Gun requirements to achieve high field spheromaks Page: 4 of 14
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2
I = j (V/B) = k (a) V/p = f ko V/ t , (1)
where f = k./X(a) is the injection efficiency [4] and , = [tj/B, k(a) being the value in the
flux core and k0 being the lowest Taylor eigenvalue. As the spheromak begins to fill the
volume, k -- initially less than ko -- saturates at X.. After that, j and B continue to increase
but j /B = k /po is constant in the spheromak. Ideally unstable flow of magnetic energy out
of the flux core causes k in the flux core to track that in the spheromak, giving f = 1 at
saturation (as was assumed in Figure 1). In fact k always sags significantly, inward during
injection and outward during decay [4]. We shall characterize the sag in X by the parameter
a used by Jarboe to characterize experimental results [4], given here by:
- a = ( k(a ) - ,X )/ X. = f 1 - 1 . (2)
Typical values inferred from magnetic probe measurements give a = - 0.3 during
injection [4]. Even so the free energy -- the difference relative to the lowest Taylor state at a
given helicity -- is of order a, hence small enough to justify taking the Taylor state as
representative of the spheromak; the actual free energy is 2% for a = - 0.3 [4]. From
Eq. (3), a = - 0.3 corresponds to f = 0.75, compared to Taylor's ideal value of unity, and
the overall efficiency f (1 - AV/V) > 0.5 for typical values, consistent with the assumption
of 50% efficiency in recent conceptual design studies for Pulsed Spheromak reactors [1].
We will return to the physics determining f in Section 3.
2. Gun Circuit -- Choosing the Voltage
Since helicity is better conserved than energy, the fundamental "circuit" equation is
that for helicity injection, given by [3, 4]:
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Fowler, T K. Gun requirements to achieve high field spheromaks, report, March 4, 1999; California. (https://digital.library.unt.edu/ark:/67531/metadc619507/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.