The Grad-Shafranov operator is ^* = R R ( 1 R R ) + ^2 Z^2

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

The Grad-Shafranov operator is Δ∗ψ=R∂∂R(1R∂ψ∂R)+∂2ψ∂Z2,\Delta^*\psi = R\frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial \psi}{\partial R}\right) + \frac{\partial^2\psi}{\partial Z^2}, and it has already been shown that Δ∗[(R2−R02)2]=8R2\Delta^*[(R^2-R_0^2)^2] = 8R^2 and Δ∗[R2Z2]=2R2\Delta^*[R^2Z^2] = 2R^2. Consider the trial flux function ψ(R,Z)=C18(R2−R02)2+C22R2Z2+C32Z2,\psi(R,Z) = \frac{C_1}{8}(R^2-R_0^2)^2 + \frac{C_2}{2}R^2Z^2 + \frac{C_3}{2}Z^2, with C1C_1, C2C_2 (units T m−2\mathrm{T\,m^{-2}}) and C3C_3 (units T\mathrm{T}) constants and R0R_0 the intended major radius. ((a)) Show, by evaluating Δ∗[Z2]\Delta^*[Z^2] directly from the operator above and combining it with the two given identities, that this ψ\psi solves the Grad-Shafranov equation Δ∗ψ=−μ0R2p′(ψ)−FF′(ψ)\Delta^*\psi = -\mu_0R^2p'(\psi) - FF'(\psi) with p′(ψ)p'(\psi) and FF′(ψ)FF'(\psi) both constants (independent of ψ\psi), and give μ0p′\mu_0p' and FF′FF' in terms of C1C_1, C2C_2, C3C_3. ((b)) Expand ψ\psi to leading order about the point R=R0R=R_0, Z=0Z=0, writing x=R−R0x = R-R_0, and show that the nearby flux surfaces are ellipses. Give their elongation κ\kappa (ratio of the semi-axis along ZZ to the semi-axis along xx) in terms of C1C_1, C2C_2, C3C_3 and R0R_0. ((c)) A compact tokamak has major radius R0=1.65 mR_0 = 1.65\,\mathrm{m} and a measured elongation near its magnetic axis of 1.51.5. Taking C1=4.0 T m−2C_1 = 4.0\,\mathrm{T\,m^{-2}}, C2=1.0 T m−2C_2 = 1.0\,\mathrm{T\,m^{-2}} and C3=2.0 TC_3 = 2.0\,\mathrm{T} as the parameters of the fitted Solov'ev-type solution, compute κ\kappa from your formula and compare it with the measured value.

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