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New Conserved Quantities Around a Magnetic Surfaces for Plasma Equilibria with Non-linear Convective Terms and Low Vorticity: A Review.

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LABORATORIO DE FÍSICA DE PLASMA

Julio Puerta, Enrique Castro and Pablo Martín

Departamento de Física, Universidad Simón Bolívar, Apdo. 89000,

Caracas 1080A, Venezuela.

Non-linear convective terms and viscosity modify the plasma confinement in tokamaks in such a way that magnetic surfaces are not anymore coincident with the isobars[1,2]. Though a complete non-linear analysis can only be performed by plasma simulation[3], however analytic results can also be obtained in the low vorticity case, where same simplifications can be carried out. In previous paper the equilibria analysis was performed here using a generalized tokamaks system of coordinates published a few years ago, with seems well suited for plasma of any shape. Here our previous analysis is extended to more general conditions, and we discuss in detail the new conserved quantities around a each magnetic surfaces, which in this case are different to the pressure. In this way a new extended Grad-Shafranov equation is derived including the non-linear terms due to convectivity and low vorticity.

As in a previous work the differences between isobars and magnetic surfaces are considered for different plasma configurations[4]. However in the present paper the analysis is extended to a more general plasma shapes than those Haas type considered previously[5,6]. Now the type of generalized surfaces recently described by other authors[7-9] are also included in our calculations. In this way some preliminary results are obtained by elliptic plasmas with different ellipticity. Shafranov shift and triangularity values are also introduced in the analytic forms describing this kind of magnetic surfaces. A new function F (,T), including pressure and kinetic terms has been found, which is preserved on the magnetic surfaces.

Tokamak equilibria and stability with arbitrary flows is the great current interest. In the usual Grad-Shafranov (G-S) equation the flows are not taken in account and isobars and magnetic surfaces are coincident [1,2]. However neutral beam injection produces large toroidal plasma rotations and the nonlinear convective terms in the momentum equation become important for equilibrium. Generalized G-S equation can be deduced in this case [3-5]. G-S type equations including toroidal flows have been solved using numerical codes based in the finite element method (FEM) and an energy formalism [6]. This new extended G-S type equations are not elliptic and very complex to be solved.

A simplified analysis in the low vorticity case has been performed recently [7]. Here we are extending this preview analysis and performing some calculation in order to show the discrepancies between isobars and magnetic surfaces in a simple way. Our analysis include viscosity terms, however for the calculations here performed those viscous terms are not taken into account. In the theoretical calculation, viscosity is considered in the simple model used by other authors for transport analysis [8,9]. However here we do not have any singularity in ν, and the results for ideal MHD are obtained when ν tends to zero.

In order to get a clear picture of the discrepancies between isobars and magnetic surfaces we have considered the simple have considered the simple sharp boundary model for a high pressure Tokamak described by F. A. Haas[10]. In this model we are making the changes due to the flow terms, and later isobars and magnetic surfaces are determined and compared. In this paper we are using generalized coordinates described in previous works [11].

In the next Section 2 a differential equation for the poloidal flux is calculated for the low vorticity including flow terms. In Section 3 the equations to obtain the plasma density ρ is derived. In Section 4, numerical calculations are performed and several figures showing isobars and magnetic surfaces are shown and the discrepancies are analyzed. Section 5 is devoted to discussion and conclusion.

Steady-state conditions by axis-symmetric toroidal geometry are assumed, thus, we have to find the solutions of the three coupled equations

and

(1)

where , and ν are the velocity, current and viscosity respectively. Montgomery [5] has shown that the viscosity can change the topology of solutions, in such a way that when viscosity approaches to zero, results of the ideal MHD can not be found as a limiting case due to the fact that ν appears in the denominator of the solution and therefore this tend to be singular. If an adiabatic or isothermal condition could be assumed, it is possible to define a function by

(2)

- where T is the temperature of a magnetic line and it change when the line changes, but is the same for each line.
- The non-linear convective therms can be decomposed using the vorticity

in the form:

(4)

(3)

- On the other hand, we have
- (5)
- By substitution in Eq.(1) we get,
- (6)
- Now defining the function through the expression
- (7)

We find the very important formula:

(8)

Neglecting vorticity and viscosity we obtain the equation:

(9)

- From this equation through scalar multiplication by and we arrive to the following boundary conditions for the function :
(10)

Consequently the surfaces family

(11)

coincides with the magnetic surfaces.

Now, in order to discuss the main problem we use generalized Tokamaks coordinates introduced in a previous paper [11]. It is easy to see in Figure 1, that the function is only a function of the generalized coordinates and not of . Thus we can write down this function in the following form

(12)

- Introducing the poloidal flux function , this is also only a function of and we write simply . Now it is possible to derive a modified Grad-Shafranov equation. In order to do that, it is convenient first to calculate along a magnetic field line. It is known that
(13)

Using well-known vector analysis identity we found

(14)

Furthermore, due to axis-symmetry, we get

(16)

By using the results in Ref. [11], we obtain

(17)

Integrating

(18)

where the subindex 1 means the value of this function for s=0, which is usually taken as the most external point [11]. Now using the definition of poloidal flux function, as defined before, we will obtain an extended Grad-Shafranov equation. The rule of thumb will be to replace the pressure function by taking care with the fact that is variable along a magnetic line. The rotational using the coordinate, already mentioned, gives

(20)

where

This is a new type of Grad-Shafranov equation, for the conditions explained before. Solving this equation we obtain , and from here we obtain numerically each isobar =constant.

In order to compare with other authors we simplify our analysis considering the particular case of magnetic surfaces described by Haas, as mentioned before [10]. There he assumes that the pressure and magnetic field are derived from the poloidal flux through

(21)

where a and b are dimensionless free parameters to be chosen defending the plasma geometry. Here Ro is the mayor radius and rois one of the toroidal coordinates. The pressure vanishes at the plasma boundary, that is, . Considering now the Eq. (20) and (9) we get

(22)

or in a compact form

(23)

For Haas the pressure is constant in a magnetic surface, but now instead of the pressure, the functionin our description, is constant along a magnetic surface. Therefore considering the similarities the new assumptions will be

(24)

where Fcis the value of the function F at the minor magnetic axes. The equation for will be

(25)

For Haas the pressure is constant in a magnetic surface, but now instead of the pressure, the functionin our description, is constant along a magnetic surface. Therefore considering the similarities the new assumptions will be

(26)

where Fcis the value of the function F at the minor magnetic axes. The equation for will be

(27)

Defining now a dimensionless densityand also the dimensionless distance x=1-R/Ro, then the isobars will be determined by the equation

(28)

The density 1 at r = rmax , wherermaxis given by equation

(29)

Where The magnetic flux surfaces are given, in this model by the relation

(30)

Now in order to compare with other model we choose a plasma model in Solov`vev type equilibrium giving in parametric form by following definitions(Grishanov et al) in quasi-toroidal coordinates(r, , )

(31)

where

(32)

with r an non - dimensional radius ( 0 < r < 1).

Now defining

(33)

and ,

,

(34)

we found for the following expression

(35)

Solving this equation for we get the isobars. Figures 9 – 13 shown the comparison between isobars and magnetic surfaces where discrepancies are proofed including triangularity and ellipticity.

In figure 2 we shown magnetic surfaces for different Q values versus isobars. In order to compare both surfaces the isobars begin at same rmax than the magnetic surfaces.. From the figure is clear that discrepancies are larger for the outer surfaces. In the inward directions decrease this effect due to the fact that the relation magnetic pressure is higher and the model is not so correct. In figure 3 the effect of the parameter is investigated. The discrepancies are larger inward in comparison with the magnetic surface for Q = 0. We think that these effect is due to the increasing of the triangularity in this direction.

The effect of changing the inverse of the aspect ratio is shown. We see that for higher aspect ratio the discrepancy decrease. With the other model the discrepancies seems to be smaller. This is shown in figures 6 – 13 several triangularities and ellipticities .

- A new extended Grad – Shafranov equation has been derivate including non – linear terms as convectivity and resistivity, ignoring the viscosity and assuming low vorticity. The form of this extended equations is simpler than previously published and it isobars and magnetic surfaces can be compared numerically. In order to perform the calculations De Haas type model and Grishanov model has been used. A clear discrepancy between isobars and magnetic surfaces is demonstrated. Though isobars are not coincident with magnetic surfaces . On the other hand, a new function F(, v, T) that include pressure and kinetic terms has been found and is conserved in the magnetic surfaces.

1.- J. Weson, Tokamaks ( Oxford Science Publications, 2nd ed., 1997 ) pp.152-154

2.- R. B. White, Theory of Tokamaks Plasmas ( North Holland, Amsterdam, 1989 ) pp.162-175

3.- M. Furukawa, Y. Nakamura, S. Hamaguchi and W. Wakatami, J. Plasma and Fusion Res. 76, 937 (2000)

4.- J. Puerta, E. Castro and P. Martin, 5th Symposium on Current Trends in International Fusion Research: A Review. Washington D. C. , U. S. A. 24-28 March 2003. ( to be published )

5.- F. A. Hass, Phys. Fluid 15, 141 (1972)

6.- P. Martin, E. Castro and J. Puerta, Rev. Mexicana Física 49, 146 (2003)

7.- L. L. Lao, S. P. Hirshman and R. M. Wieland, Phys. Fluid 28, 1431 ( 1981)

8.- H. Weitzner 1981 appendix in [7]

9.- G. O. Ludwig, Plasma Phys. Control Fusion 37, 633 ( 1995 )

Figure1.Cross-section of the magnetic surfaces, showing the reference curves (- RL and s – RL), and the new coordinate and s . The Frenet frame at poin P is also shown together with the magnetic surface section and the orthogonal line through P.

Figure 2.Magnetic Flux Surfaces at Q = 0 (green), Q = - 0.64, and Q = - 1.11 (blue). Isobar at = 1.0 begin at the point of maximum pressure with the following parameters : = 2/3 and = 1/3

Figure 3.Magnetic Flux Surfaces at Q = 0 (green), at Q = - 0.64 (red) with isobar at = 1.0 for the following parameters : = 1/3 and = 1.5, 2.0, 2.5, 2.7

Figure 5.Magnetic Flux Surfaces at Q = 0(green) and Q = -0.64 (red). Isobars at = 1.0, 0.84, 0.70, 0.65. Parameters = 1/5 and = 2/3.

Figure 6.Magnetic Flux Surfaces with Ro = 0.86 m, R1 = 0.7 m and a = 0.5 m. Isobar at = 1.59 in a Tokamak with Solov`ev type equlibrium.

Figure 10. Magnetic Flux Surfaces with Ro = 0.86 m, R1 = 0.70 m, a = 0.5 m. Triangularity = 0.0 and Elipticity = 1.5 . Isobar at = 1.59

Figure 11.Magnetic Flux Surfaces with Ro = 0.86 m, R1 = 0.70 m, a = 0.5 m. Triangularity = 0.3 and Elipticity = 1.5 . Isobar at = 1.59