Coordinate System

For this study, we adopt a field aligned coordinate system which is related to the classical 2D dipole. To begin, we state these classical coordinates (ψ, Az, z) in terms of cylindrical polar coordinates (R, φ, z) given as

ψ = B0 R2 0 R sinφ, (5.1.1) Az = A0 R0 R cosφ, (5.1.2) B = ∇ψ=∇×(ˆzAz), (5.1.3) where B0 is the background field strength at the point φ = 0, R = R0 and R0 is the

radius to the equatorial crossing point of field lineAz=A0. In this description, field lines

are contours of Az which are circles in the z =const plane, and are intersected by lines of constant ψ (also circles). Such coordinates can be derived from solutions to the 2D Laplace equation in cylindrical coordinates, which hence satisfy the solenoidal magnetic field constraint. ψ is the magnetic scalar potential such that B =∇ψ =∇×(ˆzAz). A simple schematic of the system is given in Figure 5.1.1.

Figure 5.1.1: A schematic of the classical 2D dipole coordinate system. Field lines are contours of Az, intersected by contours of ψ labelledψ1 and ψ2.

These coordinates present the issue however, that equal increments in dψ or dAz can

produce vastly different increments in path length dr. This point is well demonstrated for 3D dipole coordinates by Kageyama et al. [2006], where the implications for numerical methods using such a coordinate system are discussed. Using equal increments in each of the coordinates leaves the equatorial region massively under resolved, and hundreds of points would be required to appropriately resolve this region. Furthermore, the increasing number of points and reduced grid spacing toward the polar regions produces a hugely restrictive CFL condition. Hence, we can try to work around this resolution issue to some extent by defining a new coordinate system given by (α, β, γ), whereα=g(Az),γ =f(ψ) and β=z. In defining the new coordinates as functions of the old ones, the contours will remain unchanged in the new system.

Before defining these new coordinates, a short mention is made of deriving scale factors in curvilinear coordinate systems for the unfamiliar reader. A scale factor describes the geometric changes in the coordinates. In Cartesian coordinates, these scale factors are just 1, since the coordinate axes are straight lines. Consider the coordinates as stated above, (α, β, γ). A small length elementdr can be expressed as

dr=hαdα+hβdβ +hγdγ, (5.1.4) wherehα,hβ and hγ are the scale factors in theαˆ,βˆand γˆ directions respectively. The direction of a coordinate, for exampleα, is given by the normal to a constant plane/surface of that coordinate. This is conveniently given as∇α. Hence to find the scale factor forα

we can take the dot product of dr with∇α (which is just the length element dα) which gives

dr·∇α=hαdα(αˆ·∇α) + 0 + 0 =dα

where the other directions evaluate to 0 as they do not contribute in the αˆ direction. Cancelling thedα and seeing thatαˆ·∇α =|∇α|, yields

hα = 1

|∇α|.

This is the key result in determining the form of the scale factors in terms of the coor- dinates. At this point, the coordinates and scale factors are merely stated with a little discussion on their properties, with a full derivation given in Appendix B.1, which sum- marises a private communication with Dr Andrew Wright, January 2016. The coordinates and the corresponding scale factors are

α = R cosφ, (5.1.5) β = z, (5.1.6) γ = Rgtan−1 Rg R sinφ , (5.1.7) hα = cos2φ, (5.1.8) hβ = 1, (5.1.9) hγ = R Rg 2 + sin2φ, (5.1.10) where Rg is the radial equatorial crossing point of a field line chosen at some point in

the domain, along which hγ = 1 and hence γ corresponds to path length along the field line. To see this, consider setting α =Rg and substitute for R using equation (5.1.5) in equations (5.1.7) and (5.1.10). Away from this field line,γ no longer corresponds to path length along the field line and hence to minimise distortion of the grid inγ, it is sensible to choose Rg to correspond to the field line in the middle of the α domain.

Withβ =z, the 2D dipole corresponds to a dipole with field lines confined to a plane of constant β. The field is translationally independent of β. This implies that our model will neglect the converging of field lines inβ as the poles are approached. This will act to increase the disparity between the toroidal and poloidal eigenfrequencies compared to a model including a variation inβ (to be shown in Section 5.1.3), and is the reason we have chosen to use a 2D dipole.

From equation 5.1.5 for α, it can be seen that in the equatorial plane (φ = 0), α = R

and hα = 1, which implies that equal increments in α cover a uniform distance in the equatorial plane. This is distorted at higher latitudes, with field lines converging, which is an unavoidable property of a field aligned coordinate system. However, overall we have made a significant improvement to the resolution problems of the original coordinates. This can be easily seen when equally spaced contours of ψ,Az in the classical 2D dipole are compared with contours ofα,γ, from the new coordinates. Figure 5.1.2(a) shows these contours in the x, y plane for the classical 2D dipole coordinates, while (b) displays the new coordinates. It can be clearly seen how the uniformity of α in the equatorial plane makes a huge difference to the grid spacing. Furthermore, the changes to theγ coordinate have drastically improved the spacing along the fieldlines.

(a) (b)

Figure 5.1.2: Comparison of (a) classical 2D dipole coordinates ψ,Az and (b) the newly derived coordinates α,γ given by equations (5.1.5)-(5.1.10). Plotted are equally spaced contours of (a)ψ,Az and (b)α,γ.