Example

In this section we present an example of a situation which involves discontinuous branch switching. We choose a normalized parallel pressure function to be

Pk=eA B

B+ 1, (4.32)

where quantities have been normalized in the usual way. For convenience we introduce a variablea, called the modified magnetic flux, such thatA= ln(a), which is a simpler quantity to work with. The actual magnetic flux function can always be reconstructed by taking the logarithm of the modified magnetic flux, which we should note is never allowed to take negative values. In this simple example we will assume the discontinuity is in they-z plane, thus the normal component is in thex-direction.

Let us now define some relatively arbitrary upstream parameters by

F = 0.3, a0= 1,

B0= 0.29,

Bx= 0.075.

In other words, upstream we have a magnetic field mainly in the tangential direction (with respect to the discontinuity), a magnetic flux function of zero, and a relatively small magnetic field strength. The other upstream quantities then follow, since the shear can be deduced from (2.36), and the remaining component in the z-direction can be found from the difference of the total magnetic field strength and

other two components. They are found to be

By0=−0.28,

Bz0= 0.013.

We now substitute our upstream quantities into (4.27) and (4.31). This gives us two equations which parametrically describe two curves ink-a1space, wherek is the tangential stretching parameter anda1

is the downstream modified magnetic flux. Points of intersection between these curves correspond to solutions of both equations, and thus the allowed downstream quantities. We show the curves in Figure 4.17.

Figure 4.17: Plots of the curves defined by (4.27) and (4.31) (red and blue lines respectively) with the parameters given in Section 4.5.1.

As can be seen from the diagram, there are precisely three solutions. One of these is the trivial solution where k = 1 and a1 = 1. This represents the case where there is no sudden change in any of

the quantities when we pass through the discontinuity. The other cases, however, represents a definite change in parameters. These solutions have the downstream parameters shown in Table 4.1.

αprofile β profile

k 0.07 -1.45

a1 1.49 0.21

By1 -0.02 0.4

Bz1 0.0009 -0.02

Table 4.1: Downstream stretching parameters, modified magnetic fluxes and tangential magnetic field components.

Theαandβ profiles distinguish the two possible downstream states, and values have been rounded to two decimal places. From the new values ofkdownstream, we see that in theαcase, the tangential magnetic field has decreased significantly in magnitude whilst maintaining its direction. In theβ case, we see that the tangential magnetic field has become about one and a half times larger in magnitude than it was upstream, and has flipped in direction. The modified magnetic flux function has also changed considerably, increasing by 50% in theαcase, and dropping by 80% in theβ case.

One way of visualising the changes across the discontinuity is to look at the shear field as a function ofBp. In Figure 4.18 we have plotted the shear field profile for the upstream and downstream values of

A, with an asterisk representing the precise value of the shear field in each of the three cases. The solid red line represents the upstream shear profile, the dashed blue line represents the downstreamαprofile, and the dot-dashed green line represents theβ profile.

Figure 4.18: Plots of By againstBp in the upstream and two downstream cases (red, blue and yellow

respectively). The asterisks represent the actual values ofBy andBpand are over-plotted on cuts of the

functionBy(Bp).

We see that in the upstream case we are on branch 2, as defined in Figure 4.10. If we cross a discontinuity and jump to theα quantities downstream, we end up on branch 1. Similarly, if we cross onto the β quantities, we arrive on branch 3. Thus, one can see that it is entirely possible to switch between different values of a multi-valued shear field by crossing a discontinuity. Of course, the exact branches that it is possible to switch to will entirely depend on the pressure profile and the upstream quantities.

Chapter 5

Numerical equilibrium calculations

in 2D

In the previous sections we derived a variety of Grad-Shafranov type equations, which varied depending on assumptions such as isotropy, symmetry and whether they involved a component of magnetic shear. One then faces the problem of actually solving these Grad-Shafranov equations to recover quantities such as the magnetic flux function, which can prove to be quite a difficult task. In the literature, one can find a variety of solutions in the isotropic case after assuming idealised forms of the pressure – the Harris sheet (Harris, 1962) for instance, or the Bennett pinch (Bennett, 1934). However, once one includes anisotropic pressure, it becomes difficult to find any solutions other than in idealised 1D cases.

Therefore a numerical approach is required if one wishes to find anisotropic equilibria. In this chapter we will use a continuation method which is able to reconstruct sequences of equilibria by varying some scaling parameter that is incorporated into the parallel pressure. In this way, we can explore the effects of increasing the magnitude of the pressure for instance, or increasing anisotropy by varying an anisotropy parameter. More specifically, this chapter will focus on 2D solutions to the anisotropic Grad-Shafranov equation with both translational and rotational invariance. This allows us to use the total field formalism of Chapter 2 without the need to consider the problems that necessitated the poloidal and combined formalisms. Equilibria with a non-zero magnetic shear field component will be discussed in Chapter 6.

The structure of this chapter is as follows. Section 5.1 will give an overview of the numerical method which we use to solve the Grad-Shafranov equation, and Section 5.2 will discuss the appropriate nor- malisation of quantities. Then, in Section 5.3, we will use some 1D solutions to the Grad-Shafranov equation with translational invariance (which we have found analytically) to test the accuracy of the code. Section 5.4 will present an example which compares a mirror type anisotropy with a firehose type

anisotropy (as defined in Section 2.6.2). In Section 5.5 we will explore the isotropic pressure considered by Zwingmann (1983). We will reconstruct his results in the isotropic case, and then extend his results into an analogous anisotropic regime using a parallel pressure suggested by N¨otzel et al. (1985). Finally, we will solve the Grad-Shafranov equation with anisotropic pressure in the rotationally invariant case in Section 5.6, where we look at dipole and quadrupole type solutions. Section 5.6 will also discuss the accuracy of the technique by comparing different numerical resolutions, and explore the stability of various equilibria with respect to the firehose instability.

5.1

An overview of the numerical code

The numerical code used is based on a method by Keller (1977), which has been adapted by Zwingmann (e.g. Zwingmann, 1983; Zwingmann et al., 1985) and Neukirch (e.g. Neukirch and Hesse, 1993). It is a continuation method designed to solve non-linear elliptic partial differential equations, and has a long history, having been used with a great deal of success to solve a variety of problems in many fields. Among others, for a selection of some notable applications of this numerical technique in a space plasma context, one should consult the following papers: Zwingmann (1987); Neukirch (1993a,b); Platt and Neukirch (1994); Schr¨oer et al. (1994); Becker et al. (1996, 2001); Romeou and Neukirch (1999, 2001, 2002a,b); Kiessling and Neukirch (2003); Neukirch and Romeou (2010), and Hodgson and Neukirch (2015).

We will now give a general overview of how the numerical technique can be used to solve the Grad- Shafranov type equations derived in the previous chapters. Keller’s continuation method is, to be more specific, a so-calledpseudo-arclength continuation technique which is able to solve problems of the form

G(u, λ) = 0, (5.1)

whereuis a function (possibly a vector function),λis a parameter (or perhaps several parameters) and

Gis an operator which maps the pair (u, λ) into the same space asu. While (5.1) is extremely general, and covers quite a large range of possible types of problem, we can think of G as simply an elliptic PDE of a single variable with a single parameter. That is to say, for our purposes,Gwill be one of the Grad-Shafranov equations (depending on the symmetry),uwill be the magnetic flux functionA, andλ

will be some parameter in the parallel pressure.

For a given initial parameter λ0 and solution estimate ¯u0, the code uses a Newton-Raphson scheme

to iterate towards an actual solutionu0such thatG(u0, λ0) = 0. We then introduce a pseudo-arclength

Figure 5.1: A schematic of the pseudo-arclength predictor-corrector method. The diagram shows an initial estimate solution pair (¯u0, λ0) which is iterated towards an exact solution (u0, λ0). The predictor

step finds an estimate of the solution forλ1: the pair (¯u1,λ¯1) which lies in the tangent space of (u0, λ0).

The corrector step then iterates towards the exact solution (u1, λ1).

Now we construct the pair (¯u1,λ¯1) such that

¯ u1=u0+s du0 dσ ¯ λ1=λ0+s dλ0 dσ,

where sis left arbitrary for now. Thus the pair (¯u1,λ¯1) lies in the tangent space of the solution curve

at (u0, λ0). In general, this will not be a solution to the problem (5.1), however it can be used as an

estimate from which we can iterate towards an actual solution (u1, λ1). In this sense, we are using a

type of predictor-corrector method, which is shown diagrammatically in Figure 5.1. When solving the linear PDE’s involved in the predictor and corrector steps, the code uses a finite element discretisation with quadratic shape functions. This allows for a flexible grid-structure such that we can choose to have a higher numerical resolution in certain parts of the domain that may be of particular interest (e.g.

surrounding a magnetic null point).

By using the method above, we are able to reconstruct the entire solution branch Γ containing some pair (u0, λ0), as long as we can provide a good enough estimate of the initial solution. We shall see several

examples of this as we proceed to use the numerical continuation method to find families of solutions to the anisotropic Grad-Shafranov equations given in the previous chapters.