An equilibrium state of the plasma confined by magnetic fields must exist in order to maintain the hot plasma away from the vessel walls. While the general solution of equilibrium states of a magnetically confined plasma constitutes a complicated problem, some useful physical relations follow readily from viewing the plasma as a single fluid (Sec. 6.3) subjected to the magnetohydrodynamic (MHD) description. The equilibrium exploration suggests that one consider the steady state MHD equations from which the plasma current density j and the

magnetic field В can be self-consistently derived, i. e., in agreement with Maxwell’s Equations. Consequently, an equilibrium plasma must satisfy the previously introduced force balance (Eq.(9.2))

Vp = j xB (10.22)

and the magnetostatic Maxwell equations

VxB = ^oj (10.23)

and

V-B = 0. (10.24)

As in Sec. 9.1, Eq. (10.22) implies that j and В lie on isobaric surfaces. Note that taking the divergence of Eq. (10.23) yields

V-j = 0 (10.25)

which corresponds with the quasi-neutrality assumption Ni ~ Ne.

For an axisymmetric tokamak plasma the isobaric surfaces will be nested toroidal surfaces, as previously mentioned. We assign now to each such surface a function ¥ = constant such that 2тгіЕ represents the poloidal magnetic flux through a toroidal plane extending from the minor axis (r = 0) to the considered isobaric surface at r represented by the shadowed area in Fig. 10.8, i. e.

2^: = jBe — dre. (10.26)

Since isobaric surfaces need not necessarily have circular cross sections, here r0 denotes the radial distance of an isobaric surface at a specified poloidal position

0. Then we may approximate the flux passing through the planar ribbon between the two isobaric surfaces labeled 4* and 4* + d^ and encircling the major axis with a radius R by

2л(‘¥ + cM>) — 2iW = JBe • dre — jBe • dre — Be2nRdr (10.27)

о о

with dr representing the radial distance between the considered surfaces. From Eq.( 10.27), the following relation between the poloidal magnetic field strength and the flux surface function ¥ is now apparent:

dfV

RBe=————————————————— (10.28)

dr

or, respectively, in vectorial form according to the orientation shown in Fig. 10.8,

Bfl=4*xv*- (io-29>

к

Similarly, due to the symmetry of j and В in Eq. (10.22), there exists a corresponding poloidal current function F^) such that the poloidal component of the current density can be expressed as

j e = 4e*xVF

which-via Eq. (10.23)-determines the toroidal field as

Вф = -^ЄФ — (Ю.31)

К

Since F = constant for ‘F = constant, it is therefore also seen that B^xR is constant on a magnetic surface.

Z Fig. 10.8: Toroidal and poloidal field components in a cylindrical coordinate system, also indicating the nested flux surfaces *F = constant.

Next, we decompose the poloidal magnetic field into its components along the R and z axes of Fig. 10.8 by multiplying Eq.( 10.29) with the respective unit vectors eR and ez. Noting that

we find

1 ЭЧ*

(,0.33a)

and

(10.33b)

Using these expressions the toroidal component of Eq. (10.23) is then

d_J_W 1 92vE dR R dR R dz2

which defines the operator

dR R dR dz2

used in the following. Upon introducing the flux and the current function in the force balance, Eq. (10.22),

Vp = (j, + je)x (в* + Be) = (i* "j^e* * VF j x (вф — x VV j, (10.36a)

and recognizing the pertinent orthogonalities, we obtain

Vp(T’) = ^-VT’-^-VF(T’). (10.36b)

R R

Since Vpf’E) = VT’ and VFf’F) = VT’, Eq.(10.36b) provides the relation

which is a nonlinear elliptic partial differential equation for the magnetic flux 4* in terms of the pressure distribution and the poloidal current function F. The corresponding boundary conditions are provided by the transformer-induced poloidal magnetic field outside the plasma. In evaluating Eq. (10.38) it is seen and has been discussed earlier that the total magnetic field must also possess a uniform vertical part Bv in order to prevent the current-carrying plasma from radial expansion; only then can a toroidal plasma be maintained in equilibrium. The topological magnetic field structure effected by the imposition of Bv is demonstrated in Fig. 10.9 where, in the resulting field, the domain of closed magnetic surfaces is separated by a so-called separatrix from the open surface contours intercepting the wall of the reactor chamber.

In practice, the Grad-Schlueter-Shafranov Equation, Eq.( 10.38), is solved numerically to find the geometrical location of the magnetic surfaces and the radial distribution of the axial current density in a way that is consistent with the

experimentally measured pressure profiles and the externally applied field. That is, the functions p(‘F) and F(vf/) are given for the numerical evaluation. For plasma pressures which are not too high, solutions of Eq.( 10.38) exist in the form of closed nested flux surfaces inside the separatrix indicated in Fig. 10.9.

which is the ratio of the volume-averaged plasma pressure to the energy density of the poloidal magnetic field averaged over the magnetic flux surface at r = a. Figure 10.10 exhibits some typical features of toroidal equilibria as they are affected by the plasma pressure. A more detailed analysis shows that the toroidal

field contributes to the pressure balance by the difference } ~ (5p2)

affected by the poloidal current components which may either increase or decrease the toroidal field in the plasma. A plasma is paramagnetic when the volume-averaged square of the toroidal field strength, (B^), exceeds the outer

flux surface average, (b^ ^ and it is diamagnetic for (b^ > (В,2). Asa

consequence, the diamagnetic plasma is associated with an average plasma pressure greater than ^Вe2^ J{2jio), that is with pp > 1, while pp < 1

corresponds to a paramagnetic plasma. Whereas the total beta of a tokamak plasma is constrained to values of a few percent for MHD stability reasons, the poloidal beta appears in the range 0.1 < pp < 2.5 for the equilibrium regimes displayed in Fig. 10.10.

It is observed that the magnetic axis of the torus column is displaced outwards with increasing plasma pressure until this displacement D is about half of the minor plasma radius, which can be shown to manifest the limit of

Surfaces on which p = constant

Surfaces on which j(j, = constant

diamagnetic behaviour of plasma decreases toroidal field; occurrence of reverse current

Fig. 10.10: Axisymmetric tokamak equilibrium characteristics varying with plasma
pressure. The symbol l; appearing in b) denotes the plasma internal inductance per unit

2 n

Byrdr

™2(Be)v(a)

equilibrium. The poloidal magnetic field must be capable of supporting at least the fraction a/R0 of the kinetic pressure, that is

2 Vo Ro

which obviously restricts (3P to values

Pp<R0/a. (10.40b)

Recalling the Kruskal-Shafranov stability limit in Sec. 10.1 which suggests that the safety factor at the plasma edge should be q(a) > 2.5, and further extracting from the geometric sketch in Fig. 10.11 the obvious relation

Be (а) аАв

by using the previous definitions l = Д0(Дф = 2%) and q = 2m.. Taking the square of Eq.(10.42) and assuming the approximate equivalence of

Be (a) with (я2 ) and of Я2 with [в] ) ^, we express the averages of

both Be2(a) and Вф2 by means of their (З-value to obtain <P> . о _ a PP ^ 1 a

Вф/2ц0 * Ro q2(<*)~ 6.25 Rl

which essentially is the toroidal beta, (3,, for large aspect ratios. Hence, for example, a tokamak featuring an aspect ratio Ro/a = 3 will be limited to (3 < 5% by principal MHD stability considerations, while the poloidal beta may amount to values in the order of Ro/a.

Another important consequence resulting from the poloidal-beta limitation

becomes evident when the poloidal magnetic induction is expressed-via Ampere’s Law-by the plasma current generating the Be-field. Doing so, we introduce Eq. (9.50) into Eq. (10.42) to obtain

JL->q(a)^- (10.47)

Vo1 a

2m

That is, the plasma current cannot be increased arbitrarily in order to achieve higher temperatures through Ohmic heating. Its magnitude appears to be bounded for stability reasons. We inject to note that the above derivation of plasma-beta constraints associated with simple MHD stability limits represents just a rough assessment nonetheless demonstrating the useful and important relations in an illustrative manner. Thorough analysis would provide more accurate factors in the several inequalities at the expense of extended expressions and more detailed concepts to be introduced.