As a first fluidic description of a medium containing fusion fuel ions and sustaining fusion reactions, we consider a characterization which emphasizes particle mobility. Consider therefore an arbitrary volume V containing several time-varying particle populations Nj*(t), N2 (t), …, Nj*(t), …, each species characterized by some macroscopic kinetic property such as temperature. We take the volume’s surface to be non-reentrant and of total area A with its outward normal direction determined by the differential vector dA.

Our interest here is in the Nj (t) population which, in the case of a spatially varying number density Nj(r, t), is found from

N](t)=jNj(r, t)d3r. (6.3)

v

In general, the population Nj (t) can change with time because of various types of gain and loss reaction rates, R±j and because of inflow and outflow rates, F±j , across the total surface A. Hence, the rate equation for Nj*(t) is evidently

with the subscripts к and 1 enumerating different reaction types.

To begin, we now restrict ourselves to the case for which, in any fractional volume AV of V, the reaction contributions are zero or that the reaction gains are exactly canceled by the reaction losses; that is, we take

ад

к I

Additionally, we refer to the condition

F*+j = 0

which means there is no fueling by injection or other mechanisms, and we determine the outflow rate across the surface, accounting for global particle leakage, via the local particle current vector Jj(r, t) through

F. j = ij(r, t)-dk. (6.7)

A

As previously introduced, dA is an oriented surface element pointing outward. Obviously, particles leak out where J|A also points in this outward direction.

We now substitute Eqs.(6.3) and (6.5) — (6.7) into Eq.(6.4) to obtain for the space-time description of the j-type particle species

j-Nj(r, t)d3r = — J J j(r, t)-dA (6.8)

t V A

for the case specified by Eqs. (6.5) and (6.6). Then, in order to reduce both integrals to the same integration variable, we use Gauss’ Divergence Theorem

jjj(r. t) dA =jV-Jj(r, t)d3r (6.9)

Here, the partial derivative is used since the temporal change in Nj(r, t) is considered at fixed spatial coordinates, respectively. Since the volume V is arbitrary, we must therefore have

^-Nj(r, t) + V-Jj(r, t) = 0. (6.10b)

Э t 1

This important relation is known as the Continuity Equation and must hold everywhere in the volume of interest and for all times of relevance providing the assumptions imposed here hold. If, however, there were sources and sinks for the considered species j in the volume of interest-i. e. reactions which produce or consume j-type particles-or particle injection, the corresponding gain and loss rate densities would appear on the right hand side of Eq.(6.10b).

Though compact, Eq.(6.10b) however suffers from a severe shortcoming: it represents only one equation containing two unknown functions, the scalar particle density Nj(r, t) and the vector particle current Jj(r, t). Evidently, another relationship between these two quantities is necessary. As it turns out, this is often possible.

Many particle fluid media possess the property that the local particle current J(r, t) is proportional to the negative particle density gradient, — VN(r, t). This is often called a diffusion phenomenon and associated with the label Fick’s Law. Introducing a proportionality factor D, called the diffusion coefficient, we may write therefore everywhere in the medium

J j(r, t)=-DjVNj(r, t)

so that, by substitution into Eq.(6.10b) we get

Nj (r, t) + V • [- DjVNj (r, t)] = 0. (6.12)

Numerous reductions are now often applicable. If the medium is homogeneous and isotropic then the diffusion coefficient is a space-independent scalar, and we obtain

^-Nj(r, t)-DjV2 Nj(r, t) = 0. (6.13)

ot

Steady-state conditions in such a medium will yield Laplace’s Equation

V2 N j( r, t) = 0. (6.14)

The diffusion coefficient in Eq.(6.12), Dj, is clearly of importance in specifying the spatial variation of the particle density Nj(r, t). Considerable thought has gone into analytical characterizations of this parameter so that the diffusion processes be adequately incorporated. For example, for neutral particles in random thermal motion and in regions sufficiently far from boundaries, it has been found that, to a good approximation

Dj°cvjA, j (6.15)

where Vy is the mean speed of the j-type particles and A. j is the mean-free-path between scattering collisions among the j-type particles-and is much smaller than

(Vn/n)"1.

In contrast to neutral particles, charged particle motion in a magnetic field may involve considerable anisotropic diffusion governed by the local B-field requiring that a distinction between a direction which is parallel or perpendicular to the local В-field be made. Relationships with a good physical basis in classical diffusion are

where COg is the gyrofrequency and Tc is the mean collision time.

By analogy to the mean-time between fusion events Tfu, Eq.(4.3), we take herein

and recall that the Coulomb cross section Gs varies as v’4, Eq.(3.15).

Further, substituting for the averaged square velocity by the thermal energy E, h kT, or the kinetic temperature, respectively, and using the explicit

expression for COg given in Eq.(5.12), we rewrite Eq.(6.16a) and (6.16b) to exhibit the proportionalities Dux T5’2 (6.18a) 1 D. L x, r — ■ b24t (6.18b)

Thus, a higher temperature and an increasing magnetic field will reduce classical diffusion across the В-field lines. This conclusion must, however, be tempered by the recognition that collective oscillations in a plasma destroy some of the classical diffusion features. The plasmas of interest to nuclear fusion applications are severely affected by such collective processes through plasma instabilities which do not obey classical diffusion considerations.

The destabilizing oscillations induce turbulent phenomena which enhance diffusion across the magnetic field lines. Incorporation of these considerations leads to so-called Bohm diffusion characterized by

Db x ~~ (6.19)

D

thereby placing greater demands on magnetic confinement at higher temperature.

Finally, we add that the Continuity Equation for the particle density Nj(r, t), Eq.(6.13), translates into an equation for the mass density pj by using Pj(r, t) = Nj(r, t)mj,

^-Pj(r, t)-DjV2Pj(r, t) = 0 (6.20)

at

and is generally found to be of considerable utility, particularly in conjunction with other relationships as we will show next.