Consider one nested flux surface in a large-aspect-ratio circular tokamak

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Fluid and magnetohydrodynamic description · Fusion plasmas

Plasma Physics, Ch Holding a plasma still: pinches, flux surfaces and the Grad-Shafranov equation

Problem

Consider one nested flux surface in a large-aspect-ratio circular tokamak, at minor radius rr with ϵ=r/R0=0.30\epsilon = r/R_0 = 0.30. Along a field line on this surface the field strength varies with poloidal angle θ\theta as B(θ)≈B0/(1+ϵcos⁡θ)B(\theta) \approx B_0/(1+\epsilon\cos\theta) (weakest on the outboard side, θ=0\theta=0; strongest on the inboard side, θ=π\theta=\pi), because B∝1/RB\propto 1/R and R=R0(1+ϵcos⁡θ)R = R_0(1+\epsilon\cos\theta) on this surface. ((a)) Find the mirror ratio Rmir=Bmax⁡/Bmin⁡R_{\rm mir} = B_{\max}/B_{\min} on this surface. ((b)) A particle crossing the outboard midplane has an isotropically distributed velocity direction there. Using the mirror trapping condition from adiabatic-invariant theory, find the fraction of such particles that are trapped on the low-field side rather than passing all the way around the flux surface. (Use the standard result that, for an isotropic distribution at the weak-field point, the fraction with sin⁡2α\sin^2\alpha above the loss-cone value Bmin⁡/Bmax⁡B_{\min}/B_{\max} is 1−Bmin⁡/Bmax⁡\sqrt{1-B_{\min}/B_{\max}}.) ((c)) Explain briefly why, despite this local mirroring, the tokamak's overall confinement of the guiding centre does not rely only on μ\mu conservation and a loss cone in the way an open mirror device does.

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