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 where is a surface quantity, is an angle-like variable around the poloidal direction (short way around), and is an angle-like variable around the toroidal direction (long way around) [4,5,6]. The variable 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 could be implemented by including the convective term 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

The coordinates are chosen so that the Jacobian is a surface quantity

The angle-like variables and are assumed to be periodic with period . 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
in terms of both contravariant components
and covariant components
:

The perturbed current density is written in terms of contravariant components

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

By linearizing the equilibrium equations, Eqs. (1), (2) and (3), one obtains the equations , , and . 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 [2]

Equations (11), (12), and (13) are derived by equating Eq. (4) and Eq. (5) and taking the dot product with the gradients , , and . 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
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

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

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

Eqs. (7 - 10) for the harmonics of , , , and together with Eqs. (23 - 25) for the components , , and form a complete systems of equations.