Theory of Tokamak Equilibria with Central Current Density Reversal

It is found that, with a model current profile, the Grad-Shafranov equation can be reduced to the Helmholtz equation, which can describe a variety of equilibrium configurations. With the eigenvalue problem solved in the toroidal coordinate system, an analytical solution to the Grad-Shafranov equation is found. It is demonstrated that current reversal equilibrium configurations exist with finite radial gradient of plasma pressure and continuous current density, and that current density reversal is accompanied by pressure gradient reversal.

Figure 1

Magnetic flux surfaces, normalized current density (open circles) and normalized pressure (open triangles) for a normal configuration with $I_N(1)=0.33$, $I_N(0.7)=0.50$, and ${\beta }_V=0.01$. $\alpha ={\lambda }_{0,1}-0.06960$, (${\lambda }_{0,1}=2.406904$), $a_1=-0.04531$, and $a_2=-1.0808.$.

Figure 2

Magnetic flux surfaces, normalized current density (open circles) and normalized pressure (open triangles) for a force-free CREC with $I_N(1)=0.25$, $I_N(0.7)=0.00$, and ${\beta }_V=0.02$. $\alpha ={\lambda }_{0,2}-0.04402$, (${\lambda }_{0,2}=5.520984$), $a_1=0.0$, and $a_2=0.05852$.

Figure 3

Magnetic flux surfaces, normalized current density (open circles) and normalized pressure (open triangles) for a CREC with $I_N(1)=0.25$, $I_N(0.7)=0.00$, and ${\beta }_V=0.02$. (a) $\alpha ={\lambda }_{0,2}-0.08818$, $a_1=0.1193$, $a_2=3.1034.$ (b) $\alpha ={\lambda }_{0,2}-0.02088$, $a_1=-0.06325$, and $a_2=-1.5527$.

Figure 4

Magnetic flux surfaces, normalized current density (open circles) and normalized pressure (open triangles) for a CREC with $I_N(1)=0.25$, $I_N(0.6)=0.00$, and ${\beta }_V=0.00$. (a) $\alpha ={\lambda }_{31}-0.00586$,(${\lambda }_{31}=6.379377$), $a_1=0.09234$, $a_2=0.1721.$ (b) $\alpha ={\lambda }_{3,1}-0.00102$, $a_1=0.0758$, and $a_2=-0.2507.$ (c) $\alpha ={\lambda }_{3,1}+0.03504$, $a_1=-0.04167$, and $a_2=-3.2465$.