In addition to the particle-fluid characterization of a plasma, one may identify probabilistic methodologies which, in selected applications are most useful. We discuss here one such approach to the assessment of particle leakage from a fusion reaction volume.

Consider a volume V containing N j(t) particles of the j-type. This population changes with time because of gain and loss reaction rates, R +j and R _j, respectively, and also because of particle injection and leakage rates, F*+j and F*_j, Fig.6.2. The rate equation is evidently

^=(/?;,- — r-і)+(fIj — f:}) . (6.43)

Our interest is specifically in the particle leakage rate F _j and we wish to explore an approach which has no need for a detailed space distribution of the particle population; that is, we consider a probabilistic formulation in an integrated global sense for this arbitrary volume.

A N) / N*j At

As our underlying physical basis, we adopt the proposition that the statistically expected fractional change in the population due to particle leakage is — AN*j/N*j in an interval of time At, and is only a function of time. This statement can be expressed in algebraic form as

1 dN) N*• dt

where A, j(t) is a positive function of time to be considered subsequently. For now, we take the limit of At —> dt to write

(6.47)

However, this expression tells us little about the meaning of A-j(t) nor does it provide a suggestion on how to calculate it. This can be resolved if we introduce some probabilistic considerations.

Consider the probability density function fj(t) which describes the outcome that a particle j remains in the volume V over a time t until it escapes from the ensemble. Hence, this probability is given by the product of the following probabilities which also define fj(t):

	probability of	probability of
f j(t)dt-	non — leakage	leakage during
	until time t	time dt

(

L V о )_

Note that the substitution here follows directly from Eq.(6.44). Thus, the probability density function of interest is evidently

(6.49)

with the normalization J f j(t)dt = 1 as required.

о

With fj(t) now specified, we may compute some interesting time dependent quantities. Of particular interest is the particle mean residence time until leakage. Using x j for this quantity, we may compute it as

where in the context of nuclear fusion, x ) is often called the global particle leakage or confinement time. A good estimate of x*j can frequently be obtained from a knowledge of the mean particle speed and the mean cumulative distance a particle travels in a reaction volume. We add that by a similar derivational development, one may show that the energy leakage rate-that is power leakage-from a fusion reaction volume is given by

E(t) He*

where E*(t) is the total energy content of the reaction volume and XE* is the global mean energy leakage time or global energy confinement time.

P.=

(6.53)

Problems 6.1 Specify the conditions which reduce Eq.(6.21) to its simplest form while still retaining electric and magnetic field effects. 6.2 Consider a particle density N(x, y, z, E, 0, ф) and formulate a reduced transport equation, Eq.(6.42), for each of the following cases: (a) neutral particles, (b) monoenergetic particles, (c) only the z-direction is relevant, and (d) both the source and collision terms are given. 6.3 Consider a plasma contained between two walls xQ units apart and effectively infinite in the у and z-directions. For the case that the plasma decays by diffusion to the walls and for separability of the form N(r, t) = R(r) T(t), determine N(r, t) throughout this plasma slab region for all time. Sketch this density function.

6.4 Confirm that for a cylindrically-shaped, magnetically confined plasma wherein radial diffusion is the dominant mode of transport, assuming a constant volumetric ion source, the steady state ion density will have the following relationship:

N(r)=N(0)jl-^, (6.54)

where N(0) is the ion density at r = 0, and a is the plasma minor radius. (Note that for these conditions, Eq.(6.10b) becomes VxJ(r) = S = constant.)

6.5 Given Eq.(6.54) for the ion density radial distribution, derive the average ion density.