Shafranov shift correction to the Furth–Yoshikawa scaling of tokamak adiabatic compression

In 1970, Furth and Yoshikawa (1970 Phys. Fluids13 2593–6) introduced the scalings of adiabatic plasma compression. Basically, if the shape of the external plasma boundary and the aspect ratio are preserved during the compression, then the density, kinetic pressure, beta and current scale respectively as , , , , , where C is the linear compression ratio, that is, the ratio between initial and final major radii. In this work, we show analytically, by expanding the Grad–Shafranov equation in terms of C, that the deviation to the Furth–Yoshikawa scaling is related to the Shafranov shift that arises when beta increases at large compression ratios. There is an obvious effect of the Shafranov shift because the axis is moved to a region with larger volume element, and an indirect effect, associated to the relation between flux and radius. The latter effect adds to the first, and is of the same order of magnitude. The result is that the pressure increases less than the C5 scaling, which can have a significant impact on the fusion power achieved at maximum compression. The analytical results are backed up by equilibrium simulations carried out with the CHEASE code. Equilibria are obtained for different values of C, with conservation of the total fluxes, q profile, and entropy of the plasma. The agreement of the theory and simulations is very good when the boundary of the plasma is circular and the aspect ratio small. When the aspect ratio is close to 1, and/or the boundary not circular, the analytical result gives the gist of the reduction of compression. Finally, a pressure anisotropy approximately equal to the increase in normalized Shafranov shift is predicted.