Estructuras de convivencia actuales sin una vivienda a su medida

In this example we will use the combined formalism of Chapter 4 to solve the equilibrium problem with translational symmetry. We choose the (dimensionless) parallel pressure to be

Pk=λeA(1 +kB), (6.9)

where λ is some scaling parameter and k is an anisotropy parameter. We have chosen this parallel pressure in such a way that we can avoid having to deal with some of the more technical parts of the combined formalism, which we will highlight in due course. The free function is chosen to be the constant

F = 1, which is used in conjunction with the parallel pressure to find the shear field from the constraint equation (2.36), thus we have

By=

B

B−λkeA. (6.10)

The poloidal magnetic field strength is then found to be

Bp=B

r

1− 1

B−λkeA, (6.11)

which is shown graphically in Figure 6.5a forλ= 1 andk= 0.1. Using (6.10) and (6.11) we are able to implicitly reconstruct a plot ofBy(A, Bp), which we show in Figure 6.5a (also forλ= 1 andk= 0.1). We

find that the parallel pressure (6.9) leads to a shear field in the poloidal formalism that is single-valued and exists for all values ofAand positive values ofBp. Indeed this is one of the reasons that the parallel

pressure (6.9) was chosen in the first place, as it eliminates some of the difficulties that occur in the combined formalism (there are no branch switching possibilities, for example).

Now we are almost in a position where we can solve the Grad-Shafranov equation with the contin- uation method described in the Chapter 5, but first we must find the effective parallel pressure and its

Figure 6.5: Plots of (a) the poloidal magnetic field strength in terms ofAandB, and (b) the shear field as a function ofA andBp (i.e. in the poloidal formalism) for the parametersλ= 1 andk= 0.1.

derivatives with respect toAandBp. From (3.25), we find that the effective parallel pressure is

Pk?=λeA(1 +kB) +

F2_{B B−2λke}A

2(B−λkeA₎2 . (6.12)

However, (6.12) givesPk? in terms of the total field formalism, rather that the poloidal variablesAand Bp. Therefore, to findPk?, we will make use of an iterative scheme which has 5 steps:

1. we are given values ofA=A0 andBp=Bp0and wish to find Pk?(A0, Bp0);

2. find the initialB0 such thatBp(A0, B0) = 0;

3. increase the value ofB0incrementally (with a fixed step size) until Bp(A0, B0)> Bp0;

4. decrease the step-size and proceed to findB0such thatBp(A0, B0) =Bp0with a bisection method;

5. recover the effective parallel pressure by the equalityPk?(A0, Bp0) =Pk?(A0, B0).

We should note that this method relies on certain properties of the function Bp(A, B), for instance it

must be monotonically increasing (with respect toB), and one must be able to find an expression for the initial function B0(A) (such that Bp(A, B0(A)) = 0). Fortunately, the parallel pressure (6.9) does

lead to a poloidal magnetic field strength that is monotonically increasing with B, and one can show that the function

B0=λkeA+F (6.13)

satisfiesBp(A, B0(A)) = 0 (i.e. we do not have any problems finding the starting point for the iterative

are required for the continuation code. As an example, we require derivatives of Pk? with respect to A

andBp, which can be found in terms ofAandB by applying the chain rule, giving

∂ ∂Bp Pk?(A, Bp) = _∂B p ∂B −1 _∂ ∂BPk ?_{(A, B) (6.14)} and ∂ ∂APk ?_{(A, B} p) = ∂ ∂APk ?_{(A, B)− ∂B} p ∂B −1_∂B p ∂A ∂ ∂BPk ?_{(A, B). (6.15)}

To evaluate these for values of A and Bp, we simply use the same method as above to find the corre-

sponding value ofB.

We now solve the Grad-Shafranov equation in the poloidal formalism using the continuation method described in the previous chapter. The domain is chosen to be −1 ≤ x≤1 and −1 ≤z ≤ 1. As an initial solution to the problem when λ= 0, we choose the magnetic flux function to beA0 =x2−z2,

and we will specify that asλincreases, the magnetic flux function will be fixed on the boundaries where it will take the value ofA0.

We find that there are two solutions with the parameters λ= 1 andk= 0.1, which lie on opposite sides of a bifurcation point in the solution branch. Plots ofA,Bp andBy for each solution are shown in

Figure 6.6, where the plots on the left hand side correspond to the first solution (before the bifurcation point), and the plots on the right hand side correspond to the second solution (after the bifurcation point). The first solution has a single X-point in the centre of the domain (see Figure 6.6a), which has been compressed in thez-direction in comparison to the original X-point present in the initial solution

A0. In the second solution, the X-point in the centre of the domain has transformed into an O-point,

which is shown in Figure 6.6b. The X-point and the O-point in both cases correspond to a region of zero poloidal magnetic field strength, as shown in Figures 6.6c and 6.6d, although the contours of Bp

in the second solution are much more compact than those in the first. Plots of the shear field for both solutions are shown in Figures 6.6e and 6.6f. In the isotropic case,By would have been unity over the

whole domain, however this is not the case in the anisotropic equilibria. In the second solution,Bypeaks

in the centre of the domain at just over twice as large as the isotropic case. In the first solution there is less of a difference, but there is a strip along thex-axis where the shear field is about 20% larger than the isotropic case, which increases towards the edge of the domain. As one expects from the plotBy(A, Bp)

in Figure 6.5b, the regions where By is most different from the isotropic case coincide with regions of

low poloidal magnetic field strength and high values of the magnetic flux function.

At this point we should remind ourselves of a common simplification in the literature, namely that of the shear field being set equal to the free functionF (e.g Mercier and Cotsaftis, 1961; Clemente, 1993; Shi et al., 2006). If this simplification had been made in the examples above we would not have captured

(a) (b)

(c) (d)

(e) (f)

the detail, and indeed magnitude, of the shear field. In fact, this approximation would not just yield the wrong values for the shear field, but variations in the shear field with respect to By play a large roll in

the Grad-Shafranov equation itself. Thus, the approximationBy=F would also result in wrong values

for the magnetic flux function and poloidal magnetic field strength.

This example has highlighted the general technique which should be used when applying the combined formalism of Chapter 4. One is able to work with the parallel pressure as a function ofB, yet still make use of the poloidal Grad-Shafranov equation. In this way we do not have to rely on numerical derivatives of quantities such as ∂P_k?/∂Bp, since analytical expressions are available. We have also been able to

show that the approximation By = F is inadequate, at least for the examples of equilibria above. In

fact, this is also shown to be the case in the next example.