$\alpha$ contours on a magnetic surface

Figure 19 compares a small number of $\phi$ contours and $\alpha$ contours on a magnetic surface.

Figure 19: Comparison between a series of $\phi$ contours (left) and a series of $\alpha$ contours (right) on a magnetic surface. Here the values of $\phi$ are $\phi _j = \alpha _j = (j - 1) 2 \pi / 4 / (10 - 1)$ with $j = 1, 2, \ldots , 10$, i.e., only $1 / 4$ of the full torus. The values of $\alpha$ contours are $\alpha _j = (j - 1) 2 \pi / 4 / (10 - 1)$ with $j = 1, 2, \ldots , 10$, i.e., only $1 / 4$ of the full range $2 \pi$. Every $\phi$ and $\alpha$ contours start from the lower-field-side midplane and go one full poloidal loop. Magnetic field from EAST discharge #59954@3.03s.

As is shown in the left panel of Fig. 19, with $\phi$ fixed, an $\theta$ curve reaches its starting point when $\theta$ changes from zero to $2 \pi$. However, as shown in the right panel of Fig. 19, with $\alpha$ fixed, an $\theta$ curve (i.e. a magnetic field line) does not reach its starting point when $\theta$ changes from zero to $2 \pi$. There is a toroidal shift, $2 \pi q$, between the starting point and ending point. Therefore there is generally no periodic condition along $\theta$ since $q$ is not always an integer. A mixed periodic condition involves both $\theta$ and $\alpha$ is given in (290).

In field-line-following coordinates $(\psi , \theta , \alpha )$, a toroidal harmonic of a physical perturbation can be written as

$$\delta A (\psi, \theta, \alpha) = \delta A_0 (\psi) \cos (m' \theta + n \alpha + \alpha_0)$$ (292)

where $n$ is the toroidal mode number, $m'$, which may not be an integer, is introduced to describle the variation along a field line. The periodic condition given by Eq. (290) requires that

$$\cos (m' \theta + n \alpha + \alpha_0) = \cos [m' (\theta + 2 \pi) + n (\alpha - 2 \pi q) + \alpha_0],$$ (293)

To satisfy the above condition, we can choose

$$m' 2 \pi - n 2 \pi q = N 2 \pi,$$ (294)

where $N$ is an arbitrary integer, i.e.,

$$m' = N + n q.$$ (295)

We are interested in perturbation with a slow variation along the field line direction (i.e., along $\partial \mathbf{r}/ \partial \theta$) and thus we want the value of $m'$ to be small. One of the possible small values given by expression (295) is

$$m' = n q - \ensuremath{\operatorname{NINT}} \left( n \frac{q_{\max} + q_{\min}}{2} \right),$$ (296)

where N $\ensuremath{\operatorname{INT}}$ is a function that return the nearest integer of its argument, $q_{\max}$ and $q_{\min}$ is the maximal and minimal value of the safety factor in the radial region in which we are interested. Note that $m'$ depends on the radial coordinate $\psi$ through $q (\psi)$. Also note that $m'$ here is differnt from the poloidal mode number $m$ in $(\psi , \theta , \phi )$ coordinate system. It is ready to show that the perturbation given by Eq. (292) with $m' \sim 1$ and $n \gg 1$ has large poloidal mode number $m$ when expressed in $(\psi , \theta , \phi )$ coordinates. [Proof: Expression (292) can be written as

$$\delta A = \delta A_0 (\psi) \cos [m' \theta + n (\phi - \overline{\delta} (\psi, \theta)) + \alpha_0]$$ (297)

If $\theta$ is the staight-field-line poloidal angle in $(\psi , \theta , \phi )$ coordinate system, then $\overline{\delta} (\psi, \theta) = q \theta$ and the above eqaution is written as

$$\delta A = \delta A_0 (\psi) \cos [(m' - n q) \theta + n \phi + \alpha_0],$$

which indicates the poloidal mode number $m$ in $(\psi , \theta , \phi )$ coordinates is given by $m = m' - n q$. For the case with $m' \sim 1$ and $n \gg 1$, $m$ is much larger than one.]

[In the past, I choose $m' (\psi) = n q - \ensuremath{\operatorname{NINT}} (n q)$. However, $m' (\psi)$ in this case is not a continuous function of $\psi$ and thus is not physical.]

[In passing, let us introduce the binormal wavenumber, which is frequently used in presenting turbulence simulation results. Consider the toroidal phase in expression (292), i.e.,

$$\ensuremath{\operatorname{phase}} = n \alpha$$ (298)

Define the binormal direction $\mathbf{s}$ by

$$\mathbf{s}= \frac{\mathbf{B} \times \nabla \Psi}{\vert \mathbf{B} \times \nabla \Psi \vert},$$

which a unit vector lying on a magnetic surface and perpendicular to $\mathbf {B}$. The binormal wavenumber is defined by

$$k_{b n} =\mathbf{s} \cdot \nabla \ensuremath{\operatorname{phase}},$$ (299)

which can be written as

$$k_{b n} = \frac{\mathbf{B} \times \nabla \Psi}{\vert \mathbf{B} \times \nabla \Psi \vert} \cdot \nabla (n \alpha)$$

$$= n \frac{\nabla \Psi \times \nabla \alpha}{\vert \mathbf{B} \times \nabla \Psi \vert} \cdot \mathbf{B}$$

$$= n \frac{B^2}{\vert \mathbf{B} \times \nabla \Psi \vert}$$

$$= n \frac{B}{\vert \nabla \Psi \vert},$$ (300)

Using $B_p = \vert \nabla \Psi \vert / R$, the above equation is written

$$k_{b n} = n \frac{B}{R B_p}$$ (301)

which indicates the binormal wavenumber generally depends on the poloidal angle. For large aspect-ratio tokamak, we have $B_{\phi} \approx B$, $q \approx B_{\phi} r / (B_p R)$. Then Eq. (301) is written

$$k_{b n} \approx \frac{n q}{r},$$ (302)

which indicates the binormal wavenumber are approximately indepenent of the poloidal angle. For modes with field aligned structure (i.e., $k_{\parallel} \ll k_{\perp}$), we have $m \approx n q$, where $m$ is the poloidal mode numer along $\theta$ with $\phi$ hold fixed. In this case, the above equation is written $k_{b n} \approx m / r$, which is the usual poloidal wave number. Due to this relation, the binormal wavenumber $k_{b n}$ is often denoted by $k_{\theta}$ in papers on tokamak turbulence. In some papers the binormal wavenumber is denoted by $k_y$.]

Since $\alpha$ contours are magnetic field lines, they span out the 3D shape of the magnetic surface when there are enough numbe of $\alpha$ contours on a magnetic surface, as is shown by the left-panel of Fig. 20.

Figure 20: Comparison between a series of $\phi$ contours(left) and a series of $\alpha$ contours (left) on a magnetic surface. The $\alpha$ contours correspond to magnetic field lines. Here the $\alpha$ values of adjacent $\alpha$ contours differ by $d \alpha = 2 \pi / 20$ and each $\alpha$ contour goes one full poloidal loop. Magnetic field from EAST discharge #59954@3.03s.