A circular-cross-section tokamak is being designed with major radius

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

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

Problem

A circular-cross-section tokamak is being designed with major radius R0=3.0 mR_0 = 3.0\,\mathrm{m}, minor radius a=1.0 ma = 1.0\,\mathrm{m} and vacuum toroidal field B0=4.0 TB_0 = 4.0\,\mathrm{T} at R0R_0. The cylindrical edge safety factor is qa=2πa2B0/(μ0R0Ip)q_a = 2\pi a^2B_0/(\mu_0R_0I_p). ((a)) Find the plasma current IpI_p needed to give qa=3.0q_a = 3.0. ((b)) For this current, and taking the poloidal beta βp=0.50\beta_p = 0.50 and the normalised internal inductance ℓi=1.0\ell_i = 1.0 as design targets, find the vertical field Bv=μ0Ip4πR0[ln⁡(8R0a)+βp+ℓi2−32]B_v = \frac{\mu_0I_p}{4\pi R_0}\left[\ln\left(\frac{8R_0}{a}\right) + \beta_p + \frac{\ell_i}{2} - \frac{3}{2}\right] needed to hold the plasma ring in equilibrium. ((c)) A fuller treatment than the Solov'ev solution gives an equilibrium limit βp≲R0/a\beta_p \lesssim R_0/a beyond which the outer flux surfaces open up and no equilibrium of this shape exists. Evaluate this limit for the design and state whether the target βp=0.50\beta_p = 0.50 satisfies it.

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