASMA FOCUS

Eric J. Lerner

Lawrenceville Plasma Physics

Robert E. Terry

Naval Research Laboratory

ABSTRACT

The prospects for achieving net energy production with pB11 fuel have recently

considerably brightened. Studies have shown that the multi-GG field potentially obtainable

with modest dense plasma focus devices have the effect of reducing the flow of energy from

the ions to the electrons and thus suppressing bremsstrahlung radiation that cools the

plasma[1]. We report here on new simulations that indicate that net energy production may be

achievable in high-magnetic-field devices at peak currents as low as 2.3 MA. While these

simulations only model the dense plasmoid formed in the focus, new simulation techniques

can allow a full particle-in-cell simulation of DPF functioning over the wide range of time and

space scales needed. Such simulations will be of great value in the next round of experiments

that will use pB11 fuel.

1.

INTRODUCTION

Controlled fusion with advanced fuels, especially hydrogen-boron-11, is an extremely

attractive potential energy source. Hydrogen-boron fuel generates nearly all its energy in the

form of charged particles, not neutrons, thus minimizing or eliminating induced radioactivity.

The main reaction, p+11B-> 34He, produces only charged particles. A secondary reaction,

4He+11B-> 14N +n does produce some neutrons as the alpha particles produced by the main

reaction slow down in the plasma, but only about 0.2% of the total fusion energy is carried by

the neutrons, whose typical energy is only 2.5 MeV. Hydrogen-boron fuel also allows direct

conversion of charged-particle energy to electric power, without the expensive intermediate

step of generating steam for turbines.[1-3] While this fuel requires extremely high ion

energies, above 200 keV, there is evidence that such energies can be achieved in the dense

plasma focus [1] as well as in the z-pinch [4].

However, because of the z2 dependence and boron's z of 5, bremsstrahlung x-ray radiation

is enhanced for p11B fuel. Many analyses have indicated that fusion power can barely if at all

exceed plasma cooling by bremsstrahlung [5]. If unavoidable, this situation would eliminate

the heating of the plasma by the fusion-produced alpha particles and would require that all the

energy be recovered from the x-ray radiation.

But published analyses have overlooked an important physical effect that is especially

relevant for the use of p11B with the dense plasma focus. This effect, first pointed out by

McNally [6], involves the reduction of energy transfer from the ions to the electrons in the

presence of a strong magnetic field. This in turn reduces the electron temperature and thus the

bremsstrahlung emission.

For ions colliding with electrons with gyrofrequency ωg, energy transfer drops rapidly for

impact parameters b> vi/ ωg, where vi is ion velocity, since in that case the electron is

accelerated at some times during the collision and decelerated at others, so average energy

transfer is small. This means that the bmax is less than the Debye length, λD by a factor of vi ω

Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

Page 2

p/ vet ωg, where ωp is the plasma frequency and vet is the electron thermal velocity. So the

Coulomb logarithm in the standard energy-loss formula is reduced to Ln (mvi2/ћ ωg).

This formula is approximately valid for collisions in which ions collide with slower

moving electrons, which are the only collisions in which the ions lose energy. But for

collisions of faster moving electrons with ions, where the electrons lose energy to the ions, the

Coulomb logarithm, by the same logic, is Ln(mve2/ћωg). If ve>> vi then Ln(mve2/ћωg) can be

much larger than Ln(mvi2/ћωg) for sufficiently large values of ћωg,,

sufficiently large B. Ignoring momentum transfer parallel to field, steady state occurs when

Ti/Te = Ln(mve2/ћωg)/Ln(mvi2/ћωg) [6].

This effect has been studied in a few cases for fusion plasmas with relatively weak fields,

where is shown to be a relatively small effect [7]. It has been studied much more extensively

in the case of neutron stars [8]. However, until the present research, it has not been applied to

the DPF plasmoids, whose force-free configuration and very strong magnetic fields make the

effect far more important. The author first demonstrated the importance of this effect for the

DPF in 2003[1]. This paper reviews the earlier work and extends it with new simulations.

in other words for

2.

MAGNETIC FIELD EFFECT IN DPF PLASMOIDS

The dense plasma focus device produces hot-spots or plasmoids, which are micron-sized

magnetically self-contained configurations with lifetimes of nanoseconds to tens of

nanoseconds. It is within these plasmoids that the plasma is heated to high energy and fusion

reactions take place. Such plasmoids have been observed to have magnetic field as high as 400

MG and density in excess of 1021/cc [1,9,10,11].

As shown in [1] and recapitulated here in Section 3, much higher densities and field

strengths, into the multi-GG range, seem possible with the DPF.

To apply the magnetic effect to the DPF plasmoids, which are force-free configurations,

we first note that small-angle momentum transfer parallel to the field can be neglected in these

plasmoids, since the ion velocity lies very close to the local magnetic field direction, and ∆ppar/

∆pperp ~sin2θ, where θ is the angle between the ion velocity and the B field direction[8].

In a force-free configuration, such as the toroidal vortices that make up the plasmoids,

ions disturbed by collisions return to the local field lines in times of order 1/ωgi, so

gici ω

/

ωθ≈

(1)

Where ωci is the ion collision frequency. For a decaborane plasma, θ~2x10-8n/Ti3/2B. For

the example of the plasmoid conditions obtained in [1], ni=3x1021, B =400MG, θ= 0.01 for

Ti=60keV. For an example near break-even conditions, ni=1.4x1024, B =16 GG, θ= 0.004 for

Ti=600keV Small-angle parallel momentum transfer is significant only for combinations of

very high ni and , Ti < 60keV, which generally do not occur except during very brief early

phases of the heating and compression of dense plasmoids, as we shall see in Sec. 4.

Even more significantly, the high B in plasmoids generates a regime where mvi2/ ћωg<1.

In this case the magnetic effect is very large, the above formulae break down and quantum

effects have to be considered. Such a situation has not been studied before for fusion

applications, but has been analyzed extensively in the case of protons falling onto neutron

stars[8].

In a strong magnetic field, since angular momentum is quantized in units of , electrons

can have only discrete energy levels, termed Landau levels (ignoring motion parallel to the

magnetic field):

(6 .11/) 2/ 1(eVmcBenEb

ћ

)() 2/ 1GGBn

+=+=

(2)

Page 3

Viewed another way, electrons cannot have gyroradii smaller than their DeBroglie

wavelength. Since maximum momentum transfer is mv, where v is relative velocity, for

mv2/2< Eb almost no excitation of electrons to the next Landau level can occur, so very little

energy can be transferred to the electrons in such collisions. Again ignoring the electron's own

motion along the field lines, thus condition occurs when

bi

EmME)/(

<

(3)

For Ei =300keV, this implies B>14GG for p, B>3.5GG for α , and B>1.3GG for 11B. As

will be shown below, such field strengths should be attainable with the DPF.

If we assume that Teth>> Eb, then we have to consider the motion of the electrons along the

field lines, which can increase the relative velocity of collision, v. In the classical case, the

ions will lose energy only from electrons for which vepar<vi. Since for these collisions v<2vi,

energy loss will still be very small if Ei<1/2(M/m) Eb, which can occur for boron nuclei.

However, there is a phenomenon which prevents energy loss to the electrons from falling

to negligible levels. In the classical case, considering only motion along the line of force, an

ion colliding with a faster moving electron will lose energy if the electrons' velocity is opposite

to the of the ion, but will gain energy if they are in the same direction--the electron overtaking

the ion. In the latter case the relative velocity is less than in the former case, and since the

energy transfer increases with decreasing relative velocity, there is a net gain of energy to the

ion. For an ion moving faster than the electron, the ion overtakes the electrons, and thus loses

energy independently of the direction that the electron is moving in. Thus ions only lose

energy to electrons moving more slowly than they are.

In the situation considered here, ions in some cases can lose energy to electrons that are

moving faster than the ions. Consider the case of ions moving along the field lines colliding

with electrons in the ground Landau level. If vepar is such that m(vi+vepar)2>2 Eb, while m(vi-

vepar)2<2Eb, the energy lost by the ion in collision with opposite-directed electrons will much

exceed that gained in same-directed collisions, since in the first case the electron can be

excited to a higher Landau level, but not in the second case. In neither case can the electron

give up to the ion energy from perpendicular motion, as it is in the ground state. (So, this

consideration does not apply to above-ground-state electrons, which will lose energy to

slower-moving ions.)

From these considerations, we can calcuatle an effectivce Couulomb logarithm, which is

done in [1]. The result is present in Table 1

Table 1

T

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.8

1.0

2.0

3.0

4.0

6.0

LnΛ Λ(T)

0.536

0.525

0.508

0.491

0.48

0.473

0.47

0.474

0.488

0.631

0.803

0.964

1.244

Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

Page 4

For the heating of the ions by the much faster thermal electrons, with Te>>1, quantum

effects can be ignored and the coulomb logarithm is simply Ln(2Te).

3.

CONDITIONS IN DPF PLASMOIDS

To see what the consequences of the magnetic field effect are for DPF functioning, we

first use a theoretical model of DPF functioning that can predict conditions in the plasmoid,

given initial conditions of the device. As described by Lerner [12], and Lerner and Peratt

[13], the DPF process can be described quantitatively using only a few basic assumptions.

Using the formulae derived there, Lerner [1] showed that the particle density increases with µ

and z as well as with I, and decreases with increasing r. Physically this is a direct result of the

greater compression ratio that occurs with heavier gases, as is clear from the above relations.

Thus the crucial plasma parameter nτ improves with heavier gases.

The theoretical predictions of the formulae in [1] are in good agreement with the results

obtained experimentally ,cited in the same paper. If we use these equations to predict Bc the

magnetic field in the plasmoid, we obtain 0.43 GG, in excellent agreement with the observed

value of 0.4 GG. Similarly, the formulae yield nτ = 4.6x1013 sec/cm3 as compared with the

best observed value of 9x1013 and the average of 0.9x1013.

For decaborane with z=2.66 and u =5.166, with r= 5 cm, I =3MA, the formulae in [1]

yield B =12GG and nt =6x1015. This is of course a considerable extrapolation-- a factor of 60

above the observed values in both B and nτ. However, these conditions can be reached with

relatively small plasma focus devices.

The limit on the achievable magnetic field is set mainly by the mechanical strength of the

electrodes. Since Bc for a given fill gas is proportional to Bi, the field at that cathode, a small

cathode radius is desirable. For proper DPF functioning, the anode radius generally must be

no more that 0.3-0.5 times the cathode radius. However, the anode is subject to thermal-

mechanical stress due to the transient heating and expansion of the outer layer by the discharge

current.

Thermal-mechanical simulations of the anode show that elastic stress limits will be

reached for copper electrodes at anode field strengths above 200kG for cooper electrodes and

380 kG for beryllium electrodes. With a cathode/anode radius ratio fo 2.5, this implies a

maximum Bc of 15 GG with a beryllium anode.

4.

SIMULATION OF PLASMOIDS

To determine if net energy production is feasible with pB11, a zero-dimensional

simulation has been run that contains the relevant physics, including the magnetic field effect.

While not fully realistic, the simulation is adequate to show the impact of the magnetic effect

and the possibility for high fusion yields.

The simulation, by its zero-dimensional character, assumes that the plasma in the

plasmoid is homogenous. In addition the simulation assumes Maxwellian distributions for the

electrons, and hydrogen and boron ions. Helium ions, produced by the fusion reaction, are

assumed to cool to a Maxwellian distribution, but the fusion alpha particles are treated

separately, as described below, as they are slowed by the plasma. In accordance with

observation, it is assumed that the ions are all fully ionized.

Initial conditions include the radius and magnetic field strength of the plasmoid, the initial

density and average energy (or temperature) of electrons, hydrogen and boron ions (which are

summed to be pure B11). The model assumes a smooth, sinusoidal increase in magnetic field

Page 5

and a corresponding decrease in radius. After the magnetic field peaks, the radius is assumed

to be constant. Stable confinement is also assumed, again in accord with experimental results.

At each time step, electron and ion beams are generated, which evacuate particles from the

plasmoid and subtract energy from the magnetic field. Following the formulae above, at each

instant the current in the beam is Ic/4π2 and beam power is 5.5µ-3/4Ic/4π2 W. Beam power is

subtracted from the total magnetic field energy at each time step. All the energy in the electron

beam however, is assumed to heat the plasma electrons, so does not leave the plasmoid. This

assumption is justified by experimental results at much lower plasmoid densities than modeled

here, which indicate that nearly all electron beam energy is transferred to the plasmoid

electrons [1].

For each time step, the model calculates the x-rays emitted by the electron and the energy

exchange between the ions and electron and between the ions species, with each species

having its own temperature. The magnetic effect is included using an approximation to the

numerical values calculated above for the coulomb logarithm term. The approximation used is

3 . 15045 . 0 ln0045 . 0) (ln265 . 0) (ln 02 . 0ln

3 . 15507. 0 2729 . 0268. 00561 . 0ln

23

23

>+++=

<+−

T

+−=

forTTT

forTTTT

λ

λ

(4)

Where T is the dimensionless ion temperature, as defined above. This formula is valid up

to T=100, well beyond what occurs in these runs.

During the early part of the simulation run, when ion temperature is still relatively low,

<60keV, the average angle between ion velocity and magnetic field, θ~2x10-8n/Ti3/2B, is

significant, leading to a much larger λ. However, ion temperature rises so rapidly that θ

becomes insignificant after ~0.4 ns, out of multi-ns runs. Detailed comparisons indicate that

this factor has negligible effect on the simulation outcomes, so it has not been included.

Using the calculated temperatures and densities, the model calculates the thermonuclear

reaction rate, based on published rates and cross sections (Cox, 1991). It then calculates the

transfer of energy by individual collisions to the ions and electrons, using the Coulomb

logarithm of Ln(2Te) for the electrons.

As the fusion reactions increase the thermal energy of the ions and electrons, β, the ratio

of magnetic field energy to thermal energy decreases. When β=1, the model assumes that the

plasmoid radius remains constant and half the thermal power is converted into additional

magnetic field energy to maintain a constant radius. This is of course a gross approximation,

since we can not determine with a zero-dimension simulation whether the plasmoids will in

fact remain stable, expand slowly, or completely disrupt as the thermal energy increases.

A number of simulations were run, with a wide range of parameters. The result for a

“nominal “ run are first described, then an analysis of how these results change with various

parameters and by modifying the model assumptions. The nominal simulation has the

parameters described in Table 2

Table 2 Nominal Simulation Parameters

Peak Plasmoid magnetic field

Plasmoid core radius

Peak Electron Density

Time to peak compression

Fuel

13 GG

8.63 microns

3.8x1024/cc

0.7 ns

B10H1

Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

Page 6

These parameters were selected to produce a high fusion yield, yet be achievable with

existing large DPF devices.

As seen in Figure 1, the ratio of ion to electron temperature rises rapidly to a peak of

around 17, quite close to the figure anticipated by the ratios of the coulomb logarithm in the

above magnetic field effect calculations. As a result, by t=0.9 ns, thermonuclear power

exceeds x-ray power (at a point where the proton/electron energy ratio already exceeds 11)

and rapid heating of the ions occurs . The thermonuclear /x-ray power ratio peaks at 2.2 at t=

1.4 ns.

0

2

4

6

8

10

12

14

16

18

1.00E+001.00E+032.00E+033.00E+034.00E+035.00E+036.00E+03

Time

Proton-electron energy ratio

Figure.1. Ratio of ion to electron average energy vs time. Time is in steps of 30 ps.

As the magnetic field rises, the electron temperature actually momentarily falls, but then

rises again and reaches a peak of 57 keV at 2.0 ns, while the ion temperature passes right

through the optimal temperature of 600keV and peaks at over 940keV. Thermonuclear power

has already peaked at 1.4 ns as the ion temperature passes through the optimal 600keV level.

Ion and electron temperature start to decrease as thermal energy in the plasmoid matches

magnetic confinement energy and the particles start to do work against the magnetic field.

Most of the thermonuclear burn occurs as the plasmoid cools down and feeds its energy

into the ion beam and x-rays, so that at the peak of thermonuclear power production only 20%

of the fuel in the plasma has been fused, but by the end of significant thermonuclear burn at

around 18 ns, 82.5% of the fuel has been burned. The x-ray and beam pulses decay nearly

exponentially, but are at less than 1% of peak power by 15 ns and have a FWHM of 3 ns.

Total input energy in this example is 14.6 kJ, x-ray yield is 9.5 kJ and beam yield is 13.4

kJ, so total output energy exceeds input energy by a ratio of 1.57. Preliminary estimates

indicate that energy conversion of both the x-rays and the ion beam can reach 80% with proper

design, so that net energy production with close to 50% thermodynamic efficiency should be

Page 7

possible, if other losses in the entire system can be reduced to levels small in comparison.

Leaving the model assumptions the same, the magnetic field of the plasmoid can be varied.

This is the equivalent of varying the peak current of the DPF for fixed electrode dimensions.

The results are presented in Table 3. As shown there, the total output/input ratio continues to

rise slowly above 13 GG but the beam output starts to decline above 11 GG, due to the in-

creasing density of the plasma and thus increasing x-ray emission relative to beam power. If,

as is likely, energy conversion of the ion beam is more efficient than conversion of the x-ray

pulses, 13 GG may be more desirable than higher fields, which may also be more difficult to

achieve.

Tabl e 3

Peak I

(MA)

1.92.32.83.24.3

B (GG)9.011.013.015.020.0

Gross Input

(kJ)

7.010.514.619.534.6

X-ray/In-

put

0.490. 550.650.751.00

Beam/Input 0.880.930.920.870.68

Beam+x-

ray/Input

1.371.481.571.621.68

Larger changes in yield are obtained when some of the assumption of the model is

modified. The density of the plasmoid is set, as described in section 3, by the condition that

the electron gyrofrequency is twice the plasma frequency, so the synchrotron radiation is just

barely trapped. The nominal cases calculate the density based on the condition being met in

the case of low-energy electrons. However, since the synchrotron frequency increases as γ2,

where γ is the relativistic factor and the plasma frequency decreases as γ -0.5, the ratio of the two

frequencies increases as γ 2.5 and the density for a given B increases as γ 5.

For example for Te of 30 keV, critical trapping for the synchrotron radiation occurs at a

density 1.32 times that of the nominal cases. In this case, x-ray yield increase to 1.05x input,

Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

Page 8

beam is 0.86x input and I/O ratio increases to 1.90, an increase of 20%. It should be noted,

however that there are not experimental results that indicate the higher n/B2 ratio is actually

obtained and they are instead consistent with the nominal ratio. This is probably due to the fact

that, in the force-free plasmoid configuration, the electrons travel along the lines of force,

whose curvature is considerably greater than the gyro-radius.

On the other hand, results may be decreased if we modify the assumption that the

plasmoid will not expand once thermal energy become equal to magic energy. If we instead

assume that current in the plasmoid will neither increase nor decrease due to the increased

thermal energy, the plasmoid will gradually increase in radius, decreasing density and

quenching the fusion reaction.

In this case, x-ray yield declines to 0.44x input while beam yield is almost unchanged at

0.88x input and total yield declines to 1.32, 17% less than the nominal case and too low,

probably, for net energy production with reasonable conversion efficiencies. However, there is

good reason to believe that this constant-current case is unduly pessimistic, since the decrease

in B due to expansion would induce electric fields that will increase the current, resisting

expansion.

5. DRIFT KINETIC FLUID PARTICLE SIMULATION

The preliminary zero-dimensional simulation has significant limitations that must be

overcome to achieve accurate predictions of DPF functioning. First, since we want to

maximize the transfer of energy into the plasmoid, we need a simulation that models the whole

DPF process, including the run-down phase and the formation of the plasmoid. Second, we

need to understand how the plasmoid will actually react to the release of dynamically

significant amounts of fusion energy. For both these purposes we need 3-D simulations.

However, the DPF presents real challenges for simulators that prevent conventional

techniques from being effective. The DPF involves processes that extend over a large range of

time and space scales, from the sub-ns evolution times and micron spatial scales of the

plasmoids to the microsecond time scale and multi-cm spatial scales of the rundown and

collapse phases. The plasma is non-collisional, with gyroradius being smaller than mean-free-

path, which means that MHD approximations do not give correct answers, and cannot simulate

the main features of filament formation, pinch collapse and plasmoid formation and decay. But

particle-in-cell simulations are not only handicapped by the wide range of scales, but even

more by the need to resolve frequencies as high as the electron gyrofrequency, which can be as

high as 1017Hz.

To overcome the limitations of conventional approaches, our novel computational

approach to a more robust and physically accurate model than that explored above rests with a

new drift kinetic fluid particle (DKFP) method, grounded in what are essentially exact

solutions in the limit of Knudsen or Vlasov ows.

fl

The following discussion shows how drift kinetic uid particles are built up from

fundamental Vlasov solutions and connected to observables from other parts of a model or

from experimental data. First, we set a local drift velocity and its gradient, together with local

acceleration and its gradient,

fl

(5)

as functions of local position within a box. The velocity eld is just the familiar mean

molecular velocity, and the acceleration eld is just the local imposed force on the particle

fi

species under consideration — electromagnetic, gravitational, or whatever. Next, the single

fi

Page 9

particle distribution functions are built up as outer products on the cartesian components of

con guration and velocity space,

fi

(6)

where the box in space extends initially over a width h in each dimension and the velocity

function has a characteristic thermal speed U, perhaps distinct in each velocity dimension. As

written the distribution function in (6) solves the Vlasov operator equation

(7)

exactly, in terms of the characteristics

And

Now, any velocity moment can be written as

(8)

And, with the auxiliary variables,

and one finds a density function

(9)

Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

Page 10

Figure.2. Three fluid particles collide. The acceleration and velocity fields are everywhere proportional and the

similarity S(x,t) is everywhere constant.

The expected pro les of uid variables and the usual moments are now all now available

fi fl

in closed form on all space and time as a particle set evolves. The action of pressure gradients

is built in as these particle interpenetrate, capturing the uid pressure automatically as a

superposition of partial pressures, as shown in Fig.2.

Collisions are introduced in either of two ways. First, in a weakly collisional regime, the

interactions may be described in relaxation time approximation. The momentum transfer,

elastic scattering, or even charge exchange cross sections are transformed into collision

frequencies and imposed on the drifting particles as apparent drag forces. This method can be

viewed as a sort of partial operator spitting insofar as those collision frequencies can be

updated without ever reforming the particle population.

Alternately, in a strongly collisional regime, transport uxes are computed into a target

volume and the implied time variations on any thermodynamic variables of interest are

computed for that volume from the collision operators required by the chosen formulation.

Here a complete operator split is demanded insofar are the particle population must be

annihilated and reformed with new values for densities, mean velocities and thermal speeds.

The strength of this method in relation to the problem of rapidly evolving plasmoids is in

the great freedom available to treat a wide range of uid regimes concurrently on a wide range

of spatial scales—with the same basic formulation. We intend to develop models based on this

method to initially simulate the formation of plasmoids.

fl

fl

fl

ACKNOWLEDGEMENTS

Page 11

Work performed for the U. S. Department of Energy under Contract No. W--7405--

ENG--36.

REFERENCES

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Current Trends in International Fusion Research – Proceeding of the Sixth Symposium.

Edited by Emilio Panarella and Roger Raman. NRC Research Press, National Research Council of

Canada, Ottawa, ON K1A 0R6 Canada, 2008.

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