1. Introduction

In tokamaks the plasma configurations are described in terms of solutions of the Grad-Shafranov equation (GSE) [1-2]. Analytical solutions of the Grad-Shafranov [1-3] equation are very useful for theoretical studies of plasma equilibrium, transport and magnetohydrodynamic (MHD) stability. For tokamak operation, measurements of the poloidal beta , plasma thermal energy and plasma temperature are important. Much more plasma parameters such as plasma confinement time, plasma current density profile, and magnetohydrodynamics (MHD) instabilities can be calculated.

As we know confinement is limited by thermal conduction and convection processes, but radiation is also a source of energy loss. Since maximum energy confinement time is determined by the microscopic behavior of the plasma as collisions, and micro instabilities. Such kind of behavior ultimately leads to macroscopic energy transport which can be either classical or anomalous depending on the processes involved.

In the absence of instabilities the confinement of toroidally symmetric tokamak plasma is determined by Coulomb collisions. Since these phenomena require knowledge of individual particle motion on short length and time scales, they are usually treated by kinetic models, but including only limited geometry because of the complexity of the physics. In the absence of a theoretical understanding of confinement, and given the need to predict the confinement properties of future tokamaks, it has been necessary to resort to empirical methods. The simplest of these is to accumulate data from a number of tokamaks, each operated under a range of conditions, and to use statistical methods to determine the dependence of the confinement time on the parameters involved. This provides scaling expressions which, within some error, allow extrapolation to projected tokamaks [1-15].

Lots of lithium experiments have been carried out in tokamaks for the enhancement of plasma [4-11]. In this work we calculated the the plasma thermal energy by the Simplest Grad–Shafranov Equation (GSE) Solution [12] with Lithium limiter for Circular cross-section HT-7 tokamak. A generalized Grad-Shafranov-type equation [3], has been used. Specific functional forms of plasma internal energy and current are used. GSE is solved by considering linear source functions and fixed boundary conditions. For this, the poloidal beta [13, 14] was obtained from by analytical solution of GSE. Than we can find the plasma thermal energy [15]. It was clearly observed in [14] that the calculated internal inductance and the calculated poloidal beta depend on kind of discharge or plasma current.

It is observed, the plasma thermal energy obtained from the analytical solution of GSE by using liquid lithium limiter is larger than that using graphite limiter, which shows that the plasma performance was improved. Also, the plasma thermal energy was measured using the diamagnetic loop [15].

2. Theoretical Equation

The simplest solution of GSE can be found by assuming that [12],

(1)

where and are constant. The internal energy in extended Grad-Shafranov equation [3] is a function of . The and are two free functions, while and are the vacuum permeability and plasma current density respectively. This obviously reduces the set of possible current density profile shapes to .

If the plasma is assumed to be up–down symmetric, its shape can be described by four parameters. The equatorial innermost and outermost points, and , and the coordinates of the highest point, or equivalently, the major radius , , the minor radius , the elongation , and triangularity .

The simplest solution is given by [12]

(2)

For the determination these six coefficients, it is necessary to have six equations. We assume that the internal energy vanishes at the boundary, hence [16].

With Eq. (2), the boundary conditions and gives the following equations:

(3)

(4)

(5)

We also assume that the plasma is enclosed in a perfectly conducting toroidal boundary with circular cross section, with radius a, so the normal component of the magnetic field.

(6)

The plasma current can be clearly measured by Rogowski coil [17], so the plasma current can be written.

(7)

It is simpler to first solve for a plasma with unit current and unit major radius and use the scaling relations described above to find the final desired equilibrium. However, even in this simplest case only numerical solutions to Eqs. (3-7) have been found. The coefficients can be computed numerically, given a desired plasma description.

We also selected the constraint , because the parameter can be experimentally deduced using discrete magnetic probes [17], for circular cross section HT-7 tokamak [18],

(8)

3. Calculation of Plasma Thermal Energy

The six coefficients c₁, c_2, c₃, c_4, A₁, A₂ can be derived by solving equations [3-8]. So the poloidal magnetic flux is obtained by substituting the six coefficients.

According to the definition of the internal inductance [18]

(9)

We can acquire the internal inductance with Solovev’s assumption [18]

(10)

Figure 1. Plasma Averaged Thermal Energy, obtained analytically by the solution of GSE as a function of line averaged electron density

Also the poloidal beta with Solovev’s assumption can be determine as follow [18]

(11)

All the double integrals in the above written relations have to be performed over the total cross-section area of the plasma column for circular cross section HT-7 tokamak [17]. We can determine the volume-averaged plasma kinetic pressure , and then the plasma thermal energy U. can be determined directly from the definition of the poloidal beta [18]:

(12)

where a is the plasma minor radius. For the measurement of the plasma thermal energy, we start from the plasma state equation [15]:

(13)

where subscript ‘i’ indicates the plasma species i and E indicates the plasma thermal energy density; therefore the plasma thermal energy U is obtained [15]:

(14)

where V is the plasma volume.

Therefore, according to the above discussion, the poloidal beta [18] was obtained by analytical solution of GSE and then the plasma thermal energy is calculated. It was clearly observed in [14] that the calculated internal inductance and the calculated poloidal beta depend on kind of discharge or plasma current. It is observed from figure, the plasma thermal energy, obtained from the analytical solution of GSE by lithium limiter is larger than that using graphite limiter, which shows that the plasma performance was improved. Also, the plasma thermal energy was measured using the diamagnetic loop [11, 15, 17].

4. Result and Discussion

Here we calculated the plasma thermal energy by the Simplest Grad–Shafranov Equation (GSE) Solution with Lithium limiter for circular cross-section HT-7 tokamak. GSE is solved by considering linear source functions and fixed boundary conditions. For this, the poloidal beta [13, 14, 18] was obtained by analytical solution of GSE and then the plasma thermal energy is calculated.

It was clearly observed in [14] that the calculated internal inductance and the calculated poloidal beta depend on kind of discharge or plasma current. It is observed, the plasma thermal energy obtained from the analytical solution of GSE using liquid lithium limiter is larger than that using graphite limiter, which shows that the plasma performance was improved. These parameters included the poloidal beta and plasma thermal energy can be measured by experimental method based on a diamagnetic effect [11, 15, 17].