Study of Effective Edge Safety Factor Using Analytical Solution of Grad-Shafranov Equation for Circular Cross Section Tokamak

In this work, we present effective edge safety factor using analytical solution of the Grad-Shafranov equation based on expansion of free functions of first order and magnetic probes for circular cross section HT-7 tokamak.


   
where  is The internal energy in this extended Grad-Shafranov equation is a function of The  and  are two free functions, and where 0  and J are the vacuum permeability and plasma current density respectively.

Effective Edge Safety Factor by Solution of GSE
For circular cross section HT-7 Tokamak [16,26] (see Table 1), which is the ohmically heated tokamak, the Grad-Shafranov equation [17,[27][28], is solved by formally expanding as follows [4,29]: where is the vacuum toroidal field at , , then first order flux function [29] is: If there are external coils to produce vertical magnetic field, the boundary condition on the flux function is modified so that we have: where  is the flux function due to external vertical field coils and therefore the full toroidal correction to  is: The shift of the plasma column center from the geometrical center of vacuum chamber is given by [29]: Therefore, the first relation for plasma position [29] is where p  is the poloidal beta, i is the internal inductance of the plasma, and is the average vertical l B B v magnetic field over the vacuum chamber.We can find v from saddle sine coil [22] and the sum of the poloidal beta and half the plasma internal inductance ex- , where ( is Shafranov pa- rameter) from magnetic coils measurement [22][23][24][25]30] using following equations where We measured these local magnetic fields with magnetic probes [22] at above angles.
Therefore the effective edge safety factor at the plasma edge is given by [31]: According to this relation, and also the value of Shafranov parameter which measured with magnetic probe we plotted time history of the effective edge safety factor in target shot for circular cross section HT-7 tokamak, as shown in Figure 1.

Conclusion
We have determined the effective edge safety factor using analytical solution of the Grad-Shafranov equation based on expansion of free functions of first order and magnetic probes for circular cross section HT-7 tokamak.


is a new free function replacing  F   .In the first order solution or toroidal force balance approximation and if plasma were surrounded by a perfectly conducting shell located at

Figure 1 .
Figure 1.Edge safety factor obtained for circular cross section HT-7 tokamak by the GSE solution and magnetic probes.