Summary of the derivation

The derivation for quasi-linear saturated tearing mode equations begins with the three-dimensional static scalar plasma pressure equilibrium force balance equations

$${\bf J} \times {\bf B} = \nabla p$$ (1)

$$\nabla \times {\bf B} = \mu_{0}{\bf J}$$ (2)

$$\nabla \cdot {\bf B} = 0$$ (3)

These equations are perturbed about an axisymmetric equilibrium. The derivation is carried out using magnetic flux coordinates defined in terms of the axisymmetric unperturbed equilibrium since magnetic islands are flux tubes that form around mode-rational surfaces. Specifically, the saturated tearing mode equations are readily derived using Hamada-like coordinates $(x,\theta,\zeta),$ where $x$ is a surface quantity, $\theta$ is an angle-like variable around the poloidal direction (short way around), and $\zeta$ is an angle-like variable around the toroidal direction (long way around) [4,5,6]. The variable $x$ may represent the volume of the unperturbed magnetic surfaces or any monotonically increasing or decreasing surface quantity.

The static scalar plasma pressure equilibrium force balance equation 1 does not include the effect of differential rotation of the plasma. In reality, magnetic islands on different magnetic surfaces generally rotate with different angular velocities due to sheared plasma rotation and the radial profile of the diamagnetic drift. This effect substantially reduces the mutual coupling between magnetic islands, unless the island chains mode lock with one another. The effects of the differential rotation velocity ${\bf v}$ could be implemented by including the convective term ${\bf v} \cdot \nabla {\bf v}$ in the force balance equation. In addition, non-scalar plasma pressure and viscous stress terms could be included in the force balance equation. The simplest model, which is a static scalar pressure model is implemented in this paper.

In Hamada coordinates, the unperturbed (background) contravariant components of the magnetic field and current density are surface quantities

$${\bf B}^{0} = B^{0\theta}{\cal J}\nabla\zeta\times\nabla x + B^{0\zeta}{\cal J}\nabla x \times\nabla\theta$$

$${\bf J}^{0} = J^{0\theta}{\cal J}\nabla\zeta\times\nabla x + J^{0\zeta}{\cal J}\nabla x \times\nabla\theta.$$

The coordinates are chosen so that the Jacobian $({\cal J})$ is a surface quantity

$${\cal J} \equiv (\nabla x \cdot\nabla\theta\times\nabla\zeta)^{-1} = {\cal J} (x)$$

The angle-like variables $\theta$ and $\zeta$ are assumed to be periodic with period $2\pi$. This choice of Jacobian and periodicity leads to a generalization of the usual Hamada coordinate system, in which the Jacobian and the periods are all unity.

The derivation proceeds by writing the perturbed magnetic field ${\bf B}^{1}$ in terms of both contravariant components $(B^{1x},B^{1\theta},B^{1\zeta})$ and covariant components $(B_{x}^{1},B_{\theta}^{1},B_{\zeta}^{1})$:

$${\bf B}^{1} = B^{1x}{\cal J}\nabla\theta\times\nabla\zeta + ... ...a\times\nabla x + B^{1\zeta}{\cal J}\nabla x\times\nabla\theta$$ (4)

$${\bf B}^{1} = B^{1}_{x}\nabla x + B^{1}_{\theta}\nabla\theta + B^{1}_{\zeta}\nabla\zeta$$ (5)

The perturbed current density ${\bf J}^{1}$ is written in terms of contravariant components

$${\bf J}^{1} = J^{1x}{\cal J}\nabla\theta\times\nabla\zeta + ... ...a\times\nabla x + J^{1\zeta}{\cal J}\nabla x\times\nabla\theta$$ (6)

All perturbed variables are written as a series of Fourier harmonics in the toroidal and poloidal angle-like variables, $\theta$ and $\zeta$:

$$X^{1}=\sum_{m,n}X^{1}_{mn}(x)e^{im\theta - in\zeta}$$

By linearizing the equilibrium equations, Eqs. (1), (2) and (3), one obtains the equations ${\bf J}^{1}\times {\bf B}^{0}+{\bf J}^{0}\times {\bf B}^{1}=\nabla p$, $\mu_{0}{\bf J}^{1}=\nabla\times {\bf B}^{1}$, and $\nabla\cdot {\bf B}^{1}=0$. After some algebra, the following ordinary differential equations are derived for the mix of contravariant and covariant components of perturbed magnetic field, as well as the perturbed pressure $p^{1}_{mn}$ [2]

$$\frac{d}{dx}(-i{\cal J} B^{1x}_{mn}) = {\cal J}(nB^{1\zeta}_{mn} - mB^{1\theta}_{mn})$$ (7)

$$\begin{array}{l} \hspace{-1cm}(nq-m)B^{0\theta} \left[\displ... ...u_{0}p^{1}_{mn}}{B^{0\zeta}}\frac{d}{dx}B^{0\zeta}} \end{array}$$ (8)

$$nB^{1}_{\zeta mn} = -nB^{1}_{\theta mn}-i{\cal J} B^{1x}_{mn... ...J^{0\zeta}}{B^{0\zeta}}- \frac{m\mu_{0}p^{1}_{mn}}{B^{0\zeta}}$$ (9)

$$(nq-m)B^{0\theta}p^{1}_{mn} = -iB^{1x}_{mn}\frac{d}{dx}p^{0}$$ (10)

In Hamada coordinates, the magnetic $q$ is given by $q=B^{0\zeta}/B^{0\theta}$ [4,7]. Equations (7) and (8) form a pair of coupled ordinary differential equations for each helical harmonic of the variable $(i{\cal J} B^{1x}_{mn},B^{1}_{\theta mn})$. Equations (9) and (10) can be used to eliminate $B^{1}_{\zeta mn}$ and $p^{1}_{mn}$. The perturbed covariant and contravariant magnetic field components $B^{1}_{x}$, $B^{1\theta}$, and $B^{1\zeta}$ are determined from the algebraic relations:

$$B^{1}_{x} = \frac{B^{1x}}{\vert\nabla x\vert^{2}}- B^{1}_{\t... ...\zeta}\frac{\nabla\zeta\cdot \nabla x}{\vert\nabla x\vert^{2}}$$ (11)

$$B^{1\theta} = B^{1x}\frac{\nabla\zeta\cdot\nabla x}{\vert\na... ...) ( \nabla\zeta\cdot\nabla x)}{\vert\nabla x\vert^{2}} \right)$$ (12)

$$B^{1\zeta} = B^{1x}\frac{\nabla\zeta\cdot\nabla x}{\vert\nab... ...(\nabla\zeta\cdot\nabla x)^{2}}{\vert\nabla x\vert^{2}}\right)$$ (13)

Equations (11), (12), and (13) are derived by equating Eq. (4) and Eq. (5) and taking the dot product with the gradients $\nabla x$, $\nabla\theta$, and $\nabla\zeta$. The harmonic coupling implied by Eqs. (11) through (13) should be regarded as an upper bound estimate of the true harmonic coupling, which can be substantially weakened by differential rotation within the plasma.

The gradients of $(x,\theta,\zeta)$ and, consequently, the geometric terms in equations (11), (12), and (13) can be determined from the shape of the equilibrium flux surfaces in the following way: Suppose the shapes of the equilibrium flux surfaces are given by the equations

$$R=R(x,\theta)$$ (14)

$$Y=Y(x,\theta)$$ (15)

$$\zeta - \phi = Z(x,\theta)$$ (16)

where the major radius $R$, the vertical position $Y$, and the toroidal angle $\phi$ are given as functions of the Hamada coordinates $(x,\theta,\zeta)$. For axisymmetric equilibria, both $\phi$ and $\zeta$ are ignorable coordinates with period $2\pi$. For any value of $x$, the cross-sectional shape of the corresponding flux surface is traced out by varying $\theta$ between 0 and $2\pi$ in Eqs. (14) and (15).

Then, from the gradients of Eqs. (14) through (16)

$$\nabla R =\hat{R} = \partial_{\theta} R\nabla\theta +\partial_{x} R\nabla x$$ (17)

$$\nabla Y =\hat{Y} = \partial_{\theta} Y\nabla\theta +\partial_{x} Y\nabla x$$ (18)

$$\nabla\phi = \hat{\phi}/R$$ (19)

it follows that

$$\nabla x = \frac{\hat{Y}\partial_{\theta}R - \hat{R}\partial... ...tial_{\theta}R\partial_{x}Y - \partial_{x}R\partial_{\theta}Y}$$ (20)

$$\nabla \theta = \frac{\hat{R}\partial_{x}Y-\hat{Y}\partial_{... ...tial_{\theta}R\partial_{x}Y - \partial_{x}R\partial_{\theta}Y}$$ (21)

$$\nabla\zeta = \hat{\phi}/R +\partial_{x}Z\nabla x + \partial_{\theta}Z\nabla\theta$$ (22)

and the Jacobian is given by ${\cal J}\equiv (\nabla x\cdot \nabla\theta\times\nabla\zeta)^{-1}=R(\partial_{x}R\partial_{\theta}Y - \partial_{x}Y\partial_{\theta}R)={\cal J}(x)$. In these equations, $(\hat{R},\hat{Y},\hat{\phi})$ are unit vectors along $(R,Y,\phi)$. Application of the expressions for the gradients of $(x,\theta,\zeta)$ above in equations (11), (12), and (13), leads to the following form for the mixing equations:

$$[(\partial_{\theta}R)^{2} + (\partial_{\theta}Y)^{2}]B^{1\the... ...\theta} Y) + B^{1}_{\theta} + B^{1}_{\zeta} \partial_{\theta}Z$$ (23)

$$B^{1\zeta} = B^{1x}\partial_{x}Z + B^{1\theta}\partial_{\theta}Z + \frac{B^{1}_{\zeta}}{R^{2}}$$ (24)

$$B^{1}_{x} = B^{1x}[(\partial_{x} R)^{2} + (\partial_{x} Y)... ...\partial_{x}Y\partial_{\theta}Y) - B^{1}_{\zeta} \partial_{x}Z$$ (25)

Eqs. (7 - 10) for the harmonics of $B^{1x}$, $B^{1}_{\theta}$, $B^{1}_{\zeta}$, and $p^{1}$ together with Eqs. (23 - 25) for the components $B^{1}_{x}$, $B^{1\theta}$, and $B^{1\zeta}$ form a complete systems of equations.