Eﬃciency of tokamak magnetic ﬁeld in conﬁning plasma: Plasma beta

A.4 Eﬃciency of tokamak magnetic ﬁeld in conﬁning plasma: Plasma beta

To characterize the eﬃciency of the magnetic ﬁeld of tokamaks in conﬁning plasmas, deﬁne the plasma β, which is the ratio of the thermal pressure to the magnetic pressure, i.e.,

β = --2p---. B ∕2μ0

(536)

Since the pressure in tokamak plasmas is not uniform, the volume averaged pressure is usually used to deﬁne the beta. In tokamak plasmas, the toroidal beta β_t and the poloidal beta β_p are deﬁned, respectively, by

⟨p⟩ βt = B2-∕2μ-, t0 0

(537)

⟨p⟩ βp =---2-----, ⟨B𝜃⟩s∕2μ0

(538)

where ⟨…⟩ is the volume averaging, ⟨…⟩_s is the surface averaging over the plasma boundary, B_t0 is the vacuum toroidal magnetic ﬁeld at the magnetic axis (or geometrical center of the plasma). In tokamaks, the toroidal magnetic ﬁeld is dominant and thus the the toroidal beta β_t (not β_p) is the usual way to characterize the the eﬃciency of the magnetic ﬁeld in conﬁning plasmas. Why do we need β_p? The short answer is that β_p is proportional to an important current, the so-called bootstrap current, which is important for tokamak steady state operation. Alternatively, β_p can be understood as characterizing the eﬃciency of the plasma current in conﬁning the plasma. This can be seen by using Ampere’s law to approximately write the average poloidal magnetic ﬁeld B_pa ≈ μ₀I_p∕(2πa). Then β_p is written

⟨p⟩ βp ≈ ---2----2-2, μ0Ip∕(8π a)

(539)

which is the ratio of the pressure to the plasma current, and thus characterizes the eﬃciency of the plasma current in conﬁning the plasma.

Tokamak experiments have found that it is easier to achieve higher β_t in low B_t plasmas than in higher B_t plasmas, which indicates that the eﬃciency of the magnetic ﬁeld in conﬁning plasma is a decreasing function of the magnitude of the magnetic ﬁeld.

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